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I want to buy a philosophy of mathematics book. I have three options in mind: Philosophy of Mathematics by Øystein Linnebo, Philosophy of Mathematics: Selected Readings by Paul Benacerraf, or The Oxford Handbook of Philosophy of Mathematics and Logic by Stewart Shapiro. My main interests are mathematical platonism and logicism. Do these books cover such topics? If they do, which one, in your opinion, does it better?

P.S. Feel free to suggest any other text about Philosophy of Mathematics

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    Philosophy of Mathematics is a vast subject and there are many books about it out there. How can we suggest which is "better" if you do not say anything about what you are interested in and what you want to use the book for? – Conifold Sep 11 '20 at 20:31
  • You are right. I am going to make that clear. I am mostly interested in logicism and mathematical platonism. – user926356 Sep 11 '20 at 22:02
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    There is a difference between Mathematical Philosophy and Philosophy of Mathematics. The former being identified mainly by its method, the latter by its matter. An example of mathematical philosophy is the famous attempt of Spinoza to geometrize the philosophy of Descartes or his yet more famous attempt to construct the Ethics after the manner of Euclid. jstor.org/stable/2973657?seq=1#metadata_info_tab_contents – user47436 Sep 12 '20 at 6:10
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    The philosophy of mathematics, is primarily concerned with such unanswered questions as arise in reflecting upon the nature of mathematics and the character of its foundation. jstor.org/stable/2973657?seq=1#metadata_info_tab_contents – user47436 Sep 12 '20 at 6:12
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    Plato : Knowledge= Intersection of Beliefs and Truths slideshare.net/mobile/ReyJohnRebucas/philosophy-of-mathematics – user47436 Sep 12 '20 at 6:27
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I shall suggest that you first make sure that you fully understand basic FOL (first-order logic) and the technical details and proofs of some of the crucial theorems about it up to and including the semantic completeness of FOL and the Godel-Rosser incompleteness theorems. The reason for this is that it is impossible to have a proper discussion about the philosophy of mathematics unless you first have a proper understanding of mathematical logic itself. Numerous people make nonsensical claims about philosophy of mathematics due to a lack of understanding of such fundamental facts, and for you to know which claims are based on facts and which are based on falsehoods, you yourself need to have a solid grasp of the facts.

Furthermore, most of the true experts in foundations of mathematics write articles that expect readers to know FOL up to the theorems I stated, and often more, simply because that is the bare minimum needed to understand their views and ideas, not to say engage with them. For example, see this post about building blocks in mathematics and a brief sketch of the philosophical assumptions inherent in them, and also the related comment there citing Peter Smith's explanation of why ACA is the natural stopping point for predicative mathematics. For another example, take a look at the paper by George Boolos on the iterative conception linked from this post about whether the set-theoretic universe as given by ZFC is fixed.

Here are two examples showing that a firm foundation in mathematical logic is indispensable if you do not want to be fooled by the numerous erroneous popular accounts of various concepts in foundations of mathematics. Firstly, there is a very popular misconception that one can evade the incompleteness theorem by switching to a different foundational system. Sorry, that's false. See this post for a more or less complete explanation of the generalized incompleteness theorem that applies to every possible foundational system for mathematics both now and in the future. Take note that popular accounts of the incompleteness theorem are almost all wrong or extremely misleading, and especially read the section "Popular misconceptions about incompleteness" in that linked post.

Secondly, there is another very popular notion that mathematical platonism (every mathematical object has a platonic existence) is well-defined. Sorry, that's false too. Due to the incompleteness theorem, we know that current foundations for mathematics is either inconsistent (useless) or incomplete (does not prove every true sentence about natural numbers), and so cannot pin down a unique mathematical universe. Thus the popular definition of mathematical platonism is ill-defined, because "mathematical object" is ill-defined without reference to a specific foundational system. One can amend it to "there is a real-world interpretation of ZFC", but nobody can justify such a statement. So what then? So far all the empirical evidence that mathematics is relevant to the real world only justifies that there seems to be a real-world interpretation (maybe even merely approximately) of some very very tiny fragment of ZFC, maybe ACA or ATR or HOA (higher-order arithmetic). People who attempt to use the undisputed success of applied mathematics to justify all of modern mathematics are just wrong.

People who claim that every possible axiomatic system for mathematics have a platonic model, are even more wrong. Again by the incompleteness theorem, if we PA is consistent then PA+¬Con(PA) is also consistent, but it proves itself inconsistent! So obviously some FOL theories are simply utterly false! Worse still, there are people who claim that every mathematical structure exists. This claim is not even wrong because it is ill-defined in itself. There is no way anyone can define what mathematical structure means without reference to some formal system or some class of formal systems, and the mere definition of "formal system" needs to be based on a rudimentary amount of assumptions about finite strings.

I shall not even go into the huge confusion that pervades discussions about the notion of predicativity; the definition you can find on literally every website is completely useless. (You may have noticed that one of the linked articles I gave above is about predicativity. Indeed, that author does not suffer from the same confusion.)

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  • I will start reading Symbolic Logic by Irving M. Copi this week. It is a good book for learning FOL? – user926356 Sep 13 '20 at 2:47
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    @MarceloCoto: Unfortunately, that book is a terrible book. In my experience, it is responsible for making many students have bad misconceptions about basic FOL. I gave some suggested resources here, but if you don't have mathematical background then it may be best if you start with learning a Fitch-style deductive system for FOL that I recommend, and you are welcome to ask me in this chat-room on anything related to basic logic. – user21820 Sep 13 '20 at 3:14
  • When you say it's terrible, are you referring to the way the author presents the subject? Also, thank you for the suggestions! – user926356 Sep 13 '20 at 3:19
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    @MarceloCoto: I am unable to comment on the exact details of that book, but it's unfortunately the case that most students who used it to learn logic come to me with bad misconceptions that they attributed to that book. So I've no choice but to recommend against it. The approach I recommend is not conventional, but in my experience only students who fully understand a deductive system for FOL can have a proper grasp of logic. And the only way one can become familiar with FOL deduction is by doing it. Find me in chat if you'd like to start working on it! =) – user21820 Sep 13 '20 at 6:38

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