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I'm interested in books on Philosophy of Math written by Mathematicians. Preferably more recent writings. I consider that philosophy in a field should only be done by people that work on the field.

I saw here questions about good references on Philosophy of Math, but most books were mainly written by philosophers with non-mathematicians in mind. I'd like suggestions on books of "Philosophy of Math for Mathematicians".

The only book I've read on the topic was "Proofs and Refutations" by Lakatos.

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    $\begingroup$ Of possible interest is Thomas Tymoczko (editor), New Directions in the Philosophy of Mathematics (1986, 1998). I haven't seen the 1998 revised and expanded edition (I got the 1986 edition from, I think, a science book club purchase back in 1986 or 1987), but the 1986 edition seems well worth a look. Each of the separately-authored chapters has its own list of references/notes, which will give you a lot of literature leads. $\endgroup$
    – Dave L. Renfro
    Apr 5, 2021 at 17:55
  • $\begingroup$ The autobiography (two volumes) of Bertrand Russel is a great read! $\endgroup$
    – J Miller
    Dec 18, 2021 at 22:34

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An old one is Edmund Husserl's Philosophy of Arithmetic: Psychological and Logical Investigations.

A more recent one is Morris Kline's Mathematics: The Loss of Certainty, a book that I enjoyed a lot.

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  • $\begingroup$ Anything even more recent? I'd think that advances in proofs performed by computers might impact how we percieve the philosophy of math, hence why I'm interested in more modern takes. $\endgroup$
    – Davi Barreira
    Apr 3, 2021 at 22:19
  • $\begingroup$ Right now, I cannot think of any such book. $\endgroup$
    – José Carlos Santos
    Apr 4, 2021 at 9:46
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    $\begingroup$ The book by Kline is GREAT! It haunted me for a while. I need to read it again because those ghosts are cool. $\endgroup$
    – J Miller
    Dec 18, 2021 at 22:32
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I really liked the other references given in the other answers, and I'll probably read both Hersh and Kline. But, I just found out about the very recent book "Lectures on the Philosophy of Mathematics" by Joel David Hamkins (2 feb 2021).

The preface reads:

This book is an introduction to the philosophy of mathematics, in which we shall consider all these questions and more. I come to the subject from mathematics, and I have strived in this book for what I hope will be a fresh approach to the philosophy of mathematics—one grounded in mathematics, motivated by mathematical inquiry or mathematical practice".

So this is exactly the sort of thing I was looking for! Hence, I'm posting here, since other people might be interested.

Of course, if anyone has any more suggestions, please post another answer. I'm always looking for awesome books on mathematics.

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What is Mathematics, Really? by Reuben Hersh, a mathematician. If you haven't read that one, definitely read it. I consider it worldview-changing.

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    $\begingroup$ +1 for Hersch, in general. His book The Mathematical Experience with Davis is a classic. $\endgroup$
    – J Miller
    Dec 18, 2021 at 22:33
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Stewart Shapiro
Thinking About Mathematics: The Philosophy of Mathematics

Oxford, England: Oxford University Press (2000)

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  • $\begingroup$ Thanks for the suggestion, Dario. Is Stewart a mathematician? I was under the impression that he wasn't. $\endgroup$
    – Davi Barreira
    Dec 19, 2021 at 0:20
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Two books by Hao Wang:

From Mathematics to Philosophy, Routledge Revivals;

A Logical Journey: From Gödel to Philosophy ,MIT Press.

Wang is a logician as well as a philosopher. More importantly, these two books convey the ideas of Gödel, one of the most important mathematicians and logicians in history.

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Under some interpretation of the question, I would recommend:

Hadamard, Jacques. An essay on the psychology of invention in the mathematical field. Courier Corporation, 1954. Link (no paywall).

(The inspiration of that book was a chapter by Poincare.)

I also have on hand Philosophies of Mathematics by Velleman (and George)...

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