The Logistic Development of Mathematics

THE LOGISTIC DEVELOPMENT OF MATHEMATICS Symbolic Logic and Mathematics.—Once symbolic logic was freed from its dependence upon the forms of ordinary algebra, it became an instrument of precision for the analysis of fundamental concepts. Moreover, in achieving its ideal of generality, logic and mathematics have become indistinguishable. That is to say, all the fundamental concepts of mathematics are defined in terms which are involved in any complicated process of thought, and are thus seen not to depend upon any specific notion such as that of discrete and continuous magnitude, as had been traditionally sup posed. This position has been established in the only way in which it could have been established, viz., by the detailed working out of mathematics from purely logical concepts. This achievement is mainly due to the work of Frege, Peano, Russell and Whitehead, but it has recently been carried to a more satisfactory comple tion by Ludwig Wittgenstein and Frank P. Ramsey.

The method consists in the development of a system in which all the assumptions involved are made explicit, and all other propo sitions are deduced from these assumptions by the process of substituting for variables values of those variables, and by the use of defined equivalents. Frege laid the foundations of this logical analysis in the Grundlagen der Arithmetik (1884), Grundgesetze der Arithmetik (1893-1903) and various important articles in periodicals. In the latter book Frege developed arithmetic from purely logical premises. But his notation was so difficult that his work remained almost without influence upon subsequent investi gators. Peano began in 1894 the publication of the Formulaire de Mathematique, in which he elaborated a symbolism fitted to perform the required analysis of fundamentals. Russell, using the notation of Peano, and basing his work upon the logical analysis of Frege, produced in 1910, in collaboration with Whitehead, the Principia Mathematica, which may be said to mark an epoch in the development of logistic. In this work we find an extreme log ical rigour which results in a detailed analysis of all the concepts of mathematics. The advance on Peano consists in the fact that no postulates are required except those of logic hence, the inde pendence of various mathematical branches is no longer required. It is impossible here to do more than indicate the method employed.

In this system the negation of p, symbolised by which is read, "p is false," is taken as a primitive idea. Hence, we have p and In order that deduction may be possible, there must be such a relation between one proposition and another that, if we know that the first is true, we can infer that the second is true. That is, we assume the fundamental principle "whatever is implied by a true proposition is true." Let p and q be two propositions between which this relation holds. Then we can state this relation as follows : "Either p is false or q is true." This is the logical sum of - p and q. It is called disjunction. Assuming this relation as a second primitive idea, we can now define "implication" in would mean "Socrates is mortal." Similarly, "(x)-02" asserts that "02 is always false." Hence, there is no set of values satisfying the function "0." Thus a propositional function that is always false determines the null class.

We have so

far assumed that the symbols that can be substi tuted for x in "ck" are names, i.e., symbols for individuals. It is easy to see that there is an important difference between names and symbols which do not name an individual, but describe it, e.g., "the oldest inhabitant," "the author of Waverley" and so on. Russell calls such symbols "definite descriptions" and he compares them with such symbols as "a man," "an author," which are called "indefinite descriptions." Thus a definite description is of the form "the so-and-so"; an indefinite description is of the form "a so-and-so." The distinction between a name and a definite description is important. A name must be the name of something; that is, a name must have application. But a description need not apply, and in that case the propositional function in which the description occurs will be false. Thus "the author of the Iliad" will be a description that describes nothing, unless one, and only one, person wrote the Iliad. In this way we see that such phrases as "the author of Waverley," "the most perfect Being," "the man in the Moon" do not simply name individuals, but are descriptions which may, or may not, apply. It is clear that this analysis of descriptive phrases has an important bearing upon such traditional philosophical arguments as the ontological proof.