Aubin, Bayen, and Saint-Pierre on the science of complexity

I'm trying to teach myself viability theory using Aubin, Bayen, and Saint-Pierre's tome bearing that title. In the  part of the introduction outlining applications and motivations of the theory, the reader stumbles upon a rather remarkable philosophical challenge: to radically change the current mathematical paradigm in order to better understand complex systems. The relationship between mathematics and the physical sciences is well established. We have developed mathematical tools to study the simple pieces of nature, particularly waves and particles in motion. But what about living systems or, more complex still, societies? "Simplifying complexity," say the authors (emphasis in the original), should be the purpose of an emerging science of complexity, if such a science will emerge beyond its present fashionable status."

That's an intriguing challenge. What do they have in mind? The following paragraph needs a full quote.

Quoting from Section 1.1, p. 8:

So physics, which could be defined as the part of the cultural and physical environment which is understandable by mathematical metaphors, has not yet, in our opinion, encapsulated the mathematical metaphors of living systems, from organic molecules to social systems, made of human brains controlling social activities. The reason seems to be that the adequate mathematical tongue does not yet exist. And the challenge is that before creating it, the present one has to be forgotten, de-constructed. This is quite impossible because mathematicians have been educated in the same way all over the world, depriving mathematics from the Darwinian evolution which has operated on languages. This uniformity is the strength and the weakeness of present day mathematics: its universality is partial. The only possibility to mathematically perceive living systems would remain a dream: to gather in secluded convents young children with good mathematical capability, but little training in the present mathematics, under the supervision or guidance of economists or biologists without mathematical training. They possibly could come up with new mathematical languages unknown to us providing the long expected unreasonable effectiveness of mathematics in the social and biological sciences.

What's awfully strange about this, aside from the desire to hide children away in convents, is the implicit belief that somehow mathematics must be capable of describing social and biological phenomena in a direct way as in physics. The reader is left with the sense that he has just read not so much a discussion of the state of the art in mathematics as an eschatological vision in which mathematics will conquer all of the corners of reality. It seems rather utopian.

Indeed, the next paragraph reveals, in my opinion, why this utopianism ultimately fails. "Even the concept of natural number is oversimplifying," say the authors, "by putting a same equivalence class so [sic] several different sets, erasing thier qualitative properties or hiding them behind their quantitative ones." This is an issue I've mused on myself: is there a way to "break away" from the discrete number system which gave birth to our mathematics? There are subtle issues to be dealt with, here, but I think the way the authors have phrased it leads quickly to absurdity: if what we care about is the qualitative and not the quantitative, it seems safe to say we're no longer doing mathematics. We might as well call it--ahem--biology and economics.

Though my initial reaction is critical, I don't mean to be overly negative. On the contrary, I think the vision expressed here is important. Mathematics should remain in the tradition of seeking to clarify and extend our understanding of reality. Yet I humbly suggest that this will not require children raised by economists. It will require, as the authors suggest, a great deal of trial and error--Darwinian evolution, if you like. Ultimately, however, I think mathematics is the science of quantity, and to the extent that it describes qualitative properties, it does so by mapping them to numbers. I am not saying that the concept of number is self-evident and unchanging. History proves that it isn't. But we need not rely on some forced revolution in mathematical training in order to reimagine (and sometimes redefine) the concept of number.

Progress in the direction of mathematical life/social sciences already seems underway. The very text I am citing here is evidence of that. Another example I would point to is mean field game theory, with which the authors are surely familiar. The insight that the behavior of a multitude of rational beings can be understood quantitatively in the aggregate appears to be a profound boost to the project of mathematical social sciences, in particular. Based on my general sense of the field (from a young perspective), I would like to believe that the current interest in biological, economic, and social questions in mathematics is more than a passing fad. Over time, we may think of mathematics as intrinsically tied to these subjects just as it is to physics.

But all things in their place. Mathematics may be helpful in addressing a plethora of questions, but I very much doubt it is without limits (no pun intended).

This misnomer of "self organizing" systems

The study of self organizing is extremely important in modern science, whether in evolutionary biology, economics, or even the physical sciences. I once went to a talk on the study of self-organizing systems in the context of crime syndicates in Los Angeles. There are even some interesting applications at the basic level of image segmentation: mathematicians working in nonlinear analysis are now able to devise computer algorithms to detect different parts of a photograph using evolutionary partial differential equations. There is a key idea encompassing all of these fields of research: complex systems can "organize themselves" not according to a preset plan, but according to certain rules, with certain types of organization being "selected" over others. For instance, in image segmentation, it would be pointless to try to give a computer all the necessary preset criteria for identifying a piece of an image in a photograph. (How do you know where a mountain ends and a sky begins? Try writing down the exact criteria just to see how impossible this is.) Instead, if you give a computer general rules (dictated by a PDE) by which to allow its image segmentation to evolve, eventually an appropriate segmentation will be selected.

So why do I say "self organizing" is a misnomer? The argument can be put pretty simply in a mathematical framework. A self organizing system is nothing more and nothing less than a dynamic system. Let x(t) by the state of a system at time t, with the state space given by H, and let f be a function. Then we can write x'(t) = f(x(t)) for given initial data x(0) in H, where x' is the time derivative of x. This defines a dynamic system. For discrete time, we can simply write x(n+1) = f(x(n)). The long-time behavior of a dynamic system is, in a completely literal sense, the result of evolution. The crucial point is this: this behavior is governed by the function f. It would be absurd mathematically and otherwise to say that x(t) determines itself. On the contrary, it is clearly determined by f, over which it has no control. (I could mention stochastic dynamic systems, but the same principles apply there as well; randomness does not mean the system is somehow freer.)