Conceptual Analysis Mathematized

Conceptual Analysis Mathematized

Frege mathematized meaning by invoking the concept of a function, the central concept in modern mathematics. Concepts are conceived as functions from objects to truth-values, etc. This enables Frege to give a systematic rigorous account of the workings of language, modelled on mathematical notation. I propose to do the same for conceptual analysis: the analysandum is viewed as a function from the elements constituting the analysans. For example, knowledge is a function from belief, truth, and justification to a propositional component (intuitively, the concept of knowledge viewed as a constituent of thought). There are several ways of constructing the arguments and values of such functions, but the important point is that the function exists: it takes us from the primitive elements of the concept to the concept itself. It operates as a kind of binding function: it unites the several elements into a conceptual whole, glues them together. Thus, the concept is not just a list or conjunction; there is a natural unity. We define the function as second-level because it takes first-level functions as arguments—since belief, truth, and justification are functions from objects to something else (truth-values or states of affairs). The idea is that complex concepts like knowledge are mappings from simpler concepts, themselves mappings from one kind of entity to another (ala Frege). There is a hierarchical structure, representable by a tree diagram, linking complex concepts to simpler concepts. Of course, the concepts used in the analysis will often have analyses of their own, so that the tree will be ramified and many-branched. We could assign a number to its complexity and compare analyses in respect of cardinality (the concept of a game, say, might be twice as complex as the concept of knowledge, computed according to the number of functions in the analysis). A whole proposition (thought) would combine several analyzable concepts (“John knows what a game is”) and this entity would contain the full analysis of all the concepts that compose it. It would be mathematically quite hairy. The structure is familiar from linguistics and descriptions of conceptual constituency; I am merely adding the Fregean apparatus of functions, arguments, and values. We view the basic concepts as inputs to a function that gives as output a complex concept containing the input concepts. Think of the concept-forming function as a binding function: it fuses the elementary concepts into a conceptual whole that is available for cognitive use. Geometrically, it is defined over a space of basic concepts (also functions) that can be assembled into complex wholes; it is not unlike atoms and molecules, or words and sentences, or organs and bodies. We have a kind of mereological mathematics applied to the conceptual domain. Concepts can share parts and hence be available for intersection—one concept may incorporate an element also possessed by another concept. Set theory may thus be brought to bear. Formal methods can be applied to the domain of concepts, because of their compositional structure. We get a mathematics of the conceptual—primitive perhaps but mathematically formulated. It will yield equations like “Knowledge equals the combination of Belief, Truth, and Justification” (K=BTJ), i.e., “The thought-content of the concept of knowledge is the value of a second-level function from first-level functions corresponding to the attributes of Belief, Truth, and Justification”. This should satisfy the ambitions of mathematically inclined analytical philosophers wedded to the project of conceptual analysis.[1]

[1] Nothing says that this is the only way to mathematize conceptual analysis; I offer it as one way to impose mathematical structure on the underlying reality—illustrative not definitive.