Frege on Identity Statements

ion is no patch, for the notion of numerical equivalence onto which it is to converge must presuppose a notion of number such that distinct numbers, given by distinct numerical symbols, can be the same number. This is to be the accomplishment of abstraction, by which, Frege says, “things are supposed to become identical by being equated.” But how can abstraction, a “capability of the human mind” according to Thomae, turn two things into one? “if the human mind can equate any objects whatever, [abstraction] is especially meaningless, and the meaning of equating will also remain obscure.” Frege asks: “What do [the formalists] want to achieve by abstracting? They want well, what they This and the two previous quotations are from “Reply to Mr. Thomae’s Holiday Causerie,” pp. 344-5, in which Frege, apparently incensed by Thomae, remarks that “After reading what [Thomae] said about abstraction, I vented my feelings in this verse: Abstraction’s might a boon is found While man does keep it tamed and bound; Awful its heav’nly powers become When that its stops and stays are goneion’s might a boon is found While man does keep it tamed and bound; Awful its heav’nly powers become When that its stops and stays are gone This passage is from Frege’s unpublished “Logic in Mathematics” (p. 224), in which the argument first given in the letter to Peano appears again, although, dated (1914), it was written many years later. 36 really want is identity.” To obtain this, however, we do not want formalist obscurity rooted in their confusion of numeral and number; rather, we need, according to Frege, to carefully distinguish symbols from what they stand for; we need to attend to their semantic relation. If there can be distinct symbols that are signs for the same thing, it can be said that the “signs ‘2 + 3’, ‘3 + 2’, ‘1 + 4’, ‘5’ do designate the same number.” Equality can be construed as identity; the chaos evanesces. Of course, rejecting the formalist conception of number and accepting Frege’s points (1), (2) and (3), does not make the problem raised in Thomae’s remark go away in and of itself. What does make it go away is the recognition, all too clear to Frege, that the premise of the argument is simply mistaken identity statements of the form ja = bk do have “greater cognitive content than an instance of the principle of identity.” But whatever this greater cognitive content is to consist in, such that it separates “a = b” from “a = a,” it cannot be such that it would “prevent us from taking the equals sign . . . as a sign of identity.” (Letter to Peano, p. 126.) But what is it that satisfies these dual criteria? It is precisely at this point of the discussion that the notion of sense makes its appearance. 8. Let’s consider the issue facing Frege. If we harken back to Begriffsschrift, where there are two notions, equality and identity of content, Frege only perceived the need to associate modes of presentation with judgements of the latter sort. Frege did this in order to introduce information that was not otherwise specified by the judgement itself as part of conceptual content; modes of determination are called for with