Abstract: The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal axioms consistent with ZFC, since it appears to be `possible’ (in some sense) to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we argue that there are several conceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory.
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.
This will be a talk at the 2018 Italian Society for the Philosophy of Mathematics (Filmat 2018) meeting. Slides.
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.
This will be a talk at the NYU Philosophy Department. You can find slides here.
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!