Abstract
It is a commonly held view that Dedekind’s construction of the real numbers is impredicative. This naturally raises the question of whether this impredicativity is justified by some kind of Platonism about sets. But when we look more closely at Dedekind’s philosophical views, his ontology does not look Platonist at all. So how is his construction justified? There are two aspects of the solution: one is to look more closely at his methodological views, and in particular, the places in which predicativity restrictions ought to be applied; another is to take seriously his remarks about the reals as things created by the cuts, instead of considering them to be the cuts themselves. This can lead us to make finer-grained distinctions about the extent to which impredicative definitions are problematic, since we find that Dedekind’s use of impredicative definitions in analysis can be justified by his (non-Platonist) philosophical views.
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Notes
Tout ce qui contient une variable apparante doit être exclu des valeurs possibles de cette variable.
The literature contains many different definitions of impredicativity, but the differences among them are not the subject of this paper. But for further discussion on the variety of such definitions, see George (1987).
The term “predicative” was only coined after Dedekind’s work had already appeared, and after the theory of sets was more fully developed. But although we do not find any direct commentary by Dedekind on this matter, we will see that certain considerations he raises can easily be read as having some bearing on the question of predicativity.
A more common translation of the term “Schnitt” is cut, but Stillwell translates this as “section”, so I will follow his translation.
This lets us view the reals as a structure \(\langle {\mathfrak{R}}, <, +, -, \times, \backslash \rangle.\)
See Hallett (1984).
See Reck (2003) for more discussion of Dedekind as a logical structuralist, and a comparison of logical structuralism to other structuralist views.
In fact, in the disagreement between Kronecker and Dedekind on the correct formulation of the theory of ideal divisors, we find an analogue of the situation here.
This amounts to adopting the system ACA0. Feferman (1999) discusses this system in more detail, but we are only interested in it here as a point of contrast to Dedekind’s system.
Admittedly, a less ontologically loaded version of Platonism, such as the one articulated by Bernays, which “does not claim to be more than, so to speak, an ideal projection of a domain of thought (Bernays 1934, p. 261)” might end up being compatible with Dedekind’s views. But it is at least clear that Dedekind does not justify impredicative definitions along the more contemporary Platonist lines suggested by, for instance, Gödel.
This quote appears in a paper by Parsons in which he criticizes a version of structuralism he calls Dedekindian, but later suggests an improved structuralist view. Given the characterization in Reck (2003) which I follow here, Dedekind’s actual view seems quite similar to the improved structuralism Parsons articulates in that paper. This improved version does not try to be entirely eliminativist.
See, for instance, Edwards (1983), on Dedekind’s use of sets in ideal theory.
References
Bernays, P. (1934). On platonism in mathematics. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. Cambridge: Cambridge University Press.
Dedekind, R. (1872). Continuity and irrational numbers. In W. Ewald (Ed.), From Kant to Hilbert: A sourcebook in the foundations of mathematics (Vol. II). Oxford: Clarendon Press.
Dedekind, R. (1877). Theory of algebraic integers. Cambridge: Cambridge University Press.
Dedekind, R. (1888). Letter to Heinrich Weber. In W. Ewald (Ed.), From Kant to Hilbert: A sourcebook in the foundations of mathematics (Vol II). Oxford: Clarendon Press.
Edwards, H. M. (1983). Dedekind’s invention of ideals. Bulletin of the London Mathematical Society, 15, 8–17.
Feferman, S. (1964). Systems of predicative analysis. The Journal of Symbolic Logic, 29(1), 1–30.
Feferman, S. (1999). The significance of Weyl’s Das Kontinuum. In V. Hendricks, S. Pedersen, & K. Jorgenson (Eds.), Proof theory: History and philosophical significance. Berlin: Springer.
Feferman, S. (2005). Predicativity. In S. Shapiro (Ed.), The Oxford handbook of the philosophy of mathematics and logic. Oxford: Oxford University Press.
Feferman, S. & Hellman, G. (2000). Challenges to predicative foundations of arithmetic. In G. Sher & R. Tieszen (Eds.), Between logic and intuition: Essays in honor of Charles Parsons. Cambridge: Cambridge University Press.
George, A. (1987). The imprecision of impredicativity. Mind, 96(84), 514–518.
Gödel, K. (1944). Russell’s mathematical logic. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. Cambridge: Cambridge University Press.
Hallett, M. (1984). Cantorian set theory and limitation of size. Oxford: Clarendon Press.
Parsons, C. (1990). The structuralist view of mathematical objects. Synthese, 84, 303–346.
Reck, E. (2003). Dedekind’s structuralism: An interpretation and partial defense. Synthese, 137, 369–419.
Reck, E. (2009). Dedekind’s contributions to the foundations of mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. http://plato.stanford.edu/archives/spr2009/entries/dedekind-foundations/.
Russell, B. (1906). Les paradoxes de la logique. Revue de Metaphysique et de Morale, 14(5), 627–754.
Stein, H. (1988). Logos, logic, and Logistiké: Some philosophical remarks on nineteenth-century transformation of mathematics. In W. Aspray & P. Kitcher (Eds.), Minnesota studies in the philosophy of science: History and philosophy of modern mathematics (Vol. XI). Minneapolis: University of Minnesota Press.
Tait, W. (Ed.) (1996). Frege versus Cantor and Dedekind: On the concept of number. In Frege, Russell, Wittgenstein: Essays in early analytic philosophy (in honor of Leonard Linsky). Lasalle: Open Court Press.
Weyl, H. (1994). The continuum: A critical examination of the foundation of analysis (S. Pollard & T. Bole, Trans.). Mineola, NY: Dover Publications.
Acknowledgements
I would like to thank Jeremy Avigad and Solomon Feferman for extremely helpful discussions about this paper, as well as an audience at the Society for Exact Philosophy meeting in Vancouver at which a preliminary version of this paper was presented. Finally, I would like to thank two anonymous referees for their constructive and insightful feedback on this paper.
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Yap, A. Predicativity and Structuralism in Dedekind’s Construction of the Reals. Erkenn 71, 157–173 (2009). https://doi.org/10.1007/s10670-009-9179-5
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DOI: https://doi.org/10.1007/s10670-009-9179-5