As I've often said, most mathematicians take refuge within a specific conceptual framework, in a "Universe" which seemingly has been fixed for all time - basically the one they encountered "ready-made" at the time when they did their studies. They may be compared to the heirs of a beautiful and capacious mansion in which all the installations and interior decorating have already been done, with its living-rooms , its kitchens, its studios, its cookery and cutlery, with everything in short, one needs to make or cook whatever one wishes. How this mansion has been constructed, laboriously over generations, and how and why this or that tool has been invented (as opposed to others which were not), why the rooms are disposed in just this fashion and not another - these are the kinds of questions which the heirs don't dream of asking . It's their "Universe", it's been given once and for all! It impresses one by virtue of its greatness, (even though one rarely makes the tour of all the rooms) yet at the same time by its familiarity, and, above all, with its immutability.
When they concern themselves with it at all, it is only to maintain or perhaps embellish their inheritance: strengthen the rickety legs of a piece of furniture, fix up the appearance of a facade, replace the parts of some instrument, even, for the more enterprising, construct, in one of its workshops, a brand new piece of furniture. Putting their heart into it, they may fabricate a beautiful object, which will serve to embellish the house still further.
Much more infrequently, one of them will dream of effecting some modification of some of the tools themselves, even, according to the demand, to the extent of making a new one. Once this is done, it is not unusual for them make all sorts of apologies, like a pious genuflection to traditional family values, which they appear to have affronted by some far-fetched innovation.
The windows and blinds are all closed in most of the rooms of this mansion, no doubt from fear of being engulfed by winds blowing from no-one knows where. And, when the beautiful new furnishings, one after another with no regard for their provenance, begin to encumber and crowd out the space of their rooms even to the extent of pouring into the corridors, not one of these heirs wish to consider the possibility that their cozy, comforting universe may be cracking at the seams. Rather than facing the matter squarely, each in his own way tries to find some way of accommodating himself, one squeezing himself in between a Louis XV chest of drawers and a rattan rocking chair, another between a moldy grotesque statue and an Egyptian sarcophagus, yet another who, driven to desperation climbs, as best he can, a huge heterogeneous collapsing pile of chairs and benches!
The little picture I've just sketched is not restricted to the world of the mathematicians. It can serve to illustrate certain inveterate and timeless situations to be found in every milieu and every sphere of human activity, and (as far as I know) in every society and every period of human history. I made reference to it before , and I am the last to exempt myself: quite to the contrary, as this testament well demonstrates. However I maintain that, in the relatively restricted domain of intellectual creativity, I've not been affected (*) by this conditioning process, which could be considered a kind of 'cultural blindness' Š an incapacity to see ( or move outside) the "Universe" determined by the surrounding culture.
The rightful place of such a worker is not in a ready-made universe, however accommodating it may be, whether one that he's built with his own hands, or by those of his predecessors. New tasks forever call him to new scaffoldings, driven as he is by a need that he is perhaps alone to fully respond to. He belongs out in the open. He is the companion of the winds and isn't afraid of being entirely alone in his task, for months or even years or, if it should be necessary, his whole life, if no-one arrives to relieve him of his burden. He, like the rest of the world, hasn't more than two hands - yet two hands which, at every moment, know what they're doing, which do not shrink from the most arduous tasks, nor despise the most delicate, and are never resistent to learning to perform the innumerable list of things they may be called upon to do. Two hands, it isn't much, considering how the world is infinite. Yet, all the same, two hands, they are a lot ....
I'm not up on my history, but when I look for mathematicians who fall into the lineage I'm describing, I think first of all of Evariste Galois and Bernhard Riemann in the previous century, and Hilbert at the beginning of this one. Looking for a representative among my mentors who first welcomed me into the world of mathematics (*), Jean Leray's name appears before all the others, even though my contacts with him have been very infrequent. (**)(**)Even so I've been ( following H. Cartan and J.P.Serre), one of the principal exploiters and promoters of one of the major ideas introduced by Leray, that of the bundle . It has been an indispensable tool in all of my work in geometry. It also provided me with the key for enlarging the conception of a (topological) space to that of a topos, about which I will speak further on.
Leray doesn't quite fill this notion that I have of a 'builder', in the sense of someone who 'constructs houses from the foundations up to the rooves." However, he's laid the ground for immense foundations where no one else had dreamed of looking, leaving to others the job of completing them and building above them or, once the house has been constructed, to set themselves up within its rooms ( if only for a short time) .... .
(**)Furthermore, at the same time, and without intending to, he assigns to the earlier Universe (if not for himself then at least for his less mobile colleagues), a new set of boundaries, much enlarged yet also seemingly imperious and invisible than the ones he's replaced