On the categories email list on 5 March 2006, Ronald Brown quoted the following paragraph on mathematical speculation from a 14 June 1983 letter he had received from Alexander Grothendieck:
Your idea of writing a “frantically speculative” article on groupoids seems to me a very good one. It is the kind of thing which has traditionally been lacking in mathematics since the very beginnings, I feel, which is one big drawback in comparison to all other sciences, as far as I know. Of course, no creative mathematician can afford not to ‘speculate’, namely to do more or less daring guesswork as an indispensable source of inspiration. The trouble is that, in obedience to a stern tradition, almost nothing of this appears in writing, and preciously little even in oral communication. The point is that the disrepute of ‘speculation’ or ‘dream’ is such, that even as a strictly private (not to say secret!) activity, it has a tendency to vegetate – much like the desire and drive of love and sex, in too repressive an environment. Despite the ‘repression’, in the one or two years before I unexpectedly was led to withdraw from the mathematical milieu and to stop publishing, it was more or less clear to me that, besides going on pushing ahead with foundational work in SGA and EGA, I was going to write a wholly science-fiction kind [of] book on ‘motives’, which was then the most fascinating and mysterious mathematical being I had come to meet so far. As my interests and my emphasis have somewhat shifted since, I doubt I am ever going to write this book – still less anyone else is going to, presumably. But whatever I am going to write in mathematics, I believe a major part of it will be ‘speculation’ or ‘fiction’, going hand in hand with painstaking, down-to-earth work to get hold of the right kind of notions and structures, to work out comprehensive pictures of still misty landscapes. The notes I am writing up lately are in this spirit, but in this case the landscape isn’t so remote really, and the feeling is rather that, as for the specific program I have been out for is concerned, getting everything straight and clear shouldn’t mean more than a few years work at most for someone who really feels like doing it, maybe less. But of course surprises are bound to turn up on one’s way, and while starting with a few threads in hand, after a while they may have multiplied and become such a bunch that you cannot possibly grasp them all, let alone follow.”
In a subsequent posting (2006-03-14), Brown wrote this about rigour in category theory:
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.”
Is it because I trained as a pure mathematician that I find this lack of rigor and completeness in what should be caregorial fundamentals immensely disturbing?