great article. a topic i have often wondered about. seems related to the question of whether maths is discovered or invented
i think that providing an example of a mathematical theorem that *doesn't* accord with physical reality (as far as we know...) would be helpful. even if, as you write, it "would be dull". Is there any maths that contradicts physical reality?
unrelated - would euclid's axioms be relevant here? the axioms of maths are based in physical reality, therefore isn't everything that follows also?
(speaking as a layperson. i have an undergad degree in biology and got a 'B' at maths a-level😀)
Ha. Well I failed my first exam in maths during my physics degree so...
Yeah I think these are great questions. I try to address this by talking about when good maths (knot theory) comes out of bad physics (erroneous explanation for existence of elements)... see above for my comments on this:
Vortex theory, for instance, was an early attempt by British mathematical physicist William Thomson (Lord Kelvin) to explain why atoms came in a relatively small number of varieties. He pictured atoms as spinning rings which could be tied in intricate knots, with each knot corresponding to a different chemical element. The theory was abandoned after the discovery of the electron—but the mathematics led to the development of knot theory, the field Jones and Witten have advanced with their work.
So where does this leave us? The gist of it I think is that we've evolved as pattern spotting animals and the ones we find beautiful tend to be those that in some way or other were useful and helped us survive. So understanding the parabola of a thrown spear for example--and spotting the regularity and ultimately the geometry of it--was useful. Our brains adapted to spotting these things in nature. And sometimes we spot patterns that aren't there too--and sometimes that leads to good maths as well. But still--the patterns have to be 'out there' in some sense to be found. And we know that because life couldn't exist and i doubt much of the universe could exist with starts and planets, if there were not some intrinsic patterns/rules that nature followed.
But of course I can't hope to answer the question with a feature--though it's nice I've brought it to the attention of more people!
PS And Euclid--well of course people thought for centuries that Euclid's geometry was the ONLY geomtery possible. We know now that isn't true and in fact the fabric of spacetime is not Euclidean at all. So this was an interesting case of maths putting physics and itself onto a certain track for a long time. Only in the 19th C did we actually properly discover non-Euclidean geometry and one wonders--why not earlier?
great article. a topic i have often wondered about. seems related to the question of whether maths is discovered or invented
i think that providing an example of a mathematical theorem that *doesn't* accord with physical reality (as far as we know...) would be helpful. even if, as you write, it "would be dull". Is there any maths that contradicts physical reality?
unrelated - would euclid's axioms be relevant here? the axioms of maths are based in physical reality, therefore isn't everything that follows also?
(speaking as a layperson. i have an undergad degree in biology and got a 'B' at maths a-level😀)
Ha. Well I failed my first exam in maths during my physics degree so...
Yeah I think these are great questions. I try to address this by talking about when good maths (knot theory) comes out of bad physics (erroneous explanation for existence of elements)... see above for my comments on this:
Vortex theory, for instance, was an early attempt by British mathematical physicist William Thomson (Lord Kelvin) to explain why atoms came in a relatively small number of varieties. He pictured atoms as spinning rings which could be tied in intricate knots, with each knot corresponding to a different chemical element. The theory was abandoned after the discovery of the electron—but the mathematics led to the development of knot theory, the field Jones and Witten have advanced with their work.
So where does this leave us? The gist of it I think is that we've evolved as pattern spotting animals and the ones we find beautiful tend to be those that in some way or other were useful and helped us survive. So understanding the parabola of a thrown spear for example--and spotting the regularity and ultimately the geometry of it--was useful. Our brains adapted to spotting these things in nature. And sometimes we spot patterns that aren't there too--and sometimes that leads to good maths as well. But still--the patterns have to be 'out there' in some sense to be found. And we know that because life couldn't exist and i doubt much of the universe could exist with starts and planets, if there were not some intrinsic patterns/rules that nature followed.
But of course I can't hope to answer the question with a feature--though it's nice I've brought it to the attention of more people!
PS And Euclid--well of course people thought for centuries that Euclid's geometry was the ONLY geomtery possible. We know now that isn't true and in fact the fabric of spacetime is not Euclidean at all. So this was an interesting case of maths putting physics and itself onto a certain track for a long time. Only in the 19th C did we actually properly discover non-Euclidean geometry and one wonders--why not earlier?