What can we gain by losing infinity?

As someone who has done a whole lot of IEEE 754 floating point math:

> computers handle math just fine

strong disagree tbh

Last year I made the mistake of asking ChatGPT what the world would look like if `∞ === -∞` and it took me seriously (I think) and led me on an hours-long dance where in the end it had me trying to prove, mathematically, that `2 > 1` ... and it was at that point I realised that I'm not cut out to think in numbers and maybe it was for the best that I failed my end-of-school Maths exam

> However, an axiom of infinity is independent, it doesn’t contradict anything in standard formalizations, and so it doesn’t make sense to say “infinity is wrong”.

Suppose we start with ZFC - Infinity as our base system. Then the negation of Infinity is consistent with this system. But adding Infinity itself makes the system strictly stronger, since ZFC proves the consistency of ZFC - Inf: in particular, in ZFC, we cannot prove that Infinity is consistent with ZFC - Inf.

In other words, in principle, it might be the case that ZFC - Inf is consistent, yet ZFC itself has a contradiction. In practice, most people believe that ZFC is also consistent, but we have no way to prove it a priori without accepting even more new axioms.

I don't think it's bad writing. These people actually get angry at the idea that other people do math that might not connect to the real world. And they specially have it out for infinity.

I say do whatever math you like. It is helpful to know what math you are doing. For instance, while I don't have a "problem" with the Axiom of Choice per se I do like clean specifications of when we are using it and when we are not, because it is another example of when we detach from reality as we know it. I don't have a problem with detaching from reality as we know it, I just like there to be awareness that we have.

But plenty of math is detached from reality. Honestly we don't observe very many "mathematical entities" at all; I've never seen a graph. I've never seen hyperbolic space. I'm aware of the many places aspects of them seem to map to reality, but I've never actually seen a literal graph in the real world.

Personally I am reminded of the way that we model our computers with Turing Complete formalisms, despite the fact they are observably not Turing Complete and are technically just finite state machines. However, the observation that they are "just" finite state machines doesn't move us closer to an understanding of how our computers work, it moves us farther away. Even though computers are completely real-world phenomena, if you want to understand the issues raised by things like Turing Incompleteness and other such things in the real world, you're going to be exponentially better off using Turing Machine formalisms and simply noting that you may run out of memory or practically-available computational resources before a calculation can complete than trying to build a new set of formalisms around finite state machines. We can be in an engineering context where we are well aware of the finite nature of everything we are doing because it all comes back to real, physical machines, but it's still easier to model with infinity than without it.

In that context, the real utility of "infinity" is less "an infinite number of things" than "you will never reach for another X [byte of RAM, byte of disk, CPU cycle, incrementing counter, etc.] and be told you're out of resources". Basically we write our proofs, formal or informal, as ignoring "what if I reach for this resource and it's not there?" for every such resource and every time we reach for a resource, which is quite often. You could go through a system and add a "what if" check for every such instance, but it's way cheaper to just buy another stick of RAM or tweak the program to take fewer resources than it is to try to deal with the exponential-with-a-large-exponent explosion of states this causes mathematically.

saying infinity is a mathematical symbol we can observe is simplifying it way too much, all mathematical symbols are abstractions.

i can observe two apples. i cannot observe infinity apples.