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What well-written introductory books are there about a mathematical point of view on the philosophy of mathematics and its different school of thought?

By this I mean a book that has some mathematical treatment of the subject, for example it might describe the mathematics of constructivism, rather than just the broad ideas behind it. I'm also looking for an introductory book, as my background in philosophy is quite shallow.

Edit: I'm looking for a book that presents the philosophical ideas alongside the mathematical theories. @oneguy has linked notes about the (mathematical) foundations of constructivism, but they lack a philosophical discussion.

All of what I found about the philosophy of mathematics is very mathematically shallow, or on the other extreme (mathematical texts about logic and set theory). It would be nice to find something in between and less specialized.

The french book « Penser les mathématiques » has a few articles that fall into what I'm looking for. (http://www.amazon.fr/MATHEMATHIQUES-S%C3%A9minaire-philosophie-math%C3%A9matiques-sup%C3%A9rieure/dp/2020060612)

Edit 2: The Stanford Encyclopedia of Philosophy has very good articles on the subject, with a proper mathematical level and references (http://plato.stanford.edu/entries/mathematics-constructive/).

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  • See the answer to this post.
    – Mauro ALLEGRANZA
    Commented Sep 20, 2015 at 19:34
  • Thank you for the link. However, my question is not answered in that post. I also tried google (as is suggested in a comment on that post), and didn't find the answer.
    – Olivier
    Commented Sep 20, 2015 at 19:40
  • Check out "Where mathematics comes from" amazon.com/Where-Mathematics-Come-From-Embodied/dp/0465037712 . Definitely on the mathematics is invented not discovered side.
    – Ray
    Commented Sep 24, 2015 at 18:32

4 Answers 4

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I'd recommend starting with a straightforwardly philosophical introduction like Shapiro's Thinking about Mathematics, then once you have identified a topic or two that you are interested in, you should find primarily mathematical sources to cover the relevant mathematics separately. This is often a good idea, since it's very hard to do (i) excellent math and (ii) excellent philosophy (iii) all in a survey volume.

Good introductions to constructive mathematics would be these notes by Moschovakis, and Martin-Lof's Notes on Constructive Mathematics ((Disclaimer: I haven't read the Moschovakis notes, but I skimmed them and they look good)). If I remember right, the Shapiro has some pretty good mathematical sources included in the recommended reading, so you shouldn't have to do too much work to find mathier sources on your own once you've completed the relevant chapters.

I don't have a strong opinion about whether you should read the math or philosophy first. (I think it might be slightly better to read the math first, but that's really up to you).

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My introduction to the subject comes via Philosophy of Mathematics, Selected Readings edited by Hilary Putnam and Paul Benaceraff. Some may not consider it to be introductory, but it is serving me well so far. It is a compendium of classic papers written by leading figures in the field; both mathematicians and philosophers.

The book is presented in four parts.

Part one deals with the foundations of mathematics and includes papers by Carnap, Heyting(x2), von Neumann, Brouwer(x2), Dummett, Frege, Russell, Hilbert, Curry, and Kreisel.

Part two deals with the existence of mathematical objects and includes papers by Carnap, Bernays, Benaceraff, and Putnam.

Part three deals with mathematical truth and includes papers by Ayer, Quine(x2), Hempel, Poincare, Benaceraff, and Putnam.

The fourth and final part deals with the concept of set and features papers by Godel(x2), Boolos, Parsons, and Wang.

Each paper is typically ten to twenty pages long, with few a bit shorter and a few a bit longer. I purchased it new from Amazon in late May of this year and have (sadly) read just four papers so far. The current Amazon US prices is about $59, but Amazon market place offers used copies for about $20.

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I would point you at (in decreasing order of comprehensibility):

  • George Polya's "How to Solve It" (and/or "Mathematics and Plausible Reasoning") for a basic introduction to the Heuristic process and how mathematics 'feels' to real mathematicians. This is not philosophical, but it is hard to get the motivation to question the subject without getting the 'feel' of it first. If you are already a competent mathematician, this is still not a waste of time. Polya is good.
  • Imre Lakatos' "Proofs and Refutations" for basic questions around the language of mathematics and the establishment of mathematical meaning. This is a lovely piece of work that raises many of the right questions and objections for someone without a deep background, by focussing on a single simple problem and how various schools might approach it.
  • Arend Heyting's "Intuitionism: An introduction" for insight into one historical 'psychological' approach to mathematics, which motivated modern constructivism, and the positions it sees as its opponents. As a dialog, it lets each position speak for itself, filtered some through the prism of Heyting's fanaticism.
  • Errett Bishop's articles about the "Schizophrenia" of modern mathematics, for motivation for pursuing a constructivist approach as a way of preserving meaning in mathematical activity. His diagnosis is good, and he carefully followed up historically by proving that his proposed cure is not crazy.
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    It's nice to see you back. I hope you had a nice holiday. I think what the OP is asking for doesn't really exist. Philosophy is not mathematics and mathematics is not philosophy. It hard to imagine a presentation of, say, Group Theory with a running commentary of the philosophical issues it entails. The philosophical issues of mathematics are those underlying its foundations, the nature of truth, ontological aspects, etc...
    – nwr
    Commented Sep 21, 2015 at 17:20
  • Same here; good examples, especially Heyting.
    – Mozibur Ullah
    Commented Sep 21, 2015 at 17:42
  • @NickR See the book « Penser les mathématiques » which presents issues you might be unaware of.
    – Olivier
    Commented Sep 22, 2015 at 2:23
  • @Olivier Sounds interesting. Is "Penser" the Sharpio text "Thinking about Mathematics". Thanks for the recommendation. I do have a lot to learn in this area and often present ill-informed comments.
    – nwr
    Commented Sep 22, 2015 at 2:28
  • It's not the same text. It's a collective by known french mathematicians and philosophers (Dieudonné, Mandelbrot, ...).
    – Olivier
    Commented Sep 22, 2015 at 2:44
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The following three books are books that I have personally read and loved:

They are based on mathematics with an emphasis in philosophy and are understandable by individuals of all educational levels.

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