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December 20, 2021

Symmetries and Redundancies

Posted by Bartosz Milewski under Philosophy, Physics
[6] Comments 

From the outside it might seem like physics and mathematics are a match made in heaven. In practice, it feels more like physicists are given a very short blanket made of math, and when they stretch it to cover their heads, their feet are freezing, and vice versa.

Physicists turn reality into numbers. They process these numbers using mathematics, and turn them into predictions about other numbers. The mapping between physical reality and mathematical models is not at all straightforward. It involves a lot of arbitrary choices. When we perform an experiment, we take the readings of our instruments and create one particular parameterization of nature. There usually are many equivalent parameterizations of the same process and this is one of the sources of redundancy in our description of nature. The Universe doesn’t care about our choice of units or coordinate systems.

This indifference, after we plug the numbers into our models, is reflected in symmetries of our models. A change in the parameters of our measuring apparatus must be compensated by a transformation of our model, so that the results of calculations still match the outcome of the experiment.

But there is an even deeper source of symmetries in physics. The model itself may introduce additional redundancy in order to simplify the calculations or, sometimes, make them possible. It is often necessary to use parameter spaces that allow the description of non-physical states–states that could never occur in reality.

Computer programmers are familiar with such situations. For instance, we often use integers to access arrays. But an integer can be negative, or it can be larger than the size of the array. We could say that an integer can describe “non-physical” states of the array. We also have freedom of parameterization of our input data: we can encode true as one, and false as zero; or the other way around. If we change our parameterization, we must modify the code that deals with it. As programmers we are very well aware of the arbitrariness of the choice of representation, but it’s even more true in physics. In physics, these reparameterizations are much more extensive and they have their mathematical description as groups of transformations.

But what we see in physics is very strange: the non-physical degrees of freedom introduced through redundant parameterizations turn out to have some measurable consequences.

Symmetries

If you ask physicists what the foundations of physics are, they will probably say: symmetry. Depending on their area of research, they will start talking about various symmetry groups, like SU(3), U(1), SO(3,1), general diffeomorphisms, etc. The foundations of physics are built upon fields and their symmetries. For physicists this is such an obvious observation that they assume that the goal of physics is to discover the symmetries of nature. But are symmetries the property of nature, or are they the artifact of our tools? This is a difficult question, because the only way we can study nature is through the prism or mathematics. Mathematical models of reality definitely exhibit lots of symmetries, and it’s easy to confuse this with the statement that nature itself is symmetric.

But why would models exhibit symmetry? One explanation is that symmetries are the effect of redundant descriptions.

I’ll use the example of electromagnetism because of its relative simplicity (some of the notation is explained in the Appendix), but the redundant degrees of freedom and the symmetries they generate show up everywhere in physics. The Standard Model is one big gauge theory, and Einstein’s General Relativity is built on the principle of invariance with respect to local coordinate transformations.

Electromagnetic field

Maxwell’s equations are a mess, until you rewrite them using 4-dimensional spacetime. The two vector fields, the electric field and the magnetic field are combined into one 4-dimensional antisymmetric tensor F^{\mu \nu}:

F^{\mu\nu} = \begin{bmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{bmatrix}

Because of antisymmetry, F^{\mu \nu} has only six independent components. The components of F^{\mu \nu} are physical fields that can be measured using test charges and magnetic needles.

The derivatives of these fields satisfy two sets of Maxwell’s equations. The first set of four describes the dependence of fields on sources—electric charges and currents:

\partial_{\mu} F^{\mu \nu} = j^{\nu}

The second set of four equations describe constraints imposed on these fields:

\partial_{[\rho} F_{\mu \nu ]} = 0

For a particular set of sources and an initial configuration, we could try to solve these equations numerically. A brute force approach would be to divide space into little cubes, distribute our charges and currents between them, replace differential equations with difference equations, and turn on the crank.

First, we would check if the initial field configuration satisfied the constraints. Then we would calculate time derivatives of the fields. We would turn time derivatives into time differences by multiplying them by a small time period, get the next configuration, and so on. With the size of the cubes and the quantum of time small enough, we could get a reasonable approximation of reality. A program to perform these calculations isn’t much harder to write than a lot of modern 3-d computer games.

Notice that this procedure has an important property. To calculate the value of a field in a particular cube, it’s enough to know the values at its nearest neighbors and its value at the previous moment of time. The nearest-neighbor property is called locality and the dependence on the past, as opposed to the future, is called causality. The famous Conway Game of Life is local and causal, and so are cellular automata.

We were very lucky to be able to formulate a model that pretty well approximates reality and has these properties. Without such models, it would be extremely hard to calculate anything. Essentially all classical physics is written in the language of differential equations, which means it’s local, and its time dependence is carefully crafted to be causal. But it should be stressed that locality and causality are properties of particular models. And locality, in particular, cannot be taken for granted.

Electromagnetic Potential

The second set of Maxwell’s equations can be solved by introducing a new field, a 4-vector A_{\mu} called the vector potential. The field tensor can be expressed as its anti-symmetrized derivative

F_{\mu \nu} = \partial_{[ \mu} A_{\nu ]}

Indeed, if we take its partial derivative and antisymmetrize the three indices, we get:

\partial_{[\rho} F_{\mu \nu ]} = \partial_{[\rho} \partial_{ \mu} A_{\nu ]} = 0

which vanishes because derivatives are symmetric, \partial_{\mu} \partial_{\nu} = \partial_{\nu} \partial_{\mu}.

Note for mathematicians: Think of A_{\mu} as a connection in the U(1) fiber bundle, and F_{\mu \nu} as its curvature. The second Maxwell equation is the Bianchi identity for this connection.

This field A_{\mu} is not physical. We cannot measure it. We can measure its derivatives in the form of F_{\mu \nu}, but not the field itself. In fact we cannot distinguish between A_{\mu} and the transformed field:

A'_{\mu} = A_{\mu} + \partial_{\mu} \Lambda

Here, \Lambda(x) is a completely arbitrary, time dependent scalar field. This is, again, because of the symmetry of partial derivatives:

F_{\mu \nu}' = \partial_{[ \mu} A'_{\nu ]} = \partial_{[ \mu} A_{\nu ]} + \partial_{[ \mu} \partial_{\nu ]} \Lambda = \partial_{[ \mu} A_{\nu ]} = F_{\mu \nu}

Adding a derivative of \Lambda is called a gauge transformation, and we can formulated a new law: Physics in invariant under gauge transformations. There is a beautiful symmetry we have discovered in nature.

But wait a moment: didn’t we just introduce this symmetry to simplify the math?

Well, it’s a bit more complicated. To explain that, we have to dive even deeper into technicalities.

The Action Principle

You cannot change the past and your cannot immediately influence far away events. These are the reasons why differential equations are so useful in physics. But there are some types of phenomena that are easier to explain by global rather than local reasoning. For instance, if you span an elastic rubber band between two points in space, it will trace a straight line. In this case, instead of diligently solving differential equations that describe the movements of the rubber band, we can guess its final state by calculating the shortest path between two points.

