The unreasonable beauty of mathematics in the natural sciences*

Imagine a man and a woman, seeing and liking each other at a party or club or so. They start talking, the mutual attraction is obvious, but they want to be casual for two minutes. So they exchange informal formalities about doesn't matter what. Then he asks her: "so, what do you do?", and she replies "I'm a poet". What if the guy would say something like "I hate poetry!", or even declare proudly "I never knew how to use letters to write words and stuff, and I don't care!". Or imagine she's a musician, and he says "I hate music!". There are two things we can say about that kind of guy. First, he is very rude, he never ever deserves a second chance with that girl or any other human being for that matter. He should be isolated, kept outside society. Second, or maybe this should be first, how on earth can he be proud for being illiterate!

You probably guessed that this story is true. OK, In my case it was about math instead of poetry, and the genders are reversed. This happened to me or to anyone in the same situation quite often. There is no political correctness when it comes about math, maybe because one tends to believe that if you like math, you have no feelings, and such a remark wouldn't hurt you. And I actually was never offended when a girl said such outrageous things like that she hates math. Because whenever a girl told me she hates math, I knew she calls math something that really is boring and ugly, and not what I actually call math. Because math as I know it is poetry, is music, and is a wonderful goddess.

The story continues, years later. You talk about physics, with people interested in physics, or even with physicists. And you say something about this being just a mathematical consequence of that, or that certain phenomenon can be better understood if we consider it as certain mathematical object. It happens sometimes that your interlocutor becomes impatient and says that this is only math, and you were discussing physics, that math has no power there, and so on. Or that math is at best just a tool, and it actually obscures the real picture, or even that it limits our power of understanding.

People got the wrong picture that math is about numbers, or letters that stand for unknown numbers, or being extremely precise and calculating a huge number of decimals, or being very rigid and limited. In fact, math is just the study of relations. You will be surprised, but this is actually the mathematical definition of math. Numbers come into math only incidentally, as they come into music, when you indicate the duration or the tempo. Math is just a qualitative description of relations, and by relations we can understand a wide rainbow of things. I will detail this another time.

Imagine you wake up and you don't remember where you are, or who you are, like you were just born. You are surrounded by noise, which hurts your ears and your brain, meaningless random violent noise. You run desperately, trying to avoid it, but it is everywhere. And you finally find a spot where everything becomes suddenly wonderful: the noise becomes music, a celestial, beautiful music, and everything starts making sense. You are in a wonderful Cathedral, and you are tempted to call what you are listening "music of the spheres". The same music played earlier, but you were in the wrong place, where the acoustics was bad, or the sounds reached your ear in the wrong order, because of the relative positions of the instruments. Or maybe your ears were not yet tuned to the music. The point is that what seemed to be ugly noise, suddenly became so wonderful.

So, when someone says "I hate math!", all I hear is "I am in the Cathedral you call wonderful, but in the wrong place, where the celestial music becomes ugly violent noise!".

If you are interested in physics, you entered the Cathedral. But if you hate math, you will not last here, and maybe it is better to get out immediately! And if you are still interested in physics, come inside slowly, carefully choosing your steps, to avoid being assaulted by the music of the spheres, to allow it gently to enter in your mind, and to open your eyes. Choose carefully what you read, what lectures you watch, and ask questions. Don't be shy, any question you will ask is the right question for your current position, and for your next step.

There are some places in the Cathedral where the music is really beautiful. If you meet people there, to share the music, to dance, you will feel wonderful. If not, you will feel lonely. So you will want to share that place, you will want to invite your friends to join you.

The reason I love physics, is that I want to find these places. The reason I read blogs and papers, is that I want them to help me find such places. The reason I write papers, and I blog about this, is that I would like to  share my places with others. I attend conferences (four so far this year) because they are like concerts, where you get the chance to listen some wonderful music, and to play your own.

But these are just words. I would like to write more posts in which I show the unreasonable beauty of math in physics, with concrete examples. Judging by the statistics, I have a few readers; judging by the number of comments, I don't really touch many of them. I know sometimes I am too serious, or too brief when I should explain more, especially when mathematical subtleties are involved. I am not very good at explaining abstract things to non-specialists, but I want to learn. I would like to write better, to be more useful, so, I would like to encourage comments and suggestions. Ask me to clarify, to explain, to detail, to simplify. Tell me what you would like to understand.

