Tobias Dantzig Quotes

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Mathematical achievement shall be measured by standards which are peculiar to mathematics. These standards are independent of a crude reality of our senses. They are: freedom from logical contradictions, the generality of the laws governing the created form, the kinship which exists between this new form and those that have preceded it.

Tobias Dantzig
The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit.... The conic sections, invented in an attempt to solve the problem of doubling the alter of an oracle, ended by becoming the orbits followed by the planets... The imaginary magnitudes invented by Cardan and Bombelli describe... the characteristic features of alternating currents. The absolute differential calculus, which originated as a fantasy of Reimann, became the mathematical model for the theory of Relativity. And the matrices which were a complete abstraction in the days of Cayley and Sylvester appear admirably adapted to the... quantum of the atom.

Tobias Dantzig
In the history of mathematics, the "how" always preceded the "why," the technique of the subject preceded its philosophy.

Tobias Dantzig
The harmony of the universe knows only one musical form - the legato; while the symphony of number knows only its opposite - the staccato. All attempts to reconcile this discrepancy are based on the hope that an accelerated staccato may appear to our senses as a legato.

Tobias Dantzig
Between the philosopher's attitude towards the issue of reality and that of the mathematician there is this essential difference: for the philosopher the issue is paramount; the mathematician's love for reality is purely platonic.

Tobias Dantzig
The mathematical activity of Ancient Greece reached its peak during the glorious era of Euclid, Eratosthenes, Archimedes and Apollonius, a time when Greek letters, art and philosophy were already on the decline.... it was not Greece proper but its outposts in Asia Minor, in Lower Italy, in Africa that had contributed most to the development of mathematics.

Tobias Dantzig
With regard to the Pythagorean theorem my conjecture is that... it was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles.

Tobias Dantzig
Thales the teacher produced the first geometers, even as Thales the thinker founded the first geometry worthy of the name.

Tobias Dantzig
Banish the infinite process, and mathematics pure and applied is reduced to the state in which it was known to the pre-Pythagoreans.

Tobias Dantzig
The evolution of scientific thought is inseparable from the history of man's efforts to resolve the perplexities of his own existence.

Tobias Dantzig
Today we know that possibility and impossibility have each only a relative meaning; that neither is an intrinsic property of the operation but merely a restriction which human tradition has imposed on the field of the operand. Remove the barrier, extend the field, and the impossible becomes possible.

Tobias Dantzig
Are not most professional mathematicians spared all trouble incident to income?

Tobias Dantzig
Mathematics is not only the model along the lines of which the exact sciences are striving to design their structure; mathematics is the cement which holds the structure together.

Tobias Dantzig
And so matching by itself is incapable of creating an art of reckoning. Without our ability to arrange things in ordered succession little progress could have been made. Correspondence and succession, the two principles that permeate all mathematics - nay, all realms of exact thought - are woven into the very fabric of our number system.

Tobias Dantzig
The practical man demands an appearance of reality at least. Always dealing in the concrete, he regards mathematical terms not as symbols or thought but as images of reality. A system acceptable to the mathematician because of its inner consistency may appear to the practical man to be full of contradictions because of the incomplete manner in which it represents reality.

Tobias Dantzig
It is a remarkable fact that the mathematical inventions which have proved to be most accessible to the masses are also those which exercised the greatest influence on the development of pure mathematics.

Tobias Dantzig
The importance of infinite processes for the practical exigencies of technical life can hardly be overemphasized. Practically all applications of arithmetic to geometry, mechanics, physics and even statistics involve these processes directly and indirectly.

Tobias Dantzig
Poincaré was an artist par excellence. Estheticism with him was not a mere creed: it was a way of life.

Tobias Dantzig
One part of the dreams of the eighteenth century intellectuals was realized: the resources of nature did yield a magnificent harvest. But the thinkers who helped to tap these resources... failed to attend, detained in their studies and laboratories, lost in their dreams and calculations, seeking new fields, co-ordinating old and new, spinning abstract theories... the thinkers were unequal to the task of developing these vast resources, most of which they had themselves uncovered. The shrewd declassés, who had... the world to gain, pioneered this development and took possession of the earth.

Tobias Dantzig
Poincaré's mind was not subject to hysteresis or hibernation. He had the unique faculty of dismissing an idea from his mind, the instant the stimulus was gone, and to supplant it immediately with another creative idea.

Tobias Dantzig
Mathematics fluourished as long as freedom of thought prevailed; it decayed when creative joy gave way to blind faith and fanatical frenzy.

Tobias Dantzig
To describe means to classify, and the man Poincaré defies classification, as does indeed his philosophy.

Tobias Dantzig
The arithmetization of mathematics... which began with Weierstrass... had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics.
These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought.

Tobias Dantzig
Mathematics is the supreme judge; from its decisions there is no appeal.

Tobias Dantzig
The preface to the French edition of that work contains the following passage: To me the French edition of my work is not a mere translation, but a transcription of ideas into a language in which it should have been written in the first place... I proudly acknowledge... my master. His words are among the most brilliant recollections of my youth; his piercing wisdom and potent prose have inspired my efforts of a riper age. To the memory of Henri Poincaré, the intellectual giant who was the first to recognize the role which the idiosyncrasies of the race play in the evolution of scientific ideas, I dedicate this book.

Tobias Dantzig
Greek thought was essentially non-algebraic, because it was so concrete. The abstract operations of algebra, which deal with objects that have been purposely stripped of their physical content, could not occur to minds which were so intently interested in the objects themselves. The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and reflects all the phases under which the human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations.

Tobias Dantzig
The progress of mathematics has been most erratic, and... intuition has played a predominant role in it.... It was the function of intuition to create new forms; it was the acknowledged right of logic to accept or reject these new forms, in whose birth in had no part.... the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied.

Tobias Dantzig
But how can we avoid the use of human language? The... symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason—only thus may we hope to build mathematics on the solid foundation of logic.

Tobias Dantzig
His essays on the foundations of science are cases in point. They strike one as extemporaneous speeches rather than edited articles.... those who knew him best insisted that he rarely, if ever, would revise a manuscript, even if he was fully aware of its stylistic shortcomings.

Tobias Dantzig
Science... may be viewed as man's supreme effort to find himself in that perplexing pattern which he calls Nature.... Has he succeeded in achieving some measure of harmony with Nature? Or has he merely managed to transfer to Nature the irreconcilable duality within himself?

Tobias Dantzig