It is the later task, above all, which absorbed most of my energy - a long and patient labor on foundations, which I was the first to see with clarity and, above all, to "know in my guts". It is this which took up the major part of my time between 1958 ( the year in which one after another, the schemes and the topos made their respective appearances), and 1970, ( the year of my departure from the mathematical scene.)
It often happened also that I chaffed at the bit to be constrained in this fashion, like someone pinned down by an immovable weight, by those interminable tasks which ( once the essentials had been understood) seemed more of a routine character than a setting forth into the unknown. I had constantly to restrain the impulse to thrust forward - in the manner of a pioneer or explorer, occupied somewhere far distant in the discovery and exploration of unknown and nameless worlds, crying out for me to become acquainted with them and bestow names upon them. This impulse, and the energy I invested in them , ( partially, ,in my spare time ), were constantly held in abeyance.
However I knew very well that it was this energy, so slight, ( in a manner of speaking)in comparison with what I gave to my "duties", that was the most important and advanced; in my "creative" work in mathematics it was this that was involved; in that intense attention given to the apprehension of, in the obscure folds, formless and moist, of a hot and inexhaustibly nourishing womb, the earliest traces and shapes of what had yet to be born and which appeared to be calling out to me to give it form, incarnation and birth ... This work of discovery, the concentrated attention involved, and its ardent solicitude, constituted a primeval force, analogous to the sun's heat in the germination and gestation of seeds sown in the nourishing earth, and for their miraculous bursting forth into the light of day.
In my work as a mathematician I've seen two primary forces or tendancies of equal importance at work, yet of totally different natures- or so it seems to me. To evoke them I've made use of the images of the builder, and of the pioneer or explorer. Put alongside each other, both strike me somehow as really quite "yang", very "masculine", even "macho"! They possess the heightened resonance of mythology, of "great events". Undoubtably they've been inspired by the vestiges within me of my old "heroic" vision of the creative worker, the "super-yang" vision. Be that as it may, they produce a highly colored image, if not totally pictorial yet "standing at attention" to be viewed, of a far more fluid, humble and "simple" reality -one that is truly living.< P> However, in this "male" "builder's" drive, which would seem to push me relentlessly to engineer new constructions I have, at the same time, discerned in me something of the homebody, someone with a profound attachment to " the home". Above all else, it is "his" home, that of persons "closest" to him- the site of an intimate living entity of which he feels himself a part. Only then, and to the degree which the circle of his "close associates" can be enlarged, can it also be a "open house" for everyone.
And, in this drive to "make" houses ( as one "makes" love...) there is above all, tenderness. There is furthermore the urge for contact with those materials that one shapes a bit at a time, with loving care, and which one only knows through that loving contact. Then, once the walls have been erected, pillars and roof put in place, there comes the intense satisfaction of installing the rooms, one after the other, and witnessing the emergence, little by little, from these halls, rooms and alcoves, of the harmonious order of a living habitation - charming, welcoming, good to live in. Because the home, above all and secretly in all of us, is the Mother - that which surrounds and shelters us, source at once of refuge and comfort; and it is even ( at a still deeper level, and even as we are in the process of putting it all in place), that place from which we are all issued, which has housed and nourished us in that unforgettable time before our birth... It is thus also the Busom.
And the other spontaneously generated image, going beyond the inflated notion of a "pioneer', and in order to grasp the hidden reality which it conceals, is itself devoid of all sense of the "heroic". There once again, it is the archetypal maternal image which occurs - that of the nourishing "matrix" , and of its formless and obscure labors...
These twin urges which appeared to me as being "totally different" have turned out to be much closer than I would have imagined. Both the one and the other have the character of a "drive for contact", carrying us to the encounter with "the Mother": that which incarnates both that which is close and "known", and that which is "unknown". In abandoning myself to either one or the other, it is to "rediscover the Mother", it is in order to renovate contact with that which is near, and "more or less known", and that which is distant, yet at the same time sensed as being on the verge of being understood.
The distinction is primarily one of tone, of quantity, but not of anessential nature. When I "construct houses", it is the "known" which dominates; when I "explore", it is the "unknown". These two "modes" of discovery, or to better state the matter, these two aspects of a single process, are indissolubly linked. Each is essential and complementary to the other. In my mathematical work I've discerned a coming-and-going between these two ways of approaching things, or rather, between those moments ( or periods) in which one predominates, then the other (*)
I've no intention of wishing to minimize or denigrate either side of my nature, each essential one to the other: the "masculine" which builds and engenders, or the "feminine" which conceives, which shelters the long and obscure pregnancies. I "am" either one or the other - "yang" and "yin", "man" and "woman". Yet I'm also aware that the more delicate, the subtler in unravelling of these creative processes is to be found in the "yin" or "feminine" aspect - humble, obscure, often mediocre in appearance.
It's this side of my labor which, always I would say, has held the greatest fascination for me. The modern consensus however had tried to encourage me to invest the better part of my energy in the other side, in those efforts which affirm themselves by being incarnated in "tangible" products, if not always finished or perfected - products with well-defined boundaries, asserting their reality as if they'd been cut in stone... I can now see, upon reflection, how heavily this consensus weighed on me, and also how I "bore the weight of the accusation"-with submission! The aspect of "conception", or "exploration" of my work was accorded a meagre role by me, even up to the moment of my departure. And yet, in the retrospective overview I've made of my work as a mathematician, the evidence leaps out to me that the thing that has constituted the very essence and power of this work, has been the face which, in today's world, is the most neglected, when it is not frankly treated as an object of derision or disdainful condescension: that of the ideas, even that of dreams, never that of results.
In attempting in these pages to discern the most essential aspects of my contribution to the mathematics of our time, via a comprehensive vision that choses the forest over the trees - I've observed, not a victorious collection of "grand theorems", but rather a living spectrum of fertile ideas, which in their confluence have contributed to the same immense vision.(*)