Leibniz's Algorithmic Interpretation of Lullus' Art

ORG AN ON 4(1967)

LE 250e ANNIVERSAIRE DE LA MORT

DE G. W. LEIBNIZ

J. O. Fleckenstein (Switzerland)

LEIBNIZ’S ALGORITHMIC INTERPRETATION OF LULLUS’ ART

In 1966 we call to m ind not only the 250th anniversary of Leibniz’s death but, a t the same time, the tercen tary of his Dissertatio de arte

combinatoria, a work in w hich this young scientist, then aged 20, made his debut and, a t the same time, revealed th e significant roots of his genius. W hat we call “com binatorics” is a m athem atical discipline, requiring for its operation none of the m eans surpassing the scope of antique m athem atical knowledge. Even so, it is characteristic th a t it was not u n til th e Grand Siecle of th e Baroque, th a t com binatorics which dem ands no more th an the four fundam ental rules of arithm etic, was conceived as a new m athem atical discipline. In the fram ew ork of A ristotle’s antique logic it had been possible to p ictu re operations of thinking by operations of calculations; however, antique thinking was unable to the final step in abstraction, and to adm it logical relations no t only betw een complete logical subjects b u t betw een relations as w ell by calculating w ith relations in the same m anner as w ith num erals. The arithm etic of an tiq u ity w as not y et able to pass on to th e arithm etic of relations, th a t is, to w hat is called combinatorics. In his 1666 dissertation Leibniz even pointed out the logical superiority of relations over relating subjects; this dissertation calls attention no t only to his calculating machines of a la te r date but, likewise, to th e notion of th e logical operator in general. And it does not seem to be a haphazard occurrence th a t n either the followers of the num eral system nor practical scientists among th e m athem aticians invented calculating m achines for th e use of m erchants, b u t this was ra th e r done by the philosophers and logicians like Leibniz and Pascal, by th e use of th eir Ars combinandi. S tatistics also is an achievem ent of the 17th century; this branch of science, called A rs conjectandi, w as developed by Jakob Bernoulli, also a p u re m athem atician.

Fig. 1. 3rd and 4th logical Figure of Raymond Lullus (from a MS of th e 14th century)

Again it m ight not be by accident, th a t P ascal’s machine was inferior to th a t of Leibniz, because it was unsuitable for carrying out the last of the four fundam ental rules, the division; P ascal’s Theorie des com-

binaisons still lacked the notion of an operator which, even w hen

Leibn iz’s Interpretation of L u llu s ’ A rt 1 7 3

Leibniz was probably the one th in k er of th e Baroque who, w hile most consciously adhering to traditional thinking, aim ed a t converting, science from logically static subjects and predicates of its scholastic p ast into a science of functions of variable values of th e positivistic future, w ithout philosophical obstacles. His 1666 w riting still bore predom inantly scholastic features: quoting his predecessors he attem p ted to autom atise thinking and, in this p articu la r case, calculating. Not only did Leibniz point to N eper’s (N apier’s) rhabdology dating from 1617, by w hich this Scotch peer, know n to us as th e founder of logarithm ic calculus by means of his “rods” or “bones” (mechanical devices for carrying out arithm etical operations), was the firs t to supersede the abacus by autom atizing calculations. Leibniz also dedicated a num ber of chapters on logic to th e legendary Catalonian monk Ramon y Lull, who as early as in the 13 th century attem pted to a u to ­ m atise logical operations by means of his rotating discs. A m anuscript found by C outurat among the 80,000 papers of the heritage le ft by Leibniz m entions: “Raym undus Lullus also dabbled in m athem atics; he h it upon th e notion of the ‘science of com binations.’ This L u llu s’ a r t would undoubtedly be som ethink beautiful, w ere it n o t th a t the fundam ental expressions he uses like: goodness, greatness, duration, force, wisdom, will, virtue, fame, are vague and m erely suitable for speaking of tru th , not for detecting it.”

