Mathematical reasoning may be regarded rather schematically as the
exercise of a combination of two facilities, which we may call
intuition and ingenuity.
Mathematical reasoning may be regarded...
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In studying mathematics or simply using a mathematical principle, if we get the wrong answer in sort of algebraic equation, we do not suddenly feel that there is an anti-mathematical principle that is luring us into the wrong answers.
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Mathematics alone make us feel the limits of our intelligence. For we can always suppose in the case of an experiment that it is inexplicable because we don't happen to have all the data. In mathematics we have all the data and yet we don't understand. We always come back to the contemplation of our human wretchedness. What force is in relation to our will, the impenetrable opacity of mathematics is in relation to our intelligence.
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How happy the lot of the mathematician! He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve. No cashier writes a letter to the press complaining about the incomprehensibility of Modern Mathematics and comparing it unfavorably with the good old days when mathematicians were content to paper irregularly shaped rooms and fill bathtubs without closing the waste pipe.
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Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
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I know that two and two make four -- and should be glad to prove it too if I could -- though I must say if by any sort of process I could convert 2 and 2 into five it would give me much greater pleasure.
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Mathematicians are like Frenchman: whatever you say to them they translate Into their own language, and forthwith it is something entirely different.
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It will be found, in fact, that the ingenious are always fanciful, and the truly imaginative never otherwise than analytic.
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The analytical power should not be confounded with simple ingenuity; for while the analyst is necessarily ingenious, the ingenious man is often remarkably incapable of analysis.
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Between ingenuity and the analytic ability there exists a difference far greater, indeed, than that between the fancy and the imagination, but of a character very strictly analogous.
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Mathematics, rightly viewed, posses not only truth, but supreme beauty a beauty cold and austere, like that of sculpture.
