Eco: Infinite Songs & Locutions
“Between Lull and Bruno might be placed the game invented by H.P. Harsdörffer in his Matematische und philosophische Erquickstunden (1651: 516-9). He devises 5 wheels containing 264 units (prefixes, suffixes, letters and syllables).
This apparatus can generate 97,209,600 German words, including many that were still non-existent but available for creative and poetic use (cf. Faust 1981: 367). If this can be done for German, why not invent a device capable of generating all possible languages?
The problem of the art of combination was reconsidered in the commentary In spheram Ioannis de sacro bosco by Clavius in 1607. In his discussion of the four primary qualities (hot, cold, dry and wet), Clavius asked how many pairs they might form.
Mathematically, we know, the answer is six. But some combinations (like “hot and cold,” “dry and wet”) are impossible, and must be discarded, leaving only the four acceptable combinations: “Cold and dry” (earth), “hot and dry” (fire), “hot and wet” (air), “cold and wet” (water).
We seem to be back with the problem of Lull: a conventional cosmology limits the combinations.
Clavius, however, seemed to wish to go beyond these limits. He asked how many dictiones, or terms, might be produced using the 23 letters of the Latin alphabet (u being the same as v), combining them 2, 3, 4 at a time, and so on until 23.
He supplied a number of mathematical formulae for the calculations, yet he soon stopped as he began to see the immensity of the number of possible results–especially as repetitions were permissible.
In 1622, Paul Guldin wrote a Problema arithmeticum de rerum combinationibus (cf. Fichant 1991: 136-8) in which he calculated the number of possible locutions generated by 23 letters. He took into account neither the question of whether the resulting sequences had a sense, nor even that of whether they were capable of being pronounced at all.
The locutions could consist of anything from 2 to 23 letters; he did not allow repetitions. He arrived at a result of more than 70,000 billion billion. To write out all these locutions would require more than a million billion billion letters.
To conceive of the enormity of this figure, he asked the reader to imagine writing all these words in huge notebooks: each of these notebooks had 1,000 pages; each of these pages had 100 lines; each of these lines could accommodate 60 characters.
One would need 257 million billion of these notebooks. Where would you put them all? Guldin then made a careful volumetric study, imagining shelf space and room for circulation in the libraries that might store a consignment of these dimensions.
If you housed the notebooks in large libraries formed by cubes whose sides measured 432 feet, the number of such cubic buildings (hosting 32 million volumes each) would be 8,050,122,350. And where would you put them all? Even exhausting the total available surface space on planet earth, one would still find room for only 7,575,213,799!
With this, the conception of universal language has begun to appear, for Mersenne realizes that the answer would necessarily have to include all the locutions in all possible languages. He marveled that our alphabet was capable of supplying “millions more terms than the earth has grains of sand, yet it is so easy to learn that one hardly needs memory, only a touch of discernment” (letter to Peiresc, c. April 1635; cf. Coumet 1975; Marconi 1992).
In the Harmonie, Mersenne proposed to generate only pronounceable words in French, Greek, Arabic, Chinese and every other language. Even with this limitation one feels the shudder provoked by a sort of Brunian infinity of possible worlds.
The same can be said of the musical sequences that can be generated upon an extension of 3 octaves, comprising 22 notes, without repetitions (shades of future 12-tone compositions!).
Mersenne observed that to write down all these songs would require enough reams of paper to fill in the distance between heaven and earth, even if every sheet contained 720 of these 22-note songs and every ream was so compressed as to be less than an inch thick.
In fact the number of possible songs amounted to 1,124,000,727,777,607,680,000 (Harmonie, 108). By dividing this figure by the 362,880 songs contained in each ream, one would still obtain a 16-digit figure, whilst the number of inches between the center of the earth and the stars is only 28,826,640,000,000 (a 14-digit figure).
Anyone who wished to copy out all these songs, a thousand per day, would have to write for 22,608,896,103 years and 12 days.”
Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 139-41.