The beginnings of logistic philosophies of mathematics are to be found in the gradual application to logic of a symbolic
technique modeled upon the parallel use of symbols in mathematics. In its later stages this process was accompanied by extensive alterations in the traditional Aristotelian logic, by the introduction of many more prepositional forms than Aristotle or those who expounded his logic recognized. This in time presented fresh opportunities for the application of symbolic technique, until finally systems of symbols were invented of sufficient generality to be used in the attempt to reduce mathematics to logic.
A convenient starting point for the present brief mention of the landmarks of this process of development is made by Leibniz, whose technical researches in symbolism preceded and often inspired the long series of inventors who perfected the algebra of logic. His work contained the germ of the entire logistic conception ; it is no mere coincidence that many of the logistic philosophers find themselves sympathetic to Leibniz and inherit the characteristic atomism of his system.
The significance for our purposes of Leibniz�s studies in the algebra of logic lies in the fact that no proof with any pretensions to rigour of the thesis that mathematics can be reduced to logic is possible without a well-developed symbolism and calculus for logic itself. Statements occurring in logic must be systematically symbolized in order that their relationships to mathematical theorems should become apparent. Leibniz, a mathematician of genius as well as a philosopher, was eminently fitted to begin the task of inventing the algebra of logic and his papers show him to have made several attempts though with other motives.
Subsequent writers, of whom the most important are De Morgan (Formal Logic, 1847), George Boole (An Investigation into the Laws of Thought, 1854), E. Schröder (Vorlesungen
über die Algebra der Logik, 1890-1905), and C. S. Peirce (see bibliography), by their elaboration of the algebra of logic fulfilled Leibniz�s dream of a Characteristica Universalis, a calculus of reasoning suited for the logical analysis of concepts and the structure of scientific systems, and provided the necessary technical equipment for the logistic school. Schröder and Peirce emancipated symbolic logic not only from the Aristotelian view which permitted only the subject-predicate form for propositions but also To a great extent from the insistent preoccupation with mathematical analogies which retarded the early advance of the subject ; the way is clear for the actual analysis of mathematics. The first important work of this second period was accomplished by R. Dedekind (Was sind und was sollen die Zahlen ?, 1888), who supplied the now famous method of defining real numbers in the mathematical continuum in terms of the rational or fractional numbers. His work may be regarded as a continuation of Weierstrass�s movement to � arithmetize � mathematics, that is to reduce all pure mathematics to the study of the properties of integers ; for after Dedekind the study of irrational numbers could be replaced by the study of certain classes of fractional numbers ; and the reduction of the study of fractional numbers to that of integers presents no difficulties and had already been accomplished.
The definition of real numbers by � Dedekind section � as his method is called, although accepted by mathematicians and used as the very foundation of the modern theory of functions, had to meet serious criticism which subsequently led to attempts at improvement by the logistic philosophers.
The next works of historical importance are Frege�s Begriffsschrift, 1879, Grundlagen der Arithmetik, 1884, and Grundgesetze der Arithmetik, 1893-1903. The last two books completed the reduction of mathematics by defining the rational numbers in terms of logical entities. Unfortunately
Frege did not use Boole�s calculus of logic, (etc).
While Frege had given a philosophic analysis of the concept of number, the Italian mathematician Peano and his school (Formulaire de Mathématiques, 1895-1905), in the course of extensive researches in symbolic logic, had shown that all propositions concerning the natural numbers which are required in mathematics can be deduced from a set of five axioms.
The results of Dedekind, Frege, and Peano had covered in conjunction the whole filed of elementary pure mathematics,1 and by reducing the real numbers to integers, integers to entities occurring in logic, had supplied all the materials for the logistic thesis. There was still needed a synthesis to co-ordinate these results and remedy the imperfections of these early proofs. This was begun by Bertrand Russell in Principles of Mathematics, 1903, and continued in Principia Mathematica (first edition, 1910) written in collaboration with Alfred North Whitehead. These two books are at the apex of the second period in the logistic movement ; they profess to prove, rigorously and with the utmost detail, the identity of mathematics and logic.
The first is a philosophical and polemical discussion of the logistic theories ; the second, written except for a minimum of incidental explanation entirely in mathematical symbols, a proof of the theories.
Since Principia Mathematica little advance has been made by the logistic school and time has shown serious defects in that work, so that the third period has been one
of successive attempts to consolidate a position which at one time Whitehead and Russell appeared to have reached triumphantly.
Among the most notable of these attempts are H. Weyl�s Das Kontinuum, 1918, L. Chwistek�s Theory of Constructive Types, 1923-5 ; and F. P. Ramsey�s Foundations of Mathematics, 1927. All these defend a logistic position. In addition there remains the remarkable Tractatus Logico-Philosophicus, 1922, of L. Wittgenstein, a former pupil of Russell, whose conclusions [are] in many respects unfavourable to Principia Mathematica, (etc).
London : Routledge & Kegan Paul 1933, pp. 15-19.