Arnold On Teaching Mathematics Pdf

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.

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

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.

In the Gleick's book about complex system the author proposes a theory of why Bourbaki (the collective behind the plague called «mathématiques fondamentales») was structured: it is mainly a question of ego.

Poincaré was said to not acknowleged the «french school of mathematics» as the origin of his discoveries. And the institution especially the elite called «ENS» (forming the best teacher for university) is said to have been quite disliking his attitude.

Since Poincaré was heavily relying on geometry, it is said that since they found it unacademic they decided to change the content of math learning to avoid new «casses burnes» mathematicians.

It is very funny at this title to look at the discrepancy between the story of Mandelbrot experience whether it is written in french or english.

French biography states mandelbrot LOVED polytechnic school (another super ivy league) and english said the opposite stating that mandelbrot reproved the lack of use of geometry.

To be honest, I don't know if this is true.

PS Feynman biography makes the same statement about the irruption of this fashion of teaching «mathématiques fondamentales» in the US system.

It seems have been part of the fuel about the cargo cult science essay (What is science?)

While not coming from a "mathematics is physics" angle, this bit from "Concrete Mathematics" by Graham et. al. also warns against too much abstraction:

Abstract Mathematics is a wonderful subject, and there's nothing wrong with it: it's beautiful, general and useful. But its adherents had become deluded that the rest of mathematics was inferior and no longer worthy of attention. The goal of generalization had become so fashionable that a generation of mathematicians had become unable to relish beauty in the particular, to enjoy the challenge of solving quantitative problems, or to appreciate the value of technique. Abstract mathematics was becoming inbred and losing touch with reality; mathematical education needed a concrete counterweight in order to restore a healthy balance.