From Dedekind to Gödel pp 53-96 | Cite as

# The Mysteries of Richard Dedekind

## Abstract

Everybody loves a mystery. And I have three. Pending their resolutions, any mathematical portrait of Richard Dedekind remains unfinished. For the mysteries are mathematical veils across the face of Dedekind’s work. The first of the three involves a “proof” of something most set theorists do not prove and a conclusion most people would not allow. What set theorists normally do not prove is the axiom of infinity, the assertion that there exists an infinite set. What Dedekind offers us seems to be a mysterious “proof” of exactly that.^{1} Its logical heart is Dedekind’s assertion that the *Gedankenwelt*, his own thoughtworld, constitutes such an infinite collection. It is also Dedekind’s view that from any infinite collection one can garner the unique series of natural numbers.^{2} Can we conclude, therefore, that the one and only natural numbers, the numbers so painstakingly defined in *Was Sind and Was Sollen die Zahlen*? are to be located within the world of his — Dedekind’s — thought? Is this what Dedekind wants us to believe? The second mystery revolves around Dedekind’s famous essay on continuity, *Stetigkeit and irrationale Zahlen*. We uncover the mystery by asking, “How can Dedekind claim to have captured — in his definition of ‘real number’ — the essence of the continuum and, at the very same time, describe for us a continuum which we cannot see to satisfy that definition?” We suppose that, in the “Continuity” essay, Dedekind first unveils his definition of *the* continuum, the one formulated in terms of Dedekind cuts. We do not often remark upon the fact that, before setting out that definition, Dedekind attempts to motivate it by assuming the existence and examining the character of *another*, seemingly *distinct* continuum. This structure Dedekind calls ‘the straight line.’ Although Dedekind, in presenting reals in terms of cuts, claims to have captured the essence of *the* continuum, he makes no effort to prove a uniqueness result. He does not even assay the prospect that the two continua of his own article might be related in some fully satisfactory, mathematical way. Later — in a letter (Dedekind 1932, p. 478) — Dedekind will insist, in effect, that the two structures, the straight line and the collection of Dedekind reals, cannot be proved isomorphic. How many Dedekindian continua are there — one or two? That is the second mystery. On the face of it, the third mystery seems to be one of historical classification rather than mathematical individuation. Dedekind is generally acknowledged to be the father of classical set-theoretic algebra and, by every measure logical and mathematical, was no constructivist. Yet, Dedekind avers — often and throughout his writings — that mathematical entities are not mind-independent, Platonistic abstracta, but are literally *geistig* or mental. More troubling still, Dedekind seems to think them not sempiternal but brought into being by discrete mental episodes, acts he calls ‘*freie Schöpfung*’ — free creation. This is romantic, even revolutionary, talk and just the idiom constructivists such as Brouwer chose to herald the anticlassical revolt. “Free mathematics! A free creation of the human mind!” is a constructivist’s — and not a classical mathematician’s — rallying cry.

## Keywords

Mathematical Object Cardinal Number Infinite System Pure Reason Creative Power## Preview

