The Beauty of Physics
物理之美
In 1862 Maxwell took a number measured in electrical experiments and calculated from it a speed that came out almost the same as the speed of light that others had measured with a toothed wheel and a mirror. Two numbers from two utterly unrelated places met. Later he took down the model he had first built and kept only the equations, and the speed was still there: it is a number the equations of electricity and magnetism could not fail to give. A subject who has seen this cannot not acknowledge: the world has answered this form, and answered it exactly where the form could not be otherwise. That sentence is the beauty of physics. It has three faces: completive beauty is the world having already answered at the point of necessity; generative beauty is a form being put where the world can say no, with the answer not yet come, or come back as "no"; the beauty of the remainder is the forms that already stand and the answers the world has already given drawing a gap together, with the new form not yet come. The paper says what physics is and is not (not mathematics with a "reality" label stuck on it, not a pile of readings, not accuracy, and not the world itself); how duality lets two very different forms receive one and the same answer; what can pass for the beauty of physics (merely fitting, merely pretty forms, beauty taking the world's seat, accuracy, the big, the costly and the famous, usefulness, the isolated anomaly); and what is left to each person. It then traces where the question comes from, first separating four things beauty has been treated as among physicists, then running from the strings of Pythagoras to today's debate over whether beauty leads physics astray. It ends by handing on to the beauty of causation the question of why the world has a direction.
Keywords: beauty of physics, form, world, completive beauty, generative beauty, beauty of the remainder, duality, power of judgment, overlap, 3DD
1. Two Numbers
In 1862 Maxwell was thinking about how electricity and magnetism act on each other across space. He built a mechanical model of this action: space filled with little spinning vortices, with still smaller wheels sandwiched between them. With the model built, he asked a question: in such a medium, how fast would a disturbance travel?
The answer needed a number that had to come from electrical experiments. A few years earlier Weber and Kohlrausch had measured it with Leyden jars, wires and a galvanometer. Maxwell put it in, and the speed came out at around 310,000 kilometers per second.
The other number was somewhere else. In 1849, outside Paris, Fizeau sent a beam of light through a spinning toothed wheel to a mirror more than eight kilometers away, and back through the wheel again. At a certain speed of the wheel, the returning light was blocked exactly by the next tooth; from the speed of the wheel one could calculate the speed of light, and it too came out at around 310,000 kilometers per second.
One number came from Leyden jars and a galvanometer, the other from a toothed wheel and a mirror. Electricity and light had been two separate studies. Others had already noticed that this number was close to the speed of light; Maxwell took one step further. He wrote that the two agree so exactly that "we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena."[^1]
Two numbers meeting could be a coincidence. A coincidence surprises for a moment, and passes. This one did not pass. Two or three years later Maxwell took down all the vortices and little wheels and kept only the equations of electricity and magnetism, and the speed was still there. The equations say that an electromagnetic disturbance must travel at this speed; the speed is fixed by two constants of electricity and magnetism, and there is nowhere else in the equations to adjust it.[^2] The model was scaffolding, and came down; the number was not on the scaffolding, but at the point where the equations could not be otherwise. It was exactly there that the world answered: it could have said no, and it said yes.
That point is what this paper is about.
This paper follows the reading set out in the general outline.[^3] A personal aesthetic is each person's own and comes from his own cannot-not; aesthetics is the part that subjects with judgment cannot not acknowledge together, here called the overlap. The overlap is large; what does not overlap is larger. The overlap is not everyone being obliged to agree: whoever looks cannot not see, and whoever has seen cannot not acknowledge. When someone says "it is beautiful," he is claiming that this time he has landed on the overlap, and that claim can be right or wrong. What settles out of an act of drawing a boundary is called a construct; what it cannot contain is called the remainder. The remainder gives a place where one can go on, and a direction; it pushes no one; whether to go depends on a person's own cannot-not.[^4] A beauty has two faces: when the construct has come into place, it is completive beauty, quiet; when the construct is being chiseled out, it is generative beauty, tense. The beauty of physics has a third face as well, the beauty of the remainder: some gaps have a shape of their own, drawn by what already stands. A ray is a direction of judgment, not a classification of objects: the same planetary orbit can also be looked at from the direction of mathematics, or from the direction of causation. This paper asks only what the power of judgment sees in the one direction of form fitting the world.
2. What Physics Is and Is Not
Form can stand by its own rules alone; it needs no nod from the world. Physics begins with one thing: people take form to the world, and the world can say no.
"Can say no" is the first condition of physics. A form that fits the world however the world may be has said nothing to the world. "Tomorrow it will either rain or not rain" will never be wrong, and says nothing at all. A form in physics has to say beforehand: if the world is like this, I am right; if it is like that, I am wrong. Then it is put before the world.
So physics is not mathematics with a "reality" label stuck on it. One equation can describe how a spring oscillates back and forth, and can also describe how the current in a circuit oscillates back and forth. The equation is the same, and mathematically it is one form; but the physics is two different things: one is displacement and elasticity, the other charge and coils; the ways of measuring them differ, and so do the ranges in which they hold and the places where they fail. What makes it physics is not the equation but the road between the equation and the world.
Nor is physics a pile of readings. Readings come only through instruments: instruments have to be calibrated, data processed, errors estimated. The world does not bypass instruments to speak directly; it answers through instruments and models. But it does answer: two instruments built on different principles can measure the same thing and disagree; an anomaly can refuse to go away however one tries. The world's "no" is in these disagreements and in what refuses to go away.
Physics is not accuracy. A large enough table, a flexible enough curve, can trace known data very accurately and say nothing about why. Accuracy is the world having answered; where it answered is another matter.
