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Would it hurt graduate admissions chances if I went to a logic REU or took upper-level philosophy courses? I go to a small school that doesn't offer graduate-level mathematics. What if I applied to masters programs first, would that help?

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I think the only real answer to 'does X demographic do Y?' is 'it depends on the person'.

I've met a lot of pure mathematicians that would look down on applied mathematicians that do NOT have an interest in logic and philosophy, for seeing maths as a simple tool and not seeing the beauty of it.

I've also met a lot of applied mathematicians that think pure mathematicians are wasting their lives on abstract algebraic subjects that are too abstract to be useful.

I'll admit, as a biomedical data scientist, if I were to hire someone, I'd more likely pick the person with an applied statistics course rather than an upper-level philosophy class.

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    Some years ago I collaborated with a mathematician, an algebraist. At a certain point I mentioned that a certain ideal device could be modelled with a circulant matrix, and he replied: "Wow, I've never thought circulant matrices could have practical applications!" :-) Feb 22, 2022 at 10:28
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Quite the opposite, actually, at least for many of us. I have three math degrees and an undergraduate minor in Philosophy. I studied logic formally a couple of times both as an undergraduate and during my masters (IIRC, though it was long, long ago). A course in ethics is good for anyone, but especially so if you want to be an academic.

Formal logic is interesting and needed in math, but even old Aristotelian logic and logic puzzles stimulate the brain in some of the same ways that math does. And even some CS fields depend heavily on logic.

I can't promise that formally studying it would help more than other things, but it won't be a negative.

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