To start, I would like to write about vectors. They are so fundamentals in all areas of physics and mathematics, so I think it's a good idea to start with them. You may think they are too simple, and that you know all about them from high school, but you don't know the whole story. Later, when I will say something about quantum mechanics and relativity, they will be necessary (after all, according to quantum mechanics, the state of the universe is a vector). On the other hand, if you will understand them well, you will be around half of the way to understand some modern physics.

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* You surely guessed that the title is a reference to Wigner's brilliant and insightful lecture, The unreasonable effectiveness of mathematics in the natural sciences.

Update, October 14, 2014

I just watched an episode of the Colbert Report, where the mathematician Edward Frenkel was invited in April this year. It was about Frenkel's new book and about his movie. He discusses at some point precisely the fact that it is so acceptable to hate math, as opposed to hating music or painting. Here is what he says for The Wall Street Journal:

It's like teaching an art class where they only tell you how to paint a fence but they never show you Picasso. People say 'I'm bad at math,' but what they're really saying is 'I was bad at painting the fence.'

Also see this video:

Will science end after the last experiment will be performed?

Science is supposed to work like this: you make a theory which explains the experimental data collected up to this point, but also proposes new experiments, and predicts the results. If the experiment doesn't reject your theory, you are allowed to keep it (for a while).

I agree with this. On the other hand, much of the progress in science is not done like this, and we can look back in history and see.

Now, to be fair, making testable predictions is something really excellent, without which there would be no science. To paraphrase Churchill,

Scientific method is the worst form of conducting science, except for all the others.

I am completely for experiments, and I think we should never stop testing our theories. On the other hand, we should not be extremists about making predictions. Science advances in the absence of new experiments too.

For example, Newton had access to a lot of data already collected by his predecessors, and sorted by Kepler, Galileo, and others. Newton came with the law of universal attraction, which applies to how planets move, in conformity with Kepler's laws, but also to how bodies fall on earth. His equation allowed him to calculate from one case the gravitational constant, but then, this applied to all other data. Of course, later experiments were performed, and they confirmed Newton's law. But his theory was already science, before these experiments were performed. Why? Because his single formula gave the quantitative and qualitative descriptions of a huge amount of data, like the movements of planets and earth gravity.

Once Newton guessed the inverse square law, and checked its validity (on paper) on the data about the motion of a planet and on the data about several projectiles, he was sure that it will work for other planets, comets, etc. And he was right (up to a point, of course, corrected by general relativity, but that's a different story). For him, checking his formula for a new planet was like a new experiment, only that the data was already collected by Tycho Brahe, and already analyzed by Kepler.

Assuming that this data was not available, and it was only later collected, would this mean that Newton's theory would have been more justified? I don't really think so. From his viewpoint, just checking the new cases, already known, was a corroboration of his law. Because he could not come up with his formula from all the data available. He started with one or two cases, then guessed it, then checked with the others. The data for the other cases was already available, but it could very well be obtained later, by new observations or experiments.

New experiments and observations that were performed after that were just redundant.

Now, think at special relativity. By the work of Lorentz, Poincaré, Einstein and others, the incompatibility between the way electromagnetic fields and waves transform when one changes the reference  frame, and how were they expected to transform by the formulae known from classical mechanics, was resolved. The old transformations of Galileo were replaced by the new ones of Lorentz and Poincaré. As a bonus, mass, energy and momentum became unified, electric and magnetic fields became unified, and several known phenomena gained a better and simpler explanation. Of course, new predictions were also made, and they served as new reasons to prefer special relativity over classical mechanics. But assuming these predictions were not made, or not verified, or were already known, how would this make special relativity less scientific? This theory already explained in a unified way various apparently disconnected phenomena which were already known.

One said that Maxwell unified the electric and magnetic fields with his equations. While I agree with this, the unification became even better understood in the context of special relativity. There, it became clear that the electric and magnetic fields are just part of a four-dimensional tensor $F$. The magnetic field corresponds to the spatial components $F_{xy}$, $F_{yz}$, $F_{zx}$, and the electric field to the mixed, spatial and temporal, components $F_{tx}$, $F_{ty}$, $F_{tz}$ of that tensor. Scalar and vector potentials turned out to be unified in a four-dimensional vector potential. Moreover, the unification became clearer when the differential form of Maxwell's equations was found, and even clearer when the gauge theory formulation was discovered. These are simple conceptual jumps, but they are science. And if they were also accompanied by empirical predictions which were confirmed, even better.

