of the infinitesimal calculus had not been presented in a logically rigorous form when Hegel wrote. The care for exactness in dealing with principles is of comparative late growth in mathematics. We shall not be far wrong if we put its birth after Kant published his great Critique in 1781. I cannot find any evidence for a direct influence of Kant on Lagrange, Gauss, Cauchy, or Weierstrass; it seems that criticism was "in the air." And so, in the settlement of the logical and philosophical difficulties of mathematics philosophers have not hitherto had a large share. '.... Philosophy asks of Mathematics: What does it mean? Mathematics in the past was unable to answer, and Philosophy answered by introducing the totally irrelevant notion of mind. But now Mathematics is able to answer, so far at least as to reduce the whole of its propositions to certain fundamental notions of logic. At this point, the discussion must be resumed by Philosophy." 66 There is another aspect of the distinction between mathematicians and logicians. In modern times, from the time of Leibniz up to the middle of the nineteenth century, the only mathematicians of eminence who were also eminent logicians were John Wallis (1616-1703) and perhaps Leonhard Euler (1707-1783). About the middle of the nineteenth century there began, of course, with Boole and De Morgan, a new era for logic, in which the symbolism and methods of algebrà were used to give generality and precision to logical conclusions and to create new logical methods. “Every science,” says De Morgan,8 “that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no symbols.” Again, De Morgan in his Syllabus," says: “I end with a word on the new symbols which I have employed. Most writers on logic strongly object to all symbols except the venerable Barbara, Celarent, etc..... I should advise the reader not to make up his mind on this point until he has well weighed two facts which nobody disputes, both separately and in connection. Firstly, logic is the only science which has made no progress since the revival of letters; secondly, logic is the only science which has produced no growth of symbols.” 'B. Russell, The Principles of Mathematics, Cambridge, 1903, p. 4; cf. pp. 129-130. * Trans. Camb. Phil. Soc., Vol. X, 1864, p. 184. But De Morgan saw the advantages that would result from the use by logic of a symbolism analogous to the algebraical. In his third paper “On the Syllogism”10 he says: “As joint attention to logic and mathematics increases, a logic will grow up among the mathematicians, distinguished from the logic of the logicians by having the mathematical elements properly subordinated to the rest. This ‘mathematical logic'—so-called quasi lucus a non nimis lucendo—will commend itself to the educated world by showing an actual representation of their form of thought—a representation, the truth of which they recognize instead of a mutilated and onesided fragment, founded upon canons of which they neither feel the force nor see the utility.” At the present time the prejudice of logicians against the use of symbols that happen to have been used beforehand in mathematics has almost disappeared, thanks principally to the work of Dr. J. Venn.11 The whole objection of the old-fashioned logicians really rested on no better grounds than this: The use of x and y, which are used in mathematics, ought not to be used instead of the logical X and Y, because x and y have been used for something quantitative 12 As if the word "cabbage” had any rigid connection with the essence of the vegetable of which that word reminds us! In symbolic logic the arithmetical signs +, -, and x were used because certain logical operations have many analogies with arithmetical operations. There is no objection that can be urged against this—a fertile source of discoveries—except that when we write out mathematical theorems in symbolic logic there may be a confusion of terms. But we must be careful not to pursue the analogy too far. Logical addition, for example, and mathematical addition are not identical. There is a point where the analogy breaks down. And when we go deeply into the matter, the differences will begin to outweigh the identities in importance. Broadly speaking, we may say that modern logic is symbolic but has got beyond the rather evident analogies it has with algebra, and a man who seemed rather deep and abstract in his mathematics and logic fifty years ago now seems rather a superficial and naïf person. In one form or another, analogy probably is always guiding us in our researches, but, as we 10 Trans. Camb. Phil. Soc., Vol. X, 1864, note on page 176. Symbolic Logic, London, 1881, 2d ed. 1894. * Cf. ibid., p. ix (of either edition). 11 progress in subtlety, we become more and more convinced of the limitations of the more obvious analogies. * Let us now return to Swift. In his description of Gulliver's voyage to Laputa, he describes the mathematicians of that country as silly and useless dreamers, whose attention has to be awakened by flappers. Also, the mathematical tailor measures his height by a quadrant, and deduces his other dimensions by a rule and compasses, producing a suit of very illfitting clothes. On the other hand, the mathematicians of Laputa, by their marvelous invention of the magnetic island floating in the air, ruled the country and maintained their ascendency over their subjects. Dr. Whitehead13 says: "Swift, indeed, lived at a time peculiarly unsuited for gibes at contemporary mathematicians. Newton's Principia had just been written, one of the great forces which have transformed the modern world. Swift might just as well have laughed at an earthquake." We cannot wholly subscribe to this, for it seems not unlikely that Swift, like everybody else, could not doubt the usefulness, importance, and correctnesss of the mathematician's work, but shared, with the philosopher, a doubt of the mathematician's being able to state his principles clearly and reasonably, just as we may doubt the existence of a knowledge of thermodynamics in a man who drives a railway engine. CAMBRIDGE, ENGLAND. Philip E. B. JOURDAIN. CURRENT PERIODICALS. In the number of Scientia for August, 1915, Georges Bohn gives the second part of his article on new ideas on adaptation and evolution. It is interesting to notice that, according to the author, both Lamarck and Darwin were finalists. E. Carnevale contributes the second part of his study on democracy and penal justice. The articles concerned with questions raised by the war are by W. J. Ashley on “The Economic Conversion of England” and Charles Guignebert on the part played by the Roman Catholic Church, or what, according to him is the same thing, the Pope-in the European war. There is a short note by Federigo Enriques on the art of writing a treatise, prompted by his forthcoming book on the geometrical theory of equations and algebraic functions. There are reviews of books and periodicals, and French translations of the Italian and English articles. 13 A. N. Whitehead, An Introduction to Mathematics, London, 1911, p. 10. |