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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.

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What you are interested in is not philosophy. Any decent book on mathematical logic should cover these topics. A recent example is Fundamentals of Mathematical Logic by Peter G. Hinman. –  Andres Caicedo Sep 3 at 19:20

3 Answers 3

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/

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So what are kind of question that mathematical philosophy states and tries answer to? –  Trismegistos Sep 4 at 8:47
    
"Mathematical philosophy" questions? Or better, questions in the philosophy of mathematics? General questions like: What is the nature of numbers and other mathematical objects? If they are abstract, non-physical things, how can we know anything about them? Are we entitled to suppose that the law of excluded middle applies when we are reasoning about the infinite? More specific questions like: what justifies our selection of certain axioms for set theory? what kind of explanatory power does category theory have? what do we learn from the project of reverse mathematics? –  Peter Smith Sep 4 at 11:17
    
Why reasoning about infinity using law of excluded middle is more problematic then using same law for finite? I mean I know that reasoning about infinite is harder in general but what is specifically wrong about LOEM? –  Trismegistos Sep 4 at 19:14

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.)

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



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.

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This is an interesting comment, but has nothing to do with the question. –  Andres Caicedo Sep 3 at 22:22
    
@AndresCaicedo I am arguing Voting theory and Social Choice theory are a kind of 'Mathematical Philosophy'. Instead of a book, I offered an online course. Hmm... maybe this is not metamathematics ? –  john mangual Sep 3 at 22:26
    
Yes, clearly the title of the question is a misnomer, but the poster has specified the subject they mean. –  Andres Caicedo Sep 3 at 22:31
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The edit is nice. Again, what does it have to do with "Incompleteness theorems, Hilbert's tenth problem, the Continuum Hypothesis,..."? (It is clearly relevant to the study of foundations, of course.) –  Andres Caicedo Sep 3 at 23:00

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