Abstract:

Many people are not familiar with the theory of polytonality, nor can they read sheet music. Yet nearly everyone can enjoy the beauty of music. In the same view, most people have not been trained to interpret mathematical proofs. So, how does one appreciate the product of mathematics?

Blog:

Many people are not familiar with the theory of polytonality, nor can they read sheet music. Yet nearly everyone can enjoy the beauty of music. In the same view, most people have not been trained to interpret mathematical proofs. So, how does one appreciate the product of mathematics? Indeed, what do pure mathematicians do and why is it worthwhile? One may think that ultimately the work of a pure mathematician is proving esoteric statements and solving problems. From some perspective, this may be true. However, I believe it misses a bigger point. Mathematics is very much a human endeavour, and its supreme outcome is clarity of thought and understanding. A particularly good analogy for this is offered by David Spivak. In his view, different mathematical disciplines are like accounting systems. These systems have their own internal language, ways of tracking important objects, classifying the relationships between different objects and aggregating information. Developing a good accounting system of some phenomena both clarifies and augments our understanding.

In my project, I wanted to understand a particular example of an accounting system for dynamical systems. By this, we mean some collection of things — perhaps, they are particles, organisations or even people — with an associated notion of change or evolution through time. There is a lot to unpack, but let’s briefly describe the central object of this accounting system – a polynomial functor. In high school, we encountered polynomials like y^2 + y + 1. A polynomial functor behaves similarly to this, except we replace the exponents with sets. Moreover, instead of plugging in a number and getting back a number, we plug in a set and get back a set. Precise details aside, it turns out that this abstraction is a good way to organise and describe systems. How do we get the dynamics? This comes from specifying the relationships (morphisms) between these polynomial functors.

If you want to get a handle on some of the underlying mathematics and are not familiar with category theory, I encourage you to check out https://www.math3ma.com/blog/what-is-category-theory-anyway

For an interesting perspective on mathematics more generally, I recommend William Thurston’s essay ‘On Proof and Progress in Mathematics’ https://arxiv.org/pdf/math/9404236.pdf

Finally, to see David Spivak give his accounting analogy, I encourage you to watch https://www.youtube.com/watch?v=GswIAurBIYY

James Kelly
The University of Melbourne