This paper seeks to demonstrate how Cournot, in 1843, sought to respond to the foundational crisis then looming in the calculus of probabilities, by elevating the latter to the status of a theory in the realm of pure mathematics, while making a distinction between the subjective and objective meanings of probability, so as to give a measure of the scope and range of its applications. This leads on to a reappraisal of the representation proffered by Cournot, of mathematics and of their relationships to reality, and thus to elucidation of the connection between his probabilistic programme and his philosophy of mathematics.