Big Ideas on Modernism in Mathematics

Big Ideas is one of my very favourite talking-shops (I mean that in a nice way) amongst some stiff competition so I was delighted to be able to attend April's meeting. I was even more delighted than usual as the topic was "What is Modernism in Mathematics?" and my degree is in Mathematics.

Our guide for the evening was Jeremy Gray, Professor of the History of Mathematics at the Open University, pictured below on the right in the cute mock-Tudor bay window of the upstairs room at The Wheatsheaf in Fitzrovia.

As usual with these things, what follows is a mash-up of what the speaker said, what other people said, what I thought at the time and what I think now when writing it up. It is my summary of the topic based on this meeting but is not a summary of that meeting.

While Mathematics always had been abstract in became more so in the late 19th Century. Previously Mathematics had dealt with understandable thing like numbers and shapes but then mathematicians created new things like Sets and Fields to play with. The Algebra that was developed for numbers (2x+y=5) was extended to these new things.

Numbers themselves became from complicated with the addition of Real Numbers, i.e. numbers that are not fractions.

Euclidean Geometry was challenged and found to be lacking.

Some inherent paradoxes were discovered, e.g. does the set of all sets that do not contain themselves contain itself or not?

Euclidean Geometry starts with some basic axioms that do not need proving because they are true by definition, e.g. it is possible to draw a straight line from any point to any point. First it was shown that Geometry was still consistent if some of these were changed, e.g. parallel lines could meet, and then it was shown that any axiomatic Mathematics has inherent paradoxes.

The outcome of this was multiple domains in Mathematics where everything worked nicely in each one but they were all different and so there was no one single truth.

In trying to understand what a single truth might mean or be we discussed how much of Mathematics would we expect an alien civilisation to share. They would have numbers and shapes but would they also have Sets and Fields? And what could they have that we have not thought of? Could they have solved the one truth problem and, if so, would we be intelligent enough to understand it (you cannot teach a dog French).

The conversations we had spent a lot of time on how this new Mathematics of options was mirrored in the real world (Physics) and in Art.

Quantum Mechanics has shown us that the real world is a lot more complicated than we thought (and more complicated than we yet know). For example, particles are also waves and can be in two places at the same time. Perhaps the real world also has many truths like our Mathematics does.

Art changed around the same time and new rules were created for Impressionism, Cubism, etc. For centuries a portrait only had one rule and that was to be a reasonable likeness then other rules were invented that said that a portrait could be something else. Each school of Art has its own rules and each school is equally valid, there is no one truth for what a portrait should look like now.

It was a very animated, intelligent and thought-provoking discussion and I think we were all a little reluctant to end it after only an hour and a half, though quite a few of us stayed on for a while longer just to make that final comment or ask that final question.

I left understanding more about the multiple truths problem than when I went in (we had touched on this at university) but I think that the Physics and Arts analogies made me more relaxed about it.