It is a well-known fact that mathematics plays a crucial role in physics; in fact, it is virtually impossible to imagine contemporary physics without it. But it is questionable whether mathematical concepts could ever play such a role in psychology or philosophy. In this paper, we set out to examine a rather unobvious example of the application of topology, in the form of the theory of persons proposed by Kurt Lewin in his Principles of Topological Psychology. Our aim is to (...) show that this branch of mathematics can furnish a natural conceptual system for Gestalt psychology, in that it provides effective tools for describing global qualitative aspects of the latter’s object of investigation. We distinguish three possible ways in which mathematics can contribute to this: explanation, explication and metaphor. We hold that all three of these can be usefully characterized as throwing light on their subject matter, and argue that in each case this contrasts with the role of explanations in physics. Mathematics itself, we argue, provides something different from such explanations when applied in the field of psychology, and this is nevertheless still cognitively fruitful. (shrink)

This article examines the use of mathematical concepts in philosophy, focusing on topology, which may be viewed as a modern supplement to geometry. We show that Plato and Parmenides were already employing geometric ideas in their research, and discuss three examples of the application of topology to philosophical problems: the first concerns the analysis of the Cartesian distinction between res extensa and res cogitans, the second the ontology of possible worlds of Wittgenstein's Tractatus, and the third Leibniz's monadology. We also (...) consider the role of topology in mathematical explanations of the sort found in science, arguing that it can perform a role in philosophy that is of comparable importance. (shrink)

The aim of the present article is to make the notion of an ontology of fields mathematically rigorous. The conclusion will be that couching an ontology in terms of mathematical bundles and cross‐sections both captures many important intuitions of conventional ontologies, including the universal‐particular paradigm, the connection of universals and their ‘instantiations’, and the notion of ‘possibility’, and makes possible the framing of ontologies without ‘substrata’, bare particulars, and primitive particularizers.

The aim of the present article is to make the notion of an ontology of fields mathematically rigorous. The conclusion will be that couching an ontology in terms of mathematical bundles and cross-sections (i.e. fields) both (1) captures many important intuitions of conventional ontologies, including the universal-particular paradigm, the connection of universals and their ‘instantiations’, and the notion of ‘possibility’, and (2) makes possible the framing of ontologies without ‘substrata’, bare particulars, and primitive particularizers (a goal that trope ontologies, for (...) example, have sought to attain). (shrink)

Trope theory is the view that the world is a world of abstract particular qualities. But if all there is are tropes, how do we account for the truth of propositions ostensibly made true by some concrete particular? A common answer is that concrete particulars are nothing but tropes in compresence. This answer seems vulnerable to an argument (first presented by F. H. Bradley) according to which any attempt to account for the nature of relations will end up either in (...) contradiction, nonsense, or will lead to a vicious infinite regress. I investigate Bradley’s argument and claim that it fails to prove what it sets out to. It fails, I argue, because it does not take all the different ways in which relation and relata may depend on one another into account. If relations are entities that are distinct from yet essentially dependent upon their relata, the Bradleyan problem is solved. We are then free to say that tropes in compresence are what make true propositions ostensibly made true by concrete particulars. (shrink)

It has been a common assumption that words are substances that instantiate or have properties. In this paper, I question the assumption that our ontology of words requires posting substances by outlining a bundle theory of words, wherein words are bundles of various sorts of properties (such as semantic, phonetic, orthographic, and grammatical properties). I argue that this view can better account for certain phenomena than substance theories, is ontologically more parsimonious, and coheres with claims in linguistics.

According to the Multiple Realizability Argument, a higher-level property typically has many physical realizers, so it cannot be type-identical to any one of them. This enables the non-reductive physicalist to claim that some higher-level properties are type-distinct from physical properties. The reductive physicalist can counter with the Disjunctive Strategy: nothing prevents us from type-identifying the higher-level property with the disjunction of its realizers. Developing a powers-based ontology of properties, Shoemaker and Wilson present responses to the Disjunctive Strategy, wherein higher-level property (...) instances are token-distinct from their realizers, while instances of disjunctions are not. In this paper, I argue that such responses to the Disjunctive Strategy can be reasonably resisted, either by denying token-distinctness, or by insisting that exhaustively overlapping disjunctive properties are also token-distinct from the realizers. This secures the Disjunctive Strategy as a core component of the argument for reductive physicalism. (shrink)

Robin Le Poidevin (2007) claims that we do form perceptual beliefs regarding order and duration based on our perception of events, but neither order nor duration are by themselves objects of perception. Temporal properties are discernible only when one first perceives their bearers, and temporal relations are discernible only when one first perceives their relata. The epistemic issue remains as to whether or not our perceptual beliefs about order and duration are formed on the causal basis of an event’s objective (...) order and duration. Le Poidevin raises this issue in the form of an epistemological puzzle of time perception, from which he derives the claim that the order and duration of events do not causally contribute to our perceptual beliefs about them. Since his view is motivated by a causal truthmaker principle for grounding knowledge, it also holds that perceptual beliefs about temporal features must be caused by the features themselves in order to count as knowledge. Given these theoretical commitments, there is a puzzle concerning how such perceptual beliefs could constitute knowledge of temporal properties. In response to Le Poidevin, I argue for an account according to which order and duration are objects of perception, causally contribute to our perceptual beliefs about them, and such beliefs are capable of counting as knowledge. I conclude by showing that, on my alternative account, the epistemological puzzle dissolves and his own solution to it fails. (shrink)

In this small book logician and mathematician Jens Erik Fenstad addresses some of the most important foundational questions of linguistics: What should a theory of meaning look like and how might we provide the missing link between meaning theory and our knowledge of how the brain works? The author’s answer is twofold. On the one hand, he suggests that logical semantics in the Montague tradition and other broadly conceived symbolic approaches do not suffice. On the other hand, he does not (...) argue that the logical approach should be discarded; instead, he opts for a methodological pluralism in which symbolic approaches to meaning are combined with geo- metric ones such as Conceptual Spaces [9] and discusses ways in which these geometric accounts could be hooked up with connectionist frameworks and dynamic systems approaches in neurophysiology. (shrink)