13. Platonism and Wilderness Guides

 


Consider the following: math is a part of nature. That is - even were there no humans in the universe, indeed no life at all, mathematics would still exist; every theorem an undergrad from our world would ever encounter in class would still hold, in this lifeless universe. Consider also that mathematical beauty is real, though in a more limited sense: perhaps uniquely human aesthetics are a key part of what it means for a piece of math to be beautiful, and another species that grew up beneath the light of a different star, under different evolutionary conditions, would find our love of elegant proofs which cast light on aspects of reality and our delight in surprising second-order consequences of natural-seeming assumptions every bit as alien as we might find their sense of beauty in math.

Nonetheless, it then stands to reason that mathematical beauty is just as much a kind of natural beauty as the vastness of the Grand Canyon, or the power and splendor of Victoria Falls, or the haunting colors of the Northern Lights, or the rugged hexagonal basalt columns of Jeju Island, or every total solar eclipse, or even the shimmering wonder of the night sky. Indeed, it is more fundamental to reality than any of those: all the wonders of the world will be lost in mere millions of years, and eclipses are a lucky accident which will cease a mere billion years hence, and even the stars will one day shine their last. Indeed, mathematical beauty is a kind of natural beauty far more permanent than any of these - even after the White Cliffs of Dover have dissolved into the sea, to contemplate particularly nice proofs of the Pythagorean theorem will still bring joy; even after the last solar eclipse, Euler’s identity will still be there to light the path from complex exponentials to rotations; even once the last star has flared and faded and the universe goes dark for good, the Theorema Egregium will still cry out that local curvature measures are enough in sum to understand global curvature.

I sometimes wonder what it must be like not to be able to appreciate this beauty. It’s like watching people from a dark matter universe pass straight through the walls of the Louvre or the stone of Ayers Rock without noticing anything strange, let alone seeing any of it. Some such people passed through my classrooms with the same ghostly mien, back when I still taught math; I fancy that I managed to get at least a few of those to see the beauty that they had once drifted blindly through. Take this frame seriously: what does it mean to be a math teacher? I claim that it means that you’re like a museum curator, or like a trip sitter, or most of all like a wilderness guide - that you lead tours into the safest parts of the vast Platonic wilderness of mathematics, picking formula derivations like sweet berries here and pointing out derivation pitfalls and ravines there; splinting the twisted ankle of a bad midterm or bandaging metaphorical skinned knees and drying quite real tears; hopefully lighting trailheads and blazing trails and not mocking your charges for getting lost in the woods and getting eaten by bears. I always found it much more rewarding to hike them up my favorite trails to show them the diamond mountains glinting in the sun; the actually-infinite pits singing with shattered logic and convergent series; the toroidal rivers.

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” Bertrand Russell once said that, but if lost in the woods on some moonlit night you should come across some creature more native to the wilds of abstraction than to the physical plane, bid it speak, and try to hear some echo of these words in its polite growling, and follow its lantern-light out to safety.

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