Nor is physics the world itself. The world does not need physics; the planets were moving before anyone wrote down an ellipse. Physics is the form people take to the world, and so it can be replaced, and often is. Being replaced is not being canceled: Newton's gravity was not canceled by Einstein; it was put back into the range in which it holds, and where speeds are not too high and gravity not too strong, it still answers very well.
Nor is physics any one way of writing. The same physics can be written in very different forms; this is taken up in Section 4.
3. The Beauty of Physics
Once a subject has seen how a form fits the world, he cannot not acknowledge:
The world has answered this form, and answered it exactly where the form could not be otherwise.
That sentence is the beauty of physics. The first half is shared by everything correct in physics: the world could have said no, and did not. It is the second half that makes it beauty: the place where the world answered is exactly where the form has no room to spare. Maxwell's speed was not tuned; the equations could not fail to give it, and the world answered there.
There are three things here, and they have to be kept apart. One is the necessity in the form: once the rules are fixed, the conclusion can only be this. That is the affair of the form. The second is the world's answer: the world could have said no, and it said yes. That is the affair of the world, and no one can say it on the world's behalf. The third is the cannot-not of the one who sees: seeing necessity and answer meet at one point, he cannot not acknowledge that it is beautiful. That is the affair of the subject. Take away any one of the three, and it is not the beauty this sentence speaks of. Without necessity, the answer falls on a place that had been tuned, and that is patching things together; if the form has never been put before the world and is left with only itself, that is the beauty of mathematics, or only a wish; if no one sees, there is no one who cannot not acknowledge anything.
Why is it overlap and not preference? Someone who can follow those few equations, and who knows how those two numbers were measured, and who looks at it rather than using it to calculate something else, cannot not see the same thing: the number was not put in; it walked out of the equations by itself, and the world's answer was right there. This does not depend on nationality, or on the era, or on whether he loves physics. In physics, seeing takes two things: understanding why the form can only say this, and understanding through which hands the world's answer came. Without the first, what one sees is only accuracy; without the second, what one sees is only mathematics.
In physics classes many people say that Maxwell's equations are beautiful because the teacher says so, and because they are printed on T-shirts. That sentence does not count toward the overlap. The teacher's word can be cultivation: he points, you see for yourself how that speed walks out of the equations, and then the acknowledgment is yours. Saying it only because the teacher said it does not count. Take away the teacher's word and see: if you still acknowledge it, and can change it yourself, it is yours; if you no longer acknowledge it, it is not.
The beauty of physics has three faces. On all three, the beauty falls where the necessity of the form and the world meet. On the completive face, the world has answered there. On the generative face, the meeting is happening: the form has been put where the world can say no, and the answer has not yet come, or has come back as "no," and someone is taking it and reworking the form. On the face of the remainder, the forms that already stand and the answers the world has already given draw a gap together, and the new form has not yet come. A form to which the world has said "no" is not itself beautiful because of that meeting; the beauty is in the meeting, in the shape that "no" draws. The sentence at the start of this section speaks of the completive face; the other two faces are two other moments of the same meeting.
Completive beauty: the world has already answered at the point of necessity. Kepler spent years before he made out from the observations of Mars that it travels in an ellipse (how he made it out is told below). But why an ellipse? Kepler did not know. More than seventy years later Newton proved that if attraction falls off as the inverse square of distance, a planet going round the sun can only travel in an ellipse, with the sun at one focus.[^5] The ellipse changed from a shape observed into a shape the law of gravity could not fail to give. The same law makes stones fall, keeps the moon going round the earth and makes the sea rise and fall. One form, and the world answers it at once in many utterly unrelated places.
Another example is quieter. The conservation of energy and of momentum were once rules summed up from experience: calculated many times, they always held. In 1918 Noether proved something: in a very wide class of theories, if the laws do not change with time, some quantity is conserved, and that is energy; if the laws do not change from place to place, another quantity is conserved, and that is momentum; if the laws do not change with direction, angular momentum is conserved.[^6] From then on conservation was no longer something that happened to hold in experience, but something the symmetries of the laws could not fail to give. The world had always answered this way; Noether let people see why it could only be so.
What it can show is that the why of conservation lies in symmetry. It cannot show that the world must have these symmetries; whether it does is the world's answer, not the form's. The universe is expanding, today's universe and tomorrow's are no longer the same, and on the scale of the whole universe conservation of energy can no longer be stated as it was. This is not an exception to Noether; it is exactly the same why.
The most famous example of all is the one its fame has hidden. In 1905 Einstein started from two things: the laws of physics are the same in every frame of reference in uniform motion, and the speed of light is the same for everyone. From these two he derived that when a body gives off energy its mass must decrease, by the energy given off divided by the square of the speed of light.[^7] He had not set out to find such a relation; those two starting points could not fail to give it. Later, when atomic nuclei were weighed, a nucleus turned out a little lighter than the protons and neutrons that make it up, added together separately; the little that is missing is exactly the energy given off when they bound together, and this has now been measured to a few parts in ten million. The beauty of E = mc² is not in how many places it is printed, but here: it is a place where the form has no room to spare, and the world answered there.
Generative beauty: the form is meeting the world. This is the face of the chisel. The form has been put where the world can say no; the answer has not yet come, or has come back as "no," and someone is taking this "no" and working the form out anew.
