Space and elegance

This interesting post by Keith Lyons about Mathematical elegance inspired me to collate some sources that have intrigued me for quite a while.

1. In “Aesthetics as Pre-linguistic Knowledge” (unfortunately behind a pay wall), T. W. Allan Whitfield

‘advances the notion of aesthetics as pre-linguistic cognition, as a form of “knowing” that preceded the evolution of language. It is contended that the function of aesthetics is to elaborate the categories by which we understand the world, by attaching emotion to sensory perceptions.’

and I think that this plausibly explains the basics of the fascination with some number visualizations and mathematical proofs.

2. Keith Lyons cites Poincaré who also wrote about intuition  (French and German here) which is a similar intriguing term. The word is much more difficult to understand than the  matters which it usually refers to. In software usability, everybody uses it confidently. Its  etymology (“watch over” + “into”) suggests a promising combination of (“right-brained”) overview and (“left-brained”) diving into detail, which some might call “visual thinking”. The composition “counter-intuitive” seems to be a fixed term that has nothing more to do with intuition: In Jenny Mackness’ last 5  usages, it is even applied to unexpected insights which, IMHO, are much less abstract and disembodied than the expected opposites. And the forums discussions on the LEO.org dictionary don’t find any good translation for it.

Most confusingly, for Poncaré’s essay about mathematical intuition, a German translation was used (“Anschauung”) which amounts to visualization, illustration, exemplification, which does not at all match mathematical intuition, although the translation was certainly approved by a mathematician (the translator’s father). Interestingly, the essay describes the difference between two sorts of mathematicians:

“Some are influenced by logic and advance step by step, […] while others let themselves be guided by intuition and ‘at a stroke’ make quick conquests”. =  “Les uns sont avant tout préoccupés de la logique […] ils n’ont avancé que pas à pas […] Les autres se laisser guider par l’intuition et font du premier coup des conquêtes rapides”.

This reminds of Dyson’s  distinction of frogs and birds — and again we have the overview, or bird’s perspective.

3. Finally, Keith Harmon’s reflection about space and soccer (in his exchange with Jenny Mackness) is IMHO related, too, since it also discusses aspects of spatial visualization that one might call “right-brained”, in a context, soccer,  where you would not expect it because this is usually associated with forcefully goal-directed, “left-brained”, behavior. I have long suspected that the unique fascination of soccer has to do with the simultaneous strong engagement of both the brain’s visual overview mode, and its mode of goal-directed pursuing. I don’t dare to speculate about the difference between watching a game from an above perspective (on the couch with the beer and engaging one’s mirror neurones), vs. seeing it from the game field as a player, let alone speculate about possible gender-specific differences in such spatial interest. But I do think elegance might have to do with this spatial view.

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