The Bare Beauty of Mathematics

From my reading, Edna St. Vincent Millay seems to be endorsing a Platonic view of beauty; that is, she seems to be endorsing beauty as an unchanging, timeless abstract ideal which real, concrete things can approximate but never fully attain.

The fact that she capitalizes Beauty is one suggesting that she views it as a Platonic Form, but for me the stronger evidence comes from her use of the metaphor of light and luminosity. In Plato’s Allegory of the Cave, the sun represents the truest things, namely the Forms. St. Vincent Millay speaks of “light anatomized”, which here I interpret to mean that Euclid has perceived the true Form of Beauty, and ‘anatomized’ it by producing his Elements and the definitions, proofs, and propositions contained therein. St. Vincent Millay also speaks of how “heroes seek release From dusty bondage into luminous air”, which again can be interpreted as a reference to those people who are chained in Plato’s cave (and hence stuck in ignorance about the true nature of things), but seek to go aboveground in order to perceive the sun (and become aware of the Forms).  

Further, I think the description of Beauty as ‘bare’ has a double-meaning. First, it signals that Euclid has been able to look upon the Form of Beauty directly, instead of a particular embodiment of Beauty into a physical object. Second, it describes an aspect of the Form of Beauty, namely simplicity and clarity. 

I don’t have much experience with Euclidean geometry (at least, in its axiomatic form), but I think that there are both material and aesthetic reasons for Euclidean geometry remaining so popular for so long.  

First, Euclid was working in Alexandria, the scholarly capital of Europe (really, just Greece) and the Near/Middle East at the time. Being a prolific mathematician (or at least, a prolific mathematics teacher) in a city as intellectually and culturally significant as Alexandria probably allowed the Elements to circulate fairly widely in a relatively short window of time after it was written. If, as is claimed in the brief biography of Euclid that we read, much of the work in the Elements is simply a clarification and refinement of work that already existed at the time, then this might be another reason for its proliferation: it summarized, in a relatively simple and clear fashion, the current understanding of mathematics in Greece at the time, and so may have been used by anyone in the region who was trying to learn math. 

The discovery of Euclidean material on Elephantine Island give evidence towards this hypothesis. Located roughly 1000km from Alexandria, the fragments contain material which corresponds to book 13 of the Elements, but worked through in an order that does not follow the text of the Elements. As such, it appears that these fragments may have been evidence of somebody trying to learn mathematics from the Elements, and doing so a great distance from Alexandria.

Of course, there are probably aesthetic reasons why Euclid’s work remained popular as well. As Edna St. Vincent Millay suggests in her poem, Euclid’s Elements captures the sort of spare, cold, unchanging Beauty that is common to many great works of mathematics. Many of the mathematicians I’ve met speak of the Beauty of mathematics as a key reason why they entered and stayed in the field, and it could be reasonably expected that mathematicians throughout history thought likewise, and held Euclid’s Elements to be a paragon of such mathematical Beauty.


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