Reading List for Logic and Philosophy of Math

[Greenwood16]

I'm starting to get really interested in Logic and the Philosophy of Math (rather broadly), so what are some of the most essential (and perhaps most interesting) things that you would recommend reading? I have a pretty strong grasp on the fundamentals of Logic already (Prop Logic, FOL, a bit of Modal Logic, and some fun reading on Godel) and very little formal exposure to the Philosophy of Math. What things (readings or otherwise) would help one prepare for graduate study in either or both of those fields?

Edited April 17, 2014 by Greenwood16

[wildc4t]

Why don't you ask a relevant professor at wherever you're going?

[Establishment]

Sounds like it's time to dive into some mathematical logic, explore the areas of set theory, model theory, recursion theory, and proof theory.

 

For the latter, you can get a nice historical introduction by the classics. Pick up the Collected Papers of Gentzen and read his dissertation, Investigations into Logical Deduction, and his later papers The Consistency of Elementary Number Theory, New Version of the Consistency Proof for Elementary Number Theory, and Provability and Nonprovability of Restricted Transfinite Induction in Elementary Number Theory.

 

This will introduce you to some aspects of proof theory. Some of it may require some prerequisite knowledge of set theory, particularly of ordinals. But you'll be able to branch off from here into the different areas that interest you.

 

For the other areas, I recommend looking at: http://www.logicmatters.net/tyl/ for recommendations. (I'm not deep enough into the other areas to have my own opinions on which materials I prefer over the others.)

 

EDIT: It also depends on what you mean by first-order logic. If you've just done propositional logic and then extended to predicate logic, you'll want to check out some of the "Big Books of Mathematical Logic" listed on that above link. You'll want to explore some of the big meta-logical results of the past century. Completeness, soundness, compactness, etc. I prefer van Dalen's Logic and Structure by a lot, but the main classics seem to be either Mendelson, Kleene, or Boolos and Jeffrey.

Edited April 17, 2014 by Establishment

[maxhgns]

Frege's Foudations of Arithmetic is one of the most important starting points.

[PhiPhiPhi]

The recommended reading list on reddit is pretty decent, you can find it here: http://www.reddit.com/r/philosophy/wiki/readinglist , just look at the particular sections for logic and philosophy of math. It's not very detailed though, and I'm not an expert in those areas and don't really know if it's extensive enough.

Edited April 17, 2014 by PhiPhiPhi

[humean_skeptic]

I'm starting to get really interested in Logic and the Philosophy of Math (rather broadly), so what are some of the most essential (and perhaps most interesting) things that you would recommend reading? I have a pretty strong grasp on the fundamentals of Logic already (Prop Logic, FOL, a bit of Modal Logic, and some fun reading on Godel) and very little formal exposure to the Philosophy of Math. What things (readings or otherwise) would help one prepare for graduate study in either or both of those fields?

 

http://plato.stanford.edu/entries/philosophy-mathematics/

 

I found the Stanford article on Phil. of Math to be a good starting point, and a really nice overview. What's especially useful is the extensive bibliography. 

[Edit_Undo]

A very good introduction to Philosophy of Math: Thinking about Mathematics: The Philosophy of Mathematics by Stewart Shapiro

 

Also, beef up your knowledge on set theory, I would recommend that you start with Naive Set Theory by Paul R. Halmos

 

Once you have this down, look at the Continuum Hypothesis, then you can read Penelope Maddy's Naturalism in Philosophy.

Edited April 17, 2014 by Edit_Undo

[Edit_Undo]

Also, Professor Greg Restall has great videos on advanced logic and non-classical logic (On Vimeo, search Greg Restall)

 

You can also look at Graham Priest's 'non-classical logic'.

Edited April 17, 2014 by Edit_Undo

[Sophist]

A very good way to get into philosophy of math is Mark Balaguer's 'Platonism and Anti-Platonism in mathematics'. If you don't want to read the full book, he has a shortened paper version of it called 'Realism and Anti-Realism in Mathematics.'

For 'essential reading' in modern philosophy of math, there's Paul Benacerraf's two important papers - 'what numbers could not be' and 'mathematical truth.' Those two papers kickstarted a lot of the modern discussion on mathematical ontology.