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THE LEF T HAND OF CHAOS (MODERN AND ANC IENT INDIAN ANALYS IS )
Mi guel Angel M arti nez Iradi er

©2005, by Migu el ŕngel MartĚ nez Ir adi er

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IND EX

PR OLOGUE............................................................................. Pg. 3

F IRS T PAR T PULS E AND TIME......................................................................P g. 5

SECOND PAR T TIME AND THE MODERN SC IENCE............................................... Pg.39

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PROLOGUE

This es say is basi cally abou t philo so phy. Samkhya, which i s the ol dest Indi an philo so phy, had a decisi ve i nflu ence o n t he develo pment of Bu ddhis m. W e will u se it s vi ewpoi nt s to t ackle a wi de-r angi ng physical and mathemati cal ques tio n, r el ated to chaoti c and compl ex dynami cs. The Samkhya is an exact , mat hemati cal philo sophy, appli cabl e i n any or der o f the experience, and t her efo r e, al so it has use in t he mor e co ntroll ed experimental do mi nion. The Samkhya stu di es the waves or flu ctu atio ns of any mo vement , bei ng t he mi nd o nl y anot her for m or order of t he mo vement i n gener al. And alt hou gh t he mai n intentio n o f Pat anj ali is the self-obs er vatio n or dir ect per ceptio n of t he fluctu atio ns of o ne's o wn mi nd, wit h t he i ntentio n of r eaching the bal ance and the pos sibl e cease o f it s mo vement , al so we can appl y t hi s fr ame to behaviou rs of experi ment al and measurabl e o r der, one of whi ch i s t he s angui neous pul se. This i s what did mill enniu ms ago the ayurvedic medicine, alt hou gh, o f cour se, i n t hat case the qu antit ative co nsider atio ns wer e secondary. We ar e goi ng to take all thi s f art her , and wil l t r y to s ho w t hat t he relatio ns between t he Samkhya and t he mo der n mat hemati cal anal ysi s, i nclu di ng i n thi s one the sci ence of dynami cs , ar e real, deep and i nevit abl e. The boo k consists o f t wo parts . In t he fir st part, titl ed "Puls e and ti me", I sho w t hat t he triguna and t he tridosha, t he t hr ee mo dalit ies o f the co ndi tio ned nature i n t he co nt ext of t he Samkhya, is not hing but t he most gener al expression pos sibl e of t he t hree l aws of t he mo vement of Newto n and mo der n dynami cs ­the most gener al, becau se t hey ar e appl icabl e as mu ch to open s ys tems as to closed or ideal s ystems, bei ng thes e t he limit of the fir st. But t he s ame r elation bet ween open and clo sed sys t ems will take to us much fart her , to t he bor der s of t he anal ysi s and t he pres ent physi cs, i nclu di ng i n t he wo r d "anal ysis " even nu mber theor y, and bei ng t he utter Arit hmetic t he clos est di s ci pline to t he pure s cru ti ny of ti me. This it is the su bject of t he seco nd part , ti tled "Ti me and t he moder n science". Bei ng awar e enou gh t hat t hese subj ects are as vast as dif ficu lt, I have decided o n a expo sitio n po i nted wit h histo ri cal moti fs ; whi ch do es not mean at all t hat t hi s i s a book on t he hi stor y of the subject , but r at her that t he hi stor y, in which al ways ou r i magi natio n co ncu rs, allows u s to as si milat e t houghts, movements and current s not o nl y mor e easi l y, but al so more deepl y when we have a suit abl e pers pective. And, pr ecis ely, so me of t he t hi ngs appearing ar e so lit tle evi dent t hat even the hi stori ans have not r epaired i n t hem. But, for me at least, t he histo r y i ts elf , always ambi guous and dou bt ful, has much of nature ­of a nature t hat rarel y we manage to cont empl at e. "The l eft hand of chao s " alludes to t his r ar el y co ntemplat ed aspect of natur e, so r elat ed to ti me, beyo nd the co nventio nal s pace of dynamics . The s ame ter m "chaos " is t aken in a lar ger meaning t han t he one of t he so call ed det er mi ni sti c chaos , whi ch aros e fro m t he inst ability i n t he solutio ns of differ enti al clas sical s ys tems; i f we co nt emplat ed it s "left hand", the same chaos would be dilut ed i n somet hi ng ver y diff erent , but not necess aril y co ntro llable. The l eft hand of C hao s
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No t at any mo ment I have ent er ed det ail s o r techni cal pr eci sions , which would exceed t he i nt ention and t he reach of t he es s ay: whi ch is to i llumi nate and to hel p to concei ve new pos si biliti es i n our i dea of natur e, co nsider ing the al ways decisive limit atio ns o f our mathemati cal mo dels . A corr ect co ntempl ation will hel p u s to sur pass some of t hese li mitations, becau s e if by o ne si de we do o mis sion o f t he techni cal dif ficulties , we al so indicat e t he experi ment al fr ame in which t hese have to reveal t hems elves wit h all pro per t y. W hat the hu man mi nd can not imagi ne, off er s us t he natur e ver y oft en wit h insurmount abl e nai vet y; and i n t he pres ent l abyr i nt h of sciences of t he co mpl exit y an Ari adna t hread like t hi s one is missi ng. Natur e ignor es the anal yt ical diff icul ties, and here we ar e going to fi nd a good exampl e of so met hi ng t hat can be si mult aneousl y mu ch more si mple and mu ch more compli cat ed t hat what t he most so phi sti cat ed t echniqu es allo w u s to des cri be: it will be u p to us what part we pr ef er , kno wi ng well t hat here compl et el y new co nsi derat ions are i ntro du ced o n which i t i nvol ves t he pr edictio n and t he reconst ruct ion of t he event s. The pr esent mo del s of predi ctio n and r econst ructio n ar e clo s e to t he coll apse and it is absolu tely necessar y to ref r ame what i s po ssi ble to wait fo r of t hem, si nce often t he own expect atio n of t he pr edi ctio n or control exti nguis hes t he pos si bili t y of u nder standi ng of t he phenomena. Thi s su bject is consu bst anti al to t he develo pment of the whole text, al t hough bar el y it beco mes explicit: i t has to mani fest it self i n the mo st co ncr ete of the pl anes . Therefore, t his boo k is not mer el y s pecul ati ve. It makes a deci ded bet fo r the exper i ment al co ntr ast of its i deas. The i deas ar e diffi cult to expo se and to gr asp, but t he experi ments are easil y at tainable worl dwi de. W hat t her e is to ver if y o r to refute is a cert ai n du ali t y i n the behaviour o f the puls e. Such dualit y s eems to be treatabl e more easil y b y t he stu dy of t he corr el ation of t he pulses of bot h wri sts ; but it is not i n t hat t hat t he du alit y co nsists . This o ne si mpl y i ndicat es t he dist i nct io n bet ween prak riti and vikriti , t he i nitial and s eco ndar y co mpo nents of t he joi nt evolutio n o f t he sys t em. If we managed to defi ne t hese i n a s ati sf actory for m, we wi ll have gi ven a gr eat step, becaus e what it is at st ake i s not ano t her thi ng t hat t he fr ame of refer ence for t he evolution of dynami call y co mplex s ys t ems ; and thes e, nowadays , i nclu de pr acti call y ever yt hi ng. This text was writt en ori gi nall y i n Spani sh, my mot her l angu age; bei ng my E ngli sh ver y limit ed, I apolo gi ze for my tr anslatio n.

Au gus t, 2005

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F IR ST PAR T

PULSE AND TIME

1 It seems to be t hat i n t he W est t he o nly i dea of l egiti mat e s ci ence i s t he o ne deri ved from the Gr eeks. In t hi s line, any descri ptio n of t he worl d must be ext er nal to us, and cons equ entl y, o bjecti ve. All other des cr i ptio n would f all wit hi n t he su bjective, when not "myst ical ", i nt er pr etations . Nevert heles s, modern s cience is so met hi ng f ar away fro m t he Greek spi rit. Two are t he great events that mark t his dist ance. Fir st, t he appear ance of an experi mental pur pos e wit h r es pect to t he Natu r e, t hat arr i ved at medi eval Europe t hrough the Ar abs, and i n parti cular , through the alchemy and pr e-chemi stry alt hou gh al so t hrou gh wor ks abou t o pti cs. Seco ndl y, it was t he i ntro duction o f t he zero wit h positional not atio n, of Indi an ori gi n, t hat completel y tr ansfor med t he possi biliti es of t he calculu s and, i n t he lo ng ru n, t he whol e face of t he mat hemati cs. Each o ne of thes e el ements , compl et ely ali en to t he Greek ment ality, has inf luenced so much in mo dern science as t he own Gr eek idea o f ratio nali ty; so t hat t he vi ndi catio n of the Gr eek o ri gi ns of s ci ence, even bei ng l egiti mat e, has al ways mo re r hetor ical and romanti c element s t han we co mmo nl y admit. In XVII centu r y, and si nce Gal ileo , t he notio n of change and mo vement and t he notio n of i nf i nit e decimal expansion wer e i nt erpenetr ated and fro m it began to ou tline the concept of function . In t his co nnect io n, so met hi ng ver y import ant became f rom the insid e to the outsid e: the idea o f t he change i tself , that earlier was inter nal to t he same matt er ­or to t he su bstance, if we pr efer - happens to be exter nal and s pati al by vir tue of its mat hemati cal expr essio n. Thus , by a pecul iar t rans position, it r etur ned to become pr esent the Gr eek i deal of kno wl edge l i ke exter nal represent atio n, although i t was pr ecis el y Gr eek author s, li ke Ari stotl e, tho se who wer e challenged. But, i n an i ntangi ble and fu ndament al way, and i n t he mi ddl e o f t he mat hemati cal for malism, the notio n of di s pl acement and change was i nher ent to t he po sitio nal zero not atio n wi t hi n t he inf i nit e deci mal expansions ­and i n a so su btle, i nco mmensurabl e and elusi ve way as it coul d cor res pond to the same u se of the zero. In thi s same displacement t he i dea of t he movement r eleased of any mat eri al co nsi derat ions was alr eady i mplicated. And thus the mo der n co ncept ion of "r eal " nu mbers begi ns al so. In t hi s cru ci al r eco mbi nat io n, t he o wn i deas of s pace and ti me adopt ed a co mpl et el y new and unexpect ed charact er. So new, t hat not even to day, accu sto med alr eady i n our i magi natio n th e new rep res entati ve styl e, we can unders tand entir el y its cons equ ences. On t he one hand, t he Galil ean pri ncipl e of rel ati vit y depri ves of any meani ng t he notio n o f locatio n, of poi nt i n rest in t he s pace ­t he s ame r est no mor e exist, si nce it i s indi sti ngui s hable of the uni for m mo vement . By t he other hand, and as a r esu lt o f it, t he vari atio n i n the ti me and its rat e become the aut henti c measuri ng rul e ­ but t hi s, at t he co st of tur ni ng to ti me its elf so met hi ng s pati all y r epr es ent ed, and for t hat r easo n, co mpl et el y passi ve and depri ved. Galil eo hims elf di scovers and i nvents t he pendulum. Without r eal alt er natio n no time i s possi ble. But , bei ng t he alter natio n o f the pendulu m i ndependent of its amplitu de, t he mo del i s lent naturall y to it s i dealization, bei ng tot all y tur ned The l eft hand of C hao s
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into s pace. Eli minating any co nt ent in t he alt er natio n of t he pendulu m, what Galil eo i nvents i n fact i s the "one way pendulum". This i nco ncei vabl e obj ect is t he o ne t hat go ver ns as mu ch t he form we i magi ne t he ti me as go ver ns i n our clo cks. Or at least, i n our for m to repr es ent it: becaus e it i s difficult to see ho w it coul d aff ect to other l ayers of t he i magination t han the mo st su per ficial ones . It has been sai d that the li near ti me and t he i dea of histo r y ar e Ju deo- Chri sti ans notio ns, al though t he cert ai n thi ng i s t hat they have ar ri ved to us t hrou gh Zoro astri an Iran. Nevert heless, t he hi stori cal and irr eversibl e ti me, as we have f elt it i n t he W es ter n worl d in t he l ast centuries , mu st much, if not most, to t he idea of reversible ti me of t he mechani cs or dynami cs. How t hi s, t hat we kno w posi ti vel y cert ain, can si mply be po ss ible? B ecaus e t he "o ne way pendulu m". But i f i n physi cs the i dea of irr ever si bilit y ­t he so call ed "arrow of t he ti me"- i s not o pposed at all to t he reversi bl e ti me o f t he mechanics , no r it adds not hi ng ­ as Pri go gi ne seemed to pr esu me - , bu t it is si mpl y its ot her inevit abl e f ace, Where to watch? Our si mpl e and not ironic ans wer is: Not o nwar ds o r backwar ds , but to war ds t he si des; to our ri ght and our l eft si des . Pr ef er ably, befor e we make t he least movement . Not even mental. C ert ai nl y, i t conti nu es s eemi ng difficult.

2 "Tatah punah santoditau tulia-p ratiaiu cittasiaikagrata-pa rina mah ", may be t ransl at ed as: "The co ncent ration t akes pl ace when the ment al waves t hat ri se and fall at two dif fer ent moment s are exactly equ al ". W hat more stri king seems to u s in t he first r eadi ng of t he Yoga sutra s of Pat anj ali is its exact and mat hemati cal char act er, even t hou gh it i s s peaki ng s i mpl y o f t he spectru m of po s si bl e experi ence for any man wit h t he necess ary dis- att achment and dis ci pl i ne; not o nly exact and mat hemati cal, but, pro perl y speaki ng, analyti cal and infini tesi mal. No wo r k written with its antiquity, about t wo t hou sand years , o ff er s so met hi ng si mil ar . I woul d s ay t hat neither the mat hemat ical wor ks of Ar chi medes , nor hi s book of the sand nu mber, co ntai n a si mi lar aroma. Ar chi medes and Apolo nio did not kno w the zero, and many say t hat this was what t he main o bst acl e to attai n t he mo der n calcu lus. In f act, t hey al so needed t he o t her es s enti al el ement , a diff erent idea abou t t he mo vement and t he time. P at anj ali di d not know t he zero and it s expli cit o per atio ns; but he knew well the i dea impli ed - and, appar entl y, more enough t han t he mer e i dea. In addition, whi ch he left wr itten on t he movement and t he change is of an i mpenetr abl e dept h, but wit h an i mpeccabl e i mpl icit logi cal ar ticu lat ion. Certai nly Pat anj ali was not a mathemati ci an no r a physi cist ; not even a "ps ycholo gist ", as he is often di s patched tryi ng to i gnore what he is s ayi ng i n t he clear est for m: "t he t hi ngs exi st as mu ch if the per cei ver per cei ves t hem as if not ", to admit whi ch do es not need to take r efu ge i n any of t he pretensio ns of t he realis m. Pat anj ali is si mpl y a phi loso pher , who pref er s to leave us his co nclusio ns i nst ead of writ e to dis cuss t hem. Its su bject is t he hu man experience and what u nderlies i n t he change t hat makes it pos si ble; not hing mo r e, not hi ng l es s. W hat , natu ral l y, it i mplies t he time, t he mo vement , the nature and t he appear ance, wit h t he basi c defi nit ion of it s mi xtu r e, pr edo mi nance and pro portions . Therefore, it i nclu des ever yt hi ng wit h an extr aor di nary eco no my of means. But, pecu liarl y ­ and her e i s all t he di ff er ence wit h t he mo der n west er n thought - , t his i s o nl y o btai ned wit h a deter mined posit io n, t hat cons ist s basicall y in not to make t he s mall er di stinction bet ween int ernal and ext ernal, su bjective and obj ecti ve pheno mena. For t hat r eason it does no t make sens e to des cribe Pat anjali li ke a "ps ychologi st ". His t hou ght and vi sio n are beyond any i ngenuous nes s.

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To many ou r as so ci atio n o f capital par ts of t he clas sic Indi an t hought wi t h t he s piri t of t he anal ys is will s eem totall y u nfou nded. P erhaps t hey i gnor e t hat Indi an mat hemat ici ans li ke Madhava of Keral a di scovered the so call ed Taylor seri es ­u sed befor e by Gregory, Newto n or Lei bniz- near by t he year 1400 . As it is well known, t hese pol yno mi al s i n power s eries are t he s pine of t he calculus. Madhava was t he fir st The l eft hand of C hao s
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mat hemati cian who made t he i nfi nitesi mal pass age to t he li mit ; almost su rel y, he arri ved at t his ext reme by hi s wor k wit h conti nuous fr actions . Or t hat Ar yabhata t he gr eat , who li ved by t he year 500 and befor e Br ahmagu pta ­ who gave u s the rul es to oper ate wit h t he zero -, dis co ver ed t he co nti nuous f ractio ns, virtu all y inf i nit e, t hat was o ne of the preferr ed i nstru ment s o f Eul er o r Gaus s , and that st ill to day give many sur pris es i n pur e mat hemati cs and i n t he analys is of reso nances i n chaoti c and co mplex dynami cs . The co nti nuous fractio ns, series thems el ves , ar e in t he origi n of t he anal ysi s. Ar yabhata, almo st wit h co mpl et e certai nt y, wor ked al ready wit h t he not atio n o f the positio nal zero. Bet ween a great nu mber of achi evements , he advanced the incredi ble conjectu re t hat t he or bit s of pl anet s were elli ps es . Or t hat Sri ni vasa Ramanuj an ­who also had a noti ceable pref erence by t he co nti nuou s fr actio ns - dis co ver ed t he fu nctio nal equ atio n of t he Riemann zet a functio n ­summit and abys s of t he superior anal ys i s- i n India by hi mself and wit hout anot her sti mulu s t han t he Carr' s element ary mat hemat ical text book. Ramanujan bar el y had t he s maller i dea about t he compl ex var iabl e theor y. Su r el y, so i mpro bable fact s are not merel y coinci dences. But the mast er pi ece of P atanjal i it is the mo st forceful and po wer ful ar gu ment , for thos e who can r ead, and if we attend to t he s piri t of t he anal ysis its el f. W e fo r get easil y enou gh t hat t he i dea of the infi nit e and t he i nf i nit esi mal is i nco ncei vabl e and unmanageabl e wit hout t he zero, about whi ch t he Gr eek mat hematics are the mo st evi dent proof . And we also for get t hat t he i dea of t he i nfinit e and t he i nfinitesi mal are a doub le i dea si nce t he beginni ng, and therefor e, irr educi ble to t he si mpli cit y or u nit . Bei ng so, t he zero r emai ns for u s a ki nd of i mmanent immo vabl e motor , even withi n t he mo st neut ral of t he for mali s ms . No body says "it tends to be equ al ", but s ays "it t ends to zero ", or, on t he contr ary, "it t ends to i nfi ni te", su bsumi ng t he highl y fr agil e co ncept of equalit y wit hin t he mu ch more comprehensi ve and po werful one of t enden cy. To s pecify a tendency i s all what u nderli es to t he co ncept of functio n. Therefore, t he zero rel eas ed fo r al ways t he notio n of tendency, contr i buti ng to t he s ame met hod o f the cal culus t he idea o f po wer of r esolut ion. Ou r own i dea of t he identit y is s afeguar d by the one of t endency. And t he s ame zero allo ws us to toler ate t he ambi guit y, displ aci ng it. Thu s, it i s easy to s ee t hat t he great st ep and t he gr eat di ff er ence that s eparates t he moder n s ci ence from t he o ne of t he Gr eeks is jus t t he i mpo nder able zero, t he great Indi an co nt ri bution, so dif fi cult to est i mate al ways . Si mplif yi ng t he equ atio n: Gr eek mathematics + 0 = modern Anal ysi s That anybo dy will i nt er pr et as can. One will s ay, and it is compl et el y certai n, t hat t he Indi ans never got to develo p t he anal ys is appli ed to physical s ci ences as we underst and it, and t hat , for t he s ame reason, t hey could not develop it syst emati cally neit her wit h any co mpar abl e degree of gener alit y. But t his f act not o nl y does n't contr adi ct ou r per ceptio n of the su bj ect, but rei nfor ces it co nsi derabl y. Then what we su ggest is t hat t he anal ytical mental ity was somethi ng co nnatural to cert ai n l ayer s of t he Indi an culture, but not that t hi s o ne s ho wed i ncli natio n to a sci ence li ke mo der n dynamics , t hat must st art of f fro m t he as su mptio n of an i nertial frame. This t ype of election was co mpl etel y ali en to the Indi a cultu re, li ke to all t he other s, except fo r the west er n one. But we do not have why to confus e t he anal ysi s wit h dynami cs , alt hough for west er n peo ple such as sociation beco mes u navoi dabl e. Alr eady t he i dentifi catio n by Co per nicus of t he Sun as centr e of t he ast ro no mical syst em implies t hat inerti a like fu ndament al pro pert y has been cho s en; and it i s t he r eno vatio n o f thi s el ectio n t he one that has su ppos ed its triumph between us , u nli ke the Gr eek cont emporar ies and l ater to Arist archus. To t he anci ent peo pl es any i magi nabl e pro pert y o f the cel es tial bo di es woul d be more wort h of i nt er est t han inertia. Newto n gave an ans wer in for m of l aw to a questio n that had become so met hi ng i nt er es ti ng; but if ot her peo pl es di d not dis cover su ch laws, we mu st remember t hat t hey di d not have t he s mall er int erest by t hem eit her, nor t hey never looked for t hem u nder such clot hi ng. Sai d t his, we can r eco gnize that the anal yti cal s pirit i n t he anci ent Indian cultur e r es ponds to other int erests t hat t he o nes of mo der n dynami cs: what it i s not at all equi val ent to say t hat los es i nt er est o n the mo vement and t he time. W hat it do es not s har e i s t he out er and s pati al r epr esent atio n of t hes e. As we ju st have sai d, it i s not necess ar y to co nfuse t he s cience of dynami cs wit h t he anal ysi s. Thi s l ast o ne is a mu ch mo r e timel ess o bject , wit h it s own mo vement. The ari t hmetical fu nctio ns, like the Ri emann zet a fu nctio n, give t he mo st magni ficent example to u s . And who woul d af fir m that Ar it hmeti c not hing has to do wit h t i me? Lat er, we will tr y to s ee to war ds wher e t his l eads to u s. The l eft hand of C hao s
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Pat anj ali' s Yoga sutra s ar e t he maxi mu m expo nent of t he Samkhya. Samkh ya means analysis , numb er , measu rement, proportion, detailed scrutin y. In any cas e, t he Samkh ya was t he mor e pur ely anal yti cal of all t he Indi an philoso phi es. It exi sted develo ped befo re Bu ddha was bor n, and hi s fou nder, Kapila, coul d live at a ti me even previou s to t he birt h of P ythagor as, althou gh i n dates qu ite near to tho se of t he bir t h of the Gr eek philo so phy. The Samkhya i s t he philoso phy that under lies to the Yo ga, so t hat it s su bject onl y can be the equili br iu m or bal ance. Everybo dy i s mor e or l es s forced by lif e to mai nt ai n diff erent balances, apart f ro m to as pir e to it i n t he most gener al for m; so t hat neit her yoga nor t he Samkhya i s ali en to any of u s, even if we beli eve to i gnor e t hem compl etely. Ot herwis e, and i n t he for m of co nser vatio n princi ples, also fo r t he sci ence of dynami cs t he bal ance is t he fundamental subj ect .

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The Samkhya, whereas the stu dy of the os cil lat ions of t he mi nd t hat po ssibl y can be l ead to t he rest , i s an i ntu iti ve and per f ect l y dir ect theo ry o f t he flu ctu atio ns. But t he mo daliti es or mo ment s that detect s in t he fluctu atio ns cannot be pur el y extri nsic, but i nt er nal or i nherent , t he discer nment t hat onl y charact erize t hem opens pas sage i n as mu ch t he mi nd is co ntemplated or t esti fi ed by t he i mmo vabl e co nscious nes s wit hout attribu tes. It do es not exist i n t hi s any co nt radict ion, but co ntinuit y, si nce the char act eri sti c movement of t he mi nd i n i ts pas sages of acqu isition of knowl edge r esults fro m so many o t her pas s ages of easement and ther efo re of r el ati ve or t angenti al co nt act wit h t he s ame kno wl edge as source; we will s ee that t hi s solutio n o f co nti nuit y is t he su bject par excell ence of t he Samkhya as well as of any approach to it.

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For t he Samkh ya, all t he materi al or co ndi tio ned natur e is r educed i n last inst ance to t he fluctuatio ns of t he t hree gunas or modaliti es . Thes e always act of co nju gated for m, so t hat we cannot percei ve t hem i n its isol ated and exempt ed for m. In t he mi ddle of t hem and beyo nd t hem it i s the immut able and u nco nditioned, Purusha, t he co ns ciou s nes s wit hout qu aliti es neither att ribut es .

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The t hr ee guna s, Satt wa, Raja s, Ta mas , are, in t heir pri mar y fo r m, s ensiti vit y, activit y and i nerti a. Or if it is pref erred, bal ance ( r elati ve), mut abilit y and pot entialit y. On the matter of t he mind, i t is po ssi bl e to say t hat i t is not an exclu sive attri but e of Satt wa or t he The l eft hand of C hao s
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sensiti vit y, but that t he inher entl y associat ed as or subject t hat we want any mo ment, except by

own mi nd, alt hou gh is mor e clearl y mani fest ed u nder the aspect of Sattwa, co ntai ns mu ch the acti vi t y as t he inerti a. The thr ee guna s ar e i ncluded and present in any o bj ect to isol at e: t herefor e the gunas or pri mar y qu aliti es are thos e that cannot be isol at ed at the mo ment ar y or r elati ve do mi nio n that one of t hem al ways exer ts o n t he ot her t wo .

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Let us t ake a paradi gmati c exampl e fro m hatha yoga to concei ve in an i mmedi at e manner t he co nju gated char acter of the t hr ee gunas: the po sitio n of r emaini ng i n balance on t he head, wit h t his o ne putti ng in t he centr e o f a t ri angl e dr awn by t he for earms (si rsha sana). The no vi ce has t her efor e t he o ppo rtunit y to ret ake t he har d l ear ni ng of t he bal ance t hat as much wo r k gave hi m when babe and t hat he fo r got co mpl et el y pr eci sel y becau se it has managed to incorpo rate it. In t he beginni ng, it will be basicall y at the mercy of i nerti a; wit h so met hi ng of practice, it will l ear n to make the necessar y force to r esi st t he oscil lat io ns of t he bo dy. Thi s force will be excessive nearl y al ways and bad appli ed, u ntil graduall y it i s develo pi ng suf ficient sensitivit y and the unneces sar y ef fort s ar e avo i ded to t he maxi mu m. Aft er a vari abl e perio d of pr acti ce, a more or less o pti mal union of i nerti a, for ces and sensiti vi ty i s reached: t hen not even a s peci al al ert of sens iti vity is pr ecis e, becau s e it is co mpl etel y int egr at ed wit h t he other t wo t endenci es. A s po ntaneou s st at e of sus pension t akes pl ace fi nall y, that nevert hel ess is the fruit of all t he previous effort s and deli ber atio ns . Even i f we took t he wor k of even measu re i n det ail t he o scill atio ns t hat withi n a ci rcl e make t he bo dy of t he no vi ce and the one of an experi ment ed, and for i denti cal devi atio ns o f t he centr e i n any direction, we woul d verif y the enor mous dif f er ence t hrou ghout t he verti cal axis wit h r es pect to t he ri gidit y of t he first and t he perf ect and s ynchro nou s undul atio n t hroughout t he body of t he s eco nd for t he mo st ef fecti ve and co herent accomplis hment of the pri nci pl e of least actio n. This is an excellent illu str ation of t hat lo st dimensio n of the mo vement to t hat we allu ded; as well as o f whi ch we underst and by i ncor porable or as si milabl e knowl edge. To t hat pro cess of co nceal ment i n the i nvis ible of t he qualities i n it s tot al pr es ence i s at which it poi nts t he experience of bei ng i n the Indi an philo so phy. Such experi ence allo ws t he parti cipation i n t he extra- ment al wit h no need of met aphysi cs. In additio n, thi s exampl e o ff er s a demo nstr atio n to u s o f a pri nci pl e t hat is al ways valid i n Samkh ya: to gr eat er sens iti vity, great er st ability and l ess er co nditioni ng. But so t hat t his sensitivi t y i s so, and not an added source of pertur bations, that has had to i nt egrat e in a ct t he ot her t wo pr esent mo daliti es. This pri nci ple, hi ghl y condi tio nal, of great er stabilit y to gr eat er s ensi tivit y, i s of extr aor di nar y importance, fo r exampl e, i n Biology. But its general ity coul d ext end furt her o n.

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Anot her i mmediate fo r m to co nsider t he gunas o r basic mo daliti es i s taki ng car e of t he natur e o f our hu nger or i ncl i nat io ns . Thes e ar e r educed i n last inst ance to onl y thr ee: t he l azi ness or i nerti a, the acti vi t y and the pl easur e. A mo ment ar y pr edomi nance of o ne of t hem al ways occurs , whi ch is pro per of t hei r excludi ng charact er. The l azi nes s co rr es po nds to ta mas , t he act ivit y to Raja s, and pl easur e to sattwa . Alt hou gh sat twa is the mo r e el evated modalit y, the fact that it exi sts wit h att achment reveal s it s condit io nal , natu ral char acter. W hy t he attachment to t he pleasur e? B ecau se t he pl easur e secret l y looks for t he balance bet ween t he acti vit y and t he inactio n. This way, s ensiti vit y or satt wa becomes t he l eadi ng reason for t he evolutio n of all beings co nditioned by the t hree gunas , in s ear ch of the unco nditio ned balance, Kai valia . For the Samkh ya, t he modification of guna s is t he o nl y cau s al pr i nci ple t hat exi sts i n natu re. That is to say, t here is no ot her caus e t hat t he change o f t he guna s.

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But t he thr ee guna s or mo daliti es al so f i nd t heir tot al co rres pondence i n dynamics; and pr ecis el y i n t he thr ee laws o f t he movement and the mechani cs , enu nci at ed for t he fir st ti me by Newton. Per haps t hen it does not defi ne t he first law t he r est and i nerti a, t he seco nd t he actio n or for ce bas ed o n t he fir st t hrou gh t he mass, and t he thir d t he bal ance bet ween actio n and reactio n? Yes , Tamas is i n cor r espo ndence wit h the first law, Rajas wit h t he s eco nd, and Satt wa wit h t he t hir d. But a great dif fer ence exi st s. In cl assi cal mechani cs , t he action and r eactio n ar e simult aneous and i mmediat e. In t he Samkhya, such a thi ng it is not refus ed nor aff irmed; si mpl y, i s no t co nsi dered. The t hir d l aw i s the saf egu ar d of t he co nser vatio n pri nci ples . But , i n additio n, it is t he mean to est abli s h t he absolut e ti me l ike pri ncipl e o f glo bal s ynchronizatio n. That is to s ay, t hi s o ne is affi r med as a fourt h i mplicit pri nciple, alt hou gh it remai ns beyond any evi dence: li ke an as su mptio n of met aphys i cal charact er. Thi s way, t he Newto nian absolut e ti me pl ays t he same rol e for t he thr ee pri nci pl es that the unconditio ned consciousnes s or Purusha for t he thr ee guna s. Therefore, Newto n's t hird pri ncipl e eli minates any means o r real mediation i n benefit o f synchro nous inst antaneous reactions. Thi s way, it is consoli dated and clos ed t he pur el y mechani cal descri ptio n of t he change and the inter act ion bet ween obj ects . Lit erall y, t here i s no space ­ nor time- fo r anythi ng els e. This co mbi nat ion of t he t hir d princi pl e and t he diff erent laws of co ns er vatio n t hat ar e t hei r coroll ar y wit h t he absolu te ti me o r global synchro ni zer co nfor ms what we under stand, or rat her we gu ess , as causality. But i t i s to noti ce t hat t he caus alit y never i s defi ned expl ici tl y ­ becau se i t can not be defi ned here. The pri nci ple o f glo bal synchro ni zatio n is co mpl et el y co ns er ved in s pecial and gener al r elati vit y, by means of t he Lo renz tr ans formatio n and the not ion of co vari ance. It i s for t hat r easo n that i n t hes e t heori es is often s poken about t he "pri nci ple of cau salit y", alt hough it i s not defi ned here at all , but o nl y understood under the assu mpt io n that the events "locat ed" in t he futur e do not aff ect to tho se "located" i n t he past . In qu antu m mechanics, no longer i s po ssible to s peak of cau salit y, alt hou gh t he cons ervatio n pri nci pl es ar e assu med; t hat is to s ay, a Newto ni an i nter pr et atio n o f s yst ems co mpl et el y diff erent fro m cl assi c pro perti es r emains to a gr eat ext ent. One as sumes t hat t he t hree pri ncipl es ­wit h t he fourt h i mpli ed- are suit able preci sel y becaus e they defi ne an i deal or clos ed s ys tem, so that it i s al ways pos sibl e to be co nti nued at t he ti me of verif yi ng t he det ails . But what it happens soo n i t i s gener all y quit e di ff er ent: t he details ar e stu di ed to fit t hem wit h the pr eviou s i dea o f gener al bal ance. This way, t he behaviour of st ati sti cal mechanics or t her mo dynamics can be sent to t he part of t he qu antum mechani cs t hat i gno res tempo ral pro blems to gi ve back t he bal ance of accounts to t he clas si c mechanics wi t hout tou chi ng it. Thi s way, questio ns t hat ar e co nsi dered like ir reversi bl e wit hin the ther mo dynamics, ar e camouflaged i n a peculiar game wit h t hree bands: t he cl as si cal mechani cs, t he qu antu m mechani cs , and a t her mo dynami cs that onl y no mi nall y deal s with t he irr ever si bilit y su bj ect . If we t hou ght i n t er ms of i nfor matio n, for example, we coul d t hi nk t hat t her e is no better defi nitio n of the informatio n that t he one of i nequal it y bet ween an actio n and a r eactio n. Local as ynchrony woul d be t he mo st appro pri ate measurement of t he co mmuni catio n o r mediation.

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The t hr ee l aws of t he movement become tr ans formed, t hrou gh t he more mo der n variatio nal des cr i ptio n, develo ped amo ng other s by Fer mat , Lei bniz, t he B er nou illi, Eul er, Lagr ange, J acobi or Hamilto n, in t he pri nci pl e of least action, also deno mi nat ed of extr eme or st atio nar y action. Poincar Č, mor e than o ne hu ndred years ago, al r eady made noti ce that any ph ysical model that can be des cr i bed i n ter ms of t he l east act io n ­ a gravitational or elect ro magnet ic sys t em, fo r exampl e, al lows i n f act inf i nit e causal expl anations. It cou l d be expl ai ned by means of el asti c ru bber s o r spri ngs, or any pictures que The l eft hand of C hao s
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device t hat we want ed, pro vided it s ati sfied cert ain r equirement s ­pr eci sely t ho se of t he statio nar y actio n. That is to s ay, it do es not defi ne t he caus al r el atio n at all, since t his o ne never can be u ni que. Thi s i s t he main r easo n by which t he phys i cist s pr ef er to s peak i n t er ms of des cri ptio ns, pr edictio ns and equ atio ns; alt hou gh, finally, few o f whi ch us e t hes e t er ms wi ll doubt to have cat ch t he tru e causal nexus es. That t he pri nci ple of l east actio n is botto ml ess from t he cau sal point of view, is somet hi ng t hat cannot be str ange to us . Si nce it has been no ticed so meti mes ­ and parti cul arly by Ni kol ay Kozyr ev -, t he cl assi c or Newto nian mechanics, i n virtu e precis el y o f the thir d pri nci ple of t he mo vement , al ways co ntemplates a spa ce int erval , by ar bitr ar ily s mall t hat could be, bet ween a cause and an effect, or bet ween an act io n and a r eactio n; but it does not contemplate any time i nterval, by ar bitr ar y mi ni mum or t hat is . That' s to s ay, t hi s i s t he o nl y cause o f t he cau sal l aps e. Fo r t he Samkhya, nevert heless , a r eal time interval do es not exi st t hat i s not co nst ituted it s elf by a modification o f t he guna s ; t herefor e, it is natur al t hat vindi cat es t hese l i ke t he u ni qu e cause o f t he pheno mena. Of cou rse, qu antu m mechanics , unli ke clas si cal mechani cs, contemplates mi nimu m ti mes of r eactio n for the i nt er act ion bet ween parti cl es . But , exactl y her e, i n t he qu antu m domi nio n, it is wher e t he caus ali t y pri nci pl e is elu ded, seemi ng co mpl etel y inapplicabl e, i n benefit o f a pro babilist ic int erpr etatio n ­ alt hough accor di ng to so me aspects physi ci st s s peak of t he u nit ar y or det ermi nis t evolutio n. This way t he pro blem agai n is avoi ded, and sent to war ds another part. On the ot her hand, t he pri ncipl e of mi ni mu m o r stationar y actio n al ways seems to i nt roduce a tel eolo gical aro ma i n t he descri ptio n of a s ys tem, si nce t hi s o ne appears gui ded someho w by t he pur pos e. Nevert heles s, we mus t consi der that this uncomfor tabl e appear ance ari s es pr ecis ely from a ti me model t hat, like ou r "o ne way pendulu m", si multan eou sl y i mpo s es t he r ever sibility of t he actions and reactio ns and t he idea t hat t hes e o ccur i n an irreversi ble di rectio n. That i mpo s sibl e pendulum is t he nu cleus of all our par adoxes . The physi ci sts use t hr ee di ff er ent physi cs to better elude that impossi bi lit y.

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Let it be so , at l east for t he mo ment . It woul d be thought that the Samkh ya, bei ng a self-o bser vation di sci pli ne, does not have anyt hi ng to do wit h t he empiri cal domi nio n of t he ext er nal f acts . But t he Samkhya co ncei ved at l east o ne pur el y empiri cal s ci ence, to whi ch f ew mo der ns , neverthel es s, would give valu e enough. Thi s s cience was t he Ayur veda, the sci ence of t he care of t he li fe, no t ver y dist ant i n the s pi rit of ot her old medical art s, li ke t he Hi ppo crati c medici ne or t he C hines e. What di sti nguis hes its hu mor al t heor y is to be an appli cation of t he fr ame of guna s. Three hu mour s o r do shas , vata , pitta and kapha, or wi nd, bile and phl egm, ar e not but reacti ve for ms of t he i nher ent properti es of s enti ence, activit y and i nert ia .B y reacti ve it has to be u nderstoo d t hat its pr edo minance o n t he ot her pro perti es i s abl e to generate imbal ances and ai l ments throughou t the time; but also t hat t hey are a seco ndar y or derived fo r m from mor e o ri gi nal and les s pr ivat i ve mo dalities i n t he co ns titutio n of the su bj ect: p rana, tejas and oja s, t he vital breat h, t he brilli ance of t he co mbus tio n and t he radi cal oil t hat per meat e and nouri s h t he bo dy. We are t hen i n f ro nt o f mu ch mor e materi al qu alifi catio n of the gunas , but not for t hat reaso n l ess su bj ect to t he s ame fu ndament al lo gic. The i nfluence of t he ayurvedi c concepts in aut ho rs li ke Plato has alr eady been noticed enou gh by the schol ars, i n particul ar i n t he Ti maeu s, wher e it i s cl earl y spo ken of t he "ai r, phl egm and t he bile", apart fro m to mentio n "t he establi shed perio ds o f t he lif e... i n co nfor mit y wit h t he tri angl es of the nature o f each o ne". It do es not sto p being peculi ar to verif y ho w t his boo k of mys t er ycal root s and pol e of the enli ght ened her meti ci s m has pro vi ded a rational ins pir ation for varied r eaders of sci enti fic di e, fro m Kepl er and Gal ileo to Heisenber g. Anot her of the pl atoni c di visio ns o f the soul i n veget ati ve, s ensitive and ratio nal soul, also is of net Indi an ori gi n, or i n any cas e, it admit s at least a strictly congruent su per position wit h kapha, pitta and vata , and t he co rr es pondi ng functio ns o f vegetati ve pl asti cit y, ir rit ability and s ensitivit y. E ven t he t raditio nal co ntempl ation o f t he man li ke body, soul and s pirit , keeps a stri ct co rres pondence, if no t i n extensio n or expres sio n, yes in t he most i nt i mate nature o f it s cor relatio ns ; bei ng t hese mo des the onl y thi ng t hat matt er. W e wo n't allu de to the nu merou s t ri ads of i dentical nature, whi ch t he r eader can easil y i magi ne. The l eft hand of C hao s
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13

In pri ncipl e, any cor res ponding attempt to defi ne better t he guna s o r the t hr ee r el at ed do shas exhaust s in mer e epit het s and par aphr as es . W e can u se so me of t hem like semanti c approaches , granti ng to t he semanti cs t he neces sar y r es pect. Aft er all, our i nvesti gation can t ake us to t he co nclusio n that it does not exi st a mean to elimi nat e the "s emanti c questions " tot all y. The t hr ee guna s al so can co mfo rtabl y be cl assi fi ed i n cent ri petal, centr ifugal and or biti ng tendencies. As far as t he do shas , t he as pect r el ativel y mor e mat erial of t he pr imary qualities, pitta tal ks abou t to t he heat and t he energy, kapha to t he pl asti city and soli dity, vata to all t he pheno mena of cir culation from which t he for m i s deri ved ­t he pu rest and iso labl e as pect o f t he for m whereas i ndependent of t he dynamic or plasti c qu aliti es. It coul d be o bj ect ed t hat t his l ast o ne i s an entel echy; we will tr y to see that it is not thus . Co nt empl ating a tr ee we can s ee and tou ch char act eristi cs su ch as colou r, for ms and t exture. To s ay that t he intr insi c co mbi natio n of t hose exter nal char acterist ics i s t he r esult o r the appear ance of primar y qu aliti es li ke t he growt h, co ndens ation and for matio n coul d sou nd u nspecifi c and ar bitrar y. But if we saw the tree bu r ni ng, we coul d disti ngui sh perf ectl y bet ween that whi ch bur ns pr eci sely, somet hi ng t hat i s reduced to as hes, and somethi ng t hat r eject s bo th t hings and saves i n st eam and smo ke. Th e for m, i s what it evapor at es and vanis h, eit her we s ay it i n t he su perfici al s ense o r t he deepest. St eam and smoke i s not si mpl y t he result o f whi ch i s r eall y bur ning, bu t ver y o n the co nt rar y is what it fl ees to burn, as well as, o n t he ot her hand, fl ees mo mentaril y f rom t he gravit y. Alt hough t he scruti ny o f t hi s can t ake much mor e far , is suff ici ent to make see that, i n the mo st violent tr ansfor mation, whil e t hes e qu ali ties stil l mai nt ai n so me relatio n to each ot her, t hey al ways co ns er ve u nmi st akabl e behaviours. The for m is not an abstr actio n at all , u nl es s we u nderst and by abstr act io n, the di still ation or ci rcul atio n made by t he s ame tree and i ts natur e.

