Luca Incurvati's "How to be a Minimalist about Sets"

 

Introduction

This was an interesting paper by Luca Incurvati and I enjoyed reading it quite a bit. We read this paper as part of our weekly philosophy of maths reading group at the University of Vienna. My thanks, in alphabetical order, to Julie Lauvsland, Benjamin Marschall and Simon Weisgerber. In particular, thanks to Julie for suggesting the paper. This is, though, the first paper I've reviewed where I've generally disagreed with the author. It's not that I necessarily disagree with all the conclusions, but I think there responses that Incurvati doesn't consider and alternative arguments that are important. I'm also unconvinced by Incurvati's positive proposal.

Now one quasi-objection I have is that the paper takes place from a Platonist/realist perspective. I think that's wrong from the start and is bound to lead to confusion when we worry about which axioms or claims about sets are really correct and why. I think we should choose our concept of "set" based on practical and pragmatic considerations arising from what we (and the mathematical and scientific community) want mathematics to do. But it would be unhelpful to levy such wide-scoped objections at this paper specifically and so this is all I'll mention of them. For the purposes of this paper, I'll assume that there is such a thing as abstract mathematical reality and it's the job of mathematics to try and describe or explain the structure of this reality

In this paper, Incurvati attempts to explain a widely held view about sets: the iterative conception. The iterative conception of sets is the view that sets are arranged into a hierarchy under the membership relation. This is sometimes called the cumulative hierarchy. The transitive closure of membership is, on this view, an anti-symmetric relation: no set is a member of itself and the TC of membership never forms loop-like shapes. Sets are "built" in stages. At each stage, every possible combination of elements at that stage becomes a set. Anyone who has taken even an introductory set theory class will be very familiar with this idea.

The gauntlet thrown down in this paper is to explain why (or to give reason to think that) the iterative conception is true.

Incurvati considers two possible explanations that he calls the constructivist and platonist responses. 

The constructivist response takes the idea of sets being built at each stage very seriously. In line with classical constructivism, there is literally some mental (or similar) process by which sets can be iteratively built. The structure of the hierarchy comes from the priority in the construction order. The problem with this view, so argues Incurvati drawing on an argument by Michael Potter, is that the constructivist view does not get enough mathematics. I think the situation might be less bleak than Incurvati thinks, though this hinges on a point not considered in the paper.

The platonist view builds on earlier work by Linnebo and, again, the book by Potter. The claim is that there's some kind of metaphysical dependence relation between sets and that this explains their hierarchical nature. Incurvati argues that this relation is (a) unclear and (b) not obviously explanatory of the phenomena at hand. I agree with point (b), though have a little to add. (a) I potentially agree with, but I'm unconvinced it's that devastating problem for the Platonist. It's also worth noting that Linnebo's view has plausibly changed since his 2008 paper. In his recent book, Thin Objects (2018) Linnebo sets out a very interesting view of sets that plausibly avoids Incurvati's objections. Obvious Incurvati, writing in 2012, can be forgiven for not anticipating a view 6 years into his future! Nevertheless, Linnebo's more recent work might help expose certain issues in Incurvati's argument.

Incurvati's positive view is then his minimalist conception of a set. On this view, to be a set is just to be something that stands in a set-like position within a cumulative hierarchy. An analogy is made to what it takes to be a natural number. To be a natural number is just to sit in a certain position in a simple infinite series under the successor relation. Generally, the idea is to define mathematical kinds in virtue of some canonical structure of that kind. To be an X is to sit in an X-like position in a canonical X-structure.

My view on this is that Incurvati's definition fails to meet the burden he outlines in the paper. By packing the facts about the cumulative hierarchy into his concept of a set it becomes trivial that sets form a cumulative hierarchy. This sounds like a desirable outcome, but it simply pushes the explanatory burden back a step. The relevant explanatory burden is then to explain why there are sets, as opposed to some similar set-like but not cumulative. Incurvati fails to meet this burden, in my opinion.

Incurvati actually accepts this in his paper, opting for what I call an external explanation, but this then undermines the argument of the paper.

