Abstract
The term “structuralism” with reference to mathematics has perhaps a basic meaning as referring to the idea that mathematics is particularly concerned with structures or structure. My concern is a more specific one, with what I call the structuralist view of mathematical objects. In its general lines, it is familiar: It holds that reference to mathematical objects is always in the context of some structure, and that the objects involved have no more to them than can be expressed in terms of the basic relations of the structure. The idea is well expressed by Michael Resnik:
In mathematics, I claim, we do not have objects with an “internal” composition arranged in structures, we have only structures. The objects of mathematics, that is, the entities which our mathematical constants and quantifiers denote, are structureless points or positions in structures. As positions in structures, they have no identity or features outside of a structure.1
Reprinted from Modality, Morality, and Belief: Essays in Honor of Ruth Barcan Marcus, edited by Walter Sinnott-Armstrong in collaboration with Nicholas Asher and Diana Raffman. Copyright 1995 by Cambridge University Press. Reprinted by permission of the author and Cambridge University Press.
The penultimate version of the paper was presented to the meeting in Budapest in May 1993 of the Académie Internationale de Philosophie des Sciences. I am grateful for the comments of hearers in this and other audiences, not all of whom I remember individually. Saunders Mac Lane, Rosemarie Rheinwald, and Wilfried Sieg deserve special mention. I wish to thank Walter Sinnott-Armstrong for editorial suggestions.
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Notes
A classic instance of set-theoretic structuralism is Dedekind’s account of the natural numbers in Was sind und was sollen die Zahlen? (Braunschweig: Vieweg, 1888); an example of the effort to avoid the concept of set in stating a structuralist view is Resnik, “Mathematics as a science of patterns”. What has been variously called “if-thenism”, “implicationism”, or “deductivism” would be an example of non-set-theoretic structuralism, but in the last twenty years this approach has hardly found any defenders.
The structuralist view of mathematical objects, Synthese, 84 [1990], 303–346. The ambiguity mentioned in the text is related to but distinct from another problem in recent discussions of structuralism in the literature, whether structuralism is an alternative to, or a form of, Platonism. The view of “The structuralist view”, after what I called “eliminative structuralism” has been rejected, is that although structuralism is intended to avoid certain pictures associated with Platonism, it is in the most fundamental sense Platonistic, at least in application to strong theories such as set theory. In this I am in agreement with Resnik.
“The structuralist view”, §7. See also Mathematics in Philosophy (Ithaca, N. Y.: Cornell University Press, 1983), pp. 22, 191–192, where, however, I did not distinguish between a strictly structuralist conception of the universe of sets and the “ontological” view to be stated presently, where the latter view allows for ontological relativity and “incompleteness.”
See part III of “Genetic explanation in the roots of reference”, in Robert Barrett and Roger Gibson, (eds.), Perspectives on Quine (Oxford: Blackwell, 1990), pp. 271–290.
What I, following Frege, call extensions is what some writers call logical collections, in effect using the term “collection” as a generic term for any entity that might do duty for sets or classes. An example is Penelope Maddy, Realism in Mathematics (Oxford: Clarendon Press, 1990), pp. 102–103. That usage seems to leave us without a term for what I call a collection, at least if one distinguishes, as I argue in the text should be done, between the notion of collection and that of plurality. Maddy’s term “combinatorial collection” (ibid., p. 102) is close to my term “collection”, but her usage is more specifically tied to the iterative conception of the universe of sets.
The latter has been questioned especially for the notion of collection. Frege’s criticisms of some of his contemporaries amounted to the claim that this notion could not be clearly distinguished from that of a mereological sum. Nelson Goodman’s well-known rejection of classes is again in the first instance a rejection of collections. (Cf. “Genetic explanation”, p. 284.) See also Max Black, The elusiveness of sets, Review of Metaphysics 24 [1970–71], 614–636. In my terms, Black is severe with the idea of collection but friendly to that of plurality.
“Genetic explanation”, pp. 285–286.
