Abstract
It’s often said that set theory provides a foundation for classical mathematics because every classical mathematical object can be modeled as a set and every classical mathematical theorem can be proved from the axioms of set theory. This is obviously a remarkable mathematical fact, but it isn’t obvious what makes it ‘foundational’. This paper begins with a taxonomy of the jobs set theory does that might reasonably be regarded as foundational. It then moves on to category-theoretic and univalent foundations, exploring to what extent they do these same jobs, and to what extent they might do other jobs also reasonably regarded as foundational.
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Notes
- 1.
Items like the category of all groups or the category of all categories are exceptions. There is no set of all groups or set of all categories for the same reason that there’s no set of all sets: sets are formed in a transfinite series of stages, and there’s no stage at which all of them (or all of them that are groups or all of them that are categories) are available to be collected. The category-theoretic cases are explored in the next section.
- 2.
- 3.
Readers of Frege (1884) will recognize this as ‘the direction of a line’ and recall how it serves as Frege’s model for identifying a natural number with a collection of equinumerous collections.
- 4.
This was before the development of formal languages and deductive systems, before a clear understanding of consistency, satisfiability, and of course, before Gödel’s completeness and incompleteness theorems.
- 5.
It’s worth noting that Dedekind’s set-theoretic construction of the reals was different in character from von Staudt’s construction of imaginary points. Von Staudt was faced with a practice in good working order, but with questionable posits. Dedekind was faced with a defective practice (basic theorems of the calculus couldn’t be proved). So von Staudt’s challenge was to remove queasiness about the posits by domesticating them, while Dedekind’s was to produce a more precise replacement that would both conform to previous practice and extend it (proving those basic theorems). Thus Dedekind’s construction had a different, plausibly ‘foundational’ function (called Elucidation in [2017]). As both category-theoretic and univalent foundations are content to relegate Elucidation to ETCS, a weak category-theoretic theory of collections (see [2017], §II, and UFP (2013), p. 8, respectively), it won’t figure in the comparative analysis here.
- 6.
These are the spurious foundational virtues called Metaphysical Insight and Epistemic Source in [2017].
- 7.
The underlying methodological maxim here is to prefer non-restrictive, maximizing theories. [1997] concludes with an early attempt to formalize this notion. Various developments of this idea and alternatives to it have been suggested, but the problem remains open.
- 8.
That is, projectively definable.
- 9.
This is, the existence of such a model can be proved from the fundamental axioms.
- 10.
Another well-known example is the theory ZF + V = L(R) + AD. Again, separating the ‘mathematically worthy’ from the unworthy no doubt requires keen mathematical discernment and well-informed good judgement.
- 11.
For successful application, it’s not enough that our theory prove the existence of a suitable structure; it must exist in a context with enough mathematical tools to study and manipulate that structure. See [2011], pp. 90–96, for a related discussion.
- 12.
Because new groups or categories or mathematical X’s keep being formed at higher and higher ranks in the iterative hierarchy of sets, there’s never a rank at which they all be collected together. Recall footnote 2.
- 13.
Recall footnote 7 and surrounding text.
- 14.
Ernst shows, in particular, that a contradiction arises from the assumption that such a theory can form the category of all graphs.
- 15.
See, e.g., the work of Mathias discussed in [2017].
- 16.
Of course set theory also translates notions from outside when locating their surrogates, but set theory isn’t claiming to provide Essential Guidance.
- 17.
See Awodey (2014, p. 1).
- 18.
Awodey traces the roots of univalent foundations in traditional foundational work to Frege rather than Zermelo: ‘this new kind of … formalization could become a practical tool for the working mathematician -- just as originally envisaged by Frege, who compared the invention of his Begriffsschrift with that of the microscope, (Awodey 2016a, p. 8, see also Awodey and Coquand (2013, p. 6). While Frege does make this comparison, it involves a contrast between the microscope and the eye: ‘because of the range of its possible uses and the versatility with which it can adapt to the most diverse circumstances, the eye is far superior to the microscope’ (Frege 1879, p. 6). Frege’s formal system ‘is a device invented for certain scientific purposes, and one must not condemn it because it is not suited to others’ (ibid.). The ‘scientific purpose’ in question is to determine whether arithmetic can be derived by pure logic; the Begriffsschrift was needed ‘to prevent anything intuitive from penetrating here unnoticed … to keep the chain of inferences free of gaps’ (ibid., p. 5). It seems to me likely that Awodey’s ‘practical tool for the working mathematician’ would be analogous to the eye, not the microscope, that serving as such a practical tool is one of those purposes for which the microscope and Frege’s formal system are ‘not suited’.
- 19.
Cf. UFP 2013, p. 2: ‘univalent foundations is very much a work in progress’.
- 20.
I don’t know what Voevodsky finds lacking in category-theoretic foundations -- perhaps that it fails to provide a Generous Arena?
- 21.
Interestingly, Voevodsky (2014a) observes that ‘our intuition about types of higher levels comes mostly from their connection with multidimensional shapes, which was studied by ZFC-based mathematics for several decades’ (p. 9).
- 22.
Similarly, Awodey bemoans the ‘serious mismatch between the everyday practice of mathematics and the official foundations of mathematics in ZFC’ (Awodey 2016a, p. 2) and connects univalent foundations with structuralist tendencies in the philosophy of mathematics that frown on the extraneous features of set-theoretic surrogates.
- 23.
Cf. Awodey and Coquand 2013, p. 6: ‘the fundamental notion of a set … in univalent foundations turns out to be definable in more primitive terms’.
- 24.
Colin Mclarty was kind enough to explain to me that ‘types of h-level 2’ is just a different terminology for the ‘0-types in one or another of these universes’ in the previous paragraph.
- 25.
Cf. UFP (2013, p. 9).
- 26.
I’m not sure what these thinkers take to be wrong with ZFC, but it could be something akin to the category-theorist’s conviction that a neutral notion of ‘collection’ is better understood in top-down function-based terms (as in ETCS) rather than bottom-up element-based terms (as in ZFC).
- 27.
See also UFP (2013, p. 15).
- 28.
Many thanks to Colin McLarty, Lawrence Paulson, and an anonymous referee for very helpful explanations, discussions, and comments.
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Maddy, P. (2019). What Do We Want a Foundation to Do?. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_13
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