Hardy Math Quotes

Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

{Replying to G. H. Hardy's suggestion that the number of a taxi (1729) was 'dull', showing off his spontaneous mathematical genius} No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 13 + 123 and 93 + 103.

Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity

I was advised to read Jordan's 'Cours d'analyse'; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.

No mathematician should ever allow him to forget that mathematics, more than any other art or science, is a young man's game. … Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work later; … [but] I do not know of a single instance of a major mathematical advance initiated by a man past fifty. … A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.

Plenty of mathematicians, Hardy knew, could follow a step-by-step discursus unflaggingly—yet counted for nothing beside Ramanujan. Years later, he would contrive an informal scale of natural mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of the day, he assigned an 80. To Ramanujan he gave 100.

I do not think that G. H. Hardy was talking nonsense when he insisted that the mathematician was discovering rather than creating... The world for me is a necessary system, and in the degree to which the thinker can surrender his thought to that system and follow it, he is in a sense participating in that which is timeless or eternal.

Littlewood, on Hardy's own estimate, is the finest mathematician he has ever known. He was the man most likely to storm and smash a really deep and formidable problem; there was no one else who could command such a combination of insight, technique and power.

Repetition until Your Learning Becomes Unconscious (Outsourced to Environment) While I implemented what I learned, my teacher would watch me from a distance. He let me struggle as I tried to remember what he had just shown me. The first time, applying what he taught took a lot of time and effort. So we did it again, and again, and again. Over time, I became competent and thus confident. Learning something new is all about memory and how you use it. At first, your prefrontal cortex—which stores your working (or short-term) memory—is really busy figuring out how the task is done. But once you’re proficient, the prefrontal cortex gets a break. In fact, it’s freed up by as much as 90 percent. Once this happens, you can perform that skill automatically, leaving your conscious mind to focus on other things. This level of performance is called automaticity, and reaching it depends on what psychologists call overlearning or overtraining. The process of getting a skill to automaticity involves four steps, or stages: Repeated learning of a small set of information. If you’re playing basketball, for instance, that might mean shooting the same shot over and over. The key here is to go beyond the initial point of mastery. Make your training progressively more difficult. You want to make the task harder and harder until it’s too hard. Then you bring the difficulty back down slightly, in order to stay near the upper limit of your current ability. Add time constraints. For example, some math teachers ask students to work on difficult problems with increasingly shortened timelines. Adding the component of time challenges you in two ways. First, it forces you to work quickly, and second, it saps a portion of your working memory by forcing it to remain conscious of the ticking clock. Practice with increasing memory load—that is, trying to do a mental task with other things on your mind. Put simply, it’s purposefully adding distractions to your training regimen.