Quotes of All Topics . Occasions . Authors
Mr. Herschel... brought with him the calculations of the computers, and we commenced the tedious process of verification. After a time many discrepancies occurred, and at one point these discordances were so numerous that I exclaimed, "I wish to God these calculations had been executed by steam," to which Herschel replied, "It is quite possible."
What binds us to space-time is our rest mass, which prevents us from flying at the speed of light, when time stops and space loses meaning. In a world of light there are neither points nor moments of time; beings woven from light would live "nowhere" and "nowhen"; only poetry and mathematics are capable of speaking meaningfully about such things.
My aim is to say that the machinery of the heavens is not like a divine animal but like a clock (and anyone who believes a clock has a soul gives the work the honour due to its maker) and that in it almost all the variety of motions is from one very simple magnetic force acting on bodies, as in the clock all motions are from a very simple weight.
It is dangerous to tell the people that the laws are unjust; for they obey them only because they think them just. Therefore it isnecessary to tell them at the same time that they must obey them because they are laws, just as they must obey superiors, not because they are just, but because they are superiors. In this way all sedition is prevented.
Our imagination so magnifies this present existence, by the power of continual reflection on it, and so attenuates eternity, by not thinking of it at all, that we reduce an eternity to nothingness, and expand a mere nothing to an eternity; and this habit is so inveterately rooted in us that all the force of reason cannot induce us to lay it aside.
If the moon and earth were not retained in their orbits by their animal force or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance, and the moon fall towards the earth through the other fifty-three parts, and they would there meet, assuming, however, that the substance of both is of the same density.
The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
Foreshadowings of the principles and even of the language of [the infinitesimal] calculus can be found in the writings of Napier, Kepler, Cavalieri, Pascal, Fermat, Wallis, and Barrow. It was Newton's good luck to come at a time when everything was ripe for the discovery, and his ability enabled him to construct almost at once a complete calculus.
When Coleridge tried to define beauty, he returned always to one deep thought; beauty, he said, is unity in variety! Science is nothing else than the search to discover unity in the wild variety of nature,-or, more exactly, in the variety of our experience. Poetry, painting, the arts are the same search, in Coleridge's phrase, for unity in variety.
If theory is the role of the architect, then such beautiful proofs are the role of the craftsman. Of course, as with the great renaissance artists, such roles are not mutually exclusive. A great cathedral has both structural impressiveness and delicate detail. A great mathematical theory should similarly be beautiful on both large and small scales.
We re-make nature by the act of discovery, in the poem or in the theorem. And the great poem and the deep theorem are new to every reader, and yet are his own experiences, because he himself re-creates them. They are the marks of unity in variety; and in the instant when the mind seizes this for itself, in art or in science, the heart misses a beat.
God, possessing supreme and infinite wisdom, acts in the most perfect manner, not only metaphysically, but also morally speaking, and ... with respect to ourselves, we can say that the more enlightened and informed we are about God's works, the more we will be disposed to find them excellent and in complete conformity with what we might have desired.
When we would show any one that he is mistaken, our best course is to observe on what side he considers the subject,--for his view of if is generally right on this side,--and admit to him that he is right so far. He will be satisfied with this acknowledgment, that he was not wrong in his judgment, but only inadvertent in not looking at the whole case.
Many theories of the ancient world seem terribly childish today, a hodge-podge of fables and false comparisons.But our theories will seem childish five-hundred years from now.Every theory is based on some analogy, and sooner or later the theory fails because the analogy turns out to be false. A theory in its day helps to solve the problems of the day.
[Before the time of Benjamin Peirce it never occurred to anyone that mathematical research] was one of the things for which a mathematical department existed. Today it is a commonplace in all the leading universities. Peirce stood alone-a mountain peak whose absolute height might be hard to measure, but which towered above all the surrounding country.
We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of science are mathematics and logic; the mathematical set puts out the logical eye, the logical set puts out the mathematical eye; each believing that it sees better with one eye than with two. Note that De Morgan, himself, only had sight with only one eye.
Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted.
