Cool experiment, I hope you run more things like this!
Re: definitions, this is a serious struggle I have with the way math is presented. The definitions and notations we have now have been simplified, condensed, and iterated upon for hundreds of years, and what sticks is a matter of practical utility for mathematicians. To present the product of this process as a fait accompli without explanation or motivation is really misleading.
I mean, look how long it took to come up with the modern definition of the group! Even after it was named, for the longest time, mathematicians only considered permutation groups.
I would lay some of the blame at the feet of Platonism. Mathematical Platonism, the idea that “mathematics is out there in idea-space and we just discover it” is at best a useful metaphor or fake framework and has been taken way too far. It implies that definitions are somehow floating in thought-space and the mental move to discover them is to look really hard. In practice, the construction of definitions is more like engineering: there’s some weird thing that’s there and we’ll try to build a vessel that captures its shape as closely as possible.
Note: I tried verifying the received wisdom that “most mathematicians are Platonists” and found the answer to be much murkier than I expected. I would still say that mathematics, the way it is presented, suggests that things like chain complexes and schemes are “things out there in reality” instead of “definitions progressively engineered by people to fit reality.”
The definitions and notations we have now have been simplified, condensed, and iterated upon for hundreds of years, and what sticks is a matter of practical utility for mathematicians. To present the product of this process as a fait accompli without explanation or motivation is really misleading.
I’d like to see how “it’s conceptual engineering” vs “It’s conceptual discovery” mentalities correlate with productivity. Engineering mentality seems obviously more pragmatic and more realistic, but Discovery mentality seems much more likely to attract passion (which, for humans, fuels productivity).
Cool experiment, I hope you run more things like this!
Re: definitions, this is a serious struggle I have with the way math is presented. The definitions and notations we have now have been simplified, condensed, and iterated upon for hundreds of years, and what sticks is a matter of practical utility for mathematicians. To present the product of this process as a fait accompli without explanation or motivation is really misleading.
I mean, look how long it took to come up with the modern definition of the group! Even after it was named, for the longest time, mathematicians only considered permutation groups.
I would lay some of the blame at the feet of Platonism. Mathematical Platonism, the idea that “mathematics is out there in idea-space and we just discover it” is at best a useful metaphor or fake framework and has been taken way too far. It implies that definitions are somehow floating in thought-space and the mental move to discover them is to look really hard. In practice, the construction of definitions is more like engineering: there’s some weird thing that’s there and we’ll try to build a vessel that captures its shape as closely as possible.
Note: I tried verifying the received wisdom that “most mathematicians are Platonists” and found the answer to be much murkier than I expected. I would still say that mathematics, the way it is presented, suggests that things like chain complexes and schemes are “things out there in reality” instead of “definitions progressively engineered by people to fit reality.”
Yeah, this was one of my main motivations for asking this MathOverflow question.
I’d like to see how “it’s conceptual engineering” vs “It’s conceptual discovery” mentalities correlate with productivity. Engineering mentality seems obviously more pragmatic and more realistic, but Discovery mentality seems much more likely to attract passion (which, for humans, fuels productivity).