Aristotelianism in the Philosophy of Mathematics

Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of two heights, for example, is a perceivable and measurable real relation between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal. For example, the relation between foliage and being-a-leaf is the number of leaves on a tree,a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity

The Compositional Nature of Verb and Argument Representations in the Human Brain

How does the human brain represent simple compositions of objects, actors,and actions? We had subjects view action sequence videos during neuroimaging (fMRI) sessions and identified lexical descriptions of those videos by decoding (SVM) the brain representations based only on their fMRI activation patterns. As a precursor to this result, we had demonstrated that we could reliably and with high probability decode action labels corresponding to one of six action videos (dig, walk, etc.), again while subjects viewed the action sequence during scanning (fMRI). This result was replicated at two different brain imaging sites with common protocols but different subjects, showing common brain areas, including areas known for episodic memory (PHG, MTL, high level visual pathways, etc.,i.e. the 'what' and 'where' systems, and TPJ, i.e. 'theory of mind'). Given these results, we were also able to successfully show a key aspect of language compositionality based on simultaneous decoding of object class and actor identity. Finally, combining these novel steps in 'brain reading' allowed us to accurately estimate brain representations supporting compositional decoding of a complex event composed of an actor, a verb, a direction, and an object.Comment: 11 pages, 6 figure

Quantity and number

Quantity is the first category that Aristotle lists after substance. It has extraordinary epistemological clarity: "2+2=4" is the model of a self-evident and universally known truth. Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity

Challenging the Computational Metaphor: Implications for How We Think

This paper explores the role of the traditional computational metaphor in our thinking as computer scientists, its influence on epistemological styles, and its implications for our understanding of cognition. It proposes to replace the conventional metaphor--a sequence of steps--with the notion of a community of interacting entities, and examines the ramifications of such a shift on these various ways in which we think

Representation recovers information

Early agreement within cognitive science on the topic of representation has now given way to a combination of positions. Some question the significance of representation in cognition. Others continue to argue in favor, but the case has not been demonstrated in any formal way. The present paper sets out a framework in which the value of representation-use can be mathematically measured, albeit in a broadly sensory context rather than a specifically cognitive one. Key to the approach is the use of Bayesian networks for modeling the distal dimension of sensory processes. More relevant to cognitive science is the theoretical result obtained, which is that a certain type of representational architecture is *necessary* for achievement of sensory efficiency. While exhibiting few of the characteristics of traditional, symbolic encoding, this architecture corresponds quite closely to the forms of embedded representation now being explored in some embedded/embodied approaches. It becomes meaningful to view that type of representation-use as a form of information recovery. A formal basis then exists for viewing representation not so much as the substrate of reasoning and thought, but rather as a general medium for efficient, interpretive processing

Cutting-Edge Equivocation: Conceptual Moves and Rhetorical Strategies in Contemporary Anti-Epistemology"

An examination of conceptually and rhetorically equivocating positions among academic philosophers and other theorists who are sympathetic to constructivist epistemological developments but unwilling to relinquish key aspects of traditional understandings of truth and knowledge and/or anxious to avoid charges of relativism. A major problem with the resulting hybrid formulations is that, seeking, as they often claim, to “steer a course between Scylla and Charybdis” and being composed of essentially incompatible elements, they can do little theoretical work. While the personal-intellectual and professional-institutional pressures leading to such dubious navigations are substantial, so also, it is argued, are their intellectual costs

Is Dretske's Theory of Information Naturalistically Grounded? How emergent communication channels reference an abstracted ontic framework

By bringing together Dretske’s theory of knowledge, Shannon’s theory of information, and the conceptual framework of statistical physics, this paper explores some of the meta-physical challenges posed by a naturalistic notion of semantical information. It is argued that Dretske’s theory cannot be said to be naturalistically grounded in the world described by classical physics and that Dretske information is not consistent with Shannon information. A possible route to reconciling Dretske’s insights with Shannon’s theory is proposed. Along the way, an attempt is made to clarify several points of possible confusion about the relationships between Dretske information, Shannon information and statistical physics