Diverging Correspondences Concerning the Problem of Identity: Russell-Wittgenstein and Benjamin-Scholem

At the end of his Introduction to Mathematical Philosophy, having reached the limit of what can be done by means of "ordinary language," Bertrand Russell makes the following claim as he seeks to circumscribe the field under discussion by capturing the distinction between mathematical and non-mathematical propositions: "[the former] all have the characteristic which, a moment ago, we agreed to call 'tautology.'"¹ As for the precise meaning of the term tautology and thus the reason for the semantic agreement that he made with himself—the book was written in an English prison, to which Russell was condemned because he insulted the armed forces of the United States—he is lucidly at a loss for words: "For the moment, I do not know how to define 'tautology.' It would be easy to offer a definition that might be satisfactory for a while; but I know of none that I feel to be satisfactory, in spite of feeling thoroughly familiar with the characteristic of which a definition is wanted."² Russell agrees with himself that the term "tautology" characterizes the distinction between the two mutually exclusive types of proposition, and he is sure that he knows what characteristic should be captured by the term "tautology," but he cannot say what the term means except by way of the auto-tautological formula, "tautology is tautology," which is rather disappointing as far as definitions go. When Russell speaks of mathematics, he is in the habit of making paradoxical-sounding pronouncements. Thus, he once wrote, and was frequently quoted as saying: "mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."³ The paradoxical character of this remark—which was widely discussed and found its way into the diary of a young student of mathematics named Gerhard Scholem—can be resolved by means of straightforward exegesis: unlike other sciences, the subject-matter of mathematics is "anything," that is to say, anything as such, or anything in general, which means that mathematics is concerned with nothing in particular, and it can therefore be defined as the science in which we cannot know what we are talking about, since there is no subject there to know.⁴ The earlier quoted passage, however, is less susceptible to this kind of exegesis, above all because it is a positive statement about the nature of mathematical propositions, not a simple assertion of their difference from other kinds of utterances. Insofar as the statement concerning the tautological character of mathematical propositions does not belong to the field of mathematics, it is not itself tautological but should belong, instead, to the field of philosophy; but insofar as the "introduction to mathematical philosophy," as Russell claims in the preface to the volume—a preface that, oddly enough, solicits an editorial note because of its opacity—is never "actually dealing with a part of philosophy, except where it steps outside its province," then either the proposition quoted at the beginning of this paper is in the "province" of the field described by the introduction, in which case it may be inside of philosophy but stands outside of the introduction, or the proposition remains inside the introduction and is outside of philosophy, whereupon it would presumably be a mathematical proposition after all, since the introduction to mathematical philosophy belongs to field of mathematics and is not actually a part of philosophy.⁵ Little wonder, then, that Russell is inclined to say of tautology that he is familiar with its characteristics but cannot be satisfied with any proposition that would put this feeling into words, for the inclination toward a knowing silence responds to the convolutions of the fields that converge around the conclusion to the introduction to mathematical philosophy.

Only one thing is clear about the passage in question: not the meaning of tautology, to be sure, but, rather, the source of the term. Russell does not derive tautology from its origin in Greek grammarians and even emphasizes this fact at the beginning of the conclusion to...