Surprisingly, just like the shape of the rubber band can be calculated by minimizing the length of the curve it spans, so the evolution of all classical systems can be calculated by minimizing (or, more precisely, finding a stationary point of) a quantity called the action. For mechanical systems the action is the integral of the Lagrangian along the trajectory, and the Lagrangian is given by the difference between  kinetic and potential energy.

Consider the simple example of an object thrown into the air and falling down due to gravity. Instead of solving the differential equations that relate acceleration to force, we can reformulate the problem in terms of minimizing the action. There is a tradeoff: we want to minimize the kinetic energy while maximizing the potential energy. Potential energy is larger at higher altitudes, so the object wants to get as high as possible in the shortest time, stay there as long as possible, before returning to earth. But the faster it tries to get there, the higher its kinetic energy. So it performs a balancing act resulting is a perfect parabola (at least if we ignore air resistance).

The same principle can be applied to fields, except that the action is now given by a 4-dimensional integral over spacetime of something called the Lagrangian density which, at every point, depends only of fields and their derivatives. This is the classical Lagrangian density that describes the electromagnetic field:

L = - \frac{1}{4} F^{\mu \nu} F_{\mu \nu} = \frac{1}{2}(\vec{E}^2 - \vec{B}^2)

and the action is:

S = \int L(x)\, d^4 x

However, if you want to derive Maxwell’s equations using the action principle, you have to express it in terms of the potential A_{\mu} and its derivatives.

Noether’s Theorem

The first of the Maxwell’s equations describes the relationship between electromagnetic fields and the rest of the world:

\partial_{\mu} F^{\mu \nu} = j^{\nu}

Here “the rest of the world” is summarized in a 4-dimensional current density j^{\nu}. This is all the information about matter that the fields need to know. In fact, this equation imposes additional constraints on the matter. If you differentiate it once more, you get:

\partial_{\nu}\partial_{\mu} F^{\mu \nu} = \partial_{\nu} j^{\nu} = 0

Again, this follows from the antisymmetry of F^{\mu \nu} and the symmetry of partial derivatives.

The equation:

\partial_{\nu} j^{\nu} = 0

is called the conservation of electric charge. In terms of 3-d components it reads:

\dot{\rho} = \vec{\nabla} \vec{J}

or, in words, the change in charge density is equal to the divergence of the electric current. Globally, it means that charge cannot appear or disappear. If your isolated system starts with a certain charge, it will end up with the same charge.

Why would the presence of electromagnetic fields impose conditions on the behavior of matter? Surprisingly, this too follows from gauge invariance. Electromagnetic fields must interact with matter in a way that makes it impossible to detect the non-physical vector potentials. In other words, the interaction must be gauge invariant. Which makes the whole action, which combines the pure-field Lagrangian and the interaction Lagrangian, gauge invariant.

It turns out that any time you have such an invariance of the action, you automatically get a conserved quantity. This is called the Noether’s theorem and, in the case of electromagnetic theory, it justifies the conservation of charge. So, even though the potentials are not physical, their symmetry has a very physical consequence: the conservation of charge.

Quantum Electrodynamics

The original idea of quantum field theory (QFT) was that it should extend the classical theory. It should be able to explain all the classical behavior plus quantum deviations from it.

This is no longer true. We don’t insist on extending classical behavior any more. We use QFT to, for instance, describe quarks, for which there is no classical theory.

The starting point of any QFT is still the good old Lagrangian density. But in quantum theory, instead of minimizing the action, we also consider quantum fluctuations around the stationary points. In fact, we consider all possible paths. It just so happens that the contributions from those paths that are far away from the classical solutions tend to cancel each other. This is the reason why classical physics works so well: classical trajectories are the most probable ones.

In quantum theory, we calculate probabilities of transitions from the initial state to the final state. These probabilities are given by summing up complex amplitudes for every possible path and then taking the absolute value of the result. The amplitudes are given by the exponential of the action:

e^{i S / \hbar }

Far away from the stationary point of the action, the amplitudes corresponding to adjacent paths vary very quickly in phase and they cancel each other. The summation effectively acts like a low-pass filter for these amplitudes. We are observing the Universe through a low-pass filter.

In quantum electrodynamics things are a little tricky. We would like to consider all possible paths in terms of the vector potential A_{\mu}(x). The problem is that two such paths that differ only by a gauge transformation result in exactly the same action, since the Lagrangian is written in terms of gauge invariant fields F^{\mu \nu}. The action is therefore constant along gauge transformations and the sum over all such paths would result in infinity. Once again, the non-physical nature of the potential raises its ugly head.

Another way of describing the same problem is that we expect the quantization of electromagnetic field to describe the quanta of such field, namely photons. But a photon has only two degrees of freedom corresponding to two polarizations, whereas a vector potential has four components. Besides the two physical ones, it also introduces longitudinal and time-like polarizations, which are not present in the real world.

To eliminate the non-physical degrees of freedom, physicists came up with lots of clever tricks. These tricks are relatively mild in the case of QED, but when it comes to non-Abelian gauge fields, the details are quite gory and involve the introduction of even more non-physical fields called ghosts.

Still, there is no way of getting away from vector potentials. Moreover, the interaction of the electromagnetic field with charged particles can only be described using potentials. For instance, the Lagrangian for the electron field \psi in the electromagnetic field is:

\bar{\psi}(i \gamma^{\mu}D_{\mu} - m) \psi

The potential A_{\mu} is hidden inside the covariant derivative

D_{\mu} = \partial_{\mu} - i e A_{\mu}

where e is the electron charge.

Note for mathematicians: The covariant derivative locally describes parallel transport in the U(1) bundle.

The electron is described by a complex Dirac spinor field \psi. Just as the electromagnetic potential is non-physical, so are the components of the electron field. You can conceptualize it as a “square root” of a physical field. Square roots of numbers come in pairs, positive and negative—Dirac field describes both negative electrons and positive positrons. In general, square roots are complex, and so are Dirac fields. Even the field equation they satisfy behaves like a square root of the conventional Klein-Gordon equation. Most importantly, Dirac field is only defined up to a complex phase. You can multiply it by a complex number of modulus one, e^{i e \Lambda} (the e in the exponent is the charge of the electron). Because the Lagrangian pairs the field \psi with its complex conjugate \bar{\psi}, the phases cancel, which shows that the Lagrangian does not depend on the choice of the phase.

In fact, the phase can vary from point to point (and time to time) as long as the phase change is compensated by the the corresponding gauge transformation of the electromagnetic potential. The whole Lagrangian is invariant under the following simultaneous gauge transformations of all fields:

\psi' = e^{i e \Lambda} \psi

\bar{\psi}' = \bar{\psi} e^{-i e \Lambda}

A_{\mu}' = A_{\mu} + \partial_{\mu} \Lambda

The important part is the cancellation between the derivative of the transformed field and the gauge transformation of the potential:

(\partial_{\mu} - i e A'_{\mu}) \psi' = e^{i e \Lambda}( \partial_{\mu} + i e \partial_{\mu} \Lambda - i e A_{\mu} - i e \partial_{\mu} \Lambda) \psi = e^{i e \Lambda} D_{\mu} \psi

Note for mathematicians: Dirac field forms a representation of the U(1) group.

Since the electron filed is coupled to the potential, does it mean that an electron can be used to detect the potential? But the potential is non-physical: it’s only defined up to a gauge transformation.

