156 Mathematics Quotes

Quaternions came from Hamilton after his really good work had been done, and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.

Symmetrical equations are good in their place, but 'vector' is a useless survival, or offshoot from quaternions, and has never been of the slightest use to any creature.

Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense.

Instruction tables will have to be made up by mathematicians with computing experience and perhaps a certain puzzle-solving ability. There need be no real danger of it ever becoming a drudge, for any processes that are quite mechanical may be turned over to the machine itself.

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.

We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made.

I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days.

I don't believe Fermat had a proof. I think he fooled himself into thinking he had a proof.

I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future.

Those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or definitions, it is not true that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.

The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length... Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.

We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.

I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.

A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.

A mathematician is a blind man in a dark room looking for a black cat which isn't there.

Bear in mind the simple rule, X squared to the power of two minus five over the seven point eight three times nineteen is approximately equal to the cube root of MCC squared divided by X minus a quarter of a third percent. Keep that in mind, and you can't go very far wrong.

I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce.

The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design.

It is with children that we have the best chance of studying the development of logical knowledge, mathematical knowledge, physical knowledge, and so forth.

Do you think Arabs are dumb? They gave us our numbers. Try doing long division with Roman numerals.

Quaternions came from Hamilton after his really good work had been done, and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.

Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense.

Differential equations won't help you much in the design of aeroplanes - not yet, anyhow.

I have hardly ever known a mathematician who was capable of reasoning.

Evolution has ensured that our brains just aren't equipped to visualise 11 dimensions directly. However, from a purely mathematical point of view it's just as easy to think in 11 dimensions, as it is to think in three or four.

Equations are just the boring part of mathematics. I attempt to see things in terms of geometry.

I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.

Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings.

Equations are more important to me, because politics is for the present, but an equation is something for eternity.

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

I really believed that I was on the right track, but that did not mean that I would necessarily reach my goal.

I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.

I believe no woman could have invented calculus.

Aristotelian , Euclidian [line, space, length] and Newtonian [speed of light] systems... have a few unjustified infinities too many: when such an infinity is introduced into the denominator, it makes the whole expression vanish. In other words, faulty, insufficient observations leads to the introduction of ... fanciful 'infinities'.

Here was a problem, that I, a ten year old, could understand and I knew from that moment that I would never let it go. I had to solve it.

Young children simply aren't interested in Fermat. They just want to hear a story and they're not going to let you do anything else.

Fermat couldn't possibly have had this proof.

But what has made this problem special for amateurs is that there's a tiny possibility that there does exist an elegant 17th-century proof.

Fermat was my childhood passion.

Certainly one thing that I've learned is that it is important to pick a problem based on how much you care about it.

If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Godel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.

Its enduring value was simply a deeper understanding of the central concepts of mathematics and their basic laws and interrelationships. Their total translatability into just elementary logic and a simple familiar two-place predicate, membership, is of itself a philosophical sensation.

I will add that I have recently received from Hungary a little paper on non-Euclidean geometry in which I rediscover all my own ideas and results worked out with great elegance... The writer is a very young Austrian officer, the son of one of my early friends, with whom I often discussed the subject in 1798, although my ideas were at that time far removed from the development and maturity which they have received through the original reflections of this young man. I consider the young geometer J. Bolyai a genius of the first rank.

Gauss liked to call [number theory] 'the Queen of Mathematics'. For Gauss, the jewels in the crown were the primes, numbers which had fascinated and teased generations of mathematicians.

Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist.

The revelation that the graph appears to climb so smoothly, even though the primes themselves are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery of his formula for the number of primes up to N in terms of the logarithm function. Yet despite the importance of the discovery, Gauss told no one what he had found. The most the world heard of his revelation were the cryptic words, 'You have no idea how much poetry there is in a table of logarithms.'

Maybe we have become so hung up on looking at the primes from Gauss's and Riemann's perspective that what we are missing is simply a different way to understand these enigmatic numbers. Gauss gave an estimate for the number of primes, Riemann predicted that the guess is at worst the square root of N off its mark, Littlewood showed that you can't do better than this. Maybe there is an alternative viewpoint that no one has found because we have become so culturally attached to the house that Gauss built.

Now, to say that a lot of objects is finite, is the same as to say that if we pass through the class from one to another we shall necessarily come round to one of those individuals already passed; that is, if every one of the lot is in any one-to-one relation to one of the lot, then to every one of the lot some one is in this same relation.

