Can a Philosopher Let Herself Trust? A History Boy, 07/02/2018

After a very intense post yesterday, I want to relax a little bit. So I’m going to talk about the philosophical implications or mutual expressions of Riemannian geometry.

That's a serious joke. See, I’ve studied mathematics as someone learning about mathematics. So I know how to follow and read equations, I can see what a given operation is doing in a particular step, I can look back at the whole thing and have a decent idea of how you got from the beginning to end.

Here's how Deleuze described how Riemann changed the way people
did geometry. How he shook it to its core. Geometry had traditionally
been about the construction of different shapes. But Riemann figured
out a way to study mathematically space itself, how to construct
different spaces as well as different shapes, to see how different
shapes would behave in different spaces. A whole new axis of
complexity in geometrical possibility.

If I read the history or philosophy of mathematics, I can understand all the concepts fairly easily. I have to think a bit about it, but as long as the writer is reasonably competent, I can understand what they’re talking about.

It’s like being able to read a language, but not speak it.

I know enough mathematics to know that I’d be a terrible mathematician. When it comes to mathematics, I rely on colleagues I can trust. I run an idea by them – sometimes on a space like this blog. I ask them to comment, give a few thoughts. Tell me if I’m on the right track – definitely if I’m missing something.

I feel like university-trained researchers – I’m talking especially about the humanities and social sciences – get trapped in some paradoxes. Quite a lot of them. This one is mistaking a need to build expertise for your job, for expertise being the necessary condition of saying anything at all.

For instance, my case. If I were at a research conference, and there was a talk on some really interesting ideas in philosophy of mathematics in an otherwise lacklustre timeslot, I’d probably go. I’m pretty sure I have gone.

Sometimes – not all the time – folks would react with astonishment that I’d go see a talk in a field in which I wasn’t some level of expert. I do political theory, ethics, European thought, philosophy of ecology and biology.* Why would I go to a math theory talk?

* The people who know philosophy of ecology and biology would know that you have to understand mathematics at least as well as I do. But anyway.

There is a solitude of space
A solitude of sea
A solitude of death, but these
Society shall be
Compared with that profounder site
That polar privacy
A soul admitted to itself—
Finite infinity

My best friends in the university sector would, if that slot was lacklustre for them too, consider coming to the math theory talk with me. But others have been incredulous. As if the only thing you were allowed to be curious about anymore was what you were already an expert in.

Gilles Deleuze was a much better mathematician than I am. He studied mathematics well enough to teach it at the secondary level in France, which is more than I ever managed.

Study of Gottfried Leibniz’s work on differential calculus and Bernhard Riemann’s heterogeneous geometry helped Deleuze develop concepts of change and variety as fundamental principles of existence.

So when I read an article by someone who’s themselves become an expert in philosophy of mathematics and Gilles Deleuze’s philosophy, I’ll learn from what they say. Same way I learn from anyone who’s become an expert in what I want to learn about.

I read about how Deleuze interpreted Riemann’s work as tracing the mathematical structures of heterogeneous spaces – spaces that have entirely different properties, but are contiguous, that flow into each other. Objects can travel casually among all these spaces – ordinary as a walk between neighbourhoods – and be unaltered.

They just tumble differently. Into a valley, up a hill, around a bend, around several bends that seem like straight paths when you walk through them. Whatever. Doo bee doo.

Reality seen topologically. Perfectly legitimate – simply not quite what most of us are accustomed to. Let’s explore what makes us uncomfortable. And rely on people we trust – in their expertise and their character – to be our guides so we don’t lose ourselves.