Order:

The main purpose of science is to trace, within the chaos and flux of phenomena, a consistent structure with order and meaning. This is called the philosophy of rationalism, rational as in conforming with reason. And the purpose of scientific understanding is to coordinate our experiences and bring them into a logical system.

Thoughout history, intellectual scientific efforts have been directed towards the discovery of pattern, system and structure, with a special emphasis on order. Why? Primarily, the control of the unpredictable driven by the fear of the unknown. Those who pursue answers are known as scientists. The main occupation of a scientist is problem solving with the goal of understanding the Universe.

Science is founded on the hope that the world is rational in all its observable aspects. Its possible that there may be some facets of reality which lie beyond the power of human reasoning, that there may be things with explanations that we could never grasp, or no explanation at all, but the fact that the world is rational is connected with the fact that it is ordered.

Science is also a dialogue between mankind and Nature. Science is far from a perfect instrument of knowledge, but it provides something that other philosophies fail to, concrete results. Science is a ``candle in the dark'' to illuminate irrational beliefs or superstitions. Science does not, by itself, advocate courses of human action, but it can certainly illuminate the possible consequences of alternative courses. In this regard, science is both imaginative and disciplined, which is central to its power of prediction.

Science:

Science is any system of knowledge that is concerned with the physical world and its phenomena and entails unbiased observations and/or systematic experimentation. In general, a science involves a pursuit of knowledge covering general truths or the operations of fundamental laws of nature.

Science is far from a perfect instrument of knowledge, but it provides something that other philosophies often fail to provide, concrete results. Science is a ``candle in the dark'' to illuminate irrational beliefs or superstitions.

Science does not, by itself, advocate courses of human action, but it can certainly illuminate the possible consequences of alternative courses. In this regard, science is both imaginative and disciplined, which is central to its power of prediction.

Science can be separated from pseudo-science by the principle of falsifiability, the concept that ideas must be capable of being proven false in order to be scientifically valid. For example, crop circles -> science is truth

The keystone to science is proof or evidence/data, which is not to be confused with certainty. Except in pure mathematics, nothing is known for certain (although much is certainly false). Central to the scientific method is a system of logic.

Scientific Method:

Scientific arguments of logic basically take on four possible forms; 1) the pure method of deduction, where some conclusion is drawn from a set of propositions (i.e. pure logic), 2) the method of induction, where one draws general conclusions from particular facts that appear to serve as evidence, 3) by probability, which passes from frequencies within a known domain to conclusions of stated likelihood, and 4) by statistical reasoning, which concludes that, on the average, a certain percentage of a set of entities will satisfy the stated conditions.

The fact that scientific reasoning is so often successful is a remarkable property of th e Universe, the dependability of Nature.

To support these methods, a scientist also uses a large amount of skepticism to search for any fallacies in hypothesis or scientific arguments. In order to draw conclusions, a scientist uses the scientific method, a rigorous standard of procedure and discussion that sets reason over irrational belief. Central to the scientific method is a system of logic.

The scientific method has four steps:

Note that there is an emphasis on falsification, not verification. If a theory passes any test then our confidence in the theory is reinforced, but it is never proven correct in a mathematically sense. Thus, a powerful hypothesis is one that is highly vulnerable to falsification and that can be tested in many ways.

The goal of the scientific method is the construction of models and theories, all with the final goal of understanding.

Reductionism:

Reductionism is the belief that any complex set of phenomena can be defined or explained in terms of a relatively few simple or primitive ones.

For example, atomism is a form of reductionism in that it holds that everything in the Universe can be broken down into a few simple entities (elementary particles) and laws and interactions among them. Modern chemistry reduces chemical properties to ninety or so basic elements (kinds of atoms) and their rules of combination.

To a reductionist, once a set of equations or mathematical relations has been found to describe a system, then the behavior of the system is considered to be explained.

Reductionism is very similar to, and has its roots from, Occam's Razor, which states that between competing ideas, the simplest theory that fits the facts of a problem is the one that should be selected.

Reductionism was widely accepted due to its power in prediction and formulation. It is, at least, a good approximation of the macroscopic world (although it is completely wrong for the microscope world, see quantum physics).

