A Simple Proof Mathematics Is Discovered And Not Invented

This came up in an email thread, but for the life of me I can't find it. Since this is the second email (thread) from a colleague I have lost in as many weeks, I'm beginning to suspect my new email client (Thunderbird) is eating them.

Not that you care about that, but because I'll be rebutting this colleague's argument, I can't do so in his own words. ("Why don't you email him, Briggs?" Because I can't remember who it was.)

Anyway, the impetus was yet another anti-"white supremacist" article claiming whites were too good at math, so that we should re-define math to be what people want it to be. Like letting students hand in Tik Tok videos instead of proofs, you racists.

I naturally made a stinging witticism in answer to this, along the lines of if math is invented, then everything nitwits and the evil scream about "white supremacist" math is true. Anything goes.

My colleague thought he was disagreeing with this, because it is beyond obviously absurd to let the kiddies make up their own "math", while also insisting that math was indeed invented, not discovered.

He thought his position was saved by saying math had to align to the natural world (or just "world"), the material stuff that makes up the universe. That still sounds like a failure for "math is invention", because we discover facts about the world, we don't invent them.

Sure, our discoveries are tinged with invention, because every fact and observations fits into a theory or scheme somehow---how else do we know what facts to look for and classify them?---but that doesn't make the facts wrong. It makes them conditional. And it makes some more uncertain than others. But that's about it, logically speaking. We are always aiming at the truth of the world in science---or we used to, before it became another branch of poiltics.

If the world is only known up to a degree, and not perfectly, and math aligns with what we know of the world, then it seems math would have to change every time we learn a new or refine an old fact about the world. Yet this does not happen. Math remains constant.

This is because, of course, math begins with propositions that are true conditional on the rock-solid argument "This axiom has to be true". It proceeds from there, building step by step, expanding using, it is hoped and almost always turns out to be so, solid links in the chain.

Still, this did not satisfy my colleague, who pointed to Euclidean and non-Euclidean geometries. One was thought to fit the world, but the other fit it better. Therefore, the one that did not fit was made up. Invented.

If that is so, then it should be able to prove the not-fitting geometry false. Which can't be done by pointing to any of its theorems, or even its axioms. It works only by requiring the rule "If not fit world, then false".

What is the proof of that rule? Well, there is none. It is a desire at best.

It a false desire, too. For here is the simple proof that the proposition "Math beyond that which fits the world is invented" is false.

In math, we have the simple proof (using "successor functions") that the natural numbers run 1, 2, 3, ... and so on, all the way up. Never mind about infinity for a moment.

In the world, we have some number of objects, which can be counted as long as one defines a way to count them. Suppose this method of counting exists. Use it to number all the objects in the universe.

It will stop at some number U, for universe. This U cannot be infinite, for if it were, then the universe would be filled with stuff such that nothing could happen. There aren't, to use one example I read one, an infinite number of basketballs. If there were, that's all that we would see, given we could "see" anything when we'd be basketballs.

Infinity is not just a large number. Besides, you can't, as you'll see in a moment, invoke infinity until you first show there are an infinite number of objects.

"What about multiverses and the like, Briggs?"

Show me one. They are all, at this point, only math. And they may even be a clear case of math inventing the natural world! (Or attempting to.)

Anyway, we have U. But since U is tops, we cannot have U + 1. That number wouldn't exist. It wouldn't make answer sense to talk of U + 1 other than as invention. All math involving numbers larger than U would have to be tossed as unworthy, useful only as puzzles, like other inventions.

It's easy to see that this is nuts. Because math isn't invented. It is discovered. And it doesn't have to be that it fits only the natural world. There is also the immaterial world, in which discussions of infinity and sizes of infinities and the like make sense.

Next step is to make this into a Tik Tok.

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