Monday, October 6, 2014

Dots plus dots equal spheres

I took this photo in a bus in Pisa. We can see a pattern of spheres.

Here is how to obtain it. We overlap these dots

over a shrunk version of theirs

and we get the following pattern:

Wednesday, October 1, 2014

Living in a vector

Vectors are present in all domains of fundamental physics, so if you want to understand physics, you will need them. You may think you know them, but the truth is that they appear in so many guises, that nobody really knows everything about them. But vectors are a gate that allows you to enter the Cathedral of physics, and once you are inside, they can guide you in all places. That is, special and general relativity, quantum mechanics, particle physics, gauge theory... all these places need vectors, and once you master the vectors, they become much simpler (if you don't know them and are interested, read this post).

The Cathedral has many gates, and vectors are just one of them. You can enter through groups, sets and relations, functions, categories, through all sorts of objects or structures from algebra, geometry, even logic. I decided to show you now the way of vectors,  because I think is fast and deep in the same time, but remember, this is a matter of choice. And vectors will lead us, inevitably, to the other gates too.

I will explain some elementary and not so elementary things about vectors, but you have to read and practice, because here I just give some guidelines, a big picture. The reason I am doing this is that when you study, you may get lost in details and miss the essential.

Very basic things

A vector can be understood in many ways. One way is to see it as a way to specify how to move from one point to another. A vector is like an arrow, and if you place the arrow in that point, you find the destination point. To find the new position for any point, just place the vector in that point, and the tip of the vector will show you the new position. You can compose more such arrows, and what you'll get is another vector, their sum. You can also subtract them, just place their origins in the same point, and the difference is the vector obtained by joining their tips with another arrow.

Once you fix a reference position, an origin, you can specify any position, by the vector that tells you how to move from origin to that position. You can see that vector as being the difference between the destination, and the starting position.

You can add and subtract vectors. You can multiply them with numbers. Those numbers are from a field $\mathbb{K}$, and we can take for example $\mathbb{K}=\mathbb{R}$, or $\mathbb{K}=\mathbb{C}$, and are called scalars. A vector space is a set of vectors, so that no matter how you add them and scale them, the result is from the same set. The vector space is real (complex), if the scalars are real (complex) numbers. A sum of rescaled vectors is named linear combination. You can always pick a basis, or a frame, a set of vectors so that any vector can be written as a linear combination of the basis vectors, in a unique way.

Vectors and functions

Consider a vector $v$ in an $n$-dimensional space $V$, and suppose its components in a given basis are $(v^1,\ldots,v^n)$. You can represent any vector $v$ as a function $f:\{1,\ldots,n\}\to\mathbb{K}$ given by $f(i)=v^i$. Conversely, any such function defines a unique vector. In general, if $S$ is a set, then the set of the functions $f:S\to\mathbb{K}$ form a vector space, which we will denote by $\mathbb{K}^S$. The cardinal of $S$ gives the dimension of the vector space, so $\mathbb{K}^{\{1,\ldots,n\}}\cong\mathbb{K}^n$. So, if $S$ is an infinite set, we will have an infinite dimensional vector space. For example, the scalar fields on a three dimensional space, that is, the functions $f:\mathbb{R}^3\to \mathbb{R}$, form an infinite dimensional vector space. Not only the vector spaces are not limited to $2$ or $3$ dimensions, but infinite dimensional spaces are very natural too.

Dual vectors

If $V$ is a $\mathbb{K}$-vector space, a linear functions $f:V\to\mathbb{K}$ is a function satisfying $f(u+v)=f(u)+f(v)$, and $f(\alpha u)=\alpha f(u)$, for any $u,v\in V,\alpha\in\mathbb{K}$. The linear functions $f:V\to\mathbb{K}$ form a vector space $V^*$ named the dual space of $V$.