Dave always arrives at the right conclusions, but his mathematics is terrible. I take it home and find all sorts of errors and then have to spend the night trying to develop the correct proof. But in the end, the result is always exactly the same as the one Dave saw directly.

Reality contains not only evidence, but also the means (such as our minds, and our artefacts) of understanding it. There are mathematical symbols in physical reality. The fact that it is we who put them there does not make them any less physical. (Ch.3)

The next chapter is likely to provoke many mathematicians. This can't be helped. Mathematics is not what they think it is. (Ch. 10)

Mathematical knowledge may, just like our scientific knowledge, be deep and broad, it may be subtle and wonderfully explanatory, it may be uncontroversially accepted; but it cannot be certain. (Ch. 10)

Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing mathematics. The object of mathematics is not, and cannot be, mathematical certainty. It is not even mathematical truth, certain or otherwise. It is, and must be, mathematical explanation.

The theory of computation has traditionally been studied almost entirely in the abstract, as a topic in pure mathematics. This is to miss the point of it. Computers are physical objects, and computations are physical processes. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics.

In our time, in the period 1960-1980, we have seen particle physics emerge as the playground of group theory.

You'll have received an application from Mr Freeman Dyson to come to work with you as a graduate student. I hope that you will accept him. Although he is only 23 he is in my view the best mathematician in England.

Allow me to define more specifically the inherent limitations of our numerical knowledge which are so often overlooked. I want to do this to avoid giving the impression that I generally reject the mathematical method in economics. I regard it in fact as the great advantage of the mathematical technique that it allows us to describe, by means of algebraic equations, the general character of a pattern even where we are ignorant of the numerical values which will determine its particular manifestation. We could scarcely have achieved that comprehensive picture of the mutual interdependencies of the different events in a market without this algebraic technique. It has led to the illusion, however, that we can use this technique for the determination and prediction of the numerical values of those magnitudes; and this has led to a vain search for quantitative or numerical constants.

Les mathematiciens n'etudient pas des objets, mais des relations entre les objets; il leur est donc indifferent de remplacer ces objets par d'autres, pourvu que les relations ne changent pas. La matiere ne leur importe pas, la forme seule les interesse.

Le but principal de l'enseignement mathematique est de developper certaines facultes de l'esprit et parmi elles l'intuition n'est pas la moins precieuse. C'est par elle que le monde mathematique reste en contact avec le monde reel et quand les mathematiques pures pourraient s'en passer, il faudrait toujours y avoir recours pour combler l'abime qui separe le symbole de la realite.

La logique nous apprend que sur tel ou tel chemin nous sommes surs de ne pas rencontrer d'obstacle; elle ne nous dit pas quel est celui qui mene au but. Pour cela il faut voir le but de loin, et la faculte qui nous apprend a voir, c'est l'intuition. Sans elle, le geometre serait comme un ecrivain qui serait ferre sur la grammaire, mais qui n'aurait pas d'idees.

Toute definition implique un axiome, puisqu'elle affirme l'existence de l'objet defini. La definition ne sera donc justifiee, au point de vue purement logique, que quand on aura demontre qu'elle n'entraine pas de contradiction, ni dans les termes, ni avec les verites anterieurement admises.

Everybody firmly believes in it because the mathematicians imagine it is a fact of observation, and observers that it is a theory of mathematics.

In the interior of the atom, Bohr had tried the plan of retaining the particle-electron and modifying the classical mechanics. Heisenberg took the opposite course, his procedure amounting in effect to retaining the classical mechanics, at least in form, and modifying the electron. Actually, the electron dropped out all together, because it exists only as a matter of inference and not of direct observation. For the same reason, the new theory contains no mention of atoms, nuclei, protons, or of electricity in any shape or form. The existences of all these are matters of inference, and Heisenberg's purely mathematical theory could no more make contact with them than with the efficiency of a turbine or with the price of wheat.

The main result reached by the new theory was that the classical mechanics can be made to account for the whole range of spectral phenomena, provided entirely new meanings are given to such symbols as p and q which had hitherto been taken to describe the position and motion of the electron. ...The most significant of the new properties is that the product pq is no longer the same as the product qp - in other words the order in which the two factors are multiplied together is no longer a matter of indifference. The difference between pq and qp is found to be always the same, being Plank's constant h multiplied by a numerical multiplier.

