It was clear that what was still lacking was some new principle that could connect these geometric objects ( or "varieties", or "schemes") to the usual "well behaved" (topological) "spaces": those, let us say, whose points are clearly distinguished one from the other, whearas in the "harum-scarum" spaces introduced by Zariski, the points have a sneaky tendency to cling to one another....
Most certainly it was through nothing less than through this "new principle" that the marriage of "number and magnitude",( or of "continuous and discontinuous" geometry) could give birth to the Weil conjectures.
The notion of space is certainly one of the oldest in mathematics. It is fundamental to our "geometric" perspective on the world, and has been so tacitly for over two millenia. Its only over the course of the 19th century that this concept has, bit-by-bit, freed itself from the tyranny of our immediate perceptions ( that is, one and the same as the "space" that surrounds us), and of its traditional theoretical treatment ( Euclidean), to attain to its present dynamism and autonomy. In our own times it has joined the ranks of those notions that are most freely and universally employed in mathematics, and is familiar, I would say, to every mathematician without exception. It has become a concept of multiple and varied aspects, of hundreds of thousands of faces depending on the kinds of structures one choses to impose on a space, from the most abundant and rich, ( such as the venerable 'euclidean' structures, or the 'affine' or 'projective' ones , or again the 'algebraic' structures of similarly designated 'varieties' which generalize and extend them), down to the most 'impoverished': those in which all 'quantitative' information has been removed without a trace, or in which only a qualitative essence of "proximity" or of "limit"(*), ( and, in its most elusive version, the intuition of form ( called 'topological spaces' )), remains.
As elusive as it might appear initially, the "qualitatively pure" structure encapsulated in the notion of "space"( topological) in the absence of all quantitative givens, ( notably the metric distances between points) which enables us to relate it to habitual intuitions of "large" and "small", we have, all the same, over the last century, been able to confine these spaces in the locked flexible suitcases of a language which has been meticulously fabricated as the occasion arose. Still better, as the occasion arose, various 'weights and measures' have been devised to serve a general function, good or bad, of attaching "measures" ( called 'topoligical invariants'), to those sprawled-out spaces which appear to resist, like fleeting mists, any sort of metrizability. Most of these invariants, its true, certainly the most essential ones, are more subtle than simple notions like 'number' and 'magnitude' - often they are themselves fairly delicate mathematical structures bound ( by rather sophisticated constructions) to the space in question. One of the oldest and most crucial of these invariants, introduced in the last century ( by the Italian mathematician Betti) is formed from the various "groups" ( or 'spaces'), called the "Cohomology" associated with this space. (*)
There are numerous other "topological invariants" which have been introduced by the topologists to deal with this or that property or this or that topological space. Next after the "dimension" of a space and the (co)homological invariants, come the "homotopy groups". In 1957 I introduced yet another one, the group (known as "Grothendieck) K(X), which has known a sensational success and whose importance ( both in topology and arithmetic) is constantly being re-affirmed. A whole slew of new invariants, more sophisticated than the ones presently known and in use, yet which I believe to be fundamental, have been predicted by my "moderated topology" program ( one can find a very summary sketch of this in the "Outline for a Programme" which appears in Volume 4 of the Mathematical Reflections) . This programme bases itself on the notion of a "moderated theory" or "moderated space", which constitutes, a bit like the topos, a second "metamorphosis of the concept of space". It is at the same time more self-evident and less prodound than the latter. I predict that its immediate applications to topology "properly speaking" will be decidedly more incisive, that in fact it will turn upside down the "profession" of topological geometer, through a far-reaching transformation of the conceptual context appropriate to it. ( As was the case with Algebraic Geometry with the introduction of the point-of-view of the scheme.) Furthermore, I've already sent copies of my "Outline" to several of my old friends and some illustrious topologists, yet it seems to me that that haven't been inclined to take any interest in it....
Its my opinion that this "block" is part of a general aversion in Weil against all the global formalisms, (whether large or small), or any sort of theoretical construction. He hasn't anything of the true "builder"Êabout him, and it was entirely contrary to his personal style that he saw himself constrained to develop, starting with the 30's, the fundamentals of "abstract" algebraic geometry, which to him ( by his own dispositions), have proved to be a veritable "Procrustean bed" for those who use them.
I hope he doesn't hold it against me that I chose to go beyond him, investing my energy in the construction of enormous dwelling places, which have allowed the dreams of a Kronecker, and even of himself, to be cast into a language and tools that are at the same time effective and sophisticated. At no time did he ever comment to me about the work that he saw me doing, or which had already been done. Nor have I received any response from him about Récoltes et Semailles, which i sent to him over three months ago, with a warm hand-written personal dedication to him.
Thanks above all to the ulterior work of Jean-Pierre Serre, Leray's ideas have produced in the half century since their formulation, a major redirection of the whole theory of topological spaces, ( notably those invariants designated as "homotopic", which are intimately allied with cohomology), and a further redirection, no less significant, of so-called "abstract"Êalgebraic geometry ( starting with the FAC article of Serre in 1955). My early work in geometry, from 1955 onwards, was conceived of as a continuation of the work of Serre, and for that reason also a continuation of the work of Leray.