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Wit hi n this ayurvedic f r ame of t he doshas , t he cl inical procedur e par excell ence t ake pl ace: t he nadi vig yan, the aus cult atio n of t he pul se. Thos e who pr acti ce it wit h unequal fortune ar e deno mi nat ed vaidyas , si ght ed per sons accor di ng to t he lit eral tr anslatio n. The auscu ltator successi vel y f eels the radi al arter y of t he wri st of t he su bj ect wit h t he thr ee middl e fi nger s of t he hand, alt ernating the pr essure, i n a certai n analogous way to the tou ch of a st ringed i nstru ment. The aus cultation beco mes aft er a suffi cient int erval of rest, and in the co nditio ns of t he s mall er di stur bance pos si bl e. In t he pul s e f i ve elementar y pro per ties are r eco gni zed: r ate, f requ ency, i ntens it y, amplitu de and form. Bei ng a bio lo gical oscill ato r wit h fluctu atio ns, it is perf ectl y suit abl e to t he study of it s tempo rar y seri es ­ by means of t he cl as si c har mo nic anal ysi s, or t he most r ecent o f wavelet s, or anyone of t he sophist icat ed tools of the mo der n anal ysi s. The li mit ations of t he har moni c anal ysi s o r Fourier anal ysis for s yst ems of i rr egular profil e are well kno wn; alt hou gh t he ar teri al pul se neces saril y has accu s ed regul arit y enough, it is o ne o f tho se s yst ems i n whi ch t he s mall deviations and peculi ariti es do not f i nd eas y int erpr etatio n withi n t his t echni que. As it i s well kno wn, the har mo nic anal ys is deco mpos es any perio di c empir ical cu rve i n a su m of s ine waves o r si nusoi ds . Gi ven t he i ndependence of each one of t hes e si nusoi ds , whi ch we o btai n is a scru ti ny of stat isti cal natur e on so me of t he pro perti es of t he cur ve. Another ver y dif ferent t hing is to i nt erpr et t he dependenci es and bou nds that are rel evant i n a s ystem li ke t hi s, fo r which t he har moni c analys is never will co ntri bute expli cit proo fs. The l eft hand of C hao s
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And, of cou rs e, spect ru ms of t he pul se and o f t he el ectro car dio gram parti cul arl y have been made i n innumerabl e amou nts si nce Mo bit z, van der Pol and van der Mar k gave t he f irs t steps i n t hi s area.

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Of the f ive element ar y pro per ties t hat t he au scu ltator di stingui s hes of natur al fo r m wi th it s o wn fingers , t he four fir st ­r at e, frequency, i ntensit y and amplitu de- admit a r el ati vel y accept abl e i nt er pr et atio n i n ter ms o f har mo nic anal ysi s. The fr equ ency is t he i nvers e of t he perio d, or t he nu mber of beat s o r cycles i n a suffici ent ti me i nt er val , we say a mi nut e. The r ate i s the measur ement o f the r egularit y o r irr egul arit y of t he cycles t hroughout t he i nter val . The i nt ensi t y is t he peak altitu de r eached on t he ti ps of t he wave o f t he pu lse. Finall y, t he amplitu de i s t he tot al di ffer ence bet ween t he maxi mu ms and mi nimu ms o f t he wave. In addition to these four mor e analyzabl e el ements , we have the fo rm of t he pul se, t he tru e and evasive qui ntess ence t hat surrou nds and penet rat es t he values of t he o t her four pro perti es . In t he mo st tri vi al s ens e, t hat is to s ay, accor di ng to t he r epr esentatio n of the graph, it is cl ear that the for m i nvol ves t he ot her co mpo nent s, si nce t he form i s t he whole cur ve and it do es not have anyt hi ng el se. Nevert heles s, for t he auscult ato r ot her s hades exist i n additio n, si nce it modul at es t he pr essur e o f hi s fingers i n thr ee diff er ent poi nts. In su ch sens e, and even without co ntr adi cting t he pro perti es o f the cu r ve i n the gr aph, it extr acts so me additional i nfor matio n. Si nce t he ver bal characteri zatio n o f thi s for m of the puls e as sumes poet ic or metapho ri c char acters, many peo pl e tend to t hi nk t hat it is here where we enter a qu alit ative do minio n and we move away f ro m t he qu antifi abl e o ne. For example, it is s po ken of a t ypi cal "swan pulse" for t he mar ked predomi nance o f kapha, of a "f rog pul se" for pitta, o r a zigzaggi ng o r "s erpent pulse" for vata. But, natu rall y, thes e i ndicatio ns l eft by the cl assi c texts as a mere i ndi catio n, wit hi n a do mi nio n of conti nuous s hades ar e too si mpl e. Therefo re, we bett er not decei ve by all t hat t er mi nolo gy: we ar e tr eati ng abou t a qu anti fiable and perfectly anal yzabl e pheno meno n, and it is precisel y by it s flui dit y, by it s co nti nuity, t hat is mor e tr eatable i n anal yti cal ter ms t han for a no t very well trai ned judgement.

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It i s co mpl etel y cert ai n t hat the you ng and not so young medical pr actitio ner s of t hes e tr aditio nal medi ci nes , whether the Ayurvedic, t he Chi nes e or t he Unani medici ne, complai n the dar kness of t he classic text s and t he gr eat diffi cult y t hat ent ails to acqui re a mi nimu m of securi t y i n t he judgment for t hi s t ype o f di agnos is . In ot her wor ds, t he auscultation by t he pu lse is i n cl ear r egres sio n, becau se it is not co nsi dered pr acti cal enou gh. I am afrai d t hat t his has been al ways t he case, and t hat at any ti me have been very little do ctor s abl e to mas ter t hi s art. But her e what it i s at i ssu e i s not t he technical diffi cult y of a l ear ni ng, but t he fideli t y of t he pul s e li ke si gnal of a great er s ys tem and it s possi bl e r el evance i n ot her spheres of t he anal ysis and even dynami cs. If we u nderstood i n a legiti mat e analyti cal for m all t he i nfor mation t hat cont ai ns the pul se, and t he wider perspecti ve than can off er u s, t he pr acti ce of t hese s ame ret icent doctors would be o ne of the fir st ar eas i n benef iti ng fro m it. Bu t we t hi nk t hat t he puls e has mor e enou gh to t each to us . The pulse is t he true pendu lum t hat we are loo ki ng for.

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Li ke any purel y empiri cal s ci ence, t he diagnosi s by means o f t he pu lse has peculi ariti es t hat r es ist to bei ng t r ansf err ed to a quantifi able and exact fr ame. We will mentio n so me o f them i n passi ng. The au s cult ator obt ains pr es sur es and curves o f t he wave wi t h a di ff erent for m for t he di ff er ent points in which he lo cat es his fingers. Nevert heles s, t he mo der n doctors have stated, i ntrodu ci ng a cathet er , t hat t he pr essu re is t he s ame i n t he t hr ee al lu ded to poi nts. This has been wanted even to s ho w t hat t hi s i s a fr aud; at least for u s, t his class of f ast di s qu alifi catio ns i s one of t he mai n i ngredient s o f bad s cience. B ecau se it is evi dent that to feel a si gnal f ro m t he outsi de has littl e to do wit h i nt ro du ci ng sur gi cal sou ndi ngs, and, at l east, the po s si bility wou l d be du e to grant to t he ancient doctor of obtaini ng as mu ch dat a as po ssi ble, mai nly if he has the leas t co ncei vabl e means. The differ ences of t he for m of t he wave ar e due, of cour se, to t he gr eater pro xi mity to t he bo ne or t he s ki n; t hat is to s ay, it i s a dif ference that we mo der ns woul d consi der pheno menolo gi cal, alt hou gh t hu s we ar e not doi ng full justi ce to the su bject . Becau se, i n ef fect , t hese thr ee poi nt s are a ki nd of aver age to be able to per cei ve t he su perfi ci al , deep and i nter mediate aspects t hat are co nju gated so well wi t h the same natur e of t he tr idosha ; wit hout for getti ng t hat i n t hese medici nes, and t he same natur e of t he body, a natur al di ff er enti atio n bet ween t he bo nes , the fles h and t he s ki n exist s, of ver y si mil ar or der to whi ch at embr yolo gi c level we r eco gni ze bet ween endo der m, meso derm and ecto der m. So that, not wit hout hu mou r, we coul d say t hat what do es t he vaid ya o r its coll eagu e of other tr aditio nal medi ci nes , i s very s imil ar to empir icall y so l ve a syst em of parti al derivati ve equ atio ns, of t hose i n which we obser ved t he evolution of a vari abl e while we mai nt ain t he ot her ones co nstant. Kno wn it is t hat t he parti al deri vati ve equ atio ns appear ed f irs t i n mathemat ics i n t he prehisto ry of t he har mo nic anal ysi s wit h the approach of D' Alembert to t he problem of t he vi brant cor d ­and t hat t he measurement of r efer ence of Gali leo for hi s first o bser vat io ns of t he pendulu m was his o wn puls e. On the ot her hand, it i s cert ain enou gh t hat t he diff er ent traditio ns t hat have made u se of t he pul s e di agnos is have diff erent catego ries , but t his mai nly aff ects t he t er mi nolo gical fr amewor k, mo re t han to t he same concept s, that ar e remar kabl y ho mogenou s. The s ame clas sifi catio ns of typi cal pat holo gical puls es are even r epeat ed abu ndantl y, alt hough su ch cl assifi cations do n't pr etend to be anot her t hing that empi rical appro aches and esti mations. For exampl e, t he ayur vedi c medici ne s peech i n t er ms of the tridosha , and t he C hi nes e o f yin and yang, deficiency and excess : but bot h ar e speaki ng cl earl y of a balance. Ot her wi se, we wi ll be co mpl et ely mist aken if we t hink t hat the Chi nes e sys tem i s du ali st, or even bi nar y: t he excess and t he defi ci ency presuppose t he mean o r i nclu ded mi ddl e ­t he vacum, whi ch prevail s o n both. It coul d not be expect ed l ess f rom a cu ltur e t hat protects and values the mai n t hi ng wi t h the ell i psi s. And nevert hel es s ho w many western s cholars have not seen i n the Chi nes e vision of t he pol arit y an yt hi ng el se t han t he mor e r efi ned for m o f t he bi nary not atio n. During a lo ng ti me, the mechanical sphygmogr aphs or pulso gr aphs have not been abl e to cat ch all t he su btl eti es of t he for m of t he pul s e. If , for exampl e, we u s e a Do ppl er flu x-meter, which we will o bt ai n i s an average of t he s peed of the flo w of t he bloo d i n all t he vess el, t hat apart fro m that r es pects t he B er nouilli l aw and has a par aboli c profile, wit h gr eat er speed i n t he centre o f t he vessel and mi nor i n it s wall, and t he co ntrar y for the pr es sur e. Thi s way, har dl y we can o btai n in t he graph so me notio n of what it can be a "hollo w" pulse, or one "sli ding like a ball i n a por celai n cu p", or "rou gh l ike a worm ni bbli ng a mul berr y l eaf ". Obviousl y, t he fingerti ps t hat feel s "s ees" here mu ch more t han what t he eye can o bser ve i n aver aged and i mpo veri s hed gr aphs, alt hou gh i n fact, it is si mpl y sensiti vity and possi bilities of modul atio n. Of cours e, no w we can have multi ple classes o f s ensor s , includi ng hi gh fi delity micro pho nes and adjust able pres sure s ensors. It i s po ssi bl e t hen to s ay t hat, u ntil recentl y time, t here has been no exper i mental po ssibil ity to gat her mechanicall y t he su btl e range of i nfor matio n t hat t he physi ci an o btai n s i mpl y wit h it s fi ngers . No w t hat has sto pped bei ng pro bl em; the pro blem is t hat we ar e i ncapabl e to anal yze quantit ati vel y t hese data wit hout gro ss l y di storti ng t he o rigi nal coor di nat es t hat s erved i n hi s judgment to t he doctor. As we i ndicat ed, t her e have been made an infi nit y of s pectr al anal yses of t he puls e ­and it is to no tice t hat we s ay of the pul s e, not onl y of el ectro car diograms -, but, For what s er ves to say that below a value o f 100 ar bitr ary u nit s the pat ient s how i nsuffici ency, and that o ver t hat t hres hol d it s hows an excess o r pl etho ra of which it is wanted? Even i f it woul d be certai n, we have los t by t he way all t he lo gic t hat attended t he docto r, and we have tur ned it into inexpli cabl e nu mbers. And pr ecisel y what we look for it is to keep awake t he amplitu de and i mport ance of t hat The l eft hand of C hao s
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logi c and t hos e coo r di nat es . Quite exas perat i ng quest io ns exist soon, li ke t he fact t hat har dl y so mebo dy is put in agr eement to det ach and to lo cate t he di ff er ent parts of t he i nfor matio n o bt ai ned. Someone take t he pul se o nl y i n o ne t he wri st s, ot hers i n bo t h wrist s ; ot her s stil l as sure t hat t he meani ng of bo th hands i s r ever sed dependi ng o n t he gender. Neither in t he C hi nese medici ne t her e i s agreement wit h res pect to t he r el atio n of the meri di ans with the poi nts of t he wri st, etcet er a. It i s not to sur prise, t her efor e, t hat what i t seemed i n t he begi nni ng a relati vel y si mpl e qu es tio n becomes i n fact a dens e for est. Thi s is what has moti vat ed so many hurri ed and coars e att empt s of qu antifi catio n, wit h the r at her vain hope to tu rn t hi s t ype of diagnosi s so met hi ng mo re o bj ecti ve. One as sumes, yet , t hat it is t he kno wledge and not t he crude nu mber s what tu rn somet hi ng o bjecti ve. And nevert hel es s, it remai ns a doubtl es s fact : t hat we will no t fi nd anot her si gnal mor e fai t hful and i mmedi at e, neit her mor e globall y r epr esentati ve of t he i mmense set of f actor s t hat are u nited to for m that that we call lif e, that t he pul se.

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The pro bl em, seems to be, is t hat here we lo se our sel ves in t he forest of t he semiolo gy. The pres ent car diology makes ever y day new advances preci sel y in as much it is able to avoi d t hese s emioti c questio ns, to turn t hem functional and anato mical det ail s. Once agai n, the reductio ni st spirit s eems to prevail. But t hi s has mu ch o f il lusio n, because between what is "functio nal " and what is "anatomi cal " an enor mou s abys s r emains . As i de from the fact that the vaidya per secut es a mu ch mor e general o bj ect t han t he anato mical or fu nctio nal state of the heart: t he state of healt h of t he whol e o r ganis m is u p to hi m as well as t he det ail s mor e r el evant than co ncur i n i t, and being the stat e o f t he hear t onl y o ne of them. And s peaki ng of s emio logy, and l eavi ng apart t he f act t hat t he hu man ki nd never left to i nter pret signs , it can be sai d t hat t he puls e di agnosi s, and i n particul ar wit hi n t he Indian scheme, it i s t he fir st co nsi stent semiolo gy as much fro m t he r ational poi nt o f vi ew, as from t he experiment al one; alt hou gh it seems t hat t her e woul d be numerou s ar gument s to doubt it aft er whi ch we have s aid. Yet, if we o bser ved t he modern attempts in t his fiel d ­t he semioti c t riad of P eir ce, for example, t hat so mu ch has to do wit h t he gener al s cheme of t he triguna -, we will verif y t hat still t he ol d mo del s keep advant age, s peci all y at the ti me of t he experi ment al co nfro ntat ion wit h t he brut e facts. In addition to t his, and wit h regard to t he pretended opposit io n between t he redu ctio ni sm and the glo bal i nter pr et atio n o f t he phenomena start i ng f ro m its perfor mances, i t is wor t h the trou bl e to r emember that it is t hi s l ast what all t he men of sci ence and philoso pher s wort hy of su ch name, f ro m Newto n and Kant, have denomi nat ed with t he name of anal ysis , and no t on t he contr ar y, as sometimes t he custo m and t he for getfulnes s i ncli ne to beli eve to u s. Then to explai n t he behaviour of a syst em fro m it s basic co mpo nents, i s what i s denomi nat ed s ynth esis or co mpo sitio n; composition t hat most of the times i s so far from our reach t hat onl y takes place i n our i magination. Never t hel ess, t he i magi natio n i s the les s suit abl e that can exist to follow the behaviour of any mo der at el y co mpl ex s yst em, and we do n't say alr eady, somet hi ng wit h t he complexity o f an or gani s m. One o f t ho se glo bal anal yti cal i nt er pr et atio ns i s, pr ecis ely, t he t heo r y o f the gr avit y of Newton, that to dat e, no body has managed "to co mpo se" no r "to s ynt hesi ze". The i nt er pretat io n of the har mo nic analysi s lo cat es to us i n that same meani ng of t he t er m anal ysi s, now wit h ar bitr ary degr ees o f co mpl exit y. Befor e we s ai d ho w badly t reat s the physi cs wit h t he causal l evel of t he phenomena, and t he basic r easo n for whi ch t hi s cannot have pl ace i n it . Hume's criti c to t he caus alit y co ncept is wort h fo r a repres ent atio n li ke t he one o f t he classi c Newtoni an physi cs; but t hi s i s onl y a frame of repr esentatio ns, and ou r di rect i ntuition of t he change is not lent so easil y to t he r epr esentat io n. More o n t he co nt r ar y, may be eludes it per petu ally. The T riguna of the Samkhya is a s ystem of coo r di nates for t he change, t he si mplest , mor e dir ect and intuit i ve possi bl e. But preci sel y what we u nderst and by s yst em of coor di nates is t he clearest of t he repres entat ions , and here we ar e in t he anti podes of t he represent atio n. We must always have in mind thi s fundament al diffi cult y.

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The t hr ee doshas or i mpli ed reacti ve hu mours i n the mo vements of t he pul s e corr es po nd i n addition wit h t hr ee components or pr ecis e mechani cal moment s: i n t he most general sens e, pitta corr es po nds wit h t he co ntr actil e i mpuls e of t he heart, va ta wit h t he state of t he walls of t he vessel and its conductio n o f t he co ntr actil e wave, and kapha wit h t he fluidit y or vi s cosi t y of t he bloo d ­ we s poke, natural l y, of an element ar y corres pondence, not of a r eduction fro m one ter ms to t he ot her s. Even li ke i ndependent t erms of a pro bl em o f flui d mechanics, t hese t hr ee factor s al ready ar e suffi ci ent to ori gi nate a non- li near s ystem wit hout st able valu es in t he solutions ; but i n t he organic domi nio n, and of cou rse i n t he o ne of the Samkhya, t her e i s an intri nsi c dependency and f eedback between t hem. In t he pul s e t her e is no conti nuous t r ansitio n bet ween s ys tol e and diastol e, but t hat, even i n s pit e of t he dampi ng o f the impacts that su ppos es t he flo w, t he paus es co nti nues bei ng apprai sed, so t hat t he s cheme i s per cei ved li ke cont raction- paus e-expansion- pau s e. That is to say, t hat for many eff ects, i n additio n to the co nti nuou s si gnal of t he wave, we have a genui nel y dis cr eet sys tem, as it is deri ved from the dou btl ess f act that t he ner vou s si gnals that acti vat e and deacti vate mu scul ar f i bbers of t he heart ar e antago ni sti c and they ar e exclu ded and mutu all y blo cked. As we s ee soo n, t hi s do es not lack i mportance, si nce it i s in t he base of t he onl y co ncept of os cill ator or ci r cuit wit h an own co ntent: t hat i n whi ch t he ci r culatio n exi sts for t he o nly reason t hat two co nditio ns cannot be s ati sfi ed si mult aneous l y.

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fig 1 Of cou rse, and unli ke t he pu re s cheme o f t he mo vement in t he cl as si c mechanics, t he li vi ng bei ngs and the or ganis ms ­ot herwis e t he s ame t han the el ement ar y par ticles- have characteri sti c times of reactio n, which cannot be avoi ded. This is evi dent f rom the mor e el ementar y empiri cal poi nt of view, but it t ends to be dilut ed and to lose its s peci fi c i mpo rtance as we tr ansl at e it to habitu al anal ytical des cri ptio ns . Per haps t he mor e i mmedi at e for m to warn our s el ves of t he use of t he t hir d princi pl e, the equ alit y of action and r eactio n, is s hooting a gu n and verif yi ng i n our bo dy t he backwar d mo vement . Natur all y, t he butt do es not str ike to u s wi t h t he s ame departur e s peed o f t he bullet, but, appro xi mat el y, wit h t hat o ne which it i s quoti ent of the mas s o f t he bullet with r espect to the one of rifl e. In addition, and if we want ed to ent er detail s , al ways we will fi nd that t here have been los ses by gases of t he explo sio n. If we o nl y kno w t he exi t s peed of the proj ect ile, o r t he o ne of t he backwar d movement, we needed to know what has happened i n t he mi ddl e to pr eci se the bal ance; and kno wi ng of the action and r eaction, we even needed to kno w t he mean or appar atus to be abl e to preci se t he way i n t hat t he los s takes place. Detail i ng, we woul d al so veri f y t hat t he r eactio n i s not si mult aneous and i mmedi at e, but al ways i nclu des a time, alt hou gh for many ef fects it seems to us negli gi bl e. I ho pe t hat so elementar y exampl e will be par do ned to me; but of ten it happens t hat, when we stu di ed t he organi s m as a black bo x wit h so me or many out put dat a ­whet her t hey are anato mical or anal yt ical- we r etu r n to tur n t he s ame data anot her bl ack box to study its el f, losi ng i nt erest of t he i nputs and output s t hems elves. That is to s ay, ever ythi ng beco mes i n t he mi ddl e, u nlike the example of t he gun, and we for got what it has ent er ed and left in bot h dir ectio ns , whi ch i n t he case of t he or ganis m sur ely i nclu des somethi ng mor e than t he di et and t he depos itions . Yes , all t his rou nd about was o nly to say t hat we can appreci at e t he mean and a s ens itivit y bet ween action and r eactio n in t he global co nditions of t he or ganism wit h no need to contr adi ct t he i ndis pens able l aws of t he mechanics. How i t coul d be other wis e? Neverthel ess , Can we gras p t his i n so me way si multaneous l y el egant and di rect? Thi s alr eady seems anot her questio n.

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It coul d al so be added t hat t he human bo dy, li ke any or ganis m, is ess enti all y an o pen sys tem. And, o f course, for t he Samkhya any o bject t hat we like to consi der is an open s ystem; r easo n why t he gunas emerges like t he effi ci ent caus e ­ and not li ke mere formal l aws - of o pen s yst ems i n i nter change with t he environment , in t he same way t hat we appli ed t he three laws o f t he mo vement to i deall y clos ed sys t ems . Yet, to s peak about the or ganis m as an open s ystem does not i mpl y to r educe it i mmediatel y to a dis sipati ve s ys tem, li ke oppo sed to a co nser vati ve o ne. Si nce an organi s m has boundari es and st abilit y co nditions whose r each we onl y know i n a s mall degree, alt hou gh t hey ar e mani fest ed enough i n its effects. It is not s mall tas k to defi ne t hese di ssipative and co ns er vati ves as pect s i n the livi ng beings ; but t he pr edo minant st rategy to approach t he pro bl em at t he pres ent , t hrough t he compl exity and t he organization ­wit h t he genetics and t he biochemi stry as it bas e- har dl y can get to fulfil t hi s r el ation, i f we co nsi der t he enor mou s epi stemolo gical gap that we noti ced bet ween anal yt ical element s, co mpo nent o f a s ynthesis , (they t hems elves co mpos ed always ) and cau s es .

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As it is kno wn, in flui d dynami cs we can replace t he thr ee Newton' s laws by vari atio nal pri nciples, transfor mi ng t hem i nto t he pot enti al, ki neti c and inner ener gy o f the sys t em. While we o nly wor k wi t h t he ki neti c and the pot enti al energy, i t i s gener ally eas y to know the magnitu des t hat ar e conser ved; but when we int roduce t he i nter nal ener gy ( P + K+ I) , t hat can be expr ess ed li ke vis cosit y of t he flui d, pr essur e, o r pertur bat io ns , it is mu ch mo re diffi cult to know what is exactly what it i s co nser ved. Si mpl y, we have to pr esu me t he co nser vatio n princi pl es . This is the great difference bet ween t he hydro dynami cs and the dynami cs of punctual particl es, i n whi ch t he cons erved magnitudes r emai ns i n a expli cit fo r m. The pu ls e, t hat i s a s yst em periodi call y pertur bed by t he beat o f t he heart ­ a for ced os cill ator with power su ppl y-, admit s t hi s t ype of des cr i ptio n; o nl y t hat t he f eed- back of t he facto rs makes it still mor e co mpl ex. The t hr ee guna s and t he t hr ee doshas can be put in a corr ect cor res po ndence wit h t he ki neti c, pot enti al and i nt ernal energy of a hydro dynamic s yst em, whi ch i n fact, seems to us mu ch mo re natur al than t he pu nctu al dynami cs or mechani cs in st rict sens e. Hamilto nian dynami cs, i n additio n, su pposes t he i nco mpr essi bilit y of the space of t he flui d. Thi s i s a fundamental poi nt t hat it cannot be appli ed lit er all y to a s ystem li ke t he vas cul ar o ne, where t here ar e expansio n and co nt ract io n of t he vess el s or art eri es, and not o nl y by virtue o f it s el asti cit y, but also by mus cular fibbers o f t he walls t hat act to ner vous i mpul ses follo wi ng t he co ntr actil e rat e of t he heart . It i s to as su me t hat also the same heart and its r ate of i mpul s es has to be sensible to t he ans wers of the vascular cir cuit , and i n fact we kno w t hat t he glo bal f eedback o f t he s ys t em t akes pl ace at several l evels, ner vou s, as much chemical , not bei ng t he s ensiti vit y to t he flui dit y or vis co sit y o f t he blood t he minor of t hes e el ements. So t hat t he cir cuit and it s ans wer are a mo del example of biolo gi cal s elf-regulat io n, whos e stu dy has gro w u nceasi ngly from the pio neeri ng wor ks of Canno n and Wiener. All thi s allo ws us to r ai s e ol d and int eresting qu estio ns to us i n a new way. Truly, we kno w well enou gh and with many detai ls the exp licit physiologi cal medi ator s, su ch as ner vous, hormo nal , and bio chemical i n general . These wit hout a dou bt pl ay a fundament al and irr epl aceable rol e i n all t his s el f-r egul atio n pro ces s ­t hey wou l d not be t her e i f t hey had no utilit y. What we do n't kno w i s how they have gott en to be t here and what it has gui ded its for mation. Doi ng wit hout ans wer s of type "geneti c progr ammi ng", t hat to day we kno w perfect l y t hat it co nstitutes onl y one part of t he subj ect, and not t he great er o ne, and t hat it has to do much les s wi t h t he "progr amming" t hat with sensiti vit y and t he mo dul ation by t he enviro nment , happens anot her much mor e sti mulat i ng appro ach to us. This is bas ed o n t he co mpr essi bilit y of t he who le cir cuit and anyone of its parts, compr es si bility t hat can and mu st admit ver y variable degr ees, althou gh, of cours e, t hey must be measurable. Of cou rse, i n su ch case it is not a Hamiltoni an o r cl assi c syst em what we have; we no t o nl y co ntempl at e t he exi stence of fri ctio n, but of ot her no n expli cit variables i n i ndefi nit e nu mber. We mai nt ai n t he analo gy and t he co nnectio n wit h cl assi c sys t ems to o nl y have a refer ence o n t he cons er vative as pect s of t he s yst em. As ever ybo dy kno ws, pur e or i deal classic s ys t ems do not exis t, which do es not prevent its rout i ne appli catio n. Never t hel ess, although we are s peaking of s ys t ems r eall y f ar away from t he clo sed and The l eft hand of C hao s
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co nser vati ve models, we ar e i nt er est ed parti cul arly i n t he i nt ert wini ng of two mai n f eatu r es . We can call to t hi s "s emi -class ical sys t ems " provi sionally and o nl y fo r co mfort, alt hou gh by su ch t er ms can be u nder stoo d an inf i nit y of diff erent thi ngs . Speaki ng o f co mpr essi bilit y i n semi-clas sical s yst ems, we entered co mpl et el y which today it is known like no nli near dynami cs , o ne of who se mai n si gnatu res, alt hough not the onl y o ne, is the det ermi nist chao s. This o ne appear s i n pr i nci ple i n perf ectl y di ffer enti abl e s yst ems and wit h ini tial conditio ns of ar bitrar y pr eci sio n, and t her efore way dif fer ent from t he mo del o f the puls e. In f act, it can be s ai d t hat t he det er mi nis t chaos is an al most purel y mat hemati cal di scover y, although t heir r el atio ns with dynamics have been expli cit fro m t he fir st moment, backing do wn i n the time to Poi ncarČ and hi s stu dies of t he celesti al mechani cs and t he thr ee bodies pro bl em. In t he unit ar y evolutio n o f t he phase s pace ­t he Hamiltoni an- of i nnu mer able s yst ems, next to the i nst abilit y of t he solutio ns i n the equ atio ns, appears "basi ns" o r "attracto rs ": t he dynamic evolutio n do esn't di s per se u nifo r ml y t hrough all t he s pace availabl e, but it is compres sed wit h a cer tai n pr ef er ence i n a part of t he volu me. Thi s al ready is a cl ear example o f co mpr essi bilit y, u nles s it i s not cl ear abso lutel y what i s what caus es t hat t he s ystem is compres s ed and why. That's t he r easo n o f the co nt ro vers ial s tatu te of t his t ype of findi ngs. Dif ferent measu res can be t aken fro m t he i nstabilit y and the degree of co mpr es sion o f t he s ys tem ­ Lyapu nov expo nent s, f ract al di mensio n, entropy i ndi ces , and a lo ng list of ot hers -, without o n t he ot her hand ther e is never a cl ear di vi di ng bor der bet ween t he r ando m or stochastic component and t he deter mini st co mpo nent of t he s ystem; except fo r preci sions ad hoc of t he particul ar mo del. Thi s way, the pres ent "r evolutio n" of no nlinear dynamics i s an u nfini s hed revolution, o r so to s peak, "half a r evolution": it has acquir ed t he st atut e of t he o mni pr esence wi t hout bei ng abl e to co ntr i bute no t ype of clear co nclusions. The socall ed det er mi ni st chao s mo re s eems t he beginni ng of o ne lo ng exploration i n t he immens e t erritory of t he rel atio ns between chance and necessit y t hat a t heor y defi ned fo r anyt hi ng. Nevert hel ess, when it r eached t he thr eshold o f t he s pr eadi ng and of t he pu bli c disto rtio n al ready ti me ago it was a sus pended explo rat io n, at least about conclusi ve r esults . And t hus it t akes growi ng mo re of a qu art er of centur y, wit h no parti cul ar direction. The s ame t heo ret ical or fu ndamental phys ics , which many bl ame of stagnatio n, has much mo re s peci fic and cl ear chall enges and ur genci es ; at l east , accordi ng to the pr es ent or der of the appear ances .

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Peo pl e say t hat aft er t he f ever , t he cal m and the reason retu r ns, and life can conti nu e pro gressing wit h a more nor mal r at e. The pr esent st at e o f thi ngs o n no nli near dynami cs and t he t heor y of t he complexit y ­ dif f er ent ar eas but multi pl y asso ci at ed- are so t hat si gnifi cant advances i n a great nu mber of poi nt s coul d be expect ed, but har dl y a tr anscendent al u pset. The expert s mo ve i n the specifi c do mi nio n of t he intricat e and dif ficult, of the co ncret e appli catio ns, wher e rat her are pos si bl e to wait for techni cal r efi nements mor e t han true co nceptu al ju mps . Ever yt hi ng flows and i s mo ved, nevert hel es s. Meanwhil e, and waiti ng for resu lts of more wei ght, it is totall y certai n that no nli near dynamics has changed su bst anti all y t he way we s ee t he wo rl d. The deter mini s m it s elf has done t hat we beli eve less and less i n t he deter mi ni sm like explanatio n of t he pheno mena. Li ke t his change i s su bstantial, but still no n-fu ndament al, as if t he s pear head of our convi ctio n i n the causal explanation of t he pheno mena had been not ched, or i s even di ssol ved i n t he air, whil e t he i nerti a o f our background co nvi ctions co nti nu es mai nt ai ni ng t he s ame dir ectio n.

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W e have s poken pr eviousl y of a par tiall y co mpr essi ble cir cuit i n all its poi nt s: t he ci r culator y sys tem its elf, of which the pu ls e is t he mor e explicit dynami c si gnal. Of cou rse, i f t he ci rcui t wer e to tal l y The l eft hand of C hao s
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co mpr essi bl e, it s flo w i n a torus or ring stru ctu re woul d be r educible to t he s i mpl est of t he fo r ms . Not hi ng of that happens, o n t he co ntrar y: t here ar e ver y si gnifi cant r edu ctio ns o f t he space of flo w, but of al mo st impenetrabl e natu re. In an evi dent way, to look for t he suit able compressibl e space to t he behaviour o f t he s yst em it i s just li ke to loo k for ho w to make it comprehensi ble. Bo t h t er ms beco me t herefor e equival ent. Ian St ewart has oppor tu nel y o bs er ved t hat i t not exist any l aw o f co nser vatio n of the co mpl exit y. Of course, no w it is ver y well kno wn t hat ver y simple s ys tems can complicate t hems el ves, ju st gi vi ng t he suffici ent ti me to t hem. That was preci s ely t he lesso n of the det er mi nist chao s. And i f really t her e is no l aw o f co nser vatio n of the co mpl exit y, we have reaso ns to t hi nk t hat t his i s appli cable i n t he oppos ite sense, t hat is to say, t hat ver y compl ex behaviou rs are deri ved from ver y si mpl e laws. In fact, t his i s equ ally cert ain, and i n t he more tri vi al s ense, for many of t he sour ces of det er minist chaos , syst ems go ver ned by dif fer enti al equ atio ns as famil iar as t ho se of the cel esti al mechani cs. But if what we are s peaking of ar e n ew la ws, wit h a diff er ent des cr i ptive and expl anator y rank, st ill none has been di s covered t hat cou l d make the co mplexit y mor e co mpr essi bl e and comprehensi ble. For exampl e, the du plicatio n of per io d di s co vered by Fei genbaum gener ates complexi t y, but it do es not reduce it, except i n a mini mum thr ead of t he scheme of evolut io n i n a particul ar tr ansitio n of stat e. So that the general s ituatio n co nti nu es bei ng not iceabl y uni directio nal, wit h a bas e of fu ndament al and det er mi ni st physi cal equ atio ns t hat generat e mor e and mor e co mpl exit y accor di ng to they ar e stu died, wit hout bar el y any develo pment i n t he o pposite sens e, t he o ne o f the tru e co mpr essi bi lit y.

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The pr esence of det er mi ni st chaos i n t he acti vit y of t he heart has been and i s a co ntro ver sial subj ect also, and t his i s natural , if we co nsi der t hat t he bor ders are by definition blurr ed i n t hi s do mi nio n. What i s dou btl ess is t hat t here ar e i mpo rtant irr egul ariti es i n its t empor ar y s er ies, and that thes e o ften sho w f ract al or qu asi -fr act al structu res, that is to s ay, s elf-si milar structure wit h cor r elation o n differ ent s cal es and i n the long ter m i n gener al . Anot her t hi ng is to judge and to int erpr et t he rol e t hat such ir regu larities pl ay, if t hey have so me s pecif ic i mpor tance o r it i s si mpl e randomness. Al mo st al l thes e stu dies t al k about the elect rical acti vit y of t he heart . The pul se and its t empo rar y profil e have been studi ed much les s fro m t his pers pecti ve. In general, t he pul se i s a lo wly nonli near s ys tem; t hat is to s ay, it is not asso ciat ed di rectl y neither wit h the chaos nor wit h t he tur bulence. In f act, when t he aus cult atio n per cei ves i ndir ectl y tur bulences t hrough whi stl es or mur mur s, t hey are a cl ear si gnal of car diovas cul ar pat hologi es; i n any cas e, su ch tur bulences o nl y seem to t ake place i n t he great er ves sels , and not i n s mall ves sels li ke t he r adi al art er y. On t he ot her hand, the flow of the bloo d is not tur bul ent i n gener al, but neit her r eall y lami nar, which i s not ano t her t hi ng t han ideali zatio n. Now s peci ali sts even conti nu es stu dyi ng activel y i f the best mo del of t his flo w will not be t he o ne of heli cal or s pi ral t raj ector ies, per haps mor e eco no mi c or o pti mal. The low nonli nearit y of t he puls e is resumed i n the f act that we cannot accur at el y cal culate t he phase angle of t he beat and its wave, not even i n few cycles , alt hou gh nevert hel ess t he global regul ari ty of t he si gnal i s very gr eat , wit hout t he enormous fluctu atio ns i n t he valu es t hat can take place in diff er enti al s ystems i nfi nitel y si mpler, but chaoti c. That is to say, i t mai ntai ns a great glo bal stabilit y; ot herwi se, we woul d not be ali ve. Pr ecis el y by t his , it has been want ed to associat e the car diac f ailu re wit h a tr ans ition to t he chaos , alt hou gh t he rel atio n i s ver y far fro m bei ng cl ear .

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It i s po ssi bl e to as k to what ext ent our i nfor mation o n t he dynami cs o f t he pul se beco mes ri cher when The l eft hand of C hao s
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we tr y to catch t he s peci fic form as it cat ches t he vaidya wit h hi s fi ngerti ps , i n co mpari so n with t he schemati c and averaged fo rm of t he mechani cal registr y of the co nventional es phygmo graph. I woul d wi sh t hat it was not necessar y to say t hat t hi s s pecifi c fo rm of t he pul se i n the pal patio n not hi ng has of mys ti c or mys terious, except fo r the dif fer ences t hat al ways will exist bet ween the sens atio n and t he di rect perception and any anal yt ical r epr esent atio n i n gr aph fo rm, no matt er how exhausti ve. What dif f er enti ates t his fo rm f ro m the general and averaged profil e of the wave is , fro m t he per ceptive poi nt of vi ew, that pr ecis e qualit y that cause t hat a pulse s eems "full ", "empt y", "rou gh", "ho llow", "hard", "t ens e", et cet er a. This al r eady i s a per ceptive synt hes is of factors of dou btles s mechanical charact er, bu t they do not have why to be well s pecifi ed i n t he cur ve of a gr aph: t he i mpul se until cert ai n poi nt yes , but not the cu s hio ni ng of t he flui d, t he st at e of the art eri al wall, and less t he r elatio ns t hat can have between t hese f actor s. And all this alr eady su ppos es a preciou s s ynthesis o f i nfor matio n, by mo re confus ed t han it can seem. Not even it is hi ghpriorit y to detach it, but si mpl y to catch i t and to r egi ster it as f ait hfull y as can. Certai nl y a part of thes e qu alit ative or per cepti ve aspect s can even be i nfer red of i ndir ect for m from "t he fl at" profil e of the co nventio nal regist r y. As we alr eady s ai d befor e, no w we have adju stable pr essu re sensor s and ot her types t hat allo w us to cat ch thi s general featur e. And o f cours e that we can transf er t hi s infor matio n extended to the repres entat ions of graphs, mai nl y as thi ckness or r efi nement of t he outli ne of the wave, i nclu di ng its vari abilit y throughou t the cycl e.