Around 1600 Kepler obtained Tycho's many years of observations of Mars, the most accurate data of the time. He tried to fit them with all sorts of combinations of circles and uniform motion, at great cost of effort, and arrived at a model in which the direction of Mars in the sky differed from Tycho's observations by less than two minutes of arc (a minute is a sixtieth of a degree). But the model gave the wrong distances for Mars; correct it to get the distances right, and the direction was off by eight minutes. Eight minutes is about a quarter of the diameter of the full moon; before Tycho, the errors of observation themselves had been that large, and these eight minutes would have been let pass as error. Kepler did not let them pass. He wrote that divine goodness had given us in Tycho a most diligent observer, whose observations revealed this error of eight minutes; and because these eight minutes could not be ignored, they alone pointed the way to reforming the whole of astronomy.[^8] He gave up the circle, tried various ovals, and only years later came down on the ellipse.
This is generative beauty: the world has said "no," and this "no" has a precise size; someone takes it and works the form out, cut by cut. The tension is in those eight minutes.
Generative beauty is also in the moment a statement is put where the world can say no. On 14 September 2015 two gravitational-wave detectors in the United States, three thousand kilometers apart, recorded the same signal seven milliseconds apart: a chirp growing ever faster and higher, over in less than half a second. General relativity says that when two black holes circle each other, draw closer and closer and finally merge into one, the ripples in spacetime should look like this.[^9] But it could also have been something else: a fault in the instruments, a tremor in the ground, interference from the power grid, or a coincidence. The detectors were placed in two locations precisely to make coincidence hard. In the months that followed, the collaboration ruled out the other possibilities one by one. The beauty is not in the size of the detectors, nor only in the "first time"; it is in this road: it put a statement derived from the equations where the world could say no, and laid the road out in the open, so that others could check step by step how it was not said no to.
On such a road, even a result of "no" can be beautiful. In 1887 Michelson and Morley used an ingenious interferometer to look for the "ether wind" that the earth should meet as it passed through the ether. By the theory of the time, the interference fringes should have shifted by about four tenths of a fringe; the shift they measured was certainly less than a twentieth of that, and probably less than a fortieth.[^10] They found nothing. But the road put the place where the ether theory could not be otherwise before the world, and the world said no there; this "no" was as clear as any "yes." The beauty is in the road and in this "no," not in the ether theory.
The beauty of the remainder: the gap has a shape, and the new form has not yet come. The ellipse on which Mercury goes round the sun is itself slowly turning; its point nearest the sun moves forward a little every century. How much it moves, Newton's gravity can calculate: each of the other planets tugs at Mercury, and the tugs add up. In 1859 Le Verrier finished the calculation and found that the actual shift was larger than the calculated one, by thirty-odd seconds of arc per century; later Newcomb recalculated it, and the excess came to about forty-three seconds of arc.[^11] A second of arc is a three-thousand-six-hundredth of a degree; forty-three seconds of arc is smaller than the smallest angle the human eye can resolve.
This gap has a shape. It is not "some things cannot be accounted for," but a definite number, a definite direction, falling on a definite planet; everything known had already been taken into account, and the shape of what remained was drawn exactly by the relations that already stood together with the world's answers. Thirteen years earlier, Le Verrier had used the same method to calculate, from the anomalies of Uranus, a planet no one had ever seen, and Neptune was found near where he had calculated. This time too he calculated a planet inside Mercury's orbit, later called Vulcan. Some said they saw it; later no one ever saw it again. Others tried changing Newton's inverse square, making the two a tiny bit larger than two.[^12] The shape was there for all of them, the ways of filling it differed, and none of them filled it. The gap sat there like that for more than fifty years.
In 1915 Einstein calculated it with his newly finished general theory of relativity, adjusting nothing, and what came out was exactly forty-three seconds of arc. He later wrote in a letter that for several days he was beside himself with excitement.[^13]
But the beauty of the remainder is not the beauty of this moment; this moment is completive beauty. The beauty of the remainder lies in those fifty-odd years. Take away the fact that Einstein later solved it, take away its fame, take away the mystery that the unknown brings: an astronomer in 1890, looking at those forty-odd seconds of arc, still saw a gap drawn with complete clarity by relations that already stood. Its beauty belonged to that time, and needed no later answer to countersign it.
There are gaps like this today. The stars in the outer part of a galaxy, by the reckoning of the visible matter, should go round more slowly the further out they are; measured, they go round almost as fast.[^14] The shape of the gap is quite clear, drawn by the law of gravity together with the visible matter. The ways of filling it are of two kinds: there is invisible matter out there, as with Neptune in the past; or the law of gravity has to be changed at such scales, as with Mercury in the past. Most physicists think it is the first kind, for more reasons than the rotation alone; some hold to the second. The shape of the gap does not tell you which kind it is, but it gives a place where one can go on, and a direction: whichever way of filling it one takes has to fill out this shape, and one that cannot has gone astray. It pushes no one.
4. Duality: Two Forms, One Answer
Sometimes the world answers two very different forms at the same places.
In 1925 Heisenberg wrote down one kind of quantum mechanics. It spoke only of quantities that could be measured, and calculated with tables of numbers, one table after another; when two tables were multiplied, the order could not be swapped. The next year Schrödinger wrote down another kind: the electron is a wave, satisfying a wave equation. One proceeded in jumps, step by step; the other was continuous, and could be pictured. The two men did not think much of each other's work. Schrödinger said that the methods of transcendental algebra struck him as very difficult and lacking in anything one could picture, and that they discouraged him, "not to say repelled" him.[^15] Heisenberg wrote to Pauli that the more he thought about the physical part of Schrödinger's theory, the more repulsive he found it; what Schrödinger wrote about its picturability was "rubbish."[^16] Yet both forms calculated the set of spectral lines that hydrogen gives off, and both matched what was measured on the spectrograph.[^17] Schrödinger went on to prove that the two forms in fact gave the same answers on the problems that could then be calculated. The world had answered two forms at the same places; that proof let people see why they answered alike.
