Friday, January 10, 2014

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).

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