[Date Index] [Thread Index] [FOM Postings] [FOM Home]

*To*: fom@math.psu.edu*Subject*: FOM: FLT, 4CT, CFSG: Depth, length, and width of proofs*From*: Joe Shipman <shipman@savera.com>*Date*: Mon, 07 Dec 1998 13:34:42 -0500*Organization*: Savera systems*Sender*: owner-fom@math.psu.edu

1) Fermat's Last Theorem I have in front of me a book, "Modular Forms and Fermat's Last Theorem" (Springer 1997, eds. Cornell, Silverman, Stevens, ISBN# 0-387-94609-8) which contains (in some sense) a "proof" of Wiles's Theorem ("Fermat's Last Theorem"). It is nearly 600 pages long and chapters are written by over two dozen contributors. Chapters are: I An Overview of the Proof of Fermat's Last Theorem (Stevens) II A Survey of the Arithmetic Theory of Elliptic Curves (Silverman) III Modular Curves, Hecke Correspondences, and L-Functions (Rohrlich) IV Galois Cohomology (Washington) V Finite Flat Group Schemes (Tate) VI Three Lectures on the Modularity of /rhobar_E,3 and the Langlands Reciprocity Conjecture (Gelbart) VII Serre's Conjectures (Edixhoven) VIII An Introduction to the Deformation Theory of Galois Representations (Mazur) IX Explicit Construction of Universal Deformation Rings (De Smit and Lenstra) X Hecke Algebras and the Gorenstein Property (Tilouine) XI Criteria for Complete Intersections (De Smit, Rubin, and Schoof) XII l-adic Modular Deformations and Wiles's "Main Conjecture" (Diamond and Ribet) XIII The Flat deformation Functor (Conrad) XIV Hecke Rings and Universal Deformation Rings (De Shalit) XV Explicit Families of Elliptic Curves with Prescribed Mod N Representations (Silverberg) XVI Modularity of Mod 5 Representations (Rubin) XVII An Extension of Wiles's Results (Diamond, appendix by Diamond and Kramer) XVIII Class Field Theory and the First Case of Fermat's Last Theorem (Lenstra and Stevenhagen) XIX Remark's on the History of Fermat's Last Theorem 1844 to 1984 (Rosen) XX On Ternary Equations of Fermat Type and Relations with Elliptic Curves (Frey) XXI Wiles's Theorem and the Arithmetic of Elliptic Curves (Darmon) This is *exactly* the kind of textbook I like, which deals with a lot of different subjects in a focused way by developing each of them only to the point needed for the proof of some "big" result. Unfortunately, it is too hard for me because I am not an algebraic number theorist! Let me say this differently: I took a course in algebraic number theory in graduate school, which basically went through Serge Lang's "Red Book" on the subject. That is NOT enough of a prerequisite to understand this book! I would guess that only a third-year grad student who had been studying algebraic number theory for two years already would really be able to tackle this book and honestly say at the end that he understood the proof. To put it another way, this book would have to be combined with two other substantial ones (e.g. Lang's and its successor for the SECOND-year graduate algebraic number theory course) to have any chance of being acceptable as a proof of FLT to a mathematician who is not a specialist in the field (and even there we are assuming a solid general mathematical background which might require one more book). And that "proof" would have about the minimum level of detail permissible and require a great deal of work to verify. It is an ambition of mine to actually learn this proof someday, though it looks like it will have to wait until I get an academic job. 2) The Four-color Theorem This theorem is generally held to have a proof that is "too long" to be humanly verifiable. But the difficult part of the proof is not all that bad -- one journal article of a few dozen pages conatins sufficient information for any good mathematician to satisfy himself that if two specific algorithms report success for some input, then 4CT is true. The two algorithms (a "discharging" algorithm to generate an unavoidable set and a "reducing" algorithm to show each graph in the unavoidable set is reducible) are also supplied along with the input that is supposed to work, and the program code and input size are also quite feasible. The trace of the computer runs would be much too large to check by hand, though the computer runs themselves only take a few hours on a Sun workstation. Which of these two "proofs" is REALLY longer? Which is more reliable? This is obviously a trick question, but I think the point is clear that *some* mathematicians who actually understand both proofs will have more confidence in 4CT [because the Number Theory is so difficult and they can write their own versions of the algorithms to check], while *others* will have more confidence in FLT [like the contributors to the Springer book who have internalized all the background material and have worked with it for many years]. 3) The Classification of Finite Simple Groups I'd like to know more about this proof. My impression (anyone who disagrees please correct me) is that the "length" of the proof includes almost all the relevant background material (in other words it does not build on a lot of previous mathematics). In this way it resembles 4CT rather than FLT. Also, certain core pieces of the argument (like the Feit-Thompson odd order theorem) are both very long and very well-verified (in the case of the OOT, thousands of people have carefully read the proof but no one has been able to shorten it by very much, it is "essentially" long in some ill-defined sense), while other pieces are long but not so well-verified. The not-so-well-verified pieces are somehow considered less central and if a mistake is found in them the feeling is it can be worked around. We need more precise concepts of "depth" and "width" here, but the impression I get is that the CFSG proof is *wide* but not deep, except for a few pieces which are both deep and very well-verified. The 4CT proof is very *long* but almost all of it is extremely shallow (and it isn't particularly wide because it's pretty uniform). The FLT proof is shorter than the other two but very *deep* (and wide too, though still more unified than CFSG). These are three excellent examples! Most of the other "big" proofs I can think of don't raise any serious issues because they are not too long, not too deep, and not too wide. -- Joe Shipman

[Date Prev] [Date Next] [Thread Prev] [Thread Next]

[Date Index] [Thread Index] [FOM Postings] [FOM Home]