The Universal Library

"Now come and sit down already, Max," said Professor Wallhausen. "There's truly nothing among my papers for your magazine. What shall I pour you — wine or beer?"

Max Burkel stepped over to the table and slowly raised his eyebrows. Then he lowered his robust frame into the armchair with unhurried ease and spoke:

"Actually, I've sworn off drink. But while traveling — I see you've got a fine Kulmbacher there — ah, many thanks, dear miss — not so full! Well, to your health, old fellow, dear madam! Your health, Miss Briggen! What a tremendous pleasure to be sitting with you once more. But there's no getting around it — you simply must write something for me."

"Honestly, nothing comes to mind at the moment. Such a dreadful amount of unnecessary material is already being written — and worse yet, printed."

"You certainly don't need to tell a beleaguered editor that. The only question is which of it all is the unnecessary part. Authors and readers hold very different views on the matter. And it's always precisely what the critics declare superfluous that lands on the desk of people like me. Ha, how glad I am" — he rubbed his hands with delight — "that my deputy still has to sweat through three more weeks on my behalf."

"I'm surprised," the lady of the house began, "that you even manage to find anything new to print at all. I should have thought that by now you must have gone through just about every arrangement you could make with your handful of letters."

"That's quite true, dear Mrs. Wallhausen — one would think so — but the human mind is inexhaustible —"

"At repeating itself — is what you mean."

"Thank heaven, yes!" laughed Burkel. "But in producing new things too."

"And yet," Professor Wallhausen remarked, "everything that humanity may ever be granted — in historical experience, scientific insight, poetic power, and the lessons of wisdom — can be rendered in letters. At least, insofar as language can express it. For our books do in fact convey the knowledge of humanity and preserve the treasure that the labor of thought has accumulated. But the number of possible combinations of a given set of letters is bounded. Therefore all conceivable literature must be containable in a finite number of volumes."

"Come now, old friend, there you go once more — talking as a mathematician rather than a philosopher. How can the inexhaustible be finite?"

"Allow me — I'll calculate for you straightaway how many volumes the universal library would contain."

"Say, uncle, is this going to get very learned?" asked Susanne Briggen.

"But Suse, surely nothing is too learned for a young lady fresh out of finishing school?"

"Thank you, uncle, but really I was only asking so I'd know whether to fetch my needlework — I can concentrate better that way, you see."

"Aha, you little fox — what you really wanted to know is whether I'm going to deliver a very long speech. That is not my intention. But you could hand me that sheet of paper over there, and the pencil."

"You might as well bring the logarithm tables while you're at it," Burkel remarked drily.

"Heaven forbid," protested the lady of the house.

"No, no, that won't be needed," the professor called out. "And you needn't show off with the needlework, Suse."

"Here's something handier for your fingers," his wife said, sliding the bowl of apples and nuts toward her.

"Thanks," Susanne replied, seizing the nutcracker. "Now I'm ready to take on your toughest nuts."

"Let our friend have the first word," the professor began. "My question is this: if one keeps things compact, dispenses with special typographical niceties such as different fonts, and assumes a reader who doesn't demand too much comfort — one who cares only about meaning —"

"That reader doesn't exist."

"Well, let us suppose he does. How many distinct characters would one need for the whole of literary and popular fiction?"

"Well," said Burkel, "if we stick to the uppercase and lowercase letters of the Latin alphabet, the common punctuation marks, the digits, and — not to be forgotten — the space —"

Susanne glanced up questioningly from her nuts.

"That's the piece of type for blank intervals — the compositor uses it to keep words apart and to fill in the gaps. So that wouldn't be too many. But for scientific books! You mathematicians have an enormous heap of symbols!"

"We manage that through indices — small numbers placed above or below the letters of the alphabet, like a-zero, a-one, a-two, and so on. For that we'd only need a second and a third set of digits from 0 to 9. In fact, with a suitable convention, one could even represent any desired foreign-language sounds this way."

"Fine with me. I'd trust your ideal reader with that too. In that case, I'd estimate we need no more than about a hundred different characters to express anything conceivable in writing."

"Well, there you have it. And how thick shall we make each volume?"

"I'd say one can write quite thoroughly on a subject by filling a volume of five hundred pages. If we imagine about 40 lines per page with 50 characters each — always counting spaces, punctuation, and the rest — we'd get 40 times 50 times 500 characters per volume, which comes to — well, you'd better work that out."

"One million," said the professor. "So if we arrange our 100 characters in any order whatsoever, repeating them freely, until they fill a volume of one million characters, we produce some written work or other. And if we imagine every possible arrangement that could be assembled in this purely mechanical fashion, we end up with precisely all the works ever written or that could ever be written in the future."

Burkel clapped his friend heartily on the shoulder.

"I'm subscribing to the universal library on the spot! All the future issues of my magazine will already be sitting there, ready-made in proof copy. I won't have to chase after a single contribution. Splendid for the publisher — the author eliminated from the entire enterprise! The writer replaced by the combinatorial engine — a triumph of technology!"

"What?" cried the lady of the house. "Everything is in the library? Including the whole of Goethe? The Bible? The collected works of every philosopher who ever lived?"

