What is the "intuition" behind "brave new algebra"?
I hardly know how to begin to reduce this subject to some kind of intuitive ideas, but here are some thoughts:
* Adding homotopy to algebra allows for generalizations of familiar algebraic notions. For instance, a topological commutative ring is a commutative ring object in the category of spaces; it has addition and multiplication maps which satisfy the usual axioms such as associativity and commutativity. But instead, one might instead merely require that associativity and commutativity hold "up to all possible homotopies" (and we'll think of the homotopies as part of the structure). (It is hard to give the flavor of this if you haven't seen a definition of this sort.) This gives one possible definition of a "brave new commutative ring".
* What is really being generalized is not algebraic objects, but derived categories of algebraic objects. So if you have a brave new ring R, you don't really want to study the category of R-modules; rather, the proper object of study is the derived category of R-modules. If your ring R is an ordinary (cowardly old?) ring, then the derived category of R-modules is equivalent to the classical derived category of R.
If you want to generalize some classical algebraic notion to the new setting, you usually first have to figure out how to describe it in terms of derived notions; this can be pretty non-trivial in some cases, if not impossible. (For instance, I don't think there's any good notion of a subring of a brave new ring.)
* As for Manin's remarks: the codification of these things has being an ongoing process for at least 40 years. It seems we've only now reached the point where these ideas are escaping homotopy theory and into the broad stream of mathematics. It will probably take a little while longer before things are so well codified that brave new rings get introduced in the grade school algebra curriculum, so the process certainly isn't over yet!
This is a general phrase that refers to the direction of
- higher category theory, per Lurie (you know references)
- scheme homotopy theory, per Voevodsky
- derived spaces, per Ben-Zvi and Nadler (0706.0322, 0805.0157)
The idea is that we're again changing the fundamental nature of space — first it was something easily drawn, then topology, then schemes, then stacks. Now we're doing some infinity versions of spaces, e.g. space --> $\infty$-category, ring --> $E_\infty$ category and that's brave new (the person who wrote this was quoting somebody from the 80s — below I explain that this person may very well be not Manin). In one sentence, we're not just taking functions now, but also forms etc.
Later he actually explains that "the homotopy picture becomes more important, and if you want discrete, you need to factorize".
Note that the "brave new" phrase is absent from the Russian version of the interview linked from AMS:
И поэтому я не предвижу ничего такого экстраординарного в ближайшие двадцать лет. Происходит перестройка того, что я называю основаниями математики, не в нормативном смысле слова, а как свод подчас даже не эксплицитных правил, критериев ценности, способов представления результатов, который присутствует в мозгу у работающего математика здесь и сейчас, в каждое конкретное время. Вот это я называю основаниями математики. Их можно делать эксплицитными, при этом в нескольких вариантах, и представители разных вариантов могут начать спорить, но, поскольку это существует в мозгах работающего поколения математиков, там всегда есть нечто общее. Так вот, после Кантора и Бурбаков в мозгах, что бы там ни говорили, сидит теоретико-множественная математика.
which was translated to
And so I don’t foresee anything extraordinary in the next twenty years. Probably, a rebuilding of what I call the “pragmatic foundations of math- ematics” will continue. By this I mean simply a codification of efficient new intuitive tools, such as Feynman path integrals, higher categories, the “brave new algebra” of homotopy theorists, as well as emerging new value systems and accepted forms of presenting results that exist in the minds and research papers of working mathematicians here and now, at each particular time. When “pragmatic foundations” of mathematics are made explicit, usually in several variants, the advocates of different versions may start quarrel- ing, but to the extent that it all exists in the brains of the working generation of mathematicians, there is always something they have in common. So, after Cantor and Bourbaki, no matter what we say, set theoretic mathematics resides in our brains.
The translation is accurate except for the italicized phrase. That phrase should be translated as
The things that I call the foundation of math are being rebuilt; not in the normative meaning of that word, but rather as the codex of — not even explicit rules, but rather values, ways to represent the results that exist in the brain of a working mathematician, here and now, at every given moment of time.
(I'm going for more literal translation: the original uses present tense, "brain" rather then "mind" and there is no "codification of mathematics", rather there are "values and ways" that are "being rebuilt")
Interesting, but as you see this is referring to the general idea of change in the "homotopy" direction rather then to the specific papers. In particular, "codification" should refer to the process when this "homotopy-think" becomes firmly established in the textbooks, rather then in the recent research articles.
It's a mystery to me as to why highly intelligent people didn't notice the discrepancy when preparing the interview for publication. In some other places the words are changed, e.g. "then you factorize..." --> "then you pass to the set of connected components of a
space defined only up to homotopy", and it appears this was made to make the interview more readable and unambiguous in English — it's very informal, though understandable, in the source.
A possibility, of course, would be that Manin himself edited the English version after it was translated.
re: Manin's comments, the article says that "Manin edited this translation for publication in the Notices", so it is not surprising the English and Russian versions are different.