Saturday, November 18, 2006

the existence of mathematical entities

I've started reading The Road to Reality by Roger Penrose. I'm not far into the book at all, which stands to reason since I'm reading it slowly and in small doses. I was surprised to read that mathematical entities exist as Platonic ideals, which in effect seems to mean that their existence is neither physical nor subjective. (It reminded me of the three Worlds described by Karl Popper). In contrast, I believe that mathematics is a human creation having no independent being. That is, mathematics exists in the minds of people. It has the same claim to reality as Arda.

But you may argue, "Numbers are self-evidently real, at least because we use them to count objects and then accurately operate on those quantities". Say that I accept your argument for the time being and grant you all the whole numbers. I can do this because the whole numbers and simple 'rithmetic make up only a small sliver of modern math. Consider complex or "imaginary" numbers, which involve the square root of -1 (i), or even just negative numbers or 0. Historically speaking, these concepts did not come easily to humanity. Are these more abstract math concepts still applicable and useful, in spite of perhaps not having direct analogues to normal experience? Sure, but that's not the point. The point is to realize that from the logical standpoint of the entire math system, all numbers, whether whole or zero or negative or complex, as well as all equations and functions, have the same degree of "existence"; all belong to a system of axioms, definitions, postulates, theorems, etc. And this system was not discovered by people. It was invented, bit by bit, by starting with elementary ideas and then extending those ideas. 1 did not exist until a person wanted to represent how many noses he had.

One of the important ways in which the human creation of mathematics differs from the human creation of Rumpelstiltskin is that math proceeds logically (whereas Rumpelstiltskin isn't that logical at all). This means that mathematicians assume the truth of as few statements as they can get away with, and then show that putting those statements together leads to everything else they need. Hence the obsession of mathematicians for proofs--proofs mean that someone can rely on the truthfulness of conclusion Z13(w) provided that same someone relies on the truthfulness of some definitions and axioms. Proofs make the implicit truth explicit. The discoveries of mathematics, because the discoveries come about logically, consist of finding out what was implicitly accepted as true all along. So pi, or e, or i, are not ideal entities that eternally exist until someone finds them. That is, not like the speed of light, c. Mathematical entities, even constants, are just consequences of certain definitions. In effect, no humans created or fully comprehended the infinitely deep Mandelbrot set; they simply defined a set by defining a function that operates on complex numbers as previously defined. No human has counted every number even in the set of integers, either, but the set is infinitely large because it is defined that way.

Math does not originate or exist in the (completely objective) physical world, although the physical world displays enough order that mathematical models can correspond to it with varying degrees of accuracy. Math also does not originate or exist in a (completely objective) hypothetical world of Platonic ideals. Math is all the logical consequences of specific definitions, and the definitions in turn are/were developed by humans who used their powers of abstraction to create logical ideas out of physical reality. The same thought processes that enable generalized language (the word and representational concept of "chair" as opposed to the specific instance I'm sitting on at the moment) also enable the creation of mathematical entities. And logic, the process of combining true (abstract) statements into new true (abstract) statements, enables anyone to discover what they believed all along.