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.

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!

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!

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.

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.

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 . 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 ” 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.

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.

I am adding a few old(ish) papers to the blog section of the website for the sake of posterity. My first ever piece of published material was a little review of Michèle Friend’s Pluralism in Mathematics (you can find a pre-print here, or the journal version here). As I recall, I very much enjoyed the book. There are two issues that the book prompted me to think about and have stuck with me (I’m still trying to wrap my head around):

1. What might be the justification for holding a particular pluralism?

A first question that I think is very important is that one can have different `depths’ of pluralism. We might just be a methodological pluralists, in that we think that we should have multiple people working in multiple frameworks, as this is more effective for generating knowledge and results. (Roughly, this seems to be Chang’s position in Is Water H2O?, though he’s expressed something deeper in conversation.) However, we might have a deeper pluralism in that we might think that the structure of the world for (whatever reason) resists characterisation by a single perspective (this seems to be Kuhn’s position in some of his more `relativistic’ moments).

There are lots of challenges facing the `deep’ pluralist, but I’m especially interested in the possible differences of a deep pluralism in mathematics, versus that of science. There seems to be an important difference between the two, though we get different perspectives from both the empirical sciences and (foundations of ) mathematics, its only in the former that we have true incompatibility of our favourite theories (e.g. Beam mechanics, quantum vs. relativistic physics). In mathematics, we can instead (for many theories) simulate one within the other. For example, results discovered in the language of set theory can be translated into an algebraic theory of sets in category theory (e.g. ETCS), and conversely we can interpret any algebraic theory within the sets (possibly with the addition of large cardinals).

Despite this simulation, however, the two perspectives are very different when thought of as providing the “ontology” for mathematics. Set theory provides a system in which mathematical objects can be “built” and regards objects with very different internal structure as totally different in nature (e.g. is a very different thing from , because their transitive closures look very different, but categorially in Set they are just both terminal objects, isomorphic, and hence categorially `the same’).

With this in mind, one question I’m still struggling with is the following:

Is there a motivation that could be supplied for a deep pluralism in the case of mathematics, and is this threatened by the disanalogy with the empirical sciences?

2. How to deal with contradiction between perspectives?

A second problem occurs once we have adopted a pluralism. Presumably then, the challenge is to figure out how to transfer useful information in a meaning-preserving (in some sense) way from one perspective to another, despite any conflicts. In short, we want to `adopt’ two contradictory perspectives at once, without dissolving into triviality.

The way Friend deals with this is to just take true contradictions on the chin, and adopt Priest’s Logic of Paradox (LP) to conduct her inferences. I have two problems with this:

A: True contradictions just aren’t appealing to me. I want to understand the world as given by multiple irreconcilable perspectives, not a single incoherent one. (I appreciate that lots of great stuff has come out of the paraconsistent and dialethist literature, but I just cannot get on board with a true contradiction.)

B: LP doesn’t validate modus ponens for the material conditional. I need modus ponens to reason properly (also, I feel it should be true on any perspective).

For this reason, I’m interested in taking a step back and rather looking at the process of “information transfer”. Friend does this a little by examining Priest’s method of “chunk-and-permeate” (roughly: hive off the relevant information and move it), but I wonder whether progress can be made by analysing the essential features of a foundational language. This, however, seems like a pretty difficult task.