Quotes of All Topics . Occasions . Authors
But it is the knowledge of necessary and eternal truths which distinguishes us from mere animals, and gives us reason and the sciences, raising us to knowledge of ourselves and God. It is this in us which we call the rational soul or mind.
. . . by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.
But in the present century, thanks in good part to the influence of Hilbert, we have come to see that the unproved postulates with which we start are purely arbitrary. They must be consistent, they had better lead to something interesting.
I was interviewed on the Israeli radio for five minutes and I said that more than 2000 years ago, Euclid proved that there are infinitely many primes. Immediately the host interrupted me and asked, 'Are there still infinitely many primes?'
There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture — that it came to him from outside, and that he did not consciously create it from within.
There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture - that it came to him from outside, and that he did not consciously create it from within.
Christianity is strange. It bids man recognise that he is vile, even abominable, and bids him desire to be like God. Without such a counterpoise, this dignity would make him horribly vain, or this humiliation would make him terribly abject.
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality.
Common integration is only the memory of differentiation... The different artifices by which integration is effected, are changes, not from the known to the unknown, but from forms in which memory does not serve us to those in which it does.
The Christian's God does not merely consist of a God who is the Author of mathematical truths and the order of the elements. The God of Abraham, the God of Isaac, the God of Jacob, the God of the Christians, is a God of love and consolation.
Either there are no corporeal substances, and bodies are merely phenomena which are true or consistent with each other, such as a rainbow or a perfectly coherent dream, or there is in all corporeal substances something analogous to the soul.
When Da Vinci wanted an effect, he willed, he planned the means to make it happen: that was the purpose of his machines. But the machines of Newton ... are means not for doing but for observing. He saw an effect, and he looked for its cause.
If there is anything that can bind the heavenly mind of man to this dreary exile of our earthly home and can reconcile us with our fate so that one can enjoy living,-then it is verily the enjoyment of the mathematical sciences and astronomy.
...an incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect, just as a criminal policy is none the less criminal even if it cannot be inhibited by any court that would curb it.
My older brother was the person who got me interested in science in general. He used to tell me what he learned in school. My first memory of mathematics is probably the time that he told me about the problem of adding numbers from 1 to 100.
Adults have been brainwashed into thinking that they can't really learn about computers without being taught, so it's more difficult for them to feel comfortable with a computer. Deep down, I think they're afraid of learning about computers.
The first thing to note is that pornography and many abductions occur apart from the use of computers, and that most child abuse happens within the family. So I think the extra degree of danger that computers pose doesn't justify the frenzy.
I don't like MTV, and I don't like the culture that goes with it. It's OK in very small doses, maybe. Nevertheless, it's a social reality and has influenced how kids perceive things around them, the pace of life and the way people do things.
An announcement of [Christopher] Zeeman's lecture at Northwestern University in the spring of 1977 contains a quote describing catastrophe theory as the most important development in mathematics since the invention of calculus 300 years ago.
The standard high school curriculum traditionally has been focused towards physics and engineering. So calculus, differential equations, and linear algebra have always been the most emphasized, and for good reason - these are very important.
You know what speed is. You would not believe a man who claimed to walk at 5 miles an hour, but took 3 hours to walk 6 miles. You have only to apply the same common sense to stones rolling down hillsides, and the calculus is at your command.
I am perfectly convinced that I have both seen, and heard in a manner which should make unbelief impossible, things called spiritual which cannot be taken by a rational being to be capable of explanation by imposture, coincidence, or mistake.
Belief is a wise wager. Granted that faith cannot be proved, what harm will come to you if you gamble on its truth and it proves false? If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation, that He exists.
Whenever a man can get hold of numbers, they are invaluable: if correct, they assist in informing his own mind, but they are still more useful in deluding the minds of others. Numbers are the masters of the weak, but the slaves of the strong.
It is difficult to estimate the misery inflicted upon thousands of persons, and the absolute pecuniary penalty imposed upon multitudes of intellectual workers by the loss of their time, destroyed by organ-grinders and other similar nuisances.
