RECENT WORK ON THE PHILOSOPHY OF LEIBNIZ. 187 4he exact difference between the two. The fact is that the Ars Combinatoria, or Universal Mathematics, is more formal than the Logical Calculus : it is concerned with deductions from the assumption of a synthesis obeying such and such laws, but otherwise undefined. We may say that, in this subject, our signs of operation, our + and x and whatever other such signs we may employ, are themselves variables, subject merely to hypotheses as to their formal laws ; whereas in every other .branch of mathematics, and in the Logical Calculus itself, only the letters are variable, and the signs of operation have constant meanings. It might seem, from this account, as though Universal Mathematics were the most general of all mathematical subjects, and in a sense this is true. But it is emphatically not the logic- .ally first of such subjects, for itself employs deduction and the logical kinds of synthesis, which are explicitly dealt with in the Logical Calculus. Moreover, in order that any deductions from an assumed formal type of synthesis may have importance, it is necessary that there should be at least one synthesis of the type in question ; and this can never be proved by the Ars Combinatoria itself. This science, therefore, is logically subsequent to the Logi- cal Calculus. The matter may be stated thus : In every proposition, when fully stated, there must be constants, i.e. terms whose mean- ing is not in any degree indeterminate. When we turn our symbols of operation into variables, we do not thereby remove all constants from our propositions, for the formal laws to which our operations .are to be subjected will require constants for their statement. I have succeeded in reducing the number of indefinable terms em- ployed in pure mathematics" (including geometry) to eight (a number which may be capable of further diminution), by means of which every notion occurring throughout the whole science can be defined. Thus all mathematics is merely the study of these eight notions ; and the Logical Calculus is a name for the more elementary parts of this study. We have here precisely such a development as Leibniz desired to give to all subjects with the difference, due to the fact that propositions are synthetic, that
- he indemonstrable axioms of mathematics, instead of being
one, appear to number about twenty. l Thus Symbolic Logic is distinct from, and logically prior to, the subject which Leibniz calls Universal Mathematics. But the notion of different possible
- algorithms was very attractive to Leibniz, and the Logical Cal-
1 The only ground, in Symbolic Logic, for regarding an axiom as inde- monstrable is, in general, that it is undemonstrated ; hence there is ^always hope of reducing the number. We cannot apply the method by which, for example, the axiom of parallels has been shown to be indemonstrable, of supposing our axiom false ; for all our axioms are . concerned with the principles of deduction, so that, if any one of them be true, the consequences which might seem to follow from denying it do not follow as a matter of fact. Thus from the hypothesis that a true principle of deduction is false, valid inference is impossible.
