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Tag: Raymond Llull

Eco: Blind Thought, 2

Wittgenstein, Ludwig

Ludwig Wittgenstein (1899-1951), portrait by Moritz Nähr (1859-1945), 1930, held by the Austrian National Library under Accession Number Pf 42.805: C (1). 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 70 years or less. 

“As Leibniz observed in the Accessio ad arithmeticum infinitorum of 1672 (Sämtliche Schriften und Briefen, iii/1, 17), when a person says a million, he does not represent mentally to himself all the units in that number. Nevertheless, calculations performed on the basis of this figure can and must be exact.

Blind thought manipulates signs without being obliged to recognize the corresponding ideas. For this reason, increasing the power of our minds in the manner that the telescope increases the power of our eyes, it does not entail an excessive effort.

“Once this has been done, if ever further controversies should arise, there should be no more reason for disputes between two philosophers than between two calculators. All that will be necessary is that, pen in hand, they sit down together at a table and say to each other (having called, if they so please, a friend) “let us calculate.” (In Gerhardt 1875: VII, 198ff).

Leibniz’s intention was thus to create a logical language, like algebra, which might lead to the discovery of unknown truths simply by applying syntactical rules to symbols. When using this language, it would no more be necessary, moreover, to know at every step what the symbols were referring to than it was necessary to know the quantity represented by algebraic symbols to solve an equation.

Thus for Leibniz, the symbols in the language of logic no longer stood for concrete ideas; instead, they stood in place of them. The characters “not only assist reasoning, they substitute for it.” (Couturat 1901: 101).

Dascal has objected (1978: 213) that Leibniz did not really conceive of his characteristica as a purely formal instrument apparatus, because symbols in his calculus are always assigned an interpretation. In an algebraic calculation, he notes, the letters of the alphabet are used freely; they are not bound to particular arithmetical values.

For Leibniz, however, we have seen that the numerical values of the characteristic numbers were, so to speak, “tailored” to concepts that were already filled with a content–“man,” “animal,” etc.

It is evident that, in order to demonstrate that “man” does not contain “monkey,” the numerical values must be chosen according to a previous semantic decision. It would follow that what Leibniz proposed was really a system both formalized and interpreted.

Now it is true that Leibniz’s posterity elaborated such systems. For instance, Luigi Richer (Algebrae philosophicae in usum artis inveniendi specimen primum, “Melanges de philosophie et de mathématique de la Societé Royale de Turin,” 1761: II/3), in fifteen short and extremely dry pages, outlined a project for the application of algebraic method to philosophy, by drawing up a tabula characteristica containing a series of general concepts (such as aliquid, nihil, contingens, mutabile) and assigning to each a conventional sign.

The system of notation, semicircles orientated in various ways, makes the characters hard to distinguish from one another; still, it was a system of notation that allowed for the representation of philosophical combinations such as “This Possible cannot be Contradictory.”

This language is, however, limited to abstract reasoning, and, like Lull, Richer did not make full use of the possibilities of combination in his system as he wished to reject all combinations lacking scientific utility (p. 55).

Towards the end of the eighteenth century, in a manuscript dating 1793-4, we also find Condorcet toying with the idea of a universal language. His text is an outline of mathematical logic, a langue des calculs, which identifies and distinguishes intellectual processes, expresses real objects, and enunciates the relations between the expressed objects and the intellectual operations which discover the enunciated relations.

The manuscript, moreover, breaks off at precisely the point where it had become necessary to proceed to the identification of the primitive ideas; this testifies that, by now, the search for perfect languages was definitively turning in the direction of a logico-mathematical calculus, in which no one would bother to draw up a list of ideal contents but only to prescribe syntactic rules (Pellerey 1992a: 193ff).

We could say that Leibniz’s characteristica, from which Leibniz had also hoped to derive metaphysical truths, is oscillating between a metaphysical and ontological point of view, and the idea of designing a simple instrument for the construction of deductive systems (cf. Barone 1964: 24).

Moreover, his attempts oscillate between a formal logic (operating upon unbound variables) and what will later be the project of many contemporary semantic theories (and of artificial intelligence as well), where syntactic rules of a mathematical kind are applied to semantic (and therefore interpreted) entities.

But Leibniz ought to be considered the forerunner of the first, rather than of the second, line of thought.

