Un après-midi pluvieux avec Ali Chamseddine, Thibault Damour et Pierre Cartier
Thanks to the recent experimental confirmations of general relativity from the data given by binary pulsars  there is little doubt that Riemannian geometry provides the right framework to understand the large scale structure of space-time.The situation is quite different if one wants to consider the short scale structure of space-time... In particular there is no good reason to presume that the texture of space-time will still be the 4-dimensional continuum at such scales.
In this paper we shall propose a new paradigm of geometric space which allows us to incorporate completely different small scale structures. It will be clear from the start that our framework is general enough. It will of course include ordinary Riemannian spaces but it will treat the discrete spaces on the same footing as the continuum, thus allowing for a mixture of the two. It also will allow for the possibility of noncommuting coordinates . Finally it is quite different from the geometry arising in string theory but is not incompatible with the latter since supersymmetric conformal field theory gives a geometric structure in our sense whose low energy part can be defined in our framework and compared to the target space geometry .It will require the most work to show that our new paradigm still deserves the name of geometry. We shall need for that purpose to adapt the tools of the differential and integral calculus to our new framework. This will be done by building a long dictionary which relates the usual calculus (done with local differentiation of functions) with the new calculus which will be done with operators in Hilbert space and spectral analysis, commutators... The first two lines of the dictionary give the usual interpretation of variable quantities in quantum mechanics as operators in Hilbert space. For this reason and many others (which include integrality results) the new calculus can be called the quantized calculus’ but the reader who has seen the word “quantized” overused so many times may as well drop it and use “spectral calculus” instead.
A. Connes, Noncommutative geometry and reality, 4 avril 1995