The Beauty of Mathematics
数学之美
Two children add the numbers from one to a hundred, and both get 5,050. One adds them one by one; the other writes one to a hundred forwards in one row and backwards in another, and sees it by reading the pairs top to bottom. Both answers are right, but only the second lets one see why it could only be this number. A subject who has seen this cannot not acknowledge: this form, by its own rules alone, could not be otherwise, and it sets its why before our eyes. That sentence is the beauty of mathematics. It has three faces: completive beauty is seeing the why; generative beauty is the why being seen; the beauty of the remainder is seeing even what is missing. The paper says what mathematics is and is not (not quantity, not calculation, not notation, not any one logic, and need not fit the world); how Gödel, with mathematics' own rigor, drew the edge of what a class of formal systems cannot reach; what can pass for the beauty of mathematics (being correct, being short, being neat, being useful, difficulty and fame, a pretty picture, fitting the world); and what is left to each person. It then traces where the question comes from. "Beauty" in mathematics first meant the order of form, then the why that a proof sets out; later still, beauty entered discovery, and a problem came to have a shape before it had an answer; and today people ask whether the beauty of a proof can be measured. Along the way the paper speaks once each with Plato, Aristotle, Liu Hui, Zu Geng, Schopenhauer, Xu Guangqi, Hutcheson, Kant, Hilbert, Russell, Poincaré, Hardy, Wittgenstein, Rota and present-day research. It ends by handing on to the beauty of physics the question of how form comes to fit the world.
Keywords: beauty of mathematics, form, necessity, completive beauty, generative beauty, beauty of the remainder, Gödel, power of judgment, overlap, 2DD
1. Two 5,050s
A teacher told the class to add up the numbers from one to a hundred. Most of the children bent over their work and added them one at a time; halfway through, someone made a mistake and started again. One child handed in his answer quickly: 5,050. The teacher asked him how he had done it. He said: write one to a hundred forwards in one row, and backwards in another; pair each number with the one below it, one and a hundred, two and ninety-nine, three and ninety-eight, and every pair makes a hundred and one; there are a hundred pairs, which make 10,100; every number has been counted twice, so divide by two: 5,050.
The children who had added it all up also got 5,050. The two answers were the same, and both were right.
But the two 5,050s are not the same. The first was calculated: you believe it because no step went wrong. The second was seen: you do not only believe it, you see why it could only be this number. Change it to one to a thousand, one to ten thousand, one to any number you like, and the first child has to bend over his work again; the second does not. What he saw was not a number but a shape.
This is an old story. Among those who understand it, some feel a light go on, and some only nod; whether the light goes on is each person's own affair. But once they have understood, they cannot go back to believing without seeing. What this paper is about is that seeing.
This paper follows the reading set out in the general outline.[^1] 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.[^2] 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 mathematics 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 Pythagorean theorem can also be looked at from the direction of physics, or from the direction of perception. This paper asks only what the power of judgment sees in the one direction of pure form.
2. What Mathematics Is and Is Not
A distinction falls; this side is marked and does not immediately disperse. But the next time one points to the same thing, by what right does it still count as the same? When several distinctions are put together, how do they avoid canceling one another? Mathematics begins here: signs repeat according to rules and still count as the same when repeated; many distinctions are put together and none cancels another. Form has a law of its own.
At its minimum this law is a single sentence: once something has counted according to the rules, it cannot go on counting and be quietly canceled at the same time. The rules can be changed, but they must be changed openly, not swapped in secret. All of mathematics' rigor hangs on this one sentence. You may start from other axioms, and you may adopt another set of rules of inference; but you cannot, within one proof, go on using a result while treating it as if it did not exist.
This sentence is weaker than the rules in a logic textbook. One school of mathematicians does not accept the rule that every statement is either true or false, which logic calls the law of excluded middle; their mathematics is just as rigorous, and has its beauty all the same. So mathematics is not any one logic. It is form that keeps to that one sentence, whichever logic it is kept by.
Form has come away from matter. A triangle drawn in the sand, a triangle drawn on paper, a triangle thought of in the mind: the proof does not care. The pairing that adds one to a hundred does not care whether the hundred are apples or years. This is what "pure" means: not that it is high, but that it does not depend on matter.
So mathematics is not quantity. A topologist sees a coffee cup and a doughnut as the same thing, because each has only one hole; there is no size there, only how the shape hangs together. Mathematics is not calculation. The first child was calculating: he used form without seeing it. Mathematics is not notation. The same proof, written in the geometrical language of the Greeks or in today's algebra, is the same proof; notation is clothing. Nor need mathematics fit the world. Whether a form fits the world is not its own affair; it is the affair of the next ray.
3. The Beauty of Mathematics
Once a subject has seen why a form stands, he cannot not acknowledge:
This form, by its own rules alone, could not be otherwise, and it sets its why before our eyes.
That sentence is the beauty of mathematics. The first half belongs to everything correct in mathematics: once the rules are fixed, the conclusion can only be this; that is called necessity. Only the second half is beauty: the necessity is not only guaranteed but seen. The first child's 5,050 could not be otherwise either, but the necessity was hidden in a hundred additions, and he got only the conclusion; in the second child's 5,050 the necessity lies open on that two-row table of numbers, and whoever understands it can see it. That the why is set before our eyes is the affair of the form; whether one understands it is up to the power of judgment. And understanding is not yet seeing: what one has understood one can merely put to use; but when one looks at the thing itself, its why set before one's eyes, whoever looks cannot not see.