The fundamental intuition that lies behind Leibniz’s proposal was that, even if the numbers were chose arbitrarily, even if it could not be guaranteed that the primitives posited for the same of argument were really primitive at all, what still guaranteed the truth of the calculus was the fact that the form of the proposition mirrored an objective truth.

Leibniz saw an analogy between the order of the world, that is, of truth, and the grammatical order of the symbols in language. Many have seen in this a version of the picture theory of language expounded by Wittgenstein in the Tractatus, according to which “a picture has logico-pictorial form in common with what it depicts” (2.2).

Leibniz was thus the first to recognize that the value of his philosophical language was a function of its formal structure rather than of its terms; syntax, which he called habitudo or propositional structure, was more important than semantics (Land 1974: 139).

“It is thus to be observed that, although the characters are assumed arbitrarily, as long as we observe a certain order and certain rule in their use, they give us results which always agree with each other. (Dialogus in Gerhardt 1875: VII, 190-3).

Something can be called an “expression” of something else whenever the structure [habitudines] subsisting in the expression corresponds to the structure of that which it wishes to express [ . . . ].

From the sole structure of the expression, we can reach the knowledge of the properties of the thing expressed [ . . . ] as long as there is maintained a certain analogy between the two respective structures.” (Quid sit idea in Gerhardt 1875: VII, 263-4).

What other conclusion could the philosopher of preestablished harmony finally have reached?”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 281-4.

Eco: The Alphabet and the Four Figures, 3

12544152.0001.001-00000019

Jonathan Swift, Gulliver’s Travels, 1892 George Bell and Sons edition, Project Gutenberg. Also see Jonathan Swift, Gulliver’s Travels, A.J. Rivero, ed., New York: W.W.Norton, 2001, Part III, chapter 5. Cited in Bethany Nowviskie, “Ludic Algorithms,” in Kevin Kee, ed., Pastplay: Teaching and Learning History with Technology, Ann Arbor, MI: University of Michigan Press, 2014. 

“It follows that Lull’s art is not only limited by formal requirements (since it can generate a discovery only if one finds a middle term for the syllogism); it is even more severely limited because the inferences are regulated not by formal rules but rather by the ontological possibility that something can be truly predicated of something else.

The formal rules of the syllogism would allow such arguments as “Greed is different from goodness — God is greedy — Therefore God is different from goodness.” Yet Lull would discard both the premises and the conclusion as false.

The art equally allows the formulation of the premise “Every law is enduring,” but Lull rejects this as well because “when an injury strikes a subject, justice and law are corrupted” (Ars brevis, quae est de inventione mediorum iuris, 4.3a).

Given a proposition, Lull accepts or rejects its logical conversion, without regard to its formal correctness (cf. Johnston 1987: 229).

Nor is this all. The quadruples derived from the fourth figure appear in the columns more than once. In Ars magna the quadruple BCTB, for example, figures seven times in each of the first seven columns.

In V, 1, it is interpreted as “Whether there exists some goodness so great that it is different,” while in XI, 1, applying the rule of logical obversion, it is read as “Whether goodness can be great without being different”–obviously eliciting a positive response in the first case and a negative one in the second.

Yet these reappearances of the same argumentative scheme, to be endowed with different semantic contents, do not bother Lull. On the contrary, he assumes that the same question can be solved either by any of the quadruples from a particular column that generates it, or from any of the other columns!

Such a feature, which Lull takes as one of the virtues of his art, represents in fact its second severe limitation. The 1,680 quadruples do not generate fresh questions, nor do they furnish new proofs.

They generate instead standard answers to an already established set of questions. In principle, the art only furnishes 1,680 different ways of answering a single question whose answer is already known.

It cannot, in consequence, really be considered a logical instrument at all. It is, in reality, a sort of dialectical thesaurus, a mnemonic aid for finding out an array of standard arguments able to demonstrate an already known truth.

As a consequence, any of the 1,680 quadruples, if judiciously interpreted, can yield up the correct answer to the question for which it is adapted.

See, for instance, the question “Whether the world is eternal” (“Utrum mundus sit aeternus“). Lull already knew the answer: negative, because anyone who thought the world eternal would fall into the Averroist error.

Note, however, that the question cannot be generated directly by the art itself; for there is no letter corresponding to world. The question is thus external to the art.

In the art, however, there does appear a term for eternity, that is, D; this provides a starting point.

In the second figure, D is tied to the relative principle contrarietas or opposition, as manifested in the opposition of the sensible to the sensible, of the intellectual to the sensible, and of the intellectual to the intellectual.

