[44]The importance of "tautology" for a definition of mathematics was pointed out to me by my former pupil Ludwig Wittgenstein, who was working on the problem. I do not know whether he has solved it, or even whether he is alive or dead. We have now come to an end of our somewhat summary introduction to mathematical philosophy. It is impossible to convey adequately the ideas that are concerned in this subject so long as we abstain from the use of logical symbols. Since ordinary language has no words that naturally express exactly what we wish to express, it is necessary, so long as we adhere to ordinary language, to strain words into unusual meanings; and the reader is sure, after a time if not at first, to lapse into attaching the usual meanings to words, thus arriving at wrong notions as to what is intended to be said. Moreover, ordinary grammar and syntax is extraordinarily misleading. This is the case, e.g., as regards numbers; "ten men" is grammatically the same form as "white men," so that 10 might be thought to be an adjective qualifying "men." It is the case, again, wherever propositional functions are involved, and in particular as regards existence and descriptions. Because language is misleading, as well as because it is diffuse and inexact when applied to logic (for which it was never intended), logical symbolism is absolutely necessary to any exact or thorough treatment of our subject. Those readers, [Pg 205] therefore, who wish to acquire a mastery of the principles of mathematics, will, it is to be hoped, not shrink from the labour of mastering the symbols—a labour which is, in fact, much less than might be thought. As the above hasty survey must have made evident, there are innumerable unsolved problems in the subject, and much work needs to be done. If any student is led into a serious study of mathematical logic by this little book, it will have served the chief purpose for which it has been written. [Pg 206] INDEX Aggregates, 12 Alephs, 83, 92, 97, 125 Aliorelatives, 32 All, 158 ff. Analysis, 4 Ancestors, 25, 33 Argument of a function, 47, 108 Arithmetising of mathematics, 4 Associative law, 58, 94 Axioms, 1 Between, 38 ff., 58 Bolzano, 138 n. Boots and socks, 126 Boundary, 70, 98, 99 Cantor, Georg, 77, 79, 85 n., 86, 89, 95, 102, 136 Classes, 12, 137, 181 ff.; reflexive, 80, 127, 138; similar, 15, 16 Clifford, W. K., 76 Collections, infinite, 13 Commutative law, 58, 94 Conjunction, 147 Consecutiveness, 37, 38, 81 Constants, 202 Construction, method of, 73 Continuity, 86, 97 ff.; Cantorian, 102 ff.; Dedekindian, 101 ff.; in philosophy, 105; of functions, 106 ff. Contradictions, 135 ff. Convergence, 115 Converse, 16, 32, 49 Correlators, 54 Counterparts, objective, 61 Counting, 14, 16 Dedekind, 69, 99, 138 n. Deduction, 144 ff. Definition, 3; extensional and intensional, 12 Derivatives, 100 Descriptions, 139, 144