Fig 2 W e can see all t he phas e of sys toli c as cent of t he wave li ke t he ki neti c ener gy, and t he o ne o f reduction or diastol e l i ke caus ed by the pot enti al energy. The whol e cycl e i s already i ntri nsi call y i rr egul ar by functio nal questio ns, li ke t he attack of t he cardi ac i mpact and its lat er di ffusion; but also t he s paci ng bet ween The l eft hand of C hao s
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the beat s, and t heir corr espo ndi ng duration i s ir r egular. In addi tio n, as it can be s een i n fi gur e 1, t he di astoli c part has its own ti p of as cent, r easo n why t he wave i s f ar fro m being si mpl e. In any case, if t he as cent and reduction of t he wave ar e both t he mor e element ar y co mpo nents , all t he other s wi ll be u nder t he corres pondence of t he t hir d co mpo nent, the i nt er nal ener gy, t hat fro m t he poi nt o f vi ew o f vaidya is not another t hi ng t hat vata or s ensiti vit y bet ween actio n and reactio n. "All t he ot her s " ar e not but the i rr egul arit y of t he t emporar y series , eit her we co nsi der it i n t he way of t he "fl at " graph, o r wit h t he more det ailed gr aph that gi ves u s t he qu alit y of t he outli ne o r fo r ms of t he pul se. That i rr egul arit y includes as much t he whol e int erval as each one of t he perio ds or cycles. Evi dently, t he t hr ee factors al ways are conju gat ed, and we can measu re as much t he l i ne or outline length, as t he area i nclu ded by t he routi ne metho ds of nu meri cal int egr atio n, whi ch, i n a co mput er, i s reduced to cou nt pi xel s. Woul d it gi ve t his conju gat ed measurement so met hi ng at least si mil ar to t he bri ef est imatio n of t he quantit ati ve pro port io n and predo mi nance us t hat t he vaidya es tabl is hes when he t akes t he pu lse to a per so n? Sur el y yes, because t her e is no s pace neit her place for other consi der atio ns. Nevert heless , it r emai ns cert ai n t hat t he physi ci an does not have i n mi nd ver y fi ne qu antit ati ve esti mations, which do not need if he kno ws to gr as p the shades, and t hese ar e dilu ted of i ntuitive for m i n another type of co nsider atio ns. But t hes e shades ar e not but t ho s e variatio ns i n t he co nt i nuous do mi nion t hat to t he graphs and the cal culus gi ve u s and that the physi ci an woul d not kno w to qu anti f y ver ball y. Ther efo re, a r eal coi ncidence bet ween the qu antit ative and qu alit ati ve aspects exist s, alt hou gh i t i s onl y expres sed like a mer e par alleli s m. That is to s ay, we have decei ved t he suppos ed di ff er ence bet ween bot h as pect s, maintai ni ng t he pro per cat egori es of t he vaidya and wit hout weakeni ng t hem. It coul d not be other wi se, bei ng t hese genui nel y anal yti cal cat egori es si nce t he begi nni ng. Yet, i n t he ver bal l evel t his o nl y can be a co rres pondence, by t he same fact t hat i n t he ver bal des cri ption t he qualit y necess ari l y pr evail s o n t he qu antifi cation. In f act, t he auscu ltation of t he vaidya , it s l ist eni ng, i s an exer cis e of s ensiti vit y dif ferent as mu ch from the ver bal consi der atio ns as of t he qu antitati ve o nes, even bei ng abl e to i nclu de to bot h.

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In t he first approaches , we i dentifi ed t he variable t hi ckness of t he for m of t he pulse wit h t he s ensitivit y bet ween t he action and react io n, or t he kinetic and t he potential ener gy, i nt ro duci ng somethi ng li ke a ps eu do deri vati ve. A balance with s ensitivit y and a time of r eaction allo wed us to gi ve a ki nd of s ymbo l of t he i mplied factor s.

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Fig 3 But i t was clear that t his o nl y coul d be a symbol to i ndi cat e some ki nd of lost i nfor matio n, and no way a reali stic mo del, and even l es s mat hemati cal. Although this analogy was well clums y and even pictur esque fro m t he mat hemati cal poi nt, it is wort h the trou ble to remember that it i s no t a si mpl e su bj ect to pas s to gr aphi cal and anal ytical repr es ent ations not ions t hat apart fro m to s eem qualitati ve, are conceived beyond the repres entat ion and even, it cou l d be sai d, made s pecifi cally to decei ve t he repres ent ation. W hat we t ri ed to s how wit h that symbo l was t he neces sit y to co ntempl ate t he three princi pl es si mult aneousl y, and no t onl y t wo. In any cas e, t he global s ensiti vit y of t he s yst em o f t he puls e, t he t hird dosha whi ch we asso ciat ed wit h t he i nter nal ener gy, canno t be cau ght i mmediat el y i n any poi nt of t he wave ­it i s no t pu nctu al, neit her has derivative nor ps eu do- deri vati ve -, even t hough the pr essur e, t hat is what t he fi ngerti p feel s, can be co ncei ved as form of i nter nal ener gy by volume u nit, if we follo w t he cl as si cal Ber nouilli equ atio n. But, even doi ng wit hout ot her ess enti al f actor s li ke t he cus hio ning, flui dit y, o r t he stat e of t he ves sel , that can co ncur in t his s ynt heti c repr es entation of which trul y happens, still we woul d be i gnori ng a f act t hat has a gr eat er i mport ance: t hat t he balance between t he t hree facto rs cannot be of i nst antaneou s natur e, at least wit hi n t he f actors t hat can be repres ent ed withi n thi s so li mit ed f rame. That i s to s ay, that wit hi n t his unil ater al and i ncomplet e repr es entation, t he bal ance has to neces s aril y s ho w li ke somethi ng i n the long t er m and no n local, s catt ered on dif fer ent scal es wit hi n an int erval . That i s to say, t her e is no suffi cient space so t hat it can be made i n every mo ment . There i s no space of ti me. In eff ect, already befo re we sai d t hat a perf ectl y co mpressi ble cir cuit wou ld allo w its corr es po ndi ng and i nevit abl e element ary r edu ctio n wit hi n t he si mpl ecti c s pace. That t he ci rcui t was co mpr es si bl e woul d be equival ent, pr ecis el y, to that we could fi nd it co mpl etel y in any poi nt. And t his woul d make t he t i me unnecessar y as well, t hat woul d perf ectl y beco me mono tonous and i ncapabl e to i nt ro du ce alt eration o r no velt y. Li ke t he i deal pendulu m of the mechanics. The time woul d not have to make bal ance bet ween anythi ng, neit her t he o per ation to make t hat balance coul d cr eate or consume ti me nor develo pment at all . "To cr eate or to co nsu me", t wo f aces of a s ame pro bl em. 27

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In t he study o f electro cardio gr ams t he pr es ence of t hes e long t er m corr el atio ns has been demonstr at ed, on diff erent s cales . Al so t here is catego rical evi dence of whi ch t hes e co rr elations ar e richer and more det ail ed in a healt hy su bject , and mi nor s in as much t he pat holo gies get wor se. That i s to say, the most ir regu lar and bro ken pat ter ns ar e healthi er t han monotonous. Thi s co mes to be basi call y i n agr eement wit h the i dea and per ception of t he Samkhya of whi ch a great er s ensiti vit y makes mor e st abilit y possi bl e, and o n t he co ntr ar y. Nevert heles s, t hes e same studi es have not been abl e to go further f ro m t his. Not even is sus pect ed why reason t he car di ac i rregul ariti es would have to be more healt hful than mo no tony, what cannot be sur pris e wi t h the idea of t ime whi ch we mai nt ai n. At t he mo st , so mebody sus pect t hat it coul d have so me deli cate game of bal ance behind thes e si gnal s ­ what it i s not sus pect ed i s t hat the own si gnal s ar e alr eady an exhi bitio n of t hat bal ance. To kno w it is not ver y us eful if we do n't know t he grou nd o n whi ch that balance takes pl ace.

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Fig.4

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If we t hink fro m t he perspecti ve of t he cont i nuous vari ation, it s eems tri viall y certai n that it cannot be two exactl y equ al puls es. W hat it is alr eady mu ch mor e extr aor di nar y, and pr ecis el y wit hi n t he co nti nuous do mi nion, i s t he f act t hat i n any puls e, and at any ti me, can be disti ngui s hed t he i nnate and t he acqui red f actor s wit h su ffi ci ent clear nes s. This is a differential lu xu r y of t he natur e t hat woul d be absu rd to wast e; so much it is what must teach to us. Since even f rom t he poi nt o f vi ew o f t he genetics, to est ablis h the rel ation bet ween the int ernal and t he enviro nmental acti vatio n of the genes r emai ns li ke a pu zzle ver y far fro m bei ng u nder stoo d, and we do not s ay already of bei ng sol ved. It seems to l ack a mi nimu m r ef er ence for t hi s pro blem, so fundament al i n theoreti cal biolo gy, t he evolutio n and develo pment. The va idya di sti nguis hes bet ween p rakriti and vikrit i, biolo gi cal and t he bio graphical featur es, t he origi nal or i nnat e co nst itutio n o f t he indi vi dual and its t endenci es acquir ed t hrou ghout t he lif e. B y def inition, any l at er s tate i n t he healt h of t he su bj ect i s dependent and referr ed to t he co nstitutio nal tendenci es; so t hat t he pro du ced habit s o r alt er atio ns meanwhil e ar e, o n t he one hand, assi mil ations of t he eventualit ies to an ori ginal co nfor mat ion, and, on t he ot her hand, t hat same co mpl exio n has alr eady its slopes of reactio n before t he accidental conditio ns or vikriti . If p rakriti and vikriti co me to be like the substance and acci dents for a gi ven pul se, not for t hat r easo n it i s neces s ar y to fo rget t hat i n t he long ru n tho s e acci dent s and way of life may The l eft hand of C hao s
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become t he mo st deci si ve fo r t he destru ctio n of t he bal ance t hat represents t he healt h. But no t i n vai n it is s po ken of way of lif e, t hen. In an es senti al way, the for m o nl y i s mo difi ed by its own for m to wi nd it sel f i n t he time, and t hat t hread has to be per ceived by the vaid ya wit h t he ri ght sensiti vi ty. An ywa y we co nsi der it , it does not sto p bei ng an ext rao r di nary wonder t he fact of t hat i ndivi dual per sist ence of t he form of t he pulse t hroughout all a lif e, if we t hi nk about t he i mmense variet y of cir cumstances t hat can surround and al ter it , and about t he very sli ght nes s t hat it s si gnal r epresents. If we t hi nk about t hat t hi s little thr ead o f ti me i s goi ng to maint ain somet hi ng o wn i n t he mo st adver s e cir cu mst ances, beyond the su cces si ve loss es of identit y of t he memory, and t hat not even a viol ent deat h can ro b t he right to it to dis appear accor di ng to it s own and i nt imate l aw.

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This disti nctio n of t he t endency wit hin a t endency i s so fundament al for our stu dy as it was it the di stinction bet ween speed and accel er ation for t he bi rt h o f dynamics through the co ncept of force. A mo del of flui d dynamics ­s ay a circuit of pi pes wi th pressured water- cannot est ablis h thi s type of diff erences because the co mpo nent s ar e not co nnected from t he ori gi n, and all t he di ffer enti al charact eri stics t hat we coul d measur e lat er depend o n the ar bitrar y variatio n o f par ameter s. A pri mar y, ori ginal ref erence does not exi st. The dependency or sensiti vit y o f t he chaoti c sys t ems wit h res pect to t he i nit ial dat a i s in t he anti po des of t hi s other t ype o f "o ri gi nal co nditions ": fir st t hey are highl y u nst able befor e s mall changes , whereas t he ver y small changes i n t he pro portio n of dosha s have an equ all y small effect. But alr eady we sai d t hat t he t hr ee dosha s, their pr edo mi nance and pro portion, mu st be co ncei ved as r eacti ve slo pes or gradi ents: li ke t endenci es for mi ng a potential. Thi s pot enti al , t hei r bou ndaries, does no t have suffi ci ent fr eedo m to evol ve anyway, and a too abrupt change woul d l ead neces saril y to the deat h. Therefor e, it woul d be mo re t han desir abl e to kno w t he bou ndary condi tio ns of that pot enti al, or i n o ther words, it s eco nomy.

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The definition of a pot enti al of t hi s t ype seems extr emel y difficult. We not o nly s po ke o f an o pen s yst em, i n which t heir components alr eady are relat ed to the envi ro nment fro m t he begi nni ng, not o nl y i n fact, but in it s s ame s ens e and meani ng. Vata is not o nl y a sensiti vit y bet ween the action and r eact ion, it i s al so and at t he s ame ti me sensiti vit y to t he enviro nment . Therefor e, it is si multaneou sl y i nt er nal and exter nal sensiti vit y, and to di sti ngui s h bo th seems mo re a t as k proper of anot her poi nt of view than t his o ne. Raja s and Tama s, the eff ecti ve for ce and i nerti a, s eem mo r e i nt er nal to the sys t em, but t hey al ways are medi ated t hrou gh vata, t he s enti ence or s ensi ti vit y. This, t hat s eems a great obst acl e, is also the mai n virtu e of a sys t em or or gani zation t hus co ncei ved. Had not t hi s ambi valence, t he visio n of t he Ayu r veda or the Samkhya woul d no t off er anythi ng truly new to t he co nventio nal mechanisti c visio n, which by defi nitio n co nsi ders clos ed s yst ems , and al tered by ot her equ ally mechanical pertur batio ns. Let us r etur n agai n to t he natur e of t he change and fluctu atio ns of t he gunas o r it s associ at ed do shas. W e have alr eady s ai d t hat , for t he Samkhya, the gunas ar e anywher e as mu ch i nsi de as outsi de and t hat we can obser ve t hem if we lear n to anal yze what we s ee; t hat t here i s no t any ot her kno wabl e t hi ng i n last i nstance. The t hr ee guna s, being i n balance, are u nknowabl e and t hey ar e reabsor bed i n t he co nscious nes s wit hout qualiti es. If the existence of the gunas is bas ed o n its i nequ alit y, it cannot be on t he ot her hand mo di ficatio n or flu ctu ation wit hout t he domi nio n o f one of t he gunas o n t he ot her two, whi ch makes pos si bl e that t he mo dif icatio n is kno wable li ke moment it self . That do mi nio n has the for m o f a mo difi cation o f t he wei ght or valu e of o ne o f t he gunas that revert s on t he wei ght or value of t he ot her t wo. Thi s admits and The l eft hand of C hao s
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demands characteri sti c sequ ences t hat , beyo nd t he abst ractio n, constitut e the onl y for m of causali ty whi ch we can kno w. That 's to s ay, t here i s not another, and any ot her apparent for m of caus ali t y will be r eveal ed li ke mer e r epr es ent atio n. The r epr esent atio n of t he causalit y, ther efor e, moves away to u s of t he o nl y kno wabl e and intuit i ve caus ality. Thes e t hr ee mo daliti es ar e o nly applicabl e to t he r eal cas es, no t to imagi nary entit ies wit hout co mpariso n t er ms ; we s po ke fro m t he begi nni ng of conditio nal entiti es , not i deal or clos ed o nes . One guna or modal ity cannot exist s epar atel y, so t hat a stat e do minat ed by Satt wa, we t ake for exampl e, al ready i nclu des t he pr esence and subo rdi natio n of the ot her t wo, as mu ch for t he mos t stable cases like for the flu ctuations thems el ves . The same is vali d for t he ot her guna s . P atanj ali disti ngui s hes bet ween di ver sified st at es, u ndi vided or mo nadi c, o nl y i ndi cative, and no n-i ndicative st at es of the gunas. The s ame empiri cal or mutabl e ego i s an u ndi vided and i ndivisibl e case i n pr actice, no matt er how har d so many int ell ectuals i ns ist pres entl y on its t heor eti cal di ssolutio n; t he pur e I sens e, t he existence l i ke mer e att entio n, i s an onl y indi cati ve st ate, Linga-Matra, r easo n why t he no n-i ndi cati ve st at e, Alinga, properly i s no n- manif ested, rat her t han no nexi stent, if we want to fit expres sions that rarel y we u se wit h pro pert y. This is , to t he st at e of bal ance of guna s, it is consi der ed a met aphysi cal ter m o r not (i n whi ch met aphysi cs all t he well- kno wn rever si ble phys i cs woul d enter ), it i s no t attri bu tabl e neit her t he exist ence nor t he not -existence, and t he o nl y thi ng t hat i s po s si bl e to say is t hat it do es not gener at e i ndi catio ns no r r eferences . The fu ndament al st ates or sectio ns o f t he gunas, i n s ectio ns or grou ps of f i ve and si x, have a gr eat i mport ance if we want to deepen i n thi s do mi nion, alt hough no w we will not sto p i n it .

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If t he bal ance o f t he t hr ee gunas co nsti tutes its non- manifest ed st ate, it i s becaus e t here i s no pl ace i n them for t he di ssolutio n, gro wt h o r decr eas e, si nce t hey l ack mat eri al cau se. If we follow enough t he gro wth o r manif est decr ease o f t he gunas i n an ent ity o r s yst em it i s i nevitable to verif y ho w values that seem to be diluted can emerge fro m t heir l atency, and vice ver sa. But a value cannot si nk i n t he l at ency, i n t he r el ati vel y no n-mani fest ed, wit hout passi ng t hrou gh t he mo dali t y of t he Latency or Ret entio n, t hat is to s ay, Ta mas , i n t he same way t hat cannot emer ge wit hout actio n or Raja s, nor acquir ing cert ai n bal ance wit hout Satt wa. It must have an elo qu ent co nt i nuit y bet ween t he co ntent of its values and t he fo r m t hey are manif est ed.

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Even accepting that the gunas t hems el ves neit her i ncrease nor di mi ni sh, any action or pr edo minance of one on t he ot her s i n an empir ical tempor ary ent it y mu st be s een i n such t er ms for t he u nderst andi ng of the sequences and its evolutio n. A s equ ence begins necess aril y i n t he guna t hat it is domi nant at t he mo ment . This alr eady pro duces a r edu ndancy t hat can be det ached indi ff er entl y i n sever al ways , si nce what is i ncr ease i n o ne al ways is corr el ati ve di mi nutio n in anot her o ne. We can det ach a si mpli fi ed scheme i n co nditional for m: 1 If Sattwa or t he s ensiti vit y "i ncr eases ", t hi s ent ails a correlati ve di mi nutio n of t he acti vity of Rajas , and never its i ncr eas e ( Thi s do es not s eem evi dent and t her e is all ki nd of exper iences t hat at first si ght seems to cont r adict it ). It can not pro duce eit her "i ncr ease" or "di mi nution" of t he princi pl e of retenti veness or Tama s. 2 If Satt wa di mi nis hes , t his onl y can happen by a cor rel ated i ncr eas e of Raja s , and never by it s di minut io n. It is why t he ter m Rajas is s yno nymous of cont ami natio n or dust . Ta ma s, as in t he pr evious cas e, it is not aff ected dir ectl y b y the mo difi catio n o f Satt wa , but alr eady t he i ncr ease of Raja s oper at es it s di minut io n. 3 The dimi nutio n o f Rajas also corr espo nds to t he i ncrease of Ta ma s or i nerti a. Thus are i nclu ded t he four pos si bl e di rect movements of Rajas o r activit y, and t he t wo of Satt wa and The l eft hand of C hao s
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Tama s. The Rajas guna is i n t he i nt ermedi at e pl ace of t he t hree, and is t he o nly pos si bl e connection bet ween the ot her t wo. A defined or der i s co nser ved alt hough what it seems begi nni ng of a sequence can be s een f rom another mo dal it y li ke co ns equ ence, and the i ncr eas e, li ke cor res pondi ng di mi nution, and vice vers a. We mo ve alr eady in t he circul ar it y. W e will not enter t he for mal izatio n and qu antif ication of a mo del t hat s eems insi gnificant and t hat i n fact can be complicated indefi nit el y, having been abl e any lo gi ci an to tr eat t his ­t hat could s peak to us abou t co nnect i ves, di sjuncti ve fu nctio ns, etcet era- wit h mor e pl easu re and compet ence. We wer e s atis fi ed seei ng t he or der of t he guna s as a scale or st ai rs t hat si mult aneousl y allows the unio n by degr ees and t he separ ation or exclu sion of cert ai n i mmediat e mo vements . That i s to s ay, we have an element ar y model of medi atio n, whi ch admit s al most i nfi nite variatio ns its elf.

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The explicit di stinctio n bet ween vik riti and prak riti, the pres ent stat e and the dri ft of t he previous states, are t he u ni que t hi ng t hat coul d gi ve u s a refer ence of t he gunas, i ts co mplexio n and evolution. The onl y thi ng, t her efor e, t hat coul d make t his fi eld wort hy of a consi stent and totall y scient ifi c stu dy. Gi ven it s i mpli cit or irreduci ble i nher ent char acter, is not pos si bl e t hat the gunas or t he do shas sho w li ke expli cit or exempt co mpo nents ; but without t hi s r ef er ence, it do es not s eem po ssi ble it s ri gorous u nder standi ng and t reat ment. Alt hou gh t he i nt er preter of t he pul s e mu st make this di sti nction to have t he mini mu m of judgment cert ai nt y, it is not cl ear if he i s abl e to make it explicit anywa y. It woul d be s ai d, rat her, t hat he is bas ed o n t he co njectur e and t he supposition, i n addition to t he i ntuitio n, althou gh t his l ast can be sur pri si ngly precise and right. It can be bas ed, i n addit io n to the i dio s yncrati c pro perties of t he pul se, i n t he ju dgment on t he co nst itutio n o f t he perso n, who has cert ain har dl y variable element s t hroughout t he ti me; or i n t he colou r of t he s ki n, t he as pect of t he to ngu e and ot her semiolo gies . Nevert hel es s, all t his is not but support s to deli mit and to gi ve co nt ext to a sour ce of i nfor mation, the s ame pul se, t hat would have to be and i n fact it is mor e pr eci s e and fait hful than any ot her i ndex to co nsider . Si nce we have t he great advant age t hat t he pul se is already an el ementar y for m of t empor al s eri es, t hat woul d be r ender t he analys i s i n a mo del for m, i t would be to wi s h t hat t he ref er ence of t he past st ate or p rakri ti be equ all y dou btl es s an explicit in char act er . Su rel y, if we had all t he s eri es o f t he pulse o f a perso n from t he bir t h to its deat h ­ a nu mber of beats of t he order of 3-4 X 109 - we coul d make a quit e co mprehensi ve stu dy o f its evolutio n, its pro gr essi ve i mbalance and t he i ncr eas e of t he di sor der o r ent ro py; but t hi s i s not a r eali sti c per specti ve, si nce neit her we want to wait fo r sevent y years nor s eems very feasi bl e nor desi r able to mo nito ring to a per son alo ng t he whol e l if e. On t he ot her hand, t he moni tori ng with sensors r eleases to u s of the obl i gat ion to t ake t he readi ng i n co nditio ns o f maxi mum rest, and it allo ws us to make r eall y dynami c stu di es in real ti me, wit h regist ries or t ests of effort, pai n, pleasur e, etc. Fro m her e we can do quit e valid extr apolatio ns o n t he natur e of t he fluctu atio ns of t he pul se, balances , i mbalances , and t he fo r m t hey evo l ve throughou t the time; valid, per haps, but still quit e co nj ectur al and fr agment ar y. B y t he way t hat t his woul d be ver y f ecund at t he ti me of cr eati ng mor e soli d proto cols i n t he ps ychophysi cal stu di es and all tho s e t hat analyze t he rel atio ns bet ween t he mi nd and t he body, alt hou gh al ways i nsuffi ci ent without o ne mor e necessar y and natur al defi nitio n of t he triguna, which i s as well equi val ent to defi ne t he r el atio ns between t he potential of prak riti and t he effecti ve and pr esent valu es of vik riti. Reall y, whi ch we as ked ou rsel ves i s if bot h element s, pr esent or actu al and pot ential , ar e i ntertwi ned and present in t he ef fective si gnal or if t he pot ential onl y is a part of t he past of t he s eri es, li ke woul d be to ho pe i n a more co nvent ional mo del, wit h a li near ti me t hat does not keep memor y fro m it s past st at es . As it i s seen easil y, thi s l ast o ne woul d be quit e co ntradi ctor y, if we co ns ider t he ver y co ns i derable evidence of whi ch the bal ance of t he s yst em canno t be made pu nctuall y, but s catt eri ng itself i n a ti me i nt er val , ot her wi s e indefi ni te. Not hi ng clo ser to t he cli nical pr ecis ion t hat t he anal yti cal exacti tude, no r fu rt her mo st si mult aneousl y.

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All this woul d put i n enor mous dif fi culti es to u s ; when we tr y to anal yze valu es fro m a more natural poi nt of view, i t woul d seem to be t hat t he natur e deni es the same po ssi bilit y to u s to carr y ou t t hi s t ype of analysi s. It is not to sur pris e that so many well- i ntentio ned attempt s to co nsi der t he dynami c s ys tems wi t h another perspecti ve end inevi tably at t he f ailur e. Fo r a moment , ever yt hi ng seems to make t he ri gorou s corres pondence bet ween t he triguna and t he vari atio nal pri nci ples o f dynami cs possi bl e; soo n, quickl y, we retu rn to r ealize t hat dynami cs ends up i mposi ng it s own rul es and fo rces to was h our hands o f t he ambi guou s el ements li ke stranger s to its i nt ention. Dynami cs exclu des many co nsi derations simply becau se t hey do not make sens e for it nor have s pace wit hi n. And t hi s i s ver y dif ficult, if not i mpossi ble, to change. W hen on t he other hand we achi eve to ext end its vari abl es, not fo r t hat r easo n su ch variabl es stop to be u nder t he same exclu di ng logi c. So mus t it be: i n t he i nfi nit e s pace of t he analysi s, t her e i s no pl ace for anyt hi ng, but for t he anal ysi s.

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Finall y, t he same natu re co mes i n our ai d. And i t do es t hrou gh it s as ymmet r y; o r its s ymmet r y, becau s e bot h ar e i nt er changeable. Are s ymmet r y and as ymmetr y mutual ref er ence to each ot her? Per haps not absolu tel y: rat her all i ncli nes to us to t hi nk t hat t he s ymmetr y is a pur el y for mal r ef er ence, and t hat t he as ymmet ri es co ntri but e the real co nt ent s. Nevert hel ess , o nce we fou nd t he ter ms of s ym met r y for a pheno meno n, t he co nt ents of t his o ne s eem to volatilize i n form of a mer e par ticu lar cas e. Between s ymmetr y and as ymmetr y, t he relatio ns do not seem to be nor s ymmetri cal, nor the opposit e. An antino my: exactl y t hat cl as s o f thi ngs t hat do not have place i n the s pace o f the anal ys is . But we return to t he nature. As we already s ai d, so me of the do ctors who use the pul s e di agno sis onl y feel t he si gnal i n o ne of the wri sts, and other s i n two. Ther e woul d be good r easo ns to su ppo s e t hat a singl e wri st is enough, and t hat co ntai ns of impli cit for m all t he informatio n, havi ng i n account t he highl y s ynt heti c charact er of t he si gnal and its mor e t han pr esumable unit ar y, uni vo cal evolut ion. But that this can be so fo r an expert doctor does not eli mi nate it s i mplicit char acter, whi ch is just what we want to sur pas s i n t he dir ection of more explicit and measurabl e fo rms . Ther efor e, it i s po ssibl e to suppos e t hat the puls e o f a si ngl e wris t co ntai ns all the necess ar y i nfor mat ion, alt hough in a mor e co mpact and appar entl y co nfu s ed way, and t hat t he other wri st can be u sed precis el y li ke r ef er en ce ­ju st what we were loo ki ng for. The C hines e medi ci ne, for example, u su all y pr es cri bes t he r eadi ng of t he t wo wri sts . This , among other t hings , o beys to it s breakdown of t he t wel ve meri di ans, an es sential f eatur e of its etiolo gy, i n si x for each wri st . Whi ch, li ke almo st ever yt hi ng i n t his grou nd, do es not sto p bei ng a prescri ptio n or reci pe of heuri stic natur e, even t hough can be justified by t he result s. And li ke all t hese heuristi c f r amewor ks, t hey change accor di ng to t he t eachers and pr actitioners . What all the pr actitio ner s of t his diagno si s admit is t he mo re t han sensi bl e di ffer ence bet ween the pul s es of both wri st s. Tr adit ional ly it is sai d t hat i n t he l eft ar m t he movement of t he pul s e i s clear er , and i n t he left o ne, it s for m. This is quit e evi dent , at least for most of t he su bjects, but i t woul d not even say wh y reaso n si x meri di ans ar e as signed to o ne si de and o t her quit e diff er ent si x ar e as si gned to the ot her, even when we gr ant to these disti nctio ns a hi gh degree of r elati vit y. On the ot her hand, the co nt ri butio n o f each meridi an, o rgan or vis cera can be very diff er ent i n magnitu de. Fo r example, it i s co nsi dered that the sto mach and it s as so ci ate fu nctio ns o r ener gi es ­ and not t he heart , as we coul d believeco ntri but e i n near 4/5 part s of t he global profil e of t he puls e speaki ng i n qu antit ative t er ms; what it does not mean t hat t he ot her s ar e l es s i mpo rt ant, but t han they ar e su btl er and mor e difficult to detect . The ayur vedic medi ci ne woul d agree perf ectl y wi t h t he s ense of t his appr eci atio n i n spit e of the ter minology dif ferences. Pro babl y t he o r ganization of t he meridi ans hides a ver y i mpo rtant s ymmetr y ­t hat curiou sl y also i s gi ven in groups or s emi- grou ps o f fi ve and si x- but t his o ne cannot be loo ked for t hi s has tily and i ndependentl y of the empiri cal natu r e of the signal and t he co ncer ned values . All t hes e poi nts can sou nd quit e str ange conf ront ed The l eft hand of C hao s
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wi wi ha ne

t h t he i deas t hat we have of any cl ass of anal ysis , and of t he cli ni cal anal ysis i n parti cul ar ; nevert hel es s, I ll i ns ist o n whi ch they also ar e of unf aili ngl y and genuine anal yti cal char acter, and it i s worth t he t rou bl e to ve t hem i n accou nt at the ti me o f doi ng an i dea about what can we ho pe to fi nd. W e must o mit many ot her s cess ary preventio ns becau se o f t he br evit y.

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In order to be brief , I will s ay t hat t he pu ls e of t he ri ght hand is not o nl y di ff erent, but al so co mpl ementar y of t he o ne of t he l eft hand. What it is not anythi ng new and i s i n basi c agreement wit h t he more s pread C hinese br eakdown; al so wit h t he Indian, althou gh t hi s one oft en do es wit hout so pr ecious ref erence to i nt erpret t he brute signal of onl y o ne of the wri st s. Taki ng i nto accou nt t he di sti nctio n which we have do ne bet ween the i mplicit and explicit valu es of t he si gnal of t he puls e, thi s mu st not be co ntradi ctor y at all. The pulse i n bot h wrists includes a tot ality; but its corr el atio n al lows us to o pen an enor mou s space for t he analysi s: no w yes , also for t he quantit ati ve anal ysi s o f t he tempor ar y seri es. For t he wester n medi ci ne, t hat tries to avoi d the semioti c ambi guit ies as far as po ssi bl e ­ vai nl y, per haps -, and we do not s ay alr eady for phys ics , the idea t hat t he dynami cs of t he pul se i n bo t h wrist s can be ver y differ ent and si multaneousl y very preci sel y cor relat ed will have to seem quite out landi s h and absur d. It will t hi nk t hat , if t here are diff er ences, t hes e are redu ced bas icall y to a mi ni mum diff erence o f di stance of the art eri al tr aj ector y, and to ot her questio ns and accidents of anato mical or der . In no case it will thi nk t hat rel evant fu nctio nal dif f erences for all t he s ys tem can be r eveal ed her e ­and t hat' s what we are t al ki ng about . For t he glance o f t he anato mi st , the basi c bil at er al s ymmet r y of our body littl e means . One woul d say that it i s an aest heti c or nament of natu re; alt hou gh for so me or gans , l ike our eyes , t he advant age of t he bi nocul ar visio n i s pat ent enou gh. But even i n cases li ke this one, t hat evi dent fu nct io nal advantage s eems to close t he pass age to another type o f co nsider atio ns. That type of advant ages can be appli ed i n the mo st tri vi al for m to the ki dneys or t he lu ngs , s peaki ng about sur vival advant ages and secur it y mechanis ms. Such co nsi derat ions cont i nues bei ng entir el y superf ici al . Ho w to expl ain o f what cl ass of complement ari ness we s peak about?

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Our co nscious ness i s extremel y opaqu e to t he veget ati ve acti vit y of our organis m, alt hou gh t his vari es enormousl y fro m o ne i ndi vi du al to anot her . The cir cul ation is o ne of thos e veget ati ve acti vi ties ; also t he mai n one. The most o pposed, co mpar ati vely, it s eems to be t he br eat hi ng, whi ch u ntil t he li mit of t he emer gency, is a volu ntar y and ext remely adjustabl e activit y. Neverthel ess , and in s pit e of t his appar ent cont rast , bar ely it i s po ssi bl e to i magi ne mor e i nter dependent acti vit ies t han t hos e of t he heart and t he br eat hi ng. But since alr eady we noti ced befo re, t he pul se, i n additio n to t he o ne of t he heart , tol er ates ot her l ayers of veget ati ve activit y. Gi ven it s volu nt ar y natu re, i t i s eas y to experiment how t he vari atio ns or int erru pt io ns of t he breat hi ng do not r eco ver of i nst antaneous way: a ver y lo ng for ced r et entio n will leave i ts tr ack cus hio ned t hrou ghout many cycl es of br eathi ng. It is exactl y b y t his so or di nar y experience i n f irs t per son t hat we can u nder stand of totall y i ns ti nct i ve fo rm ho w the same has to be vali d for the acti vit y of t he heart o r t he puls e, alt hou gh o nl y indi rectly we can f eel t hem. For that r eason we coul d s peak sur el y befo re about t he puls e of bal ances def err ed throughou t the time: we onl y su ppo s e t he si milarit y wi t h what happen i n the breathi ng. So met hing we can be sure about: any pheno menon asso ciat ed to the breat hi ng will fi nd its f ait hfu l correl at e i n the car diac acti vit y and t he cir culat io n. Nevert heles s, t he br eat hing has a t ype of perfectl y well docu mented asymmet ri c alt er nation to which al most no body pays att entio n: t he pr edo mi nance of breat hi ng bet ween t he r i ght and left nostril i s alt er nat ed each hour and a half , appro ximatel y. Thi s alt er natio n was i mport ant for t he old ayu r vedi c medici ne to su ch an The l eft hand of C hao s
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ext ent t hat it was s aid t hat if t his alt ernatio n let t ake pl ace by more t han t went y- four hour s, t he su bj ect was o n the ver ge of dyi ng. What, by t he way, i s not necess ar y to i nt erpr et li ke o ne su persti tio n o r f able, but li ke an al most i rr epar able dys fu nction and i mbalance. What means t hi s alt er nation? Wester n medici ne, even kno wi ng it, has i gnor ed it wi del y, u nder pret ext of i ts irr egul arit y and var iabi lity. As if that same vari abilit y was no t t he best i ndex and sour ce o f i nfor matio n i n all t ype of anal ysis, be t hey chemical, dynami c, or r adiolo gi cal. In fact, we kno w perf ectl y t hat it is not co nt emplat ed becaus e i t do es not have place i n t hat s cheme of t hi ngs. As it is kno wn, for t he Yoga t his alternatio n is not but t he mo st visi bl e mani f est atio n of a mor e gener al underl ying pol arit y, t hat aff ects to all t he evolution and o rgani c development . In t er ms o f su btle anato my, and in addi tio n, in s ymboli c ter ms - that coul d be not merel y imagi nary, but t he mo st gener al pos sibl e- t hat polari t y is named as ida and pingala , t he i nt ermi ngl ed acti vi t y of the Su n and t he Moon, wit h su shumna l i ke virtu al axis of t he non- manif est ed. In much mo re t angi ble t erms , t hi s is translat ed i n the two fu ndament al phas es of the met abolis m: anaboli c and catabolic phas e. Or, al so, s ai d wit h ot her wor ds, wo rds , cr eatio n and destru ctio n, absor ption and eli mination, aff erent and eff erent ner vou s routes, veget ati ve for matio n and volunt ary acti vity. In sum, t hese t wo phas es woul d be t he mo st fai t hful co rr el ate of kapha and pi tta or Tama s and Raja s, co nsi dered no w not fro m t he i mmediate poi nt of vi ew of a modification or fluctu atio n, but fro m t he global eco no my of t he or ganis m. Apart fro m t hat , all t he Yoga can be resu med i n maki ng t he peace and obt aini ng a las ting bal ance bet ween our veget ati ve and volu nt ar y si des .

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The o nl y ar ea o f t he mo der n physiology in whi ch t he fu nctio nal asymmetry wit hi n an apparent s ymmetry is i nvesti gat ed activel y is the neurolo gy; ever ybo dy has hear d about t he s peci alization of t he cer ebr al hemi s pher es, and even the subject t ends to be tri vi alized, about whi ch nevert hel ess it is known s till so little. Then it is ju st her e where t he local analysis and t he s calpel ar e mor e i mpot ent . It i s so met hi ng beyo nd dou bt s t hat the speciali zatio n o f the hemis pher es exi st ; but of t he det ail s of t his s peci ali zatio n and t he correl atio ns bet ween t he fu nctio ns of both hemi s pheres , it i s ver y f ew what pres entl y can be kno wn. Thi s way, the mo der n physiology has had to as su me t he as ymmetr y o nl y for t he so call ed "no bl e o r gan" ­as if t he ot her organs wer e no t - , t here where it is mor e diff icul t to stu dy t he pheno menon, and wit hout r eali zi ng that is not but an eff ect o f a much mor e gener al pheno meno n. In eff ect, t her e i s much noi se wher eupo n i f t he left hemi s phere ­t hat by t he cros si ng of t he nervous routes cor respo nds to t he ri ght side of t he bo dy- is respo ns i ble of t he lo gi c and li ngui sti c functio ns, and t he right hemi s phere ­ corr espo ndi ng to the l eft- co ntai ns t he i magi natio n and t he s pati al repr esentatio n. Onl y of thi s, many alr eady make a mes s, as signi ng an "anal yti cal " char acter to o ne hemi s pher e and a "s ynthetic" charact er to the ot her; when the trut h is t hat the mat hemati cal anal ys is, for example, can r equi re a use of t he s patial r epr esentat io n and much mor e i ntensi ve i magination t hat i n ot her acti viti es co nsi der ed more "creative" or "artistic". And it is t hat ever ythi ng is badl y appro ached if we r emain isol at ed i n t he co gni ti ve domi nio n. Certai nl y and su rel y, bo th cerebr al hemi spheres ar e not but t he mor e el abor at ed and complex cas e o f the pol arit y or bas ic alt ernatio n that we have ai med: the one of t he volu ntar y and the vegetati ve acti vities or tendencies. These t wo aspects, as we know, have t heir parti cul ar centres and diff erentiat ed rout es rel ati vel y well known: cerebellu m, nervous pl exus es, etc. We simply affir m t hat both conti nu e reprodu cing enti rel y al so in t he whol e co gniti ve plane of t he cons ciou s li f e. W hat it s eems incr edi ble i s t hat we have still not r epair ed i n it. 39

Everyt hing whi ch will conti nu e mai ntai ni ng to us i n the per plexit y and even i n t he i ncredulit y; to su ch an ext ent our "cogniti ve st yl e" has condi tio ned and has put our i deas of t he s pace and ti me o nto t he r ails . Yes, The l eft hand of C hao s
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becau s e we ar e speaki ng al so about ti me. Once agai n we ar e for ced to r eflect on nature, and not wit h instru ment al pr etensio ns, but with s pirit of fi delity. Let us t hi nk about a tree. W e s ee a cert ai n lat eral s ymmetry, i n s pit e of appar ent diso r dered capri ces i n the ramifi catio n. The basi c l at er al s ymmet r y makes t hi nk us i mmedi atel y about t he growt h: i n f act, it i s o ne of our mor e apprai sed images at t he ti me o f repres enti ng t he u nidir ectio nal time to us, wit h all his cu mulati ve factor s. But we cannot s ee t he most remar kabl e s ymmetr y, becau se it i s sunk in t he Eart h: t he one t hat for m the roots with r espect to the branches, wi th t he grou nd li ke ir regular pl ane of refl ectio n. And just t hi s s ymmetry hides al so t he most i mpo rtant as ymmetr y, in whi ch barel y we t hi nk ­ although al l we noti ced easil y the enormous fu nctional di ff er ence t hat exi st s between t he roots and the branches. Here t he circuit and t he pol arit y ar e made evi dent and acqu ir ed t heir inevit able char acter. Exactl y the same happens i n our bo dy. Onl y t he co nditions change, bei ng ver y di ff erent . Becau se we do n't need to s ay t hat we do not have a part fro m u s to t he sun and t he other u nder eart h. It has changed t he dir ectio n of t he s ym met r y axi s, and i n additio n, all t he surrou ndi ng enviro nment. But t he pol arity and t he di ff er ence of phas es r emai n. It is not t hat the ani mal , emanci pated of t he vegetati ve sleep, becomes "tr ee wit hout root s ", except i n t he mos t tr i vial and appar ent s ens e. We ar e rat her whol e tr ees, wit h t he root s insi de. But let us t hink in addition wher e a t ree has its "head", if we do n't co nfu se it wit h t he to p. Si nce t he more "auto no mou s " and "volu ntar y" ­ willi ng, per haps- part of a tree is i n t he wedge and spearhead of i ts roots , whi ch beco me t he most s i mil ar to our br ai n. This bri ngs i nevit abl e reso nances to u s of t he great Vedic image o f t he co smi c t ree wit h i ts roots i n t he s ky and its br anches towar ds t he ground. And if so mebo dy has dou bt s on t hat it i s i n t he root s where t he t ree u nfol ds it s auto no my ­ as r elati ve as any ot her , o f cour s e -, he could t hi nk, to t he way of t he po et Will iam Blake, wher e are the "devouring" and t he "proli fi c" el ement s here: anyone under st ands at fi rst s ight t hat t he roo ts are t he devour ing el ement and the br anches, flower s and frui ts the prolifi c; o n t hi s there i s no dou bt. And it i s by t hi s t hat t he vegetal world wakes u p su ch r esonances i n our soul: becaus e, u nlike us, it is justly t he veget ati ve part t he o ne t hat el evat es it s sl eep and dreams befo re t he Su nli ght and befor e ou r own eyes.

