Tim Chow has posted a draft "beginner's guide to forcing". I very much like these opening remarks:
All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves. The proofs should be 'natural' in Donald Newman's sense: "This term ... is introduced to mean not having any ad hoc constructions or brilliancies. A 'natural' proof, then, is one which proves itself, one available to the 'common mathematician in the streets'." I believe that it is an open exposition problem to explain forcing. Current treatments allow readers to verify the truth of the basic theorems, and to progress fairly rapidly to the point where they can use forcing to prove their own independence results .... However, in all treatments that I know of, one is left feeling that only a genius with fantastic intuition or technical virtuosity could have found the road to the final result.
Leaving aside the question of how well Tim Chow brings off his expository task -- though it looks a very interesting attempt to my inexpert eyes, and I'm off to read it more carefully -- I absolutely agree with him about the importance of such expository projects, giving "natural" proofs of key results in various levels of detail: these things are really difficult to do well yet are hugely worth attempting for the illumination that they bring.
Also, Philosophy of Mathematics: 5 Questions (which I've mentioned before as forthcoming) is now out. This is a rather different kind of exercise in standing back and trying to give an overview, with 28 philosophers and logicians giving their takes on the current state of play in philosophy of mathematics (the authors range from Jeremy Avigad, Steve Awodey and John L. Bell through to Philip Welch, Crispin Wright and Edward N. Zalta). The five questions are
- Why were you initially drawn to the foundations of mathematics and/or the philosophy of mathematics?
- What example(s) from your work (or the work of others) illustrates the use of mathematics for philosophy?
- What is the proper role of philosophy of mathematics in relation to logic, foundations of mathematics, the traditional core areas of mathematics, and science?
- What do you consider the most neglected topics and/or contributions in late 20th century philosophy of mathematics?
- What are the most important open problems in the philosophy of mathematics and what are the prospects for progress?