Dear Michel,
Reading your essay is challenging, if one tries to reach beyond the pleasant dialogue effect that makes you a heir of Plato, Galileo, Lewis Caroll, Donald Knuth (Surreal numbers), or even Leslie Lamport [1] for instance, and actually see all these objects you are uncovering.
As for me, I have no other option than to trust you (as is usually the case in science --perfectly alright), with all these breathtaking connections. And though there is a lot of material I can hardly comprehend, the journey is pleasant.
So your essay proceeds by example, in the only really convincing manner, by showing new --i.e., unexpected-- occurrences of mathematics at work in uncovering new pieces of physics. (New is important, because once a connection is well known, we get acquainted to it, it vanishes into the background.)
Aren't these deep connections in fact within mathematics, since we are in a part of physics that is nearly completely mathematical? The concepts of physics we are considering are already extremely developed mathematically, and it is `natural' that they lend themselves to more mathematics (more connections, more structure). Of course, there is some selection to do because there are some experimental results that give some borders, and we naturally keep only the mathematics that speak to these facts.
Along this line you quote E. Frenkel, an if ``the Nobel prize in physics is really a Nobel prize in maths'', conversely Villani's decisive contribution that won him a Fields medal --Landau damping-- could count as a prize in physics too.
So, is it an implicit answer that you left to the reader to devise, that physics is by definition mathematical, therefore mathematics has to be efficient (reasonably or not being a matter of appreciation we don't quite have a standard to evaluate)?
I would like to try an example that (nearly) anyone can grasp: It has been shown that from any position, the Rubik's cube can be taken back to a reference position in 20 moves or less http://www.cube20.org/ (or 26, depending on what counts as a move --alas, not 24!). Given the huge size of the configuration set, its diameter being less than 20 means that the space is very highly connected. It is my best example to convey the shock of this connectivity to the layman. The many `coincidences' that we see, slowly and piece by piece, between mathematical structures initially seen as occurrences of distinct things appear to me in analogy (a sound analogy) with this high connectivity. Each time we are surprised by unexpected coincidences, hints us that we are far, far from seing the whole picture.
(Even simply when we keep on finding --or being shown-- more relationships between numbers, thus making some which look like nothing special, say 1729, very special indeed, we are in such a situation. In my comment on Wise's essay http://fqxi.org/community/forum/topic/2494, I have inserted a quotation on Lazare Carnot, to show how difficult it can be to see a structure, when you have not got properly acquainted to it, when you stick too much to initial vantage points. It is again such a simple example, I know of none which can be more convincing.)
We can also figure out that many, many things now look to us `naturally' the same, because we see groups behind them. The concept of group is so simple, so obvious, that it seems that for long, no one thought it could be of any use to take the pain to state it explicitly (certainly much like the concept of zero). But once it was there in clear, we started recognising it everywhere, with many cross-connections
So we slowly uncover `miracles', or mere connections. To begin with, we see many scattered, unorganised connections, that we begin to understand better when a structure shows, which links these connections into a whole.
Again, I cannot help but make the parallel with Bach-y-Rita experiment (see my essay), where from what at the beginning is confuse feelings in the back skin, objects, distinct, moving finally emerge, because their structure has been recognised, built up, abstracted.
So, congratulations on this moonshine explosion of new objects!
[1] Leslie Lamport, A simple approach to specifying concurrent systems, Commun. ACM, 32:1, January 1989, pp.聽32--45.
