Abstract
The Univalent Foundations of Mathematics (UF) provide not only an entirely non-Cantorian conception of the basic objects of mathematics (“homotopy types” instead of “sets”) but also a novel account of how foundations ought to relate to mathematical practice. In this paper, I intend to answer the question: In what way is UF a new foundation of mathematics? I will begin by connecting UF to a pragmatist reading of the structuralist thesis in the philosophy of mathematics, which I will use to define a criterion that a formal system must satisfy if it is to be regarded as a “structuralist foundation.” I will then explain why both set-theoretic foundations like ZFC and category-theoretic foundations like ETCS satisfy this criterion only to a very limited extent. Then I will argue that UF is better-able to live up to the proposed criterion for a structuralist foundation than any currently available foundational proposal. First, by showing that most criteria of identity in the practice of mathematics can be formalized in terms of the preferred criterion of identity between the basic objects of UF (“homotopy equivalence”). Second, by countering several objections that have been raised against UF’s capacity to serve as a foundation for the whole of mathematics.
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Notes
Although I will not be dealing with Makkai’s system directly in this paper, let me mention that in its most recent incarnation in Makkai (2013) it goes under the name of Type-Theoretic Categorical Foundations of Mathematics (TTCFM). See Marquis (2013) for an exposition as well as philosophical defence of Makkai’s views on the foundations of mathematics.
It is important to note, however, that Voevodsky’s and Makkai’s motivations for their respective projects are very different. Makkai’s motivation was at least partly philosophical, aiming to develop a language that encodes a structuralist view of mathematical objects. On the other hand, Voevodsky’s work on foundations had primarily a non-philosophical motivation behind it: to develop usable “proof assistants” that mathematicians can use to verify their theorems.
A more correct way of paraphrasing the axiom of univalence is this: identity is isomorphic to isomorphism. A fully correct way of paraphrasing the axiom of univalence is this: the (canonical) map that sends identities to isomorphisms is an isomorphism.
The clearest exposition of Kant’s views on the method of mathematics is to be found in the beginning of the Doctrine of Method and especially in [A713/B742]. There Kant explains that mathematics proceeds by constructing intuitions (e.g. specific triangles) adequate to a priori concepts, not by analyzing such concepts (which is the task of philosophy). (For a helpful overview see Carson (1999).) Russell’s very influential criticism of Kant was that a strong enough logic (e.g. first-order predicate logic) could compensate for this element of “construction” without invoking some kind of pure intuition provided by the human subject. This point of view came to be known as the “compensation thesis”. For more discussion on Kant’s views on geometry see the Parsons–Hintikka debate Parsons (1992), Hintikka (1967) as well as Friedman’s very influential recasting of the compensation thesis in Friedman (1985, (2012). It should also be made clear that the issue of whether or not Kant thought that demonstrations themselves also involved pure intuition (rather than just the construction of the concepts that they were to be applied to) remains a topic of controversy. For the latest installment, see Hogan (2015).
Strictly speaking, one must draw a distinction between topology (the study of shapes) and geometry (the study of distances)—in this very rough historical sketch I am including both under the banner of “geometry”.
Marquis (2008) has made a compelling study of the connections between category theory, Kleinian geometry and Grothendieck’s vision for algebraic geometry.
I am assuming here that the deductive system that we impose on the basic formal language comes in at stage (2)—but this is admittedly a difficult assumption to maintain in the case of MLTT where the rules of well-formedness of expressions coincide with the deductive rules. Still, even in dependent type theories used to formalize UF, there is a useful distinction to be made between “basic” type-formers (e.g. \(\varPi , \varSigma \) or W-types) and type-formers or axioms added specifically with the homotopy interpretation in mind.
That said, I will often use the terms UF and HoTT interchangeably, i.e. I will often refer by the term UF to the particular formalization of UF as HoTT. This is in keeping with standard set-theoretic practice of referring to ZFC as “set theory”.
A full technical introduction to UF is neither possible nor necessary for the purposes of this paper. What follows is something slightly more structured than a glossary, meant to give some idea of what is meant by the term “Univalent Foundations” and to introduce some terminology peculiar to the formal apparatus it employs. For more detailed technical introductions see HoTT Book (2013), Martin-Löf (1984), Voevodsky (2014), Awodey (2014).
Martin-Löf’s original name for his theory in Martin-Löf (1984) was Intuitionistic Type Theory—we shall not be concerned here with terminological issues concerning the difference between “constructivists” and “intuitionists”. Suffice it to say that although MLTT has most often been associated with constructive/intuitionistic approaches to mathematics, UF is in no way limited to constructive/intuitionistic logic. Indeed, HoTT is perfectly consistent with an appropriately stated law of the excluded middle (cf. HoTT Book 2013, Definition 3.4).
One can think intuitively of types as sets and terms as elements of sets although in homotopy type theory this way of thinking is misleading since one can prove that there are types that are not h-sets (cf. HoTT Book 2013, Example 3.1.9).
