Where can I learn about Mathematical Philosophy?

Question

This is a very vague question, but a question nonetheless. I am becoming increasingly more interested in what can be vaguely categorized as Mathematical Philosophy, or more specifically perhaps, Metamathematics, that is, the study of Mathematics itself! I.e. Theorems about theorems.

Examples of such theorems would be the Incompleteness theorems, Hilbert's tenth problem, the Continuum Hypothesis etc... I don't know what 'branch' of Mathematics to bracket these theorems into but I am sure that those reading this post know exactly what I mean.

My question is, can anybody recommend any books that would introduce me to this 'branch' of Mathematics? I don't mean Mathematical Philosophy for laymen, but a proper introduction to the theory.

Many thanks, Elie.

Answer 1

Peter Smith, who uses this forum and wrote a very accessible but rigorous book on Gödel's incompleteness theorems, has a great annotated bibliography here:

http://www.logicmatters.net/tyl/

If you already know elementary symbolic logic, you can look at the higher-level references. (And Andres, who commented on your post, is a working mathematical logician. We are blessed to have these people active in this community.)

Answer 2

I am becoming increasingly more interested in what can be vaguely categorized as ... Metamathematics, that is, the study of Mathematics itself! I.e. Theorems about theorems. Examples of such theorems would be the Incompleteness theorems, Hilbert's tenth problem, the Continuum Hypothesis etc...

Well, working backwards, the last of these questions, about the Continuum Hypothesis, is a question in set theory. Hilbert's Tenth Problem is a question about whether there is an algorithm to solve a certain class of problems, so is a question in the theory of computation. The incompleteness theorems arise from a general question about a certain class of formalized theories though we use a result about from computability theory to solve answer the question.

So what you are interested in is (not philosophy) but various branches of mathematical logic. As @symplectomorphic kindly notes, I've put together an annotated Study Guide, recommending various book options in the core branches of mathematical logic. You can find the latest version (along with other book notes) at http://www.logicmatters.net/tyl/

Answer 3

I am surprised to see what branches of math relegated to the Philosophy department or Economics department:

• Making Better Group Decisions: Voting, Judgement Aggregation and Fair Division is a course taught by a Philosophy professor at the University of Maryland. In fact... you can see the GTM books (Graduate Texts in Mathematics) behind him in the videos.

  • voting
  • fair division
  • cake-cutting
  • social choice theory

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If you are interested in Mathematical Logic, then maybe you can check out Homotopy Type Theory - as part of the Univalent Foundations Project as organized by Vladimir Voevodsky (Fields Medal 1998?)

The theory is a strange and novel mix of logic and topology and computer science. From the web site:

Homotopy Type Theory refers to a new interpretation of Martin-Löf’s system of intensional, constructive type theory into abstract homotopy theory. Propositional equality is interpreted as homotopy and type isomorphism as homotopy equivalence. Logical constructions in type theory then correspond to homotopy-invariant constructions on spaces, while theorems and even proofs in the logical system inherit a homotopical meaning. As the natural logic of homotopy, constructive type theory is also related to higher category theory as it is used e.g. in the notion of a higher topos.

To me this is a little bit easier to read than Jacob Lurie's Higher Topos Theory. In that case, he is writing to motivate some universal constructions - $\infty$-categories, etc - that appear in Topology.