Even with the most stupid video games, kids learn more about learning than they ever did before, because they want to learn codes and moves before other kids figure them out. They're motivated to seek out someone or search the Net for help. A student who makes a video game has to solve mathematical problems to make special effects happen on the screen.
[About Francis Baily] The history of the astronomy of the nineteenth century will be incomplete without a catalogue of his labours. He was one of the founders of the Astronomical Society, and his attention to its affairs was as accurate and minute as if it had been a firm of which he was the chief clerk, with expectation of being taken into partnership.
If you look at coastlines, if you look at that them from far away, from an airplane, well, you don't see details, you see a certain complication. When you come closer, the complication becomes more local, but again continues. And come closer and closer and closer, the coastline becomes longer and longer and longer because it has more detail entering in.
In order to enter into a real knowledge of your condition, consider it in this image: A man was cast by a tempest upon an unknown island, the inhabitants of which were in trouble to find their king, who was lost; and having a strong resemblance both in form and face to this king, he was taken for him, and acknowledged in this capacity by all the people.
All that we can hope from these inspirations, which are the fruits of unconscious work, is to obtain points of departure for such calculations. As for the calculations themselves, they must be made in the second period of conscious work which follows the inspiration, and in which the results of the inspiration are verified and the consequences deduced.‎
On how the motion of a planet defines its sphere: ... and thus it comes about gradually by the linking and accumulation of a great many revolutions that a kind of concave sphere is displayed, having the same center as the Sun, just as by a great many circles of silken thread, linked with each other and wound together, the dwelling of a silkworm is made.
If you have a large number of unrelated ideas, you have to get quite a distance away from them to get a view of all of them, and this is the role of abstraction. If you look at each too closely you see too many details. If you get far away things may appear simpler because you can only see the large, broad outlines; you do not get lost in petty details.
People think they don’t understand math, but it’s all about how you explain it to them. If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.
The act of imagination is the opening of the system so that it shows new connections. Every act of act of imagination is the discovery of likenesses between two things which were thought unlike. An example is Newton’s thinking of the likeness between the thrown apple and moon sailing majestically in the sky. Hence, the ‘discovery’ of the laws of gravity.
Another advantage of a mathematical statement is that it is so definite that it might be definitely wrong; and if it is found to be wrong, there is a plenteous choice of amendments ready in the mathematicians' stock of formulae. Some verbal statements have not this merit; they are so vague that they could hardly be wrong, and are correspondingly useless.
Neither you nor I nor anybody else knows what makes a mathematician tick. It is not a question of cleverness. I know many mathematicians who are far abler than I am, but they have not been so lucky. An illustration may be given by considering two miners. One may be an expert geologist, but he does not find the golden nuggets that the ignorant miner does.
If people would stop objectifying abstractions (which they probably never will), or if they would stop objectifying the abstractions they make consciously (which they might learn to do), at least half the pseudo-questions befuddling the world today - as they have befuddled it since time immemorial - would vanish. And that would be a very, very great gain.
The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.
It is impossible to discuss realism in logic without drawing in the empirical sciences... A truly realistic mathematics should be conceived, in line with physics, as a branch of the theoretical construction of the one real world and should adopt the same sober and cautious attitude toward hypothetic extensions of its foundation as is exhibited by physics.
[Science] dissipates errors born of ignorance about our true relations with nature, errors the more damaging in that the social order should rest only on those relations. TRUTH! JUSTICE! Those are the immutable laws. Let us banish the dangerous maxim that it is sometimes useful to depart from them and to deceive or enslave mankind to assure its happiness.
Our senses perceive no extreme. Too much sound deafens us; too much light dazzles us; too great distance or proximity hinders ourview. Too great length and too great brevity of discourse tends to obscurity; too much truth is paralyzing.... In short, extremes are for us as though they were not, and we are not within their notice. They escape us, or we them.
There is a virtuous fear, which is the effect of faith; and there is a vicious fear, which is the product of doubt. The former leads to hope, as relying on God, in whom we believe; the latter inclines to despair, as not relying on God, in whom we do not believe. Persons of the one character fear to lose God; persons of the other character fear to find Him.
