
V.I. Arnold, On teaching mathematics (1997) - mr_tyzic
http://pauli.uni-muenster.de/~munsteg/arnold.html
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another-one-off
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.

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stiff
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)

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homarp
Russian original here:
[http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm...](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=5&option_lang=eng)

French translation here:
[http://math.unice.fr/~rchetrit/smf_gazette_78_19-29.pdf](http://math.unice.fr/~rchetrit/smf_gazette_78_19-29.pdf)

~~~
ttflee
Chinese translation:

[http://book.douban.com/subject/3202119/discussion/1364384/](http://book.douban.com/subject/3202119/discussion/1364384/)

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julie1
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.

~~~
julie1
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?)

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Arun2009
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._

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johnaspden
It's hard to express how much I want to cheer Arnold for this!

