Online research in philosophy

Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called "internalization through a universal" based on universal mapping properties. A recently developed "heteromorphic" theory of adjoint functors allows the concepts to be more easily applied empirically. This suggests a conceptual structure, albeit abstract, to model a range of selectionist mechanisms (e.g., in evolution and in the immune system). Closely related to adjoints is the notion of a "brain functor" which abstractly models structures of cognition and action (e.g., the generative grammar view of language).

In this paper, we offer a Piagetian perspective on the construction of the logico-mathematical schemas which embody our knowledge of logic and mathematics. Logico-mathematical entities are tied to the subject's activities, yet are so constructed by reflective abstraction that they result from sensorimotor experience only via the construction of intermediate schemas of increasing abstraction. The axiom set does not exhaust the cognitive structure (schema network) which the mathematician thus acquires. We thus view truth not as something to be defined within the closed world of a formal system but rather in terms of the schema network within which the formal system is embedded. We differ from Piaget in that we see mathematical knowledge as based on social processes of mutual verification which provide an external drive to any necessary dynamic of reflective abstraction within the individual. From this perspective, we argue that axiom schemas tied to a preferred interpretation may provide a necessary intermediate stage of reflective abstraction en route to acquisition of the ability to use formal systems in abstracto.

The purpose of the paper is twofold. I first outline a philosophical theory of concepts based on conceptual role semantics. This approach is explicitly intended as a framework for the study and explanation of conceptual change in science. Then I point to the close similarities between this philosophical framework and the theory theory of concepts, suggesting that a convergence between psychological and philosophical approaches to concepts is possible. An underlying theme is to stress that using a non-atomist account of concepts is crucial for the successful study of conceptual development and change.

I show, against contemporary experimentalist criticism, that conceptual analysis has epistemic value. Its structure encourages the development of new and interesting hypotheses which are of the right form to be valuable in diverse areas of philosophy and scholarship. I show, by analysis of the Gettier programme, that conceptual analysis shares the proofs and refutations form Lakatos identified in the development of informal mathematical theorising. Upon discovery of a counterexample, this structure aids the search for a replacement. The search is guided by heuristics which constrain the potential replacements to those with the right form. The heuristics used by conceptual analysis are very similar to those used in other interesting areas of scholarship and so hypotheses generated by it are of the right form to be applicable to diverse areas. I show that the explanationist criterion in epistemology was developed and applied in just this Lakatosian way. This part of conceptual analysis' epistemic value is oblique because it is not in contributions towards the main purpose of conceptual analysis, the reduction of complex concepts to simpler, but towards the reliable development of epistemically-valuable hypotheses in philosophy and scholarship.

Carruthers labels as “too strong” the thesis that language is necessary for all conceptual thought. Languageless creatures certainly do think, but when we get clear about what is meant by “conceptual thought,” it appears doubtful that conceptual thought is possible without language.

The perennial fascination with the relationship between language and thought has generated much research across various disciplines. In recent years, commentators have called for closer examination of the connection between language acquisition and conceptual development (Bowerman & Levinson, 2001). Rather than assuming that language development always presupposes cognitive development, several researchers have started considering whether language learning could transform conceptual structure by making certain concepts available to the learner (e.g., de Villiers & Pyers, 1997; Gopnik & Choi, 1995; Bowerman, 1996).

This essay continues my investigation of `syntactic semantics': the theory that, pace Searle's Chinese-Room Argument, syntax does suffice for semantics (in particular, for the semantics needed for a computational cognitive theory of natural-language understanding). Here, I argue that syntactic semantics (which is internal and first-person) is what has been called a conceptual-role semantics: The meaning of any expression is the role that it plays in the complete system of expressions. Such a `narrow', conceptual-role semantics is the appropriate sort of semantics to account (from an `internal', or first-person perspective) for how a cognitive agent understands language. Some have argued for the primacy of external, or `wide', semantics, while others have argued for a two-factor analysis. But, although two factors can be specified–-one internal and first-person, the other only specifiable in an external, third-person way–-only the internal, first-person one is needed for understanding how someone understands. A truth-conditional semantics can still be provided, but only from a third-person perspective.

We describe a knowledge representation and inference formalism, based on an intensional propositional semantic network, in which variables are structures terms consisting of quantifier, type, and other information. This has three important consequences for natural language processing. First, this leads to an extended, more natural formalism whose use and representations are consistent with the use of variables in natural language in two ways: the structure of representations mirrors the structure of the language and allows re-use phenomena such as pronouns and ellipsis. Second, the formalism allows the specification of description subsumption as a partial ordering on related concepts (variable nodes in a semantic network) that relates more general concepts to more specific instances of that concept, as is done in language. Finally, this structured variable representation simplifies the resolution of some representational difficulties with certain classes of natural language sentences, namely, donkey sentences and sentences involving branching quantifiers. The implementation of this formalism is called ANALOG (A NAtural LOGIC) and its utility for natural language processing tasks is illustrated.

The question of how concepts are formed was central for positivist and operationalist philosophers concerned to root scientific thought directly in experience. Although the positivist program has been abandoned, the current interest in the philosophy of scientific discovery shows the need for a theory of conceptual development. This paper offers a theory of how new concepts can arise, not by abstraction from experience or by definition, but by conceptual combination. Such combination produces a new concept as a non-linear, non-definitional amalgam of existing concepts. After proposing rules intended to account for a variety of mundane cases of conceptual combination, the paper presents several illustrations of how scientific concepts have arisen through combination.

Feature-based theories of concept formation face two dilemmas. First, for many natural concepts, it is hard to see how the concepts of the features could be developmentally more basic. Second, concept formation must be guided by “abstraction heuristics,” but these can be neither universal principles of rational thought nor natural conventions.