[nextaccountic]

> Infinity is a mathematical symbol we can observe.

This is like confusing the map for the territory.

Symbols live in syntax (like the syntax of programming languages), while mathematical concepts live in semantics. Infinity is not a symbol, it's not ∞. ∞ is the symbol we use to represent infinity.

[IsTom]

There is a way to look at mathematics as just a bunch of rewrite rules for things on paper. It might not be particularly inspiring, but it's a valid way to look at things.

[nextaccountic]

Indeed, there's a way to get a semantics for free, based on the syntax alone. For example, in the first order logic this is the Herbrand interpretation

https://en.wikipedia.org/wiki/Herbrand_interpretation

The point of mathematical semantics is that for any given theory, we can have other interpretations that don't just interpret symbols as themselves.

So we could conceivably imagine an interpretation where ∞ doesn't just mean literally ∞ and nothing more.

[looneysquash]

The number 42 is also a mathematical symbol we can observe. (Or two symbols, depending on how you want to define symbol).

You can observe the symbol. You can observe 42 of some object, 42 sheep for example.

You can observe a pie chart, or an actual pie, with 42% of it missing.

You can observe a plank of wood that is 42 inches or centimeters long.

But you can't observe 42 itself.

It is not like a hill on a map, where there is a symbol, and there's an actual hill.

It is an adjective and not a noun. It's not real unless it is describing something else.

My point being that regular finite numbers are not real either. So what's wrong with infinity? Or the square root of 2, or pi?

[Avshalom]

Well importantly like scrubs points out in a sibling comment

42+1 = 43, 42 + 1 ≠ 42, ∞ + 1 = ∞

Infinity plays by very different rules than numbers.

[nextaccountic]

I actually agree, there's nothing wrong with infinity. I think finitists are silly and ultrafinitists are ultra silly.