Some interesting anecdotes and quotes:

1.  The structure of a thing is not something which it is possible for us to “invent”. We can only patiently unravel it, humbly get to know it and “discover” it. If there is any ingenuity involved in this line of work, and if we sometimes take up the role of a blacksmith or that of a tireless builder, it is never to “model” or to “construct” “structures” - they didn’t have to wait for us to exist, and to be precisely what they are! Rather, it is to express, as faithfully as we can, those things which we are in the process of scanning and discovering, the structure that is reluctant to surrender and which we attempt to grasp, fumblingly, and through a perhaps fledgling language. Thus are we constantly led to “invent” the language best suited to ever more finely express the intimate structure of mathematical things, and to “construct” by means of this language, slowly and from the ground up, the “theories” that are supposed to report what has been apprehended and seen. Underlying this process is a continual, uninterrupted back-and-forth motion between the apprehension of things and the expression of that which has been apprehended, through a language that grows finer and is created anew over time, under the constant pressure of immediate needs. As the reader will have no doubt guessed, these “theories”, “constructed from the ground up”, are but the “beautiful houses” which were discussed earlier: the houses which we inherit from our predecessors and those which we are led to build with our own hands, as we listen and follow the calling of things. Having mentioned earlier the “ingenuity” (or imagination) of the builder or the blacksmith, I should add that what lies at its heart is not the arrogance of he who asserts “I want this, and not that!” and who decides according to his whim; it is a pitiful architect who sets off with all of his plans fixed in his mind, before having even seen and felt the terrain, surveying its requirements and possibilities. What characterizes the value of the ingenuity and imagination of a researcher is the quality of his attention as he listens to the voice of things - for the things of the Universe never tire of talking about themselves and to reveal themselves to he who cares to listen. Thus, the most beautiful house, that in which the love of the builder is most evident, is not that which is larger or higher than the others. Rather, a house is beautiful if it faithfully reflects the structure and beauty hidden in things. 

     ~ A. Grothendieck, Récoltes et Semailles.

2. After some ten years, I would now say, with hindsight, that “general topology” was developed (during the thirties and forties) by analysts and in order to meet the needs of analysis, not for topology per se, i.e. the study of the topological properties of the various geometrical shapes. That the foundations of topology are inadequate is manifest from the very beginning, in the form of “false problems” (at least from the point of view of the topological intuition of shapes) such as the “invariance of domains”, even if the solution to this problem by Brouwer led him to introduce new geometrical ideas. Even now, just as in the heroic times when one anxiously witnessed for the first time curves cheerfully filling squares and cubes, when one tries to do topological geometry in the technical context of topological spaces, one is confronted at each step with spurious difficulties related to wild phenomena. For instance, it is not really possible, except in very low dimensions, to study for a given space X (say a compact manifold), the homotopy type of (say) the automorphism group of X, or of the space of embeddings, or immersions etc. of X into some other space Y – whereas one feels that these invariants should be part of the toolbox of the essential invariants attached to X, or to the pair (X, Y), etc. just as the function space Hom(X, Y) which is familiar in homotopical algebra. Topologists elude the difficulty, without tackling it, moving to contexts which are close to the topological one and less subject to wildness, such as differentiable manifolds, PL spaces (piecewise linear) etc., of which it is clear that none is “good”, i.e. stable under the most obvious topological operations, such as contraction-gluing operations (not to mention operations like X → Aut(X) which oblige one to leave the paradise of finite dimensional “spaces”). This is a way of beating about the bush! This situation, like so often already in the history of our science, simply reveals the almost insurmountable inertia of the mind, burdened by a heavy weight of conditioning, which makes it difficult to take a real look at a foundational question, thus at the context in which we live, breathe, work – accepting it, rather, as immutable data. It is certainly this inertia which explains why it took millennia before such childish ideas as that of zero, of a group, of a topological shape found their place in mathematics. It is this again which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom “wildness” is a fatal necessity, rooted in the nature of things. 

~ A. Grothendieck, Esquisse d'un Programme.