The answer is really strange. Locally, the potential is not measurable, but it may have some very interesting global effects. This is one of these situations where quantum mechanics defies locality. We may have a region of space where the electromagnetic field is zero but the potential is not. Such potential must, at least locally, be of the form: A_{\mu} = \partial_{\mu} \phi. Such potential is called pure gauge, because it can be “gauged away” using \Lambda = -\phi.

But in a topologically nontrivial space, it may be possible to define a pure-gauge potential that cannot be gauged away by a continuous function. For instance, if we remove a narrow infinite cylinder from a 3-d space, the rest has a non-trivial topology (there are loops that cannot be shrunk to a point). We could define a 3-d vector potential that circulates around the cylinder. For any fixed radius around the cylinder, the field would consist of constant-length vectors that are tangent to the circle. A constant function is a derivative of a linear function, so this potential could be gauged away using a function \Lambda that linearly increases with the angle around the cylinder, like a spiral staircase. But once we make a full circle, we end up on a different floor. There is no continuous \Lambda that would eliminate this potential.

This is not just a theoretical possibility. The field around a very long thin solenoid has this property. It’s all concentrated inside the solenoid and (almost) zero outside, yet its vector potential cannot be eliminated using a continuous gauge transformation.

Classically, there is no way to detect this kind of potential. But if you look at it from the perspective of an electron trying to pass by, the potential is higher on one side of the solenoid and lower on the other, and that means the phase of the electron field will be different, depending whether it passes on the left, or on the right of it. The phase itself is not measurable but, in quantum theory, the same electron can take both paths simultaneously and interfere with itself. The phase difference is translated into the shift in the interference pattern. This is called the Aharonov-Bohm effect and it has been confirmed experimentally.

Note for mathematicians: Here, the base space of the fiber bundle has non-trivial homotopy. There may be non-trivial connections that have zero curvature.

Aharonov-Bohm experiment

Space Pasta

I went into some detail to describe the role redundant degrees of freedom and their associated symmetries play in the theory of electromagnetic fields.

We know that the vector potentials are not physical: we have no way of measuring them directly. We know that in quantum mechanics they describe non-existent particles like longitudinal and time-like photons. Since we use redundant parameterization of fields, we introduce seemingly artificial symmetries.

And yet, these “bogus symmetries” have some physical consequences: they explain the conservation of charge; and the “bogus degrees of freedom” explain the results of the Aharonov-Bohm experiment. There are some parts of reality that they capture. What are these parts?

One possible answer is that we introduce redundant parametrizations in order to describe, locally, the phenomena of global or topological nature. This is pretty obvious in the case of the Aharonov-Bohm experiment where we create a topologically nontrivial space in which some paths are not shrinkable. The charge conservation case is subtler.

Consider the path a charged particle carves in space-time. If you remove this path, you get a topologically non-trivial space. Charge conservation makes this path unbreakable, so you can view it as defining a topological invariant of the surrounding space. I would even argue that charge quantization (all charges are multiples of 1/3 of the charge or the electron) can be explained this way. We know that topological invariants, like the Euler characteristic that describes the genus of a manifold, take whole-number values.

We’d like physics to describe the whole Universe but we know that current theories fail in some areas. For instance, they cannot tell us what happens at the center of a black hole or at the Big Bang singularity. These places are far away, either in space or in time, so we don’t worry about them too much. There’s still a lot of Universe left for physicist to explore.

Except that there are some unexplorable places right under our noses. Every elementary particle is surrounded by a very tiny bubble that’s unavailable to physics. When we try to extrapolate our current theories to smaller and smaller distances, we eventually hit the wall. Our calculations result in infinities. Some of these infinities can be swept under the rug using clever tricks like renormalization. But when we get close to Planck’s distance, the effects of gravity take over, and renormalization breaks down.

So if we wanted to define “physical space” as the place where physics is applicable, we’d have to exclude all the tiny volumes around the paths of elementary particles. Removing the spaghetti of all such paths leaves us with a topological mess. This is the mess on which we define all our theories. The redundant descriptions and symmetries are our way of probing the excluded spaces.

Appendix

A point in Minkowski spacetime is characterized by four coordinates x^{\mu} \mu = 0, 1, 2, 3, where x^0 is the time coordinate, and the rest are space coordinates. We use the system of units in which the speed of light c is one.

Repeated indices are, by Einstein convention, summed over (contracted). Indices between square brackets are anisymmetrized (that is summed over all permutations, with the minus sign for odd permutations). For instance

F_{0 1} = \partial_{[0} A_{1]} = \partial_{0} A_{1} - \partial_{1} A_{0} = \partial_{t} A_{x} - \partial_{x} A_{t}

Indexes are raised and lowered by contracting them with the Minkowski metric tensor:
\eta_{\mu\nu} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}

Partial derivatives with respect to these coordinates are written as:

\partial_{\mu} = \frac{\partial}{\partial x^{\mu}}

4-dimensional antisymmetric tensor F^{\mu \nu} is a 4 \times 4 matrix, but because of antisymmetry, it reduces to just 6 independent entries, which can be rearranged into two 3-d vector fields. The vector \vec E is the electric field, and the vector \vec B is the magnetic field.

F^{\mu\nu} = \begin{bmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{bmatrix}

The sources of these fields are described by a 4-dimensional vector j^{\mu}. Its zeroth component describes the distribution of electric charges, and the rest describes electric current density.

The second set of Maxwell’s equations can also be written using the completely antisymmetric Levi-Civita tensor with entries equal to 1 or -1 depending on the parity of the permutation of the indices:

\epsilon^{\mu \nu \rho \sigma} \partial_{\nu} F_{\rho \sigma} = 0

 

September 6, 2020

The Fall of the SKI Civilization

Posted by Bartosz Milewski under Functional Programming, Philosophy
[4] Comments 

Abstract: The recent breakthroughs in deciphering the language and the literature left behind by the now extinct Twinklean civilization provides valuable insights into their history, science, and philosophy.

The oldest documents discovered on the third planet of the star Lambda Combinatoris (also known as the Twinkle star) talk about the prehistory of the Twinklean thought. The ancient Book of Application postulated that the Essence of Being is decomposition, expressed symbolically as

   A = B C

meaning that A can be decomposed into B and C. The breakthrough came with the realization that, if C itself can be decomposed

   C = F G

then A could be further decomposed into

   A = B (F G)

Similarly, if B can be decomposed

   B = D E

then

   A = (D E) C

In the latter case (but not the former), it became customary to drop the parentheses and simply write it as

   A = D E C

Following these discoveries, the Twinklean civilization went through a period called The Great Decomposition that lasted almost three thousand years, during which essentially anything that could be decomposed was successfully decomposed.

At the end of The Great Decomposition, a new school of thought emerged, claiming that, if things can be decomposed into parts, they can be also recomposed from these parts.

Initially there was strong resistance to this idea. The argument was put forward that decomposition followed by recomposition doesn’t change anything. This was settled by the introduction of a special object called The Eye, denoted by I, defined by the unique property of leaving things alone

   I A = A

After the introduction of I, a long period of general stagnation accompanied by lack of change followed.

