My first solitary reflections, on Measure Theory and Integration, placed me without ambiguity under the rubrique of Analysis. And this remained the same for the first of the new themes that I introduced into mathematics, ( which now appears to me to be of smaller dimensions than the 11 that followed). I entered mathematics with an "analytic bias", not because of my natural temperament but owing to "fortuitous circumstances": it was because the biggest gap in my education, both at the lycée and at the university, was precisely in this area of the "analytic aspect" of things.
The year 1955 marked a critical departure in my work in mathematics: that of my passage from "analysis" to "geometry". I well recall the power of my emotional response ( very subjective naturally); it was as if I'd fled the harsh arid steppes to find myself suddenly transported to a kind of "promised land" of superabundant richness, multipying out to infinity wherever I placed my hand in it, either to search or to gather... This impression, of overwhelming riches has continued to be confirmed and grow in substance and depth down to the present day.(*)
One cannot invent the structure of an object. The most we can do is to patiently bring it to the light of day, with humility - in making it known it is "discovered". If there is some sort of inventiveness in this work, and if it happens that we find ourselves the maker or indefatigable builder, we aren't in any sense "making" or "building" these structures. They hardly waited for us to find them in order to exist, exactly as they are! But it is in order to express, as faithfully as possible, the things that we've been detecting or discovering, to deliver up that reticent structure, which we can only grasp at, perhaps with a language no better than babbling. Thereby are we constantly driven to invent the language most appropriate to express, with increasing refinement, the intimate structure of the mathematical object, and to "construct" with the help of this language, bit by bit, those "theories" which claim to give a fair account of what has been apprehended and seen. There is a continual coming and going, uninterrupted, between the apprehension of things, and the means of expressing them, by a language in a constant state improvement, and constantly in a process of recreation, under the pressure of immediate necessity.
As the reader must have realized by now, these "theories", "constructed out of whole cloth", are nothing less than the "stately mansions" treated in previous sections: those which we inherit from our predecessors, and those which we are led to build with our own hands, in response to the way things develop. When I refer to "inventiveness" ( or imagination) of the maker and the builder, I am obliged to adjoin to that what really constitures it soul or secret nerve. It does not refer in any way to the arrogance of someone who says "This is the way I want things to be!" and ask that they attend him at his leisure, the kind of lousy architect who has all of his plans ready made in his head without having scouted the terrain, investigated the possibilities and all that is required.
The sole thing that constitutes the true "inventiveness" and imagination of the researcher is the quality of his attention as he listens to the voices of things. For nothing in the Universe speaks on its own or reveals itself just because someone is listening to it. And the most beautiful mansion, the one that best reflects the love of the true workman, is not the one that is bigger or higher than all the others. The most beautiful mansion is that which is a faithful reflection of the structure and beauty concealed within things.