3. I was a visiting student from the Hebrew University and he [Nicolas Bergeron] was a young faculty at Paris Sud. I was about to give a talk at the Topology and Dynamics seminar at Orsay, my first seminar talk ever outside my home institute at Jerusalem. When I got up to leave and prepare for my talk he said “I suppose they told you about the seminar tradition”. What tradition ? “That the speaker should start with a joke. But it has to be authentic, something original or a personal story”. Instead of preparing for the talk I spent the next couple of hours trying to come up with a joke. I started my lecture by saying that I’m going to give an elementary proof for the four color theorem. The “proof” started with a reduction that any map can be drawn on a cow and ended with the fact that every cow can be colored by two colors, so we actually obtained an even stronger result. There were many famous mathematicians in the crowd, all looking at me with shocked faces, except Nicolas, who was sitting at the back completely amused and satisfied by the situation. A year later when I came to speak in the same seminar, he told me that they changed the tradition and now the speaker should tell about his first sexual experience. This time I didn’t fall into the trap. 

     ~ Tsachik Gelander, Moments with Nicolas, Hommage a Nicolas Bergeron.

4. Some fifteen or twenty years ago, while browsing through the modest volume constituting Riemann’s complete works, I was stricken by a remark he made “in passing”. He observed that the ultimate structure of space may well be “discrete”, and that the “continuous” representations which we employ may be a simplification (perhaps excessive, in the long run) of a reality that is more complex; that the “continuous” is easier to grasp for the human mind than the “discontinuous”, and that the continuous serves, as such, as an “approximation” to gain insight into the discontinuous. I find this to be a remark of surprising acuity coming from a mathematician, at a time when the euclidian model of physical space had never been put into question; in a strictly logical sense, it was rather the discontinuous which had traditionally been used as a technical mode of approach towards the continuous. 

~ A. Grothendieck, Récoltes et Semailles.

5. I can easily picture young Vladimir Abramovich (as per his fellow students' recollections) leaving Steklov Institute library building at night, stretching, and saying, "Today is not my day, proved only seven theorems."

~ V. Arnold,  About Vlamidir Abramovich Rokhlin.

6. "Mathematics today," he would say to me, "is like an exclusive aristocratic club. On top of a tremendous initiation fee, a hefty annual fee is required from its members".

~ V. Arnold,  About Vlamidir Abramovich Rokhlin.

7. In the following years, within the mathematical world which welcomed me, I had the opportunity to meet multiple people, both older and younger, which were clearly more brilliant, “gifted” than I was. I admired the facility with which they learned new notions, as if at play, juggling them as if they had known them their whole life - while I felt heavy-handed and clumsy, laboriously making my way, akin to a mole, through an amorphous mountain of important things (or so I was told) which I had to learn, despite having no sense of their ins and out. Actually, I was far from the brilliant student who aced every prestigious concours and assimilating at once the most prohibitive courses.

Many of my more brilliant peers went on to become competent famous mathematicians. In hindsight, after 30-35 years, it does not seem to me that they left a deep imprint upon the mathematics of today. They did things, often times beautiful things, in a pre-existing context which they would never have considered altering. They unknowingly remained prisoners in their imperious circles, which delimitate the Universe of a given time and milieu. In order to overcome them, they would have had to rediscover within them the ability which they had since birth, just as I did: the capacity to be alone.

The small child has no difficulty being alone. He is solitary by nature, even though he enjoys the occasional company, and knows when to ask for mom’s permission teat. And he knows, without having ever been told, that the teat is his, and that he knows how to drink. Yet often times we lose touch with out inner child. And thus we constantly miss out on the best without even seeing it...

I address myself to you, reader, as I would a person, and a person alone. It is to the person inside of you that knows how to be alone, the child, with whom I would like to speak, and nobody else. I am aware that the child is often far away. He has gone through all sorts of things for quite some time. He went hiding god knows where, and it can be hard, often times, to get to him. One could swear that he has been dead forever, or rather that he has never existed - and yet I am sure that he is there somewhere, well alive.

~ A. Grothendieck, Récoltes et Semailles.

8. Over the years I gave a number of talks at his seminar with variable success. The most disastrous was my last talk in 1985. Shortly before one of my trips to Moscow, Misha Gromov sent me a preliminary version of his now very famous paper “Pseudoholomorphic curves in symplectic geometry”, which is one of the major foundational milestones of symplectic topology. I was extremely excited about this paper and thus volunteered to talk about it at Arnold’s seminar. I think that I was at this moment the only person in the Soviet Union who had the paper. Arnold heard about Gromov’s breakthrough but had not seen the paper yet. After a few minutes of my talk, Arnold interrupted me and requested that before continuing I should explain what is the main idea of the paper. This paper is full of new ideas and, in my opinion, it is quite subjective to say which one is the main one. I made several attempts to start from different points, but Arnold was never satisfied. Finally, towards the end of the two-hour long seminar, I said something which Arnold liked. “Why did you waste our time and did not start with this from the very beginning?”, he demanded.