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## Bibliography

- Dedekind, R.: 1932,
*Gesammelte Mathematische Werke. Dritter Band*, R. Fricke, E. Noether and O. Ore (eds.), Druck und Verlag von Friedrich Vieweg und Sohn, Braunschweig, 508 pp.Google Scholar - Dedekind, R.: 1963,
*Essays on the Theory of Numbers*,*W*. W. Beman (trans.), Dover Publications, New York, 115 pp.Google Scholar - Dedekind, R.: 1967, ‘Letter to Keferstein’, in J. van Heijenoort (ed.),
*From Frege to Gödel. A Source Book in Mathematical Logic**1879–1931*, Harvard University Press, Cambridge, MA, pp. 98–103.Google Scholar - Dugac, P.: 1976,
*Richard Dedekind et les fondements des mathematiques: Avec de nombreux textes inedits*. Collection des travaux de l’Academie internationale d’histoire des sciences, no. 24. J. Vrin, Paris, 334 pp.Google Scholar - Dummett, M.: 1978,
*Truth and Other Enigmas*, Harvard University Press, Cambridge, MA, pp. lviii + 470.Google Scholar - Frege, G.: 1979,
*Posthumous Writings*, H. Hermes*et al*. (eds.), University of Chicago Press, Chicago, pp. xiii + 288.Google Scholar - Hambourger, R.: 1977, ‘A Difficulty with the Frege-Russell Definition of Number’,
*Journal of Philosophy*LXXIV(7) (July 1977), 409–414.Google Scholar - Heyting, A.: 1964, ‘Disputation’, in P. Benacerraf and H. Putnam (eds.),
*Philosophy of Mathematics. Selected Readings*, First Edition, Prentice-Hall, Inc., Englewood Cliffs, pp. 55–65.Google Scholar - Kant, I.: 1950,
*Prolegomena to any future Metaphysics*, L. W. Beck (ed.), The Liberal Arts Press, Inc., Indianapolis, pp. xxxiv + 136.Google Scholar - Kant, I.: 1965,
*Critique of Pure Reason*, N. K. Smith (trans.), St. Martin’s Press, New York, pp. xiii + 681.Google Scholar - Kant, I.: 1966,
*Kritik der reinen Vernunft*,Phillip Reclam, Stuttgart, 1011 pp.Google Scholar - Kitcher, P.: 1986, ‘Frege, Dedekind and the Philosophy of Mathematics’, in L. Haaparanta and J. Hintikka (eds.),
*Frege Synthesized*, D. Reidel Publishing Co., Dordrecht, pp. 299–343.Google Scholar - Kronecker, L.: 1899, ‘Über den Zahlbegriff’, in K. Hensel (ed.),
*Werke*, vol. 3. Teubner, Leipzig, p. 253.Google Scholar - McCarty, D. C. and L. P. McCarty: 1993, ‘Dedekind: The Greatest Art of the Systematizer’, ms., 15 pp.Google Scholar
- Moore, G. H.: 1982,
*Zermelo’s Axiom of Choice. Its Origins*,*Development*,*and Influence*, Springer-Verlag, New York, pp. xiv + 410.Google Scholar - Parsons, C.: 1983,
*Mathematics in Philosophy. Selected Essays*, Cornell University Press, Ithaca, New York, 365 pp.Google Scholar - Quine, W. V. O.: 1971,
*Set Theory and its Logic*, Revised Edition, Harvard University Press, Cambridge, MA, pp. xvii + 361.Google Scholar - Russell, B.: n.d.,
*Introduction to Mathematical Philosophy*,Simon and Schuster, New York, pp. xix + 208.Google Scholar - Salmon, N. and S. Soames: 1988,
*Propositions and Attitudes*, Oxford University Press, New York, 282 pp.Google Scholar - Stein, Howard: 1988,
*‘Logos*, Logic and*Logistiké*: Some Philosophical Remarks on the 19th Century Transformation of Mathematics’, in W. Aspray and P. Kitcher (eds.),*History and Philosophy of Modern Mathematics*, University of Minnesota Press, Minneapolis, pp. 238–259.Google Scholar - van Stigt, W.: 1979, ‘The Rejected Parts of Brouwer’s Dissertation on the Foundations of Mathematics’,
*Historia Mathematica***6**, 385–404.CrossRefGoogle Scholar - van Stigt, W.: 1990,
*Brouwer’s Intuitionism*. Studies in the History and Philosophy of Mathematics, Vol. 2, North-Holland, Amsterdam, pp. xxvi + 530.Google Scholar - Wittgenstein, L.: 1984,
*Tractatus Logico-philosophicus. Werkausgabe Band**1*, Suhrkamp, Frankfurt am Main, 621 pp.Google Scholar