Elegance, Simplicity, Beauty

Henri Poincaré, in Science And Method (1908) asks ‘What is it that gives us the feeling of elegance in a solution or a demonstration?’
His answer:

It is the harmony of the different parts, their symmetry, and their happy adjustment; it is, in a word, all that introduces order, all that gives them unity, that enables us to obtain a clear comprehension of the whole as well as of the parts.

I was reminded of this today whilst reading a paper by Semir Zeki and his colleagues in Frontiers in Neuroscience. Their study of the experience of mathematical beauty and its neural correlates concluded that ‘the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbito-frontal cortex, as the experience of beauty derived from other sources.’
James Gallagher wrote about the research for BBC’s Science and Environment page. In it he noted the impact of Euler’s identity. One professor of mathematics, David Percy, said of it “At first you don’t realise the implications it’s a gradual impact, perhaps as you would with a piece of music and then suddenly it becomes amazing as you realise its full potential.”
In the Wikipedia page on Euler’s Identity, Keith Devlin is quoted “Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.”
I have very limited mathematical intelligence but I do appreciate elegance.
Last week, I enjoyed Howard Hamilton’s review of the OptaPro Analytics Forum. He reviewed the nine projects presented at the forum and summarised the the themes that connected them:

• The start of game models in football
• The introduction of player similarity
• The usefulness of subjective data
• The appearance of statistical rigor
• The unease of the professional football analysts

I find it invaluable to have access to this kind of synthesis.
I was thinking about Howard’s observation about statistical rigor (‘What I liked about the talks was that most of the presenters were serious about statistical rigor at some level.  That’s not to say that there were proofs, but the researchers were open and precise about the processes they used to arrive at their results.’) when I came across Martin Eastwood’s Expected Goals For All post.
Martin has been collecting xy co-cordinates for shot locations to investigate the relationship between ‘the probability of scoring a goal and how far away from the goal line the shot is taken from’. His post explores how to model this relationship. One option he shares is a Power function (‘We can do this pretty easily by taking the log of the data, fitting a linear regression against it and plotting this against our non-logged data’). Martin’s model overall gives ‘a root mean square error of 8.2 goals, which seems a pretty reasonable starting point for developing the model further’.
I liked Martin’s willingness to share his thinking openly. Martin is a data scientist with a passion for football. I found his work thanks to a link from Simon Gleave.
Every time I find new expressions of mathematical and statistical insight, I marvel at the biographical influences that enable people to apply disciplined approaches to data. By coincidence, as I was concluding this post, I received a link to Lisa Harvey’s paper, Statistical power calculations reflect our love affair with P-values and hypothesis testing: time for a fundamental change. Lisa’s paper reminded me about Will Hopkins work and his conference summaries.
Thanks to Semir Zeki and his colleagues, I now have an insight into my aesthetic sense of data and analysis. I marvel that each day I have opportunities to exercise this sense. My hope is that a fascination with elegance, simplicity and beauty will help me develop my ‘less is more’ interests in observation and analysis.

Photo Credit

Maths 55/365 (Tim Geers, CC BY-SA 2.0)