~ Yakov Eliashberg, My Encounters with Vladimir Igorevich Arnold. 

9. The seminar started in 1943; I saw its later years, which coincided with the late period of the Soviet Union. After Stalin’s death the edifice of the state shrank into itself, and free space teemed with life. The ideology had lost its fulcrum, the show of democracy was simple (a single candidate to vote for, not two equally unpalatable ones), newspapers were mostly used as toilet tissue. The remaining taboos were private commerce and entrepreneurship, and political activity outside the Party’s womb. Many people shared the attitude of Pushkin’s poem “From Pindemonte” and viewed all matters political as not interesting anyway. The market in a modern sense, this incessant gavage of unneeded things, did not exist. One could quit the tarmac road to look for one’s own trail into the woods. If the trail happened to be mathematics, it would surely meet with Israel Moiseevich’s seminar.

There was a distinct inner music. The air was thin and transparent. One could hear the sound of one’s breathing, of snowflakes falling, of hoarfrost’s brush decorating the windowpane. Old villages still existed within Moscow limits, such as wonderful Dyakovo, its empty church over an ancient cemetery on a high scarp above Moscow River, wooden houses edged by deep ravines, and vast apple gardens where nightingales sang. Poetry was by far more real than social ranks - poems were rewritten by hand and learned by heart.

... In the early 1970s the high winds of the Cold War brought permission for Soviet Jews to emigrate, and many signed up for what, with hindsight, was a verification of the universality of Griboyedov’s quip that the place where it is better for us to be is where we are not. The separation from friends was deemed to be permanent (the imminent demise of the Soviet Union was anticipated then no more than that of the US is now). Dima Kazhdan, Ilya Iosifovich Piatetski-Shapiro, and Osya Bernstein, with whom we were happily doing math for his last half year in Moscow, were among those who left. No one at the seminar could replace them.

~ Alexander Beilinson, I.M. Gelfand and His Seminar - A Presence.

10. “From here it might seem,” Vladimir Abramovich used to say, “that over there in the West they have more justice, and a scientist is judged according to his scientific achievements and not according to the party membership and ethnicity, as it is done here. But on closer examination the state of things proves to be even better here because there are two distinct groups that never mix together – ambitious social climbers and true mathematicians. We know who belongs to which group, and since there could not be any real scientists in the first group, we are able to evaluate true mathematicians in a more scientific and fair manner than in the West where scientific motives cannot be distinguished from ambition-driven ones.”

~ V. Arnold,  About Vlamidir Abramovich Rokhlin.

11. Thus, this reflection/testimony/journey is not meant to be read hastily, in a day or in a month, by a reader rushed to reach the final word. There is no “final word”, no “conclusion” in Récoltes et Semailles, no more than there are any such things in my life or in yours. There is only a wine, aged over the course of a lifetime, at the core of my being. The last glass which you will be drinking will be no better nor worse than the first or the hundredth. They are all “the same”, and they are all different. And if the first glass is spoiled, so is the rest of the barrel; it is better to drink fresh water (if such can be found), than to drink bad wine.

But a good wine ought not to be drunk in haste, nor expeditiously.

~ A. Grothendieck, Récoltes et Semailles.

12. The descent into the depths always seems to precede the ascent. Thus another theologian dreamed that he saw on a mountain a kind of Castle of the Grail. He went along a road that seemed to lead straight to the foot of the mountain and up it. But as he drew nearer he discovered to his great dis­appointment that a chasm separated him from the mountain, a deep, darksome gorge with underworldly water rushing along the bottom. A steep path led downwards and toilsomely climbed up again on the other side. But the prospect looked uninviting, and the dreamer awoke. Here again the dreamer, thirsting for the shining heights, had first to descend into the dark depths, and this proves to be the indispensable condition for climbing any higher. The prudent man avoids the danger lurking in these depths, but he also throws away the good which a bold but imprudent venture might bring.

~ C. Jung, Archetypes of the Collective Unconscious.