The Constructivist View

The constructivist response takes the idea of sets being built at each stage very seriously. In line with classical constructivism, there is literally some mental (or similar) process by which sets can be iteratively built. This process might, for instance, involve a sort of mental gathering together of known elements into a set. The structure of the cumulative hierarchy is inherited from the asymmetric structure of this construction. In order to construct more complex sets, an agent is required to already have constructed its members. This means that sets fall into a natural hierarchy as agents have to first build less complex sets and then more complex ones.

There are two issues with this view: one raised by Incurvati and one of my own that I take to be a well-known "folk" objection.

Incurvati's objection is that the constructivist is committed to something weaker than ZFC. Following an argument by Potter, Incurvati argues that at any stage in the process constructivists can only construct sets specifiable by reference. This is problematic in infinite domains when there are more subsets one might want to construct than there are formulae to define them. Incurvati understands this as a weakening of separation. That seems plausible, but I personally understand this as a weakening of power, in line with Gödel's "def" function that he uses to build the constructible universe.

A brief technical diversion: the def function is the definable powerset function. It takes each set to the set of all its definable subsets. A subset S of X is definable iff there is some phi(x) such that S={x in X | phi(x)}. I suspect that Incurvati's and Potter's restriction on separation and my restriction on power will be very similar, perhaps equivalent for transitive models. I haven't properly worked out a formal comparison of the two views, though, so will refrain from any strong claims.

This, according to Incurvati and Potter, means that constructivists don't generate enough set theory. Their theory is too week in the infinite domain.

Another issue is to do with limits. The constructive hierarchy requires not just successive finite iterations of power but the taking of unions at limits (see the difference between Z and ZF). It's unclear what the mental process that does this is and, consequently, why constructivists are justified in adopting the crucial axiom of replacement that is used to break limits (similarly so with the axiom of infinity that breaks the first limit).

I have no reply on the part of the constructivists to my second objection but, if they can meet it, their situation is not as bad as Incurvati thinks. If I'm right and the constructivist's restriction can be thought of just as restricting to def rather than power and they have a solution to the limit problem, then the constructivist can happily settle themselves into Gödel's constructive universe L. This, surprisingly, actually means that they get the axiom of power back. It can be shown using ZF-Power that power holds in L. Even though some subsets are missed out at each stage of iteration, they are always recovered later in the process (or, at least, L thinks that this is true. From an objective standpoint it's indeterminate). The constructible universe has some attractive features. It gets choice for free and also gives an answer to the continuum hypothesis.

L is not seen by set theorists as the universe of sets these days. This is for a series of reasons that it would be a diversion to go into. However, I think there's at least enough of a defence of this revisionary stance, from a practical point of view, that this is probably a holdable position for the constructivist, even if it's not ideal.

Of course, all of that is only possible if the constructivist can solve the limit problem and it's not obvious that they can.

This means that I think that the argument Incurvati raises can be met, though there are other issues with constructivism not mentioned in his paper.

The Platonist View

The second approach Incurvati considers is an approach by individuals such as Linnebo and Potter, building on work on metaphysical dependence by Fine and Lowe. This approach claims that there's some kind of metaphysical dependency between sets and their members that can explain the iterative conception of sets. The broad idea is that if sets depend on the existence of their members and this dependence relation is hierarchical, then the hierarchical structure of the set-theoretic universe will be inherited from the structure of this dependence relation. A notable weakness is that this dependence relation typically remains unanalysed and basic on these views.

Several variants of this approach are discussed in the paper. There's no need here to unpack the nuances of the different views, as both Incurvati's reply and my comments work at a general level.

Now I wasn't entirely sure exactly what argument Incurvati had in mind at a few points, so I've had to do a bit of interpretive work. Hopefully, I've got the essence of his position correct.

Incurvati's response, as I understand it, is twofold. First, he builds on the fact that these views contain an unanalysed dependence relation to argue that they aren't properly explanatory. Second, he argues that metaphysical dependence doesn't necessarily have the sorts of structural features that one needs it to have (specifically regarding antisymmetry).

