I was trying to track down where the "Russian" comes from in the Russian method (as held opposed to the Bourbaki method) and have tracked it down to Vladimir Igorevich Arnol'd's opposition to Bourbaki. This essay expresses his anti-formalist stance and his view of the continuity of math and science. In fact, he opens with:

"Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap."

Given this deference to physics, I obviously disagree with much of the essay. I am also worried (maybe even offended) by the cruelty of some of his polemic (+Dan Nichol, if you thought I am confrontational in my blog posts then you should take a look at this essay). I am hoping that this is due to poor translation; I can't tell if the original was delivered in French or Russian, but the first published version of this account is in Russian. Unfortunately, I have not tracked down this original source to attempt an alternative translation. I definitely do not approve of some of the insults Arnol'd hurls (like "Mentally challenged zealots of "abstract mathematics" threw all the geometry ... out of teaching."), especially given how famous and powerful Arnol'd was by 1997.

However, the essay does offer some interesting perspectives:

"a special technique has been developed in mathematics. This technique, when applied to the real world, is sometimes useful, but can sometimes also lead to self-deception. This technique is called modelling. When constructing a model, the following idealisation is made: certain facts which are only known with a certain degree of probability or with a certain degree of accuracy, are considered to be "absolutely" correct and are accepted as "axioms". The sense of this "absoluteness" lies precisely in the fact that we allow ourselves to use these "facts" according to the rules of formal logic, in the process declaring as "theorems" all that we can derive from them.

It is obvious that in any real-life activity it is impossible to wholly rely on such deductions. The reason is at least that the parameters of the studied phenomena are never known absolutely exactly and a small change in parameters (for example, the initial conditions of a process) can totally change the result. ... In exactly the same way a small change in axioms (of which we cannot be completely sure) is capable, generally speaking, of leading to completely different conclusions than those that are obtained from theorems which have been deduced from the accepted axioms. The longer and fancier is the chain of deductions ("proofs"), the less reliable is the final result.

Complex models are rarely useful (unless for those writing their dissertations)."

The focus on intuition also meshes with my appreciation of Poincare and my advocating for viewing him as one of the grandfathers of cstheory: https://egtheory.wordpress.com/2012/11/04/connectors/ however, Arnol'd seems to depart from Poincare in prioritizing physics over the mind in such a way that seems to sideline cstheory (which I like to view as a certain idealization of mind: https://egtheory.wordpress.com/2014/09/11/transcendental-idealism-and-posts-variant-of-the-church-turing-thesis/ ). This sideline of cstheory seems to be common among Russian mathematicians, and in the case of Arnol'd (who was born a year after Turing's paper and supervised by Kolmogorov) it seems like a deliberate choice rather than a lack of exposure.

/cc +John Baez, +Abel Molina, +Alexander Yartsev
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