Previous post: Reply to JSE.
This is a reply to a comment of T. Gowers.
Yesterday I left remarks about “hard” arguments and Bott periodicity without any comments. Here are few.
First, the meaning of the word "hard" varies from person to person. There is a fairly precise definition in analysis, due to Hardy. Still, I fail to use it for classification of, say, Lars Ahlfors work: is it hard or soft? I was told once that it is not "hard analysis", but wasn't told anything meaningful why. For me, Ahlfors is the greatest analyst of the last century (let us assume that the 20th century started around 1910, at least - in order to avoid hardly relevant comparisons).
Notice that the terminology is already fairly misleading: the opposite of “hard” is not “easy” (despite the hard analysis is assumed to be difficult and hard to work in). It is “soft”. What dichotomy is understood here? Clearly, finding the right definitions is not an easy thing; more often than not it is very difficult. The defined objects may turn out to either “hard” or “soft” depending on what we wanted. For definitions I see only one meaningful interpretation: objects are hard if they are rigid (like Platonic solids), and soft if they can be easily deformed (and the space of deformations is highly dimensional) without losing their key characteristics. It seems that “hard” theories are very often the ones dealing with “soft” objects.
But I suspect that the people using the hard-soft terminology will disagree with me. At the same time in the conceptual mathematics there is no such dichotomy at all and it is impossible to acquire any experience in using it.
The example of the Bott periodicity theorem is a good testing ground. The situation with the Bott periodicity is more or less opposite to what you wrote about it. First of all, it is not a black box to be used without opening it.
The first proof was based on the Morse theory for infinitely dimensional space of curves in some classical manifolds. (A nice idea of Morse reduces the problem at once to the finitely dimensional situation.) This is result is crucial for topological K-theory; it is build into its structure. But I never saw any detailed exposition when the original theorem of Bott was used as black box for developing the topological K-theory. The theorem seems to be too weak for this, it provides for one point space the result needed for a wide class of topological spaces. Probably, the machinery of algebraic topology allows deducing this particular result just from the result for one-point space, but this definitely cannot be done without reworking the Bott theorem in order to get a more explicit result first. Atiyah in his book uses another proof (due to him and Bott), which has the advantage of being “elementary” and giving the result for all reasonable spaces without any intermediaries, and the disadvantage of being the most obscure one. Later on, the K-theory (and, hence, the Bott periodicity) were used in order to prove the Atiyah-Singer index theorem. The index theorem has an advanced version, the index theorem for families (of elliptic operators).
In fact, really useful theorem is the index theorem for families, not the original index theorem. After the second proof of the index theorem was found, which imitated Grothendieck proof of the Grothendieck-Riemann-Roch theorem and allowed to prove the index theorem for families (the first one does not), Atiyah used it to give a new proof of the Bott periodicity. One may suspect a circular argument here, but there are none. A carefully excised fragment of this proof does not depend on the Bott periodicity, and if combined with an algebraic idea due to Atiyah, led to a new proof of Bott’s periodicity. This proof turned out to be the most important one. The subject of analytic K-theory is to a big extent an attempt to use the ideas of this proof in as general situation as possible, and to apply the results. The main point here that people working in analytic K-theory are not using Bott periodicity as a black box; they are thinking about what is really inside this box. By now we have at least 8 different (substantially different) proofs of Bott periodicity, and the progress in the questions related to the Bott periodicity usually requires rethinking the theorem and its proof, not using it as a black box. Perhaps, because of all this, people prefer to speak about the phenomenon of Bott periodicity and not about Bott’s theorem.
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