It is important to note that this particular extension of MLTT is not the only homotopy type theory that formalizes UF. A recent alternative proposal is Cubical Type Theory Bezem et al. (2014), Cohen et al. (2015) designed with the specific intention of giving a constructive model of univalence. Nevertheless, for the purposes of this paper, nothing hinges on the choice of formalization and so we will identify the term HoTT with the formal system used in HoTT Book (2013).
For a more detailed exposition of a similar kind of view of homotopy types, as well as an argument for their fundamentality, see Marquis (2013).
It is an interesting question whether a choice of context determines the criterion of identity or vice versa, but I will not get into this here. Furthermore, it is clearly an oversimplification to say that mathematical disciplines are rigidly demarcated by a specific type of structure they study up to a specific criterion of identity. For instance, how would one demarcate number theory in those terms? Or algebraic geometry? Nevertheless, nothing of substance hinges on the accuracy of my sociological analysis—its purpose is merely to clarify the concepts of “theoretical context” and “criterion of identity” as they will be used below.
Everything I say applies to any standard set-theoretic foundation. I pick ZFC for this example merely for its “brand recognition”.
I am being a bit quick here, for the sake of exposition. Strictly speaking, the actual operation and identity elements of groups x, y would have to be used in defining what it is for a function f to be a homomorphism.
By “1” here I mean the set corresponding to the successor of zero in some choice of a model of arithemtic inside ZFC, e.g. the singleton set \(\lbrace \varnothing \rbrace \). By \(\mathbb {Z}\) and \(2\mathbb {Z}\) I mean the formalizations of the additive groups of integers and even integers respectively.
Of course, one could argue that there are non-standard ways of formalizing groups and group isomorphisms in ZFC such that the required invariance holds. However, it is highly doubtful that such methods would be natural (i.e. uniform and native) in the sense demanded by (SFOM).
I am glossing over the details of the exact presentation of ETCS as a many-sorted first-order theory. For a clear and concise presentation see Palmgren (2012).
As we shall see in Sect. 5, the situation is slightly more subtle.
I am indebted to an anonymous referee for raising this point.
This attitude towards mathematical objects is inspired by Burgess’ permament parameter structuralism as elaborated in Burgess (2014). What I am saying, roughly, is that mathematical objects should be understood as parameters, about whose nature we do not care as long as they behave the way we expect them to. And this allows us to speak of “alternative presentations” of such objects without committing ourselves to a criterion of identity between them.
Of course, in the context of number theory a proposition like “\(\mathbb {Z} \ne 2\mathbb {Z}\)” is highly relevant indeed since we care very much to distinguish odd from even integers, e.g. when we state Goldbach’s conjecture. But in the context of number theory group isomorphism is not, in general, the relevant criterion of identity.
More accurately, but less sonorously: there is no criterion of identity without names for entities.
As has already been noted, the precise statement of univalence asserts that identity is isomorphic to isomorphism by asserting that a canonical map from identities to isomorphisms is an isomorphism. In other words, univalence asserts that a particular (canonical) map is an isomorphism, and not merely the existence of an isomorphism.
In Awodey (2014), Awodey refers to (PI) as the Principle of Invariance but this is merely a terminological difference.
In keeping with the notation above, I will continue to use \(A \cong B\) to denote the type of homotopy equivalences, except when ambiguity might arise, as in Sect. 5, where I will use the more standard “\(\simeq \)”.
This can even be formalized in point-set topology for sufficiently nice spaces, by regarding their points as singleton subspaces and showing that equal such points are (trivially) homotopy-equivalent as singleton subspaces.
What this “class” is and how exactly it is “specified” we leave open. At the very least, there should be some consensus among practicing mathematicians that studying objects of type \(O_{\text {inf}}\) under \(\sim _{\text {inf}}\) constitutes a legitimate specialization. For instance, studying groups under group isomorphisms (“group theory”) or studying topological spaces up to homotopy equivalence (“homotopy theory”).
Just like what was said about identity types in the beginning of Sect. 4, a criterion of identity is also to be understood as a structure rather than a proposition. Of course, if \(\sim _{\text {UF}}\) is to be meaningfully regarded as a criterion of identity then we will usually assume that it is at least reflexive, symmetric and transitive (in the type-theoretic sense). But we also allow for criteria of identity that satisfy much stronger properties, e.g. identity systems in the sense of HoTT Book (2013), Definition 5.8.3.
Where “homotopy equivalences” are understood in the expanded sense outlined in Sect. 4.
S is of course itself a type, possibly in a higher universe of types.
Occasionally (e.g. in the case of (pre)categories, see below) we might be interested in more explicit descriptions of the right hand side. For example, in the case of graphs, using function extensionality and the fact that the \(E_i\) are set-valued functions, we can obtain
$$\begin{aligned} (\langle V_1, E_1 \rangle \cong _{O_{\text {graph}}} \langle V_2, E_2 \rangle ) \simeq \underset{p :V_1 \simeq V_2}{\varSigma } \,(\underset{x,y :V_1}{\varPi } E_1 (x,y) \simeq E_2 (p_* (x), p_*(y))) \end{aligned}$$which more closely resembles what we have come to expect graph-isomorphism to mean in practice, namely an isomorphism of the vertex-set that induces isomorphisms on the corresponding edge-sets. Nevertheless, even if the process of “rewriting conveniently” is not canonical, the process of obtaining \(\cong _O\) is, in fact, canonical.