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Thanks to t he t ree of our example, we ar e able to co ncei ve somet hi ng t hat i n anot her way woul d s eem more t han dou btful to us: t hat bot h sides of our bil at eral sym met r y have t hei r fu ndament al r easo n in functional dif f er ences. Therefo re o nl y this way t he bal ance invol ves so met hing real ; becau se t wo equal t hi ngs do not need balances bet ween t hem. And t his balance must involve mo re or les s regul ar osci llations and ther efo re a cir cul atio n wit h t ypi cal diff erential aspects . And thus, t he most t ypi cal mani fest atio n of t his functional as ymmet r y will appear naturall y i n t he ti me, and i ndi rectly, i n so me of it s dynami c as pect s. W e could t hi nk that t he ver y co mplementar y as pect of t he art eri al pul se i s t he venous pulse; i n t he jugul ar one, for exampl e. Willi ng to loo k for so valu abl e r ef erence, not hi ng i s to dis card. Nevert hel es s, t he venous pul se s eems to s ho w more i nfor mation o n t he ci r culator y return and t he same heart t hat o n t he joi nt bal ance of t he s ys t em. And alt hough i n t he organis m any el ement can be reveali ng o n t he set, the cert ai n t hi ng is t hat t hey seem too di ffer ent si gnal s i n or der to loo k fo r it s cor r elation, superpo sitio n and i nt er ferences. Natur ally, t he s ame heart exhi bits a bil at er al symmet ry t hat hides an evi dent functio nal as ymmet r y, t he o ne of the ar teri al flow and t he venou s reflux; t he best pos si bl e su mmar y of the ext ernal influence t hrou gh t he br eat hi ng and t he i nter nal r eflu x f ro m all t he parts o f t he bo dy. Thi s asymmetry al so i s r epro duced bet ween t he auri cl es and ventri cl es ; and the cro ssing of t hes e two dif ferent and coinci dent as ymmetri es of the ho ri zo ntal and vert ical axis is like an anato mi cal epito me of t hat di ffer ence that we find in t he dual flo w of even the t wo art eri es of t he wrist s, mor e extr eme than paral lel . The anato mi cal cro ssi ng o nly s ho ws u s an ext er nal analogy wit hout apparent caus al co nnectio n wi t h su ppos edl y parallel li nes of action; but it is jus t her e where it appear s our bli nd poi nt wit h r es pect to t he per cept io n and co nceptio n o f t he ti me ­t he s ame mind is not able to The l eft hand of C hao s
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per cei ve nor to take car e of two dif f er ent si mult aneous cour ses, but must su ppo s e t hem parallel, t hat i s to say, si mpl y spaced. Apart from t hat , one is al lowed to su ppos e t hat it is t he fu nction what creates t he o rgans, r at her that o n the cont rar y; but o nl y t he more car eful experi mental stu dy of the corr el ation and dif ferences of t he pul ses can t hro w l i ght and evi dence on t his as su rpri si ng as u nexpect ed question. If somebo dy still doubts , he may t hi nk abou t what a no nsens e woul d be t he s ymmetr y o f t he tree wit h its roots if it were a si mpl e s pace s ymmetr y. We cannot i magine anyt hi ng si mpl er , becaus e it would be si mpl y gr atuitous. In addition, t hat spati al s ymmetr y is f ar f rom bei ng r egul arl y fulfil led, becaus e o f t he ver y di ff er ent co nditions t hat exist outsi de and u nder t he grou nd. Ther efor e, t hi s s ymmetr y mai nl y exi st s i n the gro wt h and the ti me, and i n the cycl e of ci rculat ion of t he s ap t hat t he tr ee do es enti rely. We woul d say, even, that i t exis ts mai nl y in th e spirit of t he tr ee. And in t his pol ari zed game wi th t he envi ronment has t he t ree its r easo n. Let us s ay, t hen, t hat t he co mpl ementari nes s of t he arteri al and venou s ci rcul atio n fi nds its f aithful correl at e wi thi n t he same art erial cir cul atio n under t he appearance of the space s ymmet rical bifu rcatio n ­t he two r adial art eri es t hat we ado pt ed li ke mer e r ef erence, just as we coul d ado pt ot her art eri es wit h ampler si gnals . As it is well known, t he ci r culation pro cess is f ar fro m si mpl e; t he venou s r eflu x is quit e compl ex, and a cert ai n nu mber o f beats i s needed for t he whol e r enewal of the bloo d. It happens, i n additio n, t hat t he venou s cir cul atio n is, from a point o f vi ew, si mult aneous to the art erial circu lat io n, and, fro m another poi nt of view, i t is successi ve or su bsequ ent to it . Then, assu mi ng a great si mplifi cation, and att endi ng to t he mo st rel evant dif f er ences, we can ventur e that what i n the art er ial - venous pai r appears like si mult aneou s, must appear succes sivel y i n the du ali ty art er y-art er y, and on t he contr ar y: what i n the venou s- art erial pair seems successi ve, mu st have a co rrel at e of s i mult anei t y i n our ref er ence art er y- arter y. Why? Ther e is neit her pl ace nor ti me for anythi ng el se, t hat' s becau s e. That' s to say, t his is t he mai n conditio n for the balance in t he u ndi vi ded. And thu s, at a blow, a new li ght is t hrown o n the same knot , cro ss or bli nd poi nt i n our ti me- s pace co nceptio n; and, ver y probably, on many anti no mi es pr es ent i n t he different areas of t he anal ysi s. Li ght t hat, of course, we shall have to follow i n t he dir ection o f t he source. The cir cul atio n of t he tr ee has mor e to do with t he r eal ti me t hat all t he clocks i nvented si nce Gal ileo 's pendulum. And thei r roots t ake us furt her and deeper t han all t he mo der n co s mologi es, with t heir gr eat explos ions and enor mou s nu mber s wi t h a met ro no me r at e. Befor e the diff icul t y and s chi zo phr eni a t hat wou ld suppose for anyo ne of u s to co ncei ve t he "two si des of t he ti me", t he i mage of t he t ree co mes to r edeem to us and to make us sens e a uni tary ti me and rat e, co nser ving all its enigmati c char acter. I would dar e to ventur e t hat , i f for so met hi ng t he cont ent o f t hi s poor writi ng reach so me receptio n, it will be by t his fortu nat e image and it s et ernal evo cati ve power. 41

Truly, t he anci ents knew to gr as p t hi s notio n of t he ti me, so diffi cult to conceive for us; althou gh onl y the symbol s have l ast ed, so me of t hem so eloqu ent i n t heir plasti cit y: t hey ar e t he two for eheaded Janus, t he dou bl e axe, t he bi cephalou s eagle, and ot her many of i denti cal char acter, t hat expr ess more o penl y what we alr eady fou nd i n t he tr ee. To say of t hes e s ymbol s t hat t hey are allegories o f t he ti me, wit h t he pr esent mo ment wat chi ng to war ds t he past and t he futur e woul d not hel p us mu ch, becau se al most no ne o f u s makes bot h thi ngs simult aneous ly neit her at the happi est mo ment. One i gnor es t her efor e the wei ght and plexu s of the s ymbol, t hat i s to say, its al mo nd. Nobo dy will be able to r epresent t his ti me i n a poi nt, except li ke s ymbol also , bu t o nl y a fo r mal s ymbol now, and dis po ss ess ed o f its complexion - t hat i s pr eci sel y what t he ancient s ymbol allu des. But we must l eave t he mys t eri es and t r y to approach the di ffi cult y. Met apho ricall y s peaki ng, and wit hout t he small er sci entifi c pr etensio n, we can co ncei ve t he anabolis m and the for mati ve, plasti c capacity of the or ganis m li ke wat ching i n t he di rect ion of t he past, and at t he cat aboli sm li ke wat chi ng i n the di rectio n of the future. Thi s is not u nju stified i n ter ms of biologi cal development, s i nce t he fo rmative phas e must cou nt wit h which ther e i s al ready pr es ent , that it is su ppo s ed t hat "comes" fro m t he past. On t he co ntr ar y, we can co ncei ve t he catabo lis m li ke "fr ee" of t hat o bs er vance, and for t hat same reason, and becaus e it co nti nuou sl y opens a rou te of action of o ppos ed natu re, we attri bute t he o ppos ite dir ectio n to it , i n the dir ectio n of t he The l eft hand of C hao s
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futur e. As it i s s een, i ndeed we can concei ve si mult aneous acti vities her e, but we realize perf ectl y the met aphori cal s ens e of t hat wor ds: we know t hat fro m t he past nothi ng co mes, and know t hat not hi ng "goes" towar ds t he future; t hat bot h t hi ngs agr ee i n a same time and what dif fers is t he natu re of the acti vit y. But if we see wit h facili t y t hi s, we have already o btai ned somethi ng, becau se i n f act s ci ence never can see it t hu s. Of cou rse, we ar e turni ng a time do mi nion i nto a cert ai n space not well defi ned. For phys i cs , t he particl es have t he propert y o f i mpenetr abil it y i n the space, a corollar y o f the i nertial s yst ems. Nevert heles s, for the waves , t he precisel y the opposit e i s predi cat ed: two equal waves i n o ppo sit e phase and di rection beco me annull ed at a cert ai n mo ment and r etur n to emer ge followi ng t heir ori gi nal direction. It i s the call ed princi pl e of superpo sitio n of t he waves, i n t he anti po des of t he i mpenetr abilit y of parti cl es . B y this same o ne usuall y says, to si mpl if y, t hat t he mai n att ri bute of parti cles i s t he i nerti a and the one of t he waves t he energy transmi ssio n, althou gh natur all y anyone of t he t wo i mpl ies bot h. The Samkhya as su mes t he exi st ence of ato ms; but atoms of ti me, i f t hu s can be s aid. And t he onl y whi ch we can conceive li ke "ato ms of ti me" are not but t he waves ­o nl y that ti me waves, unli ke t he mor e s patial waves befo r e mentio ned. These waves o f ti me ar e, i n last inst ance, t he vrit tis , the same fluctuatio ns of the gunas. Will obey t hes e waves t he su per position pri ncipl e, will t hey have penetrabilit y? Su rel y t hey wi ll, if we co ncei ve t hem s eriousl y li ke waves. Thi s way, it is pos si ble to co ncei ve also and easil y t hat a mo ment of time can co ntai ns s ever al valu es and su pport s si mult aneousl y dif fer ent acti vities, sto ppi ng to be a pas si ve poi nt for the mere r epres entat ion i n s pace coor di nates . And of cou rse, i n t he evolutio n of a dynami c s ystem we can su per pose at ever y mo ment of ti me not t wo, but infi nit y of diff er ent vari abl es , provi ded t hey fulfil t he mi nimu m exi gencies. Cons ervative exi genci es t hat, i n last i nst ance, are equi valent to the impenetrabilit y of t he inerti al sys t ems , alt hou gh ar e not r edu ced necess aril y to it . We could s ay t hen, wit hout co ntradi cti ng t he co nser vatio n l aws , t hat an int erval of dyna mic spa ce exists when t wo co nditions cannot occur s i mult aneousl y in t he s ame poi nt ­ by t he s ame r easo n we defi ned any cir cul atio n phenomeno n befor e. Aft er all, even parti cles have "i ntri nsic" frequency and s pin, so call ed becaus e they cannot be r edu ced to t he s pace repres entation. In its gener ality, t hi s would be valid so much for a par ticle, an i nt er val of t he sangu i neous puls e or a tr ee. In thi s way ( bounded) time woul d begi n to roll by it s own rou gh ground.

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One of the mai n r easo ns to r ej ect a priori t he i nfor matio n t hat can o ff er t he anal ys is u s o f the pu lse i s to consi der that it i s a si gnal poo r, pr ecis el y, in i nfor mat ion. Really, many t hi nk t hat , even i f we accept t hat i t is s ympto mati c, t he si gnal i s exces si vel y s ynthetic. What des er ves so me co mments. Of cou rse, t he mo dern cli ni c analys is i s not pr ecisel y poor i n dat a, and even so and ever yt hi ng, it i s al ways wanted to get mor e. Is to have element s for t he judg ment ­s ynt hetic act par excellence as far as we under st and, and t hat Kant tr eated li ke a f aculty. Pr ecisel y t he ju dgment i s t he eas iest to lo se wit h t he i ncr easi ng flood of data, and I su ppose that t he do ctor s of t he modern hos pitals , not to mention t he epi demiologi st , must kno w t his enou gh - in fact , all we began to kno w t hi s enou gh, and I am afr ai d t hat we will sti ll l earn mor e. Fro m t his pers pecti ve, and at l east i n t he cli nical do mi nion, may be an exces sivel y s ynthetic i nfor mat ion can exist, but, even so , not s ynt het ic enough, yet . On t he ot her hand, alr eady we sai d t hat the brut e f eatur e or perfor mance is pro perl y t he best o bj ect of t he anal ys is, o n whi ch is not necessar y to i nsi st . It i s not pos si bl e to say t hat the puls e is poor i n i nfor mation since t he mo ment t hat we i gnor e t he factor s of it s variabilit y. Let us s ay r at her t hat we ar e i ncapable to r ead it . I will admit nevert hel ess t hat it i s ver y diffi cult to bel ieve that in t his signal we can o bt ai n t he necess ar y or su ffici ent i nfor matio n to t he judgment , s peci all y if we measure it wi t h t he st andar ds of t he mo der n exi genci es. Bu t, as it was s ai d, we t hi nk that t he corr el atio n of t he s i gnal of t he pul s e of bot h wrists not si mpl y du pli cat es, but mu lti pli es i n an enormous facto r t he po s si biliti es of t he mo st explici t, neat and detail ed anal ysis ­ becaus e the dif fer ence or co mpl ementari nes s of bot h si gnal s i s not of tri vial char acter, but extr ao r dinaril y compl ex. That i s to s ay, t he more detai led anal ysi s, wit hin t he suitable fr ame, woul d sho w t his co mpl exit y by it self, and t hi s wealt h o f The l eft hand of C hao s
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infor matio n. Let us repeat it: wit hi n the su itable fr ame; because s tu dies o f t he co rrel atio n of t he si gnal of bot h wri st s alr eady exi st in a great nu mber, wit hout havi ng r eached, as far as I kno w, to anyt hi ng essential.

Fig 5 But i t i s t hat ­apart of i gnor ing t he long dur atio n s er ies- bar ely it has been do ne anyt hing but to compar e t he pul ses , bei ng i mpos si bl e to speak o f aut henti c studi es of the cor rel ation, and not sus pecti ng it s dept h; or rather, wit hout su s pect ing t hat this corr el ation is r eveali ng exactly i n t he mo re s tri ct anal yti cal s ens e, t he physi cal and mat hemati cal o ne. Wit hout a doubt, it has co ntr i buted much to t his circu mst ance, not onl y t he s epar ation bet ween do ctor s and appli ed mathemati cians, t hat no wadays is cros s ed by nu merou s bri dges, but the spread pr ejudi ce o n t he "qualit ati ve" and "s peciou s " char acter o f t he si gnal of t he puls e and, mai nl y, its frame of int erpretat ion; preju di ce whos e di ssolution i s t he fro nt door to t he co nt ent of t hi s boo k. And it i s t hat , i n t he st yl e o f the mat hemati ci ans, we can speak o f analyti cal conti nu atio n or ext ensio n when we o pen t he s pace o f cor relation bet ween bot h pul ses . And pr acti call y i n t he same way: ext endi ng to t he co mpl ex pl ane t he dif ferent valu es of t he r eal variabl e of bot h s yst ems.

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Independentl y o f any met ho d of analys i s t hat we want to us e, and even accepti ng the sympto mati c and even r evealing char act er o f t he signal of the pul se, i t is ver y dif ficu lt to t ake off of t he mi nd t he i dea t hat t hi s si gnal has to be full of conti ngenci es, as much mor e when mor e i n detail we stu dy it: fro m anato mical i njuri es and peculi ariti es to all a s eri es o f u nco ntroll abl e factors . Si nce I am not a do ctor nor I have any direct experi ence of t he su bject, I will not dare to affir m t hat any l aw or co ntr astabl e co rr el atio n i n t hi s area cannot co ntempl at e except ions . But, admitting i n princi pl e t hi s po ss ibility, mai nly when we have not even begu n to est abli s h t he s mal ler cor rel atio n, it neces sar y to s ay t hat we t end to enor mou sly u nderesti mat e t he capacit y of adjus t ment of a glo bal signal li ke t hi s o ne. The bal ance has been there f rom t he begi nni ng, from t he first anato mical bifu rcat ions , and t he s mall er anatomi cal dif f erences will even end u p fi nding t heir fu nctio nal corres pondence and it s r eso nance t hrou gh t he s equ ence or t emporal s eri es. W hat ever it i s the co ntingency, t his will appear refl ect ed i n a doubl e way and i n the same measur e i n whi ch i t has so me i mportance for t he eco no my of t he glo bal balance of t he s ys tem; it cannot be anot her way. Natu r all y, I al ready st art fro m t he The l eft hand of C hao s
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as sumptio n that in t he pul s e an y factor t hat concurs i n t he st ate o f healt h has to be reflect ed i n dir ect pro po rtion wit h it s i mport ance: until i nfi nit esi mal extremes, if we coul d follo w t hem. The so mat ic u nit is as mu ch u ni t as t he mathemati cal u ni t, that is to s ay, t he nu mber 1 , al so kno wn like t he mo nad or u ndi vi ded totalit y by t he anci ent s ; and it i s the pro blem of t he anato mists i n ho w many part s can and want to di vide it. And if we cannot take off of t he head t he i dea t hat t he puls e, li ke t he act i vit y of t he heart and any biologi cal oscillator, has to co nti nue mai ntai ni ng u nco ntrol labl e irr egul ar it y o r r andomness , it is necess ary to r emember that t his s ame r ando mness i s alr eady its elf a measu r ement of vata, t he tot al s ensitivit y of the s ys t em - and the mo st di rect measur ement, i n additio n. So ever yt hi ng depends o n whi ch are our departur e coor di nates, and the triguna or t he trido sha, bei ng from t he begi nni ng o f glo bal natur e, can assu me all t hose conti ngenci es wit hout the small er diff icult y ­ we r ather ar e tho s e t hat have los t t he s kill to t hink thi s way. And t his way of t hinki ng is in f act mu ch mor e natur al t hat all t he cli ni cal analys is prot ected i n l ast i nstance by a for mal medi atio n, li ke for exampl e t he biochemi cal data. No bo dy happens to t hink that the sou nd o f t he motor of i ts car li es, al t hou gh i s far fro m di sti ngui s hing ever yt hi ng what it i s sayi ng. If t his happen to us wit h a car, t hat bar el y has so me thousands pieces, I do not s ee why t he human bo dy must not have expres sio ns i mmensel y more preci se and trustwort hy, wi t hout po ssi ble co mpariso n. The same we coul d s ay o f the hu man voi ce; t he probl em i s not of lack of info rmatio n, but of to what ext ent and withi n what fr ame we can anal yze i t. The perfor mance i s pr es ent , and its compo sitio n, pu rel y vi rtual. Thi s is vali d for any etiologi cal model, also t ho se of t he mo der n medi ci ne: we will always be able to fi nd an endless nu mber of i nter changeabl e causes t hat adju st to t he dat a t hat r eall y i nt er est to us . And pr eci s el y t he doctor is the one who l es s can renou nce to thi s meani ng o f the anal yti cal and t he s ynt heti c s pheres: if he do es, already renou nce to hi s very co mpet ence. Let us r emember t hat t he s ame state of heal t h, as feel s i t a su bj ect, is anot her synt heti c pheno menon par excell ence: wit h t he own lif e, t he mo st gener al perfor mance of t he o r ganism. To describe thi s si mpl y like "su bj ecti ve" s eems to me o nly attr i butable to t he awkwardness of our means of ju dgment .

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On the nature o f t he co rrelatio ns t hat we can fi nd bet ween both puls es ­ s ynchro ny and phas e lag, additi ve, multi plicative, periodi cal , etcetera- li ttl e or not hi ng we can say, unl ess t hey des erve the mo st car eful and det ail ed stu dy. This is an experi mental i nvesti gatio n, and i t i s i n t he experi ment al measur es and valu es wher e t he anal ysis tool s mu st s har pen. Not even we can advance what wou l d add t he co rr el ation of t he waves wit h t he co mpl ete for m to t he corr elatio n of fl at or si mplifi ed waves ; we cannot eit her antici pate what adds one to t he anot her at t he ti me of det ermi ni ng t he values of t he guna s or dosha s, nor to what extent we can do wit hout t hese i n ai d of a pr eci s e def i nition i n t er ms o f potenti al s. The relatio n bet ween prak riti and vikriti do es not have why to be identi cal to t he r el ation bet ween the vegetati ve and volu nt ar y as pect s, but in any cas e t he pro du ct of bot h pair s must t hro w t he s ame valu es. As f ar as t he corr elatio n between bot h pul ses and t he bil ater al al ter natio n of t he breat hi ng in t he nos e, it does not have why to gi ve a si mil ar fr equ ency or amplitu de at all , si nce fro m t he begi nning t hey are, even co nnect ed, ver y di ff er ent cycl es. All t his and much mor e o nly can be refi ned wit h the experi ment al co ntrast, t hat is what t hi s wr iti ng t ri es to sti mul ate. I t ake advantage of this poi nt to say that har dl y will be never an i nvesti gatio n t hat can o bt ai n such advances wit h so ti ny means: s enso rs to obt ain measur es and a computer to anal yze t hem. In additio n, befor e venturi ng to u s in t he su bj ect o f the co rr el ation o f both puls es, a standar d det er mi nation of t he measur ement of doshas and it s pro portio ns woul d have been seeded of difficu lti es and ar bit rari ness, i n additio n to have needed samples o f pul s e r eadi ngs o n many hu ndreds su bjects. It was , reall y, the most costl y and di s cour agi ng part of thi s t ype of investi gatio n. Now, bas ed o n t he cor relatio n of bot h puls es, I ho pe t hat it will be pos si bl e to st art to coll ect rel evant dat a fro m t he s ame begi nni ng o f the res ear ch, begi nning here also t he work of analyti cal ref i nement that thi s fi el d i s demandi ng. Of cour se, t his does not exempt to u s to do many mor e measu rements The l eft hand of C hao s
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and s ampl es, and i n particul ar t he dynami c stu di es of s eries wi t h the su bj ects under effort and acti vit y, t hat seem to be speciall y pro mis sor y and r eveali ng, also at t he ti me o f verif yi ng f ast er t he du al t er ms of bo th pul ses ; but t he f act t hat al r eady fro m t he beginni ng we can t ake t he bull by t he ho rns wou ld have to be sti mul ating.

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W hen we t r y to thi nk about what cl ass o f o r gani c ti me i s go i ng to r eveal thes e exper i ments to us, bar el y we make another thi ng t hat to co me u p agai nst ol d cat egories and ol d ways o f thi nki ng. W e coul d imagi ne fi bbers bei ng cont ract ed and dil ati ng towar ds t he past and t he futur e; or we cou l d t hi nk about an expanded pres ent whos e ent ro py equall y i ncreas es i n di rectio n to war ds t he past as i n dir ectio n to war ds t he futur e, et cet er a. All this bar ely as it is anot her t hi ng t hat to i nter mi ngl e qualiti es o f so me well- kno wn sys t ems di scar di ng ot her s to gi ve o t her possi bl e sys t ems , but it s eems to me t hat to s peak abou t past and futur e i t i s particul arl y decepti ve for t he t ype of behaviour t hat we loo k fo r ; if we truly ent er it , t hese wor ds will be u sed per haps wit h t he o nl y i ntention to u nders tand us , t hat i s to s ay, wit hout mor e valu e t han t he o ne of t he analo gy and t he metaphor. In our daily life at l east , past and futur e are no dou bt i mpo rtant r eferences , bu t few will have wo nder ed to what ext ent t hey ar e neces sar y. My opi nio n is t hat mu ch less t han what is beli eved, alt hough everyt hing depends on t he t ype of neces siti es t hat t hey want to get u s i nvol ved i n. It is more, I t hi nk t hat t he di stinction bet ween two dir ectio ns of t he ti me is of i denti cal nature t hat t he di sti nctio n between su bj ect and obj ect, and equ all y illusor y. Unl ike t he mo re s pr ead pr esent s ci enti fi c vi sio n, that al so t ri es to be beyo nd t he di stinction bet ween the past and the fu ture, but i t do es it bei ng based pr ecis el y o n the di sti nction between an obj ecti ve phys i cal worl d and a subj ecti ve and di ffu se per ceptio n. W hat of cours e it i s a fu nny incons equ ence. And fro m t his i nco nsequence, al ready o nl y remains to sear ch ho w t he su bj ectivit y can be pro du ced, how i s s ynthesized. In eff ect, when sci ence approaches s elf -or gani zatio n pheno mena li ke t he biologi cal o nes, it does not seem t hat it can do wi t hout t he questio n of t he memory anyway. At t he cellul ar l evel , many t hings s eem explai nable by means of t he geneti c inherit ance, whi ch assu mes t he rol e of i niti al mechanis m of dat a transmi ssio n t hrou gh t he t empor ar y dri ft. Yet, t his "memor y" r at her is r educed to the elaboration of bio chemi cal co mpo nents, needs to be u pdated by so met hing al ien to it and i n additio n har dl y can expl ai n its el f all t he cellular grou pi ng pheno mena of superior or der: u ntil t he poi nt that t here is always so mebo dy ready to int roduce some type of dou btful "i nher ent dynami c memor y" to t he s ys tem. For me, t he single f act t hat eit her the memo r y co ncept i s us ed implici tly, even in t he cas e of t he geneti c materi al, s eems to me somewhat sus pi ciou s yet: it is al r eady a fo r m to di s guis e t he s ame o bj ecti ve-su bjective divisio n that is pres ent i n all t he other . That is to s ay, a caus al r econst ructio n of t he su bj ect i s sear ched ­ an i magi nar y co mpositio n - , because we ar e i ncapabl e to kno w what it i s happeni ng i n a si mpl e cell i n any gi ven whil e; har dl y not hing on t he evolution i n real ti me, t hat so refr actor y seems to be the anal ysi s. B y the way all t he cell s pul s e, and not o nl y tho se of t he heart . This i ncons equ ence emer ges fi nall y, and of t he mo st s pect acul ar fo r m, i n t he su bj ect s and bu sines ses of t he great bl ack box o f t he "no ble or gan", our br ai n. The l ast front ier, as so me call it; but t hat seems to us so equ al to t he ot her s, except by t hat co mpl exit y i n whi ch we can t ake r efu ge. Here, the neurolo gi sts s peak wit h any fai t h of the det ail s o f t he s hort t er m memor y, defined more o r l es s by t he tempo ral t hres hol d i n whi ch the memor y r eall y do es not need to be memor y, but an expansio n of the speciou s pr esent. As far as t he su bj ect of the lo ng t er m memory, no bo dy will pretend to have t he s mall er i dea on where and ho w it can be sto red; surel y becau s e it i s not sto red at all . It cannot be but an expanded versio n of the expansion o f t he actual pr es ent i n whi ch consist s t he so called "s hort t er m memor y". What it does not exclu de t he f act of t he co mpl exit y of t he neuro nal plasti cit y, but rat her justi fi es it. Be cau se i f the memor y wer e r eall y stor ed, t he cer ebral architecture could be i nfinit ely simpler, asi de t hat t here woul d not have been as much pro bl em i n fi nding i t. W e are s ent then to the di ff erence and correlat io n bet ween t he veget ati ve and t he volunt eer, wher e i s t he true knot of t he su bject ; beginni ng wi t h the char act eri sti cs of t he tempor al s eri es in bot h hemi spher es , which constitut e t he The l eft hand of C hao s
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mo st global and accessi bl e of al l t he behaviour s. And alt hou gh li ke a mer e fo rmula it could s eem ver y si mpl e, we can beli eve t hat it co nt ai ns sur pr ises enough to mai nt ai n t he r es earchers bus y, as well as to remo ve unnecessar y and s puriou s co ncepts . If we are sur e of it , it will be becau se t hi s evasi ve pri nci ple cannot be reduced to a parti cul ar domi nio n.

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SECOND PAR T TIME AND THE MODER N SC IENCE

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It has been sai d, wit h t hat half trut h so u sual in t he to pi cs, t hat i f the Greek mat hemat ics wer e geo metry, aft er Newto n wer e mai nl y anal ysi s. Affi r mation t hat woul d be ri ght if it i s co nsider ed t hat t he analysi s grew mai nl y i n the applications to phys ical pro blems, but t hat it is si mpl y illusio n fro m t he chronologi cal s equ ence of t he f acts . W e ar e not speaki ng abou t t he o ld controver sy o n t he aut ho rshi p of t he calculus, t hat al ready has been set tled by t he historians: t he cal culus was not i nvent ed neit her by Newto n nor Lei bni z, but t hes e wer e rat her tho se who cro wned it after a great number of previous co nt ri butions, and litt le matt er s who was f irs t i n po sses sio n of o ne or another fo r mul a which would f all li ke matu re fruits . They t wo, wit hout a dou bt , di d mu ch by t hei r maturit y. But, i n which does to t he superseding of t he s pirit of t he geometr y, it i s to s ay t hat was not Newto n who took t he first st eps , bu t all t he oppo site. If we admitted i ts own l at er co mment ar ies, t hat we have to bel ieve, t he cal culu s met hods that cou ld u se for t he i ncu batio n and el abor atio n of Prin cipia di d no t have another di gnit y that the pur ely i nstru ment al one, wit hout gi vi ng t he s mal ler subst anti ali t y or auto nomy to them. Justl y t hu s woul d be concei vabl e hi s abs ence of pu blicatio ns o n t he su bj ect. The perf ect count er poi nt of thi s situ atio n we have it in Lei bniz himsel f, certainl y mu ch poor er of po siti ve achievements . Lei bniz concei ved indeed fro m t he beginni ng t he cal culus and t he anal ysi s like a new pl ane wit h pro per auto no my and al ready freed from geo metr y, althou gh, buri ed by t he s er vitu des, bar ely he had ti me to explor e it. What Newton took advantage of for more t han t hr ee lu stru ms to per fect indust rious l y all cl ass of t echni ques and to demand t he pr e- emi nence o f hi s aut hor s hi p. But what s ays t he previous wor ks of bot h, t her e for 1686? That t he Principia are t he most ar duous bo bbi n lace of t he histo r y o f the geo met r y, and t hat the Tr eatis e of metaph ysi cs or the Monadolog y are t he tr eati se l ess i nter pret abl e i n pl asti c ter ms of all t he histor y of t he wes tern philo so phy.

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Wit hout a dou bt bo t h Newton and Lei bniz had to pay a tribut e to their centur y; but wher eas to t he fir st that tri but e benefit ted to hi m enor mousl y, to the seco nd has i mpede any t ype of keen of r eceptio n. The fir st Fr ench review of t he P rin cipia praised thei r geometry, but t hey advi sed to t he aut hor t he impro vement of all co ncer ni ng to t he "phys i cal cau ses "; what li st ened with nowadays ears, s eems to u s t he hei ght of t he i mpertinence. But all t he per pl exit y befor e t he hypo thesis of t he remote act ion at dist ance was justifi ed enou gh. And Newton himself had to end u p realizi ng t hat t he best ju sti ficatio n of t he gr avit y law was not ot her o ne t han t he o wn s pace of the anal ys is , r el eas ed of the s er vitu de of geo metr y and t he mechanical The l eft hand of C hao s
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ar gument ations. In as much he was assu mi ng t hi s, t he autho rshi p of t he calcu lus gr ew i n i mport ance for hi m; all t his agr ees ver y well wi t h t he s equence of t he f acts. And preci s ely, it was t he f amili ari zatio n wit h t he i dea of the calculu s the one that achieved t hat , hal f a centur y lat er, when the Newto n theory became popular, peo ple as Daniel Ber nouilli or t he s ame Vo ltair e no longer were sur pri s ed at all wit h t he idea of t he remote actio n at dist ance and even found it natur al. Kno wn i t is t he phr as e o f Volt ai re o n t he matt er: "the anal ysi s i s t he art of cal cul ati ng and measuri ng with exactitu de t hat of whi ch not even we can conceive t he exis tence". It was t he anal ys i s and t he analyt ical ment alit y what gave its cr edi bil it y to t he theory of Newto n, and not o n t he co ntr ar y. And when still to day we consi dered t hat t he force of t he gr avit y has to be a const ant, we onl y pai d a tri bute to t he analyti cal exi genci es, and to no i ntuitio n at all . The i mmat eri al i deas of t he anal ys is ar e t hos e that mor e have contribut ed to our belief of a wo rl d o f i nert mass es , what is trul y antago nisti c and paradoxi cal . The cas e of Lei bni z was ri ght t he o pposi te, like it had to be for t he count er point of t his baro que fu gue that stil l l as ts extendi ng and lo si ng itself i n other s pher es . The philo sophy o f Lei bni z is , i n s pirit , enti rel y mu si cal : it is t he fir st o f the philoso phi es i n transcendi ng entirel y t he mor e g eometrico withi n the rationali sm and al so t he last o ne. But nothi ng harms mor e t he u nders tanding of Lei bniz philoso phy t han t he plasti c images , o f whi ch he hi ms el f was so met i mes involu ntar y res ponsi bl e. Then what can seem t he mo nads to us but a monstro sit y havi ng i n mi nd mi rror s and clo cks? Sti ll to day, if i nst ead of t hi nki ng about watchmaker machi neri es ad in finitu m, we r epl aced t hes e by waves, a great part o f t he wor k o f Lei bniz acquir es clarit y as i f by magic. And abou t waves he mus t have s poken, changing i n t his o ne of t he motifs of his mentor Hu ygens by the ot her o ne. Kno wn ar e t he dif ficu lti es t hat Hu ygens found for the mat hemati cal el abor atio n of his wave theor y of li ght, qu estio n thi s o ne t hat pr event ed its gr eat er dif fusion; what per haps it i s not r ecognized so oft en is t hat t he F eynman pat hs o f t he mo der n qu antu m elect ro dynamics are not but t he pri nci ple of dif fusion of t he light of Hu ygens ­f rom all the points to all the points b y all the point s- wit h a fi nite s peed for li ght. Lei bniz i s the mo st fr esh, shi ni ng and i ntu iti ve philoso pher of t he mo dern t hou ght, and t he l ess dis cur si ve o f all i n hi s best mo ments ; but it is t he mer e dis cur si venes s of moder n t hou ght, u nit ed to t he povert y of it s po wer of imagi natio n, what makes hi m to s eem li ke an ol d rusty calcul ato r. But, agai n i n tr agi comi c co ntr ast wit h Newton, Lei bni z bar el y had ti me to matur e anythi ng o f whi ch he di d and di s cerned. B een bor n i n a Germany devastat ed by the war of t he t hirt y years, for ced to loo k for t he life, hal f adventur er and half poor devil , lo s er i n all t he rou let tes of t he fortu ne and t he po st erity, it i s impossi ble not to s ympathize with him. Even t heir spiritual dis ci pl es and hei rs moved t heir look away fro m hi m dis cr et ely and tried to loo k for t he i ns pi ratio n i n his mor e i mpl acabl e ri val. But t he t hings al ways follow their cours e, i gnori ng misu nder standings .

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Lei bni z even got to challenge t he Galil ean pri nci ple of r elati vit y, allu di ng to t he i mpossi bili t y to est abli s h t he correl ati ve mo vement bet ween two mo vi ng bo dies . This o nl y acquir es s ens e wi t hi n t he prevaili ng role t hat it grant ed to t he activit y and ener gy over i nerti a. But t hese consi der atio ns moved away more and more of any pr acti cal appro ach. Ri ght it has been sai d t hat t he dynamics of Lei bniz is t he dynami cs of t he mo vi ng wit hout mo vement ­t hi s i s, wit hout a s pace cor rel at e -, wher eas t he mechani cs of Newto n i s mo vement wit hou t mo ving. The pr e-es tabl is hed har mo ny of Lei bni z, fo r exampl e, for mo st o f u s t he su mmit of t he ar bitr ar i ness , acquir es it s sens e and r elevance in as much we reali ze t hat t he absolute ti me of Newton, t he pri ncipl e of glo bal s ynchro ni zatio n, it i s no t l es s ar bitr ary nor metaphys ical at al l. And not o nl y t hat: if we percei ved the net met aphysi cal char acter of t his l ast o ne, we begi n to u nderst and or to sus pect at l east t hat to whi ch Lei bni z is allu di ng wit h t he question of t he pr e-establis hed har mo ny it i s exactl y to the po ssibi lity o f s ynchroni zing entiti es and process es t hat are as ynchro nou s thems el ves , si nce it is from t heir o wn t hat deri ve thei r l aw. But o n the ot her hand t he mo nads not even exi st i n t he s pace, but ar e lo catio n pri nci pl es , as mu ch wi t h r espect to the s pace as wit h r espect to many ot her su bs tances or pl anes. In t he same way t hat o nly for as ynchro nous The l eft hand of C hao s
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pro ces s es can have so met hi ng to s ynchroni ze, exist s the no n-lo cali zed fo r t he lo cal it y, i n a way t hat o nl y partiall y can l eave mar k i n t he s pace. Leibniz hi mself defi ned t he poi nt li ke a pur e modali ty.

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All the philoso phy of Lei bniz exist s to cons erve the ri ghts of t he co ntingency; fo r t hat r easo n it i s arti cul ated arou nd t he modali t y, which it is u nfol ded and flu ctu at es wit hi n t he s cale o f t he neces s ary, po ssibl e and t he co nti ngent. On t he contr ar y, the thr ee pri nci ples of Newton exist to exclu de absolut el y t he mo dalit y, to evacu at e it, alt hou gh it i s vir tual ly i mpo ss i ble s i nce t he necessit y alr eady is modalit y. On the ot her hand t he idea of physical co nst ant, t hat for Newto n was not cl ear and onl y instru ment al, i t is not but a superstition gener ated by t he eff ecti veness o f t he i nfi nit esi mal calculus , t hat t he l at er develo pments have onl y co ntri buted to u nderpi n. A uni ver s al co nst ant o nly s eems to just ify i tself by t he f act t hat wit hout it t he cal culu s woul d become i mpo ssi bl e; but this is mor e than suffi ci ent , so t hat actu all y it is outsi de all question, unl ess t he f acts insist o bsti natel y o n denyi ng it , i n which cas e, or can be ignored or can consi der anot her dif f erent maneu ver.

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The i dea of the mo nad conti nu es s eemi ng so str ange that sti ll to day f ew are t he o nes t hat begi n to decipher it . Becau s e it i s not t he trit e and empt y fi gu re of worl ds wit hi n worl ds ­one mor e of thi s mi sunders tandi ngs i n pl asti c t er ms- but mai nly of whi ch co exist s o f l at er al way. The mo nad is mainl y centr e or focus of act i vit y; one of its pl anes o r su bs tances can be a body li ke ou rs. This bo dy can be made of infi nit y or an u ndefi ned nu mber of part s that co ncu r in t he acti vit y of o t her monads, t hat ar e not o nl y bo di es as well , but ot her many su bstances. Of cou rs e, al ways r esult s so met hing unnatu r al and i nco nveni ent to s peak i n plur al about t he su bst ance. Here, t he substances have part s and at tri but es, but not t her efor e t he mo nads . What all t hi s is suggesti ng is not that we ar e done of many bei ngs , bu t t hat t he part s t hat apparent l y we occupy ar e and exist of natur al for m in t he s pher e of ot her mo nads. Thi s i s mo vi ng, and i n particular for t he west er n philosophy. But ho w we coul d hope that mo der n s ci ence could cont empl at e subj ects like thi s anytime? It seemed mu ch more out o f it s capacit y at t he ti me of Lebniz, and so it s eems to day.

