Posted by: Alexandre Borovik | May 28, 2009

Philosophy of the Classification of Finite Simple Groups

Adverticement:

At the Vrije Universiteit Brussel, Center for Logic and Philosophy of Science, a full-time and fully funded position for a Ph.D. student is available for a four-year period, starting at the earliest moment (as early as 1 July 2009).

The topic of the research project, funded by the FWO-Vlaanderen (Research Foundation Flanders), is: “Argumentative networks: a still missing integration of philosophical approaches to argumentation with AI-models, with an application to mathematical practice.” The intended application is the classification theorem of the finite, simple groups. We are therefore looking for a student with a philosophical training, but with the required mathematical background to handle such proofs.

Please send your CV and a letter of motivation to Jean Paul Van Bendegem (jpvbende@vub.ac.be) before June 15, 2009. A more detailed project description can be found at http://www.vub.ac.be/CL WF/activities/argumentativenetworks.pdf

A phrase from the project description:

The choice of the case study, viz., the classification theorem of
finite, simple groups, is mainly determined by the fact that no single mathematician can comprehend the entire proof, hence, it has to be ‘shared’ in the community of mathematicians.

My colleagues with experience of the Classification of Finite Simple Groups took the statement very seriously. A typical response: 

I believe that there are quite a few mathematicians involved in the  proof, e.g. AschbacherLyonsSmith,  Solomon,  Thompson, etc. each of whom understands the entire proof.

I have perhaps to add that the mathematician quoted above also understands the entire proof.

Some papers andbooks on history of CFSG:

  1. M. Aschbacher, The Classification of the Finite Simple Groups.
  2. M. Aschbacher, The Status of the Classification of the Finite Simple Groups
  3. M. Aschbacher, Highly complex proofs and implications of such proofs.
  4. D. Gorenstein, The classification of fnite simple groups, Bull. Amer. Math. Soc. (New Series) 1 (1979), 43-199. MR 80b:20015.
  5. D. Gorenstein, Finite Simple Groups: An Introduction to their Classi cation, Plenum Press, New York, 1982. MR 84j:20002.
  6. D. Gorenstein, The Classi cation of Finite Simple Groups, Plenum Press, New York, 1983.  MR 86i:20024.
  7. D. Gorenstein, The classi cation of the fnite simple groups, a personal journey: the early years. MR 90g:0103. 
  8. R. Solomon, A brief history of the classification of the finite simple groups. (Contains extensive bibliography.)
  9. J. G. Thompson, Finite nonsolvable groups.
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Responses

  1. Speaking as an outsider, my impression from what I’ve read is that there remain some gaps in the proof, in that not all of the arguments involved have actually been published.

    But even if I’m misinformed about that, a lot seems to hang on the words “comprehend” and “understand”. I can easily imagine each of these experts clearly comprehending a detailed outline of the proof, but even after compression, the full proof stretches out to several thousand pages. Does it mean that you could point to any lemma in the full proof, and expect that any one of these experts has the proof of the lemma locked and loaded in his or her head, and could reproduce it on demand?


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