I understand Incurvati's first argument as effectively accusing the platonist view of "passing the buck" with respect to explanation. We have a phenomenon that needs explaining, namely the structure of the cumulative hierarchy, and appealing to a concept that, itself, is unanalysed doesn't improve our overall explanatory position, it just pushes the explanation back a stage.

Incurvati is right, but only if this case is considered in isolation. At some point, we're going to need some conceptual primitives and these, by definition, will have to go unanalysed. I take it to generally be the case that one ought to try to keep these primitives to a minimum and that it's generally bad to have to introduce them in ad hoc ways or only for singular purposes, but this isn't the case with something like metaphysical dependence. If any relations are going to be basic, things like metaphysical dependence, causation, etc seem like good candidates. (To be clear, I think MD is analysable, it's just that I don't see too much of a problem per se with someone holding a view where it's not.)

Another way of reading this response is that Incurvati's objection is not with the unanalysed concept per se, but rather the belief that the concept has the structural properties that the platonist needs in order to use it to explain the cumulative hierarchy. A problem with any conceptually basic relation is justifying why it has the features one takes it to have. We're, plausibly, in no better position having to explain why metaphysical dependence has certain properties than we are trying to explain why the universe of sets does.

As with above, I don't necessarily think that's the case. Presumably explaining why dependence has a particular structure is a very general problem for this kind of metaphysical theory and one that they'll have to meet irrespective of what goes on in the philosophy of set theory. If all the platonist does is points out that the cumulative hierarchy is a particular case of a general phenomenon to be explained, then this is progress. The platonist has gone from having two things to explain (the structure of the cumulative hierarchy and the structure of dependence) to only having one.

But Incurvati's worry goes deeper and he expresses a general scepticism about the claim that dependence has the structure that platonists need it to. In particular, he raises worries about reflexivity and asymmetry. There's also a similar worry about transitivity that Incurvati omits (or only mentions in passing) but that I'll raise more thoroughly, as it's relevant to the reflexivity worry.

Membership, in ZFC, is an asymmetric, non-transitive, irreflexive relation (irreflexive follows from asymmetric). Dependence is reflexive, plausibly transitive and, according to Incurvati, not-antisymmetric relation. How could one give rise to the other?

Incurvati's claim that dependence is not-antisymmetric is an odd one. In the paper, he states that it's a relatively standard claim that two (presumably distinct) things can metaphysically depend on one another. Unhelpfully, no citation or footnote is accompanying this... Incurvati and I must just be reading different metaphysics because I can honestly say that my impression was the exact opposite! Now, the claim that for all phi there is some philosopher that defends phi is, sadly, a good working motto when reviewing philosophical literature. But if it is the case that symmetric dependence is a view so standard as to not even need a citation, then my summer reading list just grew quite considerably.

Thus, I think the best response here for the platonist, at least for the time being, is just to accept antisymmetric dependence as the "standard" understanding of dependence but to put the ball back into Incurvati's court. He clearly has some ordinary cases of symmetric dependence in mind. It would be interesting to hear what those are and to take things from there. If dependence is typically not-antisymmetric, then I think Incurvati's correct that this poses a problem for the platonist here.

Regarding reflexivity, this can be quite easily solved. It seems perfectly reasonable to generally understand membership via dependence but to exclude trivial self-dependence from this.

Transitivity, though, is another matter. On at least some of the views (Lowe's, plausibly Linnebo's view in 08) the aim seems to be to explain the individuation of sets via differences in their dependence. But dependence is a transitive relation and there can be non-identical sets with the same transitive closure. If the idea is to explain the identity of sets via dependence, then use the structure of the dependence relation to motivate the iterative conception, then this looks set to run into trouble when considering non-identical sets with the same transitive closure. There might be a platonist view that approaches this explanation via a different path that doesn't tie identity to dependence. This is just a pitfall that the platonist must be aware of.

