TALK: Forcing and the Universe of Sets: Must we lose insight?

This will be a talk at Methodological Approaches in the Study of Recent Mathematics: Mathematical Philosophy and Mathematical Practice, at the University of Konstanz on 21 September. You can find the slides for the talk here, and a recent draft of the paper here.

Abstract: A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems to necessitate the addition of subsets to V. We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and analyse various tradeoffs in naturality that might be made. We conclude that the Universist has promising options for interpreting different forcing constructions.

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.