Suppose for a moment that we live in an Euclidean world. Say that we performed experiments and tested the axioms of Euclid. Then, we keep performing experiments to test various propositions that result from these axioms. Would this make any sense? Yes, but not as much as it is usually implied. They already are bound to be true by logic, because they are deduced from the axioms, which are already tested. So, why bother to make more and more experiments, to test various theorems in Euclidean geometry? This would be silly. Unless we want to check by this that the theorems were correctly proven.

On the other hand, in physics, a lot of experiments are performed, to test various predictions of quantum mechanics or special relativity, or of the standard model of particle physics, which follow logically and necessarily from the postulates which are already tested decades ago. This should be done, one should never say "no more tests". But on the other hand, this gives us the feeling that we are doing new science, because we are told that science without experiment is not science. And we are just checking the same principles over and over again.

Imagine a world where all possibly conceivable experiments were done. Suppose we even know some formulae that tell us what experimental data we would obtain, if we would do again any of these experiments. Would this mean that science reached its end, and there is nothing more to be done?

Obviously it doesn't mean this. We can systematize the data. Tycho Brahe's tables were not the final word in the astronomy of our solar system. They could be systematize by Kepler, and then, Kepler's laws could be obtained as corollaries by Newton. Of course, Kepler's laws have more content that Brahe's tables, because they would apply also to new planets, and new planetary systems. Newton's theory of gravity does more than Kepler's laws, and Einstein's general relativity does more than Newton's gravity. But, such predictions were out of our reach at that time. Even assuming that Tycho Brahe had the means to make tables for all planets in the universe, this would not make Kepler's laws less scientific.

Assuming that we have all the data about the universe, science can continue to advance, to systematize, to compress this data in more general laws. To compress the data better, the laws have to be as universal as possible, as unified as possible. And this is still science. Understanding that Maxwell's four equations (two scalar and two vectorial) can be written as only two, $d F = 0$ and $\delta F = J$ (or even one, $(d + \delta)F=J$), is scientific progress, because it tells us more than we previously knew about this.

But there is also another reason not to consider that science without experiments is dead. The idea that any theory should offer the means to be tested is misguided. Of course, it is preferred, but why would Nature give us the mean to check any truth about Her? Isn't this belief a bit anthropocentric?

Another reason to not be extremist about predictions is the following. Researchers try to find better explanation of known phenomena. But because they don't want they claims to appear unscientific, they try to come up with experiments, even if it is not the case. For example, you may want to find a better interpretation of quantum mechanics, but how would you test it? Hidden variables stay hidden, alternative worlds remain alternative, if you believe measurement changes the past, you can't go back in time and see it changed without actually measuring it etc. It is like quantum mechanics is protected by a spell against various interpretations. But, should we reject an alternative explanation of quantum phenomena, because it doesn't make predictions that are different from the standard quantum formalism? No, so instead of calling them "alternative theories", we call them "interpretations". If there is no testable difference, they are just interpretations or reconstructions.

A couple of months ago, the physics blogosphere debated about post-empirical science. This debate was ignited by a book by Richard Dawid, named String Theory and the Scientific Method, and an interview. His position seemed to be that, although there are no accessible means to test string theory, it still is science. Well, I did not write this blog to defend string theory. I think it has, at this time, bigger problems that the absence of means to test what happens at Plank scale. It predicts things that were not found, like supersymmetric particles, non-positive cosmological constant, huge masses for particles, and it fails to reproduce the standard model of particle physics. Maybe these will be solved, but I am not interested about string theory here. I am just interested in post-empirical science. And while string theory may be a good example that post-empirical science is useful, I don't want to take advantage of the trouble in which this theory is now.

