The Nonlinear World by Yoshitsugu Oono is an intriguing mixture of philosophy and mathematics reminiscent of Gödel, Escher, Bach, but with much more advanced mathematics. The author says that “the main target of the book is young people who have just started to appreciate the world seriously,” though perhaps not so young as to not have seen graduate level mathematics.
Scaling is a major theme in The Nonlinear World. Sensitive dependence on perturbations is significant because the world at a scale below our perception contains a constant supply of imperceptible perturbations.
The most important feature of the nonlinear world is that disparate space-time scales (e.g. macroscopic and microscopic scales) can interfere with each other. Consequently, events of the world directly observable on our own space-time scale are, generally speaking, not closed within themselves. … The so-called chaos clearly exhibits this intrusion of the unknowable (at small scales) into the world we experience directly.
After some introduction, Oono begins by examining chaos, randomness, and related ideas. He discusses philosophical interpretations of chaos and randomness, as well as foundational mathematics: dynamical systems, measure theory, ergodic theory, and computability.
Once the reader has been thoroughly immersed in chaos and randomness, made aware of its ubiquity, Oono turns to his main theme. If nonlinear effects and uncertainty abound, why the world is nevertheless as orderly and predictable as it is? As Oono observes,
Isn’t it an amazing empirical fact that the world full of noise and nonlinearity does not give us surprise at every moment?
This is a profound question, and opposite to the question more commonly asked: rather than asking how complexity can arise out of simple systems, he asks here how simplicity can arise out of complex systems.
So how do we make sense of our nonlinear world? Through phenomenology. The subtitle of The Nonlinear World is Conceptual Analysis and Phenomenology. But what is phenomenology? Oono does not give a direct answer. He says ‘Instead of giving its definition, let us observe typical examples.’ These examples are not simple — for instance, the first example is the Navier-Stokes equations — and so the meaning of phenomenology is hard to pin down.
Oono says that large nonlinear effects
… do not show up haphazardly at arbitrary places. This is because there are (at least approximately) renormalizable structures in the broadest sense of the word. This is the secret of the world that allows us to gain its phenomenological understanding …
Phenomenology is the key to understanding the nonlinear world, and renormalization is the “secret” to phenomenology. So what is renormalization? First, Oono says that he uses the term renormalization in a broader sense than the meaning of the term in high-energy physics. Second, as with phenomenology, Oono gives examples of renormalization rather than a precise definition, and these examples are rather sophisticated.
A mathematically minded reader could enjoy The Nonlinear World by reading the substantial mathematical portions of the book, possibly without fully understanding the philosophical context. A philosophically minded reader, however, may be at more of a loss without being able to understand the mathematical examples.
John D. Cook is an independent consultant and blogs at The Endeavour.