Too much success is a dangerous thing since the reductionist philosophy led to a wider paradigm, the methodology of scientism, the view that everything can and should be reduced to the properties of matter (materialism) such that emotion, aesthetics and religious experience can be reduced to biological instinct, chemical imbalances in the brain, etc. The 20th century reaction against reductionism is relativism. Modern science is somewhere in between.

Determinism:

Closely associated with reductionism is determinism, the philosophy that everything has a cause, and that a particular cause leads to a unique effect. Another way of stating this is that for everything that happens there are conditions such that, given them, nothing else could happen.

Implicit to determinism is the fact that every event happens of necessity. It has to happen; the Universe has no choice.

Determinism also implies that everything is predictable given enough information. Since Newtonian or classical physics is rigidly determinist, both in the predictions of its equations and its foundations, then there is no room for chance, surprise and creativity. Everything is as it has to be, which gave rise to the concept of a clockwork Universe.

Laws of Nature:

The rise of science during the Age of Reason produced the idea that 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 associated 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 regularities 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.

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.

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.

If Shakespeare, Beethoven, or van Gogh had not been born its unlikely that anyone else would have ever achieved what they did. But is this true for scientists? Would someone else have discovered the classical laws of motion if there had been no Newton? Probably, because science is a collective enterprise. The solution to a scientific problem must satisfy exacting criteria and demands. These constraints do not eliminate creativity, they provoke it.

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 (such as the discovery of Hawking radiation) 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. Their 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 accompanied 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 acceptance until it forms the next paradigm. Each paradigm shift, or science revolution, leads to a major step forward in our understanding of the underlying reality.

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?

There exists two schools of thought. One that mathematical concepts are mere idealizations of our physical world. The world of absolutes, what is called the Platonic world, has existence only through the physical world. In this case, the mathematical world would be though of as emerging from the world of physical objects.

The other school is attributed to Plato, and finds that Nature is a structure that is precisely governed by timeless mathematical laws. According to Platonists we do not invent mathematical truths, we discover them. The Platonic world exists and physical world is a shadow of the truths in the Platonic world. This reasoning comes about when we realize (through thought and experimentation) how the behavior of Nature follows mathematics to an extremely high degree of accuracy. The deeper we probe the laws of Nature, the more the physical world disappears and becomes a world of pure math.

Mathematics transcends the physical reality that confronts our senses. The fact that mathematical theorems are discovered by several investigators indicates some objective element to mathematical systems. 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.

Plato's Theory of Forms:

Plato believed that there exists an immaterial Universe of `forms', perfect aspects of everyday things such as a table, bird, and ideas/emotions, joy, action, etc. The objects and ideas in our material world are `shadows' of the forms (see Plato's Allegory of the Cave).

This solves the problem of how objects in the material world are all distinct (no two tables are exactly the same) yet they all have `tableness' in common. There are different objects reflecting the `tableness' from the Universe of Forms.

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

Pythagoras, contorniate medallion engraved between AD 395 and 410

Pythagoras:

Much of the philosophical foundation to our mathematical sciences is due to the Pythagorean school. Pythagoras lived in the 6th century B.C., most of the work we attribute to the individual is more than likely a composite of work done over many centuries by his school.

The Pythagorean school presented us with our first physical models, but the role of mathematics was much more dominate than it is today. For example, the Pythagoreans believe that matter was make of numbers. Not idealized numerous fragments, but actual numbers themselves. In some sense, this is an atomic theory, but with pure mathematics replacing matter objects.

The world being composed of numbers was reached by the Pythagoreans due to their interest in gnomones. This naturally led to the development of the Pythagorean Theorem.

The discovery of the Pythagorean Theorem led to a crushing blow to the Pythagorean school since it uses the idea of irrational numbers, which was repugnant to the philosophers of the time.

Early Cosmology:

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 Euclid discovered that there exist only 5 regular solids. Plato then proposed that these five solids correspond to the four atomic elements in the Universe (earth, water, air and fire) and hypothesized that there existed a five element, called quintessence, which made up the heavenly spheres.

Plato, so impressed by the elegance 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 as that the fifth element, quintessence, made up the heavenly spheres.

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:

Mathematics went on to led the way in many scientific and technological developments over the next 2000 years. Architecture, navigation and mechanics are all examples of core elements to our civilization that depend heavily on mathematics. By the Renaissance, mathematics was wide spread in all parts of life, including art, where the development of the golden section played a strong role in painting at the time.

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.