Continuing with the themes of order, reductionism and determinism and any other 'ism ya need:

Mathematics and Science:

The belief that the underlying order of the Universe can be expressed in mathematical form lies at the heart of science and is rarely questioned. But is mathematics a human invention or does it have an independent existence?

Isn't mathematics just a language and nothing more?

The Greeks constructed their whole theory of the Universe on the concepts of numbers and shape, arithmetic and geometry. An example of this was when Empedocles discovered that there exist only 5 regular solids.

Reductionism at its best ...

Plato, so impressed by the elgence of this discovery, then proposed that four of these solids correspond to the four atomic elements in the Universe (earth, water, air and fire). Plate also hypothesized that there existed a fifth element, as yet undiscovered called ether, that corresponds to the fifth element which made up the heavenly speheres.

Each of these five elements occupied a unique place in the heavens and, thus, Plato developed the first periodic table and, at the same time, proposed the first cosmological models looked something like the following diagram:

Thus, according to Platonists, we do not invent mathematical truths, we discover them. Mathematics transcends the physical reality that confronts our senses. The fact that mathematical theorems have been discovered independently by several investigators indicates that there does exist some objective element to mathematical systems.

Or maybe that we are so unsophisticated that we can't think beyond our common language and merely rediscover the same thing in a different syntax

Since our brains have evolved to reflect the properties of the physical world, it is of no surprise that we discover mathematical relationships in Nature.

Possibly we have de-evolved, substituting mathematical precision and elegance in place of a more visceral and sensory relationship with the Universe.

The laws of Nature are mathematical mostly because we define a relationship to be fundamental if it can be expressed mathematically.

Mathematics went on to led the way in many scientific and technological developments over the next 2000 years. Architechture, navigation and mechanics are all examples of core elements to our civilzation that depend heavily on mathematics.

The historical peak of mathematics was the development of calculus by Newton as the basis for his theory of mechanics and gravitation. Calculus was so successful that the early numerological image of the Universe was replaced with a clockwork image. This clockwork model of the Universe reached its most developed form under Laplace in the late eighteenth century, who envisaged every atom in the Universe as a component in a precise cosmic mechanism.

Kepler's Second Law an example of how observation requires a new language to explain it.

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The Laws of Nature

There is a hidden order in Nature, which is mathematical in form and could be uncovered by investigation. This hidden order could be expressed in the form of mathematical principles, or laws of Nature.

Direct connections between events are usually apparent to the senses. But the underlying causes assoicated with the laws of Nature are much more subtle. Observations of events are not generally intelligible. Often phenomenon requires an abstract theoretical framework to form a context for measurements in order to link them into a framework of understanding. This framework is called a scientific theory.

The laws of Nature are attempts to capture the regularities of the world systematically. The existence of regularties in Nature is an objective fact, thus we do not impose laws onto Nature. While the form of the laws are human inventions, they reflect, albeit imperfectly, real properties in Nature. It is this absolute invariance of the laws of Nature that underwrites the meaningfulness of the scientific enterprise and assured its success.

I dissent vehemently; absolute invariance creates the detector problem

Truly basic laws of Nature establish deep connections between different physical processes. When a new law is developed, it is tested under different contexts which often leads to the discovery of new, unexpected phenomena. This demonstrates that we are determining real regularities in Nature, not imposing them with our scientific structures.

Controversial Point #1: Is what is in the red italics above really correct or do we just want to believe it?

The laws of Nature are eternal, absolute and have an independent existence outside the physical conditions of an experiment. Success in the scientific method rests on the reproducibility of the results. An experiment is repeated and the same laws of Nature apply, but the initial conditions of the experiment can be varied. There is a clear functional separation between laws and initial conditions, similar to the Platonic Forms.

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Models and Theories:

Scientific theories are essentially models of the real world (or parts of it) and the vocabulary of science concerns the models rather than reality. Often when the term `discover' is used in a scientific model or theory this, in fact, refers to a mathematical relationship that is revealed.

A true discovery would refer to the observation of the phenomenon in Nature (with respect to Hawking radiation, noone has yet directly observed a black hole).

The relationship between a theory or model and the real system represents an important distinction. For example, how do we know when a model is merely a computational device and when does it actually describe reality? Scientific theories are descriptions of reality, they do not constitute that reality.

As long as a theory sticks close to direct experience, where common sense remains a reliable guide, then there is confidence that we can distinguish between the theory and reality. Advance theories in modern physics push this boundary, for example, the use of virtual particles in quantum physics. There existence is never directly observed, so some might say that there use is a simple way of describing an unimaginable process in familiar terms.

Models or theories that are broad and encompass a significant fraction of a field of science are called paradigms. Reductionism was one of the founding paradigms of science, but was not a complete expression of the truth to Nature. However, the three hundred years of progress that accompanyed reductionism was not rooted on a misconception, for this is not the role of paradigms.

Rather a particular paradigm is neither right nor wrong, but merely reflects a perspective, an aspect of reality that may prove more or less fruitful depending on the circumstances. Science may not deliver the whole truth, but it certainly deals with truth and not dogma.

Science historian, Thomas Kuhn, argued that science moved in leaps That paradigm's form, led to many new discoveries, then become the standard in which new ideas are tested. Eventually, some new experiment or observation will not fit into the current paradigm and will led to a new theory, usually by some brilliant, young scientist.

This new theory undergoes a series of phases from disbelief to grudging acceptence until it forms the next paradigm. Each paradigm shift, or science revolution, leds to a major step forward in our understanding of the underlying reality.