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But a monad i s a pu rel y dynami c entit y, no t a su bstance who se botto m r emai ns i ndefi ni te for al ways . It i s mu ch less abst r act t han thi s one, o r that the i dea of t he s pace and t he ti me, and all t hi s o nly serves to gener ate co nfu sio ns . It i s t he whol e, undivi ded tot alit y, and when we ar e feeli ng or measuri ng t he s angui neou s pul se, we are verif ying i n a gr eater measur e the acti vit y of t he monad o ne t hat t he o wn acti vit y of the bo dy. Then, any r efl ectio n o f t he totality, by s mall or t enuous t hat is, belo ngs mor e to t he or der of t he monads t hat to the one of t he subst ances and its attribu tes. Onl y for the mo der ns could appear t his li ke i ncredible, becaus e i n fact t he mo nad i s t he mos t natur al concept . Let us t hi nk i n t hat ot her for mul a appar entl y so str ange ­ and unnatur al - of whi ch the mo nad does not have wi ndo ws . Thi s t ends to i nt er pr et i n the sense t hat t he monad i s compl et ely isol at ed o f t he envi ro nment. And nevert hel ess its r el atio ns extend to t he i nfi nite, and i n co nti nuit y. But al so t his has a fait hful r efl ectio n i n the pul s e and t he mai nt enance of its pot enti al: i f the sensiti vity of t he s ys t em, t hat is its li mit wit h the enviro nment, al so ent er s u ntil any i nter nal dept h of t he equi li briu m or bal ance. Thi s o ne is pr ecis ely t he necess ary and suff icient co ndition o f any autono my and self-or ganizatio n. The monad is t he mo st natur al co ncept io n of a syst em t hat alr eady co nt ai ns to it s enviro nment li ke par t of it s defini tion. The l eft hand of C hao s
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Dewey des cri bed wit h s kill the general t endency of t he Ger man philoso phy li ke apologeti c, and to t he one o f t he Br itis h philo sophy as scepti cal. Wit hout a dou bt it has been so ; alt hou gh as t his l ast o ne was as sumed by t he scient ifi c practi ce, t he boot 's o n the ot her foot no w. Sci ence long ago became apolo geti c, and every t i me ar e mo r e t ho se t hat are s cepti cal wit h res pect to t hei r as su mptions and just ifi catio ns. The s ame fi gur e o f Newto n su ppo sed a decisive mut atio n o f the philoso phy of t he i sl ands ; a r atio nali st co ntamination o f t he co nt i nent lodged i n an empir ical panor ama wit h vocatio n to remai n for mles s and so wit hout co mmit ments as can. W hy it i s des cri bed to Newto n li ke introdu cer o f t he empiri ci s m i n s ci ence? W hat conso li dates is exactl y t he o pposit e; i n additio n, t he experi mental met hod i n physi cs alr eady had recei ved it s bapti s m i n Galil eo. It is no wo nder t hat since Newto n t he su s cept ibility wit h r es pect to t he co nti nent i s s harpen, and t hat is tri ed to di gest t he i nflu ence until maki ng it i nvi si bl e and no n exi stent. The co ntro ver s y bet ween Newto n and Leibni z i s kno t and s ymbol of the r el ations bet ween t he co nti nent and t he isl ands to t he pr es ent ti me. Alt hou gh a ger m, st ill vi sibl e, has r emai ned wit hout digesti ng.

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At t he begi nni ng o f centur y XVIII, Her mann Bo er haave, t he mo st f amous do ctor of Euro pe at t he time, u ndertoo k t he att empt to rat io nal ize t he pr acti ce and t heo r y of t he medi ci ne to i mage and si mi larit y of an experi ment al phys i cs tot all y co nsoli dat ed si nce Newto n. Bo erhaave loo ked for yatro mechani cal pri ncipl es of the maximu m si mplicit y, even kno wing the enor mou s di ff er ence bet ween his do mi nion and t he o ne of t he mat hemati cal phys ics. Alt hough t he school of Leiden l eft t rack i n t he at tempts of renovat ion of t he moder n medi ci ne, Boer haave and t hei r follo wers had to be satisf ied wit h littl e mo re t han an appeal to the experi ment al pr acti ce, r easo n why t hey coul d have r esort ed equall y to the name o f Galil eo or t he s ame Bacon t han to t he one o f Newto n. The si mpl e pri nci pl es never coul d cr yst alli ze i n t he medi ci ne nei t her in B iolo gy. The yatro mechani cs, t he medi ci ne o f mechanical i ns pir atio n, loo ked for t he defi niti ve o vert hrow of a cert ai n medici ne o f chemi cal i ns pir atio n ­t he yatro chemi cal medici ne, whos e ori gi n s end us to P aracelso - in the same way t hat t he experiment al phys ici sts had turned o bsol et e t he s chol asti c di s put es . C hemis tr y was under stoo d du ri ng centuri es, al so i n the West, li ke t he s cience of the Natur e and t heir tr ansfor mations par excel lence, but, bei ng t hei r cat egori es too qu ali tati ve and t angl ed, t he mo ment arrived i n whi ch t hei r pri nciples seemed too vai n and s pecious i n co mpariso n wit h t he mu ch mo re net defi ned and manageabl e o f t he mechanics. The paracel sians s po ke o f t hree fundament al pr i nci ples, deduced empiri call y of t heir oper atio ns of co mbus tio n, eli mi natio n and distill atio n, and i n t his t hey agr eed wi t h the basi c philo sophy o f ot her pr evious medi cal schoo ls . Nevert hel ess, of what t hey coul d separ ate of el ementar y way o f inert su bst ances was l ittl e les s t han i mpo ssi bl e to s ay not hi ng or gani zed and reaso nable i n ter ms of act i ve and u ndi vided r elatio n. Nor we have to say that thes e t hree pri ncipl es, sulphur, mer cur y and t he s alt, were co mpl etel y equival ent of t he t riguna and of tridosha of t he ayur vedic medi ci ne, or, to s peak t he mor e sur e, t hey wer e exact l y the same.

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The t hr ee pri nci ples o f t he Newtoni an mechani cs beco me t he t hree i nher ent mo dali ties o f t he nature wit h a s ingle co ndition: that they cannot occur si multaneousl y. That i s to say, t hat any o per ation o f The l eft hand of C hao s
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reestabl is hment of t he balance consumes ti me. Natur all y, t hi s changes ever yt hi ng to it, and i n mu ch great er degree t han it i s u su all y co nsi dered accept able. And nevert heless , it i s not necess ar y to s ay t hat t he co nser vatio n l aws woul d co nti nue prevaili ng at a glo bal l evel and as it i s of ri gor . Therefor e, t hi s ver sion woul d not be an apex l ess r atio nal t han t he well - kno wn mechanics or dynami cs ; but sur el y it would be much more r ealist ic and it woul d put to us on a ver y differ ent game. And to wher e woul d go t he ti me i n that game? We have alr eady s een t hat i n hydro dynami cs t hi s has enough s ense. And it has parti cul arl y i n the qu antu m mechani cs , wher e we cannot do withou t t he ti mes of r eactio n; also local violatio ns o f t he co nser vatio n pri nci pl es occur here t hat t end to reco ver globally as soon as po ssi bl e. And i n addition, t here is to cou nt o n t he fact, fr equentl y fo r gott en, t hat t he l aw of co ns er vation of angul ar mo ment i n Newto nian mechanics has not t he same axiomatic r ank t hat t he t hr ee f amou s pri nci ples , becaus e it s tarts fro m t he co ndition i n which pu nctu al particl es has t he same li ne of actio n ­an ext remely weak co nditio n, as we can see, and t hat i n mo st of t he cas es it has not why to happen neither i n t he quantum mechanics nor i n t he st atis tics nor in t he hydro dynami cs , except by average.

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It seems to be t hat Lei bni z was t he first i n us ing the ter m fun ction, whi ch woul d not be a sur pris e taki ng i nto account that he was also the first who u nder stoo d t he po wer o f t he i dea of tenden cy. Thi s was extr emely i mport ant and even, as Bloch remar ked, t he moder n i dea o f t he di alecti cs i s u nt hi nkabl e without it. But, as it has been s ai d, the same proces s o f the cal culus and the anal ys is , by exami natio n o f t he success i ve appro aches, i s of di al ectic natur e. It do es not belong to a "normative pl ane", nor is mechani c but for t hos e for that t he so lutio ns ar e gi ven. On t his , the pr es ent mat hemati ci ans coul d s ay mor e t han the mat hematicians of any ot her ti me, but , lament abl y, li ke last co nsu mer s of t he mat hemati cal pro duct ion, bet ween us t he i nerti a pr evail s. In Newto n any di al ectic co nceptio n s hi ne by i ts abs ence, whi ch expl ai ns t hat he di d not reco gni ze t he calculus al t hou gh he was calcu lat i ng all the day and had in front of his nose t he whol e ti me. Like MoliČr e's charact er, it s po ke pro s e wit hou t knowi ng it. The histo r y of t he controversy of t he cal culu s is as si mpl e as this: To Newto n, bor n anal ys t by pur e s crupl e s piri t, and next to t heir co lleagu es Wallis, Gregor y and Bar row who alr eady al most had cros sed t he bri dge, t heir own met ho ds had to be sus pi ciou s to hi m, and even wort hy o f bei ng hi dden. To Leibniz, vi rtu all y isol ated i n t he co nti nent , the met hod its elf had to seem a wo nder to hi m ­ wit h r easo n-, and dared to publish fi rst and to generalize wit h agilit y and trust t he mo st natu ral notatio n. To thi s t he facts ar e r educed, but not t he i mplicatio ns neit her t he r esonances. And for t he i dl es t hat al ways gi ves the thi ngs by known or su pposed, we r emember that si xt y years l at er of t he appearance of arti cl es of Lei bni z, Euler r acked hi s br ain t r yi ng to defi ne t he co ncept of fu nctio n; o r t hat barel y met aphysical Gauss spo ke wit hout s hame of t he "met aphys ics of t he i nfinitesi mal s" at t he begi nni ng of the XIX centur y. Reall y, the basi c mo ment of the appear ance o f t he cal culu s i s the overco ming of geo metr y. Lei bni z was t he fi rst who gave t hat step; and to that st ep for war d i n the vacuum of t he anal ys is must all t he modern phys ical t heories their main source of cr edit. In Lei bni z a si ngle i dea of t endency, co nsi dered in al l it s ampli tu de, ser ves to i nclude pot enti all y t he physical worl d as much as t he organic o ne wit h no need of a special disti nction: t he development is i nher ent to bot h of them, althou gh not hing is gott en to s pecify and t he role o f t he ti me i s co mpl et el y ambiguous i n t hi s res pect . The pri nci ple of least actio n, to which t he Ger man gave wit h reaso n great i mport ance, its not reall y s peci fied either, alt hou gh he i nclu des potentiall y i n it a mu ch mo re vast s co pe t han t he co nt empl at ed by t he pr esent dynami cs . One of the l ess noti ced and co nsider ed aspects o f t he view of t he world of Newto n is the absolut e di sconnection of any his tori cal or evolutio nary f actor when cont empl ati ng t he Solar Syst em wit h its planets. In fact, Newto n mai nt ained a cat astro phi c visio n: God had pu t of a blo w t he worl d i n t hat state, and when he want ed that st at e woul d sto p exist. Madl y s cru pulou s, Newto n di dn' t decei ve hi msel f about t his , l ike about everyt hing of his co mpetence. Long befor e Poi ncar Č, he su s pect ed t he absolute fr agili ty of hi s l aws befor e t he The l eft hand of C hao s
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eventualit y of comets or any ot her sou rce of pertur batio ns. Wit h the great er solicitu de, Newto n knew that hi s world s li d magi call y i n the not hingness , unli ke mo st of t heir su cces sor s, i nfi nitel y more mediocre and nar coti zed alr eady by t he ef fect of t he custo m and t he i nt er ests o f t he accumulat ed capit al. To r each t he r ecent times, i n whi ch the mo st i nnocent o f t he natur alists or t he biolo gi sts arri ves t rust ed to us to expl ai n even t he origi n of t he s peci es and t he lif e by means of mechanical pri nci pl es and ever ythi ng wit hi n a r easo nabl e t erm of time.

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The attitu de o f Newto n and Lei bni z to f ace t he t raditio n of t he past cannot be more o ppos ed eit her. Lei bni z, i n the pu rest apolo geti c to ne, made al ways generou s ef fort s to conciliat e any s ci enti fic develo pment i n pers pective ­and he had a quite ample o ne- wit h the spi rit of mo st of the ol d philoso phy and wit h t he myst eries of t he r eligio n, t hat not its do gmas and do ctri nes. Not in vai n coi ned t he t erm philosophia per enni s, li ke pol e of t he wisdo m t hat woul d have to tr anscend t he circu ms tances of time and place. Hi s int egr ating effo rt was s i ncer e, and tried to gi ve new sens es to t he motifs of t he ancients and t he co nt empor ar ies fro m it s own s pontaneous, ori gi nal and superabu ndant source. Then precisel y to him, that so l ament ably had to leave dozens o f import ant concepts and i ntui tio ns to develo p, we can su ppos e les s needed t han anybo dy o f other people' s i deas. Unlike Newto n, who al ways was persu aded of th e knowl edg e o f the anci ents ­but not so of t heir wi s do m -, t hi nking t hat the mo re per spicacious amo ng t hem had transmitted it hi dden i n cryptogr ams . It i s a funny i dea t hat the anci ent s had to cont empl at e t he t hi ngs i n the same s pi rit t hat t he s cientist s of t he XVII centur y, and had to under st and t hem i n such ter ms. Co nception t hat, i n addit ion, is not very differ ent fro m t he one o f so many myster y and f antas y no vel s. That's to say, no r eal myst er y, o nl y s ecr ets to cr ack. The same aut hor of t he t hr ee pri ncipl es of the mechani cs was s ecr etl y arri an o nly to deny t he i dea of t he tri nit y, that it seemed to him a f i ddle of t he C hur ch fat hers . In thi s way, he denied any generati ve power and any possi bl e rel atio n of t hat tri nit y with t he natur e, whi ch har dl y reached t he statu s of lar ge est ate of a Lo r d. As far as his co nt empor ari es , it do es not seem coinci dence t hat he maint ained co ntro ver si es wit h al most ever y of them. The lumi nou s ri ng t hat di sco ver ed Bo yle or Hooke soo n turned t he ri ng of Newto n, and thu s wit h al most ever yt hi ng. But known it is t hat what wit h Newto n begi ns i s, pr ecis ely, t he s ci entifi c est abli s hment and t he manufactur e of i ts pr est i ge, and t his so it was needed of u nit as of t he fi gur e of a su preme po nti ff, to who m he hi ms elf co ntri but ed wi t h all his actions . The mo ment ar rived i n whi ch, li ke wit h an absolut e mo nar ch, ever yt hi ng co nver ged towards hi m, and i n as much as ever yt hi ng moved away f ro m Lei bni z. Known it i s t hat fi nall y and at t he li mit , the ver y s ame Hano ver emplo yer of Lebni z was cro wned ki ng of England, to whi ch somethi ng woul d co ntri bute t he genealo gi cal vindi cations of t he philo sopher , i n another one more o f hi s for ced ser vitu des and wor ks ; and t hat t he new ki ng for got hi m co mpl et el y for , among other t hings , no t to bot her Newton alr eady at t hat ti me tu rned tru e i nstitutio n. The detai ls of t he biogr aphi cal antit hesis of t he t wo gr eat aut hor s o f t he i nfi nitesi mal cal culus , t hat even wer e bor n under t he s ky of o pposed co nst ellations, bor der t he i ncr edi bl e, with i nfi nit esi mal as pect s i n all the mo dulat io n of t heir di alecti cs. They make r eco mmendabl e t he joi nt r eadi ng o f bot h biogr aphi es , pro babl y boring i n t hems el ves by s eparate. And it is t hat, at l east by o nce, we had to be abl e to di stinguis h t hi s deep pol arit y not o nl y i n the t ree of t he lif e, al so i n t he tr ee of sci ence: t he pol arit y of dis s emi nati ng and co ncent rati ng, t he veget ati ve and t he volunt ary, t he s peculat i ve and t he s peculator ­t he proli fic, and t he devouri ng.

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It was not coi ncidence eit her t hat t he i sl ands wer e l eft behi nd i n t he development of t he anal ysi s wit h res pect to t he conti nent throughout t he eighteen centur y and goo d part of the ni neteen centu r y, being ado pted the not atio n of Newto n i nst ead of t he Leibni z o ne. Lei bniz found dir ect dis ci pl es in t he Ber nouilli brother s, initiati ng a line that would be co nti nued wit h Daniel Ber nouilli and t he s ame Eu ler, who m among ot her t hings , develo ped t he vari atio nal princi pl es, t he las t and unavoi dabl e for m of mo der n dynami cs . Lagr ange bo ast ed of having managed t he dynamics doi ng wit hout geometry enti rel y. Newto n, we do n't need to say, was t he great ref erence for all of t hem.

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But wit h t he B ernouilli and Eul er anot her different branch o f t he anal ys i s begins also , as separate of pr acti cal problems as po ssi ble: t hat i n whi ch t he anal yti cal t echni ques begi n to approach questio ns rel ati ve to the nu mber theory. Leo nhard Eul er , i n who seems t hat t he s pi rit of Lebniz had concent rated itsel f wit h t he o nly aim to make mat hematics , fou nd a for mul a that r el ated in an exact way the i nfi nit e sum of all t he natur al nu mber s to the pro duct of all t he pri me nu mbers: t he famous Eul er pro duct for mul a. This it is a goo d exampl e o f so met hi ng i nco ncei vable t hat i t si mpl y tur ns out to be cert ai n, and even r el ati vel y eas y to demonstrat e; and t he decisi ve pr ecedent of t he anal yt ical nu mber t heo ry, s ever al gener atio ns o f di stance from its acco mpli shment. In s pite of his achievement s, t he same Eul er left written t hat the mys t eri es of the di str ibution of t he prime nu mber s woul d be beyo nd t he human u nder standi ng for ever . 59

Gaus s and Legendr e di d not have to thi nk t he same, arou nd t he change of centu r y, s peci ally t he fi rst , who was t he o ne t hat gave great er i mpul se to t he cl assi c nu mber theory, also called superior arit hmeti c: i n f act, the si ngle study of t he i mpli cations of t he Ar it hmeti c. No thi ng i s so fundamental in mat hematics, nor off ers anythi ng si mi lar r esis tance and diffi cult ies. Gaus s gave t he law of t he glo bal dist ribution of t he primes i n t he for m o f lo gari t hmi c int egr al , which too k a whol e centu r y fo r it s proof. So me co mpr ehensi bl e law for t he man was i n t he pri me nu mbers , aft er all ; to day i s call ed t he Prime Nu mber Theo r em. And in s pit e of t his great st ep, ho w little still which i s kno wn o n t he su bj ect. The s ame pr i me number t heor em o nl y fou nd its proof t hrou gh so met hi ng i nfi nit el y mo re eni gmati c: t he or der t hat t he pri me nu mbers s howed by analyt ical extensio n i n the co mpl ex pl ane. Ber nhar d Ri emann was t he fir st man who saw t hat l andscape.

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It i s wort h t he t rou ble to sto p a littl e while i n t he mo der n analyti cal number t heor y; t he o ne t hat goes in dir ect li ne fro m Eul er to Gau ss, Diri chl et and Ri emann. It is o ne o f t he gr eat achievement s of t he human s pirit , as surpri sing as t he development of physi cs and surel y wit h many degr ees mor e of dept h, but much les s appreciated and kno wn. An achi evement t hat was not born from any type of ur gency neit her of hu nti ng of t he su ppos ed s ecr ets of the nature, but from the mo st i ncorru pti ble and f ree of t he explor atio ns. But t he cir cumstances are not gr atui tous . The s pace of t he anal ys is t ends to pul verize t he obj ects , breaki ng t hem u nti l inf i nit y, or until is pos si bl e. The po wer of t he anal ysis aris es fro m t he develo pment of t he seri es ; but of all t he seri es , no ne mo re natur al t han t he o ne of t he natur al nu mbers , o r the whole numbers , wi den a litt le bit . As t he pul veri zi ng po wer of t he anal ys is gr ew, parti cul arl y for whi ch we called t he dynamic space and t he r eal s pace, the integrit y o f t he whole nu mbers, its ir redu ci bl e and di scr eet char acter , became involved i n a co mplet el y new relu ct ance and char m. Alr eady Lei bni z ­ and t his is quit e si gnif icant - had appreci at ed all t he i mpo rtance of t he The l eft hand of C hao s
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di scr ete mat hemati cs, as he hi mself denomi nated it , wit h an ample s yno ptic visio n o f whi ch now we call ed co mbi natori al anal ysi s. And alt hou gh barel y he wor ked i n nu mber t heor y, no bo dy more i ndicat ed to concei ve the new r elatio n t hat was approached here. But Gau ss s ent enced it with his habitu al s trai ghtforward st yl e: the Arithmet ic is t he s ci ence of the time, ju st as geo metr y is the science of t he s pace. For physi cs thi s af firmatio n cannot be fals er . But physi cs not o nl y has ari sen fro m t he mathematics, but i n fact is applied mat hemati cs . Wit hi n the mat hemati cs the aff ir mation of Gau ss will be unavoi dabl e for ever. Geomet r y cannot s eparate of the gi ven or empiri cal dat a; the Arit hmetic yes, and as compl et el y as can do it so met hing under our s cruti ny. It is mo re, i t woul d be pos si bl e to say t hat it i s the onl y poi nt of t he for mal sci ences in whi ch t he s cruti ny of the own i ntel lect is pos sibl e, -t he s elf-o bs er vatio n-, no matt er ho w har d t hi s i s pr evented co nstantly b y t he ver y instru ment al char act er of t he t echni ques us ed. The dynamic s pace of phys i cs is equal to t he analysi s o f geo metric origi n ( al gebr a i nclu ded) minus an inf i nit esi mal part o f t he Arit hmeti c. Thi s r esi du al su bt ractio n s till pr event s t hat we co ns i der t he physi cs li ke so met hi ng empt y, although soo n we fil l up it wit h t he mi st aken mat erial . The fil li ng i s t i me i n anyone of t he cas es . And it is pos sibl e to t hi nk t hat, i n the same measur e i n whi ch t hi s remainder gr ew, the dynami c space wit h its ghosts woul d be co ntracted.

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It i s not su r pri se t hat for so me mat hematici ans t he Ri emann zet a fu nctio n i s t he o nl y scientifi c di scovery wort hy of r es pect and admir ation: in compari son, t he best achi evement s o f physi cs ar e as magnifi cent as matu re appl es and equall y vulgar . And it is t hat t her e are r easons to t hink t hat, bei ng so str ange its behaviour i n res pect to ot her ki nd of functio ns, has had to aris e to cro sscur rent of t he habitual develo pments , i s to say: li ke o ppo sit e cu rr ent o r ref lux of t he mai n curr ent of the anal ys i s. The Ri emann zet a functio n i s cl assi fied bet ween the special functio ns; and, wit hi n t hem, it plays t he mai n paper in t he Arit hmeti c and t he s eries of t he whol e nu mbers, of whi ch deri ve all t he ot hers . The whole nu mber s ar e gener at ed by t he most element ary of t he o perations, the additio n of t he u nit . When we att end the pro du ct s, we o bser ve t hat so me nu mber s can be deco mpos ed i n pro duct s of i nf erior numbers , and o t hers not: they ar e t he pr i me number s, nu mbers o nly di visi ble by t hems el ves and t he u nit . The order and s equence o f these prime nu mbers has a glo bal regu larit y ­t he lo garit hmi c i nt egr al we al lu ded before- , but a lo cal behaviour as r andom as i t can be poss i ble. Thus, t he zeta fu nction beco mes the best kno wn for m i n mat hemati cs to u nit e these t wo ant agoni stic and appar entl y irr edu ci bl e co mpo nent s. That is to say, we cannot kno w a bond bet ween the or der and t he chance better defi ned and mor e fu ndamental t han t he o ne t han already exhi bits t he whol e nu mber s with t heir pri mes, and t he Ri emann zet a fu nctio n i s that bo nd. It is no wonder t hat such a bo nd has so endles s ramifi catio ns.

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Fro m t he very begi nni ng of the cal culu s the theory o f t he pro babiliti es begi ns to be developed; t hey ar e li ke the two faces o f a s ame cur rent . The appar ent exactitude of cert ai n laws and t he same fu nctio ns does not make but emphasize t he po ssi bi liti es of diso r der fo r i ndependent event s. And it is t hat t he fu nctio n co ncept tends to become s ynonymous o f dependency, and t he probabilit y, o f di s connectio n or i ndependence. Bet ween bot h ends it can be all kind of co mbi nations, and t hu s we fou nd fro m pro babilit y fu nctio ns to pro babilit y of a t ype o f fu nctio n wit hi n t he s pace o f the functio nal anal ys is . The anal yti cal descri ptio ns woul d be i nfi nit el y fragile if t hey di d not toler ate all t his counterpart o f t he t heo r y o f the pro babilit y, and thi s last one woul d not have been developed without t he f irst. Ther efor e, t hey are as indi vi si bl e as diff er ent: fro m t he poi nt of view of the same proces s, it can be sai d wit h all pro pert y t hat t hey ar e co mpl ement ar y, alt hou gh our u nderst andi ng The l eft hand of C hao s
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tends to diff er enti at e t hem as much as it can. A for m dif fer ent to co nsi der t he pro babi lit y, and even t he rel atio ns between necessit y and pro babilit y, is t he co mbi nato ri al anal ys is . One o f the most r emar kabl e peculi ariti es of Lei bni z like mat hemati ci an and phi loso pher it i s t hat he t hi nks i n ter ms of diff er enti al anal ys i s and co mbi nator ial anal ysi s almo st si multaneou sl y: exceptio nal char acteristic that emphasizes the mat hemat ical hi stori an E. T. Bell , but that any reader can per cei ve in philoso phical texts, sometimes dis co ncert ed, and so meti mes even wit h reluct ance. It wou l d be s ai d t hat t he hu man mi nd prefers to separate t hese t wo scopes i n dif f er ent co mpart ment s, to have mor e s pace ju mpi ng fro m o ne to t he ot her . And what it is gai ned is , preci sel y, s pace: what is lost i n t hi s o per ation is t he ti me.

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Not ver y of ten we consi der ed t hat t he u nfol ding of t he t hou ght of a perso n i s an exhi bitio n of the whol e natur e. And nevert hel ess we kno w it, al t hou gh badl y. Fo r exampl e, it i s too eas y for us to s ee other peo pl e like natural subj ects , and we even found pl ease i n it ; but o ne can not s ee o neself t he s ame way. W e alt ernat e t hus between t wo visio ns , wit hout havi ng bar el y t he s maller si gn o f a co mpl et e frame for oursel ves: the onl y o ne t hat coul d gi ve a ri ght pers pecti ve o f t he ot her peo pl e. The wor d "natu re" i s onl y a s yno nymous of t hat cor rect per s pecti ve or whol e fr ame o f t he acti vi t y of a si ngl e entit y; and we will say t hat t hat f rame and that perspecti ve ar e indist i nguis habl e o f it s s pir it and o f t he s pi rit ­ what it res po nds to t he qu estio n of it s co nti nuit y, t hat even it is not neces sary to pr esent.

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The Ri emann zet a functio n allo ws for exampl e to calcu late t he exact nu mber of "pri me number s s maller than a given nu mber ", what ever the nu mber is ; which alr eady i s mir acu lous enou gh havi ng i n accou nt the pu rel y r andom charact er of t he pri mes i n a concr et e i nter val or proxi mit y. But , o nly to arri ve at t hi s, which made t he title and t he apparent i nt entio n of t he ori gi nal Ri emann's writi ng, i t is eas y to su ppo se t hat t he functio n must cont ain much mor e i nfor matio n t han t he o ne t han it can gi ve u s i n the l imit o f one of i ts transfor mat io ns . Of cours e, it i s not informat io n whi ch l ack i n t he zet a fu nction, but rat her o n the co ntrar y: o ne woul d wis h to redu ce as mu ch as po ss i ble. Whi ch takes to t he for mul as agai n to generat e the functio n, whos e mai n disadvant age i s to be mor e or less equall y mo nol it hic and wit hout fi ssur es t han t he own ser ies o f t he nu mber s. Thi s summari zes a li ttle t he cl ass of diffi culti es t hat pr esent s t he probl em of t he f amou s "R iemann hypot hesis ". The Ri emann hypot hesi s s ays t hat all t he no n-t ri vi al zero s o f t he zet a fu nctio n i n the co mpl ex plane has a r eal part exactly equ al to 1/2 . Thi s valu e is known li ke t he criti cal li ne. The proo f o f the Gau ss prime nu mber t heor em demanded that at l east exist o ne cri tical st ri p bet ween 0 and 1 as exclu ded valu es ; what it i s not so met hi ng trivi al eit her. The zeros of t he zet a fu nctio n and t he pr ime nu mber s satisf y a cert ai n dualit y through t he spectru m and t he harmo ni c frequencies ; includi ng t hese l as t summation for mul as, t he du alit y is not of irreduci ble char acter, and we al so could s peak about a ter nar y rel atio n. And, r eall y, it is just to say t hat these t hree as pect s ­pri me number s, su mmat io ns and zeros - for m an extr aor dinar y tri angl e, o pen and in mo vement, wit h i nfinit y of trans for mations bet ween them and withi n. That 's to s ay, has a certai n resemblance, in other do mi nion, to t he o pen triangle of t he gunas i n t he Samkhya. It i s as sumed, o n t he ot her hand, t hat t his critical li ne i s t he most synt het ic part of the endless infor matio n o f t he functio n, and the easiest for m to filt er it . But, on t he ot her hand, t her e are other many int eresting qu estio ns, between whi ch o ver hang t he valu es of the i maginar y part of t he zeros , t he hei ght of thi s alo ng t he criti cal l i ne: har dl y anyt hi ng i s known on its co rr el atio n. These values o f t he i magi nar y par t ar e rel ated as well to t he "perio di c t er ms" o f t he fu nctio n, as Riemann called them, i n fact os cill ating ter ms, t hat The l eft hand of C hao s
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pl ay a ver y import ant role although oft en t hey are r el egated to t he backgrou nd. M any ver y gener al r elatio ns alt hou gh no thi ng cl ear wi t h the har mo ni c anal ysis and t he s pect ral and wave theory are deri ved from t hi s featur e.

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The zet a fu nction has been generati ng mor e asto nishment as have been made calcul atio ns o n t he nontri vial zeros , all whi ch agr ee wit h t he hypot hesis till the mo ment . To this day, i t has been verified on mo r e t han a t rillio n of zero s (1012) wit hout no cou nter exampl e. It i s mo re, almost all t he si gns s eem to i ndi cate i n t he dir ectio n t hat t his will conti nu e bei ng cert ai n u ntil number s i nco ncei vabl y more elevated, in whi ch so me of t he as pect s of t he funct ion alt er their behaviour i n r elation to t he int erval of s mall er number s and, could, per haps , gener ate so me cou nter example or r efut atio n of t he hypot hes is. The trut h i s t hat, not to have t he s mall er i dea of why t he zeros must have r eal part 1/2, t his behaviou r i s alr eady ext raor di nar y enough, even i f so meday a cou nt er example i s f ind. At t he mo ment, the criti cal line i s t her e, wit hout anot her ratio pr etends t hat its o wn obsti nate exist ence. Li ke no n mat hemati ci an, al ways has call ed my att entio n that the mat hemati ci ans who work i n t his so ter ribl y di fficult su bject speak about to demons tr at e or to r efut e t he hypot hesi s, and t hat t hey speak about under st and the functio n mu ch less. I must su ppo se t hat t hey cherish t he hope that a proof or a refutation of t he hypot hesis will al so illu mi nate t he u nder st andi ng o f t he functio n al most auto maticall y, bu t t hi s ho pe s eems to me unjusti fied; not to mention t he auto mati c charact er of t he illu mi natio n. But, to be fai r wit h t he mat hemati cians , it is necess ar y to s ay t hat t hey are t he fi rst who ki ndly expl ai n to us t hat t hey do not under st and the behaviour of t he fu nctio n; even t hey say to us t hat it i s t he most fu ndamental, el ementar y l evel, the one that i s not u nder stoo d. What it is not sur pris e, becau s e t hat el ement ar y l evel i s not ot her o ne t hat t he rel atio n between addit i ve and mult i pli cati ve pro perti es, but taken al toget her ­ glo ball y- already f rom t he ver y beginni ng. As it is eas y to see, t hi s is i n int i mate rel atio n wi t h t he double as pect, dif ferential and pro babilist ic, of t he funct io n; bei ng t he probabili stic par t ( taken "s eparatel y") bas ed o n t he independent or additive rel atio ns , and bei ng t he diff er enti al part (t aken "s eparatel y", and i nclu ding the logarit hmi c deco mpos ition i n su m of pro du ct s) bas ed in t he mul ti pli cati ve and dependent rel atio ns. Bu t t he pro bl em here is t hat t here is no separ ation at all , becaus e exactl y what does the functio n is to co mbi ne all t hese el ements of t he mos t compact for m, so t hat l ittle mo re fit s that to study t he values t hat t hro w t he ar gu ments: what we call t he "behaviou r " o f the fu nctio n.

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Si nce t he probl em t hus pr es ent ed is so i mpenetr abl e and wit hou t fis sur es , i t has been tri ed to gi ve it the tu r n and to rai s e it t he ot her way around: to tr y to i magi ne to what cl ass of phys i cal o r dynami c behaviour could cor r espo nd t he behaviour of t he Riemann zet a fu nctio n. Appar entl y, Hilbert and PŃlya al ready r ais ed t he qu est io n at the light of t he functional anal ys is and t he o perator t heo r y t hen emer gent , ther e for 1914. Thi s onl y could begi n to have so me relat io n wit h t he r ealit y when t hat same t heor y of t he o per ator s fou nd i n t he qu antu m mechani cs a suit able fi el d of applicatio ns ; and even t hen t here were the least evi dence of r el atio n. These dis pers ed evi dences have been accu mul at ed t hrou gh t he ti me. The histor y and the sequ ence ar e more or les s well- known: Selber g gave some more expl icit r elations wit h the spectr al theory; Gut zwill er made an appro ach i n t er ms of perio di c or bit s fo r a s emi- cl assi cal sys tem wit h a Pl anck co nst ant t endi ng to zero , Mo nt go mer y s ettl ed do wn t he si mil arit y bet ween t he s paci ng and r epulsion of t he neighbour s zeros and t he di stri butio n of t he ener gy level s in t he rando m matri ces t hat are appli ed to probl ems of t he i nteractio n of many bo dies or parti cles wit hi n t he qu antu m mechani cs. Onl y it was mis sed, si nce Berr y di d, to pro pos e so me t ype of quantu m sys tem that imit at e t he behaviour of the functio n; or , si nce Connes di d, even to cr eat e a specific The l eft hand of C hao s
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operator with an equi val ent behaviour. All t his is extremel y i nteresti ng, of cour se, bu t at t he mo ment do es n't hel p mu ch to u nder stand t he fu nctio n its el f: for whi ch r easo n must have or not to have t he zeros i n t he cri tical line, and even why t hey mu st have t ho s e par ticul ar valu es i n the imagi nary part.

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In s pite of t he enor mou s ext ension of t he develo ped wor k, ever yt hi ng fo llows al most exactl y jus t as i n the begi nni ng. Nevert heles s, t he relatio n wit h r eal dynamic s ys tems , wit h physi cal s ystems, does not sto p bei ng anot her extr aor di nar y fact mo re of t hi s fu nctio n, and o ne of t he mo re u nexpect ed. Becau se, what rel atio n woul d have to have pur e an arit hmetical s eri es , wit h it s pri me nu mbers and al l t he o t hers , wit h r eal a physical s ys tem? And a qu antu m s ystem, i n parti cul ar. Wel l, aft er all, t he qu antu m s yst ems have indeed an irr educi ble dis cret e co mpo nent , bu t not hing els e. The r el atio n t hat t hi s coul d have wit h t he pri me nu mbers is co mpl etel y co nj ectural. Per haps, yes, t he pri me nu mber s can be r el at ed to r esonance phenomena, in a so mewhat similar way to ho w t he r atio nal numbers play an i mport ant rol e i n the chaoti c or bits of cl assi c s yst ems ; but nothi ng of this is cl ear. The qu antu m s yst ems, apart fro m t hat , ar e of wavy natu r e; i n r el atio n to the zet a fu nct io n, t he mai n diff erence t hey i ntro duce wit h r es pect to clas sic wave dynami cs i s that t hey exhi bit a much gr eater nu mber of pos si bl e stat es. Ther efor e, it is pos si bl e to co nsider t he zet a fu nctio n i n rel atio n to dynami cs i n gener al , more t han to t he quantum mechani cs i n particul ar. Or r ather, chao tic dynami cs i n gener al: t hat ampl e and di ffuse fiel d bet ween t he pur el y random behaviour and t he deter mi nist behaviour wi t h stabilit y. Thus, for exampl e, t hi nks Michael Ber r y, who has suggested t hat t he r esolutio n of t he Ri emann hypot hesis would provi de to t he analys i s o f chaoti c s yst ems so met hi ng si mil ar to the har mo ni c anal ys is fo r t he more or di nary oscill ato rs of physi cs . Thi s woul d not be f ew; but no body can su ppos e of what co nditio ns it woul d have to start an anal ysi s wit h su ch a power o f resolution.

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The co nnectio ns o f such a her meti c obj ect as t he zet a fu nctio n with dynami cs ar e encour agi ng and sti mul ating, but , si mult aneousl y, pot enti all y mi sleading. Speaki ng so mu ch o f o per ator s and other tools appli cable to t he phys ics, o ne fo rgets for a whil e t hat i s t ryi ng to under stand a pur el y arit hmetic "obj ect ". And I writ e object i n quo tation commas because it does not seem to me suitable at all to t he arithmeti c do mi nion, whi ch, as we noti ced befor e, i s t he mor e r efi ned do minio n of t he ti me t hat we can put u nder our s cruti ny or analysi s. Let us t hi nk about t hat even it i s i mpos si bl e to say if the inf i nit e seri es of the nu mber s i s somethi ng open o r clo sed; and obviou sly I do not talk about t he pot enti al i nfi nit y o f it s enu mer atio n, bu t to t he possi bl e rel atio ns. What it woul d t ake u s no w to semanti c qu est io ns o nl y; the mat hemat ici ans settl e i n their way defi ni ng t he s et li ke an abelian s emi -group wit h dou bl e structur e of tot al or der (additi ve) and special o rder (di visi bil it y). This i s a si mpl e charact eri zatio n t hat do es not li mit al most anyt hing eit her . The per fectl y tri vial fact t hat t he fu nctio n has it s si ngul ar pol e i n the unit always makes t hink about t he pr edo mi nance in a cert ai n sense of t he to tal or der: in t he i ndefi ni te repetitio n of t he same co ndit io ns , t hat entai ls t he ot her or der . There i s no need to say t hat wi t h thi s one does no t go anywhere, but , i n any case, to consi der t hat t he functio n i n all i ts immensit y exi st s t her e just like t hat , take u s to t hat position half pl atoni c and half empiri cal t hat conforms t he more exter nal and i nstru ment al as pect t he mat hemati cs, and of t he phys i cs and our vi ew of t he wo rl d i n gener al. In any cas e, it s eems evi dent t hat t he co nfi r mation o f t he hypot hesi s i mpos es the si mplest of t he behaviour s, and t hat i ts refut atio n wou l d r ender a more complex behaviour: but o f t hi s i t i s not follo wed automatically t hat it s certif icatio n makes more t ract abl e or compressi ble t he rel ations and co mplex aspects of the seri es . That is to say, al so i t can i mpl y t he o ppo site, t hat i t i s t hat in t he criti cal l i ne all t he cir cumvolutio ns The l eft hand of C hao s
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of t he plot the cr iti cal whi ch agai desir able, i

are stret ched to t he maxi mu m and no lo nger i s po line would be t he most i mpenetr abl e. Thi s s eems n takes to t hi nk to u s that a gr eat er u nderstanding gnoring t he hypothesi s as mu ch as were pos si bl e.

ssi bl e t heir l at er reductio n. In o t her wor ds, exactl y well co mpatibl e wit h t he natur e of t he fu nction, of t his o ne ­t he fu nctio n o nl y- woul d be more And t hi s was what Riemann di d.

