Let me start by quoting John Baez.s Week 221 last paragraph. Begin quote But, someday I should really explain the ideas behind the baby "abelian" case of the Langlands philosophy in simpler terms than Frenkel does here. The abelian case goes back way before Langlands: it's called "class field theory". And, it's all about exploiting this analogy, which I last mentioned in "week218": NUMBER THEORY COMPLEX GEOMETRY Integers Polynomial functions on the complex plane Rational numbers Rational functions on the complex plane Prime numbers Points in the complex plane Integers mod p^n (n-1)st-order Taylor series p-adic integers Taylor series p-adic numbers Laurent series Adeles for the rationals Adeles for the rational functions Fields One-point spaces Homomorphisms to fields Maps from one-point spaces Algebraic number fields Branched covering spaces of the complex plane End quote. This is indeed a very beautiful but a static picture. I remember a late Chinese scholar mentioned that the western philosophers always focuses on objects, but the old Chinese wisdom is always on the relation between objects and how they convert to each other. I want to put forward another kind of analogy if it is possible. Elementary Real Analysis Homotopic/Algebraic Theory Real numbers Some objects Inequality It can compare any two objects' size. Limit It can tell one object approaches another object Continuity It can tell successive objects are not deviated from their immediate neighborhood Differentiation It can tell the "slope" of successive objects Integration (We know there is motivic integration, but it may not fit the overall picture.) Differential forms (We know there is n-gerbes, but, again, may not fit the overall picture.) I hope you can see that all these kind of concepts are to capture the ubiquitous devil, which is everywhere, but nowhere can it be caught. The devil is the change, the perpetual change, which is dormant in each object or group of objects. If one day there is another "David Hilbert" appears on the face of this planet, and he can synthesize all the current fragmentary math concepts. And, there comes another "Albert Einstein" to start building a new quantum mechanics in which "observable is relative to another observable" is achieved. (Maybe like C. Rovelli idea in Relative Quantum Mechanics or in another paper about observables of quantum spacetime, which mentioned "observables built upon observables". Frankly, as an adherent of "perpetual change", I really don't buy the no topology change in Loop Quantum Gravity because metric can be bent or stretched, so any measurement of length or area or volume is a relative measurement even in pure gravity condition. Area, and volume should be quantizable but can't be fundamental. Maybe they have never advocated that whatever quantizable is fundamental in their theory if they try to defense above criticism. I don't see LQG have conceptually demonstrated the relativity concept I put forward here or the one in C. Rovelli papers though C. Rovelli he helped build the LQG.) I found out on some websites that in some European institutes, researchers are trying to put quantum mechanics on a categorical footing, but I much heavy machinary they can bring in or invent to succeed the goal, and how much it will look like my intend theory. Second, when will Konsevich will prove his speculation that his deformation quantization theory will lead to a new class of quantum field theory. If he can do that, I may expect that a "dynamical" quantzation scheme may become realizable not in a distant future. I can foresee that if you buy this kind of concepts, it will lead you to some astounding philosophical and physical conclusions about quantum spacetime and its ramification in understanding the nature of the fundamental four forces. I save this premature thought for the future occassion. Finally, since Godel's Theorem always backs me up, I am granted to say that there is always room to add some possible math concepts to the current ones. Whether it is possible or not, it just depends on our imagination and talents. With the dominance of string theory in academic, it is very unlikely that the current talented group of people will think outside the box and even in the coming few generations. Charles Hui "I leave my thought in a region of spacetime for the physicists of future generations."