Online research in philosophy

The representation of objects and faces by neurons in the temporal lobe visual cortical areas of primates has the property that the neurons encode relatively independent information in their firing rates. This means that the number of stimuli that can be encoded increases exponentially with the number of neurons in an ensemble. Moreover, the information can be read by receiving neurons that perform just a synaptically weighted sum of the firing rates being received. Some ways in which these representations become grounded in the world are described. The issue of syntactic binding in representations, and of its value for a higher order thought system, is discussed.

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated by Herbrand’s Induction-Axiom Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata.

A number of examples are given of how localist models may incorporate distributed representations, without the types of nonlocal interactions that often render distributed models implausible. The need to analyze the information that is encoded by these representations is also emphasized as a metatheoretical constraint on model plausibility.

Many scientific discoveries have depended on external diagrams or visualizations. Many scientists also report to use an internal mental representation or mental imagery to help them solve problems and reason. How do scientists connect these internal and external representations? We examined working scientists as they worked on external scientific visualizations. We coded the number and type of spatial transformations (mental operations that scientists used on internal or external representations or images) and found that there were a very large number of comparisons, either between different visualizations or between a visualization and the scientists’ internal mental representation. We found that when scientists compared visualization to visualization, the comparisons were based primarily on features. However, when scientists compared a visualization to their mental representation, they were attempting to align the two representations. We suggest that this alignment process is how scientists connect internal and external representations.

Shanker & King (S&K) rightly stress that recent ape language research has important implications for language development and origins. But the evidence does not warrant their conclusion that we can dispense with representations. Indeed, their own discussion of the nature of communication highlights the central role that representations must play in our models of communicative competence, in and out of language.

The notion of schema has been given a major role by Recanati within his conception of primary pragmatic processes, conceived as a type of associative process. I intend to show that Recanati’s considerations on schemata may challenge the relevance theorist’s argument against associative explanations in pragmatics, and support an argument in favor of associative (versus inferential) explanations. More generally, associative relations can be shown to be schematic, that is, they have enough structure to license inferential effects without any appeal to genuine inferential processes. Associative processes are thus able to explain a number of pragmatic and linguistic phenomena which have instead been thought to require specialized inferential processes.

Mature representations of number are built on a core system of numerical representation that connects to spatial representations in the form of a ‘mental number line’. The core number system is functional in early infancy, but little is known about the origins of the mapping of numbers onto space. Here we show that preverbal infants transfer the discrimination of an ordered series of numerosities to the discrimination of an ordered series of line lengths. Moreover, infants construct relationships between individual numbers and line lengths that vary positively, but not between numbers and lengths that vary inversely. These findings provide evidence for an early developing predisposition to relate representations of numerical magnitude and spatial length. A central foundation of mathematics, science and technology therefore emerges prior to experience with language, symbol systems, or measurement devices.

Greg Chaitin is one good mathematician who also seems to be a good human being. I will be saying more about that as this review moves along. His love of math, and of humanity, comes through crystal clear — even more clear than in his previous book of interviews, Conversations with a Mathematician (which I reviewed for the Monthly) — whether he is writing about "his" math, the math of others, or not (ostensibly) about math at all. When he tells us, at the end, that this is really "a book on philosophy, not just a math book", and when, throughout the book, he talks about the philosophy of math, he means "philosophy" in the sense of "philo" — meaning love — love of math, love of humanity, and love of life and existence.