Abstract
Wittgenstein’s philosophy of mathematics has long been notorious. Part of the problem is that it has not been recognized that Wittgenstein, in fact, had two chief post-Tractatus conceptions of mathematics. I have labelled these the calculus conception and the language-game conception. The calculus conception forms a distinct middle period. The goal of my article is to provide a new framework for examining Wittgenstein’s philosophies of mathematics and the evolution of his career as a whole. I posit the Hardyian Picture, modelled on the Augustinian Picture, to provide a structure for Wittgenstein’s work on the philosophy of mathematics. Wittgenstein’s calculus period has not been properly recognized, so I give a detailed account of the tenets of that stage in Wittgenstein’s career. Wittgenstein’s notorious remarks on contradiction are the test case for my theory of his transition. I show that the bizarreness of those remarks is largely due to the calculus conception, but that Wittgenstein’s later language-game account of mathematics keeps the rejection of the Hardyian Picture while correcting the calculus conception’s mistakes.
The following abbreviations are used in this article to refer to Wittgenstein’s works: WWK: Ludwig Wittgenstein and the Vienna Circle: Conversations Recorded by Friedrich Waismann, ed. B. F. McGuinness, trans. J. Schulte and B. F. McGuinness, Oxford: Basil Blackwell, 1979; CAM I: Wittgenstein’s Lectures: Cambridge, 1930–32, ed. D. Lee, Chicago: University of Chicago Press, 1982; CAM II: Wittgenstein’s Lectures: Cambridge, 1932–35; ed. A. Ambrose, Chicago: University of Chicago Press, 1982; PG: Philosophical Grammar, ed. R. Rhees, trans. A. Kenny, Oxford: Basil Blackwell, 1974; BIB: The Blue and Brown Books, Oxford: Basil Blackwell, 1958; LFM: Wittgenstein’s Lectures on the Foundations of Mathematics: Cambridge, 1939, ed. C. Diamond, Ithaca: Cornell University Press, 1976; RFM: Remarks on the Foundations of Mathematics, ed. G. H. von Wright, R. Rhees, G. E. M. Anscombe, trans. G. E. M. Anscombe, revised ed., Cambridge: MIT Press, 1978; PI: Philosophical Investigations, ed. G. E. M. Anscombe, R. Rhees, trans. G. E. M. Anscombe, New York: Macmillan Company, 1953; Z: Zettel, ed. G. E. M. Anscombe, G. H. von Wright, trans. G. E. M. Anscombe, Berkeley and Los Angeles: University of California Press, 1970.
References to PI and Z are to remark number; references to RFM are to part number (Roman numerals) and remark number (Arabic numerals); and references to the other works are to page numbers. As the evolutionary nature of Wittgenstein’s work is an important theme of this article, following the abbreviation for the book in the text I have put in brackets the date of the book or the part of the book from which the quotation comes.
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Notes
Michael Dummett, ‘Wittgenstein on Mathematics’, Encounter 50, (March 1978), 68.
Calling the black hole view a “standard view” does not, of course, mean it is the only view. There are those who, like Dummett, read Wittgenstein as a radical in the philosophy of mathematics, but, unlike Dummett, think Wittgenstein is right. They dissent from the black hole theory for different reasons than I do. Two notable examples are Crispin Wright, Wittgenstein on the Foundations of Mathematics, Cambridge; Harvard University Press, 1980;
S. G. Shanker, Wittgenstein and the Turning-Point in the Philosophy of Mathematics, State University of New York Press, Albany, 1987.
There is ample evidence that Wittgenstein was familiar with both Hardy and his works. See John King, ‘Recollections of Wittgenstein’, in Rush Rhees (ed.), Recollections of Wittgenstein, Oxford University Press, Oxford, 1984, p. 73;
G. H. von Wright, A Biographical Sketch, in Norman Malcom, Ludwig Wittgenstein: A Memoir, Oxford University Press, Oxford, 1958, p. 6;
Wolfe Mays, ‘Recollections of Wittgenstein’, in K. T. Fann (ed.), Ludwig Wittgenstein: The Man and the Philosopher, Humanities Press, New Jersey, 1967, p. 82;
Garth Hallett, A Companion to Wittgenstein’s ‘Philosophical Investigations’, Cornell University Press, Ithaca, 1977, p. 766. References to Hardy in LFM are listed in that book’s index. Wittgenstein also mentions Hardy in CAM II [1932–33]: 215–20, 222, and 224–25; in B1B [1933–34]: 11; and quotes from ‘Mathematical Proof’ in Z [1945–48]: 273.
G. H. Hardy, A Mathematician’s Apology, Cambridge University Press, Cambridge, 1967, pp. 123–24.
Rush Rhees, ‘On Continuity: Wittgenstein’s Ideas, 1938’, in Discussions of Wittgenstein, Routledge & Kegan Paul, London, 1970, p. 109.
G. H. Hardy, ‘Mathematical Proof’, Mind 38, (January 1929), 18. Hardy continues: “This is plainly not the whole truth, but there is a good deal in it.”
See W. W. Tait, Mathematical Proof’, Mind 38, (January 1929), 343–45. Once again I am heavily indebted to this article.
See also PG [1932–34]: 89: “[L]anguage is not something that is first given a structure and then fitted onto reality”, and even more simply PG [1932–34]: 143: “It is in language that it’s all done.”
See WWK [1930]: 133;
WWK [1931]: 149 and 202;
PG [1932–34]: 369.
Charles S. Chihara, ‘Wittgenstein’s Analysis of the Paradoxes in his Lectures on the Foundations of Mathematics’, Philosophical Review 86 (1977), 369.
See also WWK [1930]: 120, 131;
WWK [1931]: 174, 208;
WWK [1932–34]: 303–05;
CAM II [1933–34]: 71;
LFM [1939]: 206–07, 211, 224–25;
RFM [1937–38]: App. III.17.
Leaves of Grass, “Song of Myself”, verse 51.
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Gerrard, S. (1991). Wittgenstein’s Philosophies of Mathematics. In: Hintikka, J. (eds) Wittgenstein in Florida. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3552-8_4
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