This is a by product of this recent question, where the concept of ultrafinitism came up. I was under the impression that finitism was just "some ancient philosophical movement" in mathematics, only followed by one or two nowadays, so It sounded like a joke to me.
But then I got curious and, after reading a bit, It seems to me that the only arguments against infinite mathematics that finitists seem to have are that "there are numbers so big that we couldn't computate in a lifetime" or the naive set theory paradoxes. The former doesn't seem like a serious argument, and the latter is not a problem now that mathematics relies on consistent axioms.
Are there some (maybe arguably) good mathematical reason to deny the existence of $\infty$ or is it just a philosophical attitude? The concept of unboundedness seems pretty natural to me, so what could be a reason to avoid it? Does this attitude even make any sense?
In short, why today-finitists have a problem with $\infty$?
Edit: First of all, thank you so much for your answers (and comments), they have been enormously illuminating. 🙂
I didn't know that "finitism vs. infinitism" was such a polemic topic. Now I myself agree this question might look as primarily opinion-based. However, It was not my intention to open a debate about "which posture is better"; I was just meaning to ask about what specific mathematical reason (argued and not-primarily opinion based) do finitists have to reject the "infinitists" use of infinity.
Based on the two excellent answers I've already had (thank you again 🙂 It is my understanding that their main problem with the use of $\infty$ is that it leads to mathematical results (like the Banach-Tarski paradox) which they don't recognize as true when looked through the glasses of our real-world experience.
Final edit: After reading every answer and comment (specially Asaf Karagila's) I've came to the conclussion that there are not strictly mathematical reason to avoid the use of infinite. That my specified question on the last edit has no answer, and that the motivation to stick to finitists or infinitists view of mathematics relies on how much one expects mathematics to describe each one's "real world". As Wildcard's answer is the one that clarifies that matter best to me, I am accepting it. Thank you all again for your answers and comments!
Best Answer
It's not known that modern set theory is consistent; in fact, by the Incompleteness Theorem, we can't ever have a system of axioms that we can prove is consistent. Which means that the only condition we can rely on for determining whether a set of axioms is "right" is whether or not it produces absurd results.
Under $ZFC$, we have different sizes of infinity - there are sets which are larger than the set of natural numbers in a precise sense. We also have a lot of weirdness involving the Axiom of Choice - for example, with the Axiom of Choice, a theorem of Banach and Tarski states that a hollow sphere can be disassembled into five pieces and then reassembled (without stretching, tearing, or otherwise deforming the pieces) into two spheres that are both identical to the first one in both size and shape. But the Axiom of Choice simply states that given a set of sets, we can "choose" one element from each set - which seems intuitively true.
A finitist's perspective on $ZFC$ is often that results like the hierarchy of infinite cardinals and the Banach-Tarski paradox are absurd - that they should count as contradictions, because they patently disagree with the intuitive picture of mathematics. The sensible conclusion is that one of the axioms of $ZFC$ is wrong. Most of them are intuitively obvious, because we can demonstrate them with finite sets - the only one we can't is Infinity, which states that there exists an infinite set. So a finitist's conclusion is to reject the Axiom of Infinity. Without that axiom, $ZFC$ becomes purely finitistic.
Now, many finitists are happy to stop here. But some are bothered by the fact that we still have an infinite collection of natural numbers; the infinite still "exists", in a sense, and gives the opportunity for the above weirdnesses to arise in the same way. So some people (including some mathematicians) subscribe to ultrafinitism and insist that there are only finitely many numbers at all. One ultrafinitist mathematician I know defines the largest integer to be the largest integer that will ever be referenced by humans.
Among mathematicians, ultrafinitists are much rarer than simple finitists. Finitists generally agree with you that "unboundedness" is a natural idea - it's essential, for example, in the definition of a limit. But they would go on to insist that this is just a formalism - that a limit, for example, is just a statement of eventual behavior, involving only finite numbers. So $\infty$ isn't an object, it's just a shorthand. This is (at least to my mind) more mathematically defensible than ultrafinitism.
EDIT: Since a lot of people seem to be having a hard time with my first sentence, I thought I'd clarify. The Incompleteness Theorem states that we cannot have a set of axioms powerful enough to express arithmetic and still be able to prove its consistency within the system. The reason I didn't include this phrase above is because it unnecessarily weakens the point. Any axiom system intended to codify all of mathematics defines the idea of "proof"; if, say, $T$ is intended to underlie all of mathematics, then by "proof" we must mean "proof in $T$". With such a $T$, we can say that $T$ cannot be proven consistent at all; because by Incompleteness, any proof of the consistency of $T$ would not be a proof from inside $T$, but $T$ is supposed to be powerful enough that all proofs are proofs from inside $T$. Thus: we can't ever have a system of axioms for mathematics that we can prove is consistent - full stop.