The issue of Philosophy of maths came up in a course on ‘Knowledge, language and Logic’ I gave at the technical university where I am based to a group of mostly engineering students. In that course I alternated between Analytic and Continental Philosophy, looking at 14 texts from Frege to Derrida. On one of the Analytic weeks, we were looking at Quine‘s ‘Two Dogmas of Empiricism’ (in From a Logical Point of View) and I got onto the topic of ontological relativity in Quine, with reference to philosophy of maths. In ‘On What there Is’ (also in From a Logical Point of View), Quine mentions three basic position in philosophy of maths as ontological position. Formalism in maths corresponds with Nominalism about names and generalities; Logicism in maths corresponds with Realism about names and generalities; Intuitionism in maths corresponds with Conceptualism about names and generalities.
The question in philosophy of maths is whether numbers, sets, and other abstract mathematical entities exist separately from symbols and from mental concepts. For the Formalist, numbers etc. only exist as symbols manipulated by rules, which corresponds with Nominalist ontology according to which general names group individual things together and do not name any kind of abstract general thing. For the mathematical Logicist, numbers etc. exist outside symbolisation and outside the mind as real abstract things, which corresponds with the Realist ontology according to which general names name an abstraction uniting the individual things coming under that abstraction. For the mathematical Intuitionist, numbers etc. exist as mental constructs, which corresponds with the Conceptualist ontology aaccoring to which general names name a mental construct that unites many individual things.
I presented the three options and asked for a vote from the students. Intutionism/Conceptualism came out first by a long way, with Formalism/Nominalism clearly preferred to Logicism/Realism which was not at all popular. I was surprised because I assumed that they would be knee jerk Realists. I get the impression that the common sense ideology of scientists, including engineers is that scientific laws are true and refer to real objects; and that mathematical laws are true and are about real objects. From what the students said, maths academics may well have that attitude towards maths. They felt it’s an inevitable consequence of being a mathematician, that you believe in the reality of mathematical objects. The engineering students had a much more instrumental attitude towards maths.
I didn’t get onto Instrumentalism, Realism and Conceptualism in science However, we did get onto Thomas Kuhn‘s The Structure of Scientific Revolutions and Michel Foucault‘s Archaeology of Knowledge, which clearly question Realism about scientific laws and theories, and even Realism about the objects of science. Students were much more sympathetic to both than I expected. The relationship with Nominalism and Constructionism, is too big to discuss here. I will just take the opportunity to suggest that we should be careful about assuming that either Kuhn or Foucault were representatives of a branch of Constructionism, know as Social Constructivism, which is how they are often taken. That is they are often taken to believe that scientific laws are social constructs. We might be better off thinking of them as
Nominalists. Foucault’s position over many stages of thought consistently includes a concern with the artifciality of categorisation, as compared with the pure physicality, or certainly unique individuality, of individual things.