When a tool maker designs a drill she does not expect it to also be useful as a hammer or a torch. You might be able to use it as either at a pinch but it would not be very effective. If you want a universal tool that can do lots of different things you need to design it with those things in mind from the start and the result may be just as complex as a box of individual tools.

The strangest thing is that this is not the case with mathematical tools. A concept or method used to solve one mathematical problem often turns out to be just as useful in solving others that are completely different and seemingly unrelated. To give a simple example, the number *pi* was first defined to quantify the ratio of the circumference to diameter of a circle, yet it appears in a whole host of mathematical and physical equations that have nothing to do with circles. The same is true for a few other special numbers such as *e*, the base of the natural logarithms. Why do the same few numbers keep coming up in mathematics instead of different ones for every problem? Other examples abound. Special functions, groups, algorithms and many more mathematical structures prove useful over and over again. Why does mathematics have this natural universality?

People who don’t know mathematics well think that mathematicians invent the methods they use in the same way that people invent stories or fashions. Most mathematicians say that this is not the case. Their experience of developing new mathematics feels more like a process of discovery rather than invention. It is as if the mathematical structures were already there before any human was aware of them. You can also invent mathematical structures like the game of chess, but chess is not regarded as important in mathematics because it is not useful for solving unrelated problems. It does not have universality and it is this universality that distinguishes the interesting mathematics from the uninteresting.

Why then is such universality so easy to find in mathematics without trying to look for it? Why does it even extend into physics where deep mathematical ideas originally used to solve problems in pure mathematics turn out to be important for understanding the laws of nature (complex numbers, differential geometry, topology, group theory etc.) ? It is even more surprising to pure mathematicians when theory developed by physicists turns to be useful in mathematics, yet even string theory has already proved useful for solving a whole host of mathematical problems that seemed otherwise intractable. It remains a huge and deep mystery why this happens.

Yesterday the Abel committee in Norway announced that it was awarding its annual prize of 6 million Kroner to Yakov Sinai for his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics. Sinai’s work covers complex dynamical non-linear systems with many variables. Naively our prior expectation for such systems would be that they are going to behave in complex unpredictable ways and the only thing they are likely to have in common is going to be randomness, but that is not what happens. Systems in statistical physics have entropy and temperature. These are macroscopic emergent quantities that follow derived laws which are common to different systems irrespective of the microscopic description of the dynamics. They have phase transitions and near these transitions you get critical phenomena that obey universal laws.

Sinai looked in particular at chaotic systems of non-linear dynamical equations where more universal emergent behavior is found and described in terms of certain Feigenbaum constants. Another area he worked in was algorithmic complexity of binary sequences and dynamical systems. All this work is highly applicable to practical problems but it is also important as a tool for understanding universality and why it arises. Perhaps one day by building on the work of Sinai we will learn much more about the unity of mathematics, the laws of physics and why these things are so beautifully connected by universality.

Congratulations to Yokov Sinai for this well deserved award which will raise the profile of such important work.

One possible, and highly natural, explanation for the universality of Sinai’s modeling of nonlinear dynamical systems is the inherent fractal self-similarity of nature.

Same explanation for Benford’s Law.

If one observes the Universe and determines that it is fractal that’s just an observation not an explanation. Sinai is not the first or the only researcher to discover universal behaviour in the phase transitions of disparate non-linear systems and the question remains: why do these universal behaviours emerge? What is the mechanism behind it?

I believe it all has something to do with algorithmic complexity and binary codes which is essentially a perspective on efficiency. Maybe it has something to do with SUSY: http://www.bottomlayer.com/PWJun10gates.pdf.

I’ll take an observation over pipe dreams any day.

Oh, hell, pull the plug out of your butt Robert! Have you never heard of the 11:11 phenomenon? Many people believe it is somehow related to fractal geometry – http://www.11phenomenon.com/. Life is a mystery . . . it’s also a comedy . . . unless you have a plug stuck in your butt!

The example of pi in formulae apparently unrelated to geometry was also used by Eugene Wigner in his 1960s paper “The unreasonable effectiveness of mathematics in the natural sciences”. Unfortunately, Wigner’s example is the Gaussian/normal distribution law, which is an example of obfuscation. Laplace (1782), Gauss (1809), Maxwell (1860) and Fisher (1915) wrote the normal exponential distribution with the square root of pi in the normalization outside the integral. But Stigler in 1982 rewrote the equation with pi in the exponent, making the formula look less mysterious because the exponent is then the area of a circle (in other words, Poisson’s exponential distribution, adapted to circular areas, with areas expressed in dimensionless form); if you think of the use of the normal distribution to model CEP error probabilities for missiles landing around a target point (Kahn 1960):

http://nige.files.wordpress.com/2011/01/gaussian-distribution-and-physics.gif

It’s a terrific example of how thinking about military applications of mathematics undermines the dogmatic prejudices that reside at the heart of so-called “pure mathematics”, which is sheer ignorance, enforced by censorship and coercion from on-high. :-)

“Perhaps one day by building on the work of Sinai we will learn much more about the unity of mathematics, the laws of physics and why these things are so beautifully connected by universality.”

I could not agree with you more. In fact, this is precisely what I did for about two decades now: showing how concepts related to nonlinear dynamics and chaos in general (and Feigenbaum’s Universality in particular) are key to understanding the family structure of particle physics and the parameters of the Standard Model.

Congrats.

Funny, I was actually writing a thesis on the Pirogov-Sinai theory because my undergrad thesis adviser Miloš Zahradník (and the co-author of our linear algebra textbook) – whom I kept when I switched to a string theory thesis (although he has no particle physics background) – is the world’s top expert in that.

http://scholar.google.com/scholar?q=pirogov-sinai&hl=en&lr=&btnG=Search

See that the first listed paper on the theory is by Zahradník. I learned what it is and how the expansion worked but I guess that I would have never found an important new result on that.

Very good, it would be interesting to hear more about his work.

BTW the first name is Yakov, not Yokov… ;-) Я́ков Григо́рьевич Сина́й

fixed thanks

this is great, I stumbled across some really investing numbers involving three Feigenbaum constants playing around with Maple after the LHC sploded the first time… came out of my wacky attempts to discover something new about the Riemann Zeta function… literalally popped out of a calculation so I put it into this Canadian inverse symbol calculator and that’s how I found out it how to get back to.the literature from string of digits

“Why do the same few numbers keep coming up in mathematics instead of different ones for every problem?” To early Greek philosophers these numbers might actually have gone against their belief in what numbers are and how they relate to an ordered universe. Turns out that most of these numbers are at odds with what was expected, e.g. by Pythagoras to be the numbers that lie behind a harmonic universe.