"With every variant reading that no one has yet conceived of, to boot. You'll find in it all the lost writings of Plato and Tacitus, along with translations. Furthermore, every future work by both of us, all the forgotten and yet-to-be-delivered Reichstag speeches, the universal treaty of world peace, the history of the wars that will follow upon it —"

"And the railway timetable, uncle!" exclaimed Susanne. "That's your favorite book, after all."

"Naturally — and every one of your German compositions for Miss Grazelau."

"Oh, if only I'd had that book while I was still at boarding school! But I thought we were always talking about an entire volume —"

"If you'll allow me, Miss Briggen," Burkel cut in, "don't forget the spaces. The tiniest little poem could have a whole volume to itself; the rest would simply be blank. And the longest works would be in there too — if one doesn't fit in a single volume, its continuation turns up in another."

"Well, good luck tracking anything down in it," said the lady of the house.

"That is exactly where the snag lies," the professor began with a grin, leaning back in his chair and lazily following the smoke of his cigar with his eyes. "It might seem as though searching would be made easier by the fact that the library must also contain its own catalogue —"

"Well then —"

"Yes, but how would you locate that catalogue? And even if you did find one volume of it, you'd be no better off, because it would contain not only the correct titles and call numbers but every conceivable wrong one as well."

"Devil take it, that's true!"

"Hmm! There are certain difficulties, of course. Take the first volume of our library, for instance. The first page is blank, the second likewise, and so on through all 500 pages. This is the volume in which the space character is repeated one million times —"

"At least there can be no nonsense in that one," Mrs. Wallhausen put in.

"Small comfort! Now the second volume — also blank, everything blank, except that on the very last page, all the way at the bottom, in the millionth character position, there sits a tentative little 'a.' In the third volume it is the same, except that the 'a' has advanced one position forward, and a space occupies the final position once again. And so the 'a' nudges one position further forward in each successive volume, through a million volumes, until in the first volume of the second million it has happily arrived at position number one. Otherwise this interesting volume contains nothing at all. And so it continues through the first hundred million of our volumes, until each of the hundred characters has completed its solitary march from back to front. The same process then repeats with 'aa,' or with any other pair of characters, in every possible placement. One volume consists entirely of periods, another entirely of question marks."

"Well," said Burkel, "one would soon spot these contentless volumes and toss them out —"

"Hmm, perhaps — but the worst only comes when you've found a volume that appears to make sense. Say you want to look something up in Faust and actually stumble upon the volume with the correct opening. You read a little further, and suddenly the text goes on: 'Pitter-patter, fiddle-faddle, nothing's here!' or simply 'aaaaa' … Or a logarithm table begins, but you cannot tell whether it is correct, because our library contains not just everything that is true but everything that is false as well. One mustn't be misled by the headings. A volume might open, 'History of the Thirty Years' War,' and then proceed: 'When Prince Blücher wed the Queen of Dahomey at Thermopylae …'"

"Oh uncle, that's right up my alley!" Susanne exclaimed delightedly. "I could write those volumes — when it comes to jumbling things up, I show genuine talent. I'm sure the library also contains the opening I once declaimed from Iphigenia:

"If that were in print, I'd be vindicated after all. And I'd surely find the long letter I wrote to both of you — the one that vanished all at once just when I was about to post it. Mika had gone and put her schoolbooks right on top of it. Oh dear!" she broke off in embarrassment, brushing the unruly brown hair from her forehead. "Miss Grazelau did tell me expressly to take care that I don't start chattering!"

"On this occasion you are entirely justified," her uncle reassured her. "Our library contains not only every one of your letters, but also every speech you've ever given or ever will give —"

"Oh, in that case please don't publish it!"

"Don't fret — they're signed not only with your name but also with Goethe's, and indeed with every possible name in the world. Our friend here, for instance, would find his own signature affixed responsibly to articles containing every imaginable press crime, so that his entire lifetime would not suffice to serve out the sentences. There would be a book by him in which every sentence is followed by a declaration that it is false, and another volume in which those very same sentences are solemnly sworn to be true —"

"All right, that'll do," cried Burkel, laughing. "I knew from the start you'd be pulling our leg. So I won't be subscribing to the universal library, because it would be impossible to sift the sense from the nonsense, the true from the false. If I find who knows how many million volumes all claiming to be the true history of the German Empire in the twentieth century, and all of them flatly contradicting one another, I might as well just read the historians themselves. I'll pass."

"Very wise of you. You'd have saddled yourself with quite a burden otherwise. And I'm not making this up, by the way. I never claimed you could extract the useful material from it — only that one can state the exact number of volumes our universal library would contain, and that among all the nonsensical ones, every piece of meaningful literature that is at all possible must also be present."

"Then go ahead and calculate how many volumes there are," said the lady of the house. "That blank sheet of paper won't give you any peace otherwise."

