Yet another post on an old paper, but one I’m trying to figure into my current work:
The rough idea (at the time) was to criticise the Multiverse view on which we regard the subject matter of set theory as given by the models of first-order . The main problem is the following: If you don’t allow yourself any determinately understood concepts, it’s very hard to see how to `get going’ in terms of specifying what you’re talking about. In the paper, I give two problems along these lines:
- Philosophically speaking, it’s unclear how one could ever come to refer to a particular universe (or set therein). This is because Hamkins identifies (first-order) models and concepts, combined with the fact that the extent of the Multiverse is dependent upon the starting universe. So, any time I have putatively referred, I can only refer based on a starting universe, but for this to be fixed I need another starting point to be fixed, but for this second starting point to be fixed, I need to fix a third, and so on. With the identification of models and concepts, it then looks like I employ infinitely many concepts when I refer.
- Because Hamkins holds that only first-order logic is allowed, there is no determinate notion of finiteness. The end result is that there’s no determinate notion of proof or well-formed formula. In fact, when I say “The Multiverse comprises all models of first-order ” it’s not determinate what “” refers to here, since it might contain a bunch of non-standard formulas.
There’s lots to be said about how one might respond to these criticisms. The response I gave in the paper, and one I think is interesting, is to let go of the idea that this radical multiversism is prescribing a particular ontology. Rather, it can be viewed as facilitating a framework in which the algebraic properties of ZFC can be discussed. The Hamkins-style multiverse might then be in a position to say that they are not telling you what set-theoretic objects exist, but rather what is true and possible to construct given an initial starting point that has certain features.
Feeding into my current projects on pluralism, we might then have the following thought: The hardcore universe-theorist and Hamkinsian multiversist aren’t really disagreeing. They rather just provide different perspectives on the same mathematical subject matter. Every mathematical entity has a concrete presentation and an algebraic presentation. For example we could think of a natural number as given by a particular concrete object (like a von Neumann ordinal), but we also could think of it as an algebraic operation (for example in a monoid), and neither is definitely the `right’ way of thinking of that natural number. Why not view sets in the same light?
Of course, that’s a rough sketch, and there’s lots to be worked out here (e.g. What should the overall logic be? Is there in actuality a unified perspective? How can we think of satisfaction from the algebraic perspective?). I think it’s an interesting start though to a pluralism that incorporates more perspectives.
Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position.