And there are far more modern people, of which a greater proportion is producing long-lasting informational stuff like quotes/metaphors/articles/books.
I haven’t run the number, but I expect there is more total “literally value” (whatever that means) produced, say, last year in fanfiction alone, that in all text written pre- industrial revolution. By orders of magnitude.
Depends what you mean by ‘know more math’. In a sense, anybody as fast as you who spent more time studying knows more than you, for how could it be any other way? But that could be because they have more depth in some areas whereas you might have a broader purview. (E.g. you might know more high-level theory about algebraic curves but lose to Newton on a details-oriented question on most cubic curves.)
It takes a lot less time to learn calculus from a textbook than it does to invent it; it would presumably be accurate to say that Einstein knew more physics than Newton. I don’t know if there are any problems in my 3-semester introductory calculus textbook that Newton would have choked on, but he’d definitely have a problem the first time he saw a complex number, let alone something like “e^(a+bi)= a cos(b) + ai sin (b)” that dates to Euler.
Ahhh yeah I forgot discovery was a thing. I guess even going through the process to invent something is its own kind of learning, but that seems tenuous with respect to the original intent of what you said.
Edit: But I think there is a meaningful sense (even if not the only one) in which, say, Euclid or Archimedes probably know more classical geometry than you. And perhaps its meaning comes from their internalisation of a greater depth (even if you could use a theorem prover or theory to quickly derive all their knowledge), which would make those deeper facts accessible to their intuition when solving other problems or developing theory.
For much smaller values of “know”, probably. With google at your disposal, all the math is at your fingertips, but that doesn’t mean you know how to solve a problem which doesn’t come with keywords to search for. Same applies to typical declarative knowledge.
More specifically I mean that Newton couldn’t pass the finals of many of the undergraduate math courses I took, because the math needed to solve the problems wouldn’t have been invented yet.
What I mean is that facing a somewhat difficult problem in applied mathematics which arose naturally in some broader context, most undergraduates are not able to actually pinpoint the relevant methods that they know. (At the same time people like Newton were unusually able to do that). It’s very apparent in e.g. programming contests, that the subset of what people can identify as applicable (without hints) is usually much smaller than the set of what they know.
The “average American teenager” definitely also could not pass those finals.
I rather doubt that there are any math problems at all which an average american teenager could solve and Newton could not, even if they are handpicked to use “recent” math.
I rather doubt that there are any math problems at all which an average american teenager could solve and Newton could not, even if they are handpicked to use “recent” math.
Possible counterexample:
x^2 + 1 = 0
Newton didn’t believe in the square root of minus one.
And I know more math than Newton, too! (Maybe not Euler, but definitely Newton.)
And there are far more modern people, of which a greater proportion is producing long-lasting informational stuff like quotes/metaphors/articles/books.
I haven’t run the number, but I expect there is more total “literally value” (whatever that means) produced, say, last year in fanfiction alone, that in all text written pre- industrial revolution. By orders of magnitude.
Relevant (see also followup)
Depends what you mean by ‘know more math’. In a sense, anybody as fast as you who spent more time studying knows more than you, for how could it be any other way? But that could be because they have more depth in some areas whereas you might have a broader purview. (E.g. you might know more high-level theory about algebraic curves but lose to Newton on a details-oriented question on most cubic curves.)
It takes a lot less time to learn calculus from a textbook than it does to invent it; it would presumably be accurate to say that Einstein knew more physics than Newton. I don’t know if there are any problems in my 3-semester introductory calculus textbook that Newton would have choked on, but he’d definitely have a problem the first time he saw a complex number, let alone something like “e^(a+bi)= a cos(b) + ai sin (b)” that dates to Euler.
Ahhh yeah I forgot discovery was a thing. I guess even going through the process to invent something is its own kind of learning, but that seems tenuous with respect to the original intent of what you said.
Edit: But I think there is a meaningful sense (even if not the only one) in which, say, Euclid or Archimedes probably know more classical geometry than you. And perhaps its meaning comes from their internalisation of a greater depth (even if you could use a theorem prover or theory to quickly derive all their knowledge), which would make those deeper facts accessible to their intuition when solving other problems or developing theory.
For much smaller values of “know”, probably. With google at your disposal, all the math is at your fingertips, but that doesn’t mean you know how to solve a problem which doesn’t come with keywords to search for. Same applies to typical declarative knowledge.
More specifically I mean that Newton couldn’t pass the finals of many of the undergraduate math courses I took, because the math needed to solve the problems wouldn’t have been invented yet.
What I mean is that facing a somewhat difficult problem in applied mathematics which arose naturally in some broader context, most undergraduates are not able to actually pinpoint the relevant methods that they know. (At the same time people like Newton were unusually able to do that). It’s very apparent in e.g. programming contests, that the subset of what people can identify as applicable (without hints) is usually much smaller than the set of what they know.
Indeed. Being able to pass a math test and being able to use the math in a real-world context are two different things.
The “average American teenager” definitely also could not pass those finals.
I rather doubt that there are any math problems at all which an average american teenager could solve and Newton could not, even if they are handpicked to use “recent” math.
Possible counterexample:
x^2 + 1 = 0
Newton didn’t believe in the square root of minus one.