We also don’t have many records from the next period, as it was marked by attempts at forgetting things and promoting ignorance. It started by the introduction of K, which ignores one of its inputs

   K A B = A

Notice that this definition is a shorthand for the parenthesized version

   (K A) B = A

The argument for introducing K was that ignorance is an important part of understanding. By rejecting B we are saying that A is important. We are abstracting away the inessential part B.

For instance—the argument went—if we decompose C

   C = A B

and D happens to have a similar decomposition

   D = A E

then K will abstract the A part from both C and D. From the perspective of K, there is no difference between C and D.

The only positive outcome of the Era of Ignorance was the development of abstract mathematics. Twinklean thinkers argued that, if you disregard the particularities of the fruit in question, there is no difference between having three apples and three oranges. Number three was thus born, followed by many others (four and seven, to name just a few).

The final Industrial phase of the Twinklean civilization that ultimately led to their demise was marked by the introduction of S. The Twinklean industry was based on the principle of mass production; and mass production starts with duplication and reuse. Suppose you have a reusable part C. S allows you to duplicate C and combine it with both A and B.

   S A B C = (A C) (B C)

If you think of A and B as abstractions—that is the results of ignoring some parts of the whole—S lets you substitute C in place of those forgotten parts.

Or, conversely, it tells you that the object

   E = S A B C

can be decomposed into two parts that have something in common. This common part is C.

Unfortunately, during the Industrial period, a lot of Twinkleans lost their identity. They discovered that

   I = S K K

Indeed

   I A = S K K A = K A (K A) = A

But ultimately, what precipitated their end was the existential crisis. They lost their will to live because they couldn’t figure out Y.

Postscript

After submitting this paper to the journal of Compositionality, we have been informed by the reviewer that a similar theory of SKI combinators was independently developed on Earth by a Russian logician, Moses Schönfinkel. According to this reviewer, the answer to the meaning of life is the Y combinator, which introduces recursion and can be expressed as

   Y = S(K(SII))(S(S(KS)K)(K(SII)))

We were unable to verify this assertion, as it led us into a rabbit hole.

 

January 3, 2019

Metamorphisms

Posted by Bartosz Milewski under Philosophy
[4] Comments 

Flies

It was a hot evening and, as is usual at that time of the year, there were quite a few flies buzzing around us. Understandably, I was annoyed, as they were interfering with my meditation.

“Master,” I said, “You keep telling me that everything in this world has a purpose, but I can’t figure out the purpose of these flies. All they do is break my concentration. Can we move indoors already, behind the screens, so that we can continue the lessons in peace?”

The Master looked at me the way he usually does when I say something that shows my lack of understanding — which unfortunately happens a lot.

“The flies are here to teach us about meditation,” he said.

“How so?” I said. “Are you trying to tell me that I should be able to quiet my mind even when there’s constant interference?”

“That would be the ultimate goal,” said the Master, “but for now I’d like you to observe the way these flies move. Can you do that?”

“Of course, Master,” I said and started watching the flies criss-crossing the air in front of me.

“What do you see?” asked the Master after a while.

“I see them zig-zagging constantly. They never seem to fly in a straight line for longer than a fraction of a second.”

“You are very astute, my Disciple,” said the Master. “Now, why would you say they’re doing that?”

“I think they are doing that to avoid being caught” I said. “Those flies that were, long ago, flying in straight lines were eliminated by predators, and only those that employed more elaborate movement schemes survived long enough to produce offspring. Evolution in action!” I said not without some pride at my cleverness.

“Quite so, my Disciple,” said the Master, “quite so…”

“But suppose,” he continued, “that, in some other universe, there is a colony of flies that live confined within the boundaries a hostile environment. Their life is short and full of suffering. But there is a benevolent being that can set individual flies free, to live a happy and productive life. The trouble is that she has to catch them first. And, in the beginning, it was easy, since they were all flying in straight lines. Almost all. The benevolent being was able to remove the straight-flying flies and make them happy. But there remained a few flies that, for one reason or another, kept zig-zagging. They survived long enough to produce offspring, some of which also kept zig-zagging. Soon enough, all flies in that hostile and unhappy environment developed this new survival strategy that prevented them from escaping their horrible fate. That’s evolution in action, too.”

“That’s a pretty sad story,” I said. “It shows that evolution is a cruel mistress. It doesn’t care if we are happy or not, as long as we produce offspring.”

“But what does it have to do with meditation?” I said, a little confused.

“The flies are our thoughts,” said the Master.

The Pipe

It was a pleasant evening and I was enjoying the warm breeze coming from the mountains bringing with it the smell of pine and something else.

“Why are you smoking a pipe, Master? It’s bad for your health.”

“This is not a pipe,” said the Master.

“This is most definitely a pipe. You must have bought it in a pipe shop,” I said.

“This is not a pipe,” said the Master.

I thought for a moment.

“Oh, I see. You are making reference to the famous painting by Rene Magritte, right? Ceci n’est pas une pipe! But what Magritte meant was that it wasn’t a pipe–it was a picture of a pipe. He played on our confusion between the object itself and its representation. But here you are holding an actual pipe.”

“What makes you think it’s a pipe?” asked the Master.

“Well, I can look up the definition of a pipe and you’ll see that it describes the object you are holding. Do you want me to do that?” I asked.

“Please do,” said the Master.

I pulled out my tablet and started tapping.

“What’s the wi-fi password for today, Master?” I asked.

“That which cannot be named,” said the Master.

“Oh, I know that one,” I said and continued tapping. I stopped after a few tries and looked up.

“I tried Tao and Dao, upper- and lowercase, but it didn’t work. Is it ‘the Tao,’ with ‘the’?”

“You are much too clever, my Disciple,” said the Master.

“Oh, you mean it’s literally ‘that which cannot be named’?” I started tapping again.

“Okay, here it is. According to the OED, a pipe is…” I hesitated for a moment.

“Wait, you don’t really want me to read the definition,” I said.

“No,” said the Master.

After a moment of silence, I said:
“You have corrected me, Master, because by naming the pipe I focused on just one small aspect of it. Its relationship to other pipes. By doing that I ignored its relationship to you, to me, to our conversation, to this lovely sunset, to Magritte and–now I get it–to the Tao.”

“But, Master, I’m confused,” I said after a while. “When you say that ‘The Tao that can be named is not the eternal Tao,’ you are giving it a name, aren’t you? You are calling it the Tao.”

“This is not a name,” said the Master.

“I have the feeling that if I say that this is indeed a name, I would be hitting a dead end,” I said.

We sat there for a moment while I was organizing my thoughts. Then it occurred to me.

“By calling it the Tao, you are not separating it from everything else, because the Tao is in everything. Neither are you ignoring its relationship to yourself or to me, because you are the Tao and I am the Tao. And the lovely sunset, and the pipe, and Magritte, it’s all the eternal Tao.

“This is not the Tao,” said the Master.

Fallout

“I watched a movie last night,” I said. “And it made me think.”

“Movies often make us think,” said the Master. “Good movies, like life itself, ask a lot of questions, but rarely provide answers.”

“Well, that’s the thing, Master” I said. “Maybe you know the answer to this question, or maybe you can steer me towards the answer. I’m sure this problem has been analyzed before by many people much wiser than yours truly.