Strictly speaking, the two quantum mechanics came to be seen as two ways of writing one theory, which is closer still than what physicists call "duality." What physicists call a duality is a dictionary between two theories that look different, translating the states, quantities and changes on one side, item by item, into those on the other. But the two quantum mechanics already let people see the thing duality has to say: the world's answer takes no account of how a theory is written.
A dictionary fixes the answer. In 1941 Kramers and Wannier studied one of the simplest models of a magnet: a plane covered with little arrows, each of which can only point up or down, with neighboring arrows preferring to point the same way. When hot, the arrows are in disarray; when cold, they line up. They found a dictionary: the description of the disarray at high temperature could be translated, sentence by sentence, into the description of the alignment at low temperature, only with the temperature changed. Hot and cold, disarray and alignment, were, astonishingly, two ways of saying the same thing. So there is one temperature that the dictionary translates back into itself; if there is only one transition point between disarray and alignment, it can only be here. This is how they fixed the temperature of the transition. Three years later Onsager solved the model exactly, and the transition point was right there.[^18] Before the model was solved, the mere fact that the two ways of saying it mirror each other fixed the point where the answer lies. This example is still within form: the transition point came from solving the model, not from the world's answer. It can show that a duality can fix where an answer lies before a model is solved; it cannot show what the world has answered. On this ray it is what duality looks like within form, and belongs to the beauty of mathematics.
Symmetry waiting for an answer. Where there are no charges, electricity and magnetism are almost symmetric in Maxwell's equations: swap electric for magnetic and magnetic for electric, add a minus sign, and the equations are unchanged. The symmetry breaks at one point: the world has lone electric charges, but no lone magnetic pole has ever been found; cut a magnet in two, and each piece still has a pole at both ends. In 1931 Dirac proved that if there is even one lone magnetic pole anywhere in the universe, every electric charge can only be a whole-number multiple of some smallest charge.[^19] Electric charges are indeed all whole-number multiples; the world answered that long ago. The lone magnetic pole has been searched for for decades and has still not been found. This is a gap with a shape: the symmetry that already stands and the answer the world has already given together draw the missing half; that half, the world has not yet answered.
Dualities still within form. String theory has deeper dualities: in some theories a space with a very large radius and a space with a very small radius give the same physics; a space with gravity and a theory without gravity on its boundary may be two ways of saying the same thing. Within form they stand very firmly, and some have been borne out by a great deal of calculation; but the world they speak of has not yet been touched by any experiment, and the world has not yet answered them at any point. On this ray they are still the beauty of mathematics; only when one day they are put before the world will they have the face of physics.
Duality lets people see two things. One: no way of writing has the world to itself. Heisenberg's tables and Schrödinger's waves: neither is more "the way the world really is" than the other. Two: different ways of writing let one see different whys. The wave lets one see why electrons interfere with each other as light does; the tables let one see why energy comes step by step. Both let one see a why, only not the same why. So the beauty here, in duality, is not in the conciseness of either way of writing, but in the one answer the two ways of writing received at the same place; which way of writing suits one's own eye is each person's personal aesthetic; saying which is more beautiful is a claim made on the overlap, taken up in Section 6.
5. What Can Pass for It
The method is still the same: take it away and see.
Merely fitting. Ptolemy described the planets with circles on circles: a small circle carried round on a large circle, with the planet moving on the small one. The method was accurate, and was used for more than a thousand years. It had several places that could be set to fit the observations: how large the big circle is, how large the small one, how far the center is set off, and about which point the uniform motion is uniform. Today it is known that with enough circles on circles, almost any periodic motion can be traced.[^20] The world answered it, but mostly at places that had been tuned; take the tuning away, and what is left says nothing about why. It is not without value; it is one of the greatest forms of antiquity. It is only that merely fitting is not beautiful.
Merely pretty forms. In his first book Kepler believed that between the six orbital spheres of the six planets then known lay exactly the five regular solids; there are only five regular solids, and so there are only six planets.[^21] It is a beautiful form, and the distances came out roughly right. The world did not answer it: Uranus and Neptune were found later. Kelvin at one time believed that atoms were knotted vortex rings in the ether, with different kinds of knot making different atoms; for this Tait classified knots one by one and compiled the first knot tables.[^22] Atoms are not like that. Take away the world's answer, and what is left can be the beauty of mathematics, and those knot tables are still in mathematics today; but it is not the beauty of physics.
Beauty taking the world's seat. A physicist's personal aesthetic can lead the way. Dirac said that it is more important to have beauty in one's equations than to have them fit experiment.[^23] His own equation was beautiful and was also answered by the world: it said there was a particle just like the electron but with the opposite charge, and a few years later the positron was found in cosmic rays. But if that sentence is taken as a rule, the personal aesthetic turns from the one who leads the way into the one who decides the case. Hossenfelder wrote a book saying that in recent decades a preference in fundamental physics for "naturalness," "symmetry" and "simplicity" led many people to expect too much in the way of new particles, and the large colliders did not deliver them.[^24] When a preference refuses the world's "no," blocks revision and still claims that "nature must be so," it is no longer a personal aesthetic leading the way; beauty has taken the world's seat. A personal aesthetic can point the way; it cannot vouch on the world's behalf.
Accuracy. Some weather forecasts made today with machine learning are, on many measures, more accurate than those calculated from the equations of physics.[^25] They fit the world, and very accurately; but they do not say why. Accuracy is the world having answered, but not at a point of necessity. This does not say they are bad, nor that no why can later be found in them; it is only that accuracy is not beauty.
The big, the costly and the famous. Each arm of the gravitational-wave detector is four kilometers long, and it won a Nobel Prize; these things draw attention to it, they do not make it beautiful. The beauty is in the road that put a statement where the world can say no; a tabletop instrument, if it walked the same road, would be just as beautiful.
