– how the dynamics of a big number of particles becomes the dynamics of a continuous system? Apart the physics BS of neglecting infinities, I know of very few mathematically correct approaches. From my mixed background of calculus of variations and continuous media mechanics, I can mention an example of such an approach in the work of Andrea Braides on the -convergence of the energy functional of a discrete system to the energy functional of a continuous system and atomistic models of solids.
– how to endow a metric space (like a fractal, or sub-riemannian space) with a theory of differential calculus? Translated: how to invent “smoothness” in spaces where there is none, apparently? Because smoothness is certainly emergent. This is part of the field of non-smooth calculus.
– how to explain the profound resemblance between geometrical results of Gromov on groups with polynomial growth and combinatorial results of Breuillard, Gree, Tao on approximate groups? In both cases a nilpotent structure emerges from considering larger and larger scales. The word “explain” means here: identify a general machine at work in both results.
– how to explain the way our brain deals with visual input? This is a clear case of emergence because the input is the excitation of some receptors of the retina and the output is almost completely not understood, except that we all know that we see objects which are moving and complex geometrical relations among them. A fly sees as well, read From insect vision to robot vision by N. Franceschini, J.M. Pichon, C. Blanes. Related to this paper, I cite from the abstract (boldfaced by me):
We designed, simulated, and built a complete terrestrial creature which moves about and avoids obstacles solely by evaluating the relative motion between itself and the environment. The compound eye uses an array of elementary motion detectors (EMDS) as smart, passive ranging sensors. Like its physiological counterpart, the visuomotor system is based on analogue, continuous-time processing and does not make use of conventional computers. It uses hardly any memory to adjust the robot’s heading in real time via a local and intermittent visuomotor feedback loop.
More generally, there seems to be a “computation” involved in vision, massively parallel and taking very few steps (up to six), but it is not understood how this is a computation in the mathematical, or computer science sense. Conversely, the visual performances of any device based on computer science computation up to now, are dwarfed by any fly.
I identified a “machine of emergence” which is in work in some of the examples given above. Mathematically, this machine should have something to do with emergent algebras, but what about the computation part?
Probably geometers reason like flies: by definition, a geometrical statement is invariant up to the choice of maps. A sphere is not, geometrically speaking, a particular atlas of maps on the sphere. For a geometer, reproducing whatever it does by using ad-hoc enumeration by natural numbers, combinatorics and Turing machines is nonsense, because profoundly not geometrical.
On the other hand, the powerful use and control of abstraction is appealing to the geometer. This justifies the effort to import abstraction techniques from computer science and to replace the non-geometrical stuff by … whatever is more of a geometrical character.
For the moment, such efforts are mostly a source of frustration, a familiar feeling for any mathematician.
But at some point, in these times of profound changes in, mathematics as well as in the society, from all these collective efforts will emerge something beautiful, clear and streamlined.