Leibniz by no means passes censure on L ullus’ intention as an a ttem p t of autom atizing processes of thinking by m achinery; however, he m erely recognizes L ullus’ “Figures” to be inadequate as logical apparatus. F or instance, in his th ird F igure Lullus depicts a model intended to illustrate, how th e 9 absolute predicates are to be combined w ith the 9 relative-predicates of his logical A lphabetum , to form judg­ ments, and how—in L ullus’ belief—one passes from the general to the specific. However, first one has to know, by w hich interm ediate notion the predicates of these judgm ents are to be combined w ith th e ir subjects. This Lullus tries to achieve by mechanizing, in his fo u rth Figure, th e arrangem ent of his 9 absolute and 9 relative p re ­ dicates in such m anner, th a t by rotating th e tw o in n er circular discs th ere can pass along each of th e 9 subdivisions of the outer disc the 9 subdivisions of the inner disc, and th a t th e subdivisions of the in te r­ m ediate disc can operate as interm ediate notions of judgm ents.

The Ars vnagna of L ullus’ a rt became not only th e w atchw ord of th e Rinascimento hum anists; it continued to be considered the Nuova

Scienza in general—un til th e tim e w hen it suffered its baroque dis­

figurem ents in Rosenkreuz’s Rota M undi w hich was believed to contain everything conceivable by science.

However, it happened to be Leibniz who, still in school, confounded his teachers by the assertion that, alongside of the scholastic table of

predicam ents by m eans of w hich concepts can be composed into tru e judgm ents, th ere m ust also exist a corresponding table of predica- m ental judgm ents by means of w hich judgm ents can be composed into tru e conclusions—so th a t th ere m ust be in existence a thought alphabet of predicam ents, by the use of w hich all kinds of tru th s can be ex­ pressed.

Fig. 2. The original of Leibniz’s calculating m achine is held By the Low er Saxonian Provincial Library at H annover (phot. L. v. M ackensen, 1966)

In youthful Leibniz this concept, p u t fo rth by Lullus, flourished w ith a significant root: to wit, as the concept of a logical automatism, w hich would exceed the previous range of definitions and advance algorithm ically to new stages. From th e tim e of A l-K hw arizm l’s 9th centu ry textbook on algebra, the a rt of calculating is called algorithm , because the L atin tran slato rs m uddled th e incipit of th e Arabic name in D ixit Algorizm i ... However, by several nom inalists this term su f­ fered in the 14th century a characteristic variation: Nicolas d’Oresme’s book Algorism us proportionum already introduced fractional exponents of roots, so th a t this abstract form ulation of root extraction is older than the radical sign of the Cossists. Even so, d ’Oresm e’s form ulation is derived from a typically algorithm ic process: the potentials which originally had been defined exclusively for integer exponents, now

Le ib n iz’s Interpretation of L u llu s ’ A rt 1 7 5

became new inverse values, th a t is, radicands, in consequence of a m erely form al transition from num bers to fractions.

The technique of an algorithm ic creation of new m athem atical form s developed in Leibniz to a high degree of artistry ; because th e Ars

Lulliana is m eant to be less a logistic A rs dem onstrandi th an ra th e r

a scientific A rs inveniendi. It should be rem em bered, th a t in 1637 D escartes had by his M ethode nouvelle subjected geom etry to algebraic calculation; in 1670 Leibniz, who la te r w as to create th e notion of “function”, already dem anded in his Physica nova—a w ork devoted m ore definitely to the fu tu re th an his baroque 1666 dissertation:—th a t G alilei’s dynam ics should be subjected to D escartes’ algebraic calcu­ lations. This la tte r scientist had steadily declined to understand G alilei’s dynamics; th e reason was, th a t th e notion of a variable transgresses th e b arrier of Clare et distincte of the C artesian finite coordinate

Fig. 3. Leibniz’s w atch as pictured in the Journal des Savan ts

system. On th e contrary, Leibniz dem anded a calculation th a t would also deal w ith variables as w ell as w ith finite values—a w ay of th in ­ king preposterous to antiquity.

