Intuitionist Engineering Students. What Engineers Really Think about Philosophy of Maths

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.

Foucault and the Body; or Why Foucault is not a Post-Modern Social Constructionist

I’m following up the last post on Derrida and Nietzsche with a briefer post on Foucault. As I emphasised in the last post, neither Foucault nor Derrida can be reduced to a cliché of ‘Post-Modern’ social constructionism which excludes the body as natural object.

Like Derrida, Foucault never sailed under the flag of ‘Post-Modernism’, or post-ism of any kind. They are rather different cases, and though Derrida was Foucault’s student at the Ecole Normale Supérieure, they appear to have had a long term falling out. This seems to have been more on Foucault’s side than Derrida’s. Despite Foucault’s expemplary qualities as thinker and libertarian social activist (future posts will return to the topic of Foucault and Libertarianism), he does seem to have been more prickly than Derrida. The prickliness seems to go all the way back to Derrida’s 1963 paper, ‘Cogito and Madness’ (collected in Writing and Difference), which is a critical but appreciative discussion of Foucault’s Madness and Civilisation.

The stereotypical view of Foucault circulated by his ‘Analytical’ philosophical critics and his ‘Post-Modern’ fans (more typically to be found in humanities and social science departments other than philosophy) is that he was a relativist who denied the existence of truth or objective, knowledge, and that he had a related assumption according to which reality only exists as a discursive social construction serving power interests of some kind. Something similar to that ‘Post-Modern’ interpretation of Foucault is also widespread in interpretation of Thomas Kuhn, a leaidng figure in Philosophy of Science in the Analytic tradition. Again very different cases, but there is no more reason to think of Foucault as a Post-Modernist than there is to think of Kuhn in that way.

There are many issues to be explored in future, but just one for today. Just a remark that one of Foucault’s most widely read books, Discipline and Punish, does refer to the discourses of power/knowledge, but it also refers to discourse as what affects the body. There is something pre-discursive in Foucault, the body. There is no ‘body’ or ‘nature’ we can identify from outside discourse for Foucault, but physicality and natural forces are there. His view of power/knowledge is just as much an attempt to think of social relations in terms of natural forces, as a discourse centred theory. The body is experienced in social and discursive contexts, but is not eliminated as a body. It is the body where there is resistance to power.