Report: The Generosity of Artificial Languages in an Asian Perspective

Amsterdam, 18-20 May 2006

Frits Staal

The workshop on "The Generosity of Artificial Languages in an Asian Perspective," was organized by Frits Staal (Berkeley/Chiang Mai) with the assistance of Wim Stokhof (IIAS), Robert van Benthem, Robert Dijkgraaf and Martin Stokhof (Amsterdam). The workshop was held from 18-20 May 2006.  The preparation, organization and logistics of the event were in the able hands of Marloes Rozing of the IIAS. The meeting took place in Amsterdam and was the second in the series Asian Contributions to the Formation of Modern Science. The Proceedings of its predecessor have been published in the Journal of Indian Philosophy, Vol. 34 (2006). Both meetings rejected the common but outdated picture that ‘science' is ‘Western,' having originated in ancient Greece with the Arabs acting as translators. Both discussed Pre-Modern, that is, Ancient and Medieval Science as expressed through the medium of classical languages such as Old-Babylonian, Chinese, Sanskrit, Greek, Arabic, and Latin. The second workshop included modern science which is largely expressed through formal or artificial languages.

An illustration of our topics of discussion is the