Wittgenstein's Lectures on the Foundations of Mathematics: Cambridge, 1939 (review)

BOOK REVIEWS 361 one hand, or realists, on the other. Dewey placed knowing within the wider view of experience ; but realists interpreted the term experience to be what the idealists called consciousness, and thus Dewey was to them a disguised idealist or at best "a half-hearted naturalist," as Santayana called him. To some idealists, Dewey was a confused realist, to others he was a materialist , and to others he was a Cartesian dualist. Dewey's account of the knowing process that developed out of the methods of experimental science had difficulties in making its way into the minds of philosophers already committed to realism or idealism. Dicker's attempt to relate Dewey to the different contexts of epistemological writing is admirable but a formidable task, a task that requires more space and more detail. Be that as it may, this reviewer will direct beginning students to Dicker's monograph as a starting place in understanding this difficult theory. S. MORalS EAMES Southern Illinois University, Carbondale Cora Diamond, ed. Wittgenstein's Lectures on the Foundations of Mathematics: Cambridge, 1939. From the notes of R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Ithaca: Corneil University Press, 1976. Pp. 300. $18.50. From 1929 through 1932 and again from 1937 through 1944 Ludwig Wittgenstein wrote extensively on the philosophy of mathematics. Manuscripts from the earlier period are close in concept to his Tractatus Logico-Philosophicus. Those of the later period, which are conceptually close to his Philosophical Investigations, have been published as Remarks on the Foundations of Mathematics (1956). The Remarks, as it turns out, do not exhaust the available literature by Wittgenstein devoted exclusively to the philosophy of mathematics. In 1939 he delivered thirty-one lectures on the foundations of mathematics at Cambridge University. These lectures, which provide extremely rich insights into Wittgenstein's thought, make up the present volume. Wittgenstein lectured without notes. In reconstructing his lectures it has been possible to draw on student notes by R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. The result is surprisingly coherent. Where student accounts stand in contradiction or are fragmentary the editor provides cautionary footnotes. References are also provided signaling closely related passages in the Remarks and Investigations. There is little in these lectures to surprise those familiar with Wittgenstein's later philosophy . The enemy here, as elsewhere, is the Iogicism of Russell and Frege, with its close ties to Platonism and Platonic interpretations of Cantorian arithmetic, and its promise of an ideal language. Hilbert's formalism is mentioned only in passing, while the intuitionism of Brouwer and others is dismissed as sheer "bosh"--a move that relieves the lecturer of having to consider how near his own position is to intuitionism. Mathematics, Wittgenstein argues, is an activity, not a series of discoveries of an increasingly known realm of mathematical "objects." The temptation to suppose that when we are dealing with Cantorian infinite sets we are confronted with something immense must be thwarted by the realization that in mathematics we are dealing with "rules of expression." The child who has learned ~ o multiplications is not confronted with something unimaginably gigantic; he has simply learned a rule for proceeding in mathematics. Mathematical concepts must be viewed as human linguistic inventions. Exchanges between Wittgenstein and his students are included by the editor. Many of these (especially the discussions with A. M. Turing, John Wisdom, and Casimir Lewy) are extremely helpful. For example: [To Turing] Before we stop, could you say whether you really think that it is the contradiction which gets you into trouble-the contradiction in logic? Or do you see that it is something quite different?-I don't say that a contradiction may not get you into trouble. Of course it may. 362 HISTORY OF PHILOSOPHY Turing: 1think that with the ordinary kind of rules which one uses in logic, if one can get into contradictions , then one can get into trouble. Wittgenstein: But does this mean that withcontradictions one musI get into trouble? Or do you mean that the contradiction may tempt one into trouble? As a matter of fact it doesn't. No one has ever yet got into trouble from a contradiction in...