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Ber nhard Riemann was a man wit h a s pecial capacit y to receive. That is to s ay, he had t hat we call recepti vit y i n a ver y hi gh degr ee. W hat is wo rt hy o f remar k, becaus e remembers to u s t hat fo r so me i mpo rtant thi ngs i t i s not enou gh wit h the mer e i nt elli gence, t he avi dit y or t he ambit io n; whol e generations o f Newto ns not even had s mel led t his su bj ect s tarti ng o ff of t hei r o wn i niti ati ve. As sure as we can be that he wor ked much in t he mani pul atio ns and tr ansfo rmatio ns of t he funct io n, we can be sur e al so that at no moment he co nsi der ed thi s su bject as t he hunt i ng of t he white whal e. W hat he had to look fo r at any mo ment was a u nio n as so li d and indi ssoluble as pos sibl e of t wo apparentl y di chotomi zi ng behaviours . And t hat sear ch and effort are exactl y its cr eatio n; becaus e what emer ged soo n t hrou gh it su rely sur pass ed hi s expectatio ns . But what Eul er and Ri emann u nit ed for ever wit h as mu ch care as fortu ne, now we want to disass emble i n pi eces li ke an art efact . Known is t hat Riemann also co ncent rat ed all it s for ces i n phys i cs and t he philoso phy, and t hat bot h co ncer ned so much or more to him o r mor e t han hi s own mat hemat ics. Shor tl y befo re it s co mmu nicatio n on nu mber t heor y, he had already elabo rated a mat hemati call y ver y refi ned t heo ry o n t he el ectro magnetism, whi ch finall y he pr eferr ed not to pu bli s h. As f ar as t he philosophy, o nl y we have had left di s per sed f ragment s, and, alt hou gh per haps li vi ng so me mor e years he had been abl e to di stil t hem, t he certai n thi ng is that we can have mor e t han dou bt s on t hat poss i bilit y: t he verbal and written express ion dis pl eas ed too much to hi m, for whi ch he had very l ittl e eas e. Nobo dy can be everyt hing si mult aneousl y, and in t he cas e of Ri emann t he physicist and t he potenti al philo so pher had to suff er the per sonal pre-emi nence of t he pur e mathemati cal thought . But, with his anal yt ical qu aliti es, so much predomi nat ed i n hi m the co ncent ratio n and s ynt hesis s pi rit, that i n t he end almost not hi ng was lost: Riemann is not o nl y the deepest of all t he mathemati cians, is also the one o f deeper i mplicatio ns for t he phys ics and t he philo sophy; because bot h t hi ngs o nly can go hand wit h hand.

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W e sai d befor e that the si mi larit y o f the behaviour of t he zet a fu nctio n with cer tai n dynami c s yst ems can be ver y mi sleading, and that i s pref erabl e to al ways have i n mi nd its arit hmetical natu re ­ pur ely tempor al- , by mor e sti mul ati ng t han t he associ atio ns can be. But we have here to i ntro duce a tu r n t hat coul d onl y be unexpect ed for thos e who i gno re ever yt hing sai d yet . We have seen t hat t he dynamic s ys t ems , al ready fro m t he s ame Galil eo, make a s electio n of t he ref erence fr ame wit h pr ef er ence fo r the i nertia; t his means t hat t he s pace its el f, or t he poi nt s i n r est do no t have any s ense and t hat i s t he mo vement what det er mines them. This means t hat t he pr evious posi tio ns si mpl y di sappear , absor bed in t he moment ary d ynami c state of t he s yst em; t hi s i s what we u nderstand by dynami c s pace, of which is i mpl ied t hat i t i nt roduces an irreduci bl e t empo rar y el ement, alt hou gh i n f act do es not make another t hi ng t hat to geometri ze the mo vement, whi ch i s co mpl et el y diff erent and even r ather the opposit e thi ng. Not hi ng of this happens i n a co mpl ete ar it hmeti cal s eries: here t he relat i ve posi tions of each o ne of t he el ements ar e not lost nor vanis h, as mu ch t hey can be nu mer abl e. E xactl y t his i s cons idered li ke t he "timeles s charact er" of t he nu mbers, t he pro pert y t hat makes o f t he mathemati cs t he i nstru ment of proo f mo re sel fsuffici ent and trust wort hy. For me it is r at her t he o pposi te: t hat not hi ng her e ­nu mbers as su ch- lose its po sitio n no r vanis h is what guarantees t hat we ar e i n ti me i ts elf as it is, u nli ke a dynami c syst em t hat defi nes The l eft hand of C hao s
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the inst ant aneous cu tti ng it out fro m all t he other as i f it had scis sor s. And i n fact, a quantum sys t em t ends to vanish a goo d part o f whi ch we consi der a classical dynami c s yst em; i n par ticu lar, it di lutes t he badl y call ed "caus alit y pri nci ple" of such s ys t ems, that on t he other hand s hi nes by it s absence and o nly t alks about t he global synchro ni zatio n pri nci pl e, t he "s ci ssors " of t he pres ent mo ment and t he mo mentar y s tat e of the syst em. Nevert hel es s, it co ntinu es being a proper dynami c s yst em, i n the sens e t hat t he co nser vation pri nciples are appli ed and pr esu med. This way, t he char acteri zatio n of the qu antu m mechani cs i n pur el y dynami c ter ms is , as i t could not be l ess, i rr edu ci bl y ambi guou s. And t he du pli cit y between t he lo cal co nnections ­wit h fi nit e speeds- and no n local co nnectio ns ­ simultaneous - it is pr acti call y i mpos si ble to tr eat: t hey can be cont empl at ed alt ernati vel y, but not si mult aneous l y i n a si ngle for m. W e can say then t hat a quantu m sys t em is dynami c, but mu ch l es s t han t he cl as si c o nes: i n all t he senses , and even alt hou gh here ther e i s apparently much mo re fluctu atio n and movement. It is by this that t he co nnectio n of a qu antu m s yst em with an arithmeti cal fu nction seems l es s sur pr ising: in as much i n whi ch t here i s less dynami c el ement t here i s mor e t empor al el ement . It i s natur al and lo gi cal, even bei ng ver y f ar st ill o f parti cul ar or expli cit relations . So met hing si milar can be s aid of chaoti c s ys tems, even bei ng t hese o f very dif ferent natur e. So me of t he al gorit hms us ed ri ght no w to explor e t he zet a fu nctio n ar e hi ghl y si mil ar to t hos e fo r t he pro bl ems o f chaoti c or bits for many bo dies i n cl as si c dynami cs , whi ch s ends us to t he pio neering approaches of Gut zwill er i n t hat dir ection. Ti me emer ges i n as mu ch we do wit hout t he cut and the sel ect io n of dynami cs. It is goo d for recogni zi ng i t, even when it do es not make t he t hi ngs easier to u s.

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This anomaly wit h r es pect to any known dynamic s yst em begi ns to be percei ved when phys i cis ts and mat hemati cians tr y to defi ne t he su ppos ed o per ator t hat woul d u nderlie to "t he zeta function dynami cs", t hat so me even call "Ri emann dynami cs " to be bri ef. It i s sai d, for example, t hat t he zeros have t he s ame statisti cal pro per ties t hat t he ener gy l evel s or eigenvalues of a "no n-i nt egr abl e Hamilto ni an not ti me-r ever s al i nvari ant , wit h not known vari ables", and quit e si mil ar t hi ngs. An y charact eri zatio n t hat we could fi nd i n t he Lit er atur e of t he subj ect will not be mor e preci se. Yet , "ext er nal" pro perti es li ke t ho se t hat Ber r y remar ks are wort hy of int erest: chaoti c, u nstabl e dynamics but wit h boundari es , inst ability of ho mo genous char act er , qu asi-o nedi mensional dynami c. But, when we want to ti e all t hes e ends, it is quit e cl ear t hat we do not kno w any dynami c s yst em li ke t hat.

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Voroni n gave a uni ver s alit y theorem wit h r espect to the zet a fu nctio n i n whi ch it demonst rated t hat, as si gni ng codes to t he numerical i nfor mat io n of the same o ne, t he fu nctio n cont ai ned any amou nt of infor matio n and for mal kno wl edge t hat t he human ki nd can r each, repeat ed i nfi nite t i mes , and wit h any ar bitrar y degr ee o f preci sio n. M at hemati cs cont ains ot her s eemed mo nsters , l ike t he fractals: but t hese ar e not inf i nit el y di ff er enti abl e nor tal k about so met hi ng as fundament al as t he s eri es of t he whol e nu mbers . Even not l i ki ng thi s type of generali zatio ns too empt y al ways , we can us e it to i magi ne anot her per specti ve: l et us i magi ne t his apparent uni ver s e, whatever i s it s di mension, wit h al l the st at es of its parti cl es , throughou t anyone of it s po ssi ble ti mes of evolutio n. Coul d be des cri bed all t hes e st at es wit hi n t he zeta functio n? Yes, naturall y, accor di ng to t he Voro nin t heor em. No w, we tr y to t hi nk a u ni verse wit h a pr eGalil ean dynamics , if it is t hat it is pos si bl e and it has so me s ens e. I am afr ai d much t hat it do es not have it, and t hat a "pr e- Galil ean dynami cs " wit hout i nertial frame of ref erence or relati vit y, aside fro m not exi sti ng, not even coul d co nsider particl es i n our ter ms; but t his now does not matt er, i s o nly a vague ment al experi ment. In t hat uni ver s e o f t he here called pr e- Galil ean dynami cs, any pr evious st ate of t he s yst em do es The l eft hand of C hao s
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not dis appear, and conti nu es adding its elf to the su cces si ve stat es wit hout we kno w if it can affect t hem i n so met hi ng or not. This way, we wou ld have an appar ent infl atio n of i nfor mation. Ho wever, we i magi ne t hat thi s pre-dynamic u niver se, still mu ch mor e i nflated of st at es that the uni ver se of f amiliar dynami cs, we had it to compres s i n a li ne or di mension, t hat is to say, i n a quasi-o ne- di mensio nal sys tem. Woul d it fit sti ll? Su rel y yes , and t he zeta fu nctio n woul d keep a goo d degr ee of s imil arit y wit h i t. And to where t he s pace wou ld have go ne i f all t he codifi cations of t he s pace can be order ed i n a line or an al mo st one-di mensional s et? Of course, I am not willi ng to res pond to t hes e so mewhat wai f qu estio ns, t hat , I am af r ai d, also r efl ect part l y our waif idea of the t ime, t he t hi ngs and t he cos mos. Nor I believe t hat it never coul d be much co mpati bilit y between dynami cal f rames and pre- dynami cal o r po st-dynami cal fr ames .

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As far as t he char acteri zat io n of an underl yi ng o per ator to t he zet a fu nction i n t er ms o f it s Hamiltoni an, t he Hilbert s paces, t he renor mali zatio n group and so o n, t hi s i s so gener al and misl eadi ng as to s peak i n dynamics t er ms. And neverthel es s , yes , it can be ver y f ruitful, and t hat i s at t he mo ment what matt er s. I beli eve t hat even i t i s absolut el y neces sar y to consi der t he analo gy wit h dynami cs wit h as much ri gor as po ssi bl e, becau se perhaps anot her way does not exi st o f whi ch we begi n to conceive t he limit s of dynami cs; limits t hat r ight no w are i nconceivable to u s. For me, t he Hamilto nian o nl y means her e t hat t he princi pl e of l east actio n, o r extr eme actio n if is pr eferr ed, is pr esent . But i n the fir st part alr eady we co nsider ed t he hypot heti cal possi bil it y to apply t his pri nci pl e to mi xed s ys t ems of a special class , o f whi ch t he pulse was a good exampl e, which tol er at es fri ctio n and cert ai n co mpr es si bilit y. That i s to s ay, we co nsi der ed t he pri nci pl e o f l east actio n in an ampl er s ense t han the one of dynami cs o r mechanics ; a s ense ver y si mi lar to t he o ne t hat Lebni z coul d have in mi nd for its s yst em of t he natur e. Kno wn it is t hat t he l ei bni zian syst em was deno mi nat ed "dynamis m" by the secondary role t hat had i nerti a and the inertial frame i n it , t hat is to say, by t he f ew t hat it had to do wit h whi ch now we call ed dynami cs . And kno wn it i s also t hat he tri ed to gener alize the pri nci pl e of l east action or of the opti mal pat hs to t he eco no m y of t he mo nad and t he o rgani zed parts of its as pects or subst ances ­ body, t hought, et c - . That we called before pre o r po st- Galil ean dynami cs woul d not be anot her t hi ng t hat t he Lebnizian dynamis m, that, o n t he ot her hand, do es not let s eem to us exces si vel y baro que. It i s t he criti cal line of t he zeta function an extr eme li ne or a line of l east act io n fo r t he u nion of t he dif f er enti als and stochasti c aspect s o f t he fu nctio n? Per haps it coul d be s poken t hus , i f we di st ance of the mer el y dynamic i nt er pr et ation of t he variational pri nci pl e. Thi s o ne is al ways ampl er than anyone of it s physical i nt er pr et ations. A disti ncti ve s eal o f t he reduced wo rk o f Ri emann i s it s unit. On the mat ter, Weyl s ai d t hat, precis el y in t he wor k of Ri emann, and mo re t han wit h any ot her mat hematician, i t was necess ary to co nsi der its u nit. Unit t hat apart fro m t hat i s too i mpl ici t and crypt ic, wi t h ample s pace for the specul atio ns. From his first great wor ks i n co mpl ex vari abl e, ther e o n 1851, Riemann was bas ed i n t he vari atio nal pri nci pl es of t he theory of t he pot enti al; i n t he call ed by hi m "Diri chl et pri nci ple", t hat Gau ss and Green al ready had u sed, and t hat so many headaches gave to him. The same Dirichlet , mento r and teacher of Ri emann, wor ked as much i n t he t heo r y of the pot enti al as in number t heor y. B et ween Lei bni z, Eul er and Ri emann t her e is a ki nd of co nti nuity li ke t he one t hat can be bet ween Newto n, M axwell and Ei nstei n; a po wer ful and persi st ent co nti nuit y, i n s pit e of t he gr eat diff er ences t hat t he evolu tion of t he main subject mar ks.

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The Ri emann zeta functio n i s t he mot her of a whol e fami ly of fu nctio ns, the auto mor phic L-functions , The l eft hand of C hao s
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deri ved fro m t he Dirichl et L-s eries. A di sti ncti ve featur e of all t hem is to have i ts o wn criti cal li ne, o n whi ch also ext ended conjectu res have been made of t he s ame natu r e t hat t he o ne o f the zet a fu nctio n. This o ne remai ns li ke t he fu ndament al by it s mo re dir ect and general connect ion wit h t he seri es of the whole number s. Not to del ay me more, I will s ay that t he idea which I have i n mi nd is t hat t he stu dy of t he ti me s eri es of t he pul s e and its i nsi de-out du al behaviou r can hel p u s to underst and some of the aspects of t he zet a fu nct io n and it s asso ci at e fu nctio ns ; aspects t hat at t he mo ment do not s eem to be co nt emplat ed neit her by the dif f erent br anches of the nu mber theo ry, nor by t he alter nat i ve dynami c i nter pretat io ns , and t hat coul d be ver y important. What it is very di ff er ent to co nst ruct a s ys tem ad hoc to tr y to replicat e t he behaviour of t he functio n. The fu nction i s enor mou s, and what we want is to isol ate some small , but deci si ve, dis ti ncti ve as pect s. We mu st remar k t hi s, becau se we ar e not s peaki ng of which the pul s e and t he fu nctio n ar e si mil ar thi ngs. The pu ls e and its du al corr elatio n can and must be stu di ed perfectl y i n a tot all y i ndependent way, wit h their o wn and si mpl er tool s of anal ysi s ; but t he approach to t he zet a functio n, wit h all t hat could i mpl y, reall y needs other references i n additio n to t ho se bett er known at t he pr es ent. I cannot ho pe t hat ever ybo dy t ake s eriou sl y t he possi bilities of t his connectio n, althou gh equ all y unexpect ed co nnectio ns i n other areas o f dynami cs have changed t he ment ality o n t he su bject remar kabl y. The rel atio ns t hat at firs t si ght we can est abli sh bet ween bot h fi el ds are, of cours e, so mewhat vagu e and acci dent al: the du al charact er of t he behaviour of puls e, st ill enti rel y unexplored; its mi xed char act er wi t h r egar d to t he di scr eet and co nti nuou s seri es , as well as to t he conser vati ves and di ssi pati ve as pect s, et c. Also we coul d s pecul ate about the tendency to the r epul sio n i n t he flu ctuations , and so me o t her analogous s i gnatures, but these si mpl y enumerat ed woul d have to be suffi cient, at l east fo r so me peopl e, t hat look for an o per ator with pro per ties and vari abl es extr emely ambi guous, if not co ntr adictory. And if we ar e s ent to the obviou s nes s that the pul s e i s a t emporal s eri es, we s ay t hat it is so in a mu ch more specif ic sense t hat what by "tempor al s eries " it is usu all y under stoo d, that i s to say, an empiri cal seri es wit h i ts co nti nuous or dis cr eet s pectru m. Trul y, I woul d s ay t hat t he pul s e has more arit hmeti cal natur e t han anyo ne of the dynami c s ys tems t hat we co ntempl at e, wher eas we co nt empl ate from t he per s pecti ve of t he dynamics ; and by t he s ame I will say, if i t i s allo wed me, t hat t his t ype of appro ach not o nl y i s not pr epo sterous, but maybe pu rer and mor e natur al t han tho se dynami cal appro aches , and even t han many of the stri ctl y mat hematical appro aches. In the zeta fu nctio n, the ext remely deli cate balance bet ween glo bal or der and flu ctu ations ar e not onl y i mport ant , but fundament al ; in f act some of the approaches wit h mo re i nt er esti ng pot enti al appli cat io ns are r ef er r ed t he t heo r y of t he stabilit y. Evi dent ly, t he ver y uncert ai n and not alr eady defined du ali t y of t he pul s e has not hi ng to do i n pri nci pl e wit h t he ver y certai n du alit y of t he functio n, but , allo wi ng a ki nd of ref er ence for the fluctuations , woul d have to s ho w fi nall y its connectio ns wit h the ot her probl em. I reco gni ze t hat an i mmense dos e of f ait h i n pur e mat hemati cs plus an equall y i mmense dos e of f ait h i n appli ed mathemati cs i s needed to beli eve i n t his po ssi bil it y. I do n't kno w if peo ple such exist i n t his worl d, but , i n any cas e, to t hem i s addr ess ed t his writi ng i n first i nstance. But t he co ncept s are, t hou gh i nco mplete, ver y natural , and wo rth bei ng explored. We start fro m t he u ndi vided, mo nadic char act er of t he Ri emann zet a. But t hi s no t i mplies that it is absolu tel y i ntr act able. Unli ke t he atom, a monad cannot be s plit ; but a monad breat hs, t hat' s t he mir acle. Li ke al most all t he dualities of t he anal ys is, t he o ne of t he pul se i s not irr edu ci bl e at all, but a comparison t er m wit h a vast grou nd wit hi n. And, i n as much we cannot "cr ack" t he fu nct io n, we will fi nd mor e and bett er si gnatur es of t hi s br eat hi ng. Natur all y, in mat hematics we o nl y can fi nd t his br eat hi ng i n t he fo r m o f pu re r el atio ns, but even so, t here i s so met hing mor e t han a metapho r here. A mo nad is pur e activi t y, exactl y li ke mat hemati cs ; pur e acti vit y, but not pur e act. Probably, t ho s e who want to cr ack t he zet a functio n beli eve much mor e i n t he plato nic heaven t han u s. Moreo ver , t hey coul d beli eve in an empiri cal plato nic heaven ­a ver y str ange place, i ndeed, t hough a gr ai n fami liar.

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If we s ay t hat t he pulse has mor e arithmeti cal natur e than anyo ne of t he u sual dynami cal syst ems, whi ch her e is allu de i s it s cu mul ati ve character , as dou btless as t he o ne o f t he s et of t he who le nu mber s, and wit h mor e t han one si milar diffi cult y. It coul d even be postul ated t hat each new cycl e of t he puls e i ntrodu ces newnes s necess aril y, but, natur all y, onl y i n t he conti nuous domi nio n coul d fi nd t hi s it s r efl ectio n. On the ot her hand, it is or certai n t hat we fou nd logarithmic s cal es o r spir als i n t he same organic r ate of develo pment , gro wt h and agi ng, and even the same experi ence of t he t i me wit h t he pace of the years: i n t hose hou rs and days that for the babe seems to be li mitl es s, and t hat so fl eeti ng became with ti me it s elf for t he el der s ­ and mor e fo r the lat e anci ent s, o ne coul d t hi nk. The s ame lo garit hmic law of Weber-F echner ­ bot h coll eagues of Riemannwhi ch rel at es very general l y t he excitation and t he stimulus, have a stro ng alt hou gh yet i ndefi nit e co nnectio n wit h t his. In t he pulse it is al so pres ent, i n addi tion, a "criti cal li ne" t hat allo ws us to disti ngu ish t he degr ee of inexor abilit y of thi s gener al tendency, t hat i s to say: the pro per degr ee o f reversi bilit y or irrever si bilit y of t he mo di ficatio ns i n cour s e. And agai n, prakr iti and vik riti are t he o nly crit erio n and fr ame of r ef er ence to s ettle thi s qu estio n. Wit hout t he r efi nement of this crit erio n, who se li mits is absolut el y co mpel ling to know, asi de fro m what r eveali ng can be in t hems el ves, any other for m to measur e t he degradatio n or entro py, the di sor der or t he agi ng of an organis m, will be so sophi sti cated technicall y as absur dl y pri miti ve i n concept .

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So per haps we do n't need to co nstru ct a co mpli cated device to i mit ate so me es sential characterist ics o f the famous zet a fu nctio n, and have t hese and ot hers eit her to our di s positio n; i n order to reali ze stri ctl y t his it is neces sar y to deepen wit h suff ici ent well-t aken car e and ri gor i n t he experi mental study of t he pulse. To under st and the natu re of its vari atio nal pri ncipl e, quite a lot ampl er t han what it i s u nder stoo d as such, but al so rat her mor e expli cit fro m t he causal poi nt of vi ew ­if we under st and caus alit y li ke t he or deri ng and sequ ence of t he fluctuat io ns . One of the many i mport ant qu estio ns t hat t her e woul d be to limit is t he l east time int erval for a fluctu atio n. Or i n ot her wo r ds , what mi nimu m i nter val we need to defi ne the sensiti vity of t he s ys tem wit hin the pot enti al of tridosha , or , vata, t hat comes al ready gi ven i n irr egul ari t y t er ms. In addition, t hi s int er val can admit diff er ent readings accor di ng to t akes t he puls e in o ne of t he wrists or consi deri ng t he correlatio n bet ween both. Not even we will tr y to enu mer ate t he most i mport ant point s that it must requir e t his stu dy, because onl y i n the experi mental ar ena coul d t hey hi ghli ght . We wil l onl y s ay t he followi ng. If so me dr eam about an analyzer of co mpl ex or chao tic dynami cs based on t he zeta function i n t he s t yle t hat Ber r y su ggest, t hey can be sur e t hat any t emporal s eri es o f t hi s cl ass cannot be r edu ced to ot her smaller element s t han t he o nes corres pondi ng to t he flu ctuatio ns of the t ridosha in t he puls e wit h i ts specifi c variatio nal dynami cs. This woul d be t he limit of "spectr al resolution" and of t he compr es si bili t y al so. But t hi s limit woul d be a bi g t heme it s elf .

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So me coul d as k to u s what coul d have t his to do wit h t he basic phys i cs . The ans wer coul d be: "Not hi ng, neit her we need it at all. A co mpl ex o r emer gent s ys t em does not have why to have explicit r el atio ns wit h t he fu ndament al phys i cs, except for t he el ementaril y i ndis pens abl e ones. " Thu s we woul d avoi d pro bl ems. Nevert heless , it i s so met hing annoyi ng al ways to l eave t he t hi ngs i n t he air, parti cularl y if we can go furt her and demandi ng it t he co nclusive char act er of the su bject . Apart from t hat , also t he relatio n bet ween qu antu m s ystems and prime nu mber s r emai ns li ke an absolut e mys ter y. Of cou rse t hat our t heme do es not s eem to have too many co nnectio ns with t he mor e spr ead visio n o f the mai nstr eam physi cs ; mainl y, when i t is consi der ed t hat t his o ne har dl y has any vi sio n, but a goo d bu nch o f equ atio ns . Leaving t hi s asi de, we will make an effor t to locat e t he su bject closer to t he ar ea of pro bl ems o f t he The l eft hand of C hao s
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physics i n it s histo rical develo pment and pr esent st at e.

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Si nce Newton we have t he pro blem of t he vacuu m i n phys ics. Namel y, t hat i n an inerti al frame wi th forces, like hi s, the emptiness or t he vacuu m cannot exi st, neit her an et her-li ke mediu m. Kno wn it is t hat own Newto n r es earched wit h regular int ermittence t he dynamic poss i biliti es of t he et her , alr eady fro m the fi rst years , r etur ni ng i n additio n on t he subj ect i n the last year s, wit h succes si ve editio ns of t he Optics ; and t hat never coul d s ettle t he quest ion. The et her or u ni ver s al medium cou nts wit h a lo ng hi stor y of illu strious attempt s to descri be it: asi de of tho se of Newto n, we have t he o wn Eul er , who gave us a hydrost ati c and al mo st Archi medean versio n of t he same o ne fo r t he gr avit y, and Ri emann, and Poi ncar Č. Ei nst ein was neit her co nvi nced at all about havi ng eli mi nated i ts neces sit y, except for his own for mul as , and Leibni z s poke of a pl enum li ke a mediu m, but still more i ntr actabl e t han t he o ne of the ot her mentio ned pro pos als. As it is s een, the pro blem of t he medium has enjoyed an enor mous attr active and presti ge, parti cul arl y bet ween t he gr eat est mathemati cians. At t his si ght, that to day we t end to fo r get, it i s more sur pri si ng t he suffici ency whereu pon we judged t he su bj ect, mai nl y i f we noti ce we are not bet ter t han befor e. It is not onl y t hat no w we have scal ar fi el ds for t he mas s es of particles, the infl atio nar y mo del and even for the flu ctuations of t he phanto m cos molo gical co nst ant, not to mentio n t he pol ari zatio n of t he vacuum and t he ener gy of t he fu ndament al level , t he so called zero poi nt . No ; all t his is or pheno menolo gy o r co mpl ement atio n o f t he pres ent disco veri es. To whi ch we t al ked about is to t he point ai med i n t he begi nning, that mo der n dynami cs exclu des by defi nitio n as mu ch t he mediu m as t he real vacuu m, so that o nl y can choos e bet ween a f als e et her and a vacuum equall y fals e. Of cours e, it always s ho ws pr eference for t his l ast o ne, becaus e it can measure it at l east and co nsi der it i n t erms of fi el ds and for ces . And to cro wn it all , the uni vers al mediu m i s made absolut el y u nnecess ar y when we have est abli s hed an absolut e ti me fro m t he begi nni ng, t hat i nvisi ble and ar bitr ar y "u ni vers al s ynchro ni zer ". But t hi s o ne it is still more u ndet ectable t han any et her ­it is absolut el y u ndetect able. Thu s, we coul d r everse t he question wit h gr eat advantage fo r the measur ement and t he experi ment atio n: i n as mu ch i n whi ch a uni vers al mediu m o r et her exist , the absolut e ti me or uni vers al s ynchronizer i s u nnecess ar y. Then, the immedi ate for m to charact erize t he et her wou l d be t ime it self.

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So met hing si milar was what Ni kolay Kozyr ev t hou ght ­ Kozyr ev, maybe t he mo st o ri gi nal experi ment er si nce t he times of Gali leo . Ko zyr ev and hi s t eam at Pul ko vo al ready made i n t he Fi fti es al l t ype of experi ments wit h pendulums and tor sio n bal ances i n o rder to measur e distortio ns of t he fo r ce of t he gr avit y. They r egul arl y measur ed t hem r epeatedl y and with enou gh homo geneit y, alt hou gh t his clas s of experi ments ar e hi ghl y dependent of so me circu ms tances . Natur ally, t he mai nst r eam phys ici st s ar e not willi ng to consi der neit her the experi ment al result s, nor the philoso phy and i deas o n which t hey wer e bas ed. Ko zyr ev co ncei ved the t ime li ke a flow in four dimensio ns wit h an additional co nst ant equi valent to a "caus al speed". Apart f rom to exis t s pace bet ween caus es and ef f ect s li ke i n dynamics, also ti me i nter vals had to exi st not ar bi tr arily s mall . Ko zyr ev made in addition consi derabl e efforts to harmo nize its t heor y wit h t he stru ctur e and data of t he r el ati vit y and quantum mechani cs, wit h whi ch it is qui te co mpati bl e. Gr eat part of all thi s l ead to the pro blem of fi tti ng t he no n-lo cal facto rs and t he si mult aneity wit h t he finit e s peeds that rel ati vi t y ­and t he vacuu m- i mpo ses ; but all t hi s fro m the per s pective of whi ch t he cl as si c mechanics has an "i nfi nit e causal s peed", and t he quantum mechani cs , a causal speed equal to zero . That i s to s ay, a brilliant reversio n of the pro bl em, co mplement ed wit h t he o nl y co nditio nal co ns er vation of t he angul ar mo ment. The ti me i n Ko zyr ev i s not t he t i me absolut ely passi ve and s pecial of dynami cs ; it is an acti ve ti me The l eft hand of C hao s
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wit h an own densit y t hat i mpo ses a cou rse and a di rectio nalit y, asi de fro m s pecif yi ng the cau s alit y. The pro bl em is t hat Ko zyr ev s eems agai n to redu ce t he ti me and its flow to a new co nstant, or a mor e or l ess li near vari abl e. The s ame title of his fi rst i ntro duction of t he su bj ect says i t openl y: "C aus al or No ns ymmetr ical Mechanics in Linear Appr o xi matio n". In additio n, after gi vi ng axio mati c rank to the necessit y of time i ntervals bet ween caus es , s eems to i gno re any ti me i nter val between t he actio n and r eactio n, whi ch es tabl is hes di vi sio ns in his o wn crit erio n. Li ke a pio neer, Kozyr ev o nl y coul d be fragmentar y; but even so is t he fir st i ntro ducer o f the pro blem o f t he real time wi t hi n phys ics , wit h alt ernatives to t he vanis hed gho st of t he dynami cs . Ko zyr ev do es not get to co nsi der of a t heor etically convi nci ng way to what ext ent t he densit y of t he time aff ects or go verns to t he co nst ant s of t he fu ndament al for ces; alt hou gh fro m it s own experi ment s its vari abilit y comes to be dedu ced. Thi s, t hat even to day s eems absolutel y i nco mpatibl e wit h t he mo der n phys ics in all t he extensio n of it s predict ive apparatus, woul d not have to be so s candalou s to us : t he so call ed "co nst ant s" of t he cou pling for ces i n t he standar d model ar e dependent of t he energy, and by the s ame, vari abl es. We do not know why reason t hi s wou l d not have to appl y for t he gravi t y.

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Zhvir blis, o ne of t he co nti nuator s o f the legacy o f Kozyr ev, s ho ws t he example of t he of Lazarev' s kolt sar ; a vessel closed i n a ri ng for m with a porou s separation, whi ch exhi bits a mo re or les s identi cal behaviour to a superco ndu cting ri ng, wit h a co nt inuous cir culatio n o f t he co ntent . As Zhvir bli s s ays , t he Lazar ev's kolt s ar is not an exper imental art efact at all, but rather an evi dent proof of t he dif f erence t hat t her e is bet ween a r eal t her mo dynami c s yst em and t he i deal sys t ems wit hout st ructu re co nt empl ated by t he clas sic ther mo dynami cs maint ained near t he t hermal equili briu m. A real t her mo dynami c s ys tem as a kolt s ar it is a meta-st abl e sys t em far f rom equili brium; t hat i s to say, i t can never sto p. The met a-stabilit y i s deri ved as well f rom the bi- st abl e character of t he si mpl est stru ctur es, and des cendi ng to t he qu antu m worl d, we s ee that the si ngle elect ron of t he molecul ar hydrogen H2 cannot be cou pl ed to o nly o ne of t he proto ns , but t hat mo ve conti nuousl y to belo ng to bo t h. The s ame t ype o f int erchange o ccur s i n t he Coo per pairs i n a super co ndu cting mat erial , so si milar i n t his to a kolt sar, and mo vi ng away of t he t hermo dynami c mo del ; nor has to s ay t hat super co nducti vit y is a pheno meno n still very poorl y well-kno wn, in s pit e of t he i ntens ive effo rt developed in t his fiel d. Si nce Lazar ev's kolts ar is i nadequ ate fo r a quantit ati ve control of t he cir cul ation, Zhvir blis propo ses a ves sel wit h os moti c barr ier, in whi ch t he t hermo dynami c demo n of kolts ar with its as so ciate par ado xes can be stu di ed bett er . In Lazarev' s kolt s ar no t onl y may be, but mu st be ci rcu latio n, becaus e of the ver y exi st ence of bot h dif f er ent st atio nar y st at es. The ener gy r eleased i n t he mo vement i s not of tri vial ori gi n; it is not follo wed fro m the laws o f Raoult and Van´t Hoof f, "rather app ear s lik e a gho st". Zhvir bl is says t hat it i s as i f so met hi ng had lost o r had l et wit hout s ay i n "so me pr e- axio mati c level ". "the circulation appears not to obtain the balance, but by the t wo diff er ent stationa ry stat es that appea r by virtu e of th e int eraction of ph ysical fo rces". The ori gi n of t ho se for ces i s not co nsi der ed at all and it is as su med as u nderstoo d li ke a "perfectl y ob vious "phys i cal realit y. Zhvir bli s s ays t his so appar entl y tri vi al t hat we have applied i n t he very diff er ent co nt ext of t he dualit y of t he pul s e: " Wh en two condition s cannot be sati sfi ed simultan eou sly, the circulation o ccu r s". Zhvir blis ends up concludi ng t hat t he probl em o f t he irr ever sibility i s an i llusio n: "the th er mod yna mi c rever sibili ty is inf erred f ro m the r ever sibilit y of the mechani cs a s it is appli ed to a koltsa r o r to th e meta- stabl e stru ctu res . Cor r espondingl y, th e ir reversibil ity of the th ermodynami cs can b e infer red fro m the mechani cs only i f the last one i s po stulat ed like ir r eversibl e, that i s what Ko zyr ev did." The questio n is that the kolt sar will remai n an o pen s yst em, i ndependentl y o f ho w much we want to isol ate it. And any other o bject t hat we consi der, if we know to appr eci ate it s fluctu ations . Zhvir blis s ho ws dif f er ent el ectri cal devi ces to u s fro m whi ch can be dedu ced a si mil ar behaviour. The au t hor conclu des t hat any real o bj ect i s met a- stabl e and o nly can mai nt ai n a defi ned structure i n as mu ch i nter changes ener gy wit h the envi ro nment "Thi s inter change not necessa rily manif est t ransfor mation s, but in the bi -stable s yst ems , with The l eft hand of C hao s
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cir culat ion, cau se p rocess es a ccompanied b y u s eful wo rk. " That is to s ay, if r ever si bl e behaviours exi st, it will be t hanks to a det er mi ned st ructur e of int erchange wit h t he mediu m, and not by t he gr ace of t he mathemati cal for malis m. However , what is what flu ctu ates i n al l thi s i nterchange? Is it ener gy, or rat her t he ener gy and t he movement are their manif est ations? Thes e qu estio ns al ways have been consi der ed i nsolu bl e; but exactl y when arrivi ng here, and if r eall y we had some new ri ght to s peak about t he causali t y, we coul d to r efr ame t he question entir ely. A kolts ar i s not but a ki nd o f mo nad, and we have many of t hem to study pro perl y.

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"Wi th each thought , so mething invi sibl e, sup er- substantial ent er s ou r spi rit. " Words t hat one would wi s h t hat entered mor e t han one s pirit li ke somet hi ng more t han i deas. The phr as e is not fro m Ril ke, but from Ri emann, t hat l eft t hem poi nt ed in o ne of its outlines fo r a ps ycho- phys ic t heor y. By t he way that he was not maki ng po etr y, but t hat tri ed a mat hematical and t echni cal defi nitio n of how t hat happens. At the mo ment, f ew woul d be able to take s eriou sl y t hes e t heories; but li ke ever yt hi ng what Riemann left to us, t he phrase has inf i nit y of r eso nances, and t hey ar e not exhaust ed i n the physical i nterpr etations. Of cour s e, Riemann is thi nki ng about a ki nd of br eat h. The pr eoccupation of Ri emann by t he gener al st atut e of t he physi cs was not precisel y a pasti me of its free s ho rt whil es li ke mat hemati ci an. As so me kno w, he wor ked mu ch i n a t heor y of t he ether i n whi ch an inf luence o r f low of t he mediu m to t he i nt erior of t he particl e too k pl ace, jus t vanis hi ng t hen; in t his also he followed t he t rack of Leibni z and Eul er, tr yi ng to r efine ar gu ments , alt hough much mor e explicitl y s peaks about Newton li ke mo del . But Riemann kno ws per fectl y wher e it is : "th e inertia is th e h ypoth esi s... ", and cert ai nl y t here i s not i n phys ics anot her hypothesis t hat t he i nerti a fro m Copernicus and Gal ileo , bei ng Newto n the one that i sol at es it o f any o t her possi bl e i nflu ence. Mo del s of flo w insi de-out of t he parti cl e have been conti nu ed r epro duci ng u ntil t he comi ng of t he qu antu m el ectro dynami cs. The s ame F eynman wor ked i n o ne of them with Wheeler , two years befor e gi ving the ver sio n that no w has become st andar d; and still today t here are phys i cis ts who co nti nue def endi ng it. The qu est io n i s not t hat i t cannot work, t he question i s t hat wit hi n t he i nerti al fr ame t his is i gnor ed by unnecessar y.

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There is no har m i n rememberi ng t he pl ace t han Ri emann o ccu pi es i n mo der n mathemati cs and physics . As far as t he mat hemati cs , it i s not dif ficu lt to s ee t hat he i s t he mai n i nflection poi nt wit h res pect to the ol d one, bas ed i n co mputatio ns and algor it hmi c; t hat is to s ay, he is t he mai n i nt ro du cer of t he present co nceptual st yl e, gener ali zi ng who le mass es of concepts so to s peak at bird' s eye. He is i n t he s ame begi nni ng of t he u nfol di ng o f t he topology, as well as of t he s et t heor y, that by many year s t he s ame C antor also call ed "t heor y of t he vari eties ". B eing his styl e not incli ned to put algebra i n fir st pl ane, all his wor k on vari eti es and surf aces are pr egnant to the maximu m of al gebr aic i mpli cations. The si gnatur e he l eft i n t he complex vari abl e and t he real vari abl e, est abli shi ng t he bri dge for all t he modern t heo ries o f i ntegratio n, is si mply i nescapabl e. But all thes e achi evements are concent r ated and enor mousl y sur pass ed by t hat s mall co mmu nication on t he pri me number s gi ven to the Berlin Academy of Science. This it i s t he o nl y writi ng t hat do es not cont ain a si ngl e geo metri c i dea of an aut hor t hat li ke geo met rici an remai ned to t he po st erit y; after t hat writ i ng, Ri emann, just at his t hir t y thr ee year s, no longer produ ced no r pu blis hed anyt hi ng. The co ntri butio n of Ri emann to the phys ics i s critical and delicatel y str at egi c. Known it is t he Ri emann t enso r, co nsu bstantial wit h t he gener al theory of r el ati vit y and r el ati vi stic co smolo gy. It is quit e worthy of i nt er est t he fact t hat t he Riemann tensor can be deco mpos ed i n ot her t wo: t he Ricci and W eyl The l eft hand of C hao s
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tensors , whi ch r es pecti vel y measur e the change of i nitial volume and the ti dal dis tortion. These t wo tensors are theor eti call y s epar at ed, but i n f act t hey can be i nter mingled and to be aff ect ed mutu ally, s ho wing wit hin t he line o f evolutio n or dir ectio n o f the co s molo gi cal s pace-time so me su ggesti ve analo gi es to us wit h o t her pot enti al du ali ties of t he s ame zet a functio n, that her e we o nl y can ai m. Al so we have t he sphere of Ri emann, to det er mi ne t he po ssi bl e states of a particle of spi n ˝, as i t i s t he case of electro ns, proto ns and neutrons ­all the "o r dinar y matt er ". That i s to say, we spo ke o f o bj ect s wit h a mi ni mu m of two di stinguis habl e st ates; i n additio n t his woul d take to cons i der the transitio n f ro m t he angular moment of parti cl es , defi ned i n t he co mpl ex pl ane, to t he angul ar moment of t he cl assi c level of descri ptio n. Known it i s that so me of t he pro babili sti c i nter pr et ations of t he zeta function beco me in t he s i mpl est t erms of t he "fai r coi n" with a pro babili t y of ˝ ; t he el ectro ns and t he ot her particl es woul d be o ther many f air coins here... A goo d reason exist s no t to believe i n t he possi bilit y of a pro babili sti c demo nstr atio n: t hat t he r ando m as pect of t he zet a fu nction is o nl y half of t he su bject , and t hat a hal f arbit raril y taken can never deci de o n a s et ari s en f ro m mo re pr eci se and ampl er co nditions si mult aneou sl y.

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Ri emann coul d not i ntro du ce t he et her ­ mediu m and su bst ance mo re tempor al t han s pat ial- i n physics becau s e t hi s o ne alr eady was born wit hout l eavi ng s pace for it ; but, i n tur n, we can say t hat he dis co vered an et her o f mat hemati cal o r nu meri cal natu re. In ef f ect , it i s po ssi ble to speak i n a cert ai n s ens e of a "nu meri cal et her ", i n a way s i mil ar to how t he engi neers s peak t he ether of f requ enci es . Nevert heles s , t hi s nu merical et her is somethi ng quite dif ferent of t he so called "C antor du st ", and even al most t he o pposite: it is so i nfi nitel y bou nded t hat it must fi nd at any pl ace a vent or a f ree li ne of action. To t he criti cal li ne woul d co rres pond thi s role.

