Research Group
in Analytic Philosophy

Abstract

ontemporary theory of illocutionary acts was originally developed inspired by Austin (1962) and further elaborated by Searle (1969, 1975, 1979) as an account of the illocutionary aspects of utterances produced in the concrete uses of language. In particular, this theory searched for a foundational account of the possibility of promises, orders, statements, suggestions, etc., and of the differences between these different acts. It was originally thought as a theory belonging only to the pragmatics and devoted solely to linguistic aspects of human actions. Later it found widespread application in the philosophy of mind, philosophy of law and, more recently, in the foundation of social sciences. However, in the philosophy of mathematics very little attention has been paid to pragmatic phenomena; indeed, pragmatic aspects of mathematical language are almost universally ignored. This is in part understandable. Mathematics is usually seen as the realm of objective truths and truth-functional propositions, and ideally its results are expressed in a purely formalized language. Typically pragmatic phenomena such as implicatures, presuppositions and illocutionary acts are ubiquitous in ordinary language but are far less evident in mathematics. However, this picture overlooks many important (and, in some cases, essential, as we shall argue) aspects of mathematical theories and mathematical practice. It is our working hypothesis that the activity of discovering and proving theorems is impregnated with some illocutionary acts perpetrated by mathematicians (either as a group or individually or through the projection of an “ideal” subject with “ideal” judgments). For instance, they must contain some initial stipulations (definitions, postulates, choice of vocabulary, rules of inference, etc.), and include in its metalanguage typically performative terms (‘therefore’, ‘we conclude’, etc.). These illocutionary acts create a network of what Searle calls “institutional facts” (i.e., non-natural facts) that do not belong originally to the mathematical realm, but interact with that realm and are used as a kind of platform for the study of that realm.

Our working hypothesis should not be understood as a defense of an anti-realist ontology of mathematical entities or propositions. Indeed, as we shall argue, this hypothesis is largely independent of any such ontology. Even if one adopts a strict realist view of mathematical entities, the discovery of these entities and of their structure depends largely on some illucutionary acts. It should also not be confounded with the trivial claim that communication among mathematicians is done in part through natural language and, as such, it is impregnated with illocutionary acts (questions, assertions, promises, praises, invitations, etc.). What we mean is that even at the level of perfectly formalized language there are some essential illocutionary acts as well as some illocutionary force indicators.