13. How corrupting boredom is, everyone recognizes also with regard to children. As long as children are having a good time, they are always good. This can be said in the strictest sense, for if they at times become unmanageable even while playing, it is really because they are beginning to be bored; boredom is already coming on, but in a different way. Therefore, when selecting a nursemaid, one always considers essentially not only I that she is sober, trustworthy, and good-natured but also takes into esthetic consideration whether she knows how to entertain children. Even if she had all the other excellent virtues, one would not hesitate to give her the sack if she lacked this qualification. Here, indeed, the principle is clearly acknowledged, but things go on so curiously in the world, habit and boredom have gained the upper hand to such a degree, that justice is done to esthetics only in the conduct of the nursemaid. It would be quite impossible to prevail if one wanted to demand a divorce because one's wife is boring, or demand that a king be dethroned because he is boring to behold, or that a clergyman be exiled because he is boring to listen to, or that a cabinet minister be dismissed or a journalist be executed because he is frightfully boring. 

... This rotation of crops is the vulgar, inartistic rotation and is based on an illusion. One is weary of living in the country and moves to the city; one is weary of one's native land and goes abroad; one is europamiide [weary of Europe] and goes to America etc.; one indulges in the fanatical hope of an endless journey from star to star. Or there is another direction, but still extensive. One is weary of eating on porcelain and eats on silver; wearying of that, one eats on gold; one burns down half of Rome in order to visualize the Trojan conflagration. This method cancels itself and is the spurious infinity. What, after all, did Nero achieve? No, then the emperor Antoninus was wiser; he says: You can begin a new life. Only see things afresh as you used to see them. In this consists the new life (Book VII, 2).

The method I propose does not consist in changing the soil but, like proper crop rotation, consists in changing the method of cultivation and the kinds of crops. Here at once is the principle of limitation, the sole saving principle in the world. The more a person limits himself, the more resourceful he becomes. A solitary prisoner for life is extremely resourceful; to him a spider can be a source of great amusement. Think of our school days; we were at an age when there was no esthetic consideration in the choosing of our teachers, and therefore they were often very boring-how resourceful we were then! What fun we had catching a fly, keeping it prisoner under a nutshell, and watching it run around with it! What delight in cutting a hole in the desk, confining a fly in it, and peeking at it through a piece ofpaper! How entertaining it can be to listen to the monotonous dripping from the roofl What a meticulous observer one becomes, detecting every little sound or movement. Here is the extreme boundary of that principle that seeks relief not through extensity but through intensity.

~ S. Kierkegaard, Rotation of Crops, a Venture in a Theory of Social Produnce.

14. The very idea of a scheme is of a childlike simplicity - so simple, so humble, that no one before me had even thought to look so low. So “silly”, in fact, that for many years and despite the evidence pointing to the contrary, many of my erudite colleagues found this whole affair “not serious”! It actually took me months of intense and solitary work, to convince myself that it indeed “worked” just fine - that this new language, so silly, that I had the incorrigible naivety to persist in testing, was after all adequate in capturing in a new light and with greater finesse, and in a common framework, some of the very first geometric intuitions attached to the prior “geometries in characteristic p”. It was the kind of exercise, considered mindless and doomed in advance by any “well informed” person, which I was without a doubt the only one, among all of my colleagues and friends, to ever dare attempt, and even (under the impulse of some secret demon...) carry to a successful end despite everybody’s expectation!

Rather than letting myself be distracted by the consensus which prevailed around me, regarding what is considered “serious” and what isn’t, I simply trusted, as I had before, the humble voice of things, and followed that within me which knew how to listen. The reward was immediate, and beyond all expectations. Within the span of a few months, without even “meaning to”, I had discovered powerful and unexpected new tools. They allowed me to not only recover old results, reputedly difficult, in a more telling light, but also to surpass them, as well as to finally tackle and solve problems in “geometry in characteristic p” which to that day seemed out of reach using the methods known at the time. 

In our probing of things in the Universe (mathematical or otherwise), we dispose of a crucial rehabilitating power: innocence. By this I mean the original innocence which we have all received at birth and which rests within us, often the target of our scorn and of our deepest fears. It alone unites the humility and audacity which allow us to penetrate into the heart of things, while also allowing these things to penetrate into us and impregnate us with their meaning.

This power does not only come as a privilege for the extraordinarily “gifted” - (say) with an exceptional intellectual power allowing them to absorb and manipulate, with ease and dexterity, an impressing quantity of known facts, ideas, and techniques. Such gifts are admittedly precious, and susceptible to generate the envy of those (like myself) who were not so gifted at birth, “beyond all measure”.

Yet, it is not those gifts, nor even the most burning of ambitions, accompanied with a relentless will, which allow us to cross the “invisible and imperious circles” which enclose our Universe. Only innocence can cross them, without noticing or even caring to, during the times where we find ourselves alone and listening to the voice of things, intensely absorbed in child’s play...

~ A. Grothendieck, Récoltes et Semailles