Mathematical Quotes

One can prove or refute anything at all with words. Soon people will perfect language technology to such an extent that they'll be proving with mathematical precision that twice two is seven.

For many years I was the youngest among my mathematical friends. It makes me melancholy to realize that I now have become the oldest in most groups of scientists.

The universe can best be pictured as consisting of pure thought, the thought of what for want of a better word we must describe as a mathematical thinker.

We will never fully explain the world by appealing to something outside it that must simply be accepted on faith, be it an unexplained God or an unexplained set of mathematical laws.

We've taught you that the earth is round, That red and white make pink, And something else that matters more - We've taught you how to think.

Ignorance and superstition ever bear a close and mathematical relation to each other.

It is the heart which perceives God and not the reason.

The mathematical expectation of the speculator is zero.

I am giving this winter two courses of lectures to three students, of which one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the onera of a mathematical profession.

Thus not only the mental and the material, but the theoretical and the practical in the mathematical world, are brought into more intimate and effective connection with each other.

The mathematical fraternity is a little like a self-perpetuating priesthood. The mathematicians of today teach the mathematicians of tomorrow and, in effect, decide whom to admit to the priesthood.

Our minds are finite, and yet even in these circumstances of finitude we are surrounded by possibilities that are infinite, and the purpose of life is to grasp as much as we can out of that infinitude.

You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.

Mathematical studies may serve for a pleasant entertainment for those hours which young men are apt to throw away upon their vices.

It is clear that economics, if it is to be a science at all, must be a mathematical science.

Whoever despises the high wisdom of mathematics nourishes himself on delusion and will never still the sophistic sciences whose only product is an eternal uproar.

Mathematical economics is old enough to be respectable, but not all economists respect it. It has powerful supporters and impressive testimonials, yet many capable economists deny that mathematics, except as a shorthand or expository device, can be applied to economic reasoning. There have even been rumors that mathematics is used in economics (and in other social sciences) either for the deliberate purpose of mystification or to confer dignity upon common places as French was once used in diplomatic communications.

The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.

If I were asked to name, in one word, the pole star round which the mathematical firmament revolves, the central idea which pervades the whole corpus of mathematical doctrine, I should point to Continuity as contained in our notions of space, and say, it is this, it is this!

The Mandelbrot set is the most complex mathematical object known to mankind.

The shortest path between two truths in the real domain passes through the complex domain.

Paradox is the sharpest scalpel in the satchel of science. Nothing concentrates the mind as effectively, regardless of whether it pits two competing theories against each other, or theory against observation, or a compelling mathematical deduction against ordinary common sense.

I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much. But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming . . . and a little mad.

We do not worry about being respected in towns through which we pass. But if we are going to remain in one for a certain time, we do worry. How long does this time have to be?

What I love about cooking is that after a hard day, there is something comforting about the fact that if you melt butter and add flour and then hot stock, it will get thick! It's a sure thing! It's a sure thing in a world where nothing is sure; it has a mathematical certainty in a world where those of us who long for some kind of certainty are forced to settle for crossword puzzles.

We know next to nothing with any certainty about Pythagoras, except that he was not really called Pythagoras. The name by which he is known to us was probably a nickname bestowed by his followers. According to one source, it meant ‘He who spoke truth like an oracle’. Rather than entrust his mathematical and philosophical ideas to paper, Pythagoras is said to have expounded them before large crowds. The world’s most famous mathematician was also its first rhetorician.

It is not certain that everything is uncertain.

Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity.

Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed. They are no exceptions to the rule that God always geometrizes. Their problems of form are in the first instance mathematical problems, their problems of growth are essentially physical problems, and the morphologist is, ipso facto, a student of physical science.

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.

It is easy to make out three areas where scientists will be concentrating their efforts in the coming decades. One is in physics, where leading theorists are striving, with the help of experimentalists, to devise a single mathematical theory that embraces all the basic phenomena of matter and energy. The other two are in biology. Biologists-and the rest of us too-would like to know how the brain works and how a single cell, the fertilized egg cell, develops into an entire organism

Mathematical Mark all mathematical heads, which be only and wholly bent to those sciences, how solitary they be themselves, how unfit to live with others, and how unapt to serve in the world.

It is more important that a proposition be interesting than that it be true. This statement is almost a tautology. For the energy of operation of a proposition in an occasion of experience is its interest and is its importance. But of course a true proposition is more apt to be interesting than a false one.

It is more important that a proposition be interesting than that it be true.

Though this be madness, yet there is method in't.

The method of differences is, in fact, a method of additions; and as it includes within its means a larger number of results attainable by addition simply, than any other mathematical principle, it was very appropriately selected as the basis on which to construct an Adding Machine, so as to give to the powers of such a machine the widest possible range.

Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.

Twice two makes four seems to me simply a piece of insolence. Twice two makes four is a pert coxcomb who stands with arms akimbo barring your path and spitting. I admit that twice two makes four is an excellent thing, but if we are to give everything its due, twice two makes five is sometimes a very charming thing too.

The principle is so perfectly general that no particular application of it is possible.

Wave particle duality is a core feature of our world. Or rather, we should say, it is a core feature of our mathematical descriptions of our world. But what is critical to note here is that, however ambiguous our images, the universe itself remains whole and is manifestly not fracturing into schizophrenic shards. It is this tantalizing wholeness and the thing itself that drives physicists onward like an eternally beckoning light that seems so teasingly near. It is always out of reach.

Sometimes I've believed as many as six impossible things before breakfast.

One can't believe impossible things.

Alice laughed. 'There's no use trying,' she said. 'One can't believe impossible things.' I daresay you haven't had much practice,' said the Queen. 'When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast. There goes the shawl again!

In the last two months I have been very busy with my own mathematical speculations, which have cost me much time, without my having reached my original goal. Again and again I was enticed by the frequently interesting prospects from one direction to the other, sometimes even by will-o'-the-wisps, as is not rare in mathematic speculations.

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.

Not only in geometry, but to a still more astonishing degree in physics, has it become more and more evident that as soon as we have succeeded in unraveling fully the natural laws which govern reality, we find them to be expressible by mathematical relations of surprising simplicity and architectonic perfection. It seems to me to be one of the chief objects of mathematical instruction to develop the faculty of perceiving this simplicity and harmony.

All the effects of Nature are only the mathematical consequences of a small number of immutable laws.

We have heard much about the poetry of mathematics, but very little of it has as yet been sung. The ancients had a juster notion of their poetic value than we. The most distinct and beautiful statements of any truth must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both.

More people have access to education today than ever before. But I cannot help but feel that the modern educational experience is not preparing us adequately to attend the rich banquet of life. Certainly the young people of today have mastered the use of technology and are capable of solving complex scientific and mathematical problems, but who and what do these serve if they cannot think for themselves? If they have no understanding of the meaning and purpose of their own lives? If they do not know who they are as individuals?

There is nothing mysterious, as some have tried to maintain, about the applicability of mathematics. What we get by abstraction from something can be returned.