"Deligne’s method was totally perpendicular to Grothendieck’s: he knew every trick of his master’s trade by heart, every concept, every variant. His proof, given in 1974, is a frontal attack and a marvel of precision, in which the steps follow each other in an absolutely natural order, without surprises. Those who heard his lectures had the impression, day after day, that nothing new was happening–whereas every lecture by Grothendieck introduced a whole new world of concepts, each more general than the one before–but on the last day, everything was in place and victory was assured. Deligne knocked down the obstacles one after the other, but each one of them was familiar in style. I think that this opposition of methods, or rather of temperament, is the true reason behind the personal conflict which developed between the two of them. I also think that the fact that “John, the disciple that Jesus loved” wrote the last Gospel by himself partly explains Grothendieck’s furious exile that Grothendieck has imposed upon himself."

"In Récoltes et Semailles, Grothendieck counts his twelve disciples. The central character is Pierre Deligne, who combines in this tale the features of John, “the disciple whom Jesus loved”, and Judas the betrayer. The weight of symbols!"

"The nice thing about mathematics is doing mathematics."

""In almost all textbooks, even the best, this principle is presented so that it is impossible to understand." (K. Jacobi, Lectures on Dynamics, 1842-1843). I have not chosen to break with tradition."

"Let me just say that Arnold was a geometer in the widest possible sense of the word, and that he was very fast to make connections between different fields."

"The axiomization and algebraization of mathematics, after more than 50 years, has led to the illegibility so such a large number of mathematical texts that the threat of complete loss of contact with physics and the natural sciences has been realized."

"Mathematics is the part of physics where experiments are cheap."

"Such axioms, together with other unmotivated definitions, serve mathematicians mainly by making it difficult for the uninitiated to master their subject, thereby elevating its authority."

"When you are collecting mushrooms, you only see the mushroom itself. But if you are a mycologist, you know that the real mushroom is in the earth. There’s an enormous thing down there, and you just see the fruit, the body that you eat. In mathematics, the upper part of the mushroom corresponds to theorems that you see. But you don’t see the things which are below, namely problems, conjectures, mistakes, ideas, and so on. You might have several apparently unrelated mushrooms and are unable to see what their connection is unless you know what is behind."

"All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA.)."

"In the last 30 years, the prestige of mathematics has declined in all countries. I think that mathematicians are partially to be blamed as well—foremost, Hilbert and Bourbaki—the ones who proclaimed that the goal of their science was investigation of all corollaries of arbitrary systems of axioms."

"A person, who had not mastered the art of the proofs in high school, is as a rule unable to distinguish correct reasoning from that which is misleading. Such people can be easily manipulated by the irresponsible politicians."

"It is almost impossible for me to read contemporary mathematicians who, instead of saying “Petya washed his hands,” write simply: “There is a t_1<0 such that the image of t_1 under the natural mapping t_1 \mapsto {\rm Petya}(t_1) belongs to the set of dirty hands, and a t_2, t_1, such that the image of t_2 under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.”"

"In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun)."

"At the beginning of this century a self-destructive democratic principle was advanced in mathematics (especially by Hilbert), according to which all axiom systems have equal right to be analyzed, and the value of a mathematical achievement is determined, not by its significance and usefulness as in other sciences, but by its difficulty alone, as in mountaineering. This principle quickly led mathematicians to break from physics and to separate from all other sciences. In the eyes of all normal people, they were transformed into a sinister priestly caste . . . Bizarre questions like Fermat's problem or problems on sums of prime numbers were elevated to supposedly central problems of mathematics."

"Here was a problem, that I, a ten year old, could understand and I knew from that moment that I would never let it go. I had to solve it."

"Originally the Kolyvagin-Flach method only worked under particularly constrained circumstances, but Wiles believed he had adapted and strengthened it sufficiently to work for all his needs. According to Katz this was not necessarily the case, and the effects were dramatic and devastating. The error did not necessarily mean that Wiles's work was beyond salvation, but it did mean that he would have to strengthen his proof. The absolutism of mathematics demanded that Wiles demonstrate beyond all doubt that his method worked for every element of every E-series and M-series."

"Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that proceed them."

"I know it's a rare privilege, but if one can really tackle something in adult life that means that much to you, then it's more rewarding than anything I can imagine."

"I had this rare privilege of being able to pursue in my adult life, what had been my childhood dream."

"Always try the problem that matters most to you."

"However impenetrable it seems, if you don't try it, then you can never do it."

"Certainly one thing that I've learned is that it is important to pick a problem based on how much you care about it."

"But perhaps that's always the way with math problems, and we just have to find new ones to capture our attention."

"I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future."

"Fermat was my childhood passion."

"But what has made this problem special for amateurs is that there's a tiny possibility that there does exist an elegant 17th-century proof."

"I don't believe Fermat had a proof. I think he fooled himself into thinking he had a proof."

"Fermat couldn't possibly have had this proof."

"Young children simply aren't interested in Fermat. They just want to hear a story and they're not going to let you do anything else."

"I think I'll stop here."

"I really believed that I was on the right track, but that did not mean that I would necessarily reach my goal."

"I realized that anything to do with Fermat's Last Theorem generates too much interest."

"I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days."

"I loved doing problems in school. I'd take them home and make up new ones of my own."

"But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem -- Fermat's Last Theorem."