However, Leibniz did not find his new solution w ithin th e theorem s of G alilei’s dynamics, w here th e problem of velocity is explained as th e tangential lim it position of th e secant betw een tw o points infinitely close to each other on th e curve of velocity, as a y e a r previously Newton had discovered by his fluxion theory. It so happened, th a t Leibniz’s intellect had to be sparked at th e point w hen th e form ally logical predom inance of a constant relation over th e infinitely disap­ pearing relationed subjects became more clearly visible th an in the problem of tangents. Leibniz arrived a t his success in 1673 w hen he w as shown, by Huyghens, th e w ritings left by Pascal: draw ing a t the tan g en t of a circle from tw o adjoining points the parallels to th e axes of the circle, the small triangle created a t th e tangent m u st always

equal the larger triangle form ed by the radius, the ordinate and the subnorm al. Even so, Leibniz perceived som ething “que l’au teu r Pascal n ’avait pas v u ” : th a t this axiom is valid for all curves, not only for the circle, and th a t from point to point of th e curve th is triangle changes, so th a t it is possible from these changes to define th e shape of the curve; and that, for this reason, this trian g le is justly called the T riangulum characteristicum.

Because therefore th e tangential triangle, be its sides as small as may be, always resem bles the large triangle—this does n o t apply to G alilei’s tangents, though—Leibniz, the champion of Ars combinatoria, takes the im portant step: in view of the logical principle of a relation being constant, he continues to w rite the lim it value of this constant relation even then as quotient, w hen th e logical m aterial subjects of this relation, i.e. th e sides of the triangle, tu rn into zero. And while, from th e Cartesian point of view, these values are no extensiva, thus

Fig. 4. The T rian gulum ch aracteristicu m m entioned by G alilei and Leibniz

w ithout meaning, Leibniz revives them as inextensive, in other words, intensive values on the basis of the relative trend of his thinking. Thus, for example, to him G alilei’s inextensive y et intensive indivisible

ds

of velocity is, as — a tru e quotient of the infinitely sm all distance ds to the infinitely small tim e dt. As soon as the two injinitesim alia, the logical m aterial of this relation, tu rn into zero, in the scholastic unity

Le ib n iz ’s Interpretation of L u llu s ’ A r t 1 7 7

of Forma and Materia (i. e., subjectum ) the form becomes infinitely dense, creating in this m anner a new subject: the derivative.

More than Newton, Leibniz adhered to th e C artesian-L ullian tra d i­ tion, b u t less th an N ew ton to the trad itio n of th e N om inalist school of Oxford and Paris; in la te m edieval times these schools im plicitly already m ade use of the concept of a function in th eir graphical presen­ tation of n a tu re ’s law—a procedure earlier anticipated by Guido d ’A rez- zo w ith his scale of tones. There exist several “codices” dated from th e 12th and 14th centuries, w here already the ecliptical latitu d es of planets were shown graphically as functions of lengths, th a t is, of time. It does not seem by haphazard, th a t th is probably oldest relation, known already, though em pirically only, to th e Babylonians—because the G reeks did not attem p t to express this relation by a v arie ty of geom etrical models—became th e basis for graphically indicating n atu ra l processes during the Middle Ages; this is evident from th e fact, th a t in Mensurae form arum the la te r C artesian ordinates w ere called Lati-

tudines and th e la te r C artesian abscissae Longitudines. Leibniz chose

to define his concept of functions w ithin the system of A ristotle’s scholastics. And it is only by th e application of the L ullian algorithm to th e concept of functions, th a t th e c e n tu ry ’s yearning for th e Scienza

Nuova was fulfilled. Leibniz asserted: “Functio est continuatio omnium

variationum form arum .” In th e late Scholastic sense of the word, Galilei called velocity Forma motus. Therefore, by continuity in time of th e change of all form s of velocity, Leibniz solved th e process des­ cribed by the function. In this m anner physics—a special case of m eta­ physics—denotes th e distance travelled as the product of process:

s = J udt =

However, for Leibniz the differential quotient is a tru e quotient—this is th e proper M etaphysique du calcul infinitesim al—so th a t dt can be cut out and thus the logical, and th e more so, th e arithm etic identity

s = J'ds results.

