Is set theory the most fundamental aspect of mathematics?

Question

First of all I apologize if this question is too out of topic but it's been bothering me for a while.

From what I've gathered after years of study it seems that set theory is essentially the foundation of modern mathematics.

Now, I've been studying formal logic for a while and it seems to me that logic is even more fundamental than set theory in the sense that we rely on it to prove things about sets. However, from what I've read we also rely on set theory to prove things about logic.

I realize that formal logic is not the same as naïve logic. But if we expect to have a solid foundation for mathematics, shouldn't we have a rigourous approach to the way we reason that is independent from set theory? Is such a thing even possible?

In a nutshell, is it possible to have a reasonable framework for logic that is rigourous and independent from set theory?

I'm sorry if this question is too vague. And maybe a lot of people don't really care as long as everything makes sense. But since modern mathematics is so concerned with consistency and rigour, how do we know logic itself isn't flawed?

I would really appreciate your insight on this topic! Thank you!

PS: Please let me know if this question isn't appropriate for this forum.

Answer 1

I am by no means an expert on the topic but in one sense, no, set theory (built on a foundation of first order logic) is no more fundamental than alternative foundations, such as type theory. In fact, type theory can encode logical operators (e.g., disjunctions and conjunctions) as types themselves, which means it (unlike set theory) can be constructed without a basis of first order logic.

For prior discussions of the most low-level foundations of math, see (Question 1334678) and (Question 121128).

Answer 2

The "from the ground up" approach to mathematics is formal string manipulation. This only requires the absolute bare minimum assumptions (that we can actually manipulate strings of symbols according to explicit rules - this basically amounts to "language is possible"). Basically, we can present first-order logic as simply a finite set of formal rules for manipulating finite strings over a finite alphabet. Expressions of the form "$\sigma$ is a sequence of rule applications yielding $p$ from $q$" are then perfectly meaningful - even computer-checkable!

We can then, if we so desire, in principle import all of mathematics into this framework. Of course this is easier said than done (to put it mildly), but for those skeptical of foundational assumptions (and there is good reason to be) this can constitute a "formalist bullwark:" it guarantees that the mathematics we actually do day-to-day does amount to something, even if one completely rejects the "naive" interpretation of it (as describing actually-existing mathematical objects, whatever that means).

Put another way, we can interpret all of mathematics as a purely symbolic "game," if we so choose.

---

But do we so choose?

The satisfactoriness of a putative foundation of mathematics depends on what we want our foundation to do. The above adopts the stance that what we really want is a framework for interpreting the work mathematicians do which allows it to remain coherent while dispensing with all unnecessary philosophical assumptions. The following objections are, to my mind, the most obvious ones:

• To the extent that one believes that mathematical objects exist in any sense, this approach utterly fails to capture their nature.

• This approach leaves open the question of why some sets of rules for manipulating strings are interesting while others are not. For the Platonist this isn't a problem (some rulesets actually do describe a universe of mathematics), but for the formalist this is a genuine issue. (On the other hand, the Platonist has to defend the idea of "mathematical objects" in the first place, which is ... nontrivial.)

This variability of end goals accounts for a lot of the confusion around what mathematics "really is" at the bottom: different people may have different approaches entirely. At the end of the day, though, one important thing to note is that mathematics is largely philosophy-independent: a Platonist and formalist can work together to prove the same theorems. (Amusingly, they can also each use this situation as a defense of their respective positions!)