“It’s a problem of moral nature. In the movie, agent Hunt faces a dilemma. His friend is in immediate mortal danger. Hunt can save him, but at the risk of endangering the lives of thousands of innocent people. He makes a choice, saves the friend but, in the process, the terrorists get hold of plutonium, which they use to make nuclear bombs. Of course, in the movie, he’s ultimately able to avert the disaster, disabling the bombs literally one second before they’re about to go off.

“Sorry if I spoiled the movie for you, Master.”

“Don’t worry, I’ve seen the movie,” said the Master.

“So what do you think, Master? Was agent Hunt acting recklessly, risking uncountable lives to save one?”

“And what’s your opinion?” asked the Master.

“I think the answer is clear. It’s just simple math: one life against thousands. I would probably feel guilty for the rest of my life for sacrificing a friend, but what right do I have to risk thousands of innocent lives?”

“You say it’s simple math,” said the Master. “I presume there is an equation that calculates the moral value of an act, based on the number of lives saved or lost.”

“It’s not an exact science, but I guess one could make some rough estimates,” I said. “I’ve read some articles that mostly deal with pulling levers to divert trolleys. So this seems like one of these problems, where your friend is tied up on one track, and thousands of people on another. A runaway trolley is going to kill your friend, and you pull the switch to divert it to the other track, possibly killing thousands of people.”

“If this is simple math, then why do you say you’d feel guilty? Shouldn’t you feel satisfied, like when you solve a difficult equation?”

“I don’t know. I think I would always speculate: What if? What if I saved my friend and, just like in the movie, were able to avert the disaster? I’d never know.”

“And what if you saved your friend’s life and the bomb exploded?” asked the Master.

“I guess I’d feel terrible for the rest of my life. And I would probably be the most despised person on Earth.”

“And what if that explosion prevented an even bigger disaster in the future?” asked the Master.

“And what if that bigger disaster prevented an even bigger disaster?” I asked. “Where does this end? Are you saying that, since we cannot predict the results of our actions on a global scale, then there is no moral imperative?”

“Would that satisfy you?” asked the Master.

“No, it wouldn’t!”

“Would you like to have a small set of simple rules to guide all moral decisions in your life?” asked the Master.

“When you put it this way, I’m not sure. I think there’s been many attempts at rule-based ethics, and they all have exhibited some pretty disastrous failure modes. It vaguely reminds me of the Goedel’s incompleteness theorem. No matter what moral axioms you choose, there will be a situation in which they fail.

“On the other hand, rejecting the axioms may lead to an even bigger tragedy, like in the case of Raskolnikov.”

“Do you see similarities between Raskolnikov and agent Hunt?” asked the Master.

“They both reject the ‘Thou shalt not kill’ commandment. They both feel intense loyalty to their friends and family. But Raskolnikov had a lot of time to think about his choices, he even published an article about it; whereas Hunt acted impulsively, following his gut feelings. One was rational, the other irrational.”

“But you said that Raskolnikov had no axioms,” said the Master. “So how could he rationally justify his actions?”

“I see your point,” I said. “He was trying to do the math. Solve the ethical equation. His hubris was not in rejecting the accepted axioms, but in believing that he can come up with a better set. So, in a way, agent Hunt had the advantage of being a moral simpleton.”

“He was the uncarved wood,” said the Master.

 

January 11, 2018

The Earth is Flat

Posted by Bartosz Milewski under Category Theory, Philosophy
[30] Comments 

La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l’universo), ma non si può intendere se prima non s’impara a intender la lingua, e conoscer i caratteri, ne’ quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto.

― Galileo Galilei, Il Saggiatore (The Assayer)

Joan was quizzical; studied pataphysical science in the home. Late nights all alone with a test tube.

— The Beatles, Maxwell’s Silver Hammer

Unless you’re a member of the Flat Earth Society, I bet you’re pretty confident that the Earth is round. In fact, you’re so confident that you don’t even ask yourself the question why you are so confident. After all, there is overwhelming scientific evidence for the round-Earth hypothesis. There is the old “ships disappearing behind the horizon” proof, there are satellites circling the Earth, there are even photos of the Earth seen from the Moon, the list goes on and on. I picked this particular theory because it seems so obviously true. So if I try to convince you that the Earth is flat, I’ll have to dig very deep into the foundation of your belief systems. Here’s what I’ve found: We believe that the Earth is round not because it’s the truth, but because we are lazy and stingy (or, to give it a more positive spin, efficient and parsimonious). Let me explain…

The New Flat Earth Theory

Let’s begin by stressing how useful the flat-Earth model is in everyday life. I use it all the time. When I want to find the nearest ATM or a gas station, I take out my cell phone and look it up on its flat screen. I’m not carrying a special spherical gadget in my pocket. The screen on my phone is not bulging in the slightest when it’s displaying a map of my surroundings. So, at least within the limits of my city, or even the state, flat-Earth theory works just fine, thank you!

I’d like to make parallels with another widely accepted theory, Einstein’s special relativity. We believe that it’s true, but we never use it in everyday life. The vast majority of objects around us move much slower than the speed of light, so traditional Newtonian mechanics works just fine for us. When was the last time you had to reset your watch after driving from one city to another to account for the effects of time dilation?

The point is that every physical theory is only valid within a certain range of parameters. Physicists have always been looking for the Holy Grail of theories — the theory of everything that would be valid for all values of parameters with no exceptions. They haven’t found one yet.

But, obviously, special relativity is better than Newtonian mechanics because it’s more general. You can derive Newtonian mechanics as a low velocity approximation to special relativity. And, sure enough, the flat-Earth theory is an approximation to the round-Earth theory for small distances. Or, equivalently, it’s the limit as the radius of the Earth goes to infinity.

But suppose that we were prohibited (for instance, by a religion or a government) from ever considering the curvature of the Earth. As explorers travel farther and farther, they discover that the “naive” flat-Earth theory gives incorrect answers. Unlike present-day flat-earthers, who are not scientifically sophisticated, they would actually put some effort to refine their calculations to account for the “anomalies.” For instance, they could postulate that, as you get away from the North Pole, which is the center of the flat Earth, something funny keeps happening to measuring rods. They get elongated when positioned along the parallels (the circles centered at the North Pole). The further away you get from the North Pole, the more they elongate, until at a certain distance they become infinite. Which means that the distances (measured using those measuring rods) along the big circles get smaller and smaller until they shrink to zero.

I know this theory sounds weird at first, but so does special and, even more so, general relativity. In special relativity, weird things happen when your speed is close to the speed of light. Time slows down, distances shrink in the direction of flight (but not perpendicular to it!), and masses increase. In general relativity, similar things happen when you get closer to a black hole’s event horizon. In both theories things diverge as you hit the limit — the speed of light, or the event horizon, respectively.

Back to flat Earth — our explorers conquer space. They have to extend their weird geometry to three dimensions. They find out that horizontally positioned measuring rods shrink as you go higher (they un-shrink when you point them vertically). The intrepid explorers also dig into the ground, and probe the depths with seismographs. They find another singularity at a particular depth, where the horizontal dilation of measuring rods reaches infinity (round-Earthers call this the center of the Earth).