This logically form al id entity was ap t to become the source of the whole m aterial essence of physics. Because now th ere w ere only to be established th e differential laws—a m a tter to be assigned to th e physi­ cists; afterw ards it w as th e m athem atician’s task to solve them. And should he be unable to do the integrating operations, com puting m achi­ nes w ould atten d to this w ith any desired accuracy. And this indeed happened to be fundam ental task of N ew ton’s physics: in practical use, N ew ton’s calculus of fluxions operated by th e developm ent of func­ tions into infinite sequences. H ere came to light the fundam ental d if­ ference in the invention of the calculus betw een Newton and Leibniz. The notorious controversy on p riority between these two scientists was

essentially a contest as to the correct notation of th e infinitesim al calculus. Both of them combined the attitu d es adopted by Descartes and by Leibniz, and this synthesis was bound to result autom atically in th e infinitesim al calculus, because ju s t as much as Descartes re­ pudiated G alilei’s continuous variables, Galilei—being a Rinascimento Italian—shrinked from an algebraization of geometry, which already had been condemned by Plato. And yet, w hile Newton accomplished this synthesis as a physicist, Leibniz did it as the algorithm ian. In the la tte r ’s work the fundam ental theorem of inverse calculus achieved its full expression in its form al perfection: if a function is to represent the “continuatio variationum functionis f = Jd f,” this should operati­ vely be w ritten:

1 = fd .

By introducing this relation Leibniz has com pletely shattered the fram ew ork of antique m athem atics; because now one can w rite d~ 1 = / and algorithm ically develop the full sequence

... d—3, d - 2, d“ 1, d°, d, d2, d3, ...

This sequence of higher differentials and integrals impressed the C artesians m erely as a m etaphysical m onstrosity: how to im agine the existence of an infinite sequence of values, all m utually infinitely large, or infinitely small?

Even more so: w hile the Newtonians discovered in th e problem of interpolation the Taylor series as an approxim ate polynomial of higher order, Leibniz deduced the Taylor series from his / = d—1 formula, by using this form ula for the n -th product-differentiation dn (uo) =

= (du + do)n.

A dm ittedly the n -th differential of a product can be w ritten symbolic­ ally as th e binom ial developm ent for th e sum of the first differential. Therefore, one can consider J f d x to be the — 1 derivative of the product of f and d r—m eaning th a t (df + d2x )—1 m ust be symbolically developed into an infinite binomial sequence. This then was indeed, as Johann B em ouilli enthusiastically claimed, th e “series universalis- sima, quae omnes integrationes exprim it,” th a t is: J / dx. To be sure, here the coefficients, u n d er developm ent tow ards higher derivations, w ere still functions; however—as pointed out by Pringsheim —this Leibniz-Bernouilli sequence can be transform ed into the Taylor series. The developm ent of the Taylor series m aterially com pleted the full scope of the theory of functions, because it also em braced the complex and, from a historical point of view, even became the foundation of the universal theory of functions. By th is achievem ent Leibniz the algorithm ian attained his suprem e trium ph: w ith a m inim um of m a­ them atical form he expressed a m axim um of m athem atical contents.

Le ibn iz’s Interpretation of L u llu s ’ A rt _{1 7 9}

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Fig. 5. Leibniz’s letter to J. I. Bernoulli (dated Dec. 16, 1694) containing the “Taylor series”

I t is characteristic, th a t u ntil into the 19th century Leibniz was looked upon as a plagiarist of Newton, reputed—at best—to have m erely invented a m ore suitable form of notation. Due to this, Leibniz’s prestige had greatly suffered among the professional philosophers. However,.

how w ere these to understand, th a t Leibniz was one of the exceptionally few who, like P ythagoras and Plato, succeeded in inventing a whole esoteric philosophy in order to describe th eir m athem atics for which, in th e ir respective time, the vocabulary, later on in common use, was still lacking. H ere it seems significant th a t th e only man, who made a stand against this official secular condem nation voiced by academies and universities, w as the chief ideologist of the Esprit positif de l’Êcole

Polytechnique in Paris. In his Cours de Philosophie positive A uguste

Comte asserts: “Des exem ples de n a tu re aussi divers sont p lu s que suffisants pour faire nettem en t com prendre en général l’immense portée de la conception fondam entale de l’analyse transcendante, telle que Leibniz l’a form ée (and here w e feel tem pted to add: “et que l’École Bâloise l’a développée”), et qui constitue sans aucune doute la plus hau te pensée à laquelle l’esprit hum ain se soit jam ais élevé ju squ ’à p résen t.”