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I've just stumbled upon Gorodentsev's upcoming textbook 'Algebra I'. The description of it claims that it's very 'Russian-style'.

This book is the first volume of an intensive “Russian-style” two-year graduate course in abstract algebra, and introduces readers to the basic algebraic structures – fields, rings, modules, algebras, groups, and categories – and explains the main principles of and methods for working with them.

What does this mean? What differs 'Russian-style' from 'American-style' mathematics?

ClassicEndingMusic
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    This is a guess, but it sounds like something akin to a style of rigor perhaps employed at Russian universities. You could perhaps read up on math and science education in the time of the Soviet Union. – Sean Roberson Oct 31 '16 at 17:31
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    One might view "Russian style" as "math that skips steps." It can be intimidating since the steps that are skipped are implicitly viewed as being "obvious." – Michael Oct 31 '16 at 17:32
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    I would understand it as elitist and intellectually demanding, the kind of course that aims at pushing the smartest 10% of the class as far as they can be pushed, even at the cost of failing 80% -- whereas a stereotypically "American-style" course would aim at making as many students as possible pass, even at the cost of the 10% brightest only learning a tenth of what they could have achieved in a course tailored for them. – hmakholm left over Monica Oct 31 '16 at 17:39
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    Kolmogorov, Chebyshev, Markov, Lusin, Suslin, Egorov, Khinchin, Vinogradov, Pontryagin, ... are all great Russian mathematicians; perhaps their works would speak for Russian style. :) – Yes Oct 31 '16 at 17:41
  • The springer page also states "Challenging amount of material thoughtfully organized for deep and fast learning" maybe this is what is meant. – PhoemueX Oct 31 '16 at 19:06
  • What is "American style"? – Geremia Dec 10 '20 at 00:16

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Russian-style should be understood not in opposition to American-style (that's cold war stuff) but rather in opposition to French-style or more precisely Bourbaki-style. The latter emphasizes formalism even sometimes at the expense of readability. The Russian style tends to focus on the essence rather than the formalism, and emphasize what is novel. A good example of accessible, popular, and rigorous writing in the Russian-style is a typical book by Vladimir Arnold; for example, his Mathematical methods of classical mechanics, an all-time favorite.

The flip side of excessive formalism is often committing errors; this was richly illustrated in the case of Bourbaki by Adrian Mathias; see e.g., his http://link.springer.com/article/10.1007%2FBF03025863 .

Mikhail Katz
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    "To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about!" — V. I. Arnold (in On teaching mathematics) – Rodrigo de Azevedo Nov 01 '16 at 12:40
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    Well this particular story doesn't go very far because a certain French mathematician stepped forward at MathOverflow and claimed that he was that child, and he meant it as a joke, so Arnold may have overused that example. At any rate the calamities of Bourbaki style are well-known and have recently been critiqued by Mathias; see e.g. http://link.springer.com/article/10.1007%2FBF03025863 @RodrigodeAzevedo – Mikhail Katz Nov 01 '16 at 12:43
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    @MikhailKatz I (and I think everybody else) would be interested in what the French mathematician said; could you provide the link?. Googling didn't yield any results. – user376034 Nov 01 '16 at 13:00
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    http://mathoverflow.net/questions/153604/the-arnold-serre-debate/153606#comment393855_153606 and http://mathoverflow.net/questions/153604/the-arnold-serre-debate/153606#comment393860_153606 – Mikhail Katz Nov 01 '16 at 13:05
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    How can a textbook claim to be for aspiring mathematicians and not be written in Bourbaki-Style? I thought formalism was the whole point of mathematics. – ClassicEndingMusic Nov 01 '16 at 13:51
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    @ClassicEndingMusic, as a professional mathematician I hurry to disabuse you of the notion that Bourbaki-style is obligatory or for that matter desirable; but perhaps this is a subject for a separate question. – Mikhail Katz Nov 01 '16 at 13:56
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    @MikhailKatz Now I am intrigued. I will read Bourbaki's Elements of Mathemtics II Algebra and afterwards Gorodentsev's book. Let's see how this compares. – ClassicEndingMusic Nov 01 '16 at 14:06
  • Is it fair to say "Russian style" is inductive whereas "Bourbaki's style" is deductive? – Geremia Jun 29 '17 at 20:26
  • And what exactly is "American style"? – Geremia Dec 10 '20 at 00:16