Being useful. Satellite navigation has to include the corrections of general relativity: the clocks on the satellites run faster than those on the ground by thirty-odd microseconds a day, and without the correction the positions would drift by around ten kilometers in a day.[^26] The beauty of general relativity is not increased by a hair on that account; without this use, it would not be diminished by a hair either.
The isolated anomaly. In 2011 an experiment reported that neutrinos sent from near Geneva to Italy arrived sixty nanoseconds earlier than light would have. The news caused a sensation. A few months later it was found that a fiber-optic connector had not been plugged in properly, and a clock was faulty too.[^27] Even at the time it could be told apart from Mercury's gap. It came from one road only; no second instrument, built on a different principle, had measured it. And it clashed with answers the world had already given: in 1987 a supernova exploded more than a hundred thousand light-years away, and its neutrinos and its light reached the earth only a few hours apart; by this anomaly the neutrinos should have arrived more than four years early. The two kinds of neutrinos differ in energy and in type, so the comparison was not conclusive, but it was enough to raise doubts. Mercury's gap was not like this: for many people, over many years, by many methods of calculation, it was there. To see an anomaly as a remainder is a judgment that can be right or wrong; when the connector was found to be at fault, what was withdrawn was that judgment. It does not mean that the beauty of the remainder has to wait for a later answer to countersign it.
Neighboring rays. The beauty of mathematics looks at how form stands by its own rules; the beauty of physics looks at how the world answers form at its point of necessity. A theorem needs no answer from the world; an equation that stands only within form, not yet answered by the world, is still in mathematics. The beauty of causation looks at the world's direction: a broken cup does not put itself back together. Most of the fundamental equations of physics do not distinguish before from after; turn time around, and they still hold. Where the direction comes from is the affair of the next ray. Mercury's gap is a difference between a form and the world, not the world's direction, so it belongs on this ray. The beauty of perception looks at how the world is seen for the first time: the orange ring in the first photograph of a black hole is pretty, and its colors were chosen when the image was made; that is a matter for the ray of perception. The beauty of physics is in the size of the ring: weigh that black hole by other means, calculate a size from general relativity, and the ring comes out about that size.[^28]
One last sentence, which this aesthetics says of every ray: beauty is not correctness, not usefulness, not importance, not understanding. Ptolemy's circles fit, satellite navigation puts relativity to use, the big machine is important, and all three have been tested above by taking them away; understanding is taken up in the section on where the question comes from, with Wigner.
6. What Is Left to Each Person
Which physics moves you is each person's own. Some love Maxwell's four lines of equations; some love thermodynamics, which hardly cares what the world is made of and yet is rarely wrong; some love experiment, the road that carries a statement before the world; some love the gaps, and spend a lifetime looking at a place not yet filled in.
That Heisenberg and Schrödinger did not think much of each other's way of writing is each person's own; that the two ways of writing received the world's answer at the same places, and that those who have seen this cannot not acknowledge it together, is the overlap. To say "Schrödinger's waves are more beautiful than Heisenberg's tables" is a judgment of beauty, and it can be discussed: has the wave really set out one more why? Discussion may show that each side has its own why, and then each keeps its own.
Einstein never liked quantum mechanics. He wrote to Born that quantum mechanics was very impressive, but an inner voice told him that it was not the real thing; at any rate, he was convinced that the "Old One" does not throw dice.[^29] That inner voice was his own. Later experiments ruled out the kind of explanation he most wanted: that particles each carry answers fixed in advance, with no need to communicate across a distance. As liking or disliking, his voice was neither right nor wrong; taken as a guess about the world, it met the world's no. If he said "quantum mechanics is not beautiful," that would be a claim made on the overlap, which can be discussed and may be wrong; if he said only "I do not like it," there is nothing there that can be wrong.
Physicists choosing theories share several standards: accuracy, consistency, broad scope, simplicity and fruitfulness. These are standards for choosing theories, not the beauty of physics; accuracy, among them, has already been tested above by taking it away. Kuhn said that these standards do not fix a single order, and different people give them different weights.[^30] There is something of each person in choosing theories, and in seeing beauty too. The overlap is large; what does not overlap is larger.
Whether to spend a life looking at why the stars in the outer parts of galaxies go round so fast is each person's own. Aesthetics says only this: whoever has seen that shape cannot not acknowledge that it is beautiful; aesthetics decides for no one whether the gap is worth a life.
7. Where the Question Comes From
Among physicists, beauty has been treated as at least four things. As a motive: Poincaré said that scientists study nature because nature is beautiful.[^31] As a guide: Dirac let the beauty of equations walk ahead of experiment.[^32] As evidence: a theory is beautiful, and so more likely to be right. As metaphysics: the world is made of beautiful forms to begin with, as Plato thought; the Pythagoreans said it is made of number and harmony. Physicists' words often slide back and forth among these four layers. The beauty of physics this paper speaks of is none of them. Motive and guide are in each person: research comes from each person's own cannot-not, and what leads the way is each person's personal aesthetic, which may lead down the right road or the wrong one; evidence this paper does not give, for beauty cannot decide the case; metaphysics this paper does not assert, for being answered is not being so. What this paper speaks of lies wholly where the necessity of a form and the world meet: the world has answered there; or the statement has been put there and the answer has not yet come, or has come back as "no"; or the forms that already stand and the answers the world has given have drawn a gap together. Whoever has seen it cannot not acknowledge it.
The layer in which beauty is taken as evidence is simply absent in some traditions. The fourth verse of the Indian Sāṃkhyakārikā says that reliable knowledge has three means: perception, inference and the word of a trustworthy person.[^33] Beauty is not among them. The verse lists them in order to distinguish reliable knowledge, not to speak of beauty. When this paper says "beauty cannot decide the case," there is no need to argue it here: beauty was never among the means to begin with.
