What is "Russian-style" mathematics?

math-history

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?


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 .

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