The featured Quanta Magazine article right now is about the connection between the colliding blocks and quantum search. If you have never watched the "surprising appearance of \(\pi\) in a counting problem", you should watch these three videos by 3Blue1Brown:
* Pi: how it appears in colliding blocks (5 minutes)A small block is moving back and forth between a wall and another rectangular block where the mass ratio is a power of 100. The collisions are elastic and you need to count how many collisions there are. You will find that the number of collisions is a number like 314159... And it goes on and on, you clearly see the pi. In the phase space, the critical points belong to a circle, a beautiful problem with a beautiful solution due to Gregory Galperin (2003).
* Why it's so (15 minutes)
* Add a beam of light (14 minutes)
The new Quanta Magazine article points out that the very same "approximation of a circle" by many points appears in Grover's algorithm, the most famous quantum computer's method to search in a database. In that algorithm, the state vector is also being gradually turned to the right direction so you're also approximating a circle. The similarity of the "quantum search" and "the billiards" was promoted in a recent preprint by Adam Brown.
It is not really surprising that approximations of circles by many points appear in the quantum algorithms. In quantum mechanics, the space of normalized states is a (complex projective) sphere, a generalization of a circle which has lots of circles in it. For this simple reason, \(\pi\) is guaranteed to be omnipresent in quantum mechanics. The same \(\pi\) was a bit more surprising to be found in the classical collision problem.
The same geometry is being used in the "colliding blocks problem" as well as in the "quantum search". I would still say that physically, those are completely different things and the presentation of these two situations as "siblings" may bring as much as confusion as helpful ideas. But it's surely true that elegant mathematical solutions exist, they may be recycled at seemingly unrelated places, and \(\pi\) appears at many unexpected (and seemingly "discrete", not circular) places.
Now, Pradeep Mutalik posted some solutions to his puzzles in the Quanta Magazine:
In October, he used some puzzles involving classical random generators to claim that probability isn't fundamental in quantum mechanics. The problem is that probability is totally fundamental in quantum mechanics – in the laws that Nature actually follows – and playing with some puzzles that aren't matched with the fundamental processes of Nature can change nothing about this conclusion.
Now, he and his readers discuss solutions to some problems that lead most of them to use some numerical methods or brute force. And Mutalik's far-reaching conclusion is as follows:
So was Wigner right?What a pile of propagandist šit. It reads just like the activists' rewriting of the history.
Yes and no. He was right that elegant... works for... However, there are areas in physics, and many more in other complex sciences, where this is not true. Perhaps Wigner was a bit of a mystic, or a “patriot of mathematics,” as a couple of readers suggested, and somewhat overstated the case.
First, Mutalik asks whether Wigner was right but he doesn't even discuss Wigner's famous article (click for the full text). To make things worse, Mutalik doesn't even quote a single statement that Wigner has made. He only quotes the verbless title of the famous paper
The Unreasonable Effectiveness of Mathematics in the Natural Sciences.I am sorry but without discussing a single statement (let alone arguments), you can't justifiably conclude that Wigner was half-wrong (or a mystic who overstated something). Mutalik makes these incredibly far-reaching claims (that Wigner was half-wrong) and wants his readers to believe that Wigner was a half-moron and Mutalik is perhaps smarter and more correct than Wigner was. And perhaps Wigner was completely unaware of the omnipresent complexity in the world.
But these statements are utterly ludicrous. Wigner was not only aware of the omnipresent complexity in the world but it is discussed in his essay in quite some detail and the character of this complexity actually plays an important role in making his statements true and profound. The cool thing is that even in the presence of complexities, there are regularities. As Wigner wrote:
...It is, as Schrödinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered. One such regularity, discovered by Galileo, is that two rocks, dropped at the same time from the same height, reach the ground at the same time. The laws of nature are concerned with such regularities. Galileo's regularity is a prototype of a large class of regularities. It is a surprising regularity for three reasons.Wigner continued by listing these three reasons. First, Galileo's "principle of equivalence" is invariant under translations in space and time and rotations – a named, recognized invariance. Second, the result is independent of a large subset of conditions (the sex of the experimenter, to pick a funny example from the text). Third, and that was most shocking for Galileo, the time the object needs to fall is independent of the material and shape of the object.
What I hate about Mutalik's whole way of talking about these matters is that he sounds as a chronic complainer who is just inventing verbal would-be justifications of the predetermined conclusion that "this can't be done by mathematics or elegant mathematics". But this whole approach is the opposite of the very purpose of physics (and mathematics).
As Wigner correctly wrote, the laws of nature are concerned with regularities. The physicist's job is to find and understand them, not to build a neverending stream of vitriol, excuses, or a whole ideology purporting to explain why they can't be found or even shouldn't be looked for. Yes, it has been proven that the trajectories in the three-body problem (three point masses affecting the motion of all three) cannot be written in terms of elementary functions.
