DOI: 10.4324/9780415249126-Y061-1
Version: v1,  Published online: 1998
Retrieved September 21, 2020, from

1. Preliminary discussion

Although the thesis that arithmetic is a part of logic was stated quite explicitly by Frege (1884, 1893, 1903) and by Russell (1903), it is surprisingly difficult to determine the exact content of this thesis; and, indeed, to determine whether it should be understood to mean the same thing to each of them. For the obvious question presents itself: exactly what are we to understand by ‘logic’? How do we tell whether a certain basic assumption, or a certain basic concept, properly belongs to logic, or not? And, since an important part of the thesis is its epistemological aspect – the claim that mathematical knowledge is purely logical in nature – one would naturally expect an account of the grounding of our knowledge of logic. On the first of these questions – how to tell what belongs to logic – Frege has little to say (although he acknowledges the importance of the issue for his philosophy, and does provide a few pregnant suggestions); Russell has some lengthy considerations, which have proved not to be very satisfactory. On the second question, Frege and Russell appear to disagree sharply. Both regard the thesis as being in opposition to Kant; but whereas Frege, in denying that arithmetic is based upon any Kantian ‘pure intuition’ and maintaining that it is entirely grounded in logic, concludes that Kant was wrong to consider arithmetical propositions synthetic, and holds by contrast that, as propositions of logic, they are analytic (roughly speaking: known to be true by ‘analysis’ of their meanings alone), Russell denies the latter, and holds that logic itself is synthetic in character:

Kant never doubted for a moment that the propositions of logic are analytic, whereas he rightly perceived that those of mathematics are synthetic. It has since appeared that logic is just as synthetic as all other kinds of truth; but this is a purely philosophical question, which I shall here pass by.

(1903: 457)

Yet even here the issue is not so clearly drawn; for one must still ask whether Frege, in affirming that logic is analytic and not synthetic, and Russell, in affirming the opposite, understood the words ‘analytic’ and ‘synthetic’ in the same way – in particular, whether their opposite formulations might none the less have expressed the same opinions. The possibility is lent some weight by the fact that in certain late writings Frege refers to a ‘logical source of knowledge’ as a distinct knowledge-source (alongside ‘sense perception’ and ‘the geometrical source of knowledge’) ([1924–5] 1979: 267, 278–9). Russell’s view that logical truth is synthetic may be essentially identical with Frege’s view that knowledge of such truth requires a ‘knowledge-source’.

Citing this article:
Stein, Howard. Preliminary discussion. Logicism, 1998, doi:10.4324/9780415249126-Y061-1. Routledge Encyclopedia of Philosophy, Taylor and Francis,
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