Alexander Grothendieck on “speculation”

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.

—an extract from a letter dated 14/06/83 from Alexander Grothendieck to Ronnie Brown (Bangor)