The idea that science will continue to exist after we will exhaust all experiments, which I am not sure describes fairly the real position of Richard Dawid, was severely criticized, for example in Backreaction: Post-empirical science is an oxymoron. And the author of that article, Bee, is indeed serious about experiment. For example, she entertains a superdeterministic interpretation of quantum mechanics. I think this is fine, given that my own view can be seen as superdeterministic. In fact, if you want to reject faster-than-light communication, you have to accept superdeterminism, but this is another story. The point is that you can't make an experiment to distinguish between standard quantum mechanics, and a superdeterministic interpretation, because that interpretation came from the same data as the standard one. Well, you can't in general, but for a particular type of superdeterministic theory, you can. So Bee has an experiment, which is relevant only if the superdeterministic theory is such that making a measurement A, then another one B, and then repeating A, will give the same result whenever you measure A, even if A and B are incompatible. Now, any quantum mechanics book which discusses sequences of spin measurements claims the opposite. So this is a strong prediction, indeed. But how could we test superdeterminism, if it is not like this? Why would Nature choose a superdeterministic mechanism behind quantum mechanics, in this very special way, only to be testable? As if Nature tries to be nice with us, and gives us only puzzles that we can solve.

Science and lottery

Ask anyone who buys lottery tickets systematically, most of them will confirm they have a system. Most of them seem to be based on birthdays, although the days of the month are a serious limitation of the possibilities. Some play random numbers, which they withdraw from a bag (this is the best "system"), but most have a sort of a system.

I don't believe there is a winning system. People tried to convince me that numbers have their own life, and they are not quite random. "Laymen" tend to believe that if you toss a coin and you get head, next time are bigger chances to get tail. If you pay attention in US movies, you will see that almost every time a number appears, its digits are unique, for example 52490173, a permutation of a subset of 0123456789. Except of course for the phone numbers, which start with 555. This is because a number like 254377 seem too special. In fact such numbers which don't have unique digits are encountered more often in real life. So I don't buy the idea that lottery numbers are not random. Some try to convince me that because the balls are not perfect, they are biased, and some numbers are more likely to be extracted than others. Even if this is the case, I don't think you can actually use this to predict the numbers.

My opinion is that from lottery only the house wins, at least on average. This doesn't mean that if you play you will not win.

Now, since almost anyone who plays systematically has a system, and since the winner will be among these guys, most winners have a system. So, what happens when you win? You will believe that finally your system turned out to be correct. You may even write a book in which you explain the system, end get even richer by selling it. But you will definitely believe that you won because of your system. While I don't believe your system. You can tell me that your system turned out to be correct, even that it is science, because it made predictions, and it was confirmed by the most difficult test: actually playing and winning in real life! But I still don't believe in your system. Because anyone who wins has a system, and he won because sometimes people win, but not because of the system.

Now, imagine a world in which

• in order for a paper to be considered scientific, its basic hypotheses have to be falsifiable by experiments
• scientists have to publish lot of original papers, otherwise they will perish

This is pretty much our world, and I think that these two conditions lead to an avalanche of predictions. Whenever an experiment will be about to be performed, scientists will bet for various outcomes. And just like in betting, they will try to cover all possible outcomes.

So, after the experiment is performed, some will win the lottery, while some will lose it. Does this ensure that the winners really cracked the laws of Nature? Did they win because of their theory, because of their system? Or just because of pure luck, and they just tend to give credit to their system?

Doesn't this mean that something is wrong with the way we define science? Making predictions is easy. Suppose that there are 5 possible outcomes, and there are 5 theories predicting them, one for each outcome. Suppose that the experiment corroborates one of them, and falsifies the other four. Why where those 4 wrong in the first place? Just because after the experiment they turned out to be wrong? Why couldn't we see the reason why they are wrong before performing the experiment? What if the fifth, which was corroborated, is correct by a coincidence, for the wrong reason? What if there are 10 possible other explanations of the same result?

Yes, it is possible for a theory to be right for the wrong reason. Consider for example the following calculation:

The result is clearly correct, but the proof is wrong.

If a theory makes a correct prediction, this doesn't mean that it is correct. This is why we never consider a theory to be proved, or even confirmed. We just say that the experimental results corroborate it. Maybe later we will find a better theory, which will make the right predictions for the right reasons.

The problem is that, if we will find another theory which makes the same predictions, it will be considered inferior. The theory will be asked to come with new experimental proposals and its own predictions, which will contradict the predictions of the previous theories. If it will not be able to make new predictions, rather than being considered equal to the currently accepted one, it will be considered inferior. Because the current one made new predictions, but the new one made the same predictions.

