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I am wondering if there's any field in mathematics that can help philosophers define things or help a philosopher make an argument for something. I am just wondering if there's any mathematics that can be relevant to a philosopher.

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    Mathematical logic. – Mauro ALLEGRANZA Feb 24 at 16:41
  • Outside of philosophy of mathematics? Kant took Euclidean geometry as a model for synthetic a priori knowledge, it still features a lot in epistemology, as does arithmetic. ZFC set theory comes up a lot in any metaphysical talk about infinity, Badiou even took it as a basis of his whole metaphysics. – Conifold Feb 25 at 5:42
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    The foundations of mathematics is impossible to distinguish from metaphysics, and Russell's paradox may be the central problem that all fundamental theories must overcome, but at higher levels I'm not sure there's much in maths that's useful. – PeterJ Feb 25 at 11:00
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    @PeterJ Metaphysics is a claim about how the world is. Mathematical foundations are simply axioms that let you derive theorems, with no claim of absolute truth. Not only CAN we distinguish metaphysics from math foundations, they're not even remotely comparable. They're two different things. You might as well say you can't distinguish metaphysics from the rules of chess. Your statement is as false as false can be. – user4894 Mar 9 at 23:23
  • @user4894 - You might like to read some mathematicians on the topic. (Hermann Weyl, Spencer Brown and Robert Kaplan come to mind) If we cannot solve Russell's paradox then we cannot have a fundamental theory. In formal terms axiomatising set-theory or the number-line is no different from axiomatising the world. There need be no claim to absolute truth, but logic demands mathematically sound axioms. . . , – PeterJ Mar 10 at 12:18
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It depends on the objectives one is trying to fulfill. Normally, the foundations of mathematics like Set Theory and Logic are the branches that people see more related with philosophy.

On the other side, if you study more geometry, like euclidean geometry (as an axiomatic system), analytic geometry etc. you will gain some intuition about the mathematical practice and history. Many examples in logic will be more clear after one understand the practice of mathematics.

Abstract Algebra is a more recent branch, but have a lot of questions that could be interesting. The notions of Group, Ring and Module are very important and some theorems are very cool. In fact, someone could put category theory as a algebraic branch, and a lot of the research in the area is going in a more categorical direction (many of pure mathematics is going in a category like direction today).

The area like Operator Algebras (analysis) could be important if one is trying to understand more about quantum mechanics, and this is important to a philosopher that wants to study the foundations of physics and the material reality. This is my personal interest, for example. I think one don't need to know the same math that a professional mathematician knows (this costs many years in study, maybe a decade), but some basic notions like a Hilbert Space, operators, theorems and some physical interpretation on that.

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I would suggest that right now complexity theory and model theory are the cutting edge of mathematics that bear on important philosophical topics.

See:

On Philosophy and Model Theory: https://global.oup.com/academic/product/philosophy-and-model-theory-9780198790396?cc=us&lang=en&

On Philosophy and Complexity Theory: https://www.scottaaronson.com/papers/philos.pdf

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Logic is probably the most important field to a philosopher. Although other fields of mathematics can also be useful, LOGIC is definitely most important. Just look around this site! Start with propositional logic, then predicate logic.

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    Logic has the most questions even, based on tag – Math Bob Feb 25 at 20:47
  • Logic is to broad though. Do you have specific fields or books in mind? – repomonster Mar 13 at 1:07
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    I would say to first learn about propositional logic, then predicate logic. Try to master both. – Math Bob Mar 13 at 16:13
  • What mathematical field does it belong to? Abstract algebra? – repomonster Mar 13 at 22:10
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    Logic is it’s own mathematical field – Math Bob Mar 14 at 23:17
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Logic is the most obviously related field, since it is still part of philosophy, and philosophy of logic is more-or-less continuous with the formalism of logic itself. However there are many areas of philosophy that draw on other areas, for philosophy of science, a strong statistics/probability background is common to get a feel for how scientists make inferences in practice and how we might improve those inferences. In logic-adjacent fields and philosophy of mathematics, mathematical logic is common (esp. set and topos theory). Other areas draw on other portions of mathematics, philosophy of mathematics often considers algebra, arithmetic, and geometry as three disciplines that provide different instances of how mathematical practice works (algebra is more formalistic on one end, and geometry appears more synthetic). Many of the foundational and philosophically interesting logical results are from the study of axioms of arithmetic, so arithmetic makes an appearance.

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