Abstract
Our first goal here is to show how one can use a modal language to explicate potentiality and incomplete or indeterminate domains in mathematics, along the lines of previous work. We then show how potentiality bears on some longstanding items of concern to Mark Steiner: the applicability of mathematics, explanation, and de re propositional attitudes toward mathematical objects.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
- 3.
We make use here of contemporary modal notions, foregoing any attempt to recapitulate what Aristotle himself says about modality.
- 4.
Strictly speaking, there are no sets, collections, or totalities that are themselves potentially infinite. But it is useful to use a count noun to talk about the kinds of “things” said to be potentially infinite. We will use “collections” for this, sometimes in scare quotes.
- 5.
See Linnebo (2013) for an account along these roughly Cantorian lines.
- 6.
See Linnebo and Shapiro (2020) for a bit more detail on these cases.
- 7.
If a potentialist did make essential use of the explication of modality in terms of possible worlds, she would, presumably, think of the worlds themselves as potentially infinite. So it is not clear that there is much of a gain in understanding, analysis, or the like. Here we make use of the usual possible-worlds semantics to obtain results about what is, and is not, derivable in the formal systems. For example, one can block an inference from (1) to (2) above by citing a model in which every world has only finitely many members (and thus only finitely many parts of the given stick), but each world has access to another with more parts. We do not directly address the interesting question concerning the extent to which a potentialist can accept our results.
- 8.
In terms of possible worlds, the relevant modality is the one that results from restricting the accessibility associated with metaphysical modality by imposing the additional requirement that domains don’t ever decrease along the accessibility relation. This restriction can be captured proof-theoretically, using the resources of plural logic (see Linnebo and Shapiro, 2019, p. 188, n. 15).
- 9.
Recall that S4 and (non-free) first-order logic entails (CBF). We can also require the accessibility relation to be well-founded, on the grounds that all mathematical construction has to start somewhere. However, nothing of substance turns on this here.
- 10.
See for example, Brauer et al. (2022). The topic there is intuitionistic free-choice sequences, which are not convergent. A given lawless sequence (for example) can take on any value for an argument on which it is not yet defined. But once it takes on some value, this value cannot later be changed.
- 11.
This is an alternative to the more familiar Gödel translation, which translates ‘∀’ as ‘\(\Box \forall \)’ (as we do), ‘∃’ as itself, and also adopts a non-homophonic translation of negation and the conditional. This translation is poorly suited to explicating potentialism. For example, the translation of ∀m∃n Succ(m, n) is \(\Box \forall m \exists n \Box \text{SUCC}(m,n)\), which, as discussed in Sect. 1.1, a potentialist would reject. For more detail, see Linnebo and Shapiro (2019).
- 12.
There are interesting issues concerning comprehension axioms in higher-order frameworks. See Linnebo and Shapiro (2019, §7).
- 13.
A salient example is the Aristotelian statement, above, rejecting the actual infinity of the natural numbers:
$$\displaystyle \begin{aligned} \neg \Diamond \forall m \exists n \text{SUCC}(m,n) \end{aligned} $$(5)This is not in the range of the potentialist translation, and so has no counterpart in the non-modal framework. Moreover, the formula, \(\neg \Box \forall m \Diamond \exists n \text{SUCC}(m,n)\), which is in the range of the translation, is the contradictory opposite of (4).
- 14.
The intuitionistic modal predicate system must be formulated with some care, since the two modal operators are not inter-definable. See Simpson (1994) for the details.
- 15.
It must also be assumed that the nearby tape factory will never run out of whatever material is used to make tape.
- 16.
This would involve adopting the potentialist translations of all the axioms of Peano Arithmetic, not just of the axiom (4) stating that necessarily every number potentially has a successor.
- 17.
Aristotle, too, was interested in “Why?” questions: “Knowledge is the object of our inquiry, and men do not think they know a thing until they have grasped the ‘why’ of it” (Physics II.3, 194b 18–20). Steiner’s innovation is to replace Aristotle’s essentialism with the notion of a “characterizing property” of a proof.
- 18.
This is implicit in Aristotle’s thesis, noted above, that matter and thus space is infinitely divisible.
- 19.
Here we move away from Steiner’s own “essentialist” account of explanatory proof toward a more unificationist account. See, for example, Kitcher (1989).
- 20.
Quine (1966b, p. 9) suggests that a vivid designator “is the analogue, in the logic of belief, of a rigid designator”.
- 21.
Thanks to Craige Roberts here.
References
Aristotle, (1941). The basic works of Aristotle. R. McKeon (Ed.). New York: Random House.
Brauer, E., Linnebo, Ø., & Shapiro, S. (2022). Divergent potentialism: A modal analysis with an application to choice sequences. Philosophia Mathematica, 30(2), 143–172.
Burge, T. (2007). Foundations of mind. Oxford: Oxford University Press.
Chomsky, N. (1957). Syntactic structure. Mouton: The Hague and Paris.