As a brief summary of my thoughts on Incurvati's response to the platonist view, I think the platonist can generally meet these worries. The only outstanding concerns they should have are (a) if Incurvati can provide a good case for ordinary symmetric dependence and (b) navigating my worry about transitivity.

Two final comments on this section.

First, this whole reply might be moot. Linnebo's view has changed since his 2008 paper and he outlines a very thoroughly argued position in his 2018 book Thin Objects. In that book, Linnebo provides a relatively detailed analysis of a notion close to dependence: metaphysical thinness. His analysis of thinness is, in my opinion, quite convincing. Linnebo also gives a neo-logicist account of sets that does seem to explain the structure of the cumulative, combining elements of both the constructivist and platonist approaches. It derives from a priority relation between stages in what he calls dynamic abstraction (repeated, predictive abstraction). I have some objections to Linnebo's position, but it certainly seems to bypass Incurvati's worries. It's not fair to criticise Incurvati for missing that position, as it wasn't out yet, but that doesn't mean that Linnebo's more recent work doesn't create a serious problem for this objection when considering it presently.

Lastly, I think there's a general lesson to be learned about how much downward dependence can actually motivate in terms of axioms of set theory. Dependence is rarely going to motivate the existence of something. If X depends on Y and Y exists, this doesn't mean that X exists. Dependence gives us necessary but not sufficient conditions on existence. The dependence claims will, at best, give us reason to think that the universe of sets is transitive, but it will never give us any "upwards" implications. This is useful but is not sufficient to motivate a strong set theory (or, even, the existence of any sets at all). This isn't unrecognised. Both Incurvati and Potter make the claim that the platonist doesn't merely want to say that a set can only exist if its members exist but that it will always exist if its members exist. It's for this reason that I think platonists should think more about sufficiency rather than dependence. They need to find a way of thinking about sets such that they get the appropriate sufficiency claims, not simply dependence. One doesn't need to imagine what such a view would look like, as this is effectively what Linnebo does in Thin Objects. Whilst this isn't a theme pulled out in his book, I think it's worth understanding that one reason why his work is more convincing (in my opinion) than previous comparable works is the move away from dependence towards sufficiency. Nothing Incurvati says contradicts that claim, this was just a more general point about the topic.

Incurvati's positive proposal: Minimalism

Incurvati's proposal is to adopt a conception of sets whereby a set is just some object that sits in an appropriate set-like position in the (or a) cumulative hierarchy. So something if a set iff it's the empty set, or it's the set of a set, or omega, etc. An analogy is made to Shapiro-style structuralism about the natural numbers. To be a natural number is just to sit in a certain position in a simple infinite series under the successor relation. Generally, the idea is to define mathematical kinds in virtue of some canonical structure of that kind. To be an X is to sit in an X-like position in a canonical X-structure.

As a brief side note, this is not to be confused with an algebraic conception of mathematical objects (see my review of Leng's "and i for an i, a truth for a truth"). The major difference is that the relational property remains unvariabalised on this view. This means that there is a privileged "orientation" within target structures. It's just that identity within that structure is determined relationally.

The view is clearly sufficient for the claim that sets form a cumulative hierarchy- it's packed the notion into the concept. However, this then gives up any attempt to explain why there is a cumulative hierarchy of sets. Just as the platonist arguably "passes the buck" and simply pushes the explanatory burden down the line, rather than meeting it, Incurvati can be accused of the same. The explanatory burden now lies on explaining why we have sets (in Incurvati's sense) rather than some other nearby possible structure schmets. Incurvati's conception simply doesn't meet the burden that the paper sets out to.

Confusingly, though, Incurvati admits as such. He argues for what he calls the "indirect strategy" (we used the term "external justification" in our discussions, as opposed to internal or conceptual justifications). On this view, one justifies a belief in sets, over schmets, via external factors such as their usefulness in mathematical practice, that it has certain theoretic virtues compared to its rivals, etc. But then the minimalist conception plays absolutely no role in this explanation.

