Treebeard's Stumper Answer

The Universal Library of Stumpers
Our perfect spring days early this week made it hard to think about stumpers, but maybe there's another way. My stumpers are all about 500 characters long, composed of the 26 letters, 10 digits, spaces, and some punctuation marks, say 60 different characters all together. My computer could pick these characters at random and make a library of every 500 character "page". Most would be gibberish, but along the way I'd get every possible stumper and answer! How many pages would there be in this universal library? Does this mean that human thought is limited?
Consider this paragraph you're reading. With 60 choices for the first letter, and 60 for the next, and 500 characters to choose, that's 60 x 60 x 60 x ... (500 times) = 60^{500} different paragraphs, including this one. That's a number with 890 digits. It's immense beyond meaning, but finite and exact. By comparison, there are only about 10^{80} electrons and protons in the known universe. Even if the universal library somehow existed, it would be impossible to use. You're as likely to find meaning by pressing random keys on your keyboard. Writing is the art of making sense, not arranging letters. It's hard!
Notes:
The Universal Stumper Library may not fit in the known universe, but here's a virtual copy that fits on this wegpage! Just click on the "Fetch page" button (if you have JavaScript ennabled), and you will retrieve a random 500 character page direct from the library! Or maybe it's from a roomful of monkeys, or the JavaScript Math.random() function. Either way, your very next try could be the real answer to this stumper, but don't count on it. The odds are beyond astronomical though not quite zero. Is it worth a shot? Want to buy a lottery number?
I put the problem of evaluating 60^{500} to my BIGNUM Basic program, which found the answer in less than 2 seconds. Note that the tail of 500 zeros is significant. (Of course, we're multiplying by ten 500 times.) This is the exact answer.
11,902,143,766,496,379,253,772,329,743,743,101,633,875,460,812,098, 010,525,744,118,168,552,096,670,094,348,705,893,653,723,004,019,017,101,498,207, 354,272,809,608,260,330,284,006,793,620,054,046,453,041,396,908,660,754,258,070, 022,124,205,655,287,315,947,548,138,853,760,010,158,699,929,528,425,934,200,448, 288,054,170,028,331,591,769,460,915,170,693,741,537,846,971,699,157,454,734,197, 157,550,545,835,854,691,697,512,477,684,161,139,586,018,663,325,985,923,122,497, 631,026,248,092,373,710,950,937,517,427,279,894,937,600,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 1.919922 seconds to figure 890 digits.Graybear managed to find the answer to 32 significant places with his calculator as 1.190214376649637925377232974374e+889 pages. At first I wondered how he did that, but then I noticed that the Windows calculator can do it right off. Useful tool.Graybear goes on:
You mention that most of these combinations would be gibberish, but many of them would be basically the same, e.g. this week's stumper could be written by spelling out the numbers as words.Also, you would have every wrong answer! You would have the answer that there is no combination, a negative number of combinations, and an infinite number of combinations. You would also have the toad answer to this stumper!
I don't know if this proves that there is a limit to human thought, but it does show that even extremely large numbers are still finite.
It's easy to make a mistake when thinking about these huge (and hugely small) numbers. Brett Watson in his essay on The Mathematics of Monkeys and Shakespeare relates this joke:
During the US government's "Strategic Defense Initiative" program, better known as "Star Wars", leading scientists on the project were asked to report their progress to the Minister of Defense... During the course of their presentation, the following exchange took place.SCIENTIST: ...and so you can see, Mr Minister, that in order to achieve an acceptable hitrate against the missiles, our instruments need to be accurate to one part in ten to the ninth. So far, the best we have been able to achieve is one part in ten to the fifth.Our poor politician completely failed to understand the meaning of the scientist's statement. Ten to the fifth is not more than half of ten to the ninth. Ten to the fifth is 100,000 and ten to the ninth is 1,000,000,000. Perhaps our scientist friend would have done better to let the politician see all those zeros  and then translate it into terms of a budget increase!MINISTER: That's tremendous! We're over half way there!