There are two kinds of "could not be otherwise" here, and they have to be kept apart. One is in the form: once the rules are fixed, the conclusion cannot be anything else, and everyone who accepts these rules has to accept 5,050, whether or not he has seen why. This is necessity; it is the affair of the object. The other is in the one who sees: once he has seen the why, he cannot not acknowledge that it is beautiful. This is the cannot-not; it is the affair of the subject. Being forced to accept a conclusion is not seeing beauty; the child who added it all up was forced by arithmetic to accept 5,050, and he saw nothing. In the order of asking, mathematics is the first place where necessity appears in form; but beauty is still on the side of the one who sees.
Why is it overlap and not preference? Because this seeing depends on no particular body, culture or canon. In the Three Kingdoms period Zhao Shuang proved the Pythagorean theorem with a single figure, the hypotenuse diagram: four identical right triangles are fitted together to form a large square whose side is the hypotenuse, leaving at its center a small square whose side is the difference between the two legs; the area of the large square equals the four triangles plus the small square in the middle, and when one works it out, it comes to exactly the square of one leg plus the square of the other.[^3] The figure does not require its reader to know Chinese, or to know when the Three Kingdoms were; someone who has learned only Euclid and understands it sees the same why that those who understood it more than seventeen hundred years ago saw. What he cannot not acknowledge is not Zhao Shuang's taste but the necessity set out in that figure. One study asked British mathematicians, Chinese mathematicians and British undergraduate mathematics students to compare a set of proofs in pairs and say which was more beautiful; agreement within the groups and between them was quite high, and neither culture, nor level of training, nor the kind of mathematical object pulled their judgments far apart.[^4] This does not prove the overlap, which is not a statistic; at the least it shows that the beauty of mathematics is not each person talking past the others.
When a mathematician finishes reading a proof and says "elegant," the word is not borrowed and it is not a metaphor. He is making a judgment of beauty: he is claiming that this time he has landed on the overlap. He may be right, or wrong, and it can be discussed; what is discussed is whether the proof has set out its why, not whether he ought to be pleased.
The beauty of mathematics has three faces.
Completive beauty: seeing the why. A proof, a theorem, stands, with nothing more and nothing less. Euclid proves that there are more primes than any given number of them: give me a list of primes, however long; multiply all the numbers on the list together and add one. The new number is greater than one, so it has a prime factor; but divide it by any prime on the list and one is always left over, so its prime factors are not on the list.[^5] Who gave the list and how long it is do not matter; it will always be outrun by the number it produces itself. The proof uses nothing but division and what is left over, and borrows nothing else. Whoever has read it sees not only "the primes are infinite" but why any finite list will be outrun by itself. Another proof, just as old, says that the square root of two is not a ratio of two whole numbers: if it were, reduce the fraction to lowest terms and square both sides, and it follows that numerator and denominator are both even, contradicting "lowest terms." Both proofs are short and both are unexpected; whoever understands them sees that it could only be so, and sees why.
"Nothing more" has to be read with care. It does not mean the shorter the better; it means that the form stands by its own rules, not by something propped up for the occasion. If, to make a conclusion hold, one adds an axiom that serves only that conclusion, the conclusion does stand, but not on its own feet; this is what alien control looks like in mathematics. Rigor is not having more: the more items there are, the more likely it is that one of them is a prop; rigor is every item holding. The more tightly the rules are kept, the more clearly the necessity can be seen.
Generative beauty: the why being seen. This is the face of the chisel: the problem resists, and then the form comes out. This is the moment when the second child wrote out the backward row. It is often unexpected at first: what could adding one to a hundred have to do with writing the numbers backwards? Then, once seen, it could not be otherwise. The unexpected followed directly by the could-not-be-otherwise is the commonest shape of generative beauty in mathematics.
The moment can come very suddenly. Poincaré says that once, just as he was about to board an omnibus, the moment his foot touched the step, an idea came to him, with nothing in his earlier thoughts preparing the way for it: the transformations used in the class of functions he had long been studying were the same as those of non-Euclidean geometry. He did not stop to verify it, took his seat and went on with the conversation, yet he was already entirely certain; only when he got home did he sit down and verify it, and it was right.[^6]
Generative beauty is also found in some very small judgments. Mathematics today does not count 1 as a prime. One of the reasons is not that 1 does not look like a prime, but this: if it were counted, the way each number breaks down into primes would no longer be unique; six could be two times three, or one times two times three, or one times one times two times three. Cut 1 out, and the form "every number has only one factorization" stands. Where the knife falls so that the form stands is the work of the power of judgment in mathematics. Rules can check whether a proof is correct, but they do not decide for anyone where the knife falls; this holds even in the most rigorous place.
One thing makes the generative beauty of mathematics different from that of other rays: a proof can be walked through again by everyone who understands it. Poincaré's moment on the step happened only once, but once his result was written down, everyone who understands it can do that seeing over again. A proof is a discovery that others can make again. This is one reason why the overlap is especially wide in mathematics.
The beauty of the remainder: seeing even what is missing. Some things in mathematics do not yet stand, yet the gap itself has a clear shape.
The further out one goes, the sparser the primes become: there are twenty-five below a hundred, and only a little over seventy-eight thousand below a million. The average rate at which they thin out was proved at the end of the nineteenth century. But the actual primes do not line up according to the average; they sway above and below the average law. How far do they sway? In 1859 Riemann linked this swaying to the zeros of a function, and conjectured that, apart from a set of zeros whose positions had long been known, all the rest lie on one straight line. It was later proved that the conjecture is equivalent to saying that the swaying of the primes is held within a very tight bound, of about the size of a square root.[^7] More than a century and a half has passed, and no one has proved it or refuted it. Computers have checked their way along, and none of the zeros checked has left that line; many theorems begin: assume that the Riemann hypothesis holds.
