On Apr 25, 2010, at 11:51 AM, Russ Abbott wrote:
In answer to Eric and lrudolph, the answer I'm looking for is not related to epistemology. It is related to the domains to which mathematical thinking is successfully applied, where successfully means something like produces "interesting' theorems. (Please don't quibble with me about what interesting mean -- at least not in this thread. I expect that interesting can be defined so that we will be comfortable with the definition.) What is it about those domains that enables that.
If I get your drift, I think it may have to do with domains with very simple definitions yet have many far reaching results. Almost a success criterion of being approachable and powerful both.
Consider graph theory: a graph G = {V,E} where V=a set of vertices, and E is a set of edges connecting two vertices. (This can be made more "mathy" but with very little gain.) A few definitions such as the degree of a vertex is the number of edges it has, and you're off to the races.
With nearly no training, many simple theorems can be proved, for example: Prove that in any finite graph, the number of vertices with odd degree is even.
Possibly part of this simplicity is how simple a proof can be in the given domain. I like Cris Moore's definition of a proof: a short essay that convinces a skeptic! For the above, for example, you can just use a few statements in english, or resort to Induction. In either case the proof actually convinces and conveys an "a-ha" experience.
I rather like what is going on in computer science, your field, in this regard. Many regard an algorithm as best if it is structured upon its proof. Indeed, all algorithms *are* theorems (their inputs, outputs and a statement on the relations between the two) and proofs (the algorithm itself).
-- Owen
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
