Mathematical reasoning is known to be rigidly structured in premises and conclusions that, to avoid infinite regress, lay down a set of axioms and postulates as starting points. In this talk, we propose to analyze the function that axioms and postulates play in structuring mathematics, using the fruitful perspective of the philosophy of language. Building on the preliminary work of Ruffino, San Mauro and Venturi (2020), we extend the application of Speech Act Theory to Mathematics, explaining the type of illocutionary forces in play with axioms and postulates in the mathematical language.
We start historically, by pointing out the different developments of axiomatic theories from such perspective. One can find substantial differences in the axiomatic methods by reading the different illocutionary forces in play, from Euclid's directive postulates to Frege's assertive axioms, and Hilbert's later fusion between axioms and definitions. We argue that this reading can also highlight the nineteenth-century triumph of mathematical ontology, as mathematics became objectual, rather than procedural.
After the historical excursion, we aim at a more recent proposal: Kit Fine's Postulationism. Fine argues that one can lay down axioms by taking postulates as imperatival, as means for introducing new entities. We evaluate Fine's proposal from the Speech Act Theory perspective, arguing that his postulates are not imperatives, but declaratives in disguise. In other terms, we claim that Fine is Hilbert's descendant, rather than Euclid's.
We conclude by proposing our own reading, that in laying down axioms and postulates one can find both assertive and declarative illocutionary forces. This not only shows a hybrid type of Speech Acts related to them but also explains their current linguistic use as consistent with the historical, mainly Hilbertian, picture of modern mathematics. (Joint work with Giorgio Venturi)