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Tag: Ars magna sciendi

Eco: Characteristica and Calculus

Gottfried Wilhelm von Leibniz, Dissertatio de Arte Combinatoria

Gottfried Wilhelm von Leibniz (1646-1716), Dissertatio de Arte Combinatoria, an excerpt from his first doctoral dissertation, Dissertation on the Art of Combinations, Leipzig, 1666. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less. 

“The theme of invention and discovery should remind us of Lull; and, in fact, Lull’s ars combinatoria was one of Leibniz’s first sources. In 1666, at the age of twenty, Leibniz composed his own Dissertatio de arte combinatoria (Gerhardt 1875: IV, 27-102). But the dream of the combinatoria was to obsess him for the rest of his life.

In his short Horizon de la doctrine humaine (in Fichant 1991), Leibniz dealt with a problem that had already troubled Father Mersenne: how many utterances, true, false or even nonsensical, was it possible to formulate using an alphabet of 24 letters?

The point was to determine the number of truths capable of expression and the number of expressions capable of being put into writing. Given that Leibniz had found words of 31 letters in Latin and Greek, an alphabet of 24 letters would produce 2432 words of 31 letters.

But what is the maximum length of an expression? Why should an expression not be as long as an entire book? Thus the sum of the expressions, true or false, that a man might read in the course of his life, imagining that he reads 100 pages a day and that each page contains 1,000 letters, is 3,650,000,000.

Even imagining that this man can live one thousand years, like the legendary alchemist Artephius, it would still be the case that “the greatest expressible period, or the largest possible book that a man can read, would have 3,650,000,000,000 [letters], and the number of truths, falsehoods, or sentences expressible–that is, readable, regardless of pronounceability or meaningfulness–will be 24365,000,000,001 – 24/23 [letters].”

We can imagine even larger numbers. Imagine our alphabet contained 100 letters; to write the number of letters expressible in this alphabet we would need to write a 1 followed by 7,300,0000,000,000 (sic) zeros. Even to write such a number it would take 1,000 scribes working for approximately 37 years.

Leibniz’s argument at this point is that whatever we take the number of propositions theoretically capable of expression to be–and we can plausibly stipulate more astronomical sums than these–it will be a number that vastly outstrips the number of true or false expressions that humanity is capable of producing or understanding.

From such a consideration Leibniz concluded paradoxically that the number of expressions capable of formulation must always be finite, and, what is more, that there must come a moment at which humanity would start to enunciate them anew.

With this thought, Leibniz approaches the theme of the apochatastasis or of universal reintegration–what we might call the theme of the eternal return.

This was a line of speculation more mystical than logical, and we cannot stop to trace the influences that led Leibniz to such fantastic conclusions.

It is plain, however, that Leibniz has been inspired by Lull and the kabbala, even if Lull’s own interest was limited to the generation of just those propositions that expressed true and certain knowledge and he thus would never have dared to enlarge his ars combinatoria to include so large a number of propositions.

For Leibniz, on the contrary, it was a fascination with the vertiginous possibilities of discovery, that is of the infinite number of expressions of which a simple mathematical calculation permitted him to conceive, that served as inspiration.

At the time he was writing his Dissertatio, Leibniz was acquainted with Kircher’s Polygraphia, as well as with the work of the anonymous Spaniard, of Becher, and of Schott (while saying that he was waiting for the long-promised Ars magna sciendi of the “immortal Kircher“).

He had yet to read Dalgarno, and Wilkins had still not published his Essay. Besides, there exists a letter from Kircher to Leibniz, written in 1670, in which the Jesuit confessed that he had not yet read Leibniz’s Dissertatio.

Leibniz also elaborated in the Dissertatio his so-called method of “complexions,” through which he might calculate, given n elements, how many groups of them, taken t at a time, irrespective of their ordering, can be ordered.

He applied this method to syllogisms before he passed to his discussion of Lull (para. 56). Before criticizing Lull for limiting the number of his elements, Leibniz made the obvious observation that Lull failed to exploit all the possibilities inherent in his combinatorial art, and wondered what could happen with variations of order, which could produce a greater number.

We already know the answer: Lull not only limited the number of elements, but he rejected those combinations that might produce propositions which, for theological and rhetorical reasons, he considered false.

