Mathematical Quotes

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We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.

Remembering... that Eratosthenes of Cyrene, employing mathematical theories and geometrical methods, discovered from the course of the sun, the shadows cast by an equinoctial gnomon, and the inclination of the heaven that the circumference of the earth is two hundred and fifty-two thousand stadia, that is, thirty-one million five hundred thousand paces.

The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C. S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard.

The soul, considered with its Creator, is like one of those mathematical lines that may draw nearer to another for all eternity without a possibility of touching it; and can there be a thought so transporting as to consider ourselves in these perpetual approaches to Him, who is not only the standard of perfection, but of happiness?

We shed as we pick up, like travellers who must carry everything in their arms, and what we let fall will be picked up by those behind. The procession is very long and life is very short. We die on the march. But there is nothing outside the march so nothing can be lost to it. The missing plays of Sophocles will turn up piece by piece, or be written again in another language. Ancient cures for diseases will reveal themselves once more. Mathematical discoveries glimpsed and lost to view will have their time again. You do not suppose, my lady, that if all of Archimedes had been hiding in the great library of Alexandria, we would be at a loss for a corkscrew?

Haavelmo was the ﬁrst to recognize the capacity of economic models to guide policies. This paper describes some of the barriers that Haavelmo's ideas have had (and still have) to overcome, and lays out a logical framework that has evolved from Haavelmo's insight and matured into a coherent and comprehensive account of the relationships between theory, data and policy questions. The mathematical tools that emerge from this framework now enable investigators to answer complex policy and counterfactual questions using simple routines, some by mere inspection of the model's structure.

In that disputable point of persecuting men for conscience sake, I see such dreadful consequences rising, I would be as fully convinced of the truth of it as a mathematical demonstration, before I would venture to act upon it or make it a part of my religion.

Perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth — whatever that may be, if anything.

Eric Temple Bell

It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profoundity, he is talking nonsense.

Alfred North Whitehead

Our sunsets have been reduced to wavelengths and frequencies. The complexities of the universe have been shredded into mathematical equations. Even our self-worth as human beings has been destroyed.

If we wish to express our ideas in terms of the concepts synthetic and analytic, we would have to point out that these concepts are applicable only to sentences that can be either true of false, and not to definitions. The mathematical axioms are therefore neither synthetic nor analytic, but definitions.... Hence the question of whether axioms are a priori becomes pointless since they are arbitrary.

Hans Reichenbach

I cannot see why it is necessary that every deduction from algebra should be bound to certain conventions incident to an earlier stage of mathematical learning, even supposing them to have been consistently used up to the point in question. I should not care if any one thought this treatise unalgebraical, but should only ask whether the premises were admissible and the conclusions logical.

Augustus De Morgan

Rohan, one of us is an unmarried man with superior mathematical abilities and no prospects for the evening. The other is a confirmed lecher in an amorous mood, with a willing and nubile young wife waiting at home. Who do you think should do the damned account books?" And, with a nonchalant wave, St. Vincent had left the office.

Idiotically he thought: Humpty Dumpty explained it. A wabe is the plot of grass around a sundial. A sundial. Time — it has something to do with time. A long time ago Scotty asked me what a wabe was. Symbolism.
'Twas brillig —
A perfect mathematical formula, giving all the conditions, in symbolism the children had finally understood. The junk on the floor. The toves had to be made slithy — vaseline? and they had to be placed in a certain relationship, so that they'd gyre and gimbel.

Mathemagical mathematics combines the beauty of mathematical structure with the entertainment value of a trick.

From the imputation of confounding axioms with assumed concepts Euclid himself, however, is free. Euclid incorporated the former among his postulates while he separated the latter as common concepts—a proceeding which even on the part of his commentators was no longer understood, and likewise with modern mathematicians, unfortunately for science, has met with little imitation. As a matter of fact, the abstract methods of mathematical science know no axioms at all.

Hermann Grassmann

But in spite of the obvious effectiveness of mathematics in physics, I have never heard of a good a prioriargument that the world must be organised to mathematical principles.

Supposing you knew — not by sight or by instinct, but by sheer intellectual knowledge, as I know the truth of a mathematical proposition — that what we call empty space was full, crammed. Not with lumps of what we call matter like hills and houses, but with things as real — as real to the mind.

The next grand extensions of mathematical physics will, in all likelihood, be furnished by quaternions.

Peter Guthrie Tait

History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.

During the last two centuries and a half, physical knowledge has been gradually made to rest upon a basis which it had not before. It has become mathematical.

Augustus De Morgan

It is not particularly satisfactory to see equations set forth as direct results of observation and experiment, where we used to get long mathematical deductions as apparent proofs of them. Nevertheless, I believe that we cannot, without deceiving ourselves, extract much more from known facts than is asserted in the papers referred to. If we wish to lend more color to the theory, there is nothing to prevent us from supplementing all this and aiding our powers of imagination by concrete representations of the various conceptions as to the nature of electric polarisation, the electric current, etc.

In the most general mathematical sense, a space is a set of elements which conform to certain postulates.

James Grier Miller

When we went to school we were told that we were governed by laws, not men. As a result of that, many people think there is no need to pay any attention to judicial candidates because judges merely apply the law by some mathematical formula and a good judge and a bad judge all apply the same kind of law. The fact is that the most important part of a judge's work is the exercise of judgment and that the law in a court is never better than the common sense judgment of the judge that is presiding.

Robert H. Jackson

Whether or not Pythagoras discovered his own theorem, it is certainly true that Pythagoras and his followers acquired a reputation for mathematical and geometrical investigations. They conducted their mathematics in a spatial way that strikes us as strange: numbers were conceived as lines, squares, triangles, and oblongs, laid out as patterns of counters or pebbles.

Catherine Rowett