Consider now two sets, $S$ and $S'$, and a field $\mathbb{K}$. The Cartesian product $S\times S'$ is defined as the set of pairs $(s,s')$, where $s\in S$ and $s'\in S'$.  The functions defined on the Cartesian product, $f:S\times S'\to\mathbb{K}$, form a vector space $\mathbb{K}^{S\times S'}$, named the tensor product of $\mathbb{K}^{S}$ and $\mathbb{K}^{S'}$, $\mathbb{K}^{S\times S'}=\mathbb{K}^{S}\otimes\mathbb{K}^{S'}$. If $(e_i)$ and $(e'_j)$ are bases of  $\mathbb{K}^{S}$ and $\mathbb{K}^{S'}$, then $(e_ie'j)$, where $e_ie'_j(s,s')=e_i(s)e'_j(s')$, is a basis of $\mathbb{K}^{S\times S'}$. Any vector $v\in\mathbb{K}^{S_1\times S_2}$ can be uniquely written as $v=\sum_i\sum_j \alpha_{ij} e_ie'j$.

Also, the set of functions $f:S\to\mathbb{K}^{S'}$ is a vector space, which can be identified with the tensor product $\mathbb{K}^{S}\otimes(\mathbb{K}^{S'})^*$.

The vectors that belong to tensor products of vector spaces are named tensors. So, tensors are vectors with some extra structure.

The tensor product can be defined easily for any kind of vector spaces, because any vector space can be thought of as a space of functions. The tensor product is associative, so we can define it between multiple vector spaces. We denote the tensor product of $n>1$ copies of $V$ by $V^{\otimes n}$. We can check that for $m,n>1$, $V^{\otimes (m+n)}=V^{\otimes {m}}\otimes V^{\otimes {n}}$. This can work also for $m,n\geq 0$, if we define $V^1=V$, $V^0=\mathbb{K}$. So, vectors and scalars are just tensors.

Let $U$, $V$ be $\mathbb{K}$-vector spaces. A linear operator is a function $f:U\to V$ which satisfies $f(u+v)=f(u)+f(v)$, and $f(\alpha u)=\alpha f(u)$, for any $u\in U,v\in V,\alpha\in\mathbb{K}$. The operator $f:U\to V$ is in fact a tensor from $U^*\otimes V$.

Inner products

Given a basis, any vector can be expressed as a set of numbers, the components of the vector. But the vector is independent of this numerical representation. The basis can be chosen in many ways, and in fact, any non-zero vector can have any components (provided not all are zero) in a well chosen basis. This shows that any two non-zero vectors play identical roles, which may be a surprise. This is a key point, since a common misconception when talking about vectors is that they have definite intrinsic sizes and orientations, or that they can make an angle. But in fact the sizes and orientations are relative to the frame, or to the other vectors. Moreover, you can say that from two vectors, one is larger than the other, only if they are collinear. Otherwise, no matter how small is one of them, we can easily find a basis in which it becomes larger than the other. It makes no sense to speak about the size, or magnitude, or length of a vector, as an intrinsic property.

But wait, one may say, there is a way to define the size of a vector! Consider a basis in a two-dimensional vector space, and a vector $v=(v^1,v^2)$. Then, the size of the vector is given by Pythagoras's theorem, by $\sqrt{(v^1)^2+(v^2)^2}$. The problem with this definition is that, if you change the basis, you will obtain different components, and different size of the vector. To make sure that you obtain the same size, you should allow only certain bases. To speak about the size of  a vector, and about the angle between two vectors, you need an additional object, which is called inner product, or scalar product. Sometimes, for example in geometry and in relativity, it is called metric.

Choosing a basis gives a default inner product. But the best way is to define the inner product, and not to pick a special basis. Once you have the inner product, you can define angles between vectors too. But size and angles are not intrinsic properties of vectors, they depend on the scalar product too.

The inner product between two vectors $u$ and $v$, defined by a basis, is $u\cdot v = u^1 v^1 + u^2 v^2 + \ldots + u^n v^n$. But in a different basis, it will have a general form $u\cdot v=\sum_i\sum_j g_{ij} u^i v^j$, where $g_{ij}=g_{ji}$ can be seen as the components of a symmetric matrix. These components change when we change the basis, they form the components of a tensor from $V^*\otimes V^*$. Einstein had the brilliant idea to omit the sum signs, so the inner product looks like $u\cdot v=g_{ij} u^i v^j$, where you know that since $i$ and $j$ appear both in upper and in lower positions, we make them run from $1$ to $n$ and sum. This is a thing that many geometers hate, but physicists find it very useful and compact in calculations, because the same summation convention appears in many different situations, which to geometers appear to be different, but in fact are very similar.