This is the beauty of the remainder. It is not "we do not know yet"; it is knowing very clearly what is missing. What is missing is exactly that line; once it is filled in, how far the primes sway is settled too. The shape is drawn by relations that have already been proved, so whether or not the conjecture turns out to be true, that shape is there; if it is refuted, the moment of refuting it would itself be a moment of generation.
To tell it apart, again take things away and see. Take away the prize money hanging over it, take away its uses elsewhere, take away the mystery that the unknown brings, take away the fact that it has been hard for over a century, and do not guess when it will be solved: the shape is still there, and so the beauty is still there. By contrast, the decimals of π have been computed very far out, and no one yet knows what some digit further along will be. That too is a gap, and what is missing is also quite clear: a digit between zero and nine. But what it lacks is only calculation; once calculated, it is there, and there is no why in it. What the Riemann hypothesis lacks is a why: relations already proved have fenced it into a shape, and no number of zeros checked can prove it. An exercise at the end of a chapter that I have not yet solved lacks my effort, not something in mathematics. The beauty of the remainder needs a place that lacks a why and has been drawn by what already stands.
Nor is the beauty of the remainder an advance on a future answer. That line is there today; its beauty is present, and needs no later proof to countersign it. Some people look at this gap and cannot walk away, and spend their lives there; that is their own cannot-not at work. Many more take one look and go. The gap shows the place where one can go on and the direction: what has to be filled in is the why on that line, which no number of zeros checked can fill in. It pushes no one.
4. Gödel: An Edge Drawn by Mathematics' Own Rigor
At the beginning of the twentieth century some mathematicians hoped to find a single set of rules for the whole of mathematics: rules that could be listed one by one, from which every true mathematical statement could be proved, and which could also prove that they did not contradict themselves. In 1931 Gödel proved two things.[^8] First, a formal system whose rules can be listed one by one and which is strong enough to do basic arithmetic, provided everything it proves is true, always has a true sentence that it can neither prove nor refute (the condition he actually used is somewhat weaker than "everything it proves is true"). Second, such a system, provided it does not contradict itself, cannot prove that it does not contradict itself.
How did he prove it? He gave every symbol, every formula and every proof a number, so that "this string is a proof of that sentence" became a statement about numbers, and arithmetic could then talk about the sentences and proofs of arithmetic. The liar says "this sentence of mine is false," which is a paradox; Gödel himself said that his argument was a close relative of the liar. The sentence he constructed says, roughly, "I cannot be proved in this system": replace "false" with "cannot be proved," and the paradox is no longer a paradox but a theorem.
The three faces of the beauty of mathematics meet here in one place. The proof itself shows the completive face: conditions, numbering and inferences are all in the open, every step can be checked, and whoever understands it sees the why. What it speaks of is a remainder, and a remainder with a precise shape: which sentence it is, and why it is out of reach, are both stated with complete clarity. It is also a moment of generation: an old paradox chiseled into a theorem. Most important of all, this edge was not drawn by an outsider. It was not someone saying from outside that "mathematics has limits"; mathematicians, using mathematics' own most rigorous methods, drew the edge of a class of formal systems. The rules, kept to the utmost, marked out the place the rules cannot reach.
Seen from the direction of philosophy, this separation of "true" from "provable" opened a great question, and that is the beauty of philosophy. Seen from the direction of mathematics, the beauty lies in how this separation was proved: by arithmetic's own means, step by step, in the open.
What it can show is this: in such systems, "true" and "provable" have come apart; such systems cannot vouch for themselves. What it cannot show is far more. It speaks of formal systems whose rules can be listed and which are strong enough to do arithmetic, not of all knowledge. It has not made the theorems already proved doubtful. It does not prove that people are cleverer than machines. Nor does it push mathematics forward: you can add that sentence to the rules and get a new system; the new system has a new such sentence; this can go on without end, but nothing forces anyone to keep going.
5. What Can Pass for It
The method is still the same: take it away and see.
Being correct. The first child's 5,050 is correct. In 1976 the four-color theorem was also proved correct: for any map drawn on the plane, so long as each region is all in one piece, four colors are enough to give any two regions with a common border different colors. But that proof needed a computer to check more than a thousand configurations one by one;[^9] later someone gave a new proof, done differently, which still could not do without a computer. It lets people believe that four colors are enough; it does not let them see why four are enough. More than thirty years before, Hardy had already said that listing cases one by one is one of the duller forms of mathematical argument.[^10] This does not mean that the computer proof is second-rate, nor that it is any less true; it is true, and correct is correct. It is only that being correct is not being beautiful. Rules can check whether a proof is correct, and so can machines; what they check is correctness, not beauty. Whether a machine can see beauty is another question, and not on this ray.
Being short. A one-line proof, if that line is "this follows at once from such-and-such great theorem," is short on paper and long elsewhere; the necessity is borrowed, and nothing can be seen here. Take away the borrowed great theorem, and nothing of this line remains. Shortness is not beauty; having nothing more than is needed is.
Being neat. Two to the first power, the second, the fourth, the eighth, the sixteenth, each plus one: 3, 5, 17, 257, 65537, all prime. It is neat enough to make one want to believe that going on this way will always give a prime; Fermat believed just that. Next comes two to the thirty-second power plus one, 4294967297, which is 641 times 6700417. Neatness is a pattern, not a why; to see a pattern is not to see that it could not be otherwise.
Being useful. Hardy wrote that no one had yet found any warlike purpose to be served by the theory of numbers, and that it seemed unlikely anyone would for many years.[^11] A few decades later, primes became the basis of encryption on the internet. The beauty of the primes did not grow by a hair on that account, nor shrink.
