Reddit mentions of Philosophy of Mathematics 2ed: Selected Readings

Depends what you mean by “Mathematical Philosophy.”

Intro to Mathematical Philosophy is kind of an abridged version of Principia Mathematica. Russell is attempting to derive mathematics from basic logical principles.

Many of the principles you read about in this book are covered in the first couple weeks of a Real Analysis class, though, Russell definitely has his own style. It reads less like a textbook and more like Russell giving a lecture. Knowledge of Analysis would help. Though I don’t think it’s absolutely necessary. If you have a fair bit of math knowledge, and you go slow, I think you’ll be fine.


If you’re interested in the Philosophy of Mathematics, I would highly recommend Philosophy of Mathematics by Paul Benacerraf and Hilary Putnam. It is an anthology of different writings on the philosophy of math and even includes portions of Russell’s intro to mathematical philosophy.

/u/LeeHyori nailed it as to where you should start. After that, follow a philosophy of mathematics course. If you can't find one, you won't go wrong by reading through Benacerraf and Putnam's antology which manages better than almost any academic anthology I know to select all/only the heaviest-hitting papers in the field, stopping about 1-2 generations ago.

You should also pump your technical background in logic, set theory, model theory, category theory and the like. Logic and set theory are absolutely essential: tackle these sooner rather than later. The rest can wait a bit longer.

Following this up I would recommend "Empiricism, Semantics, and Ontology" by Carnap, "Critique of Pure Reason" by Kant, "Philosophy of Mathematics" compiled by Putnam and Benacerraf, "Philosophy of Mathematics: Structure and Ontology" by Shapiro, "Mathematics in Kant’s Critical Philosophy: Reflections on Mathematical Practice" by Shabel, and "On the Infinite" by Hilbert.

Also I would recommend looking into the lesser works of Shapiro, Shabel, and Yablo

Edit: I forgot to mention that Aristotle and Badiou also have writings on mathematics.

This anthology is also a popular choice for use in courses on the philosophy of math. There's quite a bit of overlap with certain areas of philosophy of science, too, so you may want to look into things like the interpretations of probability and model theory. Eugene Wigner's paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is also a fun piece bridging the two fields, and raises some great questions that straddle both areas.

http://www.amazon.com/Philosophy-Mathematics-Selected-Paul-Benacerraf/dp/052129648X

This is a collection of important papers on the subject compiled by Putnam and Benacerraf (perhaps the two most prominent philosophers of mathematics alive today... though they are both retired). It's longer than what you want, but you can select and read any or all of the articles that interest you.

To understand the general history of math, you won't need to understand what you most likely consider to be math. You will, however, need to understand how to put yourself in the shoes of those who came before and see the problems as they saw them, which is a rather different kind of thinking.

But anyway, the history of math is long and complicated. It would take years to understand everything and much of it was work done on paths that are now basically dead ends. Nevertheless, here are some other resources:

• In Pursuit of the Unknown: 17 Equations That Changed the World - In general, Ian Stewart is very good for introductory explanations to math.
  
  
• Gödel, Escher, Bach: An Eternal Golden Braid - Perhaps the only explanation of Gödel's work that I would consider "good." It's notoriously lengthy and can be difficult to read, but it doesn't assume much about the reader.
  
  
• Thinking about Mathematics: The Philosophy of Mathematics
  
  
• Philosophy of Mathematics: Selected Readings
  
  
• Stanford Encyclopedia of Philosophy
  
  
• Science: A History - Math did not evolve in an intellectual bubble and much of the mathematics we use today evolved out of a sort of feedback loop between scientists and mathematicians.
  
  Of course, let's not forget what is possibly the most complete single resource available today:
  
  
• Wikipedia
  
  This is a huge area and I applaud you for being interested in it. If you decide to look into it seriously, you WILL find yourself confused, bored, and annoyed at times. Yet it's a beautiful history and one that will illuminate much of the apparently meaningless gibberish in calculus, as well as many other areas.