Now I entirely agree with Incurvati that this is how one ought to go about arguing for a particular set of axioms, but this claim seems to undermine the dialectic of the entire paper. The criticism of previous views is that they can't provide a merely conception explanation of the cumulative hierarchy. At least some of these views (though perhaps not constructivism), if given access to "external" tools, can provide some kind of a prima facie plausible explanation. Incurvati's minimalist view of sets is on a par with its rivals: it cannot provide an internal explanation of the cumulative hierarchy and needs to rely on external evidence.

I hope I'm not being uncharitable here. There is, perhaps, I more nuanced version of the argument that can be pulled out of the paper. Incurvati continually stresses that adopting very metaphysically loaded conceptions of a set fail to explain the cumulative hierarchy and that one can get by just as well with his minimal conception. It might be that Incurvati has a kind of conceptual parsimony argument working in the background. If both his view and more metaphysically loaded rivals are explanatorily on a par, then this is a problem for the more loaded concepts. Loaded concepts, claims conceptual parsimony, need to earn their keep and, all else being equal, simpler concepts are better.

If this is Incurvati's argument, then it would put his view ahead of at least the platonist view (arguably not the constructivist view which is fairly conceptually simple, it just has other flaws). However, despite its name, Incurvati's view is actually not that minimalist. There are far more minimal conceptions of sets than Incurvati's, which packages the whole of ZFC into the concept itself. Call the ultra-minimalist view the view that sets are just those mathematical objects whose identity is determined by their membership. Like Incurvati's minimalist view, the ultra-minimalist view does not explain why sets form the cumulative hierarchy. But, again like Incurvati's view, the ultra-minimalist relies on external evidence to explain the cumulative hierarchy.

Assuming that Incurvati is correct about platonists, the platonist view, Incurvati's view and the ultra-minimalist view are all in the same situation. They cannot merely conceptually explain the cumulative hierarchy and need to rely on external evidence. However, if Incurvati intends the conceptual parsimony argument, it's clear that the ultra-minimalist conception wins out over all others.

So overall I'm still unclear what the case for Incurvati's concept of a set is. If this is just supposed to be a  paper about how merely conceptual arguments for the cumulative hierarchy fail, then there's a decent (but potentially false) case for that. If the purpose is to advance Incurvati's own concept of a set, then it's unclear what the case for that concept is, given that it plays no role in the explanation at hand.

Closing remarks

This was a really interesting paper that I enjoyed reading, even if I ultimately remain unconvinced. I think of this paper in two parts: first, there's the claim about moving away from "internal" justifications of axioms towards external ones. Second, there's Incurvati's own concept of a set. I find the first part very plausible, even if I don't totally agree with the details of Incurvati's arguments and especially think they need revising in light of more recent work (especially Linnebo 2018). The second part, Incurvati's own concept, I find largely unconvincing and don't think there's a good case for it. His concept of a set kind of sits in an awkward middle. If one accepts his claims about internal/external justifications for axioms, then there are more minimal concepts of a set available. If one rejects those claims and thinks that there is a concept of a set that explains the cumulative hierarchy, then it's not Incurvati's and his view will lose out to that. So either way, Incurvati's concept of a set won't be a desirable one.

Nevertheless, the paper was interesting and it's certainly worth a read.

For anyone wishing to read this article it can be found via these link:

• https://philpapers.org/rec/INCHTB
• https://link.springer.com/article/10.1007/s11098-010-9690-1

Luca Incurvati is an Associate Professor at the department of philosophy and the ILLC in Amsterdam https://www.uva.nl/en/profile/i/n/l.incurvati/l.incurvati.html?cb

Linnebo's views can be found in his 2008 paper "Structuralism and the notion of dependence" in Philosophical Quarterly. His more recent work is in his 2018 OUP book "Thin Objects".

Potter's views can be found in this 2004 OUP book "Set theory and its philosophy"

I welcome comments or friendly criticism below, or you're welcome to email me at gareth.pearce@univie.ac.at

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The picture is attributed to FerdiBf (Diskussion), Copyrighted free use, via Wikimedia Commons. File:ProjectiveHierarchyInclusions.png - Wikimedia Commons