“Multiversism and Concepts of Set” revisited

A month ago I posted here a link to an interesting paper here by Neil Barton. There’s now a discussion exchange, which it would be a pity to leave buried unread in comments on an old posting: so here it is.

From Rowsety Moid. It’s an interesting paper, but to me it seems there are many questionable steps in its arguments, and I would like to know what people who know more than I do about set-theoretic multiverses would say.

The “algebraic” interpretation strikes me as incoherent, or else slight of hand. When he explains it on page 10, he seems to be saying it is not involved with existence and reference, but he then talks of “a group G”, of “elements” of G, and of “constructing new groups from old”, which all involve existence and reference. His way out of this seem to be to say the algebraic view is not “concerned with” such things (which is largely a matter of attitude, focus and interest), and that we can understand operations on groups “not as making any claims about existence and reference” (as if that were the only way issues of existence or reference could come in).

Or, page 11: “We do not make any claims as to what exists within the Multiverse, rather it is seen as an intuitive picture to facilitate algebraic reasoning concerning sets.” Even the sets don’t exist? “Given a structure” The structure doesn’t exist?

Also, it’s not clear whether he is addressing Hamkins’s actual view or a maximally “radical” alternative. For instance, on page 8 “we are interested in Multiversism in its most radical form” is given as a reason for assuming that every level of metalanguage is indeterminate.

Even when the aim doesn’t seem to be the maximalisation of radicalism, there are a number of questionable interpretations or restatements. On page 9: “One way to understand Hamkins’ suggestion is to hold that we refer to several universes at once via description”. By page 11, the “one way to understand” has dropped out. Hamkins saying “in this article I shall simply identify a set concept with the model of set theory to which it gives rise”, quoted on page 7, becomes “it was noted that the Multiversist thought that every model of set theory constituted a set concept” on page 13. The idea that the concept gives rise to the model has been lost, and the idea that every model constitutes a concept has been added. Between pages 7 and 8, Hamkins’s “Often the clearest way to refer to a set concept is to describe the universe of sets in which it is instantiated” becomes “each model is correlated with a set concept, and we refer (to?) this concept through a description.” Such examples can be multiplied.

On page 14, it turns out to be important for Neil Barton’s argument that every model (or, perhaps, every “cloud” of models) corresponds to a concept and so to a description. I don’t see how that could work. Aren’t there more models than descriptions? (Hamkins seemed to have it that there was a model for every concept, not that there was a concept for every model.)

Another important step in Barton’s argument is the idea that we (or at least “Hamkinsians”) can use only first-order descriptions and so “lack the conceptual resources to pin down a single universe precisely”. From that, via the “One way to understand Hamkins” mentioned above, we reach the idea that we end up referring to “clouds”. For some purposes, that restriction seems correct. But when we’re trying to ground reference and so avoid a vicious infinite regress? That’s not so clear. Natural language (English, for example) isn’t restricted to FOL, for a start.

In the background, there seems to be an ideological element to the argument. It’s difficult to pin it down, but I think it may become visible on page 15 when arguing that “the Hamkinsian can give no particular reason to focus on one stopping point rather than another” and then saying “the response that we simply stop somewhere (without being able to give any reason for a particular stopping point) seems, like Go ̈del, to ascribe unexplained powers to the human mind.”

The power to stop somewhere? Is that supposed to be mysterious? I supposed that, in a sense, it is unexplained; but only because pretty much everything about the mind is currently unexplained, if you push hard enough.

From Neil Barton First, let me say a big “Thank You!” to Peter for publicising my paper, and to Rowsety Moid for some excellent comments. Indeed, your remarks were very timely, as they highlighted a mistake that I corrected in the proofs (shameless self-promotion: the paper will come out in this volume.

You said, RM: “The “algebraic” interpretation strikes me as incoherent, or else slight of hand. When he explains it on page 10, he seems to be saying it is not involved with existence and reference, but he then talks of “a group G”, of “elements” of G, and of “constructing new groups from old”, which all involve existence and reference. His way out of this seem to be to say the algebraic view is not “concerned with” such things (which is largely a matter of attitude, focus and interest), and that we can understand operations on groups “not as making any claims about existence and reference” (as if that were the only way issues of existence or reference could come in).”

Sure: we can have the algebraic interpretation relate to issues of existence and reference. The point is just that one requires additional assumptions first. For example, if we assume that the relevant objects exist, then there is a class of structures which instantiate the relevant algebraic properties about which the algebraist talks. Indeed, then various claims she makes might be a good way of proving the existence of certain objects, and better than trying to construct these things `absolutely’ (I think non-standard models are possibility a vivid case here). But those additional assumptions are needed before her view gets off the ground.

To bring this out, imagine that nominalism about sets were true. The ontological interpretation would then be null and void, there simply is no multiverse. It seems that the algebraic interpretation might still live on, despite the fact that the relevant algebraic properties are uninstantiated. For, we could still say that IF we were given some objects satisfying the relevant properties, we would be able to do such and such operations (and similarly with other algebraic theories like group theory).

[N.B. It’s an interesting question how the algebraic interpretation relates to if-then-ism. It’s not clear that these are wholly the same because of the algebraists acceptance of indeterminacy in metalogical notions.]

