2
" Total obscurity. Bilbo in Gollum's tunnel.
A mathematician's first steps into unknown territory constitute the first phase of a familiar cycle.
After the darkness comes a faint, faint glimmer of light, just enough to make you think that something is there, almost within reach, waiting to be discovered . . .
Then, after the faint, faint glimmer, if all goes well, you unravel the thread - and suddenly it's broad daylight! You're full of confidence, you want to tell anyone who will listen about what you've found.
And then, after day has broken, after the sun has climbed high into the sky, a phase of depression inevitably follows. You lose all faith in the importance of what you've achieved. Any idiot could have done what you've done, go find yourself a more worthwhile problem and make something of your life. Thus the cycle of mathematical research . . . "
― Cédric Villani , Birth of a Theorem: A Mathematical Adventure
6
" And John Nash, my mathematical hero, revolutionized analysis and geometry with the proof of three theorems in scarcely more than five years before succumbing to paranoid schizophrenia. There is a fine line, it is often said, between genius and madness. Neither of these concepts is well defined, however. And in the case not only of Grothendieck but also of Gödel and Nash, periods of mental derangement, so far from promoting mathematical productivity, actually precluded it. Innate versus acquired, a classic debate. Fischer, Grothendieck, Erdős, and Perelman were all Jewish. Of these, Fischer and Erdős were Hungarian. No one who is familiar with the world of science can have failed to notice how many of the most gifted mathematicians and physicists of the twentieth century were Jews, or how many of the greatest geniuses were Hungarian (many "
― Cédric Villani , Birth of a Theorem: A Mathematical Adventure
12
" There is another remark to be made regarding the conditions of this unconscious work, which is, that it is not possible, or in any case not fruitful, unless it is first preceded and then followed by a period of conscious work. These sudden inspirations are never produced (and this is sufficiently proved already by the examples I have quoted) except after some days of voluntary efforts which appeared absolutely fruitless, in which one thought one had accomplished nothing, and seemed to be on a totally wrong track. These efforts, however, were not as barren as one thought; they set the unconscious machine in motion, and without them it would not have worked at all, and would not have produced anything. "
― Cédric Villani , Les mathématiques sont la poésie des sciences (La tortue de Zénon)
14
" It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations, which appear to be the result of a preliminary sifting. Does it follow from this that the subliminal ego, having divined by a delicate intuition that these combinations could be useful, has formed none but these, or has it formed a great many others which were devoid of interest, and remained unconscious? Under this second aspect, all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness. This, too, is most mysterious. How can we explain the fact that, of the thousand products of our unconscious activity, some are invited to cross the threshold, while others remain outside? "
― Cédric Villani , Les mathématiques sont la poésie des sciences (La tortue de Zénon)
17
" If Nash attracted Hollywood’s attention, it wasn’t only on account of his mathematical exploits. It was also because of the tragic story of his life. At the age of thirty he succumbed to paranoid schizophrenia. In and out of psychiatric clinics and hospitals for more than ten years, he seemed fated to live out his days as a pitiable phantom haunting the halls of Princeton, his mind an incoherent ruin. But then, after three decades of purgatory, Nash miraculously came back from the far shores of madness. Today, more than eighty years old, he is as normal as you or I. Except that there is an aura about him that neither you nor I have, an aura due to phenomenal accomplishments, strokes of pure genius—and a way of dissecting and scrutinizing problems that makes Nash a model for all modern analysts, myself most humbly among them. "
― Cédric Villani , Birth of a Theorem: A Mathematical Adventure
18
" Often when a man is working at a difficult question, he accomplishes nothing the first time he sets to work. Then he takes more or less of a rest, and sits down again at his table. During the first half-hour he still finds nothing, and then all at once the decisive idea presents itself to his mind. We might say that the conscious work proved more fruitful because it was interrupted and the rest restored force and freshness to the mind. But it is more probable that the rest was occupied with unconscious work, and that the result of this work was afterwards revealed to the geometrician exactly as in the cases 1 have quoted, except that the revelation, instead of coming to light during a walk or a journey, came during a period of conscious work, but independently of that work, which at most only performs the unlocking process, as if it were the spur that excited into conscious form the results already acquired during the rest, which till then remained unconscious. "
― Cédric Villani , Les mathématiques sont la poésie des sciences (La tortue de Zénon)
