Philosophy of Mathematics Seminar (Monday - Week 2, MT26)

Abstract: The first use of the axiomatic method in mathematics was Euclid’s axiomatisation of plane geometry.  Despite the importance of Euclid’s Elements over the ensuing 2,300 years, the first subsequent use of the axiomatic method in mathematics was not until the 19th century, and this use was to stipulate a general structure, that of a group, rather than to understand the mathematics of a particular structure, as had been done by Euclid.  The first use of the axiomatic method to understand a particular mathematical structure was by Dedekind, in 1858, published in 1872, in which he axiomatised the continuum of real numbers in terms of his notion of a cut on the rational numbers.  In 1888 Dedekind published Was sind und was sollen die Zahlen?, the title of which can be roughly translated as *On the nature of the natural numbers*, in which he axiomatised the natural numbers in full second-order logic.  I will argue for the correctness of Dedekind’s axiomatisation by appeal to Kreisel’s notion of informal rigour.  I will then focus on the first-order formal system for arithmetic, Peano Arithmetic (as it is inappropriately called), which arises intrinsically from Dedekind’s second-order axiomatisation, and consider the relationship between the first-order and second-order systems.

Registration: If you do not hold a University card which gives access to the upper floors of the Schwarzman Centre, please write to admin@philosophy.ox.ac.uk copied to daniel.isaacson@philosophy.ox.ac.uk at least two working days before a seminar to register your attendance.  

Please note: A notice will be sent out for each talk with the Zoom link.

_{Philosophy of Mathematics Seminar Convenors: Daniel Isaacson and Beau Mount}