Math in high school is primarily about memorizing and applying set recipes for problems. Math at (a serious) college level has a large proof-theoretic component: prove theorems not solve problems. Math research still involves solving problems, and proving theorems but it has a novel dimension: stating conjectures & theorem, and most importantly the search for the ‘right’ definitions.
If math in high school is like playing a game according to a set of rules, math in college is like devising optimal strategies within the confines of the rules of the game [actually this is more than an analogy!] than math research involves not just playing the game and finding the optimal strategy but coming up with novel games, with well-chosen rules that are simulataneously ‘simple & elegant’ yet produce ‘interesting, complex, beautiful’ behaviour.
Also, a good definition does not betray all the definitions that one could try but that didn’t make it. To truly appreciate why a definition is “mathematically righteous” is not so straightforward.
I thought not cuz i didn’t see why that’d be desideratum. You mean a good definition is so canonical that when you read it you don’t even consider other formulations?
Math research as Game Design
Math in high school is primarily about memorizing and applying set recipes for problems. Math at (a serious) college level has a large proof-theoretic component: prove theorems not solve problems. Math research still involves solving problems, and proving theorems but it has a novel dimension: stating conjectures & theorem, and most importantly the search for the ‘right’ definitions.
If math in high school is like playing a game according to a set of rules, math in college is like devising optimal strategies within the confines of the rules of the game [actually this is more than an analogy!] than math research involves not just playing the game and finding the optimal strategy but coming up with novel games, with well-chosen rules that are simulataneously ‘simple & elegant’ yet produce ‘interesting, complex, beautiful’ behaviour.
Seems like choosing the definitions is the important skill, since in real life you don’t usually have a helpful buddy saying “hey this is a graph”
Hah! Yes.
Also, a good definition does not betray all the definitions that one could try but that didn’t make it. To truly appreciate why a definition is “mathematically righteous” is not so straightforward.
‘Betray’ in the sense of contradicting/violating?
Hah no ‘betray’ in its less-used meaning as
unintentionally reveal; be evidence of.
“she drew a deep breath that betrayed her indignation”
I thought not cuz i didn’t see why that’d be desideratum. You mean a good definition is so canonical that when you read it you don’t even consider other formulations?