Coope, U. (2012). Aristotle on the infinite. In C. Shields (Ed.), The Oxford Handbook of Aristotle (pp. 267–286). Oxford: Oxford University Press.
Fine, K. (2005). Our knowledge of mathematical objects. In T. S. Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology I (pp. 89–110). Oxford: Oxford University Press.
Frege, G. (1994). Die Grundlagen der Arithmetik, Breslau: Koebner; The foundations of arithmetic, translated by J. Austin, second edition, New York, Harper, 1960.
Gauss, K. F. (1831). Briefwechsel mit Schumacher, Werke. Band 8, p. 216.
Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Oxford University Press.
Hellman, G., & Shapiro, S. (2018). Varieties of continua: From regions to points and back. Oxford: Oxford University Press.
Hilbert, D. (1983). Über das Unendliche. Mathematische Annalen, 95, 161–190; translated as “On the infinite”, in Philosophy of mathematics, second edition, edited by Paul Benacerraf and Hilary Putnam, Cambridge, Cambridge University Press, pp. 183–201, 1983.
Hintikka, J. (1966). Aristotelian infinity. The Philosophical Review, 75, 197–218.
Kaplan, D. Quantifying in. Synthese, 19, 178–214.
Kearns, K. (2011). Semantics. Modern linguistics series (2nd edn.). London: Palgrave Macmillan.
Kitcher, P. (1989). Explanatory unification and the causal structure of the world. In P. Kitcher & W. Salmon (Eds.), Scientific explanation. Minnesota studies in the Philosophy of science (vol. 13, pp. 410–505). Minneapolis, MN: University of Minnesota Press.
Knorr, W. R. (1982). Infinity and continuity: The interaction of mathematics and philosophy in antiquity. In N. Kretzmann (Ed.), Infinity and continuity in ancient and medieval thought (pp. 112–45). Ithaca: Cornell University Press.
Kripke, S. (2011). Philosophical troubles. Oxford: Oxford University Press.
Kripke, S. (2022). Wittgenstein, Russell and our conception of the natural numbers. In Mathematical knowledge, objects and applications. Berlin: Springer.
Lasnik, H., Depiante, M. A., & Stepanov, A. (2000). Syntactic structures revisited: Contemporary lectures on classic transformational theory (vol. 33). Cambridge, MA: MIT Press.
Lear, J. (1980). Aristotelian infinity. Proceedings of the Aristotelian Society New Series, 80, 187–210.
Lear, J. (1982). Aristotle’s philosophy of mathematics. Philosophical Review, 41, 161–192.
Linnebo, Ø. (2013). The potential hierarchy of sets. Review of Symbolic Logic 6, 205–228.
Linnebo, Ø., & Shapiro, S. (2019). Actual and potential infinity. Noûs 53, 160–191.
Linnebo, Ø., & Shapiro, S. (2020). Potentiality and indeterminacy in mathematics. Journal of Applied Logics, 7, 1201–1221.
Linnebo, Ø., & Shapiro, S. (2022). Predicativism as a form of potentialism. The Review of Symbolic Logic, 1–32. https://doi.org/10.1017/S1755020321000423
Miller, F. D. (2014). Aristotle against the atomists. In N. Kretzmann (Ed.), Infinity and continuity in ancient and medieval thought (pp. 87–111). Ithaca NY: Cornell University Press.
Parsons, C. (1977). What is the iterative conception of set? In R. Butts & J. Hintikka (Eds.), Logic, foundations of mathematics and computability theory (pp. 335–367). D. Reidel: Dordrecht.
Putnam, H. (1967). Mathematics without foundations. Journal of Philosophy, 64, 5–22.
Quine, W. V. O. (1966a). The ways of paradox and other essays. Random House: New York.
Quine, W. V. O. (1966b). Intensions revisited. Midwest Studies in Philosophy, 2, 5–11.
Simpson, A. K. (1994). The proof theory and semantics of intuitionistic modal logic, Ph. D. Dissertation, University of Edinburgh.
Sorabji, R. (2006). Time, creation and the continuum: Theories in antiquity and the early middle ages. Chicago: University of Chicago Press.
Steiner, M. (1978). Mathematical explanation and scientific knowledge. Nous, 12, 17–28.
Steiner, M. (1980). Mathematical explanation. Philosophical Studies, 34, 135–152.
Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Cambridge, MA: Harvard University Press.
Steiner, M. (2011). Kripke on Logicism, Wittgenstein, and de re beliefs about numbers. In A. Berger (Ed.), Saul Kripke (pp. 160–176). Cambridge: Cambridge University Press.
Studd, J. P. (2019). Everything, more or less: A defence of generality relativism. Oxford: Oxford University Press.
Turing, A. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 2, 230–265.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Linnebo, Ø., Shapiro, S. (2023). Potential Infinity and De Re Knowledge of Mathematical Objects. In: Posy, C., Ben-Menahem, Y. (eds) Mathematical Knowledge, Objects and Applications. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-031-21655-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-031-21655-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-21654-1
Online ISBN: 978-3-031-21655-8
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)