On August 2, Mr Harvie of the Mathematics department, addressed the Philosophical Society on the relationship between logic and mathematics.

In order to provoke discussion Mr Harvie maintained a viewpoint he did not completely hold. Mr Harvie maintained that although mathematics is logical in form it does not have its roots in logic.

Over the past hundred years the characteristic method has been the axiomatic one. In this method you postulate axioms and then argue from I hem. But do these axioms refer to anything: are there any entities to which the axioms refer. This led to the idea that mathematics should be expressed in logical terms.

However during the last 50 years working mathematicians have ignored this. They know little of the logical foundations of the subject. A mathematician has experience of a thing and he puts down axioms to make the idea precise. This is to him a satisfactory basis for asserting the axioms. Mathematics has no need for the calculations logicians are trying to force upon them. The clearest example of this is in the work of the "Bourbaki." They believe mathematics is a set of unconnected systems.

In other words logicians have made their own subject and can solve their own problems. The results do not affect mathematicians.

Mr Harvie went on to discuss various specific points arising from logic and mathematics.

Non Constructive Proofs

Intuitionists have held that non-constructive proofs are invalid. However, most mathematics rests on non-constructive theorems. Quoting an example from Stage I mathematics: Euclid's proof of the irrationality of "root two." Root two is assumed to be rational and it is shown that this leads to a contradiction and is therefore wrong. Intuitionists say that something does not exist unless it can be constructed.

Axiom System

Now any axiom system characterises at least two groups. There is more than one model for any set of axioms. Mathematicians are only interested in one system. They ignore the fact that other models are characterised by the axioms.

Mr Harvie finally asserted that mathematics is uncontainable. This allows mathematicians to hope their subject is alive after all. You are free to accept or to reject any axiom. Mathematics is an unbounded subject.

In the ensuing discussion various points inferring a relationship were raised. Professor Hughes said you cannot find the dividing line between logic and mathematics. Doctor Lundy said you can work in both fields at once without difficulty.

However the main point discussed was set theory. It was claimed that up to the "axiom of choice" set theory is logical as it is concerned with class membership. This axiom is objected to on logical not on mathematical grounds. Godel however has pointed out that the use of the axiom of choice makes mathematics no more inconsistent than before. This has done away with a lot of objections to using this axiom.

—D.F