Version: v1, Published online: 1998
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3. The logicist foundations of arithmetic
The fundamental problem for the logicist theory of the arithmetic of the ordinary whole numbers was to show how (1) the general concept of such numbers, (2) the individual numbers themselves, and (3) the modes of reasoning that suffice for the mathematical demonstrations of their properties, can all be derived from principles of logic. It will be convenient here to consider first the solution offered by Frege (1884), with which Russell’s substantially coincides; a brief comparison with Dedekind’s rather different approach will be made later.
Frege set about creating an entirely systematic and universally applicable formal system codifying all logical concepts and processes: his Begriffsschrift, or ‘concept writing’, which he characterized as ‘a formula language…for pure thought’ (1879). He therefore faced the task of making explicit, within the framework of such a formal system, how the cardinal numbers could be defined.
Now for Frege, as for Dedekind, a central logical role was played by a notion of ‘class’ – Dedekind’s ‘system’, characterized by him in the following way:
‘Such a system S…is completely determined when with respect to every thing it is determined whether it is an element of S or not. The system S is hence the same as the system T…when every element of S is also an element of T, and every element of T is also an element of S‘
( 1901: 45)
(Compare with ‘extensionality’ in Set theory, different systems of.) However, in Frege’s opinion, such notions as that of a system or class, and that of a ‘correspondence’ (‘a thing belongs to a thing’) are ‘not usual in logic and are not reduced [by Dedekind] to acknowledged logical notions’ ( 1964: 4). Frege therefore relies, for the logical ground of the notions of class and correspondence, upon the ‘acknowledged’ logical notions of a ‘concept’ (which when it is unary corresponds to a property, and when binary, ternary,…, to a two-place, three-place,…relation) and of the ‘extension’ of such a concept (in particular, the extension of a unary concept is a class). The existence of such an extension for every concept, satisfying the principle that two concepts have the same extension if and only if every object falling under either also falls under the other (see Dedekind’s characterization of ‘system’ above), is a fundamental principle of the logic on which Frege bases arithmetic.
To avoid certain verbal complexities, we now follow Russell in formulating the arithmetic notions in terms of classes. Two classes A, B are said to be ‘equinumerous’ if, for some relation R: (1) for every element x of A there is an element y of B such that x has the relation R to y, but to no other element of B; and (2) for every element y of B there is an element x of A such that x, but no other element of A, has the relation R to y. One says that R is a one-to-one relation between A and B. ‘Equinumerous’, as just defined, is a binary relation that holds between certain classes. Being equinumerous with a given class A is, then, a property which holds of certain classes. In Frege’s usage, the extension of this property [or concept] – and thus, the class of all such classes – is called ‘the number of the class A‘; thus we have succeeded, according to the premises of this whole procedure, in defining the individual numbers as ‘objects’, in purely logical terms. As to the general concept of number, that is now easy: ‘n is a number’ means that there is some class A such that n is the number of the class A.
A most crucial step remains to be taken. We have defined the concept of number – allegedly at least – in purely logical terms; but this is too wide a concept to serve as the basis for ordinary arithmetic. Indeed, ordinary arithmetic has to do with the non-negative integers, and these, in the present context, are numbers of finite sets. But no such restriction appears in our definition of number: we have defined ‘the number of the set A’ for arbitrary sets – for example, for the set of all the points on a given line. At the same time – what at first will seem a quite unrelated issue – we have yet to show how all arithmetical reasoning can be carried out by purely logical argumentation concerning the number-concept.
The two issues have in fact a deep connection; and it is one of Frege’s great accomplishments (also achieved independently by Dedekind) to have perceived the connection and found in it the simultaneous solution to both of them. A characteristic feature of reasoning in arithmetic is the ‘argument from n to n+1’, or the principle of mathematical induction:
If a certain property P holds of the number 0, and if whenever P holds of a number n it holds also of the number n+1, then P holds of all the whole numbers.
(Of course, to formulate this principle at all, one must have defined the process of adding 1 to a given number; but that is a detail, easily accomplished.) The chief problem for placing the argumentation of arithmetic on a ‘purely logical’ footing was to find a logical justification for this principle. But, speaking ‘intuitively’, it is clear that the principle in question cannot apply except to finite numbers (for the property of being finite holds of 0, and holds of n+1 whenever it holds of n; but of course holds only of finite numbers). This ‘intuitive’ argument can of course have no systematic standing so long as the notion of finite number has not been defined. Frege’s idea, however, was to use this ‘intuitive’ insight as the basis for a purely logical definition of the concept of finite number:
A number x is said to be ‘finite’ if and only if x belongs to every class C such that: (1) 0 belongs to C; and (2) whenever a number n belongs to C, so does n+1.
With this definition, not only are the finite numbers singled out, but all the remaining concepts of traditional number theory can be introduced, and the reasoning of traditional number theory can be carried out within the framework of Frege’s extended logic.
Stein, Howard. The logicist foundations of arithmetic. Logicism, 1998, doi:10.4324/9780415249126-Y061-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/logicism/v-1/sections/the-logicist-foundations-of-arithmetic.
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