I propose a theory of space with infinitesimal regions called smooth infinitesimal geometry based on certain algebraic objects, which regiments a mode of reasoning heuristically used by geometricists and physicists. I argue that SIG has the following utilities. It provides a simple metaphysics of vector fields and tangent space that are otherwise perplexing. A tangent space can be considered an infinitesimal region of space. It generalizes a standard implementation of spacetime algebraicism called Einstein algebras. It solves the long-standing problem of (...) interpreting smooth infinitesimal analysis realistically, an alternative foundation of spacetime theories to real analysis, 277–392, 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations. Against this, I argue that SIG is such a reformulation. But SIG has an unorthodox mereology, in which the principle of supplementation fails. (shrink)

Nel nostro universo, qualunque cosa, dalla più piccola particella alla più smisurata galassia, esiste in un qualche tempo e in un qualche luogo. Ma cosa significa esistere in un qualche tempo? Il fenomeno dell’esistenza temporale gioca un ruolo fondamentale nella comprensione dell’universo e di noi stessi quali creature temporali. Eppure è un fenomeno profondamente misterioso. L’esistenza temporale è da intendersi come una relazione? Che legami ha con l’esistenza dell’ontologia? L’esistenza temporale e la localizzazione spaziale sono due fenomeni essenzialmente differenti o (...) due istanze di un’unica relazione? Tutto ciò che esiste nel tempo esiste nel tempo nello stesso modo? Tutto ciò che è nel tempo è anche nello spazio? (shrink)

Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of (...) this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefinite extensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch. (shrink)

Part One of this paper is a case against classical mereology and for Heyting mereology. This case proceeds by first undermining the appeal of classical mereology and then showing how it fails to cohere with our intuitions about a measure of quantity. Part Two shows how Heyting mereology provides an account of sets and classes without resort to any nonmereological primitive.

In this paper a solution of Whitehead’s problem is presented: Starting with a purely mereological system of regions a topological space is constructed such that the class of regions is isomorphic to the Boolean lattice of regular open sets of that space. This construction may be considered as a generalized completion in analogy to the well-known Dedekind completion of the rational numbers yielding the real numbers . The argument of the paper relies on the theories of continuous lattices and “pointless” (...) topology.

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Logics of part/whole relations frequently take parthood or proper parthood as primitive, defining the remaining mereological properties and relations in terms of them. I argue from considerations involving Weak Supplementation for the conclusion that we should take fusion as our mereological primitive. I point out that the intuitions supporting Weak Supplementation also support a stronger principle, Weak Supplementation of Pluralities, and that the principle can only do the work demanded by our intuitions when formulated in terms of a notion of (...) fusion that cannot be defined merely in terms of mereological properties and relations, logic, and a membership relation. So, insofar as we think any definition of fusion must be so restricted, we have motivation to take fusion as primitive; further, we have greater insight into the motivation for our supplementation principle and which version of that principle we ought to endorse. (shrink)

An overview of contemporary part-whole theories, with reference to both their axiomatic developments and their philosophical underpinnings.

In this paper, I tackle the problem of diachronic identity. Far from providing a criterion for identity over time, the aim of this work is to understand if this issue pertains to ontology, conceived as that part of philosophy that tries to answer the question about what entities exist, or metaphysics, conceived as that part of philosophy that tries to explain, of those entities, what they are. On the face of it, only metaphysics has the task to solve this problem, (...) but I argue that this is false. Through the analysis of different theories concerning identity through time, I show how both ontology and metaphysics are concerned with the problem of diachronic identity, and how actually ontology turns out to be primary in solving the problem. (shrink)