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Blows my mind that the proof shows something as tangible as 18 types + all the sporadics. Since I have absolutely no insight into the field, does anyone know i

Blows my mind that the proof shows something as tangible as 18 types + all the sporadics. Since I have absolutely no insight into the field, does anyone know if the 18 is special in any sense? The difference to me between physics and maths is that the latter has the capacity to be fully deducible from the axioms, and in that sense, it's fascinating that somewhere buried in our system of numbers, groups, geometry etc. lies a set of characteristics dictating the existence of those 18 types.

pavpanchekha 13y ago
Abstract algebra has a tendency to reveal integers without explanation. Consider for example the exceptional Lie groups, which is an analogous problem. Lie groups, which characterize continuous groups (as in, groups with a geometry of sorts among the elements of the group), can be broken down into simple Lie groups which can then be classified into 4 infinite families and 5 exceptions. Why five? Well, it is the number of graphs with a certain type of property (Dynkin diagrams). And beyond this, I have no good answer: it just simply is the number. You can write them all down, you can verify that there are no more (this part is hard), but I do not know of any deep insight as to why. I should note that the 26 and the 18 for finite simple groups were not just computed numbers, either. A few attempts at a complete classification were made and failed because they were missing a few cases. If I recall, it was unclear for a time if the Monster group really was the largest sporadic group. I also recall a story I was once told by Andrew Sutherland. He was working on what is called a local-global principle, which would show that if a property is true of elliptic curves at every prime, it is true over the rationals (with some constraints). He found, after several failed proof attempts, that something special happened at the prime 7. Why 7? Well, it is not too large and not too small... A last connection: consider the deterministic linear-time median-finding algorithm. It begins: "break up the elements into groups of 5." In this case, the reason 5 is the size of the groups is simply that the answer must be odd; that 3 doesn't work; and that smaller is better. Small numbers are weird.
- kartikkumar 13y ago
  > Small numbers are weird. Weird and wonderful :). Thanks for all the examples, makes me wanna crack open my abstract algebra book! I decided to go into engineering when I got out of high school, but part of me was leaning towards pure maths. Still feel that yearning when I read things like this. So incredibly fascinating, simple, and yet wildly complex at the same time. I've worked a lot on chaos theory in the context of celestial mechanics and the gravitational 3-body problem (Newtonian). After having done my MSc thesis in the field and nearing the completion of my PhD in it, I still can't get my head around how such an incredibly simple problem can spawn such a wealth of complexity and beauty. Kinda envious of Poincaré when he first mapped out the tremendous complexity and structure underpinning it all. Small numbers rule.

pavpanchekha 13y ago

Abstract algebra has a tendency to reveal integers without explanation. Consider for example the exceptional Lie groups, which is an analogous problem. Lie groups, which characterize continuous groups (as in, groups with a geometry of sorts among the elements of the group), can be broken down into simple Lie groups which can then be classified into 4 infinite families and 5 exceptions. Why five? Well, it is the number of graphs with a certain type of property (Dynkin diagrams). And beyond this, I have no good answer: it just simply is the number. You can write them all down, you can verify that there are no more (this part is hard), but I do not know of any deep insight as to why. I should note that the 26 and the 18 for finite simple groups were not just computed numbers, either. A few attempts at a complete classification were made and failed because they were missing a few cases. If I recall, it was unclear for a time if the Monster group really was the largest sporadic group. I also recall a story I was once told by Andrew Sutherland. He was working on what is called a local-global principle, which would show that if a property is true of elliptic curves at every prime, it is true over the rationals (with some constraints). He found, after several failed proof attempts, that something special happened at the prime 7. Why 7? Well, it is not too large and not too small... A last connection: consider the deterministic linear-time median-finding algorithm. It begins: "break up the elements into groups of 5." In this case, the reason 5 is the size of the groups is simply that the answer must be odd; that 3 doesn't work; and that smaller is better. Small numbers are weird.

kartikkumar 13y ago
> Small numbers are weird. Weird and wonderful :). Thanks for all the examples, makes me wanna crack open my abstract algebra book! I decided to go into engineering when I got out of high school, but part of me was leaning towards pure maths. Still feel that yearning when I read things like this. So incredibly fascinating, simple, and yet wildly complex at the same time. I've worked a lot on chaos theory in the context of celestial mechanics and the gravitational 3-body problem (Newtonian). After having done my MSc thesis in the field and nearing the completion of my PhD in it, I still can't get my head around how such an incredibly simple problem can spawn such a wealth of complexity and beauty. Kinda envious of Poincaré when he first mapped out the tremendous complexity and structure underpinning it all. Small numbers rule.