Difficulty and fame. A problem that has been hard for a hundred years, carries a prize and is known to everyone draws attention to itself; that does not make it beautiful. The beauty of the Riemann hypothesis lies in its shape, not in its fame; this was already tested above by taking these things away.
A pretty picture. Pictures of the Mandelbrot set are pretty: one simple rule applied over and over draws an infinitely complicated edge. The colors in the picture were chosen by whoever drew it, and are not in the mathematics; that the picture is pretty is a matter for the ray of perception. If there is beauty of mathematics here, it has to be sought in the proofs: what has been proved about that simple rule and the edge it generates, and why. The picture does not set out these whys.
Fitting the world. In his first book Kepler believed that between the orbits of the six planets then known lay exactly the five regular solids, nested one inside another. As geometry the construction is elegant; as astronomy it is wrong. The beauty of mathematics does not vouch for the world; whether a form fits the world is the affair of the next ray. Conversely, a form that fits the world is not thereby more beautiful in mathematics.
Neighboring rays. The beauty of philosophy looks at how a distinction gives a question its shape; the beauty of mathematics looks at how form stands by its own rules. When a question takes shape, its answer need not yet have any necessity; when a proof is finished, the necessity is there. The beauty of cognition looks at how what is absent still does work, and how what has not yet come is already awaited; a proof is not a prediction, and does not wait for results to test it. Many theorems begin with "assume the Riemann hypothesis holds," which looks like something not yet come already doing work; but that is a conditional, not a prediction: it says "if so, then so," and does not wait for the day the Riemann hypothesis arrives.
One last sentence, which this aesthetics says of every ray: beauty is not correctness, not usefulness, not importance, not understanding. The four-color theorem is correct, the primes are useful, the Riemann hypothesis is important, and all three have been tested above by taking them away; understanding is left for the section on where the question comes from, where it is taken up with Rota.
6. What Is Left to Each Person
Which proof moves you is each person's own. The same equation lights one person up and leaves another cold. Feeling nothing is not a mistake. If he says "it is not beautiful, it only writes the definition another way," that is a judgment of beauty, and it can be brought out and discussed: has this equation set out a why or not? For adding one to a hundred, some prefer the pairing, because it takes in the whole at a glance; some prefer induction, proving step by step that if it holds for n it still holds for n + 1, because what they see is another why: how a relation is kept, one step at a time. Both see a why, only not the same why; which one is liked is each person's own. Some prefer pictures, a single figure that says it all, as Zhao Shuang's hypotenuse diagram does; some prefer symbols, worked down line by line. Some love most the moment of discovery; some love most a theorem long finished, standing there quietly. Some people never feel anything for mathematics in their whole lives, and there is nothing wrong with that either.
What does not overlap is larger, in mathematics as elsewhere. The study mentioned above asked people to compare proofs in pairs, and different people agreed to a good degree on which proof was more beautiful; another study asked a group of mathematicians to rate one and the same proof as a whole, and they differed widely on whether it was beautiful, whether it was cumbersome, whether it was rigorous, whether it was useful.[^12] The two studies do not contradict each other. When people are asked to compare, they often meet at the same place; when they are asked to rate a proof as a whole, each person's own things all come in. The overlap is large; what does not overlap is larger. Mathematics is a ray on which the overlap is especially wide, because everyone who understands a proof can walk through it again; but meeting on "it could not be otherwise, and one can see why" is not meeting on "it lights me up."
Whether to spend a life looking at a gap like the Riemann hypothesis 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
That mathematics and beauty go together is a very old idea; but what "beauty" means here has been rewritten several times. First it was the order of form; then the why that a proof sets out; later still beauty entered discovery, and a problem could have a shape before it had an answer; and today people have begun to ask whether the beauty of a proof is inseparable from the people who do mathematics, and whether it can be measured. What follows runs roughly in the order of these rewritings, with two further strands in between: coherence and immovability, and necessity and practice.
The order of form. Plato has Socrates say that straight lines and circles, and the planes and solids made from them with compass, ruler and square, are beautiful not relative to anything, but by their nature and always beautiful in themselves, and that the pleasures they bring are pure, unmixed with pain.[^13] This paper agrees with one thing in this: the ground of form waits for no one; the circle is that circle before anyone sees it. Where this paper parts company with him is "beautiful in themselves." The construct waits for no one; aesthetics waits for people; where there is no one who has seen, there is no one who cannot not acknowledge anything. As for that pure pleasure, it is each person's own.
Aristotle says that those who hold that mathematics has nothing to do with beauty are wrong; the chief forms of beauty are order, symmetry and definiteness, and mathematics shows these more than anything.[^14] He presses on this paper: if beauty is just these three things, the beauty of mathematics is just these three things too, and there is no need to speak of the why being seen. This paper takes the three: they are marks of completive beauty. Order is everything in its place, symmetry is the two rows in the pairing, definiteness is nothing more and nothing less. But with the three laid out, someone still has to see why they are so; he says that mathematics shows these things, and this paper adds: shows them to those who can see. He says nothing of the generative face, or of the face of the remainder.
Russell says in "The Study of Mathematics" that mathematics, rightly viewed, possesses not only truth but supreme beauty, a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.[^15] What he discerned: the beauty of mathematics does not depend on sensuous ornament. He presses on this paper: if the beauty of mathematics is supreme, other beauties rank lower. This paper parts from him on "supreme": that is a ranking, and this series does not say there is a most beautiful. Beauty on each dimension is its own, and the beauty of one dimension is not used to press down another.
