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)



War Minus the Shooting Learning from War with Shooting?

In 1945 George Orwell wrote The Sporting Spirit for the Tribune.

In it he observed that:

Nearly all the sports practised nowadays are competitive. You play to win, and the game has little meaning unless you do your utmost to win. On the village green, where you pick up sides and no feeling of local patriotism is involved. it is possible to play simply for the fun and exercise: but as soon as the question of prestige arises, as soon as you feel that you and some larger unit will be disgraced if you lose, the most savage combative instincts are aroused. Anyone who has played even in a school football match knows this. At the international level sport is frankly mimic warfare. But the significant thing is not the behaviour of the players but the attitude of the spectators: and, behind the spectators, of the nations who work themselves into furies over these absurd contests, and seriously believe — at any rate for short periods — that running, jumping and kicking a ball are tests of national virtue.

He added that:

Serious sport has nothing to do with fair play. It is bound up with hatred, jealousy, boastfulness, disregard of all rules and sadistic pleasure in witnessing violence: in other words it is war minus the shooting.

On the day that I revisited George Orwell’s article I received a PhilPapers alert to Philippe Mongin’s paper, A Game-Theoretic Analysis of the Waterloo Campaign and Some Comments on the Analytic Narrative Project.

In his paper Phillippe presents a game-theoretic model of Napoleon’s last campaign, which ended dramatically on 18 June 1815 at Waterloo. It looks in particular at the decision Napoleon made “on 17 June 1815 to detach part of his army against the Prussians he had defeated, though not destroyed, on 16 June at Ligny”.

On page 7 of the paper he proposes a model for:

Napoleon’s all-crucial decision, June 17, 1815, the day after his victory over Blucher at Ligny. That day he chose to send more than a third of his forces, under the command of Grouchy, against the retreating Prussians. All the commentators agree that this division of the French army was the key to Wellington’s victory, June 18 at Waterloo. Grouchy spent the fateful day at Wavre, baited by Blucher’s rear guard, while the advance guard marched unimpeded to join Wellington in the mist of an uncertain battle. The campaign’s greatest question, which involves Napoleon’s rationality, is whether he could have made better use of Grouchy’s detachment. The model we propose to answer this question takes the form of a simple zero-sum game between Napoleon and Blucher. Despite the absence of Grouchy as an autonomous player, it adds precision to the competing hypotheses.

Section 3 of Phillippe’s paper provides extensive detail to support the game theoretic approach. As I worked my way through his model I was fascinated by the interplay of other observations on events.

I finished the paper thinking about many sporting coaches’ interest in Sun Tzu’s The Art of War and how Phillippe might extend their understanding.

I thought too about how knowledge discovery in databases is transforming the support available for real-time decision making.

Whilst contemplating this I thought about the number of announcements being made around the world about additional funds for Olympic programs with ‘medal prospects’. The sporting world George Orwell described has access now to very powerful analysis tools that project medal success. If we can revisit world changing military encounters then it is highly likely we can extend the methodologies from the battle ground to the sport arena.

I think I will retrace my steps with a look at Will Hopkins, John Hawley and Louise Burke’s paper (1999) (Design and analysis of research on sport performance enhancement) and by dusting off my copy of Carl von Clausewitz’s On War.

Photo Credits

6000 Girls at Sokol Sports at Prague, Austria

Waterloo, Belgium