Talk: Set Theory and Structures

This will be a talk at the 2018 Italian Society for the Philosophy of Mathematics (Filmat 2018) meeting.  Slides.

Abstract:

Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a `structural’ perspective to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure.

Talk: Mathematical Gettier Cases

This will be a talk at the NYU Philosophy Department. You can find slides here.

Abstract:

Are Gettier cases possible in mathematics? At first sight we might think not: The standard for mathematical justification is proof and, since proof is bound at the hip with truth, there is no possibility of having an epistemically lucky justification of a true proposition. In this paper, we argue that Gettier cases are possible (and very likely actual) in mathematical reasoning. We do this via arguing that abductive inference and auxiliary assumptions are essential to mathematical practice. This results in the following two argumentative strands: (1.) We dispute the claim that the standard of mathematical justification is the production of an actual formal proof from obviously true premises, and (2.) We argue that even if we do accept that this is the standard of justification, there is still the possibility of luck resulting in true belief. We’ll do this by considering several examples, some more fantastical than others.

Talk: Large Cardinals and the Iterative Conception of Set, CUNY, 9 February 2018

This will be a talk in the CUNY set theory seminar. You can find the slides here.

Abstract: Large cardinals are seen as some of the most natural and well-motivated axioms of set theory. Often *maximality* considerations are mobilised in favour of consistent large cardinals: Since (it is argued) the axioms assert that the stages go as far as a certain ordinal, and it is part of the iterative conception that the construction be iterated as far as possible, if it is *consistent* to form a particular large cardinal then we should do so. This paper puts pressure on this line of thinking. We argue that since the iterative conception legislates in favour of forming all possible subsets at each additional stage and *then* iterating this as far as possible, what is regarded as ‘consistently formable’ will depend upon the nature of the subset operation in play. We present a few cases (some involving forcing) where the *consistency* of a large cardinal axiom comes apart from its *truth* on the basis of *maximality* criteria. Thus there are interpretations of maximality on which large cardinals are consistent but not true, and so maximality of the iterative conception does not clearly legislate in favour of large cardinals. We will even argue that there may be a natural, maximal, and strong version of set theory on which every set is countable!

Talk: Large Cardinals and the Iterative Conception of Set: Is every set countable?

Here you can find slides for my talk at the Forcing and Philosophy workshop, held at the University of Konstanz on the 18th and 19th of January.

Abstract:

Large cardinals are seen as some of the most natural and well-motivated axioms of set theory. Often *maximality* considerations are mobilised in favour of consistent large cardinals: Since (it is argued) the axioms assert that the stages go as far as a certain ordinal, and it is part of the iterative conception that the construction be iterated as far as possible, if it is *consistent* to form a particular large cardinal then we should do so. This paper puts pressure on this line of thinking. We argue that since the iterative conception legislates in favour of forming all possible subsets at each additional stage and *then* iterating this as far as possible, what is regarded as ‘consistently formable’ will depend upon the nature of the subset operation in play. We present a few cases (some involving forcing) where the *consistency* of a large cardinal axiom comes apart from its *truth* on the basis of *maximality* criteria. Thus there are interpretations of maximality on which large cardinals are consistent but not true, and so maximality of the iterative conception does not clearly legislate in favour of large cardinals. We will even argue that there may be a natural, maximal, and strong version of set theory on which every set is countable!

 

Maximality and Ontology: How axiom content varies across philosophical framework

Something I worked on a bit last year, and continue to work on, is how *what an axiom says* is dependent on the ontological framework in which it is evaluated. In particular, when an axiom of set theory uses *extensions* of universes in order to evaluate its content (as is the case with Friedman’s Inner Model Hypothesis [which states that any parameter-free sentence true in an extension is true in the universe]), the axiom can have quite a radically different meaning based on philosophical perspective. This is because while axioms might not have their `intended’ content (in the case of the Inner Model Hypothesis for example, you might not believe that there are extensions of the set-theoretic universe), they can quite often be coded. You then get an interesting phenomenon where what an axiom says varies according to whether it states a property of a universe by asserting a relationship between actually existing bona fide universes, or alternatively whether it states that a universe behaves a certain way with respect to kinds of coding mechanism (dependent upon one’s philosophical views). In fact, interpretation of certain complicated axioms in set theory turn out to have very similar features to standard debates between anti-realists and realists in the philosophy of mathematics.

You can find more details here: Journal, Pre-print.

Abstract:

Discussion of new axioms for set theory has often focussed on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular ways) face complementary problems. The latter view is unable to use maximality axioms that make use of extensions, where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology.

Independence and Ignorance: How agnotology informs set-theoretic pluralism

A little while ago a wrote a piece on agnotology (the study of ignorance) and set-theoretic pluralism (journal, academia.edu, preprint).

The rough idea was that the ways in which we might be ignorant of set-theoretic claims might lead us to accept a methodological pluralism concerning set theory, even if we think there’s just one universe.

What do I mean by methodological pluralism concerning some discipline X? Simply that we should investigate a wide range of diverse theories pertaining to X. So in the case of set theory, proponents of the one universe view, even those with a preferred foundational theory, should still want people investigating other `competing’ theories.

Why so? Well, one obvious reason is that technical insight obtained by researching a particular theory can often be transferred across to ones own foundational theory. However, more philosophically, the point (made in detail in the paper) is that research into competing theories (thereby completing the theoretical picture they provide) provides evidence that we are not in a case of ignorance concerning our own foundational theory. In particular, it is only through the research of other programmes that we can be convinced that we are not in a state of unconscious ignorance (i.e. a state where we don’t know that we don’t know something) with respect to a particular mathematical question. So, we should actually want research into theories we think are false, in order to bolster the case for our own preferred theory actually being true.

A substantial question left open here is whether these ideas can be fed into a wider pluralistic framework. For instance, there’s some interesting technical facts about how category theory can be used set-theoretically (see Bagaria and Brooke-Taylor, `On Colimits and Elementary Embeddings’), but can this be fed into a wider epistemological/agnotological story elucidating the two kinds of foundation?

Here’s the abstract for anyone interested:

Much of the discussion of set-theoretic independence, and whether or not we could legitimately expand our foundational theory, concerns how we could possibly come to know the truth value of independent sentences. This paper pursues a slightly different tack, examining how we are ignorant of issues surrounding their truth. We argue that a study of how we are ignorant reveals a need for an understanding of set-theoretic explanation and motivates a pluralism concerning the adoption of foundational theory.

Multiversism and Concepts of Set: How much Relativism is acceptable?

Yet another post on an old paper, but one I’m trying to figure into my current work:

A little while ago, I wrote a paper on Joel Hamkins’ paper `The Set-Theoretic Multiverse’ (get Joel’s paper here or here).

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 \mathbf{ZFC}. 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:

  1. 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.
  2. 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 \mathbf{ZFC}” it’s not determinate what “\mathbf{ZFC}” 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.

You can read the full paper here or here.

Abstract:

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.