Metaphysics is supposed to tell us about the metaphysical nature of our world: under what conditions composition occurs; how objects persist through time; whether properties are universals or tropes. It is near orthodoxy that whichever of these sorts of metaphysical claims is true is necessarily true. This paper looks at the debate between that orthodox view and a recently emerging view that claims like these are contingent, by focusing on the metaphysical debate between monists and pluralists about concrete particulars. This (...) paper argues that we should be contingentists about monism and pluralism, and it defends contingentism against some necessitarian objections by offering an epistemology of contingent metaphysical claims. (shrink)

The Caesar problem arises for abstractionist views, which seek to secure reference for terms such as ‘the number of Xs’ or #X by stipulating the content of ‘unmixed’ identity contexts like ‘#X = #Y’. Frege objects that this stipulation says nothing about ‘mixed’ contexts such as ‘# X = Julius Caesar’. This article defends a neglected response to the Caesar problem: the content of mixed contexts is just as open to stipulation as that of unmixed contexts.

There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants. Taking a cue from some work in linguistics and (...) the philosophy of language, I suggest that i functions like a parameter in natural deduction systems. This may require some rethinking of the role of singular terms, at least in mathematical languages. (shrink)

I argue for a conception of essence that does not rely on distinctions of metaphysical fundamentality.

ABSTRACTAnte rem structuralists claim that mathematical objects are places in ante rem structural universals. They also hold that the places in these structural universals instantiate themselves. This paper is an investigation of this self-instantiation thesis. I begin by pointing out that this thesis is of central importance: unless the places of a mathematical structure, such as the places of the natural number structure, themselves instantiate the structure, they cannot have any arithmetical properties. But if places do not have arithmetical properties, (...) then they cannot be the natural numbers. The self-instantiation thesis turns out to be crucial for the identification of mathematical objects with places in structures. Unfortunately, we have no reason to believe that the self-instantiation thesis is true. (shrink)

The ante rem structuralist holds that places in ante rem structures are objects with determinate identity conditions, but he cannot justify this view by providing places with criteria of identity. The latest response to this problem holds that no criteria of identity are required because mathematical practice presupposes a primitive identity relation. This paper criticizes this appeal to mathematical practice. Ante rem structuralism interprets mathematics within the theory of universals, holding that mathematical objects are places in universals. The identity problem (...) should be read as challenging this claim about universals. However, what mathematical practice presupposes is only relevant to what is true according to the theory of universals, if one takes it for granted such a theory offers the best interpretation of mathematics. In the current context, taking this for granted begs the question. Therefore, the appeal to mathematical practice in response to the identity problem is illegitimate. (shrink)

As a new foundational language for mathematics with its very different idea as to the status of logic, we should expect homotopy type theory to shed new light on some of the problems of philosophy which have been treated by logic. In this article, definite description, and in particular its employment within mathematics, is formulated within the type theory. Homotopy type theory has been proposed as an inherently structuralist foundational language for mathematics. Using the new formulation of definite descriptions, opportunities (...) to express ‘the structure of’ within homotopy type theory are explored, and it is shown there is little or no need for this expression. (shrink)

I show that a particular version of Hume's Dictum together with the falsity of Composition as Identity entails an incoherency, so either that version of Hume's Dictum is false or Composition as Identity is true. I conditionally defend the particular version of Hume's Dictum in play, and hence conditionally conclude that Composition as Identity is true. I end by suggesting an alternative way out for a persistent foe of Composition as Identity, namely mereological nihilism.

In this work I first develop, motivate, and defend the view that mereological composition, the relation between an object and all its parts collectively, is a relation of identity. I argue that this view implies and hence can explain the logical necessity of classical mereology, the formal study of the part-whole relation. I then critically discuss four contemporary views of the same kind. Finally, I employ my thesis in a recent discussion of whether the world is fundamentally one in number.