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Intuiting the infinite

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Abstract

This paper offers a defense of Charles Parsons’ appeal to mathematical intuition as a fundamental factor in solving Benacerraf’s problem for a non-eliminative structuralist version of Platonism. The literature is replete with challenges to his well-known argument that mathematical intuition justifies our knowledge of the infinitude of the natural numbers, in particular his demonstration that any member of a Hilbertian stroke string ω-sequence has a successor. On Parsons’ Kantian approach, this amounts to demonstrating that for an “arbitrary” or “vaguely represented” string of strokes, we can always “add” one more stroke. Critics have contested the cogency of a notion of an arbitrary object, our capacity to vaguely represent a definite object, and the role of spatial and temporal representation in the demonstration that we can “add” one more. The bulk of this paper is devoted to demonstrating how to meet all extant criticisms of his key argument. Critics have also suggested that Parsons’ whole approach is misbegotten because the appeal to mathematical intuition inevitably falls short of providing a complete solution to Benacerraf’s problem. Since the natural numbers are essentially and exclusively characterized by their structural properties, they cannot be identified with any particular model of arithmetic, and thus a notion of intuition will fail to capture the universality of arithmetic, its applicability to all entities. This paper also explains why we should not reject appeal to mathematical intuition even though it is not itself sufficient to fully “close the gap” on Benacerraf’s challenge.

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Notes

  1. The Evelyn/eleven example is due to Burgess (1999).

  2. As philosophers of language and linguists, we have a stake in upholding the thesis, for overturning it comes at a cost. Without a naïve semantics of mathematical discourse, we cannot preserve systematicity and uniformity in our semantic analyses, and our semantic theorizing cannot proceed in relative isolation from epistemic issues, for epistemic constraints would dictate or at least seriously impinge upon our semantic analyses. This is hardly a decisive consideration, however. Many a philosophical-great regarded epistemological considerations as foundational and sufficient reason for abandoning a naïve semantics.

  3. As Benacerraf developed the problem in his (1973), it focused on our ability to know mathematical truths, as opposed to merely think of the truth-makers of our mathematical discourse. Benacerraf also maintained that our knowledge had to be accounted for on a causal theory of knowledge. Most contemporary developments of the problem have shed these assumptions, neither of which is essential. Cf., Hale and Wright (2002).

  4. “Eliminative” and “non-eliminative” are used by Parsons (1990, 2004, 2008); “eliminative”, “in rebus” and “ante rem” by Shapiro (1997, 2011); “hard-headed” and “mystical” by Dummett (1991); and “pure” and “abstract” by Hale (1996). Though numerous formulations have been offered, some substantive to the main debate, nothing hinges on them here.

  5. Shapiro (1997, 2011) and Parsons (2008) have been explicit that their non-eliminativist structuralist theories are Platonist. Cf., also Hale (1996).

  6. Gödel (1964).

  7. Cf., Parsons (1990, 2008).

  8. Parsons (2008, p. 34).

  9. Parsons (2008, p. 36).

  10. Though Parsons does not appeal to any empirical evidence about perception to back the point, research programs on visual perception and object tracking underwrite the point that we are capable of perceiving objects, and tracking them, without distinguishing them with a conceptual representational content. Cf., Pylyshyn (2002, 2003).

  11. Is seeing as sufficient for constituting an intuition of? Does seeing a cat as a cat constitute intuition of? Parson would deny this. Intuitions of must be of objects that do not occur in space and time and so you do not intuit the cat, even if you see the cat as a cat. But it is interesting (and natural) that seeing as can occur in perception of concrete objects as well as (it must) in intuition.

  12. Cf., Maddy (1990), who claims that we directly perceive sets in perceiving configurations of objects. I have just transferred the point to one about numbers.

  13. Frege (1884, §46). Frege’s point applies to both numbers and sets.

  14. A full justification of Axiom of Infinityss requires the justification of two additional claims in addition to PA3ss.

    1. (a)

      we can indefinitely iterate the operation of adding one new string

    2. (b)

      the new string of strokes obtained from adding a new stroke is in fact new, i.e., is of a different type from all previous strokes.

    Since PA3ss is essential, showing that it is grounded on intuition will suffice to show that so too is Axiom of Infinityss.

  15. I simply assume here the widely accepted non-finitist view. For the opposing finitist perspective, see Tait (1981).

  16. Parsons (1971, p. 46).

  17. Parsons (1980, p. 156).

  18. Page (1993).

  19. Parsons (1993).

  20. Page (1993, p. 229).

  21. Page (1993, p. 243).

  22. Parsons (1993) describes the thought experiment in this way, exhibiting the ambiguity discussed above.

  23. Hale and Wright (2002, p. 107).

  24. Page (1993, p. 230).

  25. Parsons (1980, p. 158).

  26. For fuller discussion, see Parsons (1993), who makes both points.

  27. Hale and Wright (2002, p. 107).

  28. Parsons (2008, p. 174, footnote 68).

  29. This animation was created by John Blackburne.

  30. Shapiro (2011, p. 130).

  31. Thanks to Bob Hale for these concerns regarding intuitions of types of quasi-concrete types.

  32. Parsons (2008, p. 233).

  33. This brief explanation of why I do not regard the limitation in the range of intuition as devastating draws on some points in Parsons (2008, especially Chaps. 6 and 9).

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Acknowledgments

I presented an ancestor of this paper at the 2013 NYU La Pietra workshop on the a priori. Portions of the dialectic were discussed in a colloquium at the University of Texas, Austin in 2002 and in my philosophy of mathematics seminar at Yale University in 2004. Warm thanks to Crispin Wright, Christopher Peacocke, Tim Williamson, and Jane Friedman. I am especially grateful to Bob Hale for excellent criticisms of an ancestor of this paper.

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  1. School of Philosophy, University of Southern California, 3709 Trousdale Parkway, Los Angeles, CA, 90089, USA

    Robin Jeshion

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Jeshion, R. Intuiting the infinite. Philos Stud 171, 327–349 (2014). https://doi.org/10.1007/s11098-013-0274-8

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