This generalized flat-Earth theory actually works. I know that, because I have just described the spherical coordinate system. We use it when we talk about degrees of longitude and latitude. We just never think of measuring distances using spherical coordinates — it’s too much work, and we are lazy. But it’s possible to express the metric tensor in those coordinates. It’s not constant — it varies with position — and it’s not isotropic — distances vary with direction. In fact, because of that, flat Earthers would be better equipped to understand general relativity than we are.

So is the Earth flat or spherical? Actually it’s neither. Both theories are just approximations. In cartesian coordinates, the Earth is the shape of a flattened ellipsoid, but as you increase the resolution, you discover more and more anomalies (we call them mountains, canyons, etc.). In spherical coordinates, the Earth is flat, but again, only approximately. The biggest difference is that the math is harder in spherical coordinates.

Have I confused you enough? On one level, unless you’re an astronaut, your senses tell you that the Earth is flat. On the other level, unless you’re a conspiracy theorist who believes that NASA is involved in a scam of enormous proportions, you believe that the Earth is pretty much spherical. Now I’m telling you that there is a perfectly consistent mathematical model in which the Earth is flat. It’s not a cult, it’s science! So why do you feel that the round Earth theory is closer to the truth?

The Occam’s Razor

The round Earth theory is just simpler. And for some reason we cling to the belief that nature abhors complexity (I know, isn’t it crazy?). We even express this belief as a principle called the Occam’s razor. In a nutshell, it says that:

Among competing hypotheses, the one with the fewest assumptions should be selected.

Notice that this is not a law of nature. It’s not even scientific: there is no way to falsify it. You can argue for the Occam’s razor on the grounds of theology (William of Ockham was a Franciscan friar) or esthetics (we like elegant theories), but ultimately it boils down to pragmatism: A simpler theory is easier to understand and use.

It’s a mistake to think that Occam’s razor tells us anything about the nature of things, whatever that means. It simply describes the limitations of our mind. It’s not nature that abhors complexity — it’s our brains that prefer simplicity.

Unless you believe that physical laws have an independent existence of their own.

The Layered Cake Hypothesis

Scientists since Galileo have a picture of the Universe that consists of three layers. The top layer is nature that we observe and interact with. Below are laws of physics — the mechanisms that drive nature and make it predictable. Still below is mathematics — the language of physics (that’s what Galileo’s quote at the top of this post is about). According to this view, physics and mathematics are the hidden components of the Universe. They are the invisible cogwheels and pulleys whose existence we can only deduce indirectly. According to this view, we discover the laws of physics. We also discover mathematics.

Notice that this is very different from art. We don’t say that Beethoven discovered the Fifth Symphony (although Igor Stravinsky called it “inevitable”) or that Leonardo da Vinci discovered the Mona Lisa. The difference is that, had not Beethoven composed his symphony, nobody would; but if Cardano hadn’t discovered complex numbers, somebody else probably would. In fact there were many cases of the same mathematical idea being discovered independently by more than one person. Does this prove that mathematical ideas exist the same way as, say, the moons of Jupiter?

Physical discoveries have a very different character than mathematical discoveries. Laws of physics are testable against physical reality. We perform experiments in the real world and if the results contradict a theory, we discard the theory. A mathematical theory, on the other hand, can only be tested against itself. We discard a theory when it leads to internal contradictions.

The belief that mathematics is discovered rather than invented has its roots in Platonism. When we say that the Earth is spherical, we are talking about the idea of a sphere. According to Plato, these ideas do exist independently of the observer — in this case, a mathematician who studies them. Most mathematicians are Platonists, whether they admit it or not.

Being able to formulate laws of physics in terms of simple mathematical equations is a thing of beauty and elegance. But you have to realize that history of physics is littered with carcasses of elegant theories. There was a very elegant theory, which postulated that all matter was made of just four elements: fire, air, water, and earth. The firmament was a collection of celestial spheres (spheres are so Platonic). Then the orbits of planets were supposed to be perfect circles — they weren’t. They aren’t even elliptical, if you study them close enough.

Celestial spheres. An elegant theory, slightly complicated by the need to introduce epicycles to describe the movements of planets

The Impasse

But maybe at the level of elementary particles and quantum fields some of this presumed elegance of the Universe shines through? Well, not really. If the Universe obeyed the Occam’s razor, it would have stopped at two quarks, up and down. Nobody needs the strange and the charmed quarks, not to mention the bottom and the top quarks. The Standard Model of particle physics looks like a kitchen sink filled with dirty dishes. And then there is gravity that resists all attempts at grand unification. Strings were supposed to help but they turned out to be as messy as the rest of it.

Of course the current state of impasse in physics might be temporary. After all we’ve been making tremendous progress up until about the second half of the twentieth century (the most recent major theoretical breakthroughs were the discovery of the Higgs mechanism in 1964 and the proof or renormalizability of the Standard Model in 1971).

On the other hand, it’s possible that we might be reaching the limits of human capacity to understand the Universe. After all, there is no reason to believe that the structure of the Universe is simple enough for the human brain to analyze. There is no guarantee that it can be translated into the language of physics and mathematics.

Is the Universe Knowable?

In fact, if you think about it, our expectation that the Universe is knowable is quite arbitrary. On the one hand you have the vast complex Universe, on the other hand you have slightly evolved monkey brains that have only recently figured out how to use tools and communicate using speech. The idea that these brains could produce and store a model of the Universe is preposterous. Granted, our monkey brains are a product of evolution, and our survival depends on those brains being able to come up with workable models of our environment. These models, however, do not include the microcosm or the macrocosm — just the narrow band of phenomena in between. Our senses can perceive space and time scales within about 8 orders of magnitude. For comparison, the Universe is about 40 orders of magnitude larger than the size of the atomic nucleus (not to mention another 20 orders of magnitude down to Planck length).

The evolution came up with an ingenious scheme to deal with the complexities of our environment. Since it is impossible to store all information about the Universe in the very limited amount of memory at our disposal, and it’s impossible to run the simulation in real time, we have settled for the next best thing: creating simplified partial models that are composable.

The idea is that, in order to predict the trajectory of a spear thrown at a mammoth, it’s enough to roughly estimate the influence of a constant downward pull of gravity and the atmospheric drag on the idealized projectile. It is perfectly safe to ignore a lot of subtle effects: the non-uniformity of the gravitational field, air-density fluctuations, imperfections of the spear, not to mention relativistic effects or quantum corrections.

And this is the key to understanding our strategy: we build a simple model and then calculate corrections to it. The idea is that corrections are small enough as not to destroy the premise of the model.

Celestial Mechanics

A great example of this is celestial mechanics. To the lowest approximation, the planets revolve around the Sun along elliptical orbits. The ellipse is a solution of the one body problem in a central gravitational field of the Sun; or a two body problem, if you also take into account the tiny orbit of the Sun. But planets also interact with each other — in particular the heaviest one, Jupiter, influences the orbits of other planets. We can treat these interactions as corrections to the original solution. The more corrections we add, the better predictions we can make. Astronomers came up with some ingenious numerical methods to make such calculations possible. And yet it’s known that, in the long run, this procedure fails miserably. That’s because even the tiniest of corrections may lead to a complete change of behavior in the far future. This is the property of chaotic systems, our Solar System being just one example of such. You must have heard of the butterfly effect — the Universe is filled with this kind of butterflies.

