Proofs really should give us insight into why statements are true, and this one really doesn't. That's not to knock on the accomplishment: this clearly took a lot of effort, some very smart work, and some clever use of group theory and very skilled programming.
This comment reminds me that it's not what you have, it's what you do with it. Sometimes you hear about an athlete that he or she has "an extra gear" in the heat of battle. I went to school with a lot of smart people. The median smart person would sometimes make a lazy statement of sentiment such as this one that would never have passed the lips of my classmates with the hard-baked intellectual edge. Hard-baked was part talent, but mostly attitude: people who just thought that the lazy use of "should" was beneath their level of intellectual determination (as it should be, in my personal opinion).
Obviously the landmark results in mathematics are the ones which forge a deep connection between branches of mathematics formerly distinct. Every proof should be one of those. Or at least that's how the coke addict would phrase it. Mathematics as Willy Wonka's chocolate factory. Who needs peas? No candy cane construction permitted by the Chocolate Port Authority if less intriguing that Dessin d'enfant [wikipedia.org].
This discovery, which is technically so simple, made a very strong impression on me, and it represents a decisive turning point in the course of my reflections, a shift in particular of my centre of interest in mathematics, which suddenly found itself strongly focussed. I do not believe that a mathematical fact has ever struck me quite so strongly as this one, nor had a comparable psychological impact. This is surely because of the very familiar, non-technical nature of the objects considered, of which any child?(TM)s drawing scrawled on a bit of paper (at least if the drawing is made without lifting the pencil) gives a perfectly explicit example. To such a dessin we find associated subtle arithmetic invariants, which are completely turned topsy-turvy as soon as we add one more stroke.
I arrived at this page yesterday evening beginning my tour with a question about the provability of reachable states, the mechanism of temporal logic, Zermelo's contribution to set theory, the Hilbert epsilon operator, the Bourbaki group (before Sheldon Cooper there was Jean Dieudonne), and finally to Grothendieck. I have a fairly clear recollection of reading a long piece about Grothendieck several years ago which lamented the loss to mathematics when he devoted the bulk of his career to elaborating a program in algebraic geometry instead of cracking one hard problem after another, which it seemed some people thought he could do. He was regarded by some as much too brilliant for the pedestrian task of assembling an overarching synthesis.
All mathematicians should be more like Grothendieck should have been. Doesn't that sentiment become quickly cloying once you engage the mental clutch?
A year ago another tour took me to Knuth's algorithm of dancing links, which I compiled out of curiosity, then modified the decision step with the next most obvious heuristic. I was interested to watch the famous dancing links during a back-tracking step, so I searched the internet for a famously hard Sudoku example, found one, then single-stepped through the solution process in the debugger. I was disappointed: it reached solution without once backtracking. I think it made three guesses in total, either binary or trinary. I vaguely recall the odds of it guessing correctly all the way to solution was about ten to one. I loaded some other hard problems. On these it actually backtracked from time to time, but not as often I would have presumed. Even hard problems fall quickly to structured guess-work. It's only when you map Sudoku into a logic inference framework that hard problems are hard.
In the Kolmo
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