Arthur Cayley

Arthur Cayley (August 16 1821 - January 26 1895) was a British mathematician. He helped found the modern British school of pure mathematics.

Philosophy

To Cayley's presidential address we are indebted for information about the view which he took of the foundations of exact science,

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and the philosophy which commended itself to his mind. He quoted Plato and Kant with approval, J. S. Mill with faint praise.

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Although he threw a sop to the empirical philosophers at the beginning of his address, he gave them something to think of

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before he finished.

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He first of all remarks that the connection of arithmetic and algebra with the notion of time is far less obvious than that of

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geometry with the notion of space; in which he, of course, made a hit at Hamilton's theory of Algebra as the science of pure time.

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Further on he discusses the theory directly, and concludes as follows:

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:Hamilton uses the term algebra in a very wide sense, but whatever else he includes under it, he includes all that in contradistinction to the Differential Calculus would be called algebra. Using the word in this restricted sense, I cannot myself recognize the connection of algebra with the notion of time; granting that the notion of continuous progression presents itself and is of importance, I do not see that it is in anywise the fundamental notion of the science. And still less can I appreciate the manner in which the author connects with the notion of time his algebraic couple, or imaginary magnitude, a+bsqrt{-1}.

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So you will observe that doctors differ---Tait and Cayley---about the soundness of Hamilton's theory of couples. But it can be shown that a couple may not only be represented on a straight line, but actually means a portion of a straight line; and as a line is unidimensional, this favors the truth of Hamilton's theory.

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As to the nature of mathematical science Cayley quoted with approval from an address of Hamilton's:

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:These purely mathematical sciences of algebra and geometry are sciences of the pure reason, deriving no weight and no assistance from experiment, and isolated or at least isolable from all outward and accidental phenomena. The idea of order with its subordinate ideas of number and figure, we must not call innate ideas, if that phrase be defined to imply that all men must possess them with equal clearness and fulness; they are, however, ideas which seem to be so far born with us that the possession of them in any conceivable degree is only the development of our original powers, the unfolding of our proper humanity.

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It is the aim of the evolution philosopher to reduce all knowledge to the empirical status; the only intuition he grants is a kind of

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instinct formed by the experience of ancestors and transmitted cumulatively by heredity. Cayley first takes him up on the subject

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of arithmetic:

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:Whatever difficulty be raisable as to geometry, it seems to me that no similar difficulty applies to arithmetic; mathematician, or not, we have each of us, in its most abstract form, the idea of number; we can each of us appreciate the truth of a proposition in numbers; and we cannot but see that a truth in regard to numbers is something different in kind from an experimental truth generalized from experience. Compare, for instance, the proposition, that the sun, having already risen so many times, will rise to-morrow, and the next day, and the day after that, and so on; and the proposition that even and odd numbers succeed each other alternately ad infinitum; the latter at least seems to have the characters of universality and necessity. Or again, suppose a proposition observed to hold good for a long series of numbers, one thousand numbers, two thousand numbers, as the case may be: this is not only no proof, but it is absolutely no evidence, that the proposition is a true proposition, holding good for all numbers whatever; there are in the Theory of Numbers very remarkable instances of propositions observed to hold good for very long series of numbers which are nevertheless untrue.

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Then he takes him up on the subject of geometry, where the empiricist rather boasts of his success.

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:It is well known that Euclid's fifth axiom, even in Playfair's form of it, has been considered as needing demonstration; and that Lobatschewsky constructed a perfectly consistent theory, wherein this axiom was assumed not to hold good, or say a system of non-Euclidean plane geometry. My own view is that Euclid's fifth axiom in Playfair's form of it does not need demonstration, but is part of our notion of space, of the physical space of our experience---the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all external experience. Riemann's view before referred to may I think be said to be that, having in intellectu a more general notion of space (in fact a notion of non-Euclidean space), we learn by experience that space (the physical space of our experience) is, if not exactly, at least to the highest degree of approximation, Euclidean space. But suppose the physical space of our experience to be thus only approximately Euclidean space, what is the consequence which follows? Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.

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In his address he remarks that the fundamental notion which underlies and pervades the whole of modern analysis and geometry is that of imaginary magnitude in analysis and of imaginary space (or space as a locus in quo of imaginary points and figures) in geometry. In the case of two given curves there are two equations satisfied by the coordinates (x, y) of the several points of intersection, and these give rise to an equation of a certain order for the coordinate x or y of a point of intersection. In the case of a straight line and a circle this is a quadratic equation; it has two roots real or imaginary. There are thus two values, say of x, and to each of these corresponds a single value of y. There are therefore two points of intersection, viz., a straight line and a circle intersect always in two points, real or imaginary. It is in this way we are led analytically to the notion of imaginary points in geometry. He asks, What is an imaginary point? Is there in a plane a point the coordinates of which have given imaginary values? He seems to say No, and to fall back on the notion of an imaginary space as the locus in quo of the imaginary point.

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