Tobias Dantzig - Mathematical Quotes

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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
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
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
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 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