Ode to Math
Ode to Math
Mathematics, as such, does not exist in nature. At its basic, simplest level the relationship between arithmetic and natural phenomena seems obvious. For example, if a farmer has two cows in one field and three in another, then he can see how many he has altogether. If he wants to convey this information to another person, he can establish a one-to-one correspondence between his fingers and the cows, holding up two fingers and then another three, to produce a result for the total, symbolically represented by the five fingers. When he runs out of fingers, he has to start over, while keeping in mind the ten that he has already counted. If he wants to record this calculation, he can make two marks on papyrus to represent the two cows, or fingers, etc. When he gets tired of making all of those marks, it's natural that he will adopt a different symbol for two marks (ll - 2), (lll - 3), etc.
So, we can see how the development of the simplest math was derived and motivated by natural situations. But to make the leap from that beginning to the present level of mathematics and the highly complex physical situations that it models challenges the imagination. But that relationship between math and physics is only part of the story. There exist relationships between different mathematical entities that are sometimes surprising, and often useful. For example, in most high school algebra texts we find a chapter on convergent series. At a higher level, we find a relationship between that theory and the theory of vectors in an infinite dimensional space. In turn, we find a relationship between that vector theory and the theory of characteristic functions (eigenfunctions) that represent the solutions of certain linear differential equation problems. Furthermore, that vector theory can also be applied in the theory of statistics, because if we consider the m elements of two random distributions to be independent variables in an m-dimensional space, then the correlation coefficient of the two distributions is the cosine of the angle between the two vectors. Of course, you don't have to understand any of those terms to get the general idea. It is a kind of thrilling experience to discover one of these unforeseen relationships, and then it is satisfying to think it through to understand why it exists.
But the guy that thought of a short, efficient way to represent numbers by replacing II with 2 and III with 3 didn't discover that scheme. He invented it. Similarly, Algebra was invented. Newton and Leibniz invented calculus independently. These are just a few of the many inventions that revolutionized mathematics.
So: mathematics is both discovered and invented – which is in itself remarkable, because both terms (discovery and invention) are usually applied to actual physical entities rather than to a set of abstract symbols on paper.
Another remarkable fact about mathematics is its use of symbols for things that don't exist to solve problems about something that does exist. For example, in the discussion above about series we found that the m terms of a series can be treated as if they were independent variables in an m-dimensional space – even if m is infinite, provided that the series converges. But there is no such thing as an m-dimensional space. Our minds cannot even conceive of a space of higher dimension than 3. But Mathematics doesn't care, and it provides us with simple, elegant solutions of practical problems in our mundane 3-d world by pretending that these abstract spaces do exist.
One more example of a number that doesn't correspond to any real thing is the square root of -1, which is denoted by i. Any number, whether positive or negative, when multiplied by itself, must be positive, and therefore there can be no number that, when squared, equals -1. But that's okay. As long as we bear that fact in mind, we can build up a powerful field of mathematics. Starting with i, we can multiply it by 2 to get an imaginary number of magnitude 2, so that (2i x 2i = -4), and so wind up with a complete set of imaginary numbers. Then we can combine real and imaginary numbers, say 2+3i, to form a set of complex numbers. Then we can define complex variables, x+iy, and develop an algebra and a calculus of complex variables – which provides an approach to many physical problems.
Matter and mathematics exist almost independently of each other. Once we had a basic system to count physical objects, math could be developed without any reference to the laws of nature. However, when we set out to use math to describe and model the activities of nature, what do we find? Most of the fundamental behavior of nature fits into incredibly simple equations. E.g., Newton’s laws all involve integer powers of the variables. Why does the gravitational attraction of 2 masses vary as 1/r**2 , where r is the distance between them? Why not 1/r**2.396426441. . . . , or some other irrational number? The other physical laws also possess amazingly simple mathematical expressions. The perfect gas law is the essence of brevity and contains only the first powers of each of its variables. One could almost conclude that the physical world was designed in consultation with a mathematician.
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