Leibniz, however, was interested in a logica inventiva (para. 62) in which the play of combinations was free to produce expressions that were heretofore unknown.

In paragraph 64 Leibniz began to outline the theoretical core of his characteristica universalis. Above all, any given term needed to be resolved into its formal parts, the parts, that is, that were explicitly entailed by its definition.

These parts then had to be resolved into their own components, and so on until the process reached terms which could not, themselves, be defined–that is, the primitives. Leibniz included among them not only things, but also modes and relations.

Other terms were to be classified according to the number of prime terms they contained: if they were composed from 2 prime terms, they were to be called com2nations; if from 3 prime terms, com3nations, and so forth. Thereby a hierarchy of classes of increasing complexity could be created.

Leibniz returned to this argument a dozen years later, in the Elementa characteristicae universalis. Here he was more generous with his examples. If we accept the traditional definition of man as “rational animal,” we might consider man as a concept composed of “rational” and “animal.”

We may assign numbers to these prime terms: animal = 2, and rational = 3. The composite concept of man can be represented as the expression 2 * 3, or 6.

For a proposition to be true, if we express fractionally the subject-predicate (S/P) relationship, the number which corresponds to the subject must be exactly divisible by the number which corresponds to the predicate.

Given the preposition “all men are animals,” the number for the subject (men), is 6; the number for animals is 2; the resulting fraction is 6/2 = 3. Three being an integer, consequently, the preposition is true.

If the number for monkey were 10, we could demonstrate the falsity of either the proposition “all men are monkeys” or “all monkeys are men:” “the idea of monkey does not contain the idea of man, nor, vice versa, does the idea of the latter contain the former, because neither can 6 be exactly divided by 10, nor 10 by 6” (Elementa, in Couturat 1903: 42-92). These were principles that had all been prefigured in the Dissertatio.

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 271-5.

Eco: A Dream that Refused to Die, 2



Athanasius Kircher (1602-80), the philosophical tree, from Ars Magna Sciendi, 1669, digitized in 2007 and published on the web by the Complutense University of Madrid. This illustration courtesy of Stanford University. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less. 


“Again apropos of the crusty old myth of Hebrew as the original language, we can follow it in the entertaining compilation given in White (1917: II, 189-208).

Between the first and ninth editions of the Encyclopedia Britannica (1771 and 1885), a period of over one hundred years, the article dedicated to “Philology” passed from a partial acceptance of the monogenetic hypothesis to manifestations of an increasingly modern outlook in scientific linguistics.

Yet the shift took place only gradually–a series of timid steps. The notion that Hebrew was the sacred original language still needed to be treated with respect; throughout this period, theological fundamentalists continued to level fire at the theories of philologists and comparative linguists.

Still in 1804, the Manchester Philological Society pointedly excluded from membership anyone who denied divine revelations by speaking of Sanskrit or Indo-European.

The monogeneticist counterattacks were many and varied. At the end of the eighteenth century, the mystic and theosophist Louis-Claude de Saint-Martin dedicated much of the second volume of his De l’esprit des choses (1798-9) to primitive languages, mother tongues and hieroglyphics.

His conclusions were taken up by Catholic legitimists such as De Maistre (Soirées de Saint Petersburg, ii), De Bonald (Recherches philosophiques, iii, 2) and Lamennais (Essai sur l’indifférence en matière de religion).

These were authors less interested in asserting the linguistic primacy of Hebrew as such than in contesting the polygenetic and materialist or, worse, the Lockean conventionalist account of the origin of language.

Even today, the aim of “reactionary” thought is not to defend the contention that Adam spoke to God in Hebrew, but rather to defend the status of language itself as the vehicle of revelation.

This can only be maintained as long as it is also admitted that language can directly express, without the mediation of any sort of social contract or adaptations due to material necessity, the relation between human beings and the sacred.

Our own century has witnessed counterattacks from an apparently opposite quarter as well. In 1956, the Georgian linguist Nicolaij Marr elaborated a particular version of polygenesis.

Marr is usually remembered as the inventor of a theory that language depended upon class division, which was later confuted by Stalin in his Marxism and Linguistics (1953). Marr developed his later position out of an attack on comparative linguistics, described as an outgrowth of bourgeois ideology–and against which he supported a radical polygenetic view.

