Both sheets of paper end with 5,050.
The task was to add the numbers from one to a hundred. On the first sheet, someone has worked through the additions, recording running totals. On the second, there are only a few lines: pair 1 with 100, 2 with 99, 3 with 98. Fifty pairs, each totaling 101.
If we are marking answers, both deserve credit. The first person has made no mistake. Taking longer is not an offense.
But leave the papers on the table a moment before collecting them. The second offers something besides the answer. It gives the sum a shape in which we can see why the answer comes out as it does. What one side of a pair loses, the other gains. A long queue of additions has become fifty equal totals.
“Oh. You can do it that way.”
The discovery may bring delight, or irritation at not having thought of it. Either way, we can return our attention to the paper. What makes the arrangement work is not the cleverness of the person who wrote it.
Speed is not the point
A calculator could display 5,050 before either person picked up a pen. It would be faster. We might trust it completely.
Speed would not show us the fifty pairs.
Even pairing can become a procedure we follow without looking: add the first and last terms, multiply by the number of terms, divide by two. Memorize the instruction and it becomes a useful tool. Correct use does not guarantee that we see the reason every time. A discovery that once stopped us in our tracks can eventually become something we reach for automatically.
There is nothing wrong with that tool. Here, though, I want to look at it before putting it to work.
Replace a hundred with ten. Pair 1 with 10, 2 with 9: five pairs of 11. The numbers have changed, but the relation holds. Replace ten with nine, and a 5 remains in the middle. Now we must pay attention rather than repeat the instruction about pairing unchanged.
We could write the whole sequence twice, once forwards and once backwards. Pair the terms vertically. There are nine pairs of 10, totaling 90. We used the original sequence twice, so its sum is 45.
Returning to the hundred, we can see that the second sheet did more than save labor. It showed which relations survive a change of length. We do not have to acquire a separate confidence in a separate answer every time the sequence changes.
“Beautiful,” here, is not a prize for finishing first. It names a clarity we can enter again.
A short explanation can conceal a long one
Some solutions consist of a single sentence: “This follows from the theorem.”
For a reader who knows the theorem, that may be enough. The missing work is available to be unfolded. For someone else, one unexplained thing has merely been handed over to a larger unexplained thing.
Fewer words do not necessarily bring the reason closer. An explanation willing to take a few more lines may make the relations much clearer. Where did this step come from? Why can the next one follow? The reader does not have to leap a gap and pretend to have understood.
A neat sequence is not necessarily an explanation either. A pattern may work for the first few cases and then fail. Neatness can attract us, quite legitimately. But attracting us to look for a proof is different from already being a proof.
Beauty may encourage an investigation; it cannot excuse us from checking. A wonderfully promising idea may be wrong. If it is, we need to revise the judgment, not quietly hide the inconvenient case to preserve the appearance.
The distinction made in the source paper is straightforward: establishing a result and bringing its reason into view are not identical achievements. Correctness, brevity, difficulty and reputation cannot stand in for the second.
A path someone else can walk
An old proof asks why prime numbers never run out. A prime is an integer greater than 1 divisible only by 1 and itself: 2, 3, 5 and 7 are examples.
Begin with someone presenting a finite list and saying, “These are all of them.”
Multiply the listed primes together and add one. Divide the resulting number by any prime on the list and the remainder is one. But the new number is greater than 1, so it must have a prime divisor. That divisor cannot be on the list.
The new number need not itself be prime. This qualification matters. Starting with 2, 3 and 5 gives 31, which is prime. Starting with 2, 3, 5, 7, 11 and 13 gives 30,031, which equals 59 times 509. We do not need the construction always to produce a prime. We need it to produce a number with a prime divisor the list missed.
Once that step is visible, the length of the list ceases to be the issue. Add the missing prime, declare the revised list complete, and the same argument still works. The list supplies the material for going beyond itself.
This may take patience. Not seeing it immediately is not an aesthetic failure. Try a shorter list. Do the divisions. Then return to the argument.
When it does become clear, however, the reason no longer belongs exclusively to the person explaining it. You need not accept it on the authority of a famous mathematician. You can retrace the steps and see where a substitution would break them. Whoever first found the path does not have to remain at its head forever.
That is a distinctive possibility in mathematics. The original occasion of a discovery cannot be repeated. The relations that sustain it can be encountered anew. We need not settle for hearing how excited somebody else once felt.
Unfinished is not automatically more beautiful
Does this leave beauty only to things already proved? The source paper also considers another possibility: an answer is missing, but established relations have given the missing reason a definite shape.
It discusses the Riemann hypothesis in this connection. Following that discussion takes more mathematical preparation than this essay asks for; a few suggestive metaphors would not be a substitute. What we can preserve here is a distinction. There are many things we do not know. Their being unknown does not, by itself, make them beautiful.
An unfinished exercise may mark a gap in my training. Digits not yet calculated may mark work that has not been done. Neither becomes profound merely because it is incomplete.
The paper is interested in a more articulated absence: relations already established reveal precisely what kind of reason is missing. There can be something to attend to now, without borrowing all its value from a future solution. But that interest cannot advance us the proof we have not yet supplied.
Nor does it outrank the finished pairing argument. One person may love the tension of a search. Another may prefer a result that stands quietly, still clear each time it is revisited. We need not disparage completed work to prove that we value discovery.
At the table, the person who wrote the shorter solution turns it around and passes it to the other. The other picks up a pen, draws a line between 1 and 100, then another line.
There is no need to say, “It's so simple. How did you miss it?”
That would turn a path the reader can now walk into another examination of whether the reader is clever enough.
Let the lines be drawn. There may be a smile, or only a nod. The person may continue not to like mathematics. That is allowed.
Today, a relation between these numbers became visible. That can be a complete occasion in its own right.