A History of the Infinite III: The Infinite in Mathematics
This lecture is concerned with the mathematics of the infinite, most of which has developed relatively recently. I outline the principal ideas of the brilliant German mathematician, Georg Cantor, who not only showed that we can distinguish between different infinite sizes, but who devised infinitely big numbers to measure them and who showed how to perform calculations with these numbers. Cantor’s work was both revolutionary and profound. But it greatly polarized opinion amongst his late nineteenth- and early twentieth-century contemporaries, contributing to a complete breakdown in Cantor’s mental health. His work also gave rise to several new paradoxes that contributed to another breakdown of sorts – a breakdown in work on the foundations of mathematics. I expound some of these paradoxes, including Bertrand Russell’s famous paradox of the set of all and only those sets that do not belong to themselves, and I discuss how mathematicians reacted to them. In particular, I talk about Gottlob Frege’s reaction, which was exacerbated by the fact that Russell’s paradox looked as though it had completely destroyed his life’s work, namely his attempt to provide mathematics with rigorous and secure foundations. I also discuss subsequent developments in mathematics that exploit some of these paradoxes, including some of Kurt Gödel’s work. I conclude that, by subjecting the infinite to formal scrutiny, mathematicians have ended up creating more problems for themselves than they have been able to solve and have found themselves having to reckon with some extraordinarily deep puzzles at the very heart of their discipline.
[1] Cantor, Georg, Letter to K.F. Heman, dated 21 June 1888, quoted by Joseph W. Dauben, in his Georg Cantor: His Mathematics and Philosophy of the Infinite (Harvard University Press, 1979). p. 298.
[2] Cantor, Georg, Quoted by Michael Hallett, in his Cantorian Set Theory and Limitation of Size (Oxford University Press, 1984), p. 13.
[3] Frege, Gottlob, ‘Letter to Russell’, trans. Beverly Woodward, in Jean van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879 – 1931 (Harvard University Press, 1967), pp. 127 - 128.
[4] Frege, Gottlob, Grundgesetze der Arithmetik [Basic Laws of Arithmetic] (Jena: H. Pohle, 1893 and 1903).
[5] Gödel, Kurt, ‘1951 J. W. Gibbes Lecture’, quoted by Hao Wang, From Mathematics to Philosophy, (Humanities Press, 1974), p. 324.
[6] Hilbert, David, ‘On the Infinite’, trans, Stefan Bauer-Mengelberg, in Jean van Heijennort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879 – 1931 (Harvard University Press, 1967).
[7] Kronecker, Leopold, Quoted in the obituary ‘Leopold Kronecker’, by Heinrich Weber, in Jahresbericht der Deutsche Mathematiker Vereinigung, 1893, p. 19.
[8] Lucas, J. R., ‘Minds, Machines and Gödel’, reprinted in A. R. Anderson (ed.), Minds and Machines (Prentice-Hall, 1964).
[9] Moore, A. W., The Infinite, third edition (Routledge, 2019).
[10] Russell, Bertrand, ‘Letter to Jean van Heijenoort’, quoted by Jean van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879 – 1931 (Harvard University Press, 1967), p. 127.
[11] Russell, Bertrand, Our Knowledge of the External World as a Field for Scientific Method in Philosophy (Allen & Unwin, 1926).
[12] Wittgenstein, Ludwig, Lectures on the Foundations of Mathematics, ed. Cora Diamond (Harvester Press, 1976).
[13] Wittgenstein, Ludwig, Remarks on the Foundations of Mathematics, third edition, ed. G. H. von Wright, R. Rhees, and G. E. M. Anscombe, and trans. G. E. M. Anscombe (Blackwell, 1978).