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The cu rse of phys i cs has depended i n an evi dent way o n the mat hematical t echni ques availabl e. A s pect acul ar exampl e o f t his is t he ro le pl ayed by the har mo ni c anal ys is i n t he qu antu m mechani cs ; to su ch an ext ent, t hat one never kno ws at all if t he li mi ts t hat are attri but ed to it as i nherent ar e reall y of experi mental or techni cal natur e, bei ng bot h so u nit ed as pect s. The s ame Planck's co nst ant, t he quantum of actio n, i s not si mpl y an ener gy per s eco nd, but an energy per cycl e per second ­t he cycle i s gener ally mis si ng. And, natur all y, it can be s ai d the same for the Heisenber g' s u ncert ai nt y li mit , t hat it is not mo r e t han a t echni cal limit of the spectr al anal ysi s . This way t he fr equency, t he most pur el y tempo r al phenomeno n, i s empti ed of any co ntent and become a mer e integr ant part of the fo r mul as. W hat i s here what tu r ns? Why it turns? Thes e qu est io ns are co nsi der ed t he cus p of the absu rd o r t he no ns ense, but t he phys i cist s do not gi ve mu ch bett er ans wer s: "what os cill at es it is an o scillator " (may be a particle), "and what vi br at es, it i s a vi brator". Of t he cycle, li ke of t he cir cl e, we have l ear ned ho w round it i s, and t hat is ever yt hi ng; but t hi nk fo r a moment t hat we ar e speaki ng of a do mi nio n ­t he qu antu m mechani cs- i n whi ch t he t emporal seri es fi nd t heir apot heo sis. And in t his domi nio n, t he ti me i s what s hines by its abs ence. W e can o bt ai n r esolut io ns of phas e great er t han t he o nes of t he s pectr al anal ysis by means of diff erent t ypes of phas e- s ensi bl e fr equency met er s, so me of t hem ar e so common t hat t hey ar e in t he colour t elevisions. It does not seem to have ano ther t echni cal li mit t hat the noi se, and woul d be ver y int eresti ng to co nsi der u nder what assu mptio ns and conditio ns coul d be useful t hi s cl ass o f i nfor matio n. If a cycl e is so met hi ng more t han a mer e u nit, we woul d have to be abl e to ask if it is not po ssi ble or necess ary a tr i- mo dal defi nitio n of t he qu antu m o f action, whi ch al ready s eems ju stifi ed si nce it s i niti al defi nitio n: wo r k per cycl e per ti me. W hil e I am writing this I s ee a r ai nbow by t he windo w, and I t hi nk t hat t he mere pheno meno n of t he colour is absolutel y impossi bl e wit hout breat hi ng it self, t hat colour it self it i s al ready a br eat h t hat penetr ates The l eft hand of C hao s
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wit hout li mit u ntil t he deepest. And I t hi nk, al so, t hat i n spite of t he magnificent s pectral t heor y, we do not kno w much about t he colou r. The Pl anck's const ant is an i nt egr al, and if we coul d u nder stand t he fi ne stru ctur e co nst ant i n appro pri at e ter ms of ti me, we woul d u nderst and to what ext ent t he s pect ral anal ysi s put limits to u s. Thi s is al so inti mat ely rel at ed to t he Ri emann zeta, whos e zero s have been want ed to rel at e to t he zero point of t he fu ndament al ener gy l evel. But t he zero poi nt, besi des t he vacuu m, woul d be al so absolute dar knes s. An y cycl e mu st i nvol ve so met hi ng, of cou rs e. A cycle i s no t but ano t her wor d fo r which i n ot her co ntexts we denomi nat ed circul atio n: t hat what occurs when two co nditions cannot be s atisfi ed si mult aneousl y, as Zhvirbli s remember ed to us. And the int erval that medi ates between an absor ptio n and a emi ssio n, s eems t hat it woul d have to be decisi ve to di s cer n bet ween lo cal and no n-lo cal actio n, t hat, i n t he same way t hat a wave and a particl e, ar e tempo rall y excluded and do not manif est si mu ltaneousl y. It i s sai d t hat so meo ne asked Ei nst ei n near t he end o f hi s lif e, when t he quantum elect ro dynamics alr eady had been establi shed, what was what he would l i ke mor e to u nderst and. The ans wer was t hat he wou l d like to underst and a si ngle electron. The mo der n theori es of fiel ds have an extrao rdi nar y predict i ve r each, to such an ext ent t hat t he u nderst andi ng s eems to have made u nneces s ary. Of cours e, it is cert ai n t hat we do no t under st and a si ngl e el ectro n, and i n t hi s res pect to s peak about t he vacuu m woul d not impro ve t he t hi ngs pr eci sel y. What is what flu ctu at es? "What fluctu at es , it i s a fluctuator ". And what o perat es, an o per ator ; natur all y.

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Gr adu ally, phys ics has been appro achi ng nu mber t heor y and Ar it hmeti c. So me r esearchers still apart fro m t he mai nst ream, li ke Pitkanen or Volo vich, alr eady put it i n the centre of t heir consi der atio ns. It is even s po ken alr eady about t he nu mber t heo r y li ke t he ulti mat e physi cs ; and here, "ult i mat e" wou l d have to be under stoo d si mult aneousl y it i n the dou ble sens e o f l as t and i nevitabl e i n the develo pment of t he f acts and fir st by its ir redu ci bl e char acter . As we s ay, t his s eems to be quit e f ar of t he mai nstream and it s daily agenda; but, as it is obs erved wit h Connes's develo pment s and no n- commut ative geo metr y, i t is quit e logi c to ju mp fro m the al gebr ai c as pect s of the topolo gy and diff erent ial geo metry to t he al gebr ai c t heo r y o f nu mber fiel ds, to wher e it seems t hat the i nco mpatibilities woul d find its l ast i nst ance.

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In t he s ame way which t he zet a fu nctio n cannot be u nderstoo d li ke a si mpl e partit io n fu nctio n, we mu st u nder stand t hat t he cor rel atio n that we can measur e i n t he puls e i t i s not bor n from a mere di vision of t he whol e s ys tem. It i s no t t ri vi al pr eci sel y becaus e it i s not a di visio n at all. We have alr eady s ai d t hat t he du aliti es we s peak about ar e not irr educi ble, bu t onl y t er ms of a bal ance t hat presuppos es a mean to be defined. Wit hout t his mean, not even t here woul d be po s sibility of tr ansl atio ns neither of transfor mation of ter ms ­ no br eat h at all. That is to s ay, co nsider ed from a t er nar y poi nt of view t hey i nvolve necess aril y TIM E, a real time, u s mu ch r eal for u s as the time of comput atio n, no matt er how har d the pu rely abs tr act time coul d r emai n unaff ected. What allows us to make a curiou s reflect io n. As it i s kno wn, t he pri ncipl e of t he glo bal s ynchro ni zer i n the rel ativi t y and t he t heori es of fi el ds depends o n an i magi nar y time agai nst whi ch t he "r eal time" st ands out ; t hat i s to s ay, t he order pri nci pl e depends o n a non or dered set of nu mbers . Wit h t hi s peculi ar cir cumstance f ew can be do ne si nce the poi nt o f vi ew o f dynami cs. But a si mil ar cir cums tance yes admit s an insi de-out r eversio n i n t he s ame Arit hmeti c, in t he zeta functio n for exampl e, so t hat we can consi der the t ime of comput atio n that co nsu me t he dif ferent t ypes of tr ansfor matio ns o f t he fu nctio n, t he preci sion i n t he imagi nary part of t he zero s and any ot her as pect. That is to s ay, we can make an algor it hmi c approach as sumi ng diff er ent co mput atio n al gorit hms and t ranslati ng t hem in t erms of s ynchr o ny and real and virtual The l eft hand of C hao s
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time; appro ach t hat has extr aor dinarily i nteresti ng possi bilit ies and t hat her e onl y we can leave ai med. E ven the mo st i ntract able as pect s becaus e of t he s ame magnitu de o f t he nu mbers , mo ment s and i nt erval s, must fi nd si gnifi cant cor r elates i n ter m of co mpu tatio n ti me and nu mber of operat ions i nvol ved. This way we can gener ate i nf init y o f fu nctio ns wi th totall y new char acteristi cs around the mai n fu nctio n and it s crit ical li ne. The same crit erion for t he ti me of co mputation and t he al gorithms al ready is a gr eat subj ect i ts elf , but , in any cas e, whi ch i s wi thi n thi s fu nctio n, gi ven its natu re, al so has to be out side, and the mo re dir ect is t he mode of transl ation, mor e will be t hus. We beli eve t hat t his ki nd of approach is t he most strai ght forward, even invol vi ng many co ncentri c and eccentri c s pher es . On t he ot her hand, it woul d be gr eat iro ny t hat t he o ld al gorit hmic appro aches had seen i n t he l engt h of t he co mput atio ns t he main o bst acl e, instead of t he r eal pat h, and t hat t hose who ar e t he whol e day maki ng co mputations had not evalu at e the valu e of the nu mber of his own o per ations . May be, fi nall y, we begin to t ake seriou sl y t he i dea t hat all t hi s has to do wit h t he ti me and t he acti vit y i n t he most dir ect way pos si bl e. Onl y i n as much we renounce to the idea that we ar e in f ront of an obj ect, the r eal t hi ng wil l beco me fo r u s a t hread. The zet a fu nction always i mpels to us to entir el y r eframe t he analysi s i n t he measur e of our capacit y, t hu s reveali ng its el f mo re us efu l than any pos si bl e ut ilit y. If we s poke before wit h regar d to t he hydro dynami cs about t he i nter nal ener gy of the syst em like a thi r d r el evant el ement , i s neces sar y to r eali ze t hat, i n ter ms of t he t ridosha, vata penetr at es i nto t he most int ernal , wit hout bei ng for t hat r easo n s epar ated of t he mediu m o r enviro nment . F ro m her e ari ses a so rt of "auto mor phis m" t hat we have deno mi nated mo nadi c. The qu estion o f t he co mpr essi bilit y or i nco mpr essi bilit y of t he vari atio ns t hrou ghout the ti me arises, al so ­what it is t he l ess abstr act o f t he questions, si nce is inti mat el y associat e wit h t he r eversi bilit y o r reestablishment of t he or ganic balance or it s i rr evers i ble det erior atio n. In t hese conditio ns, any r eal s ys tem, t hat i s to s ay, any co mpl ex o ne, must have its o wn li ne of s peci fic t empor al cancellations , well dif ferent f rom t he cancell ations that appear , fo r exampl e, i n the fiel d theori es. Ot her wis e, it woul d o nly be a fugu e or es cape li ne for t hr ee ar bitrar y pri nci ples , whi ch i s no t t he case. And t hus , t he criti cal li ne o f an ent it y, its u nion of p rakri ti and vikriti, to speak i n t er ms of t he Samkhya and t he Ayur veda, f ar fro m being an escape li ne, is t he su btlest and int i mate axis of its breat hi ng, it s unfol di ng and i ts desti ny. The monad i s indi visi ble, but not u nco ndi tional . Al l what exi sts i s breat hi ng. A mo nad is anal ys is plus breat hi ng, and t hen, we natu rall y ent er t he Samkhya domi nio n. The zeta fu nction coul d be t he most unconditio nal o f t he ent iti es or mo nads, but even so , it i s po ssi bl e t hat it mu st r emai n necess aril y open, and per haps o nl y thus can be r el at ed to t he co nti ngency, if not thr eaded wit h it . Thi s mo de, per haps all real s ys tem ar e related to zet a o f t he most i ntimate form; but, i n any cas e, more t han to worr y about t his , t here is to tr y to concei ve the real sys t ems i n its purer tempor al for m. The zeta fu nctio n s eems to br eat h i n t he unli mi ted; if so meo ne coul d close t his u nli mited br eat hi ng i n a closed fi eld, may be all t he t hi ngs would not br eat h anymo re. Do n't we need hazard i n or der to breat h, by an y chance?

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It i s sai d t hat Davi d Hil bert as serted anyti me t hat t he Ri emann hypot hesis was no t t he most i mportant qu est io n of the mat hemati cs , bu t ju st t he most i mpo rtant qu estio n. Assert t hat we do n't need to agree wit h, but that s eems perf ectl y co mprehensi bl e and justifi able wit hi n t he mi cro co s m o f the sci entifi c mentality, and not onl y t he mathemati cal o ne. Becaus e the world woul d be equ all y mys terious wit hout t he f amous function, bu t the coi nci dence is t hat t he same fu nctio n al so s eems to be i nher ent to t he world, and fro m her e all it s wo nder and i nt erest . Independentl y of t he hypot hes es, t he achi evements of Eul er and Ri emann ar e absolut e achievements o f it s ki nd. The experience seems t hat it i s s ho wing to us, and will still s ho w more, t hat it i s much mor e viabl e to su bsu me behaviour s wit hi n t he zet a fu nctio n t han to der i ve t hi s o ne fro m ot her ar eas and obj ect s. And t hu s it has to be, i f we admit ted t he fu ndament al char acter of t he Arit hmeti c. In fact, it s eems easier t hat thi s one fit s The l eft hand of C hao s
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the nut s to all t he al gebr a, t he geometr y and the anal ys is , t han o n the co nt rar y; nevert hel ess it is natu ral and necess ary t hat t hes e t hree adjust and r eadju st around the bi g su bj ect as t he o nly poss i ble for m of beco mi ng awar e. It happens her e exactly just li ke it happens to the gunas i n r el atio n to Puru sha wit hi n t he Samkhya, and her e t he co mpariso n i s not a met aphor . The pro bl em is what cl ass o f t hi ngs we try to su bsu me, and how. W e have alr eady s een t hat here t he gr eat er obst acl e is preci sel y the co mmo n meani ng o f dynami cs wit h its s ys t emati c evasio n of the t empor alit y and it s equ all y s ys t ematic subst itution. W e have tr ied to s how t hat t he zet a functio n is not si mpl y a chall enge for t he anal ys i s and t he analys t s, but that i s t he chall enge of t he other par t of t he anal ysis s ecularl y i gnor ed by co nsider atio ns and i nt er est s pr esu mabl y mo re pr agmati c or at least mo r e immedi at e. The si ngle i ntuition alr eady s ays to us t hat t his is the cas e, but all cl ass o f ar gument s co mes to co nfir m it. Mat hemati ci ans ar e full y wit hi n their ri ght to explo re t heir su bject s as t hey pleas e and can; bu t it s eems that what with t he bles si ng o f t he natur e of t he t hings has u nited the zet a fu nct io n, it wil l not separ at e t he man wit h al l hi s ef fort s. The ato ms and t he parti cles ar e i nfi nit el y more docile to our mani pul atio n. The zet a fu nction and its i mpli catio ns appear to us li ke o ne of t he mo st l egiti mat e and fu ndament al su bject s o f t he philoso phy, t hat no w, i n additio n, it i s bei ng loaded of all type of practical co nnot atio ns and urgenci es . Then it s su bj ect i s t he maxi mum po ss ible ext ensio n of t he glo bal-local r el atio ns, t he chance and t he necess it y - maxi mu m extension and maxi mu m exis tent ial i mpli catio n. B e cause it has been s poken about t he chance and the necessit y too mu ch and fli ppantl y enou gh, and from t he s ci entifi c ment ali ty pr eci sel y, as si gni ng t hem co mfo rt able co mpart ment s for what we want to emphasi ze and what we want to i gno r e. So t hat even it is su rpri si ng that t hi s great motif of t he mat hematics has not pro voked a genui ne philo so phi cal i nt erest befor e, to whi ch wit hout a dou bt it has contr ibuted t he hermeti c charact er of t he subject . Writi ng li ke gener alist o r mer e philo so pher, I must s ay t hat I have not lost anyt hi ng i n t he i nner worl ds and i nfi nit e co mpl exiti es o f the fu nctio n; what o n t he co nt rar y i t calls my att entio n and surpr ises to me i s t he same f act t hat t his so gener al su bj ect has gotten to co nsi der wit hi n t he mathemati cs of a so el egant, so net and so clo sed form, and wit h su ch an archet ypi cal pu reness , as well as t he f act o f it s pract icall y su bt erranean statute f ro m t he ti mes of Euler to no w, and bei ng extremel y significant its emer gency i n t he present . And of course, it woul d be int er esti ng to know why reason a so general subj ect i s so indi gesti bl e for ou r actu al met ho ds and so di s sonant wit h our i nter est s ; and why a so gener al su bj ect i s not abl e to fi nd a mo re natur al pl ace i n our u nder st andi ng o f ho w t he t hings ar e. E ven if our u nderst andi ng coul d be far away fro m it, t he zet a functio n is t he s har pest t hi ng t hat exist, and it is near er to u s than ou r o wn ju gul ar vei n. Sai d t hi s, we l eave t he Arithmet ic alr eady and we r etur n to the Samkh ya, t he u niversal t heor y of t he flu ctu ations, and to the pul se li ke experi ment al example.

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An y prognosi s t hat can make a doctor when is taki ng t he pul se depends o n t he s har pness and pr ecisio n of t he diagnosis ; natur ally, t hi s is equall y valid for any t ype of medi cine. Nevert hel ess, when we consider ed the pu rely anal yti cal character o f t he t i me seri es of t he puls e, t hi s r ais es to us ot her qu estions on t he natur e o f the predict ion i n general . In phys ics, it is wort h to say i n its mor e linear as pect s and behaviour s, t he preci sion i n t he pr ediction exempts to us o f any det er mini ng caus al u nder standi ng; havi ng a mo del wit h it s par amet ers, vari ables and co nst ants i n a manageable for m, t he qu estio n about t he cau s es can be r el egat ed to t he li mbo of t he met aphysi cs. We have alr eady seen in s everal o ccasions that the mo st po wer ful r easo ns exi st for t hi s, reasons that are in t he s ame co nstitu ent plane of their activit y. W hen in phys ics it is want ed to better u nderst and so met hi ng and mor e pr eci sely ­ say, for exampl e, t he hypot het ical quantu m natur e of the gr avit y -, all the effort i s made to ext end the do minio n of t he pr edi ctions, and not hi ng el s e. Then i n phys i cs, or at l east fo r that to which it is co nfor med, t he u nderst anding is s yno nymous o f an absolut el y cont roll abl e pr edictio n. And I The l eft hand of C hao s
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wo n't be who tr y to change to t he r each and co mpet ence of t he est ablis hed phys i cs. This al l changes r adicall y when we approach co mpl ex, nonli near and no n co nt rollabl e s ystems. Here, it is co nti nu ed supposi ng t hat t he more well - kno wn l aws wit h t heir mor e li near expressio n remai n t he same, but , apart of t hem, it beco mes neces sar y in gener al to also su ppose anot her t ype o f co nditions that assur e t he stabilit y and ot her t ypi cal characteri sti cs of t he behaviours. In su ch conditio ns, we eit her r es i gn our s el ves to a stu dy ad hoc of the mult iple particul ar details of the sys t em, or look for anot her clas s of mo re general si gnatures ; and in f act, bot h thi ngs occur to get her u suall y. Neverthel es s, it is co nt inued looki ng for s pecifi c levels of causality for getti ng t hat i n the fundament al level t he caus es have never been s pecifi ed. Wit hi n the f ramework of t he Samkhya or t he Ayur veda t hi s pro blematic i s not even consi der ed. An y pro ces s is r edu ced to t he for m of it s tempor al s er ies ; but thi s one not even i s r educed to a space of coo rdi nat es . There is no need to say that for us a ti me s eri es i s so met hing ver y dif fer ent. Thi s i s vali d also fo r t he ass embl y of organs and anato mi cal char acteri sti cs which her e we s ee i ndeed li ke agent s o f the "causalit y": all o f t hem ar e pro ducts of thei r res pective s eries and s equ ences, no mat ter how har d t hey are co nnected. W hen we s ee a face, we ar e seei ng t he pro duct and t he last for m of a ti me s eries. We thi nk about it ver y r ar el y, alt hou gh i t i s evident. For t he Samkhya, the same flu ctu atio ns of gunas or dosha s ar e equi val ent to s mall phas e tr ansitions towards t he i nterior and t he ext erior to t he s ystem, wit h a mediator t hat penetr at es and go es beyo nd t hos e apparent ends: i n ter ms o f t he o r ganis m, t hi s goes fro m t he s ki n to the bo nes, wit h a sensiti vit y t hat sur rou nds to the as sembl y and s hade of f, but t hat co ntinues being cl ear i n the for m ­ and not another way occur s in t he puls e, wit h t he fo r m o f it s tempo ral profil e. The s ame soli d, li qui d and gaseou s phases mai nt ai n a corr espo ndence wit h t he guna s t hat go es beyo nd t he analo gy. W e say too oft en t hat t he man al ways has wished to ext end his pr edi ctio ns, but not hi ng f als er than that. In the past he was mu ch mor e wor ri ed t han us about t he interpr etation o f t he r eal it y, that not of its pr edi ction. The pro gno sis of t he vaidya cannot be r emot er of t he pr ediction of t he physi ci st: t he vaid ya t ries to under st and the reasons of t he u nco ntrol labl e, and t he physi ci st of what he can i solate and co ntrol . The advantage of t he fi rst visio n is evi dent i n everyt hing co ncer ning t he behaviour of co mplex s ys tems. W hat we have not explor ed nearl y at all are t he relatio ns bet ween bot h met ho ds; becau se i f it is certai n that they ar e int erpenetr ated, it is not less cert ai n eit her t hat t hey are exclu ded.

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The sto ck mar ket i s the maximu m expo nent of a sys t em of si gnals u nder the mos t i nt ensi ve scruti ny. Known it i s t hat t he fi nanci al anal ys ts and t he agent s co nti nuou sl y s peak about "to t ake t he puls e to t he mar ket ". M et aphor or not, t her e is no bett er expression to descri be so met hi ng t han no analys t kno ws i n what co nsi st s it exactly, beyo nd t aki ng t he most direct co ntact po s si bl e wit h t he flo w and t he fluctu atio n o f the stocks or valu es. Natu rall y, the mo st di rect contact wit h t his flo w is to partici pat e in i t, to unit e o nes elf to t he torr ent of buyi ng and s ell i ng, o bser ving the r eactio ns and t he "to ne" of t he mar ket. The effi ci ent mar ket hypot hesis says t hat i n a certai n mo ment ­ at any mo ment t he pri ces of t he stocks alr eady refl ect all t he r elevant i nfor mation o n t he market ; or in mor e att enu at ed ver sions , at l east t he mai n of the informatio n. Si nce t he r elevant i nfor mation i n t his r espect can be uni magi nably vari ed, it i s dif fi cult to fi nd in t he dail y lif e a si mi lar po stul ate of s ynthesis, update or r eduction; even remember us, i n t he furt her sphere, the famous "wave functio n r edu ctio n" o f t he quantu m mechanics . This postul at e of actu alization cannot be more r elat ed to the ho pe o f t he vaid ya t hat whi ch i n t he puls e all the di sti ncti ve el ement s of t he healt h and t he di seas e ar e yet pr es ent ­ wit h t he s ensi bl e di ffer ence that no vaidya will pret end t hat his anal ysi s, di agnosi s o r reduction can be co nfi ned to a mo ment or purel y mo mentar y state. A sui tabl e t empor al s ampl e, an auscult atio n wit h a mi ni mu m i nt er val of ti me will be needed. The effi ci ent mar ket hyphot esis is consi der abl y mor e hypot het ical and r adi cal and t han t he vi ndicatio n of vaidya, and nevertheless we give much mor e credit to the fo r mer t han to t his l ast o ne. In our wor l d it woul d not be sur pris e t hat we were more s ensi ble and consi der ed wit h t he mo ney t hat wit h our o wn bo dy. The hypot hesis its el f can be co mpar ed wit h t he i dea of t he "r andom wal k", alt hou gh o n t he ot her hand t he The l eft hand of C hao s
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pr esu med assi mil ation of t he i nfor matio n tur ns t hi s randomness t he mo st relati ve and ambi guou s t hi ng we could i magi ne. On t he other hand, it is qui te evi dent that this hypot hesi s is not but a r epri nt o f t he suf ficientl y ambi guou s "i nvisi bl e hand" of Adam Smit h, which i nvit es u s to tru st i n t he eff ectiveness of t he f ree co mpetitio n li ke a new i ncar nation of t he provi dence. Preci s el y, t he t heor y of t he effi ci ent mar ket i s a for m to do wit hout all t he s pu rious s pecul atio ns of t he so call ed t echni cal anal ysi s, that is not anot her t hing that the int erpr etatio n o f the graphs by t hemsel ves. In opposition to it we have t he so call ed fu ndament al anal ysi s o n t he r eal st ructu re of t he co mpani es that sus tai n the sto cks . This las t one do es not tr y to predi ct , but to u nderstand what t her e i s i n game. The cli nical anal ysi s that t he vaidya r eali ze, his diagnosi s, woul d agr ee pr eci s el y wi t h thi s ki nd of fu ndament al analys is. But, o n t he other hand, whi ch makes bu yer s and s ell er s at ever y mo ment , whet her t hey cons i der a t ype of analys is or another one, it cannot stop bei ng s pecul ati ve i n a hi gh degree, and being the mar ket whi ch i s ­and aff ecti ng more and mor e the cur ves of t he pr ice to t he int ernal dynami cs of t he co mpanies ; so we have anot her species of i nes capable duality, t hat t he t heor eti ci ans , u nt il no w, have onl y been able to co nt empl ate i n ambi guit y ter ms. The i nsi de-out t ype duali ty bet ween t he t echnical and the fu ndament al as pect s is i n f act irr edu cibl e in its own ter ms, and nor to s ay t hat it i s o f hi ghl y simil ar natu r e to which exis ts bet ween p rakrit i and vikriti. This is not t he place for doi ng a detailed analys i s o f t he paradoxes of the of t he effi ci ent mar ket hypot hesis , thi ng t hat o n t he ot her hand alr eady has been made o ver and over again and wit h so littl e fru it. In the co nt ext of our writi ng, it is enou gh to s ay t hat t he dynami cs of t he market i s not punctual except i n t he gr aphs: the acqui sition and t he s al e of s tocks, li ke any tr ans actio n, ar e si mult aneous by defi nitio n; but not so the or ders of acqui sitio n and sal e, with it s added tendencies. The par ado x of t he effi cient mar ket , rai s ed ju st at the begi nni ng of t he t wenti eth century ­t he centur y of t he parado xes , and no t onl y t heor etical - i s not but an exempl ar amo ng ot hers many o f i denti cal o r ver y s i mil ar stru ctu re, gener at ed all by a pur el y binar y posi ng i n ter ms o f yes or not, wit h exclu ded middl e, i n a s ystem conceived alt ernativel y fro m an i nt er nal o r ext er nal per specti ve. This coul d be appli ed as much to t he parado xes of t he s et t heor y, r el ati vity, quantum physi cs, and ther mo dynami cs, and ot her many. And t hou gh appar entl y t heor etical, all has co ntri buted in not littl e measur e to i nst all to us i n t he ambi guit y; unt il t he poi nt i n which t he cont emporar y worl d has beco me a s yst emati c runni ng of t hi s ambi guit y, so appar entl y fou ght wit h the exi genci es of t he scient ific exactitu de. And we t hink that t here ar e i n all t his su bstr at e o f ambiguit y and par adoxes not si mpl y theoreti cal pro blems to sol ve, also a gr eat i nerti a to mo bili ze. It does exist for the mar ket , no t onl y an "i nvisi ble hand" li ke the one of Smi t h, but a left hand next to a right o ne as we have o bser ved i n t he pul se? Not anatomi call y, cert ai nly. But t her e is a du alit y of compo nent s that co ntri but e to so met hi ng ver y si mi lar, and perhaps al most i denti cal. Of cours e, t he co nditions of st ability i n a mar ket do not have anyt hi ng to do nor are co mpar able wit h t hose o f an or ganis m; but t her e i s an ass imil atio n and an el i mination; a cycl e of cr eation and dest ruction, and not o nl y at a level, but to many l evel s. W e do not have nor t he s mall er i dea if it i s po s sibl e to deco mpo se natur all y a rough cur ve as t he one of t he stock mar ket in t wo as happens i n t he case of the pul s e, or i f i t i s simply i mpos si bl e; but gi ven the si milarit y of co mpo nent s impli cated, it is pos si bl e to l ear n ver y much of t he approach of bot h mo del s, r es pecti ng all it s diff er ences . We al ways spo ke o f t he most s ati sf actor y character izatio n of t he flu ctu ations. The cu r ve of a sto ck or co mpany never is isolat ed, but co ntai ns t he hi ghest degr ee of sensiti vity to t he outer circu mst ances and t he behaviour of all the ot her curves ; and t his as mu ch t hrou ghout t he ti me as al most si mult aneousl y. Of cour se, t here is all cl ass of combi nat ions of graphs. But i n any cas e, i f we had so me fr ame of r ef er ence fo r the opti mal st at e o f the valu es as it can be deduced i n t he pulse, t hi s woul d i mpl y ver y l ittl e pr edi ction; t hat is to say, t he pr edi cti ve compo nent woul d be mo re i mpli cit t han expli cit , and ever yt hi ng what it woul d t end to o bt ai n woul d be a mo ment ary adju st ment arou nd t he o ptimal values ­r educi ng t he s pecul ati ve co mpo nent near to t he mi ni mu ms. It is t hi s possi bl e? Do n't we conti nu e mo ving i nsi de t he s ame parado x, t hat s peaks us o f ho w t he o wn kno wledge af fect s to t he s ensiti vit y o f the sys tem and i ts evo lutio n? This is a parado x o nly for t he pr edi cti ve-specul ative met ho d that char act eri zes to us. In t he f rame of the gunas and t he Samkhya, t he sensit i vity and t hrou gh it t he knowledge for m an int egr ant part of t he cur ve and t he tempor al s eries . But this exclu des as well gr eat part fro m whi ch we consi der ed predictio n, and even dilutes t he i nvariant character o f t he tempor ary axi s by means of the du alit y and its f an of po ssi ble co mbi natio ns. Ther e i s no possi bl e magic i n this res pect ; t he onl y t hi ng t hat woul d change i s t he cent re of the The l eft hand of C hao s
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att entio n. But t his is mu ch and mo r e t han what it is bel ieved. Of bei ng vi abl e, t hi s metho d of anal ys is of t he fluctu atio ns woul d be applicabl e to t he econo mi c behaviour i n gener al and woul d impos e new cri teri a: becau s e by for ce it would r adio gr aph and clarif y so many el ements t hat to day are opaque and t hat even t here i s not too much i nter est to bri ng to t he li ght. The question i s ho w much t he dar k we can allo w ou rs el ves . On the ot her hand, it i s not i mprobabl e t hat t hi s new approach t hrows li ght o n all that nou ris hed body of mo der n for mal par adoxes.

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The exampl e of t he stock mar ket is parti cul arl y useful becau se i n it t he mor e pu rely quantit ati ve as pect s, and t he co ntents and t he s emiology cannot be more unit ed bot h; but not i n ot her way happen wit h t he organi c fu nctio ns . We already can put t he hear t i n t he billfol d, t hat t he billfol d will go to t he heart . It coul d be t hou ght t hat the scheme of t he gunas of t he Samkhya is si milar to the one of t he di alecti cs, the well known scheme thesis- antit hes is-s ynthes is ; but di al ecti cs is i deali sti c, even i n it s po pu lar and natur ali sti c ver sio ns ver y pr evious to t he i dealis m, pr eci s el y by it s appear ance of pro duction o f t he reali t y. Al so t he mo der n anal ysi s, even bei ng appli ed f aithfull y to experi mental dat a, is i dealisti c si nce t he mo ment t hat believes to pro duce t he mo vements f ro m closed fu nctio ns wit h changes i n t he valu es or co effi ci ent s. No thi ng o f thi s has to do with t he Samkhya. Alt hou gh we even fou nd characteri sti c mat hemati cal stru ctur es for t he fluctu atio ns with r es pect to t he bal ance, we woul d onl y have a sel ector of r eadi ngs, a tu ner that allows us to li st en bett er what t her e is pr es ent . A co mplete and closed descri pt io n as t he one of many analyti cal functions would be of ten irrel evant, woul d make t hat list ening impos si bl e. In f act, if t her e i s so met hi ng fu ndamental t hat co rr es po nds wit h t he fundamental nature o f t he gunas, the sensit i vity, t he actio n and r eact io n, is t he same arti culatio n of t he language, wit h its t hree gr ammar peopl e and t hei r equi valent su bject- ver b- predi cate flow. Not i n vai n it i s adjudge to t he name o f Pat anj ali a tr eati se o n sanskrit gr ammar , i n additio n to another one on medi ci ne. The puls e of t he cat tru ly woul d s ay "I- eat -sar dines " if it had not much mor e vari ed and subtl e t hi ngs to cou nt . B y t he way t hat t hat l angu age of t he pul se has so mu ch an articul ated and di s creet as pect , in addition to co nti nuous t he mo st apparent o ne. Thi s s ends to u s to the ol d Indi a disti nctio n bet ween vaikharÎ and madh yamÄ, t he articul at ed di s cur si ve l anguage and t he inarti culat e and deep l anguage of nature and thi ngs. Onl y i n a fr ame as t hi s one coul d have some value and s pecific wei ght t he no n deci dabl e qu estio ns of the last centur y on s emanti cs and s ynt ax of t he badl y call ed anal yti cal philosophy. To s peak about t he langu age and t he langu ages of t he natur e coul d not be a met aphor ­r ather t he hu man l anguage has to be a poo r met aphor of t hat ot her langu age, t he Adami c langu age o f cr eatio n.

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W hen t he ver b, the br eath and t he ti me dis appear o f ou r per ception o f t he natur e, the mi ddl e, also the entity of t he t wo ot her i nst ances is diluted. Somethi ng more t han a di mensio n of t he "pro blem" vani s hes ther efo re, and somethi ng mor e t han it s "dept h"; it di sappear s al so t he man and hi s pri vil ege of bei ng t he mi ddl e and the means , o r even just a means i n t he mi ddl e. Those who put at t he top t he humanis m and science jointl y would have to medit at e o n so a dou bt ful pedest al . In additio n, the exu ber ant growt h o f patt er ns t hat we beli eve to s ee i n t he natur e, its abstr act int ell ectual reco gnitio n, has result ed i n t he pro portio nal i mpoveri shment of our s ensitivit y to per cei ve them, The l eft hand of C hao s
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until t he poi nt t hat to s peak about any di rect per ception mu st sou nd to u s li ke a f abl e unavoi dabl y, o r to a clu ms y tr ansl ation o f so me underl yi ng mechanis m or hi ghl y refi ned patt ern. Refi nement and dulli ng can go perfectl y hand wit h hand, and meanwhil e, we ar e co ndemned to loo k for in s eas of t he abst ractio n what we thi nk t hat t her e we have lo st.