Given a basis, we can define the inner product by choosing the coefficients $g_{ij}$. And we can always find another basis, in which $g_{ij}$ is diagonal, that is, it vanishes unless $i=j$. And we can rescale the basis so that $g_{ii}$ are equal to $-1$, $1$, or $0$. Only if $g_{ii}$ are all $1$ in some basis, the size of the vector is given by the usual Pythagoras's theorem, otherwise, there will be some minus signs there, and even some terms will be omitted (corresponding to $g_{ii}=0$).

Quantum mechanics

Quantum particles are described by Schrödinger's equation. Its solutions are, for a single elementary particle, complex functions $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}$, or more general, $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}^k$, named wavefunctions. They describe completely the states of the quantum particle. They form a vector space $H$ which also has a hermitian product (a complex scalar product so that $h_{ij}=\overline{h_{ji}}$), and is named the Hilbert space (because in the infinite dimensional case also satisfies an additional property which we don't need here), or the state space. Linear transformations of $H$ which preserve the complex scalar product are named unitary transformations, and they are the complex analogous of rotations.

The wavefunctions are represented in a basis as functions of positions, $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}^k$. The element of the position basis represent point particles. But we can make a unitary transformation and obtain another basis, made of functions of the form $e^{i (k_x x + k_y y + k_z z)}$, which represent pure waves. Some observations use one of the bases, some the other, and here is why there is a duality between waves and point particles.

For more elementary particles, the state space is the tensor product of the state spaces of the individual particles. A tensor product of the form $|\psi\rangle\otimes|\psi'\rangle$ represents separable states, which can be observed independently. If the system can't be written like this, but only as a sum, the particles are entangled. When we measure them, the outcomes are correlated.

The evolution of a quantum system is described by Schrödinger's equation. Basically, the state rotates, by a unitary transformation. Only such transformations conserve the probabilities associated to the wavefunction.

When you measure the quantum systems, you need an observable. One can see an observable as defining a decomposition of the state space, in perpendicular subspaces. After the observation, the state is found to be in one of the subspaces. We can only know the subspace, but not the actual state vector. This is strange, because the system can, in principle, be in any possible state, but the measurement finds it to be only in one of these subspaces (we say it collapsed). This is the measurement problem. The things become even stranger, if we realize that if we measure another property, the corresponding decomposition of the state space is different. In other words, if you look for a point particle, you find a point particle, and if you look for a wave, you find a wave. This seems as if the unitary evolution given by the Schrödinger's equation is broken during observations. Perhaps the wavefunction remains intact, but to us, only one of the components continues to exist, corresponding to the subspace we obtained after the measurement. In the many worlds interpretation the universes splits, and all outcomes continue to exist, in new created universes. So, not only the state vector contains the universe, but it actually contains many universes.

I have a proposed explanation for some strange quantum features, in [1, 2, 3], and in these videos:

Special relativity

An example when there is a minus signs in the Pythagoras's theorem is given by the theory of relativity, where the squared size of a vector is $v\cdot v=-(v^t)^2+(v^x)^2+(v^y)^2+(v^z)^2$.

This inner product is named the Lorentz metric. Special relativity takes place in the Minkowski spacetime, which has four dimensions. A vector $v$ is named timelike if $v\cdot v < 0$, spacelike if $v\cdot v > 0$, and null or lightlike if $v\cdot v = 0$. A particle moving with the speed of light is described by a lightlike vector, and one moving with an inferior speed, by a timelike vector. Spacelike vectors would describe faster than light particles, if they exist. Points in spacetime are named events. Events can be simultaneous, but this depends on the frame. Anyway, to be simultaneous in a frame, two events have to be separated by a spacelike interval. If they are separated by a lightlike or timelike interval, they can be connected causally, or joined by a particle with a speed equal to, respectively smaller than the speed of light.

