The Aperiodical 

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That which we call an identity

By Andrew Stacey. Posted November 27, 2019

I’m grateful to Jemma Sherwood and Rob Low for reading an early draft of this and for their comments thereon. All opinions are, of course, my own.

This post is inspired by something that I see crop up now and again in discussions with other Maths teachers. It usually manifests itself as a rallying cry to use ≡ in place of = in identities and reserve = for equations. My standard response is to mutter something about identities being equations and leave it at that. But in the latest round, Jemma Sherwood challenged me, in the nicest possible way, to explain a bit further. This is that explanation.

Although I’m going to state my case here, I’m well aware that there are different opinions. In matters of opinion, such as this, agreement and disagreement is less important than that all sides think. So if what I write seems to you wrong, that’s fine so long as it makes you think about why you think that it is wrong.

The neuroscience of mathematical beauty, or, Equation beauty contest!

By Peter Rowlett. Posted February 13, 2014

Neuroscientists Semir Zeki and John Paul Romaya have put mathematicians in an MRI scanner and shown them equations, in an attempt to discover whether mathematical beauty is comparable to the experience derived from great art.

They’ve detailed the results in a paper titled “The experience of mathematical beauty and its neural correlates”. Here’s a bit of the abstract:

We used functional magnetic resonance imaging (fMRI) to image the activity in the brains of 15 mathematicians when they viewed mathematical formulae which they had individually rated as beautiful, indifferent or ugly. Results showed 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 (mOFC), as the experience of beauty derived from other sources.

BBC News puts it: “the same emotional brain centres used to appreciate art were being activated by ‘beautiful’ maths”. This is interesting, according to the authors, because it investigates the emotional response to beauty derived from “a highly intellectual and abstract source”.

As well as the open access paper, the journal website contains a sheet of the sixty mathematical formulae used in the study. Participants were asked to rate each formula on a scale of “-5 (ugly) to +5 (beautiful)”, and then two weeks later to rate each again as simply ‘ugly’, ‘neutral’ or ‘beautiful’ while in a scanner. The results of these ratings are available in an Excel data sheet.

This free access to research data means we can add to the sum total of human knowledge, namely by presenting a roundup of the most beautiful and most ugly equations!