Fact and why. In the Posterior Analytics Aristotle also says that knowing the fact and knowing the why are two different things. He takes the planets as his example: to infer from the planets not twinkling that they are near is to know the fact; to infer from their being near that they do not twinkle is to know the why.[^16] This is a very early version of this paper's two 5,050s. He presses on this paper: he is speaking of knowledge, not of beauty. This paper's answer lies in seeing, and is taken up below in the paragraph on Rota.
Liu Hui, of the state of Wei in the Three Kingdoms period, wrote a commentary on the Nine Chapters on the Mathematical Art. The Nine Chapters is a book about use: measuring fields, dividing grain, digging canals and building walls. In his preface Liu Hui says that things of the same kind can be derived from one another, "though the branches divide, they share one trunk," and that it is enough to understand how they issue from one end. He also says that his commentary "analyzes principles in words and dissects bodies with diagrams," seeking to be "concise yet comprehensive, coherent yet not tedious," so that "those who read it will have grasped more than half."[^17] Concise yet comprehensive, branches sharing one trunk: this meets, at one point, what this paper calls nothing more and nothing less, and things far apart turning out to be one form. In commenting on the area of the circle he also says that when a polygon inscribed in the circle is cut finer and finer, "the finer the cutting, the less is lost; cut and cut again until it can be cut no more, and it merges with the circumference, and nothing is lost." This is the generative face: approaching step by step until the form closes. He wrote to comment on a book about use, yet he discerned something that goes beyond use.
Zu Geng (called Zu Gengzhi in the commentary of Li Chunfeng and others), seeking the volume of a sphere, left one sentence: "When layers are stacked to make a volume, if the areas of the sections are the same at every height, the volumes cannot differ." Stack small blocks layer upon layer; since the area of the cross-section is the same at every height, the volumes cannot be different. The sentence is found in the commentary by Li Chunfeng and others of the Tang on the method for the sphere in the Nine Chapters, where they quote Zu Gengzhi.[^18] What he discerned: once the prior relations are fixed, the result cannot be otherwise. This is how mathematics says "cannot differ"; he said it to ground an algorithm, not as a judgment of beauty. This paper meets him in "could not be otherwise."
Schopenhauer was dissatisfied with Euclid. He says that what Euclid proves, one is forced by the principle of contradiction to admit is so, without knowing why it is so; one finishes almost with a feeling of discomfort, as after a conjuring trick. He also says that in proofs like that of the Pythagorean theorem lines are drawn without our knowing why, and it turns out later that they were snares that close unexpectedly. He says that Euclid's proof of the Pythagorean theorem walks on stilts, is even underhanded, and abandons us just when we reach the why. He singles out proof by contradiction too: one door after another is shut, until only one is left, and one has no choice but to go through it.[^19] What he discerned: to be forced to admit a conclusion is not to see. This is exactly what Section 3 said: being forced to accept a conclusion is not seeing beauty. This paper parts from him over where the why comes from. The why he wants is intuitive, a figure taken in at a glance; the why of this paper is set out by the form, and can be a figure or a step of reasoning. In Section 3, Euclid's proof that the primes outrun any list rests on a step of reasoning, and whoever understands it sees why. The proof about the square root of two is a proof by contradiction; what whoever understands it sees is not only that the doors have all been shut, but also why they are shut: in a fraction in lowest terms numerator and denominator cannot both be even, yet squaring both sides and working it through shows that they both are. Some proofs by contradiction only shut the doors one by one, and some also set out the why; which kind a proof is has to be looked at one proof at a time.
Coherence and immovability. Xu Guangqi and Matteo Ricci together translated the first six books of Euclid's Elements; the translation was published in 1607. In his "Miscellaneous Remarks on the Elements" Xu Guangqi says that the book "seems most obscure, yet is most clear," "seems most complicated, yet is most simple," "seems most difficult, yet is most easy"; and he says of it that "one cannot leave anything out of it, cannot refute anything in it, cannot cut anything from it, cannot rearrange its order."[^20] The second of these is very nearly what this paper calls nothing more, nothing less, and could not be otherwise. He has both. When he says "in sum, its wonder lies in clarity alone," he is praising the form's own clarity; he also says that it can "with its clarity make clear what is most obscure in other things," and that is for use, taking it to light up other learning. Both are reported as they are: in looking at it for its own sake, this paper walks with him; in looking at it for use, he takes another road, which is no fault.
Hutcheson's Inquiry into the Original of Our Ideas of Beauty and Virtue has a section devoted to the beauty of theorems. He says that the beauty of theorems is very different from other beauties, yet in no other kind of beauty is uniformity amidst variety seen so astonishingly: in one theorem, with the most exact agreement, infinitely many particular truths are contained; the pleasure that comes of this is great, and has nothing to do with any further advantage.[^21] What he discerned: a theorem gathers countless particular truths in one place. He presses on this paper: if beauty is the pleasure of an internal sense, the beauty of theorems is on the side of the person, not of the form. This paper parts from him over the ground: he relies on an internal sense, and beauty is that sense's pleasure; the ground of this paper lies in the why that the form sets out.