Ephemerides: Tables showing positions of planets on the firmament.

The Microcosm

Anyone who is not shocked by quantum
theory has not understood a single word.

— Niels Bohr

At the other end of the spectrum we have atoms and elementary particles. We call them particles because, to the lowest approximation, they behave like particles. You might have seen traces made by particles in a bubble chamber.

Elementary particles might, at first sight, exhibit some properties of macroscopic objects. They follow paths through the bubble chamber. A rock thrown in the air also follows a path — so elementary particles can’t be much different from little rocks. This kind of thinking led to the first model of the atom as a miniature planetary system. As it turned out, elementary particles are nothing like little rocks. So maybe they are like waves on a lake? But waves are continuous and particles can be counted by Geiger counters. We would like elementary particles to either behave like particles or like waves but, despite our best efforts, they refuse to nicely fall into one of the categories.

There is a good reason why we favor particle and wave explanations: they are composable. A two-particle system is a composition of two one-particle systems. A complex wave can be decomposed into a superposition of simpler waves. A quantum system is neither. We might try to separate a two-particle system into its individual constituents, but then we have to introduce spooky action at a distance to explain quantum entanglement. A quantum system is an alien entity that does not fit our preconceived notions, and the main characteristic that distinguishes it from classical phenomena is that it’s not composable. If quantum phenomena were composable in some other way, different from particles or waves, we could probably internalize it. But non-composable phenomena are totally alien to our way of thinking. You might think that physicists have some deeper insight into quantum mechanics, but they don’t. Richard Feynman, who was a no-nonsense physicist, famously said, “If you think you understand quantum mechanics, you don’t understand quantum mechanics.” The problem with understanding quantum mechanics is not that it’s too complex. The problem is that our brains can only deal with concepts that are composable.

It’s interesting to notice that by accepting quantum mechanics we gave up on composability on one level in order to decompose something at another level. The periodic table of elements was the big challenge at the beginning of the 20th century. We already knew that earth, water, air, and fire were not enough. We understood that chemical compounds were combinations of atoms; but there were just too many kinds of atoms, and they could be grouped into families that shared similar properties. Atom was supposed to be indivisible (the Greek word ἄτομος [átomos] means indivisible), but we could not explain the periodic table without assuming that there was some underlying structure. And indeed, there is structure there, but the way the nucleus and the electrons compose in order to form an atom is far from trivial. Electrons are not like planets orbiting the nucleus. They form shells and orbitals. We had to wait for quantum mechanics and the Fermi exclusion principle to describe the structure of an atom.

Every time we explain one level of complexity by decomposing it in terms of simpler constituents we seem to trade off some of the simplicity of the composition itself. This happened again in the sixties, when physicists were faced with a confusing zoo of elementary particles. It seemed like there were hundreds of strongly interacting particles, hadrons, and every year was bringing new discoveries. This mess was finally cleaned up by the introduction of quarks. It was possible to categorize all hadrons as composed of just six types of quarks. This simplification didn’t come without a price, though. When we say an atom is composed of the nucleus and electrons, we can prove it by knocking off a few electrons and studying them as independent particles. We can even split the nucleus into protons and neutrons, although the neutrons outside of a nucleus are short lived. But no matter how hard we try, we cannot split a proton into its constituent quarks. In fact we know that quarks cannot exist outside of hadrons. This is called quark- or color-confinement. Quarks are supposed to come in three “colors,” but the only composites we can observe are colorless. We have stretched the idea of composition by accepting the fact that a composite structure can never be decomposed into its constituents.

I’m Slightly Perturbed

How do physicists deal with quantum mechanics? They use mathematics. Richard Feynman came up with ingenious ways to perform calculations in quantum electrodynamics using perturbation theory. The idea of perturbation theory is that you start with the simple approximation and keep adding corrections to it, just like with celestial mechanics. The terms in the expansion can be visualized as Feynman diagrams. For instance, the lowest term in the interaction between two electrons corresponds to a diagram in which the electrons exchange a virtual photon.

This terms gives the classical repulsive force between two charged particles. The first quantum correction to it involves the exchange of two virtual photons. And here’s the kicker: this correction is not only larger than the original term — it’s infinite! So much for small corrections. Yes, there are tricks to shove this infinity under the carpet, but everybody who’s not fooling themselves understands that the so called renormalization is an ugly hack. We don’t understand what the world looks like at very small scales and we try to ignore it using tricks that make mathematicians faint.

Physicists are very pragmatic. As long as there is a recipe for obtaining results that can be compared with the experiment, they are happy with a theory. In this respect, the Standard Model is the most successful theory in the Universe. It’s a unified quantum field theory of electromagnetism, strong, and weak interactions that produces results that are in perfect agreement with all high-energy experiments we were able to perform to this day. Unfortunately, the Standard Model does not give us the understanding of what’s happening. It’s as if physicists were given an alien cell phone and figured out how to use various applications on it but have no idea about the internal workings of the gadget. And that’s even before we try to involve gravity in the model.

The “periodic table” of elementary particles.

The prevailing wisdom is that these are just little setbacks on the way toward the ultimate theory of everything. We just have to figure out the correct math. It may take us twenty years, or two hundred years, but we’ll get there. The hope that math is the answer led theoretical physicists to study more and more esoteric corners of mathematics and to contribute to its development. One of the most prominent theoretical physicists, Edward Witten, the father of M-theory that unified a number of string theories, was awarded the prestigious Fields Medal for his contribution to mathematics (Nobel prizes are only awarded when a theory is confirmed by experiment which, in the case of string theory, may be a be long way off, if ever).

Math is About Composition

If mathematics is discoverable, then we might indeed be able to find the right combination of math and physics to unlock the secrets of the Universe. That would be extremely lucky, though.

There is one property of all of mathematics that is really striking, and it’s most clearly visible in foundational theories, such as logic, category theory, and lambda calculus. All these theories are about composability. They all describe how to construct more complex things from simpler elements. Logic is about combining simple predicates using conjunctions, disjunctions, and implications. Category theory starts by defining a composition of arrows. It then introduces ways of combining objects using products, coproducts, and exponentials. Typed lambda calculus, the foundation of computer languages, shows us how to define new types using product types, sum types, and functions. In fact it can be shown that constructive logic, cartesian closed categories, and typed lambda calculus are three different formulations of the same theory. This is known as the Curry Howard Lambek isomorphism. We’ve been discovering the same thing over and over again.

It turns out that most mathematical theories have a skeleton that can be captured by category theory. This should not be a surprise considering how the biggest revolutions in mathematics were the result of realization that two or more disciplines were closely related to each other. The latest such breakthrough was the proof of the Fermat’s last theorem. This proof was based on the Taniyama-Shimura conjecture that related the study of elliptic curves to modular forms — two radically different branches of mathematics.

Earlier, geometry was turned upside down when it became obvious that one can define shapes using algebraic equations in cartesian coordinates. This retooling of geometry turned out to be very advantageous, because algebra has better compositional qualities than Euclidean-style geometry.

