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I am neither a philosopher nor a mathematician but I am assuming that philosophers don't know a lot of math. So, does that keep them from doing similar research in mathematical logic as mathematicians do or are they just interested in the philosophical aspects? And if they do research at that same level, how come a lack of math knowledge does not cause problems in their research (especially in subjects like Model Theory which I've been told requires a lot of math knowledge)?

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  • Quite few: Putnam, Quine. Dec 9, 2023 at 9:56
  • Classical philosophers were often experts in many academic areas, which was presumably easier at the time as there were not that many academic books one needed to know. Contemporary philosophers typically remain within the confines of the non-scientific areas like hermeneutics, ethics. Else they become mathematicians, biologists, etc, as today much more lifetime can be invested into deeply learning a scientific area.
    – tkruse
    Dec 9, 2023 at 10:08
  • Most of the interesting work in logic is done by computer scientists and philosophers, not mathematicians. Dec 9, 2023 at 12:03
  • @DavidGudeman That is what I am asking. How come a lack of mathematical knowledge does not keep philosophers back?
    – user56417
    Dec 9, 2023 at 12:16
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    If a philosopher has an advanced degree in mathematics obtained as a result of learning and practicing advanced mathematical logic, and publishes in a mathematics journal because her philosophy involves advanced research in mathematical logic and philosophy journals provide poor options for peer-review and an audience which will understand the articles, does she cease to be a philosopher?
    – g s
    Dec 9, 2023 at 16:55

2 Answers 2

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Some do, but of course not all. For a prime example, see Hamkins and Loewe[05], which is well-informed by philosophical work on modal logic (they show that the modal logic of forcing, such as they work it out, is of the type S4.2). Or see about Boolos' work on plural quantification (Boolos has a peculiar intuition about the usefulness of alephs beyond the first fixed point of the aleph function, but never mind that for the purposes of this answer to your question). Quine's New Foundations (NF) theory is also a deep analysis of various problems in mathematical logic (see also about his system named Mathematical Logic (ML), which differs from NF in having proper classes as well as sets). NF and its offspring NFU (which has ur-elements, which are elements but don't have elements, not even in the sense of being empty sets) oddly doesn't prove the existence of ℶω without an add-on, and then presumably requires more to be added on to go on to prove the existence of ℶω (see another section of the same Wikipedia article from the preceding link).

Historically, philosophical logicians such as Frege and Russell were pathbreakers in the arena of mathematical logic and set theory. Or check out the history of the Lvov-Warsaw school, unfortunately desolated by the Nazi ravage in Poland but definitely a solid mix of philosophy and mathematics modulo research into logic.

Another zone in which occupants of the vague overlap between mathematics and logic appear, is the theory of truth values. Paoli[20] is an example I am particularly fond of. Whether Paoli counts himself more as a philosophical logician than a mathematical one, I don't know, but I do know that the Stanford Encyclopedia of Philosophy has a detailed article on truth values. Indeed, the SEP has many such articles on logic; see e.g. Humberstone[10] or McNamara and Ven De Putte[21].

For some other philosophically vibrant takes on logic, see also Moretti[09], the work of Jean-Yves Béziau, and (much more clearly on the mathematician's side of things) the work of Jean-Yves Girard.

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A minor addendum to Kristian Berry's answer:

I agree with him if the area of study is logic theory but not if the area of study is anything like complex number theory, representation theory, the theory of complex spaces, group theory, noneuclidean geometry, and the like. To play in those particular sandboxes requires complete mastery of mathematical abstractions, so you can accomplish something more than simply sifting cat turds out of the sand.

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