## Abstract

Frege’s famous definition of number (in)famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle (where Frege does not use extensions). I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part of his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a *lingua characterica universalis*.

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## Notes

I am grateful to anonymus reviewer for pointing out that J.L. Austin here translates the expression “sei gleichbedeutend mit”.

“Equal” is the term used by J L. Austin to translate Frege’s “gleichzahlig”. I have preferred the term of “equinumericity” for the relation of one-to-one correspondence.

According to each prespective and problem, the above definition (“an odd kind of definition”, as Frege himself addmitted) has received several names and ways of referring to it in the literature. Probably the most famous denomination is ‘Hume’s Principle’, due to the fact that Frege attributes his ideea to Hume, as a predecessor; it appears under this name throughout the disputes regarding the Neo-Fregeean attempts to revive some of the logicist program (Boolos 1987; Heck 1998; Demopoulos 1998, 2003; Schirn 2017). However, in Wright (1983) the label “N

^{=}” is used. Michael Dummett, in an appendix to his article “Neo-Fregeeans: in Bad Company?”, protests against the ‘Hume’s Principle’ label of the above definition on the ground that Hume was referring to something else (Frege was mistaken to quote him there) and, besides, the term ‘Hume’s Principle ‘would blur a vital distinction between two principles covered by the same name: equinumericity and abstraction principle. Dummett (1991) will use the name of one of the two principles or the term ‘contextual definition’ as opposed to the second, ‘explicite definition’—in terms of extensions—given by Frege. The term “contextual definition “is also used in*Frege: Making Sense*(Beaney 1996). On historical grounds, Beaney (2005) also calls it the “Cantor–Hume Principle”. The name I will use for this assertion will be also Hume’s Principle or “the first attempted definition of number”.“The dominant opinion throughout this century has been that Russell’s Paradox totally exploded Frege’s approach, showing that the nature of numbers, and our knowledge of them, cannot be explained and justified an anything like Frege’s way.

Thirteen years ago I published a short book (henceforward

*Frege’s Conception*) which challenged this view, arguing that Frege’s Platonism—the conception of numbers as specific objects providing the proper subject matter of arithmetic, but given to us independently of any special ‘intuition’—admitted of an epistemologically responsible, powerful defense along the very lines suggested by Frege’s writings, and that—the paradox notwithstanding—(something to the purpose of) the logicism too was salvageable as a programme, at least as far as umber theory was concerned, and maybe further” (Wright 1998a, b, p. 340).“None of this, however, so far makes it any clearer how the identification of numbers with extensions is supposed to solve the Caesar problem. Frege thinks it does so presumably because he views the concept of extension as a

*purely logical*concept, so a concept which, in a reduction of arithmetic to logic, we can take for granted. And if we already have a sortal notion of extension, as part of our general mastery of logic, then we presumably already know whether or not Julius Caesar is an extension. To take the numbers to be (particular) extensions will settle the truth-conditions of mixed identity statements involving numerical terms by treating them as a special case of mixed identity statements involving extension terms: and, these, Frege seems to be supposing, we already understand.This is completely unsatisfying. To begin with, it is not clear whether extension really is a purely logical concept, or even what ground Frege thinks he has for supposing that it is.” (Wright 1983, p. 111).

This thesis is supported by the analogy (made also by Frege) between a function and its graph, on the one side and a concept and its extension on the other side: “For example, take the functions (x + 1)(x − 1) and x² − 1. Each one of them performs different operations on numbers. When mathematicians say that they are the same function, what is meant, according to Frege, is that they have the same graph. Thus the identity-assertion, if it is to be construed as free from category mistakes, says simply that these functions have the same graph, and the respective graph is what is meant by ‘the function f’. The graph is an (abstract) object that represents the function.”(Ruffino 2003, p. 66).

My italics.

“Frege required a characterization of the numbers that would single out one ω- sequence from all the rest. This is what the explicit definition in terms of equivalence of classes of concepts was supposed to accomplish. If successful, it would have achieved this—

*not because there is greater clarity about the claim that Julius Caesar is not an extension than that he is not a number—but because the definition in terms of equivalence classes of concepts characterizes the numbers by reference to their use in ordinary applications*, and therefore, fixes the reference of numerical singular terms to the degree that the reference of*any*singular term is fixed. By contrast, the numbers which are shown to exist on the basis of the contextual definition are characterized purely internally, with respect to their predecessors in the sequence of numbers. The construction of numbers which Frege gives in the proof of his theorem must have this internal character if it is to serve he purpose of a proof; but the explicit definition in terms of equivalence of classes of concepts is needed if the account of number is to be complete in the sense of subsuming all our application of the numbers. In short, it is precisely because the explicit definition purports to comprehend*all*our uses of number—both pure and applied—that it can claim to give*the*reference of our numerical singular terms.” (Demopoulos 1998, p. 491).My italics.

van Heijenoort (1967), in “Logic as Calculus and Logic as Language”, places Frege and Leibniz on the side of the tradition maintaining the view named “logic as language”.

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Grigore, N. Extensions, Numbers and Frege’s Project of Logic as Universal Language.
*Axiomathes* **30**, 577–588 (2020). https://doi.org/10.1007/s10516-019-09468-5

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DOI: https://doi.org/10.1007/s10516-019-09468-5