But Mutalik and some of his readers insanely generalize this fact and contradict the following important points, among others:
* Not every question in physics is morally equivalent to the three-body problem
* When you haven't found a closed solution to a problem in X minutes, it doesn't prove that such a solution doesn't exist
* When you and a bunch of people around you have failed in this task, it still fails to prove that the solution doesn't exist
* It even fails to prove that others (perhaps smarter or more hard-working or more lucky people) won't find it
* When the trajectories are functions that aren't "elementary", it doesn't mean that "it's over". Even with new, previously non-elementary functions, mathematics may continue and one may find and prove many exact properties of these functions
* The homework problems assigned by Mutalik aren't really similar to the problems in the fundamental laws of physics
* Numerical calculations are ultimately mathematical activities, too
The fundamental difference is in the underlying normative statements. Wigner obviously assumes that the right normative statement addressed to physicists is "never give up searching for deeper regularities and relationships" (because, you know, Wigner liked that kind of work and he was good at it) while Mutalik's normative statement is "give it up and switch to the job of chronic complainers".
But chronic complainers who get quickly disoriented in the mathematics – and who produce demagogy trying to discourage others as well – are inadequate parasites who simply cannot be considered physicists. A good physicist is someone who has contributed something nice to the amazing discoveries in recent 40 years (and who is rightfully proud about it because this outcome wasn't quite a coincidence but a result of a highly intentional activity), not someone who wants to persuade others that nothing has been found in 40 years.
You know, lots of things exist in mathematics. Mathematics isn't just the three basic operations – addition, subtraction, and Feynman's path integral. ;-) Advanced enough mathematicians – which should include theoretical physicists – are persistently extending the list of structures and regularities that they consider "elementary".
The basic elementary functions of the complex variables may be written as a composition of addition, subtraction, the exponential, and the logarithm (yes, you can write multiplication using addition, exponentiation, and logarithms). However, centuries ago, people added some reasonably close functions such as Bessel's functions, Gamma function, Riemann zeta function, and more. They have also extended the list of friends of sequences (Bernoulli numbers, various solutions to counting problems which are coefficients from partition sums...) and many other things (topological invariants and polynomials associated with manifolds, shapes, and other structures). You may give symbols to these "previously non-elementary" functions and if you're active and hard-working, you will find out it was a good idea to do so rather soon. These new functions have lots of properties, regularities, and relationships as well. It is still mathematics. In fact, it is often cooler and more mathematical mathematics than what you had before.
The solution to the three-body problem may be considered an ugly new special function. But if you choose a name for this multi-parameter function, you will still be able to ask some questions (e.g. how many times the distance between the three bodies exceeds some huge threshold, asymptotically), and many such questions will have nice and compact answers that you will be able to prove in many cases, too.
In some of his paragraphs, Mutalik wrote "pretty much the correct thing" but decorated it with some serious inaccuracies as well as an irrationally negative, anti-mathematical emotional flavor. For example, he wrote:
...The entire field of thermodynamics and statistical mechanics occurs because physical systems such as an ice cube in a bath of water are too complex to represent mathematically for every single water molecule...This statement resembles some correct propositions we can make. But it is just wrong and misleading about so many important details. First, thermodynamics didn't arise as a simplification of some overly complex situations. Thermodynamics – which indeed existed before statistical mechanics – captured people's experimentally measured properties of heat and work. And they were very simple!
Historically speaking, statistical physics wasn't invented as a tool to simplify the equations governing a large number of atoms. Instead, statistical physics was invented as a microscopic explanation of the apparently simple, macroscopic laws of thermodynamics that were previously found, largely by direct experiments. It was an amazingly deep physics discovery by the likes of Ludwig Boltzmann that the thermal phenomena (hot, cold, warming...) are explained in terms of statistical properties of atoms. In the real world, statistical physics as a field also uses lots of approximations (and expansions) that are valid in the thermodynamic limit (i.e. for asymptotically many atoms). But the very term "statistical physics" doesn't make approximations mandatory.
On top of that, the existence of the closed solution for the approximations in the thermodynamic limit is a great news about the power of mathematics, not bad news as Mutalik ludicrously paints it. The cool thing is that even intractable problems become tractable in most limits such as the thermodynamic one: mathematics has placed Her "guards" to most of the important places so that She has physics, the world, and the set of possible problems under control. Wigner discusses these things far more intelligently than Mutalik. I find it crazy for people like Mutalik who are clearly not in Wigner's league to be encourage to post arrogant articles unjustifiably making wrong claims such as "Wigner was half-wrong and I am smarter than he was".
This article by Mutalik is seemingly "just a page with puzzles and their solutions" but it greatly encourages all the readers to lower their standards, to increasingly dislike physics and mathematics, to nurture increasingly irrational beliefs that "mathematics and physics breaks down here and there and perhaps almost everywhere", and to replace the proper approach of a deep physicist of mathematician – which also includes the humility towards the things that "I" don't understand and even those that "we" haven't understood yet – by some group think of lazy and sloppy mediocre non-thinkers who haven't looked carefully enough and who want to sell this collective defect of theirs as a virtue.