Number is answered by the world first. The earliest story is on a string. The Pythagoreans found that when a string is taken at half its length, the sound rises an octave; at two thirds, a fifth; at three quarters, a fourth. What the ear hears as harmonious is exactly the simple whole-number ratios. This is the earliest well-known case of a form being answered by the world, and they went a long way from it: Aristotle says that they took the elements of number to be the elements of all things, and the whole heaven to be a harmony and a number.[^34] There is also a small episode here. The story goes that Pythagoras, passing a smithy, heard that the sounds of several hammers were harmonious, weighed them, and found the hammers' weights in simple ratios too. The story circulated for two thousand years, and it is wrong: pitch does not follow the weight of the hammer. At the end of the sixteenth century the musician Vincenzo Galilei, Galileo's father, tried strings hung with weights, and found that to raise the pitch an octave the weight had to be four times as heavy, not twice.[^35] The world answered the ratio of string lengths, not the ratio of hammers; where a form fits the world has to be tried one place at a time. What the Pythagoreans discerned: the world can answer number. They press on this paper with the phrase "all things are number": if the world has answered form, does that mean it is form to begin with? This paper parts here: being answered is not being so.
The Mohist Canon has an entry on the pinhole image: "The image is inverted. At the crossing there is a point, together with the length of the image. The explanation lies in the point."[^36] The image is upside down because light travels in straight lines and all of it crosses through that one small hole: light from above falls below, and light from below falls above. An inverted image is explained by a why. This is a very clean instance, in the ancient Chinese books, of a form fitting the world. It does not use the word "beauty," nor does it dwell here; it records this in order to explain how shadows and images come about. What it shows is that a form fitting the world, and the why being visible, belong to no single culture; looking at it for its own sake is another matter.
In the Timaeus Plato says that this cosmos is the most beautiful of things that have come to be, because its maker made it after the best model; he assigns fire, air, water and earth to four of the regular solids.[^37] He voiced one of people's expectations of the world: that the world should be made of beautiful forms. The pressure this expectation puts on this paper is heavy: is beauty the guide to truth? Where this paper parts: the expectation is people's; the answer is the world's. That the elements are regular solids, the world did not answer.
Saving the phenomena, or seeing the why. Ptolemy's circles on circles belong to the tradition of "saving the phenomena": if it can calculate what is seen, a model counts as a success. When Copernicus's book was published in 1543, it opened with an unsigned preface, added by a theologian named Osiander, which said that these hypotheses need not be true, nor even probable; it is enough that the calculations agree with the observations.[^38] This is one of this paper's strongest opponents: physics need only fit, and need not see why, still less be beautiful. Copernicus himself did not think so. In the preface dedicating the book to the Pope he says that the astronomy of his predecessors was like someone taking hands, feet, a head and other members from different places, each very well drawn, but not drawn from a single person; put together, they make a monster rather than a man. He also says that with the sun placed at the center one sees a wonderful symmetry of the universe, and a definite harmony between the size of the orbits and their motions, such as can be found in no other arrangement.[^39] The further a planet is from the sun, the longer it takes to go round: in Ptolemy there was no could-not-be-otherwise relation between the order of the planets and their periods; in Copernicus there was. Yet his system was no more accurate than Ptolemy's, and it too needed many small circles. What Copernicus discerned was exactly the half this paper speaks of: the answer falling on the point of necessity. This paper parts from him only slightly: he thought this harmony already enough to show that the sun is at the center; a fuller answer from the world had to wait for Kepler, Galileo and Newton.
Beauty leads the way; the world decides the case. Kepler is the most moving figure on this road. As a young man he believed that the regular solids were nested between the planetary orbits, an idea he was proud of all his life; in middle age he would not let Tycho's eight minutes pass, and gave up the circle; in 1618 he found his third law: the square of the time a planet takes to go round the sun is proportional to the cube of its mean distance from the sun; the following year he wrote it into a book on the harmony of the cosmos.[^40] The same man, chasing beauty, came away with fantasies and also with laws. He discerned that the world's "no" can have a precise shape, and that what the world finally answers is not necessarily the most perfect curve. The pressure he puts on this paper: when beauty leads the way, how can one tell whether it is leading toward a law or toward a fantasy? This paper's answer is not a rule, but to put each of two things back in its place: chasing beauty was his own cannot-not; what decides the case is the world; and neither pushes the other.
Galileo says that philosophy is written in this great book, the universe, which lies always open before our eyes; the book is written in the language of mathematics, and its letters are triangles, circles and other geometrical figures.[^41] He discerned that form is the language in which the world answers. This paper parts from him on a single point: who wrote the book. This paper says that form is written by people and taken to the world as a question; the world says only yes or no. In the same book he also proposed that comets might be only an optical phenomenon, whose parallax cannot be measured; he says he never asserted this, only raised it as a doubt for consideration.[^42] Comets are not like that. Reading nature in the language of mathematics does not exempt any particular reading from error.
Newton gave the why of the ellipse. In the Principia he laid down a rule for the study of nature: no more causes should be admitted than are both true and sufficient to explain the phenomena, for nature is simple and does not indulge in superfluous causes. He also says that this most elegant system of the sun, planets and comets could only have arisen from the design and dominion of an intelligent and powerful being.[^43] This paper meets him in the first sentence: simplicity is a good guide. It parts from him in two places: simplicity is a guide, not a law of nature; and beauty does not prove a designer either: it is what those who see cannot not acknowledge, and it does not say on the world's behalf what else lies behind it.
