Cool Logic

Sam Vincent Adam-Day (ILLC)

Non-mathematical Content by Mathematical Means

November 9th at 19:00, in F1.15

Never worried about the validity of using mathematics in philosophical arguments about mathematics? Well, here's your chance! I will talk about the use of mathematical results in arriving at conclusions with some non-mathematical content, with the view that in general such usage requires a little justification. The Continuum Hypothesis states that the size of the set of all real numbers is the next biggest infinite cardinality after that of the set of all natural numbers. Kurt Gödel and Paul Cohen famously showed that this statement is not decided one way or the other by the first-order axioms of ZFC set theory. However, Georg Kreisel argued in 1965 that the Continuum Hypothesis is in fact determined by the axioms. To do so he made use of a technical result concerning their second-order versions. I will look at Thomas Weston's 1976 reply, and on the basis of this offer Kreisel's argument as a warning example against the unsupported use of mathematical results in philosophical arguments. After this, and in a more positive mood, I will take a particular perspective which very much embraces the distinction between mathematical and non-mathematical content, and from it highlight one route we might take to justify the employment of mathematical results. I will consider one particularly strong form this justification might take, using the ideas of Kurt Gödel and Juliette Kennedy on 'formalism independence'.