The Growth of Symbolism in Mathematics

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PrefaceMathematical Logic is a necessary preliminary to logical Mathematics. “Mathematical Logic” is the name given by Peano to what is also known (after Venn ) as “Symbolic Logic”; and Symbolic Logic is, in essentials, the Logic of Aristotle, given new life and power by being dressed up in the wonderful–almost magical–armour and accoutrements of Algebra. In less than seventy years, logic, to use an expression of De Morgan’s, has so _thriven_ upon symbols and, in consequence, so grown and altered that the ancient logicians would not recognize it, and many old-fashioned logicians will not recognize it. The metaphor is not quite correct: Logic has neither grown nor altered, but we now see more _of_ it and more _into_ it.The primary significance of a symbolic calculus seems to lie in the economy of mental effort which it brings about, and to this is due the characteristic power and rapid development of mathematical knowledge. Attempts to treat the operations of formal logic in an analogous way had been made not infrequently by some of the more philosophical mathematicians, such as Leibniz and Lambert ; but their labors remained little known, and it was Boole and De Morgan, about the middle of the nineteenth century, to whom a mathematical–though of course non-quantitative–way of regarding logic was due. By this, not only was the traditional or Aristotelian doctrine of logic reformed and completed, but out of it has developed, in course of time, an instrument which deals in a sure manner with the task of investigating the fundamental concepts of mathematics–a task which philosophers have repeatedly taken in hand, and in which they have as repeatedly failed.First of all, it is necessary to glance at the growth of symbolism in mathematics; where alone it first reached perfection. There have been three stages in the development of mathematical doctrines: first came propositions with particular numbers, like the one expressed, with signs subsequently invented, by “\(2+3=5\)”; then came more general laws holding for all numbers and expressed by letters, such as\[“(a+b)c=ac+bc”;\]lastly came the knowledge of more general laws of functions and the formation of the conception and expression “function”. The origin of the symbols for particular whole numbers is very ancient, while the symbols now in use for the operations and relations of arithmetic mostly date from the sixteenth and seventeenth centuries; and these “constant” symbols together with the letters first used systematically by Viete (1540-1603) and Descartes (1596-1650), serve, by themselves, to express many propositions. It is not, then, surprising that Descartes, who was both a mathematician and a philosopher, should have had the idea of keeping the method of algebra while going beyond the material of traditional mathematics and embracing the general science of what thought finds, so that philosophy should become a kind of Universal Mathematics. This sort of generalization of the use of symbols for analogous theories is a characteristic of mathematics, and seems to be a reason lying deeper than theerroneous idea, arising from a simple confusion of thought, that algebraical symbols necessarily imply something quantitative, for the antagonism there used to be and is on the part of those logicians who were not and are not mathematicians, to symbolic logic. This idea of a universal mathematics was cultivated especially by Gottfried Wilhelm Leibniz (1646-1716).Though modern logic is really due to Boole and De Morgan, Leibniz was the first to have a really distinct plan of a system of mathematical logic. That this is so appears from research–much of which is quite recent–into Leibniz’s unpublished work.The principles of the logic of Leibniz, and consequently of his whole philosophy, reduce to two1: (1) All our ideas are compounded of a very small number of simple ideas which form the “alphabet of human thoughts”; (2) Complex ideas proceed from these simple ideas by a uniform and symmetrical combination which is analogous to arithmetical multiplication. With regard to the first principle, the number of simple ideas is much greater than Leibniz thought; and, with regard to the second principle, logic considers three operations–which we shall meet with in the following book under the names of logical multiplication, logical addition and negation–instead of only one.Courturat, La Logique de Leibniz d’apres des documents indéits, Paris, 1901, pp. 431-432, 48.”Characters” were, with Leibniz, any written signs, and “real” characters were those which–as in the Chinese ideography–represent ideas directly, and not the words for them. Among real characters, some simply serve to represent ideas, and some serve for reasoning

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