"It's quite simple — I can do it in my head. We just need to think about how we'd produce our library. First we set down each of our hundred characters once. Then we append each of the hundred characters to every one of those, producing a hundred times a hundred groups of two characters. Adding each character a third time gives us a hundred times a hundred times a hundred groups of three characters apiece, and so on. And since we have a million positions available per volume, the total number of volumes equals the number you get by multiplying 100 by itself a million times. Because 100 is nothing other than ten times ten, the same result is obtained by writing ten as a factor two million times. That is nothing more than a 1 followed by two million zeroes. Here it is: 10^2,000,000"

The professor held the paper aloft.

"Sure," his wife said, "that's the easy way out. Try writing the whole thing out."

"I wouldn't dream of it. I'd need at least a fortnight of writing, day and night, without stopping. Printed out, the number would stretch roughly four kilometers."

"Goodness," said Susanne. "How would you even say it out loud?"

"There is no name for it. In fact, there is simply no way to make it even remotely tangible to the imagination — so enormous is this quantity, even though it is demonstrably finite. Whatever vast magnitudes one might care to name, they all shrink to nothing beside this numerical monstrosity."

"What if you expressed it in trillions?" asked Burkel.

"A trillion is a perfectly respectable number — a thousand billion, a 1 with 18 zeroes. If you divide our volume count by a trillion, you strike off exactly 18 of the two million zeroes. That leaves you with a number carrying 1,999,982 zeroes, which conveys just as little to the mind. But hold on a moment —" The professor jotted a few figures on the paper.

"I knew it," sighed his wife. "The calculating has begun after all."

"Already finished. Consider what this number means for our library in practical terms. Suppose each volume is two centimeters thick, and we line them all up in a single row — how far do you suppose that row would reach?"

He glanced around in triumph as silence fell.

Then Susanne burst out: "I know! May I?"

"Go right ahead, Suse!"

"Twice as many centimeters as there are volumes in the library."

"Bravo, bravo!" the others cheered. "Perfectly sufficient."

"Quite so," said the professor, "but let us look a bit more closely. You all know that light covers 300,000 kilometers in a single second — roughly ten billion kilometers in a year, which amounts to a trillion centimeters. So if our librarian were to fly along our row of books at the speed of light, it would still take him two years merely to pass a single trillion volumes. To traverse the entire library would therefore require twice as many years as there are trillions contained in the total volume count — that is, as we said before, a 1 with 1,999,982 zeroes. What I mean to illustrate is this: the number of years that light would need to travel the length of the library is just as unimaginable as the number of volumes itself. And that shows as plainly as anything that it is a hopeless enterprise to form any intuitive picture of this number, finite though it is."

The professor was about to set the paper aside when Burkel spoke up: "If the ladies will indulge me a moment longer, I have just one more question. I have a sneaking suspicion you've computed a library for which there isn't enough room in the entire world."

"We'll settle that in short order," said the professor, and fell to calculating once more. Then he announced:

"If we were to pack the whole library so tightly that a thousand volumes occupied a single cubic meter, then in order to house it all, the entire visible universe — out to the most distant nebulae — would have to be duplicated so many times that even that number of stuffed universes would have only about 60 zeroes fewer than the 1 with two million zeroes representing our volume count. So there it is — by no route whatever can we get any nearer to this colossal figure."

"There, you see," said Burkel. "I was right to call it inexhaustible."

"Not at all. Subtract the number from itself and you have zero. It is finite; as a concept it is sharply defined. The only startling thing is this: we can write down the number of volumes in which this seeming infinity of all possible literature is contained using barely a handful of digits. Yet the moment we attempt to absorb those contents into our actual experience — to picture concretely, say, what it would be like to seek out a single volume from our universal library — we stand before this limpid creation of our own intellect as though facing something infinite and unfathomable."

Burkel nodded gravely. "The reach of reason is infinitely greater than the grasp of comprehension."

"What does that enigmatic remark mean?" asked the lady of the house.

"Simply this: we are able to think correctly about infinitely more than we can ever truly know through experience. Logic is infinitely mightier than sensation."

"That is precisely what is sublime," Wallhausen observed. "What the senses apprehend passes with time, but logic stands outside of time altogether — it holds universally. And since logic is nothing other than human thought itself, we partake, through this timeless possession, of the unchanging laws of the divine, of the destiny of the boundless creative power. That is the foundation upon which mathematics rests."

"Granted," said Burkel. "The laws instill in us a confidence in truth. But we can put them to use only once we have filled their empty form with the living substance of experience — that is, only once we have tracked down the particular volume we need from the library."

Wallhausen nodded in agreement, and his wife recited softly:

"The great poet strikes home," said the professor. "Yet without the logical law there would be nothing sure to raise us to the stars and beyond them. Only we must not abandon the solid ground of experience. It is not in the universal library that we ought to search; the volume we require we must produce ourselves, through steady, earnest, honest labor."

"Chance plays; reason builds," exclaimed Burkel. "And for that reason you will write down tomorrow what you improvised today, and I shall carry off my article after all."

"That much I can do for you," Wallhausen laughed. "But I warn you now — your readers will be convinced it belongs among the superfluous volumes. What are you up to, Suse?"

"I intend to do something rational," she declared with great solemnity. "I am going to fill form with substance."

And she topped up everyone's glass.