In Newtonian mechanics, the laws remain unchanged to translations and rotations in  space, translations in time, and inertial movements of the frame - together they form the Galilei transformations. However, electromagnetism disobeyed. In fact, this was the motivation of the research of Einstein, Poincaré, Lorentz, and FitzGerald. Their work led to the discovery of special relativity, according to which the correct transformations are not those of Galilei, but those of Poincaré, which preserve the distances given by the Lorentz metric.

Curvilinear coordinates

A basis or a frame of vectors in the Minkowski spacetime allows us to construct Cartesian coordinates. However, if the observer's motion is accelerated (hence the observer is non-inertial), her frame will rotate in time, so Cartesian coordinates will have to be replaced with curved coordinates. In curved coordinates, the coefficients $g_{ij}$ depend on the position. But in special relativity they have to satisfy a flatness condition, otherwise spacetime will be curved, and this didn't make much sense back in 1905, when special relativity was discovered.

General relativity

Einstein remarked that to a non-inertial observer, inertia looks similar to gravity. So he imagined that a proper choice of the metric $g_{ij}$ may generate gravity. This turned out indeed to be true, but the choice of $g_{ij}$ corresponds to a curved spacetime, and not a flat one.

One of the problems of general relativity is that it has singularities. Singularities are places where some of the components of $g_{ij}$ become infinite, or where $g_{ij}$ has, when diagonalized, some zero entries on the diagonal. For this reason, many physicist believe that we should replace general relativity with a theory of quantum gravity, like string theory or loop quantum gravity. But until we will know what is the right theory of quantum gravity, general relativity can actually deal with the singularities. I will not describe this here, but you can read my articles about this, and also this essay, and these posts about the black hole information paradox [1, 2, 3]. And watch this video

Vector bundles and forces

We call fields the functions defined on the space or the spacetime. We have seen that fields valued in vector spaces are actually vector spaces. On a flat space $M$ which looks like a vector space, the fields valued in vector spaces can be thought of as being valued in the same vector space, for example $f:M\to V$. But if the space is curved, or if it has nontrivial topology, we are forced to consider that at each point there is another copy of $V$. So, such a field will be more like $f(x)\in V_x$, where $V_x$ is the copy of the vector space $V$ at the point $x$. Such fields still form a vector space. The union of all $V_x$ is called a vector bundle. The fields are also called sections, and $V_x$ is called the fiber at $x$.

Now, since $V_x$ are copies of $V$ at each point, there is no invariant way to identify each $V_x$ with $V$. In other words, $V_x$ and $V$ can be identified, for each $x$, up to a linear transformation of $V$. We need a way to move from $V_x$ to a neighboring $V_{x+d x}$. This can be done with a connection. Also, moving a vector from $V_x$ along a closed curve reveals that, when returning to $V_x$, the vector is rotated. This is explained by the presence of a curvature, which can be obtained easily from the connection.

Connections behave like potentials of force fields. And a force field corresponds to the curvature of the connection. This makes very natural to use vector bundles to describe forces, and this is what gauge theory does.

Forces in the standard model of particles are described as follows. We assume that there is a typical complex vector space $V$ of dimension $n$, endowed with a hermitian scalar product. The connection is required to preserve this hermitian product when moving among the copies $V_x$. The set of linear transformations that preserve the scalar product is named unitary group, and is denoted by $U(n)$. The subset of transformations having the determinant equal to $1$ is named the special unitary group, $SU(n)$. The electromagnetic force corresponds to $U(1)$, the weak force to $SU(2)$, and the strong force to $SU(3)$. Moreover, all particles turn out to correspond to vectors that appear in the representations of the gauge groups on vector spaces.

What's next?

Vectors are present everywhere in physics. We see that they help us understand quantum mechanics, special and general relativity, and the particles and forces. They seem to offer a unitary view of fundamental physics.

However, up to this point, we don't know how to unify
  • unitary evolution and the collapse of the wavefunction
  • the quantum level with the mundane classical level
  • quantum mechanics and general relativity
  • the electroweak and strong forces (we know though how to combine the electromagnetic and weak forces, in the unitary group $U(2)$)
  • the standard model forces and gravity

Saturday, September 27, 2014

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.


* 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:

Thursday, September 25, 2014

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.

Wednesday, September 24, 2014

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.