David Ruelle

This is a curious book. Curious because some of the topics are hard to grasp. Curious because Ruelle tells some rather colourful stories about some of the characters in it ...such as Godel and Theophile de Donder. And curious because of the sheer breadth of subjects/topics that are covered. These range from mathematics and physics, to the probabilities of coin tosses, sensitive dependence, turbulence...and Ruelle's role in developing the ideas of chaos; economics, quanta, entropy and irreversibility, information theory and Godel's theorem. Each of these sections is written more or less like a stand-alone essay...though there is some dependence on previous sections. Anyway, It's hard to summarise and do it justice. But I'll pluck out a few gems. Because of sensitive dependence slightly different initial conditions can lead to vey different outcomes ....illustrated really well with a billiard ball analogy. But scaled-up this means that with meteorology, it takes about a week for errors to become unacceptable and Ruelle claims that we already know that we shall not be able to predict the weather for more than one or two weeks in advance. I notice that, in 1968, Ruelle was trying to teach himself hydrodynamics by reading the book "Fluid Mechanics" by Landau and Lifshitz. I was trying to do the same thing at more or less the same time ....maybe a bit earlier .....but certainly with less success. (A difficult book).
A great story about Godel. Somebody had briefly occupied his office while he was away and had left a polite note saying "I hope to have the chance to get to know you more intimately on a later occasion"....he received a response from Godel which consisted of the note with that sentence underlined bye Godel and, added, in pencil, the question "Exactly what do you mean?" Ruelle describes him as "a small man yellowish, and emaciated and he wore cotton plugs in his ears". Another story about the French mathematician Jean Leray who told Ruelle that his inspiration for his great 1934 paper on hydrodynamics was watching the vortices of the River Seine as it flows past the piles of the Pont Neuf in Paris. And I've often pondered over the idea that all the molecules in a box....if you wait long enough will randomly arrange themselves all into one corner. Ruelle, does much the same with a layer of hot water poured over a layer of cold....they begin to mix into luke warm water but in principle could return to the two layers of hot and cold. How long would we have to wait....Ruelle just says too long ...and that seems to be much longer than the current age of the universe.
I like the reconciliation between chance and determinism cited by Poincare: "A very small cause, which escapes us, determines a considerable effect which we cannot ignore, and we then say that this effect is due to chance". (Though there was no quantum uncertainty in Poincare's day). And Ruelle has some harsh advice for Economists: "Legislators and government officials are thus faced with the possibility that their decisions, intended to produce a better equilibrium, will, in fact, lead to wild fluctuations, with possibly quite disastrous effects". (I think our Reserve Bank and the Federal Reserve should take notice...and I recall a friend of mine, then working in the Treasury Department in about 1990 saying that there were room-fulls of economists there who had run out of ideas about what action could be taken to fix the high inflation and high unemployment ).
As I said at the start....a curious book. But interesting and as I've been re-reading it to write this review, I've come to appreciate it more...even if it is difficult to summarise. I give it 5 stars.… (meer)

Indeholder "Foreword", "Introduction", "Part I. Steps to a deterministic interpretation of chaotic signals", " 1. Descriptions of turbulence", " 2. A bit more on turbulence. The Navier–Stokes equation", " 3. The Hénon mapping", " 4. Capacity and Hausdorff dimension", " 5. Attracting sets and attractors", " 6. Extracting geometric information from a times series", "Part II. The ergodic theory of chaos", " 7. Invariant probability measures", " 8. Physical measures", " 9. Characteristic exponents", " 9.1 The multiplicative ergodic theorem", " 9.2 Characteristic exponents in localisation theory", " 9.3 Characteristic exponents in differentiable dynamics ", " 9.4 The spectrum of characteristic exponents as a classification tool", " 9.5 Parameter dependence", " 9.6 Experimental determination of characteristic exponents", " 10. Invariant manifolds", " 11. Axiom A and structural stability", " 12. Entropy", " 12.1 Entropy and characteristic exponents", " 12.2 SRB measures", " 13. Dimensions", " 13.1 Strange attractors as inhomogeneous fractals", " 13.2 Information dimension, entropy and characteristic exponents", " 14. Resonances", " 15. Conclusions", "References", "Bibliography", "Index".

Lidt om kaosteori og strange attractors.… (meer)

The man who coined the term 'strange attractor' provides a contemporary and a personal look at mathematics in an easy to read fashion. This book is a little bit eclectic which may be considered as a positive point for people outside the world of mathematics, and Ruelle does not adhere to a linear organization, preferring to jump from one subject to another but manages to provide good connections and insights.

Among the mathematicians he writes about, I found the case of Alexander Grothendieck very remarkable, inspiring, sad and hilarious [1]. This is a very interesting part of the history of mathematics which includes important lessons about organizations, politics and power relations.

Ruelle's discussion on some messy parts of math and proof-checking is very good and he poses important questions about proofs getting longer and longer and formalisms required to handle things as rigorously as possible.

The closing chapters are devoted to Ruelle's area of expertise and he writes very strongly on mathematical physics and give very good examples how diverse scientific fields help each other.

1- See my blog entry 'Corporatism in Science and Math: Mathematician Missing - Part 2':

http://ileriseviye.org/blog/?p=2223… (meer)