In the course of the comment thread on this post, we got into some territory that interests me: namely, the reality of social constructs and mental creations. In what sense is a unicorn, for example, “real”? It was created in the mind of humanity, so without humanity the idea would not exist. However, “unicorn” is now “real” in the sense that it does exist among mankind’s instructions, with books, tapestries, and paintings depicting it, quite apart from the dolls and cartoons.

Another creation of the mind of humanity is mathematics. The epigraph for Kershner and Wilcox’s *The Anatomy of Mathematics* is a sonnet by Clarence R. Wylie, Jr.:

PARADOX

Not truth, nor certainty. These I forswore

In my novitiate, as young men called

To holy orders must abjure the world.

“If . . ., then . . .,” this only I assert:

And my successes are but pretty chains

Linking twin doubts, for it is vain to ask

If what I postulate by justified,

Or what I prove possess the stamp of fact.

Yet bridges stand, and men no longer crawl

In two dimensions. And such triumphs stem

In no small measure from the power this game,

Played with the thrice-attenuated shades

Of things, has over their originals.

How frail the wand, but how profound the spell!

Mathematics tracks the real world (the world as it would exist if humanity were not present) to the degree that some people believe that mathematics is discovered rather than invented. The inevitability of the conclusions that follow from a set of premises and postulates accounts for some of that. Leopold Kronecker declared, “God created the integers. All the rest is the work of Man.” But my commenter felt that the full structure of mathematical systems reflects reality.

I don’t think so. Start with the positive integers (1, 2, 3, . . .). Some argument can be made that these are derived pretty directly from the world—e.g., counting how many goats you have. Zero is naturally invented for the poor soul who had 1 goat and then ate it, and now needs a number for how many goats he has.

Then the negative integers are derived by observing that a + x = b sometimes has a solution (2, in the case where a=5 and b=7) and sometimes not (where a=7 and b=5). That’s very unsatisfactory: a strong feeling developed that the equation should always have a solution, and when a is 7 and b is 5 the solution is a new kind of number, –2.

So we now have all the integers. Fractions perhaps also are related to the real world and quickly derived from it. For example, you have a bolt of cloth twice as long as the bolt of cloth I have: if my bolt represents 1, yours represents 2. But what if yours represents 2? What is mine? Thus we get 1/2 and the rational numbers. You can also approach it as wanting solutions for equations: a * x = b. That’s easily solved if a=3 and b=6—x is then clearly 2. But what about when a=6 and b=3. We want to solve that equation as well, and the new kind of number 1/2 works.

So now we have the full set of rational numbers. That’s great, but even the ancient Greeks could prove that the square root of 2 (or, the side of a square whose area is 2) is not a rational number. The proof is by contradiction: you assume that the square root is indeed a rational number, but then you are forced into a contradiction. (If you’re interested, I’ll do the proof in the comments.)

These new numbers can be seen as the determination to have numbers to solve equations, such as x^{2} = 2. We can simply define a new number x and say that it satisfies this equation. Similarly we get the complex numbers, by defining a number to solve the equation x^{2} = –1.

But some very weird irrational numbers exist, such as the ratio of the diameter of a circle to its circumference: π, which turns out not to be the root of any equation with rational coefficients but can only be approached as the limit of a series.

If we throw in all these, so that every point on a continuous line is associated with a number, we get the real number system: a set of numbers that demonstrates the property of continuity.

But here we have moved beyond reality. Nature, so far as we can tell, is definitely NOT continuous: it exists as quanta—little discrete parts. Even time is thought by some not to be continuous but to move in tiny little jumps.

So mathematics is definitely not “real” in the sense that it corresponds with something in nature. Math uses continuity regularly, and while this will work as an approximation of reality, it is not reality. Reality is discontinuous.

So at some point, math is not real, just a creation of the human might: perhaps a special kind of social construct.

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