The same second figure also shows that D forms a triangle with B and C. The question also began with utrum, which appears at B under the heading Questiones in the tabula generalis. This constitutes a hint that the solution needs to be sought in the column in which appear B, C and D.

Lull says that “the solution to such a question must be found in the first column of the table;” however, he immediately adds that, naturally, “it could be found in other columns as well, as they are all bound to each other.”

At this point, everything depends on definitions, rules, and a certain rhetorical legerdemain in interpreting the letters. Working from the chamber BCDT (and assuming as a premise that goodness is so great as to be eternal), Lull deduces that if the world were eternal, it would also be eternally good, and, consequently, there would be no evil.

“But,” he remarks, “evil does exist in the world as we know by experience. Consequently we must conclude that the world is not eternal.” This negative conclusion, however, is not derived from the logical form of the quadruple (which has, in effect, no real logical form at all), but is merely based on an observation drawn from experience.

The art may have been conceived as the instrument to use universal reason to show the Averroist Muslims the error of their ways; but it is clear that unless they already shared with Lull the “rational” conviction that the world cannot be eternal, they are not going to be persuaded by the art.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 62-4.

Eco: The Elements of the Ars Combinatoria

Ramon_Llull, Ars Magna, Fig_1

Raymond Llull (1232-1316), Figure 1 from Ars magna, 1300. 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. 

“Given a number of different elements n, the number of arrangements that can be made from them, in any order whatever, is expressed by their factorial n!, calculated as 1.*2*3. . . . *n.

This is the method for calculating the possible anagrams of a word of n letters, already encountered as the art of temurah in the kabbala. The Sefer Yezirah informed us that the factorial of 5 was 120.

As n increases, the number of possible arrangements rises exponentially: the possible arrangements for 36 elements, for example, are 371, 993, 326, 789, 901, 217, 467, 999, 448, 150, 835, 200, 000, 000.

If the strings admit repetitions, then those figures grow upwards. For example, the 21 letters of the Italian alphabet can give rise to more than 51 billion billion 21-letter-long sequences (each different from the rest); when, however, it is admitted that some letters are repeated, but the sequences are shorter than the number of elements to be arranged, then the general formula for n elements taken t at a time with repetitions is n1  and the number of strings obtainable for the letters of the Italian alphabet would amount to 5 billion billion billion.

Let us suppose a different problem. There are four people, A. B, C, and D. We want to arrange these four as couples on board an aircraft in which the seats are in rows that are two across; the order is relevant because I want to know who will sit at the the window and who at the aisle.

We are thus facing a problem of permutation, that is, of arranging n elements, taken t at a time, taking the order into account. The formula for finding all the possible permutations is n!/(n-t)In our example the persons can be disposed this way:

AB     AC     AD     BA     CA     DA     BC     BD     CD     CB     DB     DC

 Suppose, however, that the four letters represented four soldiers, and the problem is to calculate how many two-man patrols could be formed from them. In this case the order is irrelevant (AB or BA are always the same patrol). This is a problem of combination, and we solve it with the following formula: n!/t!(n-t)! In this case the possible combinations would be:

AB     AC     AD     BC     BD     CD

Such calculuses are employed in the solution of many technical problems, but they can serve as discovery procedures, that is, procedures for inventing a variety of possible “scenarios.”

In semiotic terms, we are in front of an expression-system (represented both by the symbols and by the syntactic rules establishing how n elements can be arranged t at a time–and where t can coincide with n), so that the arrangement of the expression-items can automatically reveal possible content-systems.

In order to let this logic of combination or permutation work to its fullest extent, however, there should be no restrictions limiting the number of possible content-systems (or worlds) we can conceive of.

As soon as we maintain that certain universes are not possible in respect of what is given in our own past experience, or that they do not correspond to what we hold to be the laws of reason, we are, at this point, invoking external criteria not only to discriminate the results of the ars combinatoria, but also to introduce restrictions within the art itself.

We saw, for example, that, for four people, there were six possible combinations of pairs. If we specify that the pairing is of a matrimonial nature, and if A and B are men while C and D are women, then the possible combinations become four.

If A and C are brother and sister, and we take into the account the prohibition against incest, we have only three possible groupings. Yet matters such as sex, consanguinity, taboos and interdictions have nothing to do with the art itself: they are introduced from outside in order to control and limit the possibilities of the system.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 54-6.

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