In Paris in 1744 Maupertuis said that the path light takes is the one for which the action is least; in Berlin in 1746 he extended this into a general principle: whenever any change takes place in nature, the quantity of action needed for it is as small as possible. He also said that when people know that a few laws, most wisely established, suffice for all motions, the spectacle of the universe becomes greater, more beautiful and more worthy of its Author, and only then can people form a fitting idea of the power and wisdom of the Supreme Being.[^44] The later principle of action speaks of a stationary value, not necessarily a least one.[^45] What he discerned: nature seems to economize. That expectation was his, and his era's; the world answered, but at a place more exact than the one he first named, and it did not vouch for him that a designer stands behind it. This is the best instance of this paper's "the expectation is people's; the answer is the world's."
Putting the statement where the world can say no. At the beginning of the eleventh century Ibn al-Haytham wrote his Book of Optics, testing, one step at a time, how light travels, with dark rooms, small holes and screens, rather than only deducing it from the books of the ancients. In a book questioning Ptolemy he says that whoever studies the writings of earlier authors, if his aim is to find the truth, has to make himself the opponent of everything he reads.[^46] He discerned the experimental side: a statement has to be put where the world can say no. He puts no pressure on this paper; this paper only adds one sentence along his road: the road itself can be beautiful.
At the beginning of the fourteenth century al-Fārisī wrote a commentary on Ibn al-Haytham's Optics, and in it studied the rainbow afresh; he did so to understand how the rainbow comes about. Using a transparent glass sphere filled with water as a model of a raindrop, he saw that sunlight entering a raindrop is refracted twice, with one or more reflections between the two refractions.[^47] He put the statement where the world could say no, using a model anyone could build again. At about the same time Dietrich of Freiberg reached similar results in Europe; there is no known connection between the two.
Shen Kuo records two things in the Mengxi bitan. One says that the images of objects in a concave burning mirror are all inverted, because there is an "obstruction" (ai 礙) in between, which the mathematicians call geshu 格術; when a kite (the bird) flies past a crack in a window, its shadow moves the opposite way; it is like rowing with a scull: where the oar is obstructed at its fulcrum, when the outside goes this way the inside goes that way.[^48] One form runs through the mirror, the crack in the window and the oar. The other says that when the masters of techniques rub the point of a needle with a lodestone, the needle points south; yet it often inclines slightly to the east and does not point entirely due south.[^49] He did not explain this slight inclination; he only recorded it: it has a direction, it has a size, and it appears again and again. This is a gap with a shape, recorded more than three hundred years before similar records in Europe; its why had to wait until people understood that the earth itself is a great magnet. Shen Kuo does not use the word "beauty" either; the two entries are placed in the sections "Correcting Errors" and "Miscellaneous Notes," and he wrote them to make things clear and to record what he had seen.
Motive and representation. Poincaré says that the scientist does not study nature because it is useful; he studies it because he takes delight in it, and he takes delight in it because it is beautiful. A sentence later he says that he does not mean the beauty that strikes the senses, but a more inward beauty, which comes from the harmonious order of the parts.[^50] This is beauty as motive. Where this paper parts: he says that one studies because of beauty; this paper says that research comes from the scientist's own cannot-not; his personal aesthetic can lead the way, rightly or wrongly, and the world decides the case. Duhem goes to the other end. He says that a physical theory is not an explanation; it is a system of mathematical propositions, deduced from a small number of principles, which aims to represent a set of experimental laws as simply, as completely and as exactly as possible.[^51] This is the strongest representational view: theory does not uncover reality; it only represents laws economically. Where this paper parts: he puts the goodness of a theory in the simplicity, completeness and exactness of the representation; the beauty this paper speaks of lies not in simplicity but in the world's having answered the form where the form could not be otherwise. On one point this paper agrees with him: being answered is not being so.
The beauty of a few lines of equations. In the preface to the second volume of his lectures on Maxwell's theory, Boltzmann opens with a line from Goethe's Faust: was it a god who wrote these signs?[^52] Maxwell's equations gave many nineteenth-century physicists this feeling: a few lines had taken hold of electricity, magnetism and light. In a lecture in 1933 Einstein said that experience so far justifies us in believing that nature is the realization of what is mathematically simplest; in the same passage he also says that experience of course remains the sole criterion of whether a mathematical construction is of use to physics, but the truly creative principle resides in mathematics.[^53] He said both things. This paper meets him in the half about the criterion: what decides the case is the world. It parts from him in the half about creation: the drive to create lies not in mathematics but in a person's own cannot-not; mathematics only gives roads that can be walked.
Weyl told Dyson, half in jest, that his work had always tried to unite the true with the beautiful, and that when he had to choose one or the other, he usually chose the beautiful.[^54] Dyson himself later gave an example of this: in 1918 Weyl proposed a theory unifying gravity and electromagnetism, which was beautiful but was shown by Einstein not to fit the world; ten years later, when quantum mechanics came, the same idea was moved to another place and became the foundation of the theory of electromagnetism.[^55] A beautiful form was refused by the world at one place and answered at another. Dirac went further, putting beauty in equations ahead of their fitting experiment, as already mentioned.[^56] Chandrasekhar said that the most shattering experience of his forty-five years of scientific life was the realization that an exact solution of general relativity, found by the New Zealand mathematician Kerr, describes exactly the untold numbers of massive black holes in the universe.[^57] In old age Heisenberg lectured at the Bavarian Academy of Fine Arts in Munich and quoted an ancient definition: beauty is the proper conformity of the parts to one another and to the whole.[^58] That definition has only parts and whole in it, and no world; by this paper's reckoning it speaks of the beauty of form in itself, and only when the world answers at the point of necessity is there the beauty of physics. These few put the heaviest pressure on this paper: can beauty be trusted before the world has answered? This paper's answer: the beauties they trusted came to be remembered as the beauty of physics because the world later answered at the point of necessity; a form the world has not answered stays, in itself, in the beauty of mathematics, or is only a wish. It is just that many are remembered and many are forgotten: Kelvin's vortex rings and Kepler's polyhedra were trusted in their day too.
