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Mathematics is completely distinct from physics; it just happens that if you use mathematical ideas to model physical reality then the predictions are startling

Mathematics is completely distinct from physics; it just happens that if you use mathematical ideas to model physical reality then the predictions are startlingly accurate. It is entirely reasonable for mathematics to have no basis in physical reality. Mathematics is all about relations between things (if this has that property; it follows that...) and this has implications far beyond physics (eg, statistics and probability have implications for physics, but applications almost literally everywhere - these ideas are bigger than physics). This essay is relevant to teaching mathematics to people who are more comfortable with physical examples than with abstract logic. Fair enough, realise that this is calling for 'mathematics for physicists' rather than a better way of understanding maths. Making the connection that all the mathematics we do is done in the context of reality doesn't make mathematics a subset of physics any more than history is a subset of physics. If every theory we have about the universe was proven wrong tomorrow then physics people would need to get very busy indeed - but mathematicians wouldn't even notice.

stiff 12y ago
This is a manifest about how Arnol'd views mathematics, and it's meaning becomes more clear if you read some of his books and see how he does actual math. It doesn't mean his is the only view possible, but it is the view of an absolute giant of mathematics and physics, developed after a lifetime of practice in both research and education, so you would better spent some time thinking more deeply about why he says what he says, instead of brushing it off with what frankly are platitudes for anyone interested in the subject. For a start, you might want to think about the intertwined history of the two subjects, about how mathematical ideas might at all arise in human cognition, about what are the available criteria for choosing mathematical theories in the huge space of theories that are possible and true, and finally about what are the productive ways of searching for theories that have any value. Then you might want to ask yourself if school pupils are really best served by immediately being presented with the most abstract presentation of each subject, while the abstractions themselves were often a result of a whole sequence of generalizations from some first very intuitive basis. Those are complicated questions, and there are no clear answers, but Arnolds view was likely rather more sophisticated than you think. This lecture of his might help to interpret the article in a more productive way: https://www.msri.org/workshops/390/schedules/2714 https://www.msri.org/workshops/390/schedules/2714
musername 12y ago
you shouldn't think the issue here is with an overbearing attitude in physics, the problem is just the categorisation of maths. I'd put it vice versa. since mathematics means the art of learning, literally, physics is a subfield of applied mathematics. Numbers aren't just represantations, unlike numerals. numbers are only of any value, if they have context, even if that is a fictive for the purpose of learning. but teachers frequently use examples from applied mathematics, too. Math's called a structural science for a good reason, it can be used to find or build structures in physics, as well as any other natural science. A strict distinction in it's root seems unnessacary, since logic and all is based in philosophy and engineering, back in a time when polymaths didn't plow just one field. Even nowadays subfields in sciences require vastly different knowledge, so if these subfields count as part of physics, it's fair to say physics is maths and maths is physics.

stiff 12y ago

This is a manifest about how Arnol'd views mathematics, and it's meaning becomes more clear if you read some of his books and see how he does actual math. It doesn't mean his is the only view possible, but it is the view of an absolute giant of mathematics and physics, developed after a lifetime of practice in both research and education, so you would better spent some time thinking more deeply about why he says what he says, instead of brushing it off with what frankly are platitudes for anyone interested in the subject. For a start, you might want to think about the intertwined history of the two subjects, about how mathematical ideas might at all arise in human cognition, about what are the available criteria for choosing mathematical theories in the huge space of theories that are possible and true, and finally about what are the productive ways of searching for theories that have any value. Then you might want to ask yourself if school pupils are really best served by immediately being presented with the most abstract presentation of each subject, while the abstractions themselves were often a result of a whole sequence of generalizations from some first very intuitive basis. Those are complicated questions, and there are no clear answers, but Arnolds view was likely rather more sophisticated than you think. This lecture of his might help to interpret the article in a more productive way: https://www.msri.org/workshops/390/schedules/2714 https://www.msri.org/workshops/390/schedules/2714

musername 12y ago

you shouldn't think the issue here is with an overbearing attitude in physics, the problem is just the categorisation of maths. I'd put it vice versa. since mathematics means the art of learning, literally, physics is a subfield of applied mathematics. Numbers aren't just represantations, unlike numerals. numbers are only of any value, if they have context, even if that is a fictive for the purpose of learning. but teachers frequently use examples from applied mathematics, too. Math's called a structural science for a good reason, it can be used to find or build structures in physics, as well as any other natural science. A strict distinction in it's root seems unnessacary, since logic and all is based in philosophy and engineering, back in a time when polymaths didn't plow just one field. Even nowadays subfields in sciences require vastly different knowledge, so if these subfields count as part of physics, it's fair to say physics is maths and maths is physics.