In the Critique of the Power of Judgment Kant admires the circle: so simple a figure, and yet it contains the ground for solving a whole series of problems. He grants that people are in the habit of calling such properties of geometrical figures and of numbers beautiful, but says that this is not an aesthetic judgment but an intellectual one; it should be called relative perfection, not beauty, and the expression "intellectual beauty" cannot be allowed at all; what one feels toward it is a justified admiration. But he then goes on to say that it is rather the demonstration of such properties that may be called beautiful, because in the demonstration the understanding and the imagination both feel themselves strengthened.[^22] So he will not call the completive face beauty, but leaves the generative face to beauty. This paper meets him on the generative face, and parts from him in two places. One is the ground: he puts the beauty of a demonstration in the feeling that one's capacities are strengthened, which is something happening in the one who looks; this paper puts it in the why that the demonstration sets out. The other is the completive face: what he describes, a simple form holding the ground of a whole series of problems, is exactly what is here called completive beauty. His beauty is tied to the free play of imagination and understanding without concepts, which is beauty at the level of perception; beauty on each dimension is its own, and the beauty of one dimension cannot be used to measure another. When a mathematician says "elegant," it is not a metaphor; he is speaking on this one dimension. One further question can be pressed on Kant's behalf: if brevity, clarity, completeness and unity can all be accounted for by understanding, where does the aesthetic begin? This paper's answer is again in seeing: those who have understood do not necessarily look, and those who look cannot not see; this is taken up again in the paragraph on Rota.
Beauty enters discovery, and problems take shape. In 1900 Hilbert spoke on mathematical problems at the International Congress of Mathematicians in Paris. He said that as long as a branch of science offers an abundance of problems, it is alive; a lack of problems foreshadows its extinction or the end of its independent development. A mathematical problem should be difficult, in order to attract us, yet not completely out of reach, lest it mock our efforts.[^23] He listed twenty-three problems, and at the congress presented only ten of them aloud. What he discerned: a problem, before it has an answer, already has weight of its own: difficult, yet not out of reach. This paper's beauty of the remainder takes one step further from here: what is missing has already been drawn by what stands. He spoke of good problems and living problems, not of beauty; to report it as it is, this paper meets him here: a problem, before it has an answer, already has weight. It parts from him over "attract": whether a problem holds a person depends on that person's own cannot-not; what a problem gives is a place and a direction.
Poincaré's moment on the step of the omnibus has already been told.[^24] He also says that mathematical discovery is not trying out every combination; rather, an aesthetic sense acts like a fine sieve, sifting out the harmonious, symmetrical combinations that lead on to more, and handing them to consciousness to be checked. This paper agrees that the sieve is real: it is in the person doing mathematics, his personal aesthetic, his cannot-not, at work. Where this paper parts company with him is that what the sieve lets through is only what is worth checking, not what is already right. He himself says in the same essay that the feeling of complete certainty that inspiration brings sometimes deceives, and one finds this out only on sitting down to write the proof. Beauty can lead the way; it does not vouch for the proof.
Hardy says that beauty is the first test, and that there is no permanent place in the world for ugly mathematics. The examples he gives are exactly Euclid's proof about the primes and the proof about the square root of two; in such proofs, he says, there is a very high degree of unexpectedness, combined with inevitability and economy.[^25] Of his three, unexpectedness belongs to the generative face, and inevitability and economy to the completive face; this paper takes all three. It parts from him in three places. The first is "no permanent place for ugly mathematics": the proof of the four-color theorem may not be beautiful, yet its place is permanent; what is correct does not need beauty to keep its place. The second is that he took true mathematics to be useless, and was proud of it; what came afterward showed that being useful or not neither adds to nor takes from its beauty, and uselessness is no part of beauty. The third is that he believed mathematical reality lies outside us, that the mathematician's task is to discover it, and that the theorems we prove are only notes of our observations.[^26] This presses on this paper: the structure waits for no one; does its beauty wait for anyone? This paper's answer points back to the paragraph on Plato: the construct waits for no one; aesthetics waits for people, and so does beauty; what lies in the structure is the ground, while beauty is what those who have seen cannot not acknowledge together. He says the mathematician discovers, and this paper meets him here too: seeing does not wait for anyone's nod.
Necessity and practice. Wittgenstein says: "A mathematical proof must be surveyable." What we call a proof is a structure, and reproducing it is a task that can easily be carried out.[^27] What he discerned: a proof must be something a person can look over, and can do again. This goes the same way as what Section 3 said: "a proof is a discovery that others can make again." He presses on this paper: is the necessity of mathematics inseparable from the practice in which people prove things together and use rules? This paper's answer is in Section 2: the rules can be changed, but they must be changed openly, not swapped in secret; while they are not changed, what follows is not up to anyone.
Today. Steiner separated two kinds of proof: both correct, one explaining why, the other only proving that it is so.[^28] This is where today's topic of "explanatory proof" starts. It presses on this paper most directly: if a proof already explains why, what is "beauty" for? This paper's answer is again in seeing: to have explained why is a matter of knowledge; once one has understood, one can put it to use and need not look again. The why set before one's eyes, which a person who looks cannot not see, is a matter of beauty.
Rota says that when mathematicians call a theorem beautiful, what they really mean is that it is enlightening; "mathematical beauty" is an expression mathematicians invented to admit obliquely the fact of enlightenment without having to admit how fuzzy it is.[^29] This is the view closest to this paper: what this paper calls "seeing the why" is almost what he calls being enlightened. This paper parts from him in two places. One: enlightenment is something that happens in a person, deeper or shallower; the overlap of which this paper speaks falls on what is seen in the enlightenment: the form setting out its own why. Understanding and seeing are not the same, either. A student before an exam masters a proof thoroughly, only in order to use it; years later he comes across it again, learns nothing new from it, yet for the first time stops to look at it and sees that it is beautiful. His understanding has not changed; what has changed is that this time he is looking. Two: the beauty of the remainder is not enlightenment. Looking at the Riemann hypothesis, no one knows whether it is true, let alone understands why it is true; what is seen is only what is missing. So beauty is not understanding: to see completive beauty one first has to understand, but having understood is not the same as having seen, while the beauty of the remainder does not require understanding first.
