Three Kantian Strands in Frege’s View of Arithmetic


  • Gilead Bar-Elli



On the background of explaining their different notions of analyticity, their different views on definitions, and some aspects of Frege’s notion of sense, three important Kantian strands that interweave into Frege’s view are exposed. First, Frege’s remarkable view that arithmetic, though analytic, contains truths that “extend our knowledge”, and by Kant’s use of the term, should be regarded synthetic. Secondly, that our arithmetical (and logical) knowledge depends on a sort of a capacity to recognize and identify objects, which are given us in particular ways, constituting their senses (Sinne). Third, that there is a sense in which Frege’s view of definitions and explications gives new substance to Kant’s leading idea of analyticity, namely, the containment of a truth or a concept in another. In all these, Frege’s view does not endorse the Kantian strands as they are, but gives them special and sometimes quite sophisticated twists.


Frege’s Writings:

Begriffsschrift (1879), tr. Bauer-Mengelberg in: From Frege to Goedel, ed. J, van Heijenoort, Harvard, 1967.

Die Grundlagen der Arithmetik (Breslau, 1884)—The Foundations of Arithmetic, tr. J. Austin, Oxford, 1950 (FA).

Grundgesetze der Arithmetik (Jena, 1893, 1903) Georg Olms, 1998, tr. M. Furth as Basic Laws of Arithmetic, Berkley, 1964 (BL).

Nachgelassene Schriften, Herausg. Hermes et. al. Felix Meiner,
Hamburg, 1969, tr. P. Long and R. White as Posthumous Writings, Chicago University Press, 1979 (PW).

Philosophical and Mathematical Correspondence, Basil Blackwell, 1980 (PMC).

Collected Papers on Mathematics Logic and Philosophy, ed. B.
McGuinness, Oxford, 1984 (CP).

“Boole’s Logical Calculus and the Concept-Script” (BLC), in PW,

“Ueber Sinn und Bedeutung” (1892), tr. M.Black as “On Sense and Reference” in Translations from the Philosophical Writings of Frege, trs. P. Geach and M. Black, Blackwell, 1952, 56-78

“Funktion und Begriff” (1892), tr. P. Geach as “Function and
Concept”, ibid. (FC).

“Begriff und Gegenstand” (1892), tr. P. Geach as “Concept and
Object”, ibid. (CO).

“Gedankegefuege” (1923), tr. P. Geach as “Compound Thoughts”, in CP 390-406.

Kant, I: Kritik der reinen Vernunft, translated as Critique of Pure Reason, N. Kemp Smith (trnsl.) Macmillan, 1933 (CPR).

Other Writers

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