Today: can beauty be trusted. In 1960 Wigner said that the appropriateness of the language of mathematics for formulating the laws of physics is a miracle, a wonderful gift which we neither understand nor deserve.[^59] Just a little before this, he also said: so their accuracy may not prove their truth, nor prove their consistency.[^60] The same man marveled at the fitting, and would not let the fitting vouch for truth. This paper does not solve the miracle. It says only this: to see that the world has answered at the point of necessity is beauty; to understand why mathematics fits the world is another matter. That beauty is not understanding is clearest here: one can see the first without understanding the second.
McAllister says that scientists' aesthetic preferences are induced, time after time, from theories that succeeded in the past: circles succeeded, so people loved circles; and a scientific revolution often also breaks a set of aesthetic canons, as Kepler broke the circle.[^61] When quantum mechanics arrived, Schrödinger's complaint that it lacked anything one could picture was also an old set of canons speaking. This presses exactly on one point of this paper: a person's cannot-not has a history. This paper accepts this. A physicist's taste is trained, and has a history; having a history is not the same as being colonized, and may also be cultivation: teachers and past successes help him grow his own eyes, and he can still change: when the world says eight minutes, he gives up the circle. When he can no longer change, and uses that set of aesthetic canons to block the world's "no," only that is being colonized: a set of past successes has taken the place of his own law.
Hossenfelder is today's strongest opponent: beauty has led physics astray. On her most important point this paper agrees with her: beauty cannot decide the case. Where it parts: the beauty of physics this paper speaks of requires the world's answer from the start; a form that has never been put before the world and is trusted only because it is pretty does not count as the beauty of physics on this ray. The kind of beauty she criticizes, this paper does not speak for.
8. The Question Left Open
Most of the fundamental equations of physics do not distinguish before from after. Run a stretch of planetary motion backward, and it still obeys the law of gravity; run a collision of two particles backward, and it mostly still obeys the laws. But the world distinguishes before from after: a smashed cup does not put itself back together, heat flows from hot places to cold ones, the moss on a stone grows thicker year by year and does not grow thinner year by year. The direction the equations do not give is everywhere in the world.
Physics does not push anyone to ask about this direction. A person can spend a whole life looking only at how form fits the world; if he cannot not ask why the world has a before and an after, that is another ray. This ray stops here.
The question left open is: where the world's direction comes from, and why the traces it leaves are beautiful. That is the beauty of causation.
9. What Would Make This Paper Wrong
If any one of three things held, this paper would be wrong. First, "answered at the point of necessity" contributes nothing to beauty: a patched-together model and Maxwell's equations, so long as they fit the world equally accurately, are equally beautiful to whoever understands them; then the beauty of physics would be only another name for accuracy. Second, once the beauty of equations within mathematics, fame, usefulness and pretty images are taken away, nothing is left of what physicists call beauty; then the world's answer would add nothing to beauty, and the beauty of physics would be only borrowed beauty of mathematics. Third, once the later answer, the fame and the mystery are taken away from a gap like Mercury's, its shape goes with them; then the beauty of the remainder would be only a romance of the unknown. This paper does not think any of these will happen, but it writes them here, because a paper about "the world can say no" has first to make clear where the world can say no to it.
Acknowledgments
This paper is the fourth ray paper in the aesthetics branch of the SAE (Self-as-an-End) philosophical system and is published under the sole name of Han Qin. The author made the final decision on all philosophical arguments and choices, and alone is responsible for the final text.
Both drafting assistance and review were carried out with large language models. Drafting and internal precision audit were done by Claude (Anthropic); the structural gate and pre-publication sign-off were handled by ChatGPT (OpenAI). The material on the history of ideas used in Section 7, "Where the Question Comes From," was first gathered by ChatGPT's deep research, and its sources were then checked item by item. This paper, together with the second edition of the general outline and the other twelve ray papers, passed final review by both the internal precision audit and the structural gate, and passed a key-sentence acceptance check. The author thanks these systems and the research and engineering teams behind them. The author ruled on their comments one by one, and both acceptances and rejections are on record; the record of revisions and withdrawals across rounds is not reproduced here.
Notes
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[^56]: Dirac, "The Evolution of the Physicist's Picture of Nature."
[^57]: S. Chandrasekhar, Truth and Beauty: Aesthetics and Motivations in Science (Chicago: University of Chicago Press, 1987), 54.
[^58]: Werner Heisenberg, "Die Bedeutung des Schönen in der exakten Naturwissenschaft," Physikalische Blätter 27, no. 3 (1971): 97–107; Werner Heisenberg, "The Meaning of Beauty in the Exact Sciences," in Across the Frontiers, trans. Peter Heath (New York: Harper & Row, 1974), 166–183.
[^59]: Eugene P. Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications on Pure and Applied Mathematics 13, no. 1 (1960): 1–14, at 14.
[^60]: Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," final section, "The Uniqueness of the Theories of Physics."
[^61]: James W. McAllister, Beauty and Revolution in Science (Ithaca, NY: Cornell University Press, 1996).
References
Works of the SAE system cited here are archived on Zenodo, with the Chinese and English versions in the same record. The DOIs cited are concept DOIs, which always resolve to the latest version, so version numbers and access dates are not given. All are single-authored (Han Qin). Drafting and review assistance is described in the Acknowledgments.
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