Present-day research comes at the question from several directions. McAllister describes how mathematical beauty and the standards of proof have changed together over history.[^30] Thomas says that there is more to aesthetics in mathematics than beauty alone.[^31] Zeki and his colleagues asked fifteen mathematicians to look at formulas and rate them, and the ratings of beauty correlated with activity in one region of the medial orbitofrontal cortex.[^32] Earlier, Birkhoff proposed a measure of the aesthetic: order divided by complexity.[^33] The comparison study of Section 3 and the whole-proof rating study of Section 6 also belong to this line. This paper takes in what they can show: personal aesthetic responses have a history, change over history, can be measured, and can have a stable distribution within a group; having a history is not the same as being colonized, and may also be cultivation. What this paper does not take in: none of this amounts to beauty being a quantity that can be measured. What is measured is what happens in the one who looks; what is seen is for the form to say.
8. The Question Left Open
Form has come away from matter and can stand on its own; it needs no nod from the world. Yet one day people take it to the world, and, remarkably, it fits: the planets travel along ellipses, and the speed of light can be calculated from equations. Form can also fail to fit: Kepler's regular solids did not. When form settles its account with reality, why is it beautiful when it fits, what then when it does not, and is fitting alone enough to be beautiful?
Mathematics does not push anyone toward the world. A person can stay in form all his life; if he cannot not take form to the world, that is another ray. This ray stops here.
The question left open is: when people bring form into contact with reality, why is that fitting beautiful? That is the beauty of physics.
9. What Would Make This Paper Wrong
If any one of three things held, this paper would be wrong. First, "seeing the why" contributes nothing to beauty: every correct proof is equally beautiful to whoever understands it, and the two 5,050s do not differ in beauty; then completive beauty would be only another name for correctness. Second, once fame, usefulness, difficulty and perceptual attractiveness are taken away, nothing is left of what mathematicians call elegant; then Kant would be entirely right, and the beauty of mathematics would be only a word put in the wrong place. Third, once the prize, the uses, the mystery and the difficulty are taken away from a gap like the Riemann hypothesis, 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 that says "the rules can be changed, but they must be changed openly" has to state openly the conditions under which it could itself be replaced.
Acknowledgments
This paper is the third 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
[^1]: Han Qin, SAE Judgment and Aesthetics, 2nd ed. (2026), §5.1, "How to Read the Rays," DOI: 10.5281/zenodo.19296710.
[^2]: Han Qin, SAE: The First Critique (2026), DOI: 10.5281/zenodo.21439120.
[^3]: Qian Baocong 錢寶琮, ed., Suanjing shishu 算經十書 (Beijing: Zhonghua shuju, 1963), Zhoubi suanjing 周髀算經, juan shang 卷上, Zhao Shuang's 趙爽 commentary, "Gougu yuanfang tu" 勾股圓方圖.
[^4]: Rentuya Sa, Lara Alcock, Matthew Inglis, and Fenner Stanley Tanswell, "Do Mathematicians Agree about Mathematical Beauty?" Review of Philosophy and Psychology 15, no. 1 (2024): 299–325.
[^5]: G. H. Hardy, A Mathematician's Apology, with a foreword by C. P. Snow (Cambridge: Cambridge University Press, 1992), §§10, 12, 18, 28. First published 1940.
[^6]: Henri Poincaré, Science et méthode (Paris: Flammarion, 1908), livre I, chap. 3, "L'invention mathématique."
[^7]: Bernhard Riemann, "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse," Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1859): 671–680; Enrico Bombieri, "Problems of the Millennium: The Riemann Hypothesis," official problem description (Cambridge, MA: Clay Mathematics Institute, 2000).
[^8]: Kurt Gödel, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I," Monatshefte für Mathematik und Physik 38 (1931): 173–198.
[^9]: Kenneth Appel and Wolfgang Haken, "Every Planar Map Is Four Colorable. Part I: Discharging," Illinois Journal of Mathematics 21, no. 3 (1977): 429–490.
[^10]: Hardy, A Mathematician's Apology.
[^11]: Hardy, A Mathematician's Apology.
[^12]: Matthew Inglis and Andrew Aberdein, "Diversity in Proof Appraisal," in Mathematical Cultures: The London Meetings 2012–2014, ed. Brendan Larvor (Cham: Birkhäuser, 2016), 163–179.
[^13]: Plato, "Philebus," trans. Dorothea Frede, in Plato: Complete Works, ed. John M. Cooper (Indianapolis: Hackett Publishing, 1997), 51b–d.
[^14]: Aristotle, Metaphysics, trans. C. D. C. Reeve (Indianapolis: Hackett Publishing, 2016), M.3, 1078a31–b6.
[^15]: Bertrand Russell, "The Study of Mathematics," in Mysticism and Logic and Other Essays (London: George Allen & Unwin, 1917), 60.
[^16]: Aristotle, Posterior Analytics, trans. Jonathan Barnes, in The Complete Works of Aristotle, ed. Jonathan Barnes (Princeton: Princeton University Press, 1984), I.13, 78a22–b3.
[^17]: Liu Hui 劉徽, preface to the commentary on the Jiuzhang suanshu 九章算術, following the text in Wikisource, which reads 「知，發其一端而已」 after 「同本幹」; and Liu Hui's commentary on the method for the circular field in juan 1, "Fangtian" 方田, in Qian Baocong, ed., Suanjing shishu.
[^18]: Jiuzhang suanshu, juan 4, "Shaoguang" 少廣, method for the sphere (kai liyuan shu 開立圓術), commentary by Li Chunfeng 李淳風 et al. quoting Zu Gengzhi 祖暅之, following the edition collated by Guo Shuchun 郭書春.
