Research themes

Much of my research has been devoted to two broad themes at the intersection of metaphysics and the philosophy of logic and mathematics. Each theme draws inspiration from a great thinker. 

Ontology and the logical concept of an object

Objects can be “carved out” by specifying a coherent application of first-order logic. Crucially, some such applications can be specified without any mention of the objects in question. For example, conditions for correct talk about directions can be specified solely in terms of lines and parallelism, with no mention of directions. The possibility of such “reductive” correctness conditions ensures that some applications of first-order logic don’t require much of reality. Thus, my approach permits “thin” (or metaphysically cheap) objects. Inspired by Frege, I also emphasize the importance of criteria of identity in the individuation of objects.


See research contributions # 1, 2, 5, and 10 below. 


Gottlob Frege (1848-1925)

Aristotle (384-322 BC) 

Potentiality and incompletability in mathematics and metaphysics 

Taking inspiration from the ancient concept of potential infinity, my research develops useful “successor concepts”. I study incompletable processes where available objects are used, again and again, to define more objects. For example, we can use available sets to define ever more sets. It is also fruitful to consider nested definitions of intensional entities such as properties or propositions. I develop logical-mathematical analyses of these incompletable processes of iterated definitions, for example, using modal logic, a “critical” form of plural logic, or a (semi-)intuitionistic logic. This research has applications far beyond philosophy:


See research contributions # 2 - 5 and 7 - 9 below. 

Ten key research contributions

Looking back at a quarter century of hard work, I thought it might be useful to identify some central themes in my research. So, here are what I regard as my ten most important research contributions.

There is a legitimate epistemological challenge to “platonistic” mathematics. The best response is to view mathematical objects as “thin” or metaphysically undemanding. This “thinness” enables us to explain how reference to, and knowledge of, abstract mathematical objects is possible.

 

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2. Every plurality defines a set

Given any objects, xx, we have, I argue, what it takes to define their set, {xx}. Of course, the paradoxes show that not every condition A(x) defines a set. Thus, either some conditions fail to define some objects (“a plurality”), or some pluralities fail to define a set. Somewhat heretically, I defend the first option.

 

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3. The potentialist translation and the “mirroring” theorem

Potentialism regards mathematical existence as inherently potential. This suggests that ordinary mathematical existential and universal generalizations be understood as “possibly there is” and “necessarily for all”, respectively. Call this the potentialist translation. Amazingly, this translation turns out to be logically well-behaved. I proved in 2008 that the translation yields a faithful interpretation (or “mirroring”) of classical first-order logic in the modal logic S4.2 and some “stability” assumptions. Since the needed assumptions are plausible from a potentialist point of view, this result paves the way for a precise modal-logical analysis of the ancient idea of potential infinity, as well as wealth of potentialist approaches in the foundations of mathematics.

 

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4. Critical plural logic:  the logic of "joint availability"

Mathematical objects can be successively defined in terms of each other. For example, Cantor argued that any jointly available (“co-existent”) objects xx define a set {xx}. What is it for some objects to be jointly available to figure in the definition of another object, such as their set? Salvatore Florio and I argue that this notion of joint availability is simply that of a plurality of objects. We also develop a new “critical” plural logic that serves as a logic of joint availability. As a proof of concept, we combine critical plural logic with Cantor’s operation of gathering many objects xx into one set {xx} and retrieve (nearly) all of ZFC set theory.

 

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5. Grounded abstraction

Frege and neo-Fregeans study abstraction principles, which provide criteria of existence and identity for mathematical objects. This approach faces a “bad company” problem: many abstraction principles are inconsistent or otherwise problematic. I have long sought a systematic response to the problem, based on a philosophical analysis of what goes wrong in the “bad” cases. My guiding idea is to view abstraction as a “process” of successive definition of objects, each definition using only resources available at the relevant “stage”. This “grounded” approach to abstraction can be implemented using modal logic, with each “possible world” corresponding to a stage of the process of successive definition. A simpler and more elegant option is to use critical plural logic (idea #4), with “critical” pluralities serving as stages. As a proof of concept, we obtain a strong and liberal theory of grounded abstraction, which yields a natural response to the bad company problem.

 

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6. Structuralism as collective abstraction

The mathematician Richard Dedekind was a pioneer of mathematical structuralism. He proposed that an abstract structure is obtained from an appropriate system of particular objects and relations by purging all "foreign" properties, retaining only structural ones. I develop an account of “Dedekind abstraction” as a form of collective abstraction. This sheds light on non-eliminative mathematical structuralism, reveals relations of mutual dependence among mathematical objects, and identifies an abstractionist core that is shared by Fregean and structuralist approaches to mathematics, which are normally regarded as rivals.

 

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7. Non-instantial generality

What features of reality are responsible for the truth of a universal generalization? The orthodox answer proceeds via the instances of the generalization. Everything is F because a is F, b is F, and so on, plus (perhaps) the fact that these are all the objects. I show that the orthodoxy needs to be supplemented with (wholly or partially) non-instantial explanations. E.g., we can explain why everything crimson is red or why every object has a singleton set without invoking any instances of these generalizations. Although non-instantial generality is familiar from mathematical intuitionism, I divorce the idea from the intuitionistic philosophy and show how it can be put on a robustly realist footing (say, in terms of Finean essences). With non-instantial generality on board, all the truths of intuitionistic (but not classical) first-order logic turn out to have a trivial truthmaker.

 

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8.  Invariance and the definition of intensional entities

Suppose we wish to define objects successively, each definition relying only on resources available at the relevant stage. How, then, can we define intensional entities such as a property (or a class) capable of gaining instances (or members) as more and more objects are defined? The great mathematician Henri Poincaré suggested an answer: invariance. That is, a definition of an intensional entity is permissible provided it never “changes its mind” about the classification of an object as an instance of the property (or a member of the class) as ever more objects are defined. Developing this suggestion, I show that invariance yields a better explication of predicativity than the more traditional vicious circle principle, not least because invariance gives rise to better generalizations and liberalized forms of predicativity.

 

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9. Absolute but indefinite generality: a third way in the debate about absolute generality

Is it possible to quantify over absolutely everything? Absolutist say yes; relativists, no. I defend a third way. Every extensionally specified domain (e.g. a set or plurality) can be surpassed be an even more inclusive such domain, much as relativists say. But there can be an absolute and unsurpassable domain—provided this is understood as merely intensional or indefinite as to extension. To make sense of quantification over this absolute but indefinite domain, we need non-instantial generality (idea #7), which in turn means that the logic becomes semi-intuitionistic.

 

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10. Unification of universes (the lifting of type distinctions)

 Higher-order languages go back to Frege and Russell but have recently had a great resurgence. The universe is seen as stratified into orders, and every quantifier, as ranging over only one order (or, possibly, over one order and all lower ones). This points to a common feature of higher-orderism and generality relativism: an inability to generalize over absolutely all entities. I explore ways to lift the type distinctions and admit all-purpose variables and quantifiers, which range across all orders. This "unification of universes" yields an even more absolute form of absolutism.

 

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