http://www.ics.uci.edu/~eppstein/cgt/ <http://www.ics.uci.edu/~eppstein/cgt/> —Dan
On Aug 10, 2015, at 12:15 PM, James Propp <jamespropp@gmail.com> wrote:
I'm re-reading Gardner's article "Conway's Surreal Numbers" (in conjunction with writing my own essay on the subject for my blog), and one passage from Gardner's piece jumped out as me as being, not exactly wrong, but deserving of a certain sort of reply.
Gardner quotes Knuth as saying that he wrote "Surreal Numbers" in part "to teach how one might go about developing such a theory". But Knuth's protagonists Alice and Bill, much like Knuth himself, were developing, from the ground up, a theory from someone else's axioms, namely Conway's. It may be worth saying that most math isn't like this, nor is this what Conway himself did.
What most math is about is proving new theorems not just from definitions, but from earlier theorems, and using a stockpile of known examples on which one can test out new ideas.
But (and here's my main point) what Conway himself did is neither what most mathematicians do (as per the previous paragraph) nor what Alice and Bill and Knuth did (as per the paragraph before that). Conway had to figure out what the right axioms and definitions were, which is an even higher-level mathematical activity than proving theorems.
There's a lot that could be said about how mathematical theory-building works, and how it's been practiced by the architects of great theories. One can imagine a book of case-studies that attempts some sort of historical accuracy, in the spirit of David Corfield's notion of bringing philosophy of mathematics into closer alignment with mathematical practice.
But that's not exactly what I'm interested in today (though I may have time to be interested in it in the future). What I want to know today is, has anyone written a pedagogically-motivated "Whig history" of the development of the theory of numbers and games, based upon (but not 100% faithful to) what Conway actually did, showing how a Conway *could* have developed the theory, starting from nothing but the observation that some games (such as Go endgames) behave kind of like numbers, and pushing that idea in the directions the idea "wants" to go, and not in directions it doesn't (and having the wisdom to know the difference)?
I am thinking of what Lakatos did with the idea of Euler characteristic in "Proofs and Refutations", showing the intense interactions between definitions and theorems and examples (especially but not exclusively counterexamples).
Relatedly, has anyone created a "text" on combinatorial game theory that consists basically of a list of problems to be solved, such that working through the problems leads the student to develop the theory on his/her own?
If such works exist, I'd like to refer to them in my piece.
Jim Propp
On Mon, Jul 20, 2015 at 11:03 PM, James Propp <jamespropp@gmail.com> wrote:
A smart high school student whom I'm mentoring has expressed interest in learning about combinatorial game theory. My first thought was to recommend "Winning Ways", but I think it might be better to start her on something shorter and more focused, and then suggest "Winning Ways" if the shorter book whets her appetite for more. Any suggestions?
Jim Propp
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