I think that it is a relatively good approximation to truth — which is much too complicated to allow anything but approximations — that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is ... governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.
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"The Mathematician", in The Works of the Mind (1947) edited by R. B. Heywood, University of Chicago Press, ChicagoJohn von Neumann
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The definition of random in terms of a physical operation is notoriously without effect on the mathematical operations of statistical theory because so far as these mathematical operations are concerned random is purely and simply an undefined term. The formal and abstract mathematical theory has an independent and sometimes lonely existence of its own. But when an undefined mathematical term such as random is given a definite operational meaning in physical terms, it takes on empirical and practical significance. Every mathematical theorem involving this mathematically undefined concept can then be given the following predictive form: If you do so and so, then such and such will happen.
Walter A. Shewhart
Fundamental ideas play the most essential role in forming a physical theory. Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulae, are the beginning of every physical theory. The ideas must later take the mathematical form of a quantitative theory, to make possible the comparison with experiment.
Albert Einstein
The fact that randomness requires a physical rather than a mathematical source is noted by almost everyone who writes on the subject, and yet the oddity of this situation is not much remarked.
Brian Hayes
Mr. Dodgson had a mathematical, a logical and a philosophical mind; and when these qualities are united to a love of the grotesque, the resultant fancies are sure to have a quite peculiar charm, a charm so much the greater because its source is subtle and eludes all attempts to grasp it. Sometimes he seems to revel in ideas which are not merely illogical but anti-logical.
Lewis Carroll
Mr. Dodgson had a mathematical, a logical and a philosophical mind; and when these qualities are united to a love of the grotesque, the resultant fancies are sure to have a quite peculiar charm, a charm so much the greater because its source is subtle and eludes all attempts to grasp it. Sometimes he seems to revel in ideas which are not merely illogical but anti-logical.
Charles (Lewis Carroll) Dodgson
Neumann, John von
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