[ithkuil]

What about negative numbers? Or complex numbers? They are only tools, which can be quite useful to build models of the world with predictive powers but shouldn't be confused for the underlying reality.

Even whole numbers are an abstraction that makes sense only when you can clearly define what is the thing you're counting.

[tsimionescu]

Whole numbers can be defined and proven to be necessary to describe the world pretty easily. From there, rational numbers are trivial to define. Negative numbers are somewhat more abstract, but they have very intuitive definitions in many domains, such as accounting. It may be possible to avoid them in a theory of physics, though.

The complex numbers (well, at least those with a rational imaginary part and a rational real part) have been recently proven to be necessary to describe the universe[0] (assuming quantum theory is correct).

The irrational numbers are then are the only numbers that are harder to pin down, and I'm not sure that there is a way to prove that any physical quantity has an irrational value, vs a rational value that is arbitrarily close to that irrational value.

[0] https://arxiv.org/abs/2101.10873

[ithkuil]

Infinity may be also "necessary to describe the world". But like every tool, you need to know its limits.

[tsimionescu]

I'm not sure that it could be, actually. You can't use a finite amount of evidence to verify that something is infinite, so any infinity can always be replaced with a huge (or minuscule) number and the theory would make the same measurable predictions.

[ithkuil]

I'm not talking about proving that there exist infinite things.

I'm talking about using the abstract concepts of infinity as a useful mathematical tool to produce predictions. Notable example: calculus

[tsimionescu]

Sure, but calculus makes infinity sufficient, but not necessary for describing the physical world. Integers, rationals, and apparently complex numbers (presumably those with rational components) are actually necessary for describing the physical world, given our current understanding. Irrational numbers and infinities are extremely useful, but not strictly necessary.

[ithkuil]

Sorry, I'm not getting it. A large use case for complex numbers is describing things that rotate, literally or not, like oscillations, waves etc. Trigonometry lies deeply in that math and the irrational number pi pops out left and right. An approximation of pi wouldn't cut it, would it?