Discovery and Invention
God created the integers all the rest is the work of man.
Leopold Kronecker (1886)
Moral law, like space, time, and the categories, is a thing which we do not make but find. There is truth, no doubt, in saying that we impose the moral law freely on ourselves; but the truth is not that in any way duty has its origin in our will.
David Ross (1954)
It has often been asked whether the new systems elaborated from time to time by mathematicians should be described as inventions or as discoveries.
William Kneale (1955)
The contrast of invention and discovery raises perennial arguments, and appears in many fields: in mathematics, as in Kneale’s statement above; in ethics as in Ross’s claim; and most accessibly in the case of arithmetic, for which the above photograph seeks to be an illustration. If a duck had three offspring, and a fox took one, would the mother ‘see’ the difference? Given 19, as here, we might easily believe the loss would go unnoticed. From such thoughts comes the idea that while we may
discover
Another way that we become puzzled about what there is independently of ourselves.
small numbers, larger ones need human imagination to invent. Kronecker, a conservative mathematician, wanted to allow the notion of discovery for any
integer,
Are numbers like tables, objects to be found, and if they are, do we really discover them?
making such entities as π and i unequivocally human
inventions;
But maybe numbers are symbols?
if not actually the work of the devil. But discovering large integers, even moderate ones like 19, without involving any mental
creativity,
Then again could numbers be a kind of pattern?
does indeed seem a very odd idea. And does finding the ‘moral law’, or indeed space and time, not seem equally odd?
Kronecker’s biographer Heinrich Weber records the occurrence of this phrase at a lecture he attended. Ross’s book Kant’s Ethical Theory, published by Oxford, has this quote on page 26. W. C. Kneale phrased the problem in that way on page 85 of his talk The Idea of Invention which was published in the Proceedings of the British Academy Vol 41 pp 85-108.
I do not recall the presence of these chicks’ mothers - they must look tempting morsels to any fox.
Above, hovering on blue introduces a link: click to go, move away to stay.
Saturday 23rd November 2024
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