The idea of an infinitely small amount or infinitesimal is central to calculus. This stumper suggests we need something else, an almost infinitesimal. The set of paragraphs that differ from my answer in just one character is a huge number, but it vanishes when considered as part of the whole. The chance of picking out just one of all possible paragraphs is just
1 / 60^{500} , or0.000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000 000000840184776472725199950638623235...The chance of picking any of a google (10^{100} ) of paragraphs from the total is still astronomically less than the chance of picking out a single electron from the known universe. If this is a limit to thought, it's a limit I can live with. It's much more generous than living with Bill Gates' limited wealth!The idea of the Universal Library has appeared many times. Some trace the idea back to the Ars Magna of Ramon Lull, a Spanish theologian in the 13th century, who first developed the idea of a logic machine. Giordano Bruno, Liebnitz, and Pascal all played with the idea. Kurd Lasswitz coined the name in his story "The Universal Library" published in 1901. The most memorable expression of the idea is Jorge Luis Borges' The Library of Babel (1941). Borges describes the library from the inside as "The universe (which others call the Library)":
When it was proclaimed that the Library contained all books, the first impression was one of extravagant happiness. All men felt themselves to be the masters of an intact and secret treasure. There was no personal or world problem whose eloquent solution did not exist in some hexagon. The universe was justified, the universe suddenly usurped the unlimited dimensions of hope... but the searchers did not remember that the possibility of a man's finding his Vindication, or some treacherous variation thereof, can be computed as zero... As was natural, this inordinate hope was followed by an excessive depression. The certitude that some shelf in some hexagon held precious books and that these precious books were inaccessible, seemed almost intolerable.Sometimes the World Wide Web feels like the universal library. I can spend hours searching for an answer I know is out there somewhere if I can just put together the right search terms to find it. But at some point, I have to think for myself and create the answer!
Borges imagines a universal library off all books. I stuck with short paragraphs, but those paragraphs could be combined in different orders to produce all possible books of any length if we put them together in the right combinations. It would be possible to make an even smaller universal library of all possible 80 character lines. That's just
60^{80} combinations, and by arranging those strips of characters we could say everything that can be said in every language that can be transliterated into those characters! But why stop there? I can do the same thing just by pressing the right keys on my keyboard. I can even do it all with just a "1" and a "0" if I put them together in the right sequence! The same thinking can find every possible image (just pixels), and every possible sound (just samples), and even every possible universe (just particles)!I like this quote from Victor Hugo about Shakespeare: "Genius is a promontory jutting into the infinite." That is the writer's art, to find just the right words out of all the myriad possibilities. Writers know this freedom can bring both joy and despair.
Here are some links for further research:
 Martin Gardner has a chapter on Ramon Lull in his out of print book Logic Machines and Diagrams. Kurd Lasswitz' story "The Universal Library" and the Postscript by Willy Ley are reprinted in the classic Fantasia Mathematica, edited by Clifton Fadiman. This and the companion volume The Mathematical Magpie are still in print and worth getting. I've had these books since I was a kid! Two new collections of mathematical fiction are Rudy Rucker's Mathenauts (out of print) and William Frucht's new Imaginary Numbers. Rudy Rucker has lots to say about very big numbers in his interesting books Mind Tools and Infinity and the Mind. His scifi novel White Light: Or, What Is Cantor's Continuum Problem is unique if you can find a copy.
 Jorge Luis Borges' The Library of Babel (1941) is available on the Web (just click). It's also available in print in many collections, including Ficciones, Labyrinths, and Collected Fictions. No one writes like Borges. There's an appreciation here.
 The American philosopher Willard van Orman Quine has the last word on the idea of the universal library in this paper. Philosopher Daniel C. Dennett considers the idea in his APA address In Darwin’s Wake, Where am I?. Brett Watson discusses the math in his essays on The Mathematics of Monkeys and Shakespeare and More Monkey Business. There's more at Ask Dr. Math and a wordengine at Breed Your Own Text.
 The Internet could be the real universal library, the good, the bad, and the ugly! This is discussed at Vision for the Universal Library and The Internet and the revival of the myth of the universal library.
 My BIGNUM program can figure
60^{500} in less than two seconds. It's available with BASIC source code from Treebeard's BASIC Vault. It's a DOS program, but it runs fine in Windows. Here is the Readme file.
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