Fake Numbers

the infinitarian conspiracy

There’s a biggest number.

This is an unpopular opinion. There are many plausible-sounding counterarguments, spread by fools who think they are wise, who do not properly understand how mathematics and reality are related. Before we consider them, preliminaries.

“This sentence is false.” If it’s true, then it’s false. If it’s false, then it’s true. In either case one has a contradiction, and one can use a contradiction to prove anything. An effective formalized system of reasoning can’t simply regard statements such as “This statement is false” as true or false, it must regard them as meaningless, or at least ungrammatical, because being able to express such an idea suffices to make a formal system incapable of distinguishing truth from falsehood.

The lesson is that the ability to refer to mathematical concepts is a great power, dangerous and easy to misuse. Even though the English sentence, “This sentence is false” seems logically formalizable, it is not.

Mathematical theorems apply even to things which are imaginary: notice that you cannot imagine two balls and two balls coming together to make four balls. But there are things which you can imagine imagining which you can’t actually imagine. If I try to consider a counterexample to the Riemann hypothesis, since the Riemann hypothesis is probably true, I am performing an act of fake imagination: perceiving something in my mind with clarity insufficient to resolve the details that would explain to me why it cannot exist.

Now that we have an understanding of the rules of reference and why they are necessary, we can discuss the common counterarguments to the existence of the biggest number, and why they fail.

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Q1: “What’s the biggest number plus one, dumbass?”

That’s an illegal question. “The biggest number” is an ungrammatical construction much like “This sentence is true,” or “the least integer which can not be uniquely referenced in fewer than 1000 English words.” You’re performing a nonsense manipulation of symbols and using that to critique a system that you merely fail to understand.

Q2: “Fine. Which number is the biggest?”

I don’t know. As a matter of fact, I can’t know. If I knew for sure the biggest number was (for example) 73, I could add one to 73, resulting in a number biggest than 73, refuting the theory.

This is not all that different from the situation of infinity.

Q3: “Then why believe there’s a biggest number, if you can’t know what it is?”

Minimalism. There’s too many numbers, gotta pare down, get rid of the garbage that doesn’t spark joy.

I won’t get into the motivating technical details, but if there’s a biggest number I think 2^(2^(2^100)) is above it, and therefore a fake number: an underspecified cognitive object with no referent, as in the Riemann hypothesis example earlier.

Q4: “Do you actually believe this?”

Not completely. This is one of the possible philosophies of mathematics I believe could be Literally True, whatever that means in this context. Just imagine I put hedges everywhere in this post, if that makes you feel better.

This position (is one example of an idea) called ultrafinitism.