Although a similar kind of condition was considered already by Hofmann and Streicher in their groupoid model for MLTT, cf. Hofmann and Streicher (1998).
But there is also a strong relation between them: roughly, every precategory gives rise to a univalent category (called its Rezk completion in HoTT Book (2013)) that is “weakly equivalent” to it. For the categorically-minded reader: the obvious forgetful “functor” from univalent categories to precategories has a “left adjoint”. Perhaps someone may wish to claim that this relation between the two notions shows that there is a kind of equivalence between them. This would challenge the points I go on to make below. But this is the wrong conclusion to draw. Similar relations (“forgetful-free adjunctions”) are borne by classes of structures that on no reasonable account of “equivalence” should we wish to call equivalent. For example, groups and (bare) sets bear this relation.
Accepting the distinction between categories up to isomorphism and categories up to equivalence depends on some level on accepting that groupoids are more fundamental than categories. I argue that this is indeed the case on purely philosophical grounds in Tsementzis (2016b). Although it is by no means a trivial or obvious thesis, I will say no more about it here.
An anonymous referee has correctly pointed out that the notion of a strict category, which is a precategory where the type of objects is an h-set (cf. HoTT Book 2013, Definition 9.6.1.), serves perhaps as a better formalization of the informal notion of category-up-to-isomorphism.
For a precise and thorough explanation of the way in which HoTT/UF is a synthetic theory of \(\infty \)-groupoids see Shulman (2016).
It is important to note that among such constructions are also those of homotopy types that can be understood as models of set theory as traditionally conceived. For example, the type \(\mathbf Set _{\mathscr {U}}\) of h-sets without any further assumptions on HoTT is a model of a weak predicative set theory (a “\(\varPi W\)-pretopos”) and if we assume LEM then a model of ZFC can also be constructed as a higher inductive type (cf. HoTT Book 2013, 10.5).
This is because it does not even make sense to consider synthetic \(S^1\) and \(S^2\) “up to homeomorphism”. The very statement “\(S^1\) is not homeomorphic to \(S^2\)” cannot be stated in UF-regarded-as-a-synthetic-theory-of-\(\infty \)-groupoids. It must be stated with \(S^1\) and \(S^2\) defined topologically, i.e. as h-sets with appropriate extra properties and structure mirroring the way in which these spaces are usually defined in point-set topology.
Marquis makes a very similar point in arguing for the fundamentality of homotopy types: “I submit that the notion of [homotopy equivalence] at work here is philosophically fundamental: we are dealing with entities that can be continuously transformed into one another.” (Marquis 2013, p. 2151)
For a precise description to the problem, as well as an illuminating introduction to possible methods that could solve it, cf. Shulman (2014).
The case of traditional topological spaces as a challenge to my general method was raised by an anonymous referee, to whom I clearly owe this entire discussion.
Here \(\lnot S(b)\) can be thought of as notation for \(S(b) \rightarrow 0\).
More precisely, for any other object c an isomorphism \(f :a \cong b\) induces a bijection \(f^* :\text {Hom}(c,a) \cong \text {Hom}(c,b)\) by transporting morphisms along f and similarly for \(\text {Hom}(-,c)\).
What the above-described situation certainly does prevent us from doing is a Rezk-completion-style construction. The reason that Rezk completion works in the case of categories is exactly because we can use a given isomorphism to induce a bijection (i.e. h-equivalence) between hom-sets. This means that isomorphisms “already” respect hom-sets. But we have no reason to expect this to be the case in general. So the Rezk completion construction certainly does not generalize to arbitrary structures, as of course one ought to expect.
One is inevitably reminded of Grothendieck’s beautiful ruminations in his Esquisse d’un Programme Grothendieck (1997):
[W]hen one tries to do topological geometry in the technical context of topological spaces, one is confronted at each step with spurious difficulties related to wild phenomena. [...] This situation, like so often already in the history of our science, simply reveals the almost insurmountable inertia of the mind, burdened by a heavy weight of conditioning, which makes it difficult to take a real look at a foundational question, thus at the context in which we live, breathe, work—accepting it, rather, as immutable data. [...] It is this again which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom “wildness” is a fatal necessity, rooted in the nature of things (pp. 258–259).
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Acknowledgments
I would like to thank (in random order) John Burgess, Paul Benacerraf, Hans Halvorson, Steve Awodey, Mike Shulman, Chris Kapulkin, Vladimir Voevodsky, Colin McLarty, David Corfield, Richard Williamson, Urs Schreiber, Harry Crane, as well as audiences in Philadelphia, London and Princeton. Lastly, I would like to single out in thanks an anonymous referee, who provided extremely detailed and illuminating comments that resulted in many significant improvements.
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Tsementzis, D. Univalent foundations as structuralist foundations. Synthese 194, 3583–3617 (2017). https://doi.org/10.1007/s11229-016-1109-x
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DOI: https://doi.org/10.1007/s11229-016-1109-x