Why would you file proof theory under philosophy as opposed to mathematical logic? Proof theory isn't any less mathematical than, say, the integers. You start with a fully formal set of ground rules specifying the system you're studying, and then use a specified mechanism of formal deduction to determine features of that system that are a product of that specification.
You can use proof theory to strengthen/weaken claims in the philosophy of mathematics, like how Godel's incompleteness theorems removed the possibility of a universal logical framework for all possible mathematics. But you're doing the same when you shoot down Aristotle's claims with the physical sciences, so I don't see why you would that would be a reason to file it under philosophy.
If you are going to exclude anything that could be considered mathematics or science from the domain of philosophy, then trivially, you will find that philosophy concerns things which can't be precisely quantified or empirically proven.
Proof theory does, however, originate from the work of philosophers (Aristotle being a salient example), and issues relating to proof theory are central to epistemology and metaphysics.
One problem that philosophy has is that its most conspicuously successful branches often become disciplines in their own right, so that philosophy doesn't get any credit.
There's a difference between excluding them from the domain of philosophy, and co-opting them as developments of philosophy. The distinction here is that the methods used to develop proof theory look exceedingly similar to how mathematics is developed, and quite alien to the methods of philosophy both of that time and now. Frege didn't develop his proof theory by spending years doing literary reviews of previous philosophers, or by writing papers and books based on suppositions.
You're right that proof theory, like mathematics and science, branched off from philosophy some time in the past. But looking at the successes of these branches as evidence that you should read old philosophy or engage in the mainstream methods of contemporary philosophy when their methods are worlds apart does not make much sense.
>The distinction here is that the methods used to develop proof theory look exceedingly similar to how mathematics is developed, and quite alien to the methods of philosophy both of that time and now.
Logical reasoning has always been one of the methods of philosophy. It seems quite bizarre to say that the development of proof theory has nothing to do with philosophy, which for millennia prior had been interested in the question of what distinguishes good arguments from bad arguments and which arguments are formally valid.
Also, proof theory does not look that much like the rest of math. It certainly doesn't look much like the math that people were doing in Frege's time. So I don't really buy the argument that proof theory is distinctively "mathematical" as opposed to "philosophical" (to the extent that these terms are meaningful in the first place).
>Frege didn't develop his proof theory by spending years doing literary reviews of previous philosophers,
I'm not exactly sure what "doing literary reviews" is supposed to refer to, or why you think that this is more characteristic of philosophy than logical argumentation, but Frege certainly read previous philosophers. To a significant extent Frege's work is a reaction to the limitations of Aristotelean logic. And it's pretty clear from reading Aristotle that he didn't think of the study of logic as a branch of mathematics.
>But looking at the successes of these branches as evidence that you should read old philosophy or engage in the mainstream methods of contemporary philosophy when their methods are worlds apart does not make much sense.
Could you elaborate? I'm not sure which methods you're referring to.
> Logical reasoning has always been one of the methods of philosophy. It seems quite bizarre to say that the development of proof theory has nothing to do with philosophy, which for millennia prior had been interested in the question of what distinguishes good arguments from bad arguments and which arguments are formally valid.
Logical reasoning in the form it takes in the philosophical works of any of the philosophers you mentioned and the kind done in mathematics are worlds apart. Comparing Spinoza's "proofs" which are supposed to follow the style of geometric proofs with actual geometric proofs reveals a pretty stark contrast. More generally, philosophers rarely phrase their logic in a formal language (although they do on occasion), while mathematics has been done only in a formal logic since the advent of proper axiomatic foundations in the early 20th century. Informal reasoning is often used to assist formal arguments at an intuitive level, but no mathematician takes it seriously on its own.
How many philosophical works do you see start with a list of fully precise axioms and then continue only use those as assumptions, making their uses clear? There are a few I can think of, but they are far from the norm.
> Also, proof theory does not look that much like the rest of math.
How so? Even early results like Godel's incompleteness theorems look strikingly like other mathematical proofs. He lays out the precise definition of a formal system, and even sets up an encoding for sentences of that language in the natural numbers. A significant part of it is also pretty much straight out of Cantor's diagonal argument.
More modern proof theory looks even more like conventional mathematics, with a lot of it being phrased in the language of category theory.
> And it's pretty clear from reading Aristotle that he didn't think of the study of logic as a branch of mathematics.
Yes, because the mathematics of Aristotle's time was not equipped to handle something like proof theory. Frege (as well as the others you listed) at the very least thought the right way to study logic was through mathematics.
> Could you elaborate? I'm not sure which methods you're referring to.
Okay, pick 10 papers at random from a philosophy journal of your choice. How many of them are yet another examination of the one of the works of "old philosophy" the original article was imploring us to read, and how many are exploring something new? I don't mean "something new" as in standing completely alone and without any references to past works (I've yet to see such a paper in math or science), but simply drawing conclusions that are something other than a response to an age old argument. And as I said before, how many use formal logic as opposed to informal logic?
>More generally, philosophers rarely phrase their logic in a formal language (although they do on occasion), while mathematics has been done only in a formal logic since the advent of proper axiomatic foundations in the early 20th century.
I don't see any distinction between the two disciplines here. Philosophers make use of formal logic when it makes sense to do so (lots of discussion of the ontological argument makes uses of formal logic, for example [1]). Mathematicians follow exactly the same strategy. It's quite false to say that mathematics is now done only in a formal logic. The vast majority of mathematical proofs are presented informally (even in textbooks on logic!)
>How so? Even early results like Godel's incompleteness theorems look strikingly like other mathematical proofs.
I guess the best example is Frege, whose work was ignored by mathematicians and logicians alike because it looked completely alien. But I was thinking more of, say, proofs of completeness for systems of propositional logic, which are highly notation-dependent and consist in running through a bunch of cases in a boring mechanical way. Peirce and Gauss seem worlds apart.
> Frege (as well as the others you listed) at the very least thought the right way to study logic was through mathematics.
I'd say he thought that the right way to study mathematics was through logic. In other words, he was trying to put mathematics on proper logical foundations. (This is the project that Russell tried, and essentially failed, to complete.)
>how many are exploring something new? I don't mean "something new" as in standing completely alone and without any references to past works (I've yet to see such a paper in math or science), but simply drawing conclusions that are something other than a response to an age old argument. And as I said before, how many use formal logic as opposed to informal logic?
Most conclusions in philosophy are responses to age old arguments for the very simple and obvious reason that people have been thinking about philosophical problems for a long time. Occasionally people do come up with new problems, of course. The philosophy of language would be one example where lots of genuinely new questions have arisen and been given interesting answers (partly as a result of developments in mathematical logic).
It's quite common for philosophy papers to use formal logic, but only when it is helpful. In exactly the same way, mathematical proofs are presented formally on occasion, when it is helpful, but not as a matter of course.
You can use proof theory to strengthen/weaken claims in the philosophy of mathematics, like how Godel's incompleteness theorems removed the possibility of a universal logical framework for all possible mathematics. But you're doing the same when you shoot down Aristotle's claims with the physical sciences, so I don't see why you would that would be a reason to file it under philosophy.