The question is sometimes raised as to how it is that mathematics, which is a creation of the human mind, without any empirical reference to external reality, should match reality so well. When we make the distinction between the reality we experience and the underlying reality, the correlation between mathematics and reality is not so surprising.
Science takes our observations of the external world and seeks to understand how they occur and to discover underlying patterns and principles. In doing so, it inevitably draws upon experience. When atoms were first imagined, they were thought of as small solid balls of matter, a model clearly drawn from everyday experience. Then, as physics realized that atoms were composed of more elementary particles (even the word “particle” contains an implicit assumption as to their nature), the model shifted to one of a central nucleus surrounded by orbiting electrons, again based on experience at the human level. Now, as we try to interpret quantum theory, we inevitably draw upon other concepts derived from our perception of reality. We interpret them as waves or bundles of energy, possessing “spin” and mass. Yet every model we come up with, fails in some way or another to capture the essence of the underlying reality.
At first we might find it surprising that the conclusions of modern physics are so far removed from our experience or reality. But it is not actually that surprising at all. All scientific models and theories have their roots in human experience. They are all based on the way the human mind interprets the incoming sensory, which is itself based on our particular, and partial, perception of the world around us. What would be far more surprising would be to find that the image of reality created in the human mind was indeed a faithful representation of the thing-in-itself.
Mathematics on the other hand is purely a creation of the mind. Mathematics is that body of knowledge that is arrived at by pure reason, and does not rely upon any observations of the phenomenal world. It is free from the limitations imposed by the particular way human minds create their experience of the underlying. As such it is probably the closest the human mind can come to understanding the thing-in-itself.
The only thing that pure mathematics depends upon is the notion of distinction. If I experience two apples I am experiencing two phenomena that can distinguished one from the other; I can eat one and keep the other. I can distinguish between the black ink and the white paper of this page. Even in the underlying reality there is distinction; we may not know what the thing-in-itself is really like, but we can measure its separation in the spacetime interval from another thing-in-itself. If there was no distinction in the cosmos, there would be no difference of any kind. No experience whatsoever. The existence of distinction is as undeniable as the existence of experience itself
If there are distinctions, we can count them. The base of the counting may vary. We use ten (probably because we have ten fingers), computers use two, the Babylonians used sixty (which is why we count sixty seconds in a minute and sixty minutes in an hour), other cultures have used five, twelve or twenty as their base.
From counting comes the concept of number, and all the integers. We can add numbers together, leading to multiplication of numbers, and the their opposites, subtraction and division. From this simple arithmetic comes the concept of nothing, zero; and beyond zero, the negative numbers, not part of our direct experience, but a concept we readily accept and are quite happy to work with. In between the integers we discover fractional numbers, numbers such as a half, or two thirds, which can be expressed as the ratio of two integers. Hence their name, the rational numbers.
Counting all the numbers, we arrive at the notion of infinity. And between the rational numbers we discover an infinity of irrational or transcendental numbers that can be expressed as the ration of two integers. Numbers such as “pi”, the ratio of the circumference of a circle to its diameter, or “e”, the base of natural logarithms. They can be defined, but never written down exactly as a number for they go on forever, to an infinite number of decimal places. All this from the notion of distinction.
And there is more. Any positive number has a square root, the number that when multiplied by itself produces that number. The square root of one is one; of four it is two; and of eight it is 2.828… (another irrational number that goes on forever). But what, asked mathematicians of negative numbers, what multiplied by itself gives minus one? Nothing in the range so far discovered; no integer, positive or negative when multiplied by itself results in minus one. So they defined the square root of minus one to be a totally new number, an “imaginary” number, not part of the range or “real” numbers, and gave it the symbol “i”. From this arose a new and even larger set of numbers, the so-called “complex” numbers, that were a combination of real and imaginary numbers. And these, it turned out were invaluable in helping mathematicians solve equations that had no solution in the realm of real numbers. Moreover, the solutions applied to the real world.
Out of this panoply of numbers a most remarkable and intriguing relationship appeared. The irrational number “pi”, the irrational number “e”, and the imaginary number “i”, come together in one of simplest equations ever; “e to the power of i times pi = -1”.
Many mathematicians have eulogized over the significance and beauty of this equation. Out on the very edge of number theory a relationship is discovered that seems to show it is all in some way pre-ordained. Little wonder that some mathematicians feel that God is to be found in the beauty and perfection of mathematics.
That these three seemingly unconnected numbers should be related in such a simple way was startling enough; but even more was in store. This simple equation is the basic equation of any wave motion. Every wave from a wave on water, the air waves coming from a violin string, to light waves, can be expressed as a combination of simple equations of this form. It also expresses the orbits of the planets, the swing of a pendulum and the oscillation of an atom. In fact, every single motion in the cosmos can ultimately be reduced to an equation of this form. The whole of quantum physics depends upon it. If mathematicians had not discovered this most remarkable relationship, the strange story of the quantum would never have been told.
And all of this without a single empirical observation. No wonder then, that in the end all science comes down to mathematics. The very fact that it is not based upon phenomena, is why it is probably the best approximation to the underlying reality we have.
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