RM: “Even the sets don’t exist? “Given a structure” The structure doesn’t exist?”

We say things like this all the time though. “If space-time is discrete, then such and such will hold, and it is theoretically possible to construct such and such kind of object.” That seems like a perfectly valid claim to make, even if space-time is not discrete. Similarly with sets. Give me a structure, and I will be able to do these sorts of operations. I make no claim on whether the structure exists.

[N.B. As someone who is generally of a realist persuasion, I tend to think that this sort of response depends on the existence of the structures anyway. But this is just a refusal to engage with the position, not a dialectically convincing response.]

RM: “Also, it’s not clear whether he is addressing Hamkins’ actual view or a maximally “radical” alternative. For instance, on page 8 “we are interested in Multiversism in its most radical form” is given as a reason for assuming that every level of metalanguage is indeterminate.”

You raise a good point here, that I think is a general feature of the debate: the exact positions on the table are unclear. The Hamkins paper, though both highly interesting and ingenious, is notoriously slippery when it comes to being fully precise about the commitments of his view. So: am I addressing Hamkins or a radical alternative? I don’t know: you’d have to ask Joel how well the view put forward coheres with his (who, it has to be mentioned, was very helpful in discussing the paper and I’ve found very approachable). However, the views I present in the paper are ones that can be extracted from some of the things that he says, and he’s often keen to embrace the radical consequences of his view. He’s very clear that he thinks that indeterminacy infects the metalanguage, and there is no definite concept of natural number.

RM: “there are a number of questionable interpretations or restatements.”

I think there are going to have to be with the literature as it stands. Nowhere is Hamkins explicit about the kinds of epistemology he envisages, or the full metaphysical character of his view. My paper is intended to be just as much filling out possible ways of taking Hamkins’ view as a criticism of some of the things he says. I would welcome it if there are alternative interpretations out there, or I have got something wrong in exegesis—that way we can be more precise about what views are available and tenable. But I need to see these additional interpretations before I can weigh them up against the ones I have put forward.

RM: “The idea that the concept gives rise to the model has been lost, and the idea that every model constitutes a concept has been added. Between pages 7 and 8, Hamkins’s “Often the clearest way to refer to a set concept is to describe the universe of sets in which it is instantiated” becomes “each model is correlated with a set concept, and we refer (to?) this concept through a description.” Such examples can be multiplied.

On page 14, it turns out to be important for Neil Barton’s argument that every model (or, perhaps, every “cloud” of models) corresponds to a concept and so to a description. I don’t see how that could work… (Hamkins seemed to have it that there was a model for every concept, not that there was a concept for every model.)”

This was sloppy on my part, and I made some alterations in the proofs as a result. You are right, we should say that we refer to the concept through the model. However, this is done through *description*: we refer to the concept by *describing* the model. However, since we can only use first-order descriptions, we can’t pin down a single model, so our reference must be indeterminate, but this requires fixing some other model (in which some concept is instantiated; I take it that every model instantiates a concept) and so on.

I think there’s a lot more to be said here (in fact it doesn’t seem impossible to me that we get a loop of concepts or models), but the challenge at least represents an invitation for the Hamkinsian Multiversist to be clear on their commitments, and explain why there’s no such problem. As it stands, the Ontological Interpretation is not developed in sufficient detail to explain how these problems are avoided.

RM: “Aren’t there more models than descriptions?”

That depends on where you live for the Hamkinsian. Since every universe’s multiverse is countable from the perspective of some other universe, the universes of a multiverse are bijective with the descriptions from a suitable perspective (though not through the natural mapping of a universe with a description of it, and not within any particular multiverse). In any case, to press the challenge I only require infinitely many concepts being used, so countable is enough. There’s also the question of whether or not Hamkins is allowing parameters, which would in turn make the issue a whole lot more complicated (given the emphasis on ultrapowers, I’m guessing he is allowing parameters for the ultrafilters), as then we could have proper-class-many descriptions. Again, this would be another area I would like clarification from the Hamkinsian.

RM: “But when we’re trying to ground reference and so avoid a vicious infinite regress? That’s not so clear. Natural language (English, for example) isn’t restricted to FOL, for a start.”

I’m in full agreement here! In fact I find the restriction to FOL excessive. Again though, this is a case where the dialectic with Hamkinsian is important. If he/she wants to admit non-FOL, s/he has to explain what is acceptable and what isn’t. Why is it okay for him/her to use non-FOL resources in giving an account of reference, yet an account of the semantic referents of the natural numbers or set theory in terms of properties/plurals/the ancestral relation/sets is not allowed? I want to know what the rules of the game are by which the Hamkinsian has a stronger position compared to the Universist.

RM:“but only because pretty much everything about the mind is currently unexplained, if you push hard enough.”

Sure! But here we’re seeing if the Hamkinsian has a decent response to Benacerraf’s challenge through description. So, given that this is the background for the paper, we can demand an explanation.

I think in general the paper shouldn’t be viewed as an all out attack on the Hamkinsian, but rather a request for him/her to clarify a number of points.

Thank you ever so much for the comments! I found them very helpful. Best Wishes, Neil

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