Axioms of set theory, p. 323. In a footnote Shoenfield states that “before” is to be understood “in a logical rather than a temporal sense”. It is possible, given other assumptions about stages, to derive the statement that they are well-ordered (which will yield the axiom of foundation) from the assumption that they are partially ordered. See ibid., p. 327. (This was first shown by Dana Scott.) For fuller discussion see George Boolos, Iteration again, Philosophical Topics, 17 [1989], 5–21. I doubt that this somewhat remarkable technical fact is of much philosophical significance, since it is hard to see how the existence of a partially ordered sequence of stages is more evident than the existence of a well-ordered sequence.
I take this to be the intention of the somewhat sceptical discussion of such justifications by Maddy, Believing the axioms, The Journal of Symbolic Logic 53 [1988], 481–511, 736–764. Similarly Boolos, “Iteration again”, sees some axioms as coming naturally from the “iterative conception of set” and others quite independently from the idea of limitation of size.
In Collected Works, Volume II: Publications 1938–1974, Solomon Feferman et al., eds., (Oxford University Press, 1990). The article mentioned in the text was first published in 1947 and reprinted, revised and expanded, in 1964.
This view is the one most explicitly expressed by Quine and the dominant view before The Roots of Reference (La Salle, Ill.: Open Court, 1974). The latter work makes some concessions to more usual ways of looking at set theory. See my “Genetic explanation”.
Cantor, Gesammelte Abhandungen, ed. Ernst Zermelo (Berlin: Springer, 1932), pp. 204, 443 and elsewhere; Wang: From Mathematics to Philosophy (London: Routledge, 1974), pp. 281–284. The choice of language already seems to force a choice as to whether one will regiment the plural by the singular. The more conservative choice is not to do so, and I will try to avoid it.
Boolos, To be is to be the value of a variable (or to be some values of some variable), Journal of Philosophy 81 [1984], 430–449, and Nominalist Platonism, Philosophical Review 94 [1985], 327–344. See also David Lewis, Parts of Classes (Oxford: Blackwell, 1991), §3.2.
Mathematics in Philosophy, pp. 315–318.
Lewis, Parts of Classes;, see also Mathematics is megethology, Philosophia Mathematica (series 3)1 [1993], 3–23.
Adorns of Set Theory, p. 326.
As already argued in Mathematics in Philosophy, p. 280, n. 16.
From Mathematics to Philosophy, p. 186; cf. Mathematics in Philosophy, pp. 279–280.
“The present situation in the foundations of mathematics”, lecture given to the Mathematical Association of America, in: Collected Works, volume III: Unpublished Essays and Lectures, Solomon Feferman et al., eds., Oxford: Oxford University Press, 1995, pp. 45–53.
Cf. Ernst Zermelo, Über Grenzzahlen und Mengenbereiche, Fundamenta Mathematicae, 16 [1930], 29–47.
Such an appeal to plenitude is implicit in Shoenfield, Axioms of set theory, p. 326. It is embodied in axiom IX of George Boolos, The iterative conception of set (first published 1971), in Paul Benacerraf and Hilary Putnam, (eds.), Philosophy of Mathematics: Selected Readings, 2d ed. (Cambridge: Cambridge University Press, 1983), p. 494.
Sur le platonisme dans les mathématiques, L’enseignement mathématique 34 [1935], 52–69, p. 276 of the translation in Benacerraf and Putnam, Philosophy of Mathematics.
Shaughan Lavine argues persuasively that the power set was not assumed in the theory of sets and transfinite numbers as Cantor originally worked it out and that reasoning that would require the power set axiom appears in his work only after 1890. In Lavine’s view, Cantor’s assuming at that point of the equivalent of the power set axiom led to the consequence that the real numbers are a set, something which Cantor had earlier hoped to prove by, in effect, proving the Continuum Hypothesis. See Understanding the Infinite, Cambridge, Mass.: Harvard University Press, 1994, chapter IV.
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Parsons, C. (1997). Structuralism and the Concept of Set. In: Agazzi, E., Darvas, G. (eds) Philosophy of Mathematics Today. Episteme, vol 22. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5690-5_10
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