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> Most of the time, it turns out to reduce to group theory I'd be really interested if you could justify this! Sure, group theory has a huge number of applicat

by taejo 14y ago

> Most of the time, it turns out to reduce to group theory I'd be really interested if you could justify this! Sure, group theory has a huge number of applications across mathematics, but to say that it reduces to group theory "most of the time" seems completely implausible.

lmm 14y ago
What I mean is: if you try, starting from nothing, to make up the definition of some new kind of object, and then start proving things about it, most of the time you find that you've defined a group or something that reduces to a group. I don't have a quantitative measure of this (I can't really define a probability distribution for "random objects you think up"), it's just personal experience. Put another way: I find it quite hard to think up a new object that doesn't turn out to just be a group. Have you tried thinking up new mathematical objects? What do you tend to come up with? (Not snarky, I'm genuinely interested - maybe some people's creativity works differently to mine)
- balsam 14y ago
  funny. yesterday i was trying to explain to my friend, a physics grad student like myself, why i wanted to study algebraic geometry. i said, it's because physicists only know about groups, because everything seems to be groups. but rings have more structure than groups and fields have more structure than rings and that's what algebraic geometry is about. this comment is probably too late. you are right that most things turn out to be a group, but not "just a group".
- taejo 14y ago
  > Have you tried thinking up new mathematical objects? Yes, I have. Off the top of my head, the objects I'm interested in are often monoidal, but usually lack inverses. Strings don't have any natural group structure (though as free monoids there is an obvious way to extend them into free groups), nor metric spaces or context-free grammars or finite state machines. Of course groups are ubiquitous, but I think your claims of universality are vastly exaggerated.

by taejo 14y ago

lmm 14y ago

What I mean is: if you try, starting from nothing, to make up the definition of some new kind of object, and then start proving things about it, most of the time you find that you've defined a group or something that reduces to a group. I don't have a quantitative measure of this (I can't really define a probability distribution for "random objects you think up"), it's just personal experience. Put another way: I find it quite hard to think up a new object that doesn't turn out to just be a group. Have you tried thinking up new mathematical objects? What do you tend to come up with? (Not snarky, I'm genuinely interested - maybe some people's creativity works differently to mine)

balsam 14y ago
funny. yesterday i was trying to explain to my friend, a physics grad student like myself, why i wanted to study algebraic geometry. i said, it's because physicists only know about groups, because everything seems to be groups. but rings have more structure than groups and fields have more structure than rings and that's what algebraic geometry is about. this comment is probably too late. you are right that most things turn out to be a group, but not "just a group".
taejo 14y ago
> Have you tried thinking up new mathematical objects? Yes, I have. Off the top of my head, the objects I'm interested in are often monoidal, but usually lack inverses. Strings don't have any natural group structure (though as free monoids there is an obvious way to extend them into free groups), nor metric spaces or context-free grammars or finite state machines. Of course groups are ubiquitous, but I think your claims of universality are vastly exaggerated.