## Metadata - Author: [[G. H. Hardy and C. P. Snow]] - Full Title: A Mathematician's Apology - Category: #books ## Highlights - This is a story of human virtue. Once people had started behaving well, they went on behaving better. It is good to remember that England gaveRamanujan such honours as were possible. The Royal Society elected him a Fellow at the age of thirty (which, even for a mathematician, is very young). Trinity also elected him a Fellow in the same year. He was the first Indian to be given either of these distinctions. He was amiably grateful. ([Location 239](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=239)) - `It is never worth a first class man's time to express a majority opinion. By definition, there are plenty of others to do that.' ([Location 318](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=318)) - `For any serious purpose, intelligence is a very minor gift.' ([Location 321](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=321)) - `Sometimes one has to say difficult things, but one ought to say them as simply as one knows how.' ([Location 323](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=323)) - In these days of conflict between ancient and modern studies, there must surely be something to be said for a study which did not begin with Pythagoras, and will not end with Einstein, but is the oldest and the youngest of all. ([Location 527](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=527)) - A MATHEMATICIAN, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. ([Location 570](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=570)) - The proof is by reductio ad absurdum, and 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 gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game. ([Location 631](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=631)) - It seems that mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below. The lower the stratum, the deeper (and in general the more difficult) the idea. Thus the idea of an `irrational' is deeper than that of an integer; and Pythagoras's theorem is, for that reason, deeper than Euclid's. ([Location 725](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=725)) - In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail-one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many `variations' in the proof of a mathematical theorem : `enumeration of cases', indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way. ([Location 742](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=742)) - It is indeed rather astonishing how little practical value scientific knowledge has for ordinary men, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to its reputedutility. It is useful to be tolerably quick at common arithmetic (and that, of course, is pure mathematics). It is useful to know a little French or German, a little history and geography, perhaps even a little economics. But a little chemistry, physics, or physiology has no value at all in ordinary life. We know that the gas will burn without knowing its constitution; when our cars break down we take them to a garage; when our stomach is out of order, we go to a doctor or a drugstore. We live either by rule of thumb or on other people's professional knowledge. ([Location 770](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=770)) - THE contrast between pure and applied mathematics stands out most clearly, perhaps, in geometry. There is the science of pure geometry*, in which there are many geometries, projective geometry, Euclidean geometry, non-Euclidean geometry, and so forth. Each ofthese geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern. It is a map or picture, the joint product of many hands, a partial and imperfect copy (yet exact so far as it extends)of a section of mathematical reality. But the point which is important to us now is this, that there is one thing at any rate of which pure geometries are not pictures, and that is the spatio-temporal reality of the physical world. It is obvious, surely, that they cannot be, since earthquakes and eclipses are not mathematical concepts.This may sound a little paradoxical to an outsider, but it is a truism to a geometer; and I may perhaps be able to make it clearer by an illustration. Let us suppose that I am giving a lecture on some system of geometry, such as ordinary Euclidean geometry, and that I draw figures on the blackboard to stimulate the imagination of my audience, rough drawings ofstraight lines or circles or ellipses. It is plain, first, that the truth of the theorems which I prove is in no way affected by the quality of my drawings. Their function is merely to bring home my meaning to my hearers, and, if I can do that, there would be no gain in having them redrawn by the most skilful draughtsman. They are pedagogical illustrations, not part of the real subject-matter of the lecture.Now let us go a stage further. The room in which I am lecturing is part of the physical world, and has itself a certain pattern. The study of that pattern, and of the general pattern of physical reality, is a science in itself, which we may call `physical geometry'. Suppose now that a violent dynamo, or a massive gravitating body, is introduced into the room. Then the physicists tell us that the geometry of the room is changed, its whole physical pattern slightly but definitely distorted. Do the theorems which I have proved become false? Surely it would be nonsense to suppose that the proofs of them which I have given are affected in any way. It would be like supposing that a play of Shakespeare is changed when a reader spills his tea over a page. The play is independent of the pages on which it is printed, and `pure geometries' are independent of lecture rooms, or of any other detail of the physical world.This is the point of view of a pure mathematician. Applied mathematicians, mathematical physicists, naturally take a different view, since they are preoccupied with thephysical world itself, which also has its structure or pattern. We cannot describe this pattern exactly, as we can that of a pure geometry, but we can say something significant about it. We can describe, sometimes fairly accurately, sometimes very roughly, the relations which hold between some… ([Location 810](https://readwise.io/to_kindle?action=open&asin=B000TPEPTS&location=810))