Ronnie Brown, of the University of Wales at Bangor, UK, recently posted some very interesting remarks on mathematical speculation to the categories list. With his permission, I am reposting them here:

“The situation is more complicated in that what could be classed as speculation may get published as theorem and proof. For example, in algebraic topology, sometimes proofs of continuity are omitted as if this was an exercise for the reader, yet the formulation of why the maps are continuous (if they are necessarily so) may contain a key aspect of what should be a complete proof. This difficulty was pointed out to me years ago by Eldon Dyer in relation to results on local fibration implies global fibration (for paracompact spaces) where he and Eilenberg felt Dold’s paper on this contained the first complete proof. I have been unable to complete the proof in Spanier’s book, even the second edition. (I sent a correction to Spanier as the key function in the first edition was not well defined, after Spanier had replied `Isn’t it continuous?’) Eldon speculated (!) that perhaps 50% of published algebraic topology was seriously wrong!van Kampen’s original 1935 `proof’ of what is called his theorem is incomprehensible today, and maybe was then also.

Efforts to give full details of a major result, i.e. to give a proof, are sometimes derided. Of course credit should be given to the originator of the major steps towards a proof.

Grothendieck’s efforts to develop structures and language which would reduce proofs to a sequence of tautologies are notable here. Colin McLarty’s excellent article on `The rising sea: Grothendieck on simplicity and generality ‘ is relevant.Some scientists snear at the mathematical notion of rigour and of proof. On the other hand many are attracted to math because it can give explanations of why something is true. But `explanations’ need a higher level of structural language than for what might be called proofs.

I can’t resist mentioning that one student questionaire on my first year analysis wrote `Professor Brown puts in too many proofs.’ So I determined to rectify the situation, and next year there were no theorems, and no proofs. However there were lots of statements labelled `FACT’ followed by several paragraphs labelled `EXPLANATION’. This did modify the course because something labelled `explanation’ ought really to explain something! I leave you all to puzzle this out!

In homotopy theory, many matters, such as the homotopy addition lemma, had clear proofs only years after they were well used.Surely much early algebraic topology is speculative, in that the language has not yet been developed to express concepts with rigour so that a clear proof can be written down. It would be a curious ahistorical assumption that there is not at this date another future level of concepts which require a similar speculative approach to reach towards them.”

*(Ronnie Brown, posted 2006-03-14 to the categories list).*

People interested in this should also check out Barry Mazur’s article “Conjecture”, which appeared in the May 1997 issue of

Synthese. In fact, that whole issue has interesting stuff on related topics. Here is an interesting review of the issue.