Sweet and gentle and sensitive man With an obsessive nature and deep fascination For numbers And a complete infatuation with the calculation Of ?.

I'm sure a mathematician would claim that 0 and 1 are both very interesting numbers. :-)

Hey, if pi == 3, and three == 0, does that make pi == 0? :-)

Das hochste Leben ist Mathematik.

Reine Mathematik ist Religion.

With regard to this question modern physics takes a definite stand against the materialism of Democritus and for Plato and the Pythagoreans. The elementary particles are certainly not eternal and indestructible units of matter, they can actually be transformed into each other. ... The elementary particles in Plato's Timaeus are finally not substance but mathematical forms.

So, ultimately, in order to understand nature it may be necessary to have a deeper understanding of mathematical relationships. But the real reason is that the subject is enjoyable, and although we humans cut nature up in different ways, and we have different courses in different departments, such compartmentalization is really artificial, and we should take our intellectual pleasures where we find them.

Mathematics is not real, but it feels real. Where is this place?

The same equations have the same solutions. (Thus when you have solved a mathematical problem, you can re-use the solution in another physical situation. Feynman was skilled in transforming a problem into one that he could solve.)

NETWORK - Any thing reticulated or decussated, at equal distances, with interstices between the intersections.

'Mathematics and the Love of God: An Introduction to the thought of Simone Weil' by Scott Taylor (a PDF document)

Einstein is the only figure in the physical sciences with a stature that can be compared with Newton. Newton is reported to have said 'If I have seen further than other men, it is because I stood on the shoulders of giants.' This remark is even more true of Einstein who stood on the shoulders of Newton. Both Newton and Einstein put forward a theory of mechanics and a theory of gravity but Einstein was able to base General Relativity on the mathematical theory of curved spaces that had been constructed by Riemann while Newton had to develop his own mathematical machinery. It is therefore appropriate to acclaim Newton as the greatest figure in mathematical physics and the Principia is his greatest achievement.

I have hardly ever known a mathematician who was able to reason.

A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformation of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs.

The Discrete and the Continuous

...a detailed mathematical discussion shows that whatever kind of wave-packet we select to represent the electron, the product of the two uncertainties of position and momentum can never be less than h, which is precisely what Heisenberg found...

If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash. For it is not unknown that Lactantius, the writer celebrated in other ways but very little in mathematics, spoke somewhat childishly of the shape of the earth when he derided those who declared the earth had the shape of a ball. So it ought not to surprise students if such should laugh at us also. Mathematics is written for mathematicians to whom these our labors, if I am not mistaken, will appear to contribute something even to the ecclesiastical state the headship of which your Holiness now occupies.(Author's preface to de revolutionibus)

Light and matter are both single entities, and the apparent duality arises in the limitations of our language. It is not surprising that our language should be incapable of describing the processes occurring within the atoms, for, as has been remarked, it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. Furthermore, it is very difficult to modify our language so that it will be able to describe these atomic processes, for words can only describe things of which we can form mental pictures, and this ability, too, is a result of daily experience. Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme - the quantum theory - which seems entirely adequate for the treatment of atomic processes; for visualisation, however, we must content ourselves with two incomplete analogies - the wave picture and the corpuscular picture.

Historically, Marxism was born with the ambition of explaining everything in the world scientifically. It is known, that Marx even dealt with mathematics. Although he could not solve problems which are nowadays clear for even asinine pupils, Marx left behind for the future generations his smart tips.

The study of mathematics is apt to commence in disappointment... We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it.

The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit.

'Nobody before the Pythagorean had thought that mathematical relations held the secret of the universe. Twenty-five centuries later, Europe is still blessed and cursed with their heritage. To non-European civilizations, the idea that numbers are the key to both wisdom and power, seems never to have occurred.'

To the pure geometer the radius of curvature is an incidental characteristic - like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely incidental to the grin. Physics is concerned with interrelatedness such as the interrelatedness of cats and grins. In this case the 'cat without a grin' and the 'grin without a cat' are equally set aside as purely mathematical phantasies.

We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about 'and'.

Schrodinger's wave-mechanics is not a physical theory, but a dodge - and a very good dodge too.

Proof is the idol before whom the pure mathematician tortures himself.

The mathematics is not there till we put it there.