This means that from two theories making the same predictions, the one that was proposed earlier will have some advantages over the one that was proposed at a later time. Even if the latter is conceptually superior, or simpler, or have other advantages.

Castiglioncello

Last week I was in Castiglioncello. It is a small but very beautiful town in Tuscany, Italy, somewhere not far from Pisa. Between the plane and the train, I had time to detour and see the leaning tower.

 (also check this optical illusion inspired by the leaning tower.)

The reason of my visit to Castiglioncello was a physics conference :

It is organized every other year, mainly by Thomas Elze, a brilliant physicist and a dear friend who invited me. It takes place in Castello Pasquini:

Castello Pasquini

I will not list now all about 150 participants, excellent physicists from various countries and areas of research. I will mention for the moment just the Nobel prize winner Gerard 't Hooft, Tom Kibble, who also deserves a Nobel prize for co-discovering the Higgs boson, and the Fields medalist  Alain Connes. It was an excellent opportunity to finally meet in person other people, with whom I just communicated via Internet, or whom I knew only by their research papers, and also to meet again people whom I knew from other conferences.

Initially, I thought there was no beach, because people were sun bathing on the cliffs,

but asking a sandy girl, she pointed me to some stairs

leading to a real beach

It was so difficult to decide which of the talks to skip to visit Castiglioncello, or to swim in the sea. Luckily, I could swim in the night.

I staid at Hotel Leopoldo, where the breakfast was made by an excellent Romanian cook named Florin, and at the desk was a cute girl named Valentina.

At lunch, we ate at the restaurant Il Peschereccio.

Courtesy of Yaron Hadad.

A great place to have some drinks and eat a good pizza is Ghostbuster.

The conference dinner took place at Grand Hotel Villa Parisi,

a wonderful place owned by Francesca, a friendly beautiful girl.

Are sciences and arts perversions?

According to Wikipedia, perversion is

a type of human behavior that deviates from that which is understood to be orthodox or normal.

Now consider the human mind. We evolved so that we find food, make children, avoid predators etc. All these are just means that serve to our survival in a universe that is trying to kill us. Or, even better, they serve to the selfish gene, to its replication.

So, the human mind shouldn't care about things that don't serve this purpose. What evolutionary purpose can be in doing math and physics? Indulging yourself in such activities doesn't serve the purpose of your survival and replication. One may say that, at least for some, science is their job, they earn money, and they survive. But researchers know better that jobs in the industry are safer and better paid, and with better success rate. And better success at ladies (although artists are doing even better). But anyway, sciences and arts are recent, so they can't be the product of mutation and selection. So, science and arts are perversions of the original purpose of the brain.

While they are not the product of evolution, they may be a byproduct. In order to survive, our ancestors had to identify patterns around them, use these patterns to make predictions. To anticipate when a wild animal will attack them, to recognize comestible fruits, to identify a sexual partner with good potential, all these require pattern recognition and the ability to make predictions. And this is why we became intelligent. So, even if we are using our intelligence to other purposes like sciences, this is a byproduct of evolution.

Nature has a way to reward you when you do something good for your genes. This is why we like to eat and to have sex. This is why we feel proud and happy when our children accomplish tasks or acquire new skills. But the blind gene doesn't know the future, so she can't reward us for actually doing something good for her. Instead, she rewards us for guessing patterns. We feel happy when we guess a pattern, and especially when a long anticipated prediction is confirmed. We identify patterns in sounds and drawings too, so this is why we like music and other arts. Even literature, builds on our predictions and anticipation. During anticipation, the brain produces the drugs that will make us happy. Building anticipation and suspense is the craft of accumulating this happiness in the consumer of one's art.

We are surprised when we make predictions which we consider safe, and turn out to be wrong. Sometimes, the anticipation accumulates the feel-good drug, and the surprise makes it explode. So this is how we laugh. Jokes are just clever ways to manipulate us into making predictions that turn out to be wrong in an unexpected and usually harmless way.

OK, so evolution explains all these as perversions of our mind, as byproducts. However, we know that science helped us to survive better. As a result of science and of its progeny, technology, we live longer, in better conditions, we find and produce food easier, we can take care better of our children and it also helps the children of our children.

So, science really helps the replication of the selfish gene.

I can't help but asking, did the selfish gene have a secret plan all this time?