mainstream mathematics

$n$lab should not be, in my understanding, about scientific politics, but about content, to focus us on more important issues, rather than to distract us to externalities. However, we are interested in philosophy of mathematics, and its general **architecture**. Here I expose my basic attitudes on some related issues, and they are of course, not pretending to be authorative, but rather my own point of view.

Some programs like most of Grothendieck’s work, and much of category theory has architectural ambitions, by making the unifying effort by seeing the tendencies, by devising a common language and by formulating deeper, simpler and more natural versions of mathematical knowledge. Many efforts of that kind, for example of the Nicolas Bourbaki are controversial as they limited themselves to massive types of activities whose value is great in their own project, but is detrimental as copied outside of their own realm (here I allude to the employment of formal approaches in education in mathematics which is partly due Bourbaki’s influence). While the opus of Bourbaki is a valued piece for many users, one should have in mind that (according to Dieudonne) the limitation of Bourbaki to dead and stable mathematics was a natural field of exposition, and this self-imposed choice should be respected. However supplementary works on motivations, applications and history, were scarce. Now we witness a period where higher category theory simplifies and unifies much of modern mathematics and $n$lab is prompted by the existence of that phenomenon.

In mathematics, as in every other field, there is a little difference between being popular and being mainstream. The only difference is that popular refers to the attitude of anybody in the community equally while **mainstream mathematics** normally refers to the mathematics appreciated and practices in leading mathematical communities.

This distinction I see between popular and mainstream is very important nowdays. For example, because there is a surge in appearance of new journals fostering massive publication of rather irrelevant mathematical research; for example one writes axioms for some strange class of general topological spaces and proofs some properties, than one changes axioms and goes on. Many mediocre communities have administrative control systems which rely on publish and perish criteria, and gives big rewards to those who can write lots of articles of doubtful quality. This results in having big schools and even tendencies of writing irrelevant mathematics, usually in some typical marginal problematics, where the ways of hyperproduction are devised or propagated. Thus they are often popular.

On the other hand, it is possible and often the case, that even mainstream mathematics in the narrow sense, suffers from fads, and the status the mainstream mathematics has in leading communities often leads to career rejections to original mathematicians proceeding along new and unrecognized paths and problematics of research. While we often find excited about finding out of a rather unknown work of great depth and significance, once we are aware of its value, we should promote it even more than the recognized mainstream research; to correct the uneven position of the valuable research. Even in the works of some of the most well known researchers, often easy aspects get well known, while some of the deepest papers get hidden, as their understanding requires more technical knowledge which is not that easy to pick up by most of the mathematical community. This is in sharp difference from most other sciences, especially experimental. If you have a top technique, even if the technique is not acquired by others, usually the final outcome (like experimental discovery) is. This is often also in mathematics, for example, when one comes with a proof to the well-known conjecture, or solution to a popular open problem. However much of mathematical works, specially those of programmatic nature, and bring new foundational techniques, can hardly be appreciated by their mere ready final applications, but their true value is in the inherent power of the theory. Such moments are often picked up meaningfully by the community only several decades after the publication (for example the fundamental works from 1940-s of Petrovskii on lacunas of differential equations, picked up by Garding and Atiyah only around 1970).

I advocate and try to help bringing to the light deep, but less known work, more than the one which is mainstream. Of course, one should not dedicate its own focus too much to unusual formalisms when they have little to offer, this makes me less ready to understand things I face from mainstream mathematicians on the daily basis.

On the other hand, there is lots of denial. Some people claim that no understanding comes if one does not show all calculations in local coordinates for example. They dedicate to rederive standard results in such approach and despise those who used “advanced theory”. Their effort has some value, but the attitude does not, in my opinion.

Finally, there are many more unusual denials, sometimes going to irrational extremes. I would call it CPOV (crackpot point of view) and this entry is a witness of the term for future use on these pages. I also hope once to write an essay which would be an expansion of ideas outlined here and with lots of examples making it plastic to those who do not see it that way.

Last revised on April 11, 2010 at 21:26:06. See the history of this page for a list of all contributions to it.