What a wonderful and amazing Scheme have we here of the magnificent Vastness of the Universe! So many Suns, so many Earths, and every one of them stock’d with so many Herbs, Trees and Animals, and adorn’d with so many Seas and Mountains! And how must our wonder and admiration be encreased when we consider the prodigious distance and multitude of the Stars?
There are also two kinds of truths, those of reasoning and those of fact. Truths of reasoning are necessary and their opposite is impossible, and those of fact are contingent and their opposite is possible. When a truth is necessary its reason can be found by analysis, resolving it into more simple ideas and truths until we reach those which are primitive.
It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word.
A considreable portion of my high school trigonometry course was devoted to the solution of oblique triangles... I have still not had an excuse for using my talents for solving oblique triangles. If a professional mathematician never uses these dull techniques in a highly varied career, why must all high school students devote several weeks to the subject?
Man is so great that his greatness appears even in the consciousness of his misery. A tree does not know itself to be miserable. It is true that it is misery indeed to know one's self to be miserable; but then it is greatness also. In this way, all man's miseries go to prove his greatness. They are the miseries of a mighty potentate, of a dethroned monarch.
[writing to Stirling in 1740] ... an unlucky accident happened to some of the French mathematicians in Peru. It seems that they were shewing French gallantry to the natives' wives, who have murdered their servants destroyed their instruments and burnt their papers, the Gentlemen escaping narrowly themselves. What an ugly article this will make in a journal.
I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways."
The discovery in 1846 of the planet Neptune was a dramatic and spectacular achievement of mathematical astronomy. The very existence of this new member of the solar system, and its exact location, were demonstrated with pencil and paper; there was left to observers only the routine task of pointing their telescopes at the spot the mathematicians had marked.
If my false figures came near to the facts, this happened merely by chance ... These comments are not worth printing. Yet it gives me pleasure to remember how many detours I had to make, along how many walls I had to grope in the darkness of my ignorance until I found the door which lets in the light of the truth ... In such manner did I dream of the truth.
Inventive genius requires pleasurable mental activity as a condition for its vigorous exercise. "Necessity is the mother of invention" is a silly proverb. "Necessity is the mother of futile dodges" is much closer to the truth. The basis of growth of modern invention is science, and science is almost wholly the outgrowth of pleasurable intellectual curiosity.
One of the high points of my life was when I suddenly realized that this dream I had in my late adolescence of combining pure mathematics, very pure mathematics with very hard things which had been long a nuisance to scientists and to engineers, that this combination was possible and I put together this new geometry of nature, the fractal geometry of nature.
The God of Christians is a God of love and comfort, a God who fills the soul and heart of those whom he possesses, a God who makes them conscious of their inward wretchedness, and his infinite mercy; who unites himself to their inmost soul, who fills it with humility and joy, with confidence and love, who renders them incapable of any other end than himself.
Nothing is so insufferable to man as to be completely at rest, without passions, without business, without diversion, without study. He then feels his nothingness, his forlornness, his insufficiency, his dependence, his weakness, his emptiness. There will immediately arise from the depth of his heart weariness, gloom, sadness, fretfulness, vexation, despair.
Let there be two possible things, A and B, one of which is such that it is necessary that it exists, and let us assume that there is more perfection in A than in B. Then, at least, we can explain why A should exist rather than B and can foresee which of them will exist; indeed, this can be demonstrated, that is, rendered certain from the nature of the thing.
D'Alembert was always surrounded by controversy. ... he was the lightning rod which drew sparks from all the foes of the philosophes. ... Unfortunately he carried this... pugnacity into his scientific research and once he had entered a controversy, he argued his cause with vigour and stubbornness. He closed his mind to the possibility that he might be wrong.
This common and unfortunate fact of the lack of adequate presentation of basic ideas and motivations of almost any mathematical theory is probably due to the binary nature of mathematical perception. Either you have no inkling of an idea, or, once you have understood it, the very idea appears so embarrassingly obvious that you feel reluctant to say it aloud.