#im usually aware enough to know but dedinitely not always this kind of thinking dominates by default thought patterns
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I'd like to add a bullet (number?): what does it mean to actually prove something
Still similar and closely related to (1), but I don't think it's quite the same!
The two most important things (higher) math teaches you, IMO, are
how to think carefully about the assumptions of an argument,
that trivial examples of an object are still examples of that object,
where "how to properly define your terms" is an alter-ego of (1).
#re: the skeptical comment#i am extremely certain that these are things that higher math are *suppose* to teach you to do well#but does that mean taking some higher math classes (or even learning higher math concepts) mean you definitely gained these skills?#no not really#its not that hard to get a whole ass degree in something while missing some of the key tenants of the area unfortunately#math is kinda on the harder side to do that all in all? you can do it at the undergrad level fairly easily at a lot of universities though#more difficult to get a grad degree without at least mostly learning them#also by higher math I mean strictly beyond intro linear algebra and diff eqs#which i think counts among the intermediate math classes#also i think what can happen with some frequency is mathematicians dont know when to apply the same type of thinking#to situations outside their studies and when and how to modify their thinking (or just not use it)#im usually aware enough to know but dedinitely not always this kind of thinking dominates by default thought patterns
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