Ironically, however, Marr’s polygeneticism (based upon a rigid notion of class struggle) in the end inspired him–again–with the utopia of a perfect language, born of a hybrid of all tongues when humanity will no more be divided by class or nationality (cf. Yaguello 1984: 7, with a full anthology of extracts).”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 113-5.

Eco: The Alphabet and the Four Figures, 2

Raymond Llull, Combinations, Strasbourg ed 1598

Umberto Eco, The Search for the Perfect Language, 1995, pg. 60. Figure 4.2, a page of combinations from the Strasbourg edition of the Ars Magna of Raymond Llull, 1598. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less.  

Taken in groups of 3, 9 elements generate 84 combinations–BCD, BCE, CDE, etc. If, in his Ars breu and elsewhere, Lull sometimes speaks of 252 (84*3) combinations, it is because to each triple can be assigned three questions, one for each of the letters of the triple (see also the Jesuit Athanasius Kircher, Ars magna sciendi, p. 14.


Athanasius Kircher, Ars Magna Sciendi, Amsterdam, 1669. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less. 

Each triple further generates a column of 20 combinations (giving a table of 20 rows by 84 columns) because Lull transforms the triples into quadruples by inserting the letter T. In this way, he obtains combinations like BCDT, BCTB, BTBC, etc. (See examples in figure 4.2, at the top of this page).

The letter T, however, plays no role in the art; it is rather a mnemonic artifice. It signifies that the letters that precede it are to be read as dignities from the first figure, while those that follow it are to be read as relative principles as defined in the second figure.

Thus, to give an example, the quadruple BCTC must be read: B (= goodness) + C (= greatness) and therefore (switching to the second figure) C (=  concordance).

Looking at the tabula generalis, we further notice that combinations with an initial B take the question utrum, those with an initial C take quid, etc. This produces from BCTC the following reading: “Whether goodness is great inasmuch as it contains in itself concordant things.”

This produces a series of quadruples which seem, at first sight, embarrassing: the series contains repetitions. Had repetitions been permissible, there would have been 729 triples instead of 84.

The best solution to the mystery of these repetitions is that of Platzek (1953-4: 141). He points out that, since, depending on whether it precedes or follows the T, a letter can signify either a dignity or a relation, each letter has, in effect, two values.

Thus–given the sequence BCTB–it should be read as BCb. The letters in upper case would be read as dignities, and the one in lower case as a relation. It follows that, in his 84 columns, Lull was not really listing the combinations for three letters but for six. Six different elements taken three at a time give 20 permutations, exactly as many appear in each column.

The 84 columns of 20 quadruples each yield 1,680 permutations. This is a figure obtained by excluding inversions of order.

At this point, however, a new question arises. Given that all these 1,680 quadruples can express a propositional content, do they all stand for 1,680 valid arguments as well?


Athanasius Kircher, Ars Magna Sciendi sive Combinatoria, Amsterdam, 1669. Frontispiece. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less.  

Not at all, for not every sequence generated by the art is syllogistically valid. Kircher, in his Ars magna sciendi, suggests that one must deal with the resulting sequences as if they were anagrams: one starts by forming a complete list of all the possible arrangements of the letters of a particular word, then discards those that do not correspond to other existing words.

The letters of the Latin word ROMA, for example, can be combined in 24 different orders: certain sequences form acceptable Latin words, such as AMOR, MORA, RAMO; others, however, such as AOMR, OAMR, MRAO, are nonsense, and are, as it were, thrown away.

Lull’s own practice seems to suppose such a criterion. He says, for example, in his Ars magna, segunda pars principalis that in employing the first figure, it is always possible to reverse subject and predicate (“Goodness is great” / “Greatness is good”).

It would not, however, be possible to reverse goodness and angel, for while angel participates in goodness, goodness does not participate in angel, since there are beings other than angels which are good.

In other words, angel entails goodness but not vice versa. Lull also adds that the combination “Greed is good” is inherently unacceptable as well. Whoever wishes to cultivate the art, Lull says, must be able to know what is convertible and what is not.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 60-2.

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