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So me i nvesti gator s , li ke t he biologi st Zvet ko v , find si gnifi cant or at least i ntrigui ng t hat t he r atio bet ween t he di astoli c i nt er val and t he s ystoli c o ne of the car dio gr am tends to co me very near to t he valu e of the Gol den Sectio n, that, li ke was made lo ng ago , we pr efer to call Conti nuou s Proportio n. Thi s same ratio tends to occur also wit h goo d appro ach bet ween the aver age di astoli c and s ys toli c pr essu re of the aort a. The cert ai n thi ng i s t hat we find the Co ntinuou s Pro portion i n all cl as s of pl aces and aspects of t he natur e, wit hout on t he other hand t hes e manif est atio ns have anot her charact er t hat t he purel y pheno menologi cal , when it i s not si mpl y ar bitrar y as soci atio ns; and often, bar ely t here is a cr iterio n to separat e bot h t hi ngs. W e do not have to t he dat e t he s mall er i dea of whi ch can be t he i mport ance o f t he Co nti nuou s Pro portio n i n t he natu r e, nor could eit her def ine the pl ane t hat corr es po nds to it . Nevert heless cer tai n pro per ties exi st that ar e wo rthy of mention. In t he fir st pl ace, that has not been t he s maller expli cit r el atio n o f thi s co nstant with any s phere of the dynami cs, and t hat t her efor e cannot be deri ved f rom it. Thi s i s preci sely what rel egat es it to a value completely s eco ndar y or ver y s mall ; and nevert heless , i t i s t hi s s ame t hi ng what turns it so met hi ng as gratu itous as myst erious. B e caus e, aft er all, t his co nst ant appears wit h perf ect cl ear ness in t he phyllot axis o f t he veget al groupi ngs , evi dent expr essio n o f develo pment and movement . W hat today we call ed Fibo nacci nu mbers , and t hat t hi s one too k wit hout a dou bt fro m Ar ab mat hemati ci ans , it was des cri bed for the fir st ti me by Go pal a and Hemachandra, arou nd 1150, wit h regar d to pro bl ems of exact and o ptimal packi ng fo r obj ect s of l engt hs 1 and 2; t he l ast o ne of t hese even appl ies it to the metr ic of t he sanskrit po etr y. In t hi s sens e, it would not s eem str anger that t he su nflowers and so many ot her pl ant s fi nd t he solutio n for so met hi ng t hat t her e i s not a pro blem fo r t hem. Nevert heles s, t hi s do es not leave t he s maller track i n t he dynami c descri ptio n of t he pro ces s, and fo r a good r easo n: t he cell s r epro duce and accu mul at e o f di s cret e for m. Preci sel y thi s accumulatio n o f di scr ete or der t endi ng to a rat io in t erms of r eal numbers or conti nuous fractio ns, i s o ne o f t he most remar kabl e char act eri sti cs of t his co nst ant . Oleg Bo dnar , fo r exampl e, has gi ven a mo del o f veget al phyl lotaxis i n four di mensions wit h a hyper boli c tur n i n t he related functio ns. Thi s s ame pro per t y of union bet ween t he conti nuous and di screte domi nio n has call ed t he attention of mat hemati ci ans and i nfor matio n t heor eti cians li ke Alexey St akhov, t hat have demo nstrat ed t he wealt h o f al gorit hmic and co mbi natori al pos si biliti es o f t he co nst ant with r efi ned met hods . In some of its numerical ext ensions , ph i per mit s a conversio n or opti mal tr ansl ation between analogi cal and digit al expr essio ns, fo r t he s ame rat io alr eady poi nt ed. Their s ame al gebr ai c and quadrat ic pro perti es, element aril y si mple, allo w i nfinit y of co nnect io ns as well . E duar d Soroko has sho wn it s r elevance in probl ems of stru ctur al and fu nctio nal stabilit y wit h many co mpo nents , s ettli ng down valu able bri dges wit h the i nformatio n theo ry and its measurement ; his own i nfor matio n t heor y has a purel y mo nadi c st arti ng point . The same St akhov has developed an al gorit hmi c and as ymmetri c t heor y of t he measurement...W e coul d gi ve so me examples mor e of import ant co nt ri butions on t he subject , whi ch nevert hel ess , remains al mo st as di s co nnect ed and mar ginal al most as al ways. Thi s no dou bt o beys to its absence o f rel atio n with dynamics. W hat al so makes to thi nk t hat per haps wou l d be so me more luck wit hi n t he number t heor y, to which the su bj ect seems quite open. The s ame St akho v and ot her s have develo ped t he hyper boli c fu nctions of Fibonacci and Lucas, who ext end t he Fi bo nacci nu mber s to t he conti nuous domi nio n, i n a way so mewhat si mil ar , alt hou gh i n a much mor e r educed do mi nio n, to t hat of t he anal yti cal nu mber t heory. This would have to be very i mport ant, pr eci sely to isolat e t he les s dynami cal aspect s o f t he more gener al arit hmeti cal funct io ns , like t he Riemann zeta: and we have alr eady seen to what ext ent t hese l es s dynami c as pects ar e i mpo rtant f ro m our perspecti ve. The s ame veget al phyl lot axis exhi bits t he primar y ratio bet ween phi and pi, first of t he gr eat co nst ants o f t he cal culu s, i n the so call ed angle gol den, equi valent to / 2. P erhaps, o n t he ot her hand, it i s The l eft hand of C hao s
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po ssi bl e to explor e to what ext ent phi is ant ago nisti c of t he other great co nst ant of t he analys i s, t he nu mber e, bas e of the natural logarit hms and expo nent of t he proportio nali t y o n t he fu nctio ns. The r atio of t his ant ago nis m or mi x-up can be r edu ced to the fact that phi ori ginat es t he most element ary and parti cul ar as soci atio n of additi ve and multi pli cati ve properti es , wher eas e gives r is e to t he most general , al so t he more pro per of dynamics and pro babilit y. Sai d anot her way, phi per mits di scr ete and gener ati ve r elatio ns , wher eas e tends by its natur e to pulverize t hem. But t his woul d al so t ake to su bject s of number fiel ds t hat ar e out of my reach. In addit ion to t he mentio ned, phi coul d have i mportance l i ke const ant by other fu ndament al charact eristics: it i s the si mpler mat hemati cal for m o f recur r ence and sel f-r ef erence, it express t he relatio n bet ween a part and t he whol e and i n additio n do es it sho wi ng t he mo re s li ppery t ernar y r elatio n, defi ned by t he equ alit y o f rat ios AB/ B C = BC/ AC . And fi nall y, but no t les s import ant , it su ppo s es t he more element al co nnect io n bet ween s ymmet ry and mor e el ementar y as ymmetr y. On the ot her hand, t he Fibo nacci nu mbers appear i n Kl ei n grou ps and f ract al s. The s ame Kl ei n co nsi dered t he r egul ar i cos ahedron like t he most i nvol vi ng and i nter connected moti f of t he whol e mat hemati cs. This is r el at ed as well to t he equall y m ys terious and more r ecent emer gency of t he ADE cl as sifi catio ns, t hat appear i n t he mo st di ver gent and unsus pect ed fi el ds , wit h ver y f ew well kno wn o n it s po ssi bl e r el atio ns: Lie grou ps, criti cal point s of non- mo dul ar fu nctio ns , algebrai c si ngul arities , cr yst allogr aphi c refl ect ion grou ps, and many ot hers. The same pl ato nic soli ds or r egular pol yhedrons ar e t he ar chet ype and mot her of t hese cl assi fi catio ns , al ways r ef er red to si mpl e finit e o bj ect s. These clas sifi catio ns ar e, apart from t hat, in t he mor e ali en as pect s i n res pect wit h dynamics, or at least i n t he mor e li mitati ve and bor der as pects of it. It woul d be necess ar y to s ee if all t his is a chance. It is even po ssi ble to ask i f thi s could not have so me r el ation wit h the groups of si x and fi ve el ement s t hat appear i n t he di visio n o f the meri di ans of the Chi nes e medi ci ne or i n t he cano ni cal s ectio ns of the gunas ; all t his i s hi ghl y s pecul ati ve, but we have the po ssi bil it y of experi mental cont rast at l east . Fro m a st ati sti cal point of view, it i s obs er ved an i ncreasing appear ance of pheno mena t hat seem to res po nd to pot enti al l aws or scal e l aws, of which t he Par eto distri bution ­ no n- Gauss ian dis tri bution- i s pr ecedi ng and archetype. This t ype of distri butio n or 1/f nois e, mor e or less i ndependent of t he s cal e and also of an expli cit dynamics, mai nt ai ns necessar ily a co nt ro versi al st atut e. It seems as if t he co ncurrence of s pace and ti me, released of t he parti al adjust ment s of dynami cs, acqui red a magical characteris tic, t hat i s to say, the liberation of t he causalit y. But t his "cau salit y" i s o nl y t he o ne o f dynami cs, because t he exis tence of int eractio n wit hin t hese syst ems is o ft en cl ear and evident. Onl y one car eful s eparatio n of t hese super pos ed el ements coul d throw li ght and l aw on whi ch to day remains like mer e pheno menon. Let us clos e t hese consi der atio ns wit h a refl ectio n o n t he pri nci pl e of least actio n that per haps fit s her e. In phys ics t he pri ncipl e of l eas t action i s particularl y bou nded with t he ti me; for exampl e, the mi nimu m paths of Fer mat are t he t r ajectori es for which t he light spends les s time i n i ts tr aj ector y. In a si mil ar way t he rel ati vi sti c phys ics has defi ned geo desi c or mi ni mu m li nes for t he co nti nuou s s pace-t i me. Thi s i s enough str ange if o ne t hinks t hat t he nature, and speci all y t he i norgani c natur e, must have t he mo st co mpl et e indi ff er ence wit h respect to the time and it s savi ng, t hat sur el y does not pro vi de an i nter ests r at e to it . What it seems to demo nst r ate o nce agai n and fro m a dif ferent pers pecti ve t hat t he ti me o f t he physi cs is an al mo st entirel y s patial devi ce. The or gani c growt h s hows to us t he o pposit e face of t he coi n, where t he ti me i s i ndeed decisi ve and co nsti tuent by natu re. Precisel y t he pr esence o f the Co ntinuou s Pro port ion i n i nfinit y of biologi cal r atios , l i ke i n the su nflo wers , it i s tri ed to int er pr et not in t er ms o f t he pri ncipl e of l east act ion, but rat her of a pri ncipl e of mat erial eco nomy. That is to say, what it has cu mul ati ve and irreduci ble t emporal co mpo nents we want to explai n it by pri nci ples of mat er ial and s pace eco no my, and what it is al most pur ely s pace we tr y to or der it by pri nci ples o f t emporal eco no my. It is not t his t he best evi dence of t hat her e an inver sion and a str ange cro ssi ng has taken pl ace here, and t hat so met hi ng ess enti al it i s lost i n t he operatio n? Not hi ng woul d take to us mor e f ar t han t he resolutio n of this pro bl em. 93

To what ext ent the t empor ar y s eries can gi ve a r ecor d of the space co nstrictio ns of a s yst em? At f irs t The l eft hand of C hao s
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si ght t her e ar e enor mou s di ffer ences bet ween s ystems and s yst ems . In t he cas e of t he pul s e co nti nue exi sti ng necess ary connect io ns ; i n t he stock mar ket also, but it i s necessar y to loo k for t he pertinent di mensio ns o f ver y dif f er ent way. If we s aid befo re t hat t he anal yt ical s pace of dynami cs conti nued bei ng geo met r y and al gebr a except a part of Arit hmetic, also it woul d be po ssible to hope that the tempor al s eries suitabl y stu died ­and t he notions of t he Samkhya are our mai n refer ence here- had co mpo nent s o r irr educi ble dimens io ns wit hi n t he perti nent si mpl ectic s tru ctur e; and per haps it is possi ble to r each t hat taki ng s teps i n t he suitabl e di r ect ion. The pulse is an exampl e of whi ch we al ready have to ou r di s posi tio n the cl ass of mo del t hat combines these dynamic and ari t hmetic as pects su itabl y, but u nti l we do not study it properly, we do no t know what can be t her e. Not even we kno w i f o f t he co rrel atio n of t he puls es coul d be dedu ced wit h fi delit y somet hi ng so basic i n t he or gani c bal ance and eas y to measur e li ke t he al kali nity or aci dit y of t he bloo d; but i f t hat is po ssi bl e, surel y that ot her many dat a of t he cli ni cal anal ysis coul d be deduced wit h t he suit abl e extens io n of the fr ame of r ef erence.

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To what ext ent the co nfus io n with r es pect to science ar ri ves , and wi t h co nt empor ar y sci ence i n particul ar , i s illust r ated wit h t he to pi c so fr equentl y repeated that it comes to say t hat t he Newto ni an s ci ence was "hierar chic and t heolo gical ", whereas t he pr es ent o ne "destroys hierarchies and finds o r der i n t he chaos ". The o pposit e is al mos t t he cert ai n t hing, alt hou gh, not to si mpli f y i n excess , it i s po ssi bl e to be det ached: "Newto nian s ci ence" is the l ess hi er ar chic s cience t hat exis ts , and i n that exactl y its absoluti s m is bas ed. We have alr eady s een why t he t hr ee pr i nci ples o f t he mechanics exist . The moder n s cience to which t he topic tal ks about, t he het eroclit e disci plines o f t he emer gency, t he evolutio n and the co mpl exit y, whi ch does pr ecis el y is to creat e hier ar chi es , l evels and met a-l evel s where mor e li kel y t her e ar e not, and ever yt hi ng because t hey ar e incapable to accede to an appar entl y s i mpl e co ns titu ent l evel li ke t he Newto nian one; as f ar as t hat of the or der in t he chaos , as to day t he t hi ngs ar e rais ed, we bett er do n't talk about i t. But the certai n t hi ng is t hat as much one pos ing as t he ot her runs li ke chickens without head. It is necess ary to leave t he soft fantasy o f which bot h met ho ds ar e co mplemented, as if t hey wer e t he clas sicis m and t he ro manti ci s m. No, because a si ngle meeti ng poi nt do es not exi st , but li nes that ar e su per po sed to dif f er ent l evel s. Let us s ee, For what we want so mu ch "teet h and cl aw" if t he t hir d l aw of Newto n not even allo ws t he fri ction? A mechani st explanatio n of l if e, it is sai d. Wit hout a dou bt t he evolutioni s m and all t he phases of romanti c sci ence, cos molo gy i nclu di ng, have benefit ed enor mousl y from t he creduli t y gener at ed by t he ampl e enough empti ness of t he analysi s , si mult aneousl y that the fait h o f t hi s one incr eas ed i n vi ew o f the foo d t hat t he co ntents or the tempor al matt er pro mis ed to i ncor po rat e, wit hout havi ng the small er fou ndation for it . It i s alr eady ti me to co ntr ast t hese hopes, to s ee what they weigh i n t he balance. A delicat e o peration.

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Ti me agai nst t he f ield. The mo der n s ci entific view of t he worl d, l i ke t he man of Shankar a t hat confu ses t he cor d wit h a ser pent, li ves t he illusio n o n t hi s su per pos ition o f pl anes: everyt hing what cannot be expl ai ned by the dynami c analysi s, becau se it does not have i nter nal s pace for it, we gave it to t he domi nio n o f t he time and the evolution, t hat as well , i t has to be gover ned by i dentical dynami c pr i nci ples. Soo n, and to co mpl ement t he evi dent empti ness , one r esor ts to t he pro babili sti c el ements , t hat al ways are dif fus e and u nco ntroll abl e, to extr act of t hem the mo st co nvenient el ement s withi n the pr eviou s co nt ext . The probabili stic fi el d, as diffu se as can be wi s hed, may be t he sur vi val of t he s peci es o r the eco nomi c co mpetit io n: at no mo ment t he co nfli ct s ar e The l eft hand of C hao s
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co nsi dered of direct and local way, but t hat t hes e ar e stu mped and s catt er ed i n t he maxi mum po ssible gener ality. Al so i n physics we have exampl es of the same: Oers ted made hi s fu ndament al dis co very t hat pro ved the r el atio ns bet ween el ectri cal and magneti c pheno mena, coi ni ng t he t er m el ectro magneti sm. Peculi arl y, us ed t he ter m "confli ct " to s peak of t hes e r el atio ns; al l t hi s was abso r bed soo n i n t he concept of "fi eld". We ar e specialis ts i n di sso l ving t he diffi cult ies i n t he l i mitl es s and general s pace, whi ch will s end us as well to ot her s paces . If t he i nter pret atio n of t he t empor al seri es advanced in it s o wn s ens e and di rectio n, and not l i ke a poor as si stant, we woul d begin to s ee all t hese t hi ngs mo re cl earl y. P erhaps it is to this to whi ch s ecr etl y we as pir ed s peaki ng mor e and mor e ever y day of t he "r eal time" pro ces ses . There i s so met hi ng her e that sou nds like an inescapabl e demand, no matter ho w har d oft en it i s r eplaced by t he virtu al repr es ent atio n, exactl y the opposit e, again.

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Far aday, fo r exampl e, who was t he one t hat develo ped t he i dea and t he concept of fi eld, did not need mat hemati cs to develo p experi ment all y a co mpl etel y new visio n of t he el ectromagnetis m, and i n a ti me i n whi ch some o f t he great est mat hemati ci ans of all t he ti mes , like t he same Gaus s, barely wer e abl e to penetrat e in t he par adoxi cal for est of t he elect ro magnetic phenomena. What Faraday war ned was t he i mpo rtance of the dynami c process to advance i n t he i nvesti gat io n of t hes e pheno mena: s hi fts of positio n of t he magnet or changes o f i nt ensit y o f anot her curr ent . This was t he deci si ve t hi ng, and fro m t hat M axwell coul d pro du ce a totall y consist ent t heor y. Without a dou bt to Far aday hel ped hi s i ngenuou s ment alit y as much as his i nces sant experi ment atio n. Maybe so met hi ng si mil ar must happen i n t he ar ea of t he theory of t he co mpl exit y and no nli near dynami cs . Of cour se, t he fi eld is st ill mu ch mor e vast and multifor m her e, alt hou gh it assumes t hat we loo k for as general pri ncipl es as it i s po ssibl e. M y experi ment al i nvit atio n i s of t his so rt ; when we have t he suit able experi ment al proto cols and we l ear n to wor k wit h t hem, inevitably t he M axwel l of t he moment will arri ve to gi ve u s so me o f the equ atio ns. We did not need mat hemati cs to s ee what is mis si ng here, but we need them indeed to explore it and to wei gh it pro perl y. Cl ear t hat t hi s i s a mer e par all eli sm to encourage t hi s cl ass o f research, because t he i mpli catio ns i n t his ar ea, it is not necessar y to s ay it, ar e co mpl etel y di ff er ent. 97

One i s i ncli ned to t hink that in science exi st defi ned cycles o f productio n, a littl e in t he st yl e of t he Ko ndr ati ev eco no mic cycl es fo r the pr edomi nant t ypes of i ndustr y. In t hi s case, t he cycl es woul d co me to be about si xt y years, appro xi mat el y. Wit hout a dou bt between 1887 and 1947 a ver y well defined ci rcl e is clos ed, whi ch goes fro m t he experi ment o f Michel so n- Morl ey, t he first o bs ervation o f t he photo elect ric eff ect and t he veri ficatio n of t he t rans mi ssio n of t he el ectromagnetic waves at t he s peed of t he light by Hert z, to the pres ent ver sio n of t he quantum el ect ro dynami cs of To monaga, F eynmnan and Schwi nger . This l ast o ne supposes the beginni ng o f t he quantu m t heori es o f f ields i n whi ch still ar e embar ked t he t heor eti cal phys i cist s. This pr esent cycle woul d co me to close itself i n t hes e years , mo re or les s agr eei ng wit h t he st arti ng of the l ar ge hadro n collider of Geneva, wai ted for t he 2007 . And pecul iarl y, if t he cycle of 1887 began wit h t he experiment al co ntr adictio n of t he theory o f t he et her or mediu m as ref er ence fr ame and wit h t he r ei nfo rcement of t he s peed of t he li ght li ke co nst ant, also now, before a cycl e t hat seems to clos e its elf, t he statute of this co nst ant is qu est io ned, as well as t he phantom o f t he co smologi cal constant, and an experi ment al resolution of t he vacuu m li ke fr ame of refer ence for t he mass es . So we can s ee i mpor tant si milariti es, poi nti ng al most all o f them at pro bl ems o f definit ion of t he mediu m and fr ame of r ef er ence, al t hou gh we will not s pecul ate on t he results . But t he fu ndament al phys i cs is not ever ythi ng at all , and even it s eems t hat it s i mport ance i s The l eft hand of C hao s
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di mini s hi ng decidedly befor e t he much mo re i mmediat e challenges o f t he co mplexit y. And trul y, the mo der n vi sion o f t he co mpl exit y also begi ns i n t he s tarti ng o f t hi s s eco nd cycle, t here by the 47 o r t he 48 , wi t h t he appearance of t he computer, t he transi stor, and wor ks o f Shanno n and Wiener. The computer and the new notion of i nfor matio n ar e t he i nstru ment al vector s of our mo der n i dea of the co mpl exit y. Even t he mo der n geneti cs, t hat was bor n shortly aft er , has been tot all y co nfor med, and to u ndesi rabl e and absu rd li mits , by the met aphor s of t he worl d of t he i nfor mation and t he progr amming. Let us hope t hat t hi s cycle also i s next to close itsel f, of which we coul d s ho w mo re i ndi catio ns t han t hos e that at fir st si ght appear . The metaphors of t he i nfo r matio n have arri ved u nti l a point so t hat even abou nd t he phys i cis ts and mat hemati cians now who co nceive t he u ni vers e li ke a gi gantic computer ; t hat is to s ay, t hat so much time has been bo ast i ng science to su r pas s t he ant hro pomorphi sms to f all i n co ar sest and vul gar t hat it is possi bl e to imagi ne, deri ved from a mere ar tefact . At l east , t he anci ent s knew well t he r ank of t heir myt hs and metaphors , and aft er all it was not about t he man t he o nl y t he s poke, no r even t he es sent ial often; but thes e wi se peo ple of the last hou r get to beli eve i n t heir metapho rs of the mo st l iteral way, and t hey even des pise somewhat to t ho se that have still not found ou t of what the su bj ect goes. They are, t her efor e, t he best i ncar natio n o f t he li mit atio n of t heir own pro gram. But wer e they o r not an expo nent of o ne mor e gener al bar baris m, t his does no t have to wor ry to us , becaus e t he man cr eat es new devices more qui ckl y of whi ch co st s to us to i nter nalize t hem. I ventur e, t herefo re, t hat enou gh befo re t hey begi n to fulfil t he dr eams and ni ght mares of many on t he digit al world, wil l change t he o per ati ve co ncept and t he r eference o f t he s ame i nfor mation, begi nni ng with t he co mput atio n and it s use of t he ti me. Be caus e t here i s too mu ch to explor e; and because t he worl d is for t he co mput er a gr eater chall enge t han t he co mputer for t he worl d. The s equence of t he hu man genome was t he last gr eat r edu ctio ni st proj ect t hat was gi ven to t he mo der n s ci ence; si mpl y, t here are no lo nger anymor e si mpl e object s of i mpo rtance t hat we coul d tr eat i n t he same way. Not even i n phys ics , where fi el ds and par ticles begi n to s how a dependency of t he vacuu m t hat i s impossi ble to r educe to si mpl e cas es: even t he s cal ar fi el ds of t he so call ed Higgs mechanis m requi re a multi co mpound stru ctu re. The r edu ctioni sm is col lapsing visi bl y, i n t he s ame way that the ol d crit eri a of predictio n; but it wor ked so well, and for t he pr edi ctio n indeed, that it i s ver y dif fi cult to res i gn our s elves to i ts decomposit io n. The reductioni s m nowadays is not mor e than pur e i nerti a, but t his f act do es not mean that her e sci ence fi nishes and we must be reduced to t he t echnical pro blems and to t he particul ar det ail s; o n t he co ntr ar y, we t hi nk t hat what can begin now is much mo re i nt er esti ng and deep t hat all t he pr eviou s develo pments . And in addition, this does no t co nsi st in o ne mor e tu r n of the s cr ew i n the do minio n of the abstr act io n, but rat her o n t he co ntr ar y, al so. What it is not neces saril y i n co nflict wit h the fact that t he mat hemati cs ar e more usefu l when mo re abstract t hey ar e. Therefore, we do not t hi nk t hat t he wo rl d o f the futur e will be do ne nor of ato ms , nor of bit s, but of mo nads . Leibni z was t he great precurso r as mu ch o f the notion o f t he co mplexi t y as of t he mo der n co mput atio n; never t hel ess , t he mo nad, his cent ral and i ndestructi bl e co ncept , became dar ken and was made dou btfu l i ndeed wit h t he su per pos ition o f t he badl y cal led moder n "dynami cs", t hat he hi ms elf tri ed to avoi d, but not to negotiate. I have t ri ed to s ho w i n what s ense has to be understood t hat t he mo nad i s clo sed - i n t he same o ne, i ndeed, i n which it is in r elation wit h t he environment si nce t he begi nning, and exactl y by t he natur al ext ensio n o f t he pri nci pl es of t he dynami cs t hat Newton constrai ned. And t hu s, t he monad fi nds it s br eat h and beco mes the most natur al and ro bust of all t he co ncepts. The s ame co ncept o f i ndi vi du al will change as a fu nctio n of the understandi ng of the mo nad concept ; becaus e t here i s no t to s ay t hat t he no mi nali sti c co ncept of indi vi du al , unli ke t he I-s ense, i s co mplet el y deri vati ve, and not origi nal at all. An oafi sh concept i ndeed, t hat even can not include the i dea of r eci pro cit y wit hi n.

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And s peaki ng about ou r devices , no ne li ke t he clock its el f, of which we have been deri vi ng all of them The l eft hand of C hao s
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until to arri ve to t he s ame comput er . And wi th t hem, also our ir repres si bl e vi ews of t he worl d. If what is want ed is to vindicat e the "r eal ti me", what can we u nderst and as such? It i s po ssi bl e to ho pe that t hat r eal ti me i nclu des somethi ng mor e t han the li ve images and today news? And what is actualit y? The images ar e not , and les s fro m day to day. W hat i n f act we look fo r alr eady i s at t he ot her side of t he mirror . Real ti me, present or actual ti me, i s fo r exampl e the gr adient on which i s bas ed t he cli mat e and t he cli mati c change. Of whi ch barel y we know anyt hi ng, by the way. The paleo - cli matology have verif y r ecentl y changes and abrupt cli mati c i nvers io ns , in l ess t han ten years , i n the gu lf of Mexico , Greenl and, and other many places; befor e it was t hou ght t hat an i mportant change requir ed a term of centuries . Wit h t he probl ems of gr adi ent and s tabi lit y, t hat i nclu de all sort o f co mpl ex behaviou rs , f ro m eart hqu akes to our o wn devices, we retu rn to t he su bject of t he pul se and t heir "dynami c" coordi nat es , t he for m mor e reasonabl e t han is happened to me to approach t hi s quest io n, that i s to s ay, t he temporal refer ence for a gr adi ent , somethi ng t hat is to loo k for wit hin t he s ame syst em. A paradigm in t he o wn s ens e of the wor d, if we cons ider that never has been expos ed o penl y and that si nce t he s ame Patanj ali r at her always has been gi ven as su ppo sed. This r eal ti me can concern t he anal ysis as to our sensiti vity as much; what o f cours e it i s not viable i s a dir ect r epr esent atio n; and even of t he i ndir ect r epr esentat io ns o nl y i s po s sibl e to s ay t hat eit her t hey ar e a mean or t hey ar e an o bst acl e; that is to s ay, exactl y ju st li ke it alr eady happens wi t h t he pr es ent dail y r epresentat io ns . Nevert heles s, i n as mu ch i n whi ch we are able to escape fro m t he predict ive and specul ati ve hori zon usu al no w, we es cape al so of unnecessar y repr esentat ions .

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It has been sai d thous and ti mes t hat Galileo mo ved away to u s of t he cent re of t he u ni ver s e, t hat t he theor y of t he evolution clear ed to u s of t he cus p of t he creation, and t hat t he psychoanal ys is deni ed to us t he do mi nion o n ou r o wn mind. Bu t, bei ng alr eady l ame the fi rst of t hes e t hree concentr ic r evolutio ns, t he ot her two , t hat tri ed to model thems el ves to i mage and si mil arity of t he mechanics, have been mor e event s of t he pu bli c opi nion and t he i maginar y t hat true advances of t he knowl edge. Neit her t he mi nd, nor the lif e, no r t he mechanics have been expl ai ned no r under stood. The mechanics conti nu es closi ng to us t he pass age any su perior u nderst anding, and thus it is u nderstoo d t he innumerabl e amou nt of no nsens e and i ncredi ble artif ices that are pos tulated like candidat es to explai n t he co nscious ness, i n whi ch is pret ended li ke fourt h and defi nit i ve "r evolutio n". Not hi ng of t his has t he s mall er pl au si bilit y, and t he o nl y goo d t hi ng of so many des per ate at tempts is to co nvi nce to us t hat somethi ng extr emel y bas ic has been avo i ded fro m t he ver y beginni ng. Many are t hos e that loo k for t he magical co ncept to day t hat cou l d gi ve po tent iali t y to thi s las t tur n: ideas arou nd t he co mputatio n, t he i nfo r mation, t he det er mi nist chao s, t he qu antu m t heor y and t hi ngs of t he sort . Ideas of l ast hour t hat put i n qu estion t he mini mum of which i s co ns idered consoli dated. This is quit e nor mal if o ne t hinks t hat t he co nscious nes s i s t he l ast and t he most co nditio nal t hi ng. Ou r per ception, on t he co ntr ar y, i s that ther e is ver y li ttle or not hi ng to say on it, and t hat t hose o ther l ayer s are what would benefit fro m a new pri nci pl e; si nce not hing reall y has been expl ained by them. E ven the pr edictio ns o f t he phys i cs, so pr eci se, on obj ects t hat ver y pro babl y ar e not u nder stoo d, ar e bas ed o n last i nst ance i n t he cor res pondence of the pro portion, and not of t he caus alit y - fro m here t he "i nexpli cabl e eff ecti veness of t he mat hematics". This can happen perf ectl y without we make anot her t hi ng that to cut layer s o r flat fil ms without any ot her t hickness than t he pos si bl e and i magi ned r el ations wit h ot her o bjects, whi ch ar e al so defi ned i n so to s ay bi- di mensional layer s. The tri ple concept of flu ctuatio n wit h r es pect to t he bal ance can break t hrou gh fro m t he most fundament al layer of t he phys i cs to t he i mmedi ate su bjecti ve experiences : f ro m t he measurabl e movement to the qu alit ies, and from a s et to it s detail s. In the l andscape o f present kno wl edge har dl y we will fi nd anot her bri dge like t hi s one, and t his will not change while t he modern s cience, t hat histo ri cal co nti ngency, exist . Ot her approaches not even can s cr at ch t he r eal t hi ng ­ it' s li ke make photogr aphs.

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The empt y space of the anal ys i s is s yno nymou s of f reedom for t he ar bi tr ar y i nterchange, redu ci ng to the mi nimu m t he i nter nal bounds of its obj ects. And thu s , ever yt hi ng becomes po ssi ble o bj ect of t he anal ysi s . One of tho se possi ble o bject s s eemed to be t he seri es of t he number s, but t he bou nds tur ned out her e to be stro nger t han all means of deco mposit io n avail able; t hes e, o n t he co ntr ar y, di d not make anot her t hi ng t hat to manif est t hem. Maybe, du ri ng a lo ng ti me, and stil l today, we have confu sed the roots wit h t he br anches . The har mo nic o r Fouri er anal ys is was t he tri gger of t he mo der n concept of fu nction. Exactl y when t hi s co ncept became gener al , reveali ng u ntil where it s ar bit rari ness wit h r es pect to t he empiri cal cur ves ar rived, we began to co nfus e t he i dea of fu nctio n with t he o ne of an auto mati c executio n ­wit h t he auto mati s m its elf , when t his s eemed l ess justi fied t han ever. To day it s eems more t han pro babl e t hat it is an opposit e t endency the one that gener at es a new fu nctio n co ncept, whi ch o n the ot her hand is acqui ri ng l evel s of necessit y. So met hing says t he same slownes s of t hese pro ces s es of the dept h of the changes . W e will not fi nd an equival ent of t he harmo nic anal ysi s for compl ex syst ems , but r at her t he anal ysi s will end up adapting to t he natur e of whi ch resi st s to us . There is i n Ri emann an invi si bl e gestu r e, t hat o ne who s e willingl y gi ves back a key to it s legiti mate owner. Mor e t han an off eri ng, it i s a pi et y gestur e i n the sens e o f the anci ent s, co nfir med by t he modest y of his aut hor and i nco ncei vabl e withi n the vor acious modern s cientifi c spi rit. That gesture cannot be i n gr eater co ntr ast and i n gr eat er agreement with t he ti me at which it happened, when t he ni net eent h centu r y po siti vi sm reached t he cu s p of the vo ciferat io n and the vul garit y. In the same mont h of No vember of 1859 t hat R iemann gave it s paper o n t he pri me nu mbers , appear ed o n s ale t he fir st of t he i nnumerabl e editio ns o f "The origi n of the species ". Also t he st ati stical mechani cs and t he mo dern s pectro scopy rai sed. 101

The lo gic o f t he Samkh ya l eads to u s with an ir r esi stibl e for ce i n as much we kno w to r esist to t he appli catio n of t he force. So i t seems to be t he et ernal crit erio n, i mpos ed and to ou r di s pos itio n.

102

The su bject of t he Samkhya or t he Yoga is to ext end the int ernal space tr yi ng not to establis h the s maller di ff er enti al wi t h t he ext ernal s pace t hat necess arily i s to be r espect ed. Thus t he difference bet ween t he int erio r and the ext erior , and t he fi el d of ref erence fo r t he mo vement sto ps by t he mo vement itsel f. Yo ga i s to penetr ate i n t he unli mited us ing the limit s t hat are gi ven to us, no t t r ying to tr ans gres s t hem or annul li ng t hem, but wit h reciproci t y. The Samkhya is t he co ntempl ation of t he balance and t he Yoga its pract ice, not existi ng true difference bet ween bot h, and being the co nt emplatio n t he gr adu al movement of t he puri t y i n the partici patio n. In all the t heor eti cal cont empl atio n t hat we have made we have tri ed to gui de u s by thi s same pr acti ce.

103

The su bject of t he ti me and its s equ ence is si mpl y t he su bject of our i ns ertio n i n t he r ealit y.

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The s ame mo dalities, or t he gunas, are also onl y obj ect s of kno wledge for purel y co nditio nal pur pose and t herefore i mper manent: that i s already it s fo rm o f evolution fro m t he beginni ng and even wit hout beginni ng. Therefor e, o ne can complet el y tu rn t he back to t hem r eali zi ng them at any mo ment; no sort of kno wl edge fo rces to us to cont empl ate t hem, and fur ther fro m it t he one t hat t he Samkhya pro po s es as its cro wn. This woul d have to make us thi nk abou t t he i ndes cri babl e been o f f reedo m i n whi ch t he cons ciou snes s dwells , bei ng t he natur e of t he gunas t he exi st ence wit h t he appear ance of t he i ndivi dual , and bei ng t he co nscious nes s t he pur e uni ver s al exi st ence. And if met aphys i cs is ju st t he postul atio n of t he extr a- mental like an obj ect , si nce the extr a- ment al is pos tulated here deci dedl y, but not at all it s co ndition of o bject , t his has to do wit h met aphys ics as few as wit h ps ycho logy.

105

Co ns cious nes s has no memor y.

106

At no mo ment we r ej ect t he apti tu de of dynamics to deal wit h an ampl e rank of behaviou rs, and i n fact our as sumptio ns and i nf er ences wit h r es pect to t he puls e are dynami c i n a gr eat measur e. The vari atio nal pri nci pl es are al ways o f glo bal natu re and i gno re t he causali t y; when we applied t hem to an o pen sys tem t hat admit s an expli cit r efer ence of t he i nterchange bet ween t he environment and the i nt erio r of t he s ystem by vir tue of its peculiar du alit y, t he ti me and t he caus ality have to al so admit a new and mor e o r less expl ici t expres sio n, but not merel y suppos ed as it happens wit h dynami cs i n general . The same i dea of t he bal ance has to beco me also mor e or les s expli cit , becaus e to no other t hi ng r efers this co ncept of cau s alit y. And t he s ame must happen i n t he co gnitive pl ane with t he existi ng duali t y between t he predi ctio n and the memo r y, o per ations of o pposed natu re t hat cannot be s i mult aneous and t hat by t he same r equire t he adjoi ned assist ance of its co mpl ementar y i n the phas e of pr edo mi nance of anyo ne of t hem. It would be to ho pe that t his fou nd its necessar y dynami c cor rel ates i n r eal and complex vari abl e. Sur el y t his is much less abst ract than what now we can i magine, and experi ment al r esul ts i ncomprehensi ble today but verifi ed t hous ands of times, like t ho s e of Kor nhu ber r el ati ve to t he t empor al del ay of t he volu ntar y reactio n, with mor e of a second or s eco nd and a half of i nter val, will fi nd t heir i nter pretatio n in t his cont ext.

107

The Samkhya already i nco r por ates t he ext er nal medium i n t he proc pro bl em of the r ef erence. To define fro m t he out si de qu estio ns of evolution of precisio n, it s eems t he i mpro per t hi ng. No w it i s to see u ntil wher e t akes the mo vement and t he mo difi catio ns adheri ng to whi ch preci sel y cir cul at es The l eft hand of C hao s
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own measurement of t hi s o ne. P er haps t hu s we can locat e to us and our i ntel lect i n t he mi ddl e o f t he co nditions, or as near it as we can. Anyway, the dis ti nctions bet ween t he mechani c or not mechanic, t he det er mi ni s m and t he i ndet er mi nism, become ar bitr ar y and irr el evant, becau s e t hey admit indefi ni te degr ees that never are i ncu mbent o n that questio n, t hat al ready i s r ai s ed f ro m out side. The i nt ell ect i s one mo r e of tho se degr ees , simultaneously t han its r efi nements , whi ch pr ecis ely leaves back t he complexit y of t he st ates, if it is certai n that the es s ence of t he di still atio n is t he ess ence. As f ar as t he co ns ciou s nes s, it is sti ll certai n that it is not the obj ect of any di s ci pline. If some gr ant po ssi bil it y to t hat deli rium, it is because they su ppos e it l ike t he mo re hi ghl y co nditio nal, while t hey assu me the unconditio nali t y of t he phys i cal l aws . But t he situatio n i s exactl y the opposite: any phys i cal data, i nclu di ng the measur ement of the rest , alr eady i nvol ves a mo difi catio n, an as ymmetr y of t he s pace and the time. The Vedant a coul d not express it better: If t he co ns ciou s nes s can not explai n it s elf, what t hi ng coul d explai n it?

108 The r eal i s t he exist ence. The exist ence is t he co ns cious nes s. Co nscious ness i s t he r eal t hi ng. That i s the tri pl e af fir matio n i n whi ch it i s summar ized t he Vedant a and the Samkhya, a tri pl e i dent it y as exis tent ial as absolu te. E mpt y? M aybe: cons ciou s nes s i s complet ely i ndiff er ent to the full ness or t he vacuit y. Co ns ciou s nes s, al r eady we sai d it , has no memo r y, not at all a parado xi cal affir mat io n t hat anyo ne can verif y by means o f t he po rtio n that corres ponds to hi m. An aff ir matio n in whi ch u nder lies ver y consi derabl e po ssi bil iti es . If t he co ns cious ness does not have memor y, in t hat exact measur e t he co ns cious ness is will ; if t he co nscious nes s i s wil l, i n t hat exact measur e t he cons ciousness do es not need to kno w. That exact measur ement that we are speaki ng abou t i s t he co natio n, whi ch we do n't need to confu se neces saril y with t he will ; t hen if ther e is co natio n, t he co ns ciou s ness is o nl y i nt ell ection, and if t her e i s no ne, t he cons ciousness r emai ns undivi ded i n t hese t er ms and i ndiff erent to t hem. Thi s i s an exampl e of somet hi ng which not even needs to be thought ; t hen t he ter ms fall her e by t hemsel ves , wi t hout t hi nki ng about how we coul d t ake part i n them. And exact l y in t he same way but in ver y diff erent measur e it happens to us with t he equations. W e ar e not invit ed to t hi nk about it, but to veri f y it . Co nscious nes s has no memor y. Wit h so mu ch di sori ent atio n, we can not thi nk in a better i ndicato r of t he "locatio n" o f t he consciousness, fo r t hos e who want to loo k fo r it For tho s e who please to qu anti f y, an uno bj ectio nabl e defi nit ion of t he co nscious nes s is t hat it fol ds t he thr ee modal iti es i n not hi ng, co rr es pondi ng t hi s way to t he t ri pl e pr eviou s affirmatio n. Thi s tr i ple i dentit y does not have s equ ence, and t herefor e it i s u natt ai nabl e for t he lo gic o r t he mat hemati cs, t hat sometimes beli eve to be i n t he ult imate degr ee of the simpli cit y, t he u nco ndi tionality o r even t he el egance. In fact , t he mat hemati cal for ms are o nl y appeal to ot her for ms , and woul d be enou gh to have t he suffi cient att entio n, with no need of a mat hemati cal i ntell ect, to r eal ize t hat t hey are not mor e near fro m t he et ernit y t han t he fir st o r the last of t he wor ds pro nou nced by t he man. The opposit e i s a pr esu mpt io n, moti vat ed nevert hel es s by the impl ici t mo vement i n t he as pir atio n, mo re t han by t he execut io n or t he assu mpt io n of co nsci ence; ther efo re, and li ke all mo vement of t he langu age, it has an excus able necessit y. So that the tri pl e i dentit y o f the Vedant a off ers u s li ke a s pace for t he co ntemplat ion to our whole di s posit io n, si nce i n no cas e it i mposes not hi ng to u s. It i s t he maxi mum and the mi ni mu m t hat can be s ai d. But i t i s al so a proclamatio n and t he testi mo ny of t he mo st l egit i mate of t he conquest s, becau se it has been made i nsi de t he pur e co nditionalit y, of whi ch all t he for ms depend. In any cas e, to mentio n the co ns ciousness is not absolut ely i no pportu ne preci sel y becau s e o f ho w acute begi ns to be our cons cience of t he insu ffi ci ency of all t he for mal s yst ems, and by our very jus tified di stru st on t he f as ci nation o f all the t heoreti cal abys s es . We can assu me t hi s i n o pposite directio n of t he rhetoric and t he brutalit y, whi ch al ready are made pr esent i n the same i nt ell ect. Bei ng t he i ntellect adhesio n befor e any o ther t hi ng, i f i t does not real ize t hat onl y l ets t ake, he is alr eady lo st , and o nl y t hrou gh t he experi ence it will be able to r ecover of that lo ss , whenever it i s made, and no n o nl y with t he i ntell ect, t he suf ficient effort. Thi s is a fu ndament al su bj ect fo r the Samkhya, because t he The l eft hand of C hao s
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mo dalities exi st indi ff er entl y li ke li ber ation and experi ence sour ce, wit hout exclu di ng bot h t erms except by t he own modes. The Samkhya affi r ms t hat t he li ber atio n is t he i dentity of t he i nt ell ect and t he co nscious nes s: but thi s i denti t y do es not belo ng to t he i nt ell ect, becau se all t hi s , at least for it, is co nditio ned by defi ned degr ees or mo des . Pro perl y s peaki ng, cons ciousness does not belo ng to me, I can o nly hope to cor respo nd wit h it. And far fro m all rhetori c, her e takes roo t any po ssi ble f reedo m. That i s to say, t her e is no anot her one. The rest is onl y co mpu lsio n, and t he lu ci dit y of the int ellect, always lent , out of the co ns ciou sness already is su peri mpos itio n and bli ndness what ever will be the l evel i n which it appear s. Therefo re t he consciou s nes s is neit her a r hetori cal subj ect nor a pro bl em o f lo catio n, but t he onl y r ef er ence and t he o nl y means of evolution i n the real ti me. Real ti me is cons cious ness wit h no memor y.

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SOME R ECOMM ENDAB LE RE ADINGS

"Yoga philoso phy o f P atanj ali wit h bhas vati " Swami Hari har ananda Ar anya Uni ver sit y of C al cutt a, 2000 Institut e of ti me natur e explorations (R elat ed to the l egacy of N. A. Ko zyrev) htt p:/ /www.chronos .msu.ru/ V. E. Zhvir bli s: "Stars and kolt s ar s" htt p:// www.c hr onos. msu.r u/E REPORTS / zhvir blis_st ars/zhvir blis_sta rs.ht m N. A. Kozyr ev: "E xperi ment al study of time" htt p:/ / members .tri pod. com/ ~jtooker/ ko zyr ev.t xt Matt hew R . W atki ns ho me page Nu mber t heor y and physi cs ar chive (t he best r esour ces about t he Ri emann zeta I kno w i n the web) htt p:// www.maths.ex.ac. uk/ ~mwatki ns/ TGD- Topo lo gical Geo met rodynamics- Matti Pit kanen home page htt p:// www.physics. he lsi nki.fi/ ~mat pit ka/t gd. ht ml htt p:// bio . naga oka ut. ac. jp/~mats uno/ pre pri nt s/ RES URRE C. ht m Paul M ar met Absur diti es in mo dern phys i cs: a solutio n. Su bjectivit y of Hei s enber g´s u ncert aint y r el atio nshi p htt p:/ /www. newto nphys ics .on. ca/ HEISE NBER G/ C hapter3.html Koi chiro M at su no: Resurr ectio n o f Cart esi an P hys ics htt p:// bio . naga oka ut. ac. jp/~mats uno/ pre pri nts/ RES URRE C.ht m Museu m of Har mo ny and Gol den Sectio n ( Al exey Stakhov s ite) www. gol denmu seu m. co m "R iemanni ana s el ecta" EdiciŃ n y estu dio i ntro ductorio de Jos Č Ferr eirŃ s Co ns ejo Su perior de Investi gaciones ci entĚ fi cas, D. L. 2000 . "El cuerpo y lo ot ro. Intro du cciŃn a una teorĚa general de l a cul tura. " Alexi s Jar dines E ditor ial de ci enci as so ci al es , 2004 . "El eni gma del mo vi mi ento " Alexi s Jar dines Bi bl ioteca Nu eva, 2000

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"La hija del capit Ŕn Star bu ck ( Hur qual ya) " Mi guel Iradier Bi bliot eca Nu eva, 2001 "Cuaderno s " Si mone W eil Editori al Trott a, 2001 "Does Go d pl ay di ce? The new mat hemati cs of Chaos" Ian St ewart Pengui n Books Lt d. 1997 "Pri me int rodu Jo hn D Plume, Obs ession. Ber nhar d Ri emann and t he Great est Unso lved Pro bl em i n Mat hemati cs". An excellent ct io n to t he Ri emann zeta fo r no n mathemati cians. erbyshir e Pengui n Group, 2004

"C yber net ics o r Co ntrol and Co mmuni catio n i n t he Ani mal and t he Machi ne" Nor bert Wi ener M. I.T., 1948, 1961

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