Varieties of Mathematical Understanding(ams.org)
ams.org
Varieties of Mathematical Understanding
https://www.ams.org/journals/bull/2022-59-01/S0273-0979-2021-01726-5/viewer/
4 comments
I'm not convinced by the way the author regards 'complexity' and 'depth'
What the author calls complexity I would call "difficulty", because I find (or maybe hope?) that (structural) mathematical concepts which provide depth are in the end simple objects but (and this is the source of the difficulty) incredibly abstract.
As I understand, this stems from the fact that (as the author says) these notions are reduced from many many different contexts (hence mathematical results are also the most broadly applicable) which compounds the difficulty in learning them.
What's clear to me (and also the author) is that computers are responsible for changing the nature of mathematics. IMO computer science is no more a science than mathematics themselves.
Furthermore, as I understand (up to now) during the 19th century, (and because "as the subject evolves, it has a prodigious capacity for reinterpreting its past") mathematics reinterpreted logic out of itself. And logic became a field withing the philosophy of math (this was enabled by the axiomatic methodology).
> The iconic trope of a mathematician standing before a blackboard covered with symbols and arrows may be giving way to the image of a mathematician at a keyboard, coaxing mathematical understanding from calculation, simulation, and search. That should give us something to think about.
This reinforces a thought that the mathematics up until the advent of the computer were build on paper (consider the fact that mathematicians use pencils or blackboards which are easy to erase) but now, after some 10k years of writing we now have something that truly revolutionizes the reach of the writing (taken as technology)
What the author calls complexity I would call "difficulty", because I find (or maybe hope?) that (structural) mathematical concepts which provide depth are in the end simple objects but (and this is the source of the difficulty) incredibly abstract.
As I understand, this stems from the fact that (as the author says) these notions are reduced from many many different contexts (hence mathematical results are also the most broadly applicable) which compounds the difficulty in learning them.
What's clear to me (and also the author) is that computers are responsible for changing the nature of mathematics. IMO computer science is no more a science than mathematics themselves.
Furthermore, as I understand (up to now) during the 19th century, (and because "as the subject evolves, it has a prodigious capacity for reinterpreting its past") mathematics reinterpreted logic out of itself. And logic became a field withing the philosophy of math (this was enabled by the axiomatic methodology).
> The iconic trope of a mathematician standing before a blackboard covered with symbols and arrows may be giving way to the image of a mathematician at a keyboard, coaxing mathematical understanding from calculation, simulation, and search. That should give us something to think about.
This reinforces a thought that the mathematics up until the advent of the computer were build on paper (consider the fact that mathematicians use pencils or blackboards which are easy to erase) but now, after some 10k years of writing we now have something that truly revolutionizes the reach of the writing (taken as technology)
I happen to have started reading Varieties of Religious Experience within the last hour. Is this title a nod to that book or is it just a coincidence (or coinkydink as my mom says)?
I wouldn't be surprised if it is, article is a nice blend of philosophical and pragmatic. Though there are also several different mathematical objects called varieties (oddly enough algebraic varieties are completely different from varieties of algebras)
Re: the remark in the bracket
The analogy is a bit stretched, but I think it's due to both being described by systems of equations, in a sense.
The analogy is a bit stretched, but I think it's due to both being described by systems of equations, in a sense.
"This essay considers ways that recent uses of computers in mathematics challenge contemporary views on the nature of mathematical understanding. It also puts these challenges in a historical perspective and offers speculation as to a possible resolution."
I am more worry about those "relax" phrases. Not trusting computer to prove maths, may be I am old style. IT is my career. Maths is my love.
https://www.ams.org/journals/bull/2022-59-01/S0273-0979-2021...