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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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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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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Stewart Shapiro
Thinking About Mathematics: The Philosophy of Mathematics
Oxford, England: Oxford University Press (2000)
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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)...
What is Mathematics, Really? by Reuben Hersh, a mathematician. If you haven't read that one, definitely read it. I consider it worldview-changing.