Posted this a little too early in the morning.

I was thinking about how in science if something can't be observed then we say it doesn't exist, but in math even if we can't reason about it, somehow it does still exist. https://twitter.com/Tuplet/status/1388094268942729221

I'm a fictionalist and an intuitionist when it comes to mathematics. The fictionalist means I don't think mathematial things like pi are real, they're just stories we tell. The intuitionist part means I don't think pi is a particularly good story.

I'm not quite a finitist, but that's a good place to start describing the other things because a finitist is both and also doesn't believe in infinity.

A finitist would even scoff at really big numbers. What good is a googol if there aren't even that many things in the universe? If you can't count that high before the end of time. If you can't do that many things?

Importantly, you could only hope to actually *name* a very small fraction of all the numbers between 0 and a googol by writing them down or writing a computer program to compute some of them, or even come up with a proof that some other unimaginably big object has that many parts

But the vast majority will never be used or named. How can they exist? How can they be real? How can we even say they are mathematics? So, a finitist rejects them.

I don't go that far.

I think if I can come up with a procedure that would eventually list the number I need or described how to build all the parts of an unimaginably complex object then I have somehow effectively captured everything there is to know about that object.

So I'm OK with saying that anything you can construct using a computer program, or your mind, is mathematics. That is, it is something which I think is a sensible subject to teach as mathematics the same way we teach about thermodynamics and not phlogiston.

So, even I believe that (some) numbers "exist" even if nobody has ever actually conjured them into existence yet by actually using them. How does that make me different than most people who do math?

Basically I do not believe in actual infinity and I do not believe that if you prove something is not not true that the "nots" cancel out and the thing is true.

You have to actually do something to prove it's true. Not merely that it's impossible that you can't do it.

So like, sure, you *could* count to a googol. Or you could just write it down (I just did!). Or you could invent any number of ways to write down even bigger numbers. And all this is mathematics.

But all the numbers you never wrote down only exist in the same way that you know Peter Parker's high school must have hundreds of students but we only know like, Gwen Stacy and Jessica Jones.

We infer their existence because we know how High Schools work.

So now maybe you can guess why I think pi is not a good character in the story of mathematics. It is because it is a number with an infinite number of digits. An actual infinity.

Like, I'm fine that we can compute pi or that you've memorized many of it's digits, but nobody could start writing down pi and ever finish (well, you can write down just "pi" but there's a difference!)

I can write a computer program that relies on any number of digits of pi I want, or I can write a program that does algebra with the name "pi" and spits out a answer like 13pi.

What I reject is that pi is anything more than that.

An algorithm.
A symbol.
A story we tell about circles.

So, are mathematicians (or myself) delusional when we talk about actual infinity or actual pi?

Kind of.

Having a replicator that can make anything you want is kind of the same thing as having everything isn't it?

If you had a machine that could make you anything you could describe and would never run out before you died then you could come up with a set of rules for reasoning about the things you "have" as if you actually had them.

Constantly reminding yourself that what you actually have is a replicator would actually be a hinderance to the elegance and simplicity of your system!

But this isn't the whole problem. As an intuitionist I could accept this. It's perfect actually.

The problem is what happens when you can't ask the machine for something because you can't describe it.

You insist it must exist because somehow you've shown that it can't not exist, but that does the replicator no good at all.

Do you really "have" that thing?

When we ask the machine for actual "pi" or actual "infinity" what we really get is a toy that behaves like what we've reasoned "pi" or "infinity" to be (we can describe those things) and then we play with that model.

You might have caught that I'm only asserting that mathematics *should* be like the replicator, and rightfully protest that it is something more, but that's not true, because we actually have the replicator already and the replicator is *our own minds* or *a computer*

No, John. You are the replicators.

I started this thread because I was thinking about this principle in science that something is "real" if it is needed to describe a phenomenon. i.e., you cannot account for certain facts completely unless your description accounts for the "real" thing in some way.