Poincaré [was] the last man to take practically all mathematics, pure and applied, as his province. ... Few mathematicians have had the breadth of philosophic vision that Poincaré had, and none in his superior in the gift of clear exposition.
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.
A deliberate choice on my part was for the player to continue to find new possibilities in the early Attic rooms far into the game. I think this builds atmosphere, though it means there's no neat division of the prologue from the middle game.
The air in a man's lungs 10,000,000,000, 000,000,000,000 atoms, so that sooner or later every one of us breathes an atom that has been breathed before by anyone you can think of who has ever lived - Michelangelo or George Washington or Moses.
... You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS math.
In the meantime, most noble Sir, you have assigned this question to the geometry of position, but I am ignorant as to what this new discipline involves, and as to which types of problem Leibniz and Wolff expected to see expressed in this way.
Until we in the community have made up our minds that what we really want is expiation, or removal, or reform, or or the discouragement of potential criminals, we shall get none of these, but only a confusion in which crime breeds more crime.
Working with the computer gives rise to many opportunities to transcend asocial behavior, because it produces exciting and visually interesting things to share, whether it's by creating video games, computer art or sharing exciting Web sites.
The danger to society is not merely that it should believe wrong things, though that is great enough; but that it should become credulous, and lose the habit of testing things and inquiring into them; for then it must sink back into savagery.
The two principles of truth, reason and senses, are not only both not genuine, but are engaged in mutual deception. The senses deceive reason through false appearances, and the senses are disturbed by passions, which produce false impressions.
Now this connection or adaption of all created things with each, and of each with all the rest, means that each simple substance has relations which express all the others, and that consequently it is a perpetual living mirror of the universe.
By the worldly standards of public life, all scholars in their work are of course oddly virtuous. They do not make wild claims, they do not cheat, they do not try to persuade at any cost, they appeal neither to prejudice nor to authority . . .
So far as hypotheses are concerned, let no one expect anything certain from astronomy, which cannot furnish it, lest he accept as the truth ideas conceived for another purpose, and depart from this study a greater fool than when he entered it.
Intelligent design is a modest position theologically and philosophically. It attributes the complexity and diversity of life to intelligence, but does not identify that intelligence with the God of any religious faith or philosophical system.
The fact of the religious vision, and its history of persistent expansion, is our one ground for optimism. Apart from it, human life is a flash of occasional enjoyments lighting up a mass of pain and misery, a bagatelle of transient experience.
Conventional wisdom, fooled by our misleading "physical intuition", is that the real world is "continuous", and that discrete models are necessary evils for approximating the "real" world, due to the innate discreteness of the digital computer.
Thought must never submit, neither to a dogma, nor to a party, nor to a passion, nor to an interest, nor to a preconceived idea, nor to whatever it may be, save to the facts themselves, because, for thought, submission would mean ceasing to be.
That men should live honestly, quietly, and comfortably together, it is needful that they should live under a sense of God's will, and in awe of the divine power, hoping to please God, and fearing to offend Him, by their behaviour respectively.
Every writer must reconcile, as best he may, the conflicting claims of consistency and variety, of rigour in detail and elegance in the whole. The present author humbly confesses that, to him, geometry is nothing at all, if not a branch of art.
Progress imposes not only new possibilities for the future but new restrictions. It seems almost as if progress itself and our fight against the increase of entropy intrinsically must end in the downhill path from which we are trying to escape.
It is very humbling to receive the Fields Medal. The words of a Fields Medallist carry a lot of weight within mathematics - for instance, in framing future directions of research - which means that I have to watch what I say more carefully now!
There is nothing that can be said by mathematical symbols and relations which cannot also be said by words. The converse, however, is false. Much that can be and is said by words cannot successfully be put into equations, because it is nonsense.
That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
The purely formal language of geometry describes adequately the reality of space. We might say, in this sense, that geometry is successful magic. I should like to state a converse: is not all magic, to the extent that it is successful, geometry?
As professor in the Polytechnic School in Zürich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic.