MacTutor
But mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform.
Although this may seem a paradox, all exact science is dominated by the idea of approximation.
At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mathematics was my chief interest and my chief source of happiness.
A good notation has a subtlety and suggestiveness which at times make it almost seem like a live teacher.
If I were a medical man, I should prescribe a holiday to any patient who considered his work important.
Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say.
With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway about the flux. A little of this, but not much, I have achieved.
At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it.
Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be.
The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.
A habit of basing convictions upon evidence, and of giving to them only that degree or certainty which the evidence warrants, would, if it became general, cure most of the ills from which the world suffers.
The method of "postulating" what we want has many advantages; they are the same as the advantages of theft over honest toil.
Aristotle maintained that women have fewer teeth than men; although he was twice married, it never occurred to him to verify this statement by examining his wives' mouths.
[Upon hearing via Littlewood an exposition on the theory of relativity:]
To think I have spent my life on absolute muck.
"But," you might say, "none of this shakes my belief that 2 and 2 are 4." You are quite right, except in marginal cases -- and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a meter. Two must be two of something, and the proposition "2 and 2 are 4" is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. "Well, at any rate there are four animals," you may say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. "Well, then living organisms," you say. But there are things of which it is doubtful whether they are living organisms or not. You will be driven into saying: "Two entities and two entities are four entities." When you have told me what you mean by "entity," we will resume the argument.
I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
Men who are unhappy, like men who sleep badly, are always proud of the fact.
Work is of two kinds: first, altering the position of matter at or near the earth's surface relatively to other such matter; second, telling other people to do so. The first kind is unpleasant and ill paid; the second is pleasant and highly paid.
A sense of duty is useful in work but offensive in personal relations. Certain characteristics of the subject are clear. To begin with, we do not, in this subject, deal with particular things or particular properties: we deal formally with what can be said about "any" thing or "any" property. We are prepared to say that one and one are two, but not that Socrates and Plato are two, because, in our capacity of logicians or pure mathematicians, we have never heard of Socrates or Plato. A world in which there were no such individuals would still be a world in which one and one are two. It is not open to us, as pure mathematicians or logicians, to mention anything at all, because, if we do so we introduce something irrelevant and not formal.
The desire to understand the world and the desire to reform it are the two great engines of progress.
It can be shown that a mathematical web of some kind can be woven about any universe containing several objects. The fact that our universe lends itself to mathematical treatment is not a fact of any great philosophical significance.
Every living thing is a sort of imperialist, seeking to transform as much as possible of its environment into itself... When we compare the (present) human population of the globe with... that of former times, we see that "chemical imperialism" has been... the main end to which human intelligence has been devoted.
Almost everything that distinguishes the modern world from earlier centuries is attibutable to science, which achieved its most spectacular triumphs in the seventeenth century.
The whole problem with the world is that fools and fanatics are always so certain of themselves, but wiser people are so full of doubts.
The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it.
Boredom is a vital problem for the moralist, since at least half the sins of mankind are caused by the fear of it.
Drunkenness is temporary suicide: the happiness it brings is merely negative, a momentary cessation of unhappiness.
Few people can be happy unless they hate some other person, nation or creed.
No one gossips about other people's secret virtues.
Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture.
Physics is mathematical, not because we know so much about the physical world, but because we know so little. It is only the mathematical properties that we can discover.