Helmholtz, Cohen, and Frege on Progress and Fidelity by Teri Merrick

Author:Teri Merrick
Language: eng
Format: epub
ISBN: 9783030572990
Publisher: Springer International Publishing

There is also textual evidence that Frege is giving a fairly accurate description of what the formalists were proposing. Thomae, for instance, likens the tangible signs for numbers to the pieces in a chess-game. In both cases, their only significant properties and relations are those “assigned to them by the rules of the game” (FG, 115).

So, for formalists in the narrow sense, the justification for asserting the existence of a number manifesting the right sort of properties involves three steps: (1) introduce a sign (e.g., √2); (2) use this sign in calculations where the ordinary rules of algebra apply and obtain a list of properties that are mutually compatible and do not generate a contradiction with the properties ascribed to the other numerical signs; and (3) ascribe to the sign those properties and only those properties (e.g., the property of yielding 2 when multiplied by itself). While Frege may be slightly overstating their alleged mistake when he says that they assume “freedom of contradiction amounted straight way to existence,” given the additional step of producing the tangible sign, it is certainly fair to say that the formalists considered lack of contradiction sufficient for asserting the existence of an object having “properties that it has not already got” (CP, 139).64

Dedekind, Otto Stolz, and Hilbert are similarly scolded for thinking that defining a logically consistent concept or system of concepts suffices for asserting the existence of objects bearing the marks of those concepts as properties. In his critique, “The Construction of New Objects, according to R. Dedekind, H. Hankel, O.Stolz”, Frege is at pains to distinguish the views of Dedekind and Stolz from those of Thomae. Dedekind and Stolz do not equate the numbers with signs and then appeal to creative definition for the purposes of ascribing to those signs the relevant properties to that sign. Instead, Stolz appeals to creative definition to order to construct an entirely “new- at any rate non-sensible—thing”, which he then “supplies with a sign” (FR, 276).65 Frege also presents one of Stolz’ creative definitions: 1.Definition .If in this case (D1) no magnitude of System (I) satisfies the equation b °x = a, then it shall be satisfied by one and only one new thing not found in (I); this may be symbolized by a U b, since this symbol has not yet been used. We thus have

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