Abstract
Ordinality and cardinality, in the finite domain, are ordinarily considered as mere aspects of the very same objects, the natural numbers. Yet Steiner (Mathematics – application and applicability. In: Shapiro S (ed) The Oxford handbook of philosophy of mathematics and logic. Oxford University Press, 2005) draws attention to the intricate interplay between them, which is made implicit by this conception of them. In this chapter, I present a fitting cognitive framework and use it to account for how this situation comes to be. The framework concerns basic principles of object representation in the cognitive system, and particularly when and how different “aspects” of objects come, developmentally, to be integrated into a deeper, combined representation. Our coming to master the natural numbers is then presented as the process of such integration of ordinals and cardinals, based on their well-behaved interaction in the finite domain. This regular connection in the finite domain between ordinality and cardinality, however, is grasped automatically by the cognitive system as a statistical pattern rather than being stated explicitly, mathematically. Philosophically, this case study thus serves as a warning against ignoring cognition’s inner workings. The upshot is a novel, cognitively motivated account of the natural numbers.
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Notes
- 1.
- 2.
For a methodological discussion, see (Keren, 2018), Chap. 4.
- 3.
For musings in similar spirit by a historically notable figure, see (Poincaré, 1905), Chap. 4.
- 4.
My full account can be found at (Keren, 2018), Chap. 6. The main omissions in this current paper are: (1) The cognitive-scientific material that gives this framework substance, (2) A substantial section on the developmental dynamics, (3) More elaborate connections specifically to mathematics.
- 5.
I do not suggest even the beginning of an approach to the age-old question of how intentionality could arise here. The point is just to point to the level at which such an account would have to begin.
- 6.
I leave open the further issue of when and why we come to identify different (cognitive) object-types as of the same type.
- 7.
Even if it technically were (i.e., without strawberries, etc. in the world), then even though technically a watermelon (say) could be peeled, it would not be relevant as a cognitive object-type, for which mastering the preparation of a watermelon typically goes through other means.
- 8.
- 9.
For a richer discussion see (Keren, 2018), p. 156, and section 6.5.
- 10.
Though in cognition, the boundaries may ordinarily be graded or ill-defined (Rosch et al., 1976).
- 11.
See (Keren, 2018 p. 170).
- 12.
Much more importantly for this story, the objectification could in principle land directly on something larger. As will become clear when we get to it soon, this may particularly be ℕ, the system never coming to individuate anything like the object-type TOrd or some reduct along the way. This possibility, however, is not theoretically reasonable nor empirically evident. For a more detailed discussion of these developmental complexities, see (Keren, 2018), Section 7.7.
- 13.
Though there is certainly alot going on here too:
-
1.
“all that the successor procedure ever could generate from 1” is a higher-order reflection on a procedure (within the context of the rest of the array), not just its usage by another procedure (which is standard). For a preliminary cognitive account of (some) infinitary ordinals, see (Keren, 2018 pp. 235–237).
-
2.
A different amalgalm, between natural numbers and the continuum, serves here to provide the model in which this post-finite element can be seen to exist.
-
1.
- 14.
“Visual ‘pop-out’ refers to the phenomenon in which a unique visual target (e.g. a feature singleton) can be rapidly detected among a set of homogeneous distractors”. (Hsieh et al., 2011).
- 15.
- 16.
This is not to rule out the possibility for amalgamation of natural numbers as an object-type with new object-types yet (nor the possibility that there are more abstract types integrated into them to begin with). To the extent that such a further amalgamation would keep fixed the ontology of individuated objects (rather than restrict it, e.g., to the even numbers, or extend it, e.g., to the rationals), we would take that object-type to still be about natural numbers (in the mathematical world), of which we have learned something new. But this is just a special case within the broader formal structure, and the finer distinctions that my cognitively grounded framework makes must not be ignored.
- 17.
Keren (2018), Section 6.5.
- 18.
Keren (2018 p. 168).
- 19.
See (Keren, 2018), Section 7.7.
- 20.
(Keren, 2018), Section 6.5.
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Keren, A. (2023). Ordinals vs. Cardinals in ℕ and Beyond. 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_10
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