After all, the whole purpose of science is not technology - God knows we have gadgets enough already; science also explores depths of a universe that will never, by any stretch of the imagination, be visited by human beings or affect our material existence. So we shall attend also to some of the things which the great mathematicians have considered worthy of loving understanding for their intrinsic beauty.
Lagrange believed that a mathematician has not thoroughly understood his own work till he has made it so clear that he can go out and explain it effectively to the first man he meets on the street.
... the first time something new is studied the details seem too numerous and hopelessly confused, and no coherent impression of the whole is left on the mind. Then, on returning after a rest,... (查看原文)
It is probably correct to say that as a class they (mathematicians) have tended slightly to the left in their political opinions.
An impartial account of western mathematics, including the award to each man and to each nation of its just share in the intricate development, could be written only by a Chinese historian. He alone would have the patience and the detached cynicism necessary for disentangling the curiously perverted pattern to discover whatever truth may be concealed in our variegated occidental boasting.
... the nineteenth century, prolonged into the twentieth, was, and is, the greatest age of mathematics the world has ever known. Compared to what glorious Greece did in mathematics the nineteenth century is a bonfire beside a penny candle.
From the earliest times two opposing t... (查看原文)
Geometry partakes of both the continuous and the discrete.
A major task of mathematics to-day is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both.
If we care to inspect the symbols in nature's great book through the critical eyes of modern science we soon perceive that we ourselves did the writing, and that we used the particular script we did because we invented it to fit our own understanding. Some day we may find a more expressive shorthand than mathematics for correlating our experiences of the physical universe - unless we accept the creed of the scientific mystics that everything is mathematics and is not merely described for our convenience in mathematical language. If 'Number rules the universe' as ... (查看原文)
'Mathematics, rightly viewed, possesses not only truth but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.' (查看原文)
The nineteenth century, on this scale, contributed to mathematical knowledge about five times as much as was done in the whole of preceding history.
More important than the technical algebra of these ancient Babylonians is their recognition - as shown by their work - of the necessity for proof in mathematics. Until recently it had been supposed that the Greeks were the first to recognize that proof is demanded for mathematical propositions. (查看原文)
Mathematics then has had four great ages: the Babylonian, the Greek, the Newtonian (to give the period around 1700 a name), and the recent, beginning about 1800 and continuing to the present day. Competent judges have called the last the Golden Age of Mathematics. (查看原文)
Of all the ancients Archimedes is the only one who habitually thought with the unfettered freedom that the greater mathematicians permit themselves to-day with all the hard-won gains of twenty-five centuries to smooth their way, for he alone of all the Greeks had sufficient stature and strength to stride clear over the obstacles thrown in the path of mathematical progress by frightened geometers who had listened to the philosophers. [...] Had the Greek mathematicians and scientists followed Archimedes rather than Euclid, Plato, and Aristotle, they might easily have anticipated the age of modern mathematics, which began with Descartes (1596~1650) and Newton in the seventeenth century, and the age of modern physical science inaugurated by Galileo (1564~1642) in the same century, by 2,000 yea... (查看原文)
... the common whole numbers 1,2,3 ... are insufficient for the construction of mathematics even in the rudimentary form in which he (Pythagoras) knew it. [...] This was what knocked his theory flat: it is impossible to find two whole numbers such that the square of one of them is equal to twice the square of the other. [...] Actually Pythagoras found his stumbling-block in geometry: the ratio of the side of a square to one of its diagonals cannot be expressed as the ratio of any two whole numbers. This is equivalent to the statement above about squares of whole numbers. In another form we would say that the square root of 2 is irrational, that is, is not equal to any whole number or decimal fraction, or sum of the two, got by dividing one whole number by another. (查看原文)
Archimedes made his own occasions. Sitting before the fire he would rake out the ashes and draw in them. After stepping from the bath he would anoint himself with olive oil, according to the custom of the time, and then, instead of putting on his clothes, proceed to lose himself in the diagrams which he traced with a finger-nail on his own oily skin. (查看原文)
In short he (Archimedes) used his mechanics to advance his mathematics. This is one of his titles to a modern mind: he used anything and everything that suggested itself as a weapon to attack his problems. (查看原文)
It must not be imagined that the sole function of mathematics - 'the handmaiden of the sciences' - is to serve science. Mathematics has also been called 'the Queen of the Sciences.' If occasionally the Queen has seemed to beg from the sciences she has been a very proud sort of beggar, neither asking nor accepting favours from any of her more affluent sister sciences. What she gets she pays for.
... 'the three problems of antiquity': to trisect an angle; to construct a cube having double the volume of a given cube; to construct a square equal to a circle. None of these problems is possibly with only straightedge and compass, although it is hard to prove that the third is not, and the impossibility was finally proved only in 1882. (查看原文)
All constructions effected with other implements were dubbed 'mechanical' and, as such, for some mystical reason known only to Plato and his geometrizing God, were considered shockingly vulgar and were rigidly taboo in respectable geometry. Not till Descartes, 1,985 years after the death of Plato, published his analytical geometry, did geometry escape from its Platonic straightjacket (查看原文)