[^19]: Arthur Schopenhauer, Die Welt als Wille und Vorstellung, vol. 1 (1819), §15; Arthur Schopenhauer, The World as Will and Idea, trans. R. B. Haldane and J. Kemp, 7th ed., vol. 1 (London: Kegan Paul, Trench, Trübner, 1909), §15.
[^20]: Xu Guangqi 徐光啟, "Jihe yuanben zayi" 幾何原本雜議, in Xu Guangqi ji 徐光啟集, ed. Wang Zhongmin 王重民 (Beijing: Zhonghua shuju, 1963), juan 2.
[^21]: Francis Hutcheson, An Inquiry into the Original of Our Ideas of Beauty and Virtue, 2nd ed. (London, 1726), treatise 1, sect. 3.
[^22]: Immanuel Kant, Critique of the Power of Judgment, ed. Paul Guyer, trans. Paul Guyer and Eric Matthews (Cambridge: Cambridge University Press, 2000), §62, 5:362–366.
[^23]: David Hilbert, "Mathematical Problems," trans. Mary Winston Newson, Bulletin of the American Mathematical Society 8, no. 10 (1902): 437–479, at 438; Charlotte Angas Scott, "The International Congress of Mathematicians in Paris," Bulletin of the American Mathematical Society 7, no. 2 (1900): 57–79, at 68.
[^24]: Poincaré, Science et méthode, livre I, chap. 3.
[^25]: Hardy, A Mathematician's Apology.
[^26]: Hardy, A Mathematician's Apology.
[^27]: Ludwig Wittgenstein, Bemerkungen über die Grundlagen der Mathematik, Werkausgabe 6 (Frankfurt am Main: Suhrkamp, 1984), III, §1.
[^28]: Mark Steiner, "Mathematical Explanation," Philosophical Studies 34, no. 2 (1978): 135–151.
[^29]: Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty," Synthese 111, no. 2 (1997): 171–182.
[^30]: James W. McAllister, "Mathematical Beauty and the Evolution of the Standards of Mathematical Proof," in The Visual Mind II, ed. Michele Emmer (Cambridge, MA: MIT Press, 2005), 15–34.
[^31]: R. S. D. Thomas, "Beauty Is Not All There Is to Aesthetics in Mathematics," Philosophia Mathematica 25, no. 1 (2017): 116–127.
[^32]: Semir Zeki, John Paul Romaya, Dionigi M. T. Benincasa, and Michael F. Atiyah, "The Experience of Mathematical Beauty and Its Neural Correlates," Frontiers in Human Neuroscience 8 (2014): 68.
[^33]: George D. Birkhoff, Aesthetic Measure (Cambridge, MA: Harvard University Press, 1933).
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.
Appel, Kenneth, and Wolfgang Haken. "Every Planar Map Is Four Colorable. Part I: Discharging." Illinois Journal of Mathematics 21, no. 3 (1977): 429–490.
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Bombieri, Enrico. "Problems of the Millennium: The Riemann Hypothesis." Official problem description. Cambridge, MA: Clay Mathematics Institute, 2000.
Gödel, Kurt. "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I." Monatshefte für Mathematik und Physik 38 (1931): 173–198.
Hardy, G. H. A Mathematician's Apology. With a foreword by C. P. Snow. Cambridge: Cambridge University Press, 1992. First published 1940.
Hilbert, David. "Mathematical Problems." Translated by Mary Winston Newson. Bulletin of the American Mathematical Society 8, no. 10 (1902): 437–479.
Hutcheson, Francis. An Inquiry into the Original of Our Ideas of Beauty and Virtue. 2nd edition. London, 1726. First published 1725.
Inglis, Matthew, and Andrew Aberdein. "Diversity in Proof Appraisal." In Mathematical Cultures: The London Meetings 2012–2014, edited by Brendan Larvor, 163–179. Cham: Birkhäuser, 2016.
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Qin, Han. SAE Judgment and Aesthetics. 2026. DOI: 10.5281/zenodo.19296710.
———. SAE: The First Critique. 2026. DOI: 10.5281/zenodo.21439120.
Riemann, Bernhard. "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse." Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1859): 671–680.
Rota, Gian-Carlo. "The Phenomenology of Mathematical Beauty." Synthese 111, no. 2 (1997): 171–182.
Russell, Bertrand. "The Study of Mathematics." In Mysticism and Logic and Other Essays, 58–73. London: George Allen & Unwin, 1917. Written 1902; first published 1907.
Sa, Rentuya, Lara Alcock, Matthew Inglis, and Fenner Stanley Tanswell. "Do Mathematicians Agree about Mathematical Beauty?" Review of Philosophy and Psychology 15, no. 1 (2024): 299–325.
Schopenhauer, Arthur. Die Welt als Wille und Vorstellung. Vol. 1. Leipzig: Brockhaus, 1819.
———. The World as Will and Idea. Translated by R. B. Haldane and J. Kemp. 7th edition. Vol. 1. London: Kegan Paul, Trench, Trübner, 1909.
Scott, Charlotte Angas. "The International Congress of Mathematicians in Paris." Bulletin of the American Mathematical Society 7, no. 2 (1900): 57–79.
Steiner, Mark. "Mathematical Explanation." Philosophical Studies 34, no. 2 (1978): 135–151.
Thomas, R. S. D. "Beauty Is Not All There Is to Aesthetics in Mathematics." Philosophia Mathematica 25, no. 1 (2017): 116–127.
Wang Zhongmin 王重民, ed. Xu Guangqi ji 徐光啟集. Beijing: Zhonghua shuju, 1963.
Wittgenstein, Ludwig. Bemerkungen über die Grundlagen der Mathematik. Werkausgabe 6. Frankfurt am Main: Suhrkamp, 1984.
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