The Movement of Thought: Wittgenstein on Time, Change and History

Abstract

20therefore, what we do-in terms of our illustration-is to use the lines already given in a new way for the purpose of demarcating an area. Nothing essentially new, however, emerges in the process.But the more fruitful type of definition is a matter of drawing boundary lines that were not previously given at all. What we shall be able to infer from it, cannot be inspected in advance; here we are not simply taking out of the box what we have put into it. The conclusions we draw from it extend our knowledge, and are therefore, on Kant's view, to be regarded as synthetic; and yet they can be proved by purely logical means, and are thus analytic. The truth is that they are contained in the definitions, but as plants are contained in their seeds, not as beams are contained in a house." (FA §88)Frege concludes this lengthy passage with the remark: "Often we need several definitions for the proof of some proposition, which consequently is not contained in any one of them alone, and yet does follow purely logically from all of them together." (ibid.) 25 And this is, of course, exactly what Frege sets out to do in the Basic Laws of Arithmetic, published nine years after the Foundations had been laid down. 26 In this later work, the Kantian vocabulary of 'analytic' and 'synthetic' would be abandoned; nonetheless, Frege's emphasis on a genuinely productive analytical combination of definitions follows through with the same immanency of natural necessity described earlier. It is important to note, however, that although these definitions "draw boundary lines that were not previously given all"-in other words, though they give rise to new knowledge-Frege is extremely cautious to delimit the degree of creative freedom exhibited there. As in the reference above, Frege once again claims that the definitions he is proposing in this later work do not "create anything new" in and of themselves. (BA, p. 2) Rather, they mark for him the limit of logical regression, that point at which the propositions of logic and mathematics "are not 25 As Beaney notes, this passage suggest that there are two kinds of definitions: "The first kind are genuinely stipulative definitions, which do serve as abreviatory devices, and which generate straightforwardly analytic judgements in Kant's original sense […] The second kind are Frege's 'fruitful' definitions, which start from a given proposition and yield not the concepts originally 'thought into it' but new concepts." (M.