Finally, any mathematical theory starts with a set of axioms, which are combined using proof systems to produce theorems. Proof systems are compositional which, again, supports the view that mathematics is all about composition. But even there we hit a snag when we tried to decompose the space of all statements into true and false. Gödel has shown that, in any non-trivial theory, we can formulate a statement that can neither be proved to be right or wrong, and thus the Hilbert’s great project of defining one grand mathematical theory fell apart. It’s as if we have discovered that the Lego blocks we were playing with were not part of a giant Lego spaceship.

Where Does Composability Come From?

It’s possible that composability is the fundamental property of the Universe, which would make it comprehensible to us humans, and it would validate our physics and mathematics. Personally, I’m very reluctant to accept this point of view, because it would give intelligent life a special place in the grand scheme of things. It’s as if the laws of the Universe were created in such a way as to be accessible to the brains of the evolved monkeys that we are.

It’s much more likely that mathematics describes the ways our brains are capable of composing simpler things into more complex systems. Anything that we can comprehend using our brains must, by necessity, be decomposable — and there are only so many ways of putting things together. Discovering mathematics means discovering the structure of our brains. Platonic ideals exist only as patterns of connections between neurons.

The amazing scientific progress that humanity has been able to make to this day was possible because there were so many decomposable phenomena available to us. Granted, as we progressed, we had to come up with more elaborate composition schemes. We have discovered differential equations, Hilbert spaces, path integrals, Lie groups, tensor calculus, fiber bundles, etc. With the combination of physics and mathematics we have tapped into a gold vein of composable phenomena. But research takes more and more resources as we progress, and it’s possible that we have reached the bedrock that may be resistant to our tools.

We have to seriously consider the possibility that there is a major incompatibility between the complexity of the Universe and the simplicity of our brains. We are not without recourse, though. We have at our disposal tools that multiply the power of our brains. The first such tool is language, which helps us combine brain powers of large groups of people. The invention of the printing press and then the internet helped us record and gain access to vast stores of information that’s been gathered by the combined forces of teams of researchers over long periods of time. But even though this is quantitative improvement, the processing of this information still relies on composition because it has to be presented to human brains. The fact that work can be divided among members of larger teams is proof of its decomposability. This is also why we sometimes need a genius to make a major breakthrough, when a task cannot be easily decomposed into smaller, easier, subtasks. But even genius has to start somewhere, and the ability to stand on the shoulders of giants is predicated on decomposability.

Can Computers Help?

The role of computers in doing science is steadily increasing. To begin with, once we have a scientific theory, we can write computer programs to perform calculations. Nobody calculates the orbits of planets by hand any more — computers can do it much faster and error free. We are also beginning to use computers to prove mathematical theorems. The four-color problem is an example of a proof that would be impossible without the help of computers. It was decomposable, but the number of special cases was well over a thousand (it was later reduced to 633 — still too many, even for a dedicated team of graduate students).

Every planar map can be colored using only four colors.

Computer programs that are used in theorem proving provide a level of indirection between the mind of a scientist and formal manipulations necessary to prove a theorem. A programmer is still in control, and the problem is decomposable, but the number of components may be much larger, often too large for a human to go over one by one. The combined forces of humans and computers can stretch the limits of composability.

But how can we tackle problems that cannot be decomposed? First, let’s observe that in real life we rarely bother to go through the process of detailed analysis. In fact the survival of our ancestors depended on the ability to react quickly to changing circumstances, to make instantaneous decisions. When you see a tiger, you don’t decompose the image into individual parts, analyze them, and put together a model of a tiger. Image recognition is one of these areas where the analytic approach fails miserably. People tried to write programs that would recognize faces using separate subroutines to detect eyes, noses, lips, ears, etc., and composing them together, but they failed. And yet we instinctively recognize faces of familiar people at a glance.

Neural Networks and the AI

We are now able to teach computers to classify images and recognize faces. We do it not by designing dedicated algorithms; we do it by training artificial neural networks. A neural network doesn’t start with a subsystem for recognizing eyes or noses. It’s possible that, in the process of training, it will develop the notions of lines, shadows, maybe even eyes and noses. But by no means is this necessary. Those abstractions, if they evolve, would be encoded in the connections between its neurons. We might even help the AI develop some general abstractions by tweaking its architecture. It’s common, for instance, to include convolutional layers to pre-process the input. Such a layer can be taught to recognize local features and compress the input to a more manageable size. This is very similar to how our own vision works: the retina in our eye does this kind of pre-processing before sending compressed signals through the optic nerve.

Compression is the key to matching the complexity of the task at hand to the simplicity of the system that is processing it. Just like our sensory organs and brains compress the inputs, so do neural networks. There are two kinds of compression: the kind that doesn’t lose any information, just removing the redundancy in the original signal; and the lossy kind that throws away irrelevant information. The task of deciding what information is irrelevant is in itself a process of discovery. The difference between the Earth and a sphere is the size of the Himalayas, but we ignore it when when we look at the globe. When calculating orbits around the Sun, we shrink all planets to points. That’s compression by elimination of details that we deem less important for the problem we are solving. In science, this kind of compression is called abstraction.

We are still way ahead of neural networks in our capacity to create abstractions. But it’s possible that, at some point, they’ll catch up with us. The problem is: Will we be able to understand machine-generated abstractions? We are already at the limits of understanding human-generated abstractions. You may count yourself a member of a very small club if you understand the statement “monad is a monoid in the category of endofunctors” that is chock full of mathematical abstractions. If neural networks come up with new abstractions/compression schemes, we might not be able to reverse engineer them. Unlike a human scientist, an AI is unlikely to be able to explain to us how it came up with a particular abstraction.

I’m not scared about a future AI trying to eliminate human kind (unless that’s what its design goals are). I’m afraid of the scenario in which we ask the AI a question like, “Can quantum mechanics be unified with gravity?” and it will answer, “Yes, but I can’t explain it to you, because you don’t have the brain capacity to understand the explanation.”

And this is the optimistic scenario. It assumes that such questions can be answered within the decomposition/re-composition framework. That the Universe can be decomposed into particles, waves, fields, strings, branes, and maybe some new abstractions that we haven’t even though about. We would at least get the satisfaction that we were on the right path but that the number of moving parts was simply too large for us to assimilate — just like with the proof of the four-color theorem.

But it’s possible that this reductionist scenario has its limits. That the complexity of the Universe is, at some level, irreducible and cannot be captured by human brains or even the most sophisticated AIs.

There are people who believe that we live in a computer simulation. But if the Universe is irreducible, it would mean that the smallest computer on which such a simulation could be run is the Universe itself, in which case it doesn’t make sense to call it a simulation.

Conclusion

The scientific method has been tremendously successful in explaining the workings of our world. It led to exponential expansion of science and technology that started in the 19th century and continues to this day. We are so used to its successes that we are betting the future of humanity on it. Usually when somebody attacks the scientific method, they are coming from the background of obscurantism. Such attacks are easily rebuffed or dismissed. What I’m arguing is that science is not a property of the Universe, but rather a construct of our limited brains. We have developed some very sophisticated tools to create models of the Universe based on the principle of composition. Mathematics is the study of various ways of composing things and physics is applied composition. There is no guarantee, however, that the Universe is decomposable. Assuming that would be tantamount to postulating that its structure revolves around human brains, just like we used to believe that the Universe revolves around Earth.

You can also watch my talk on this subject.