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.

My thoughts: Before ZF we need first order logic, and before first order logic we need classical propositional logic, and before classical propositional logic we need classical implicational propositional logic, and before classical implicational propositional logic (continued) — Dave L. Renfro, Mar 17 '20 at 18:34
we need various fragments of implicational propositional calculus (such as intermediate logics and positive implicational logic), and before various fragments of implicational propositional calculus we need natural language, and before natural language we need $\ldots$ — Dave L. Renfro, Mar 17 '20 at 18:42
@DaveL.Renfro Could you adapt this into an answer please? I don't really understand how propositional logic necessarily requires implicational propositional logic as a foundation. — Jam, Mar 17 '20 at 18:44
@DaveL.Renfro That's not really accurate: we can just introduce FOL ex nihilo as a set of strings and rules for manipulating them. All this requires is our ability to handle strings of symbols appropriately, and of course we can't get more parsimonious than that. — Noah Schweber, Mar 17 '20 at 18:55
Perhaps requires is too strong a word. I'm thinking that one can view propositional logic as a kind of "higher level programming language" of implicational logic where the operators "and", "or", etc. are defined in terms of certain implication-only well-formed formulas. Anyway, my main point is that the OP's question is like a rabbit hole that seemingly has no bottom. — Dave L. Renfro, Mar 17 '20 at 18:56
@DaveL.Renfro But we can already do that with FOL - there's no need to introduce propositional logic at all if we don't want to. — Noah Schweber, Mar 17 '20 at 18:58

score 4 · Answer 1 · answered Mar 17 '20 at 19:16

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!)

even computer-checkable --- This seems to be based on a philosophical view that mathematics is empirical! :) My views are along the lines are expressed here, and basically I just punt on these kinds of questions. — Dave L. Renfro, Mar 17 '20 at 19:29
@DaveL.Renfro I was going for universality rather than empiricism. That is, one might quibble over the notion of what an "explict rule" is ... but the fact that we can implement them unambiguously on a well-behaved device (as opposed to our fuzzy-wuzzy brains) is in my opinion solid evidence of coherence. So really we're (or I'm) leveraging empirical observations to defend idealism. :P — Noah Schweber, Mar 17 '20 at 19:34
To the OP (see their comment below Jam's answer): Unfortunately I don't know much literature on formalism that I actually like. The general response seems to be: "We start with an intuitive idea about what mathematics should be, and look for a formalism which appropriately codifies at least the basic ideas of that in a reasonably clear way." We can then further argue that that intuitive idea is really just a surprisingly successful attempt to understand the "rules" of the physical universe we actually experience. This is satisfying to a certain extent, but I don't think it's wholly satisfying: — Noah Schweber, Mar 17 '20 at 23:18
it raises (without answering) the question of why "anti-physical" mathematical ideas are still coherent, let alone useful in more "physical" mathematics. There is also a more subtle question about the extent of mathematical meaningfulness which remains troublesome. So while in my opinion clearly correct, this only really addresses the boring part of the question. That said, I think some of the texts found here may be of interest, although I doubt they go as far as you want. — Noah Schweber, Mar 17 '20 at 23:21
@Modesto Rosado: Regarding Noah Schweber's "I think some of the texts found here ...", maybe also look at the answer and comments to Where to begin with foundations of mathematics (and note the EDIT part of the question at the end). See also Book on the Rigorous Foundations of Mathematics- Logic and Set Theory AND Highly Rigorous Logic Book AND (continued) — Dave L. Renfro, Mar 18 '20 at 05:54
Foundation of Formal Logic. This last question asks if we can formalize string manipulation (I didn't originally realize this was being asked when I first answered the question). Somewhat related are Is Logic Empirical? AND Can there be a universe with different mathematics? (surprisingly the J.S. Mill stuff I cited elsewhere here is not mentioned). — Dave L. Renfro, Mar 18 '20 at 06:08
surprisingly the J.S. Mill stuff I cited here is not mentioned --- Oops, it actually is. Conifold's answer mentions Mill’s Psychologism, although not explicitly the possible empirical nature of something like $2+2=4$ that is discussed here and in the middle of the left column of p. 842 here. — Dave L. Renfro, Mar 18 '20 at 06:20

Jam · Answer 2 · 2020-03-17T19:08:25.760

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).

edited Mar 17 '20 at 19:08

answered Mar 17 '20 at 19:03

Jam

10,325

@Noah That pretty much answers my question. I consider myself a formalist and when I encountered this set theoretic approach to logic I started to wonder whether there was a way to do it with minimal assumptions. In particular, relying only in a language and a list of rules rules that describe how to manipulate the symbols; like a game as you said. And derive all of mathematics from there. Then the problem reduces to deciding which symbols and what rules we should use. But that depends entirely on what your purpose is. Do you know of any literature that elaborates on these ideas? Thank you. – Modesto Rosado Mar 17 '20 at 23:05
1

@ModestoRosado I think you meant to comment on my answer. I'll respond there, so that Jam doesn't get pinged unnecessarily. – Noah Schweber Mar 17 '20 at 23:12
@Noah Yes, I wanted to do that but I had yet to figure out how to notify another user of my reply. – Modesto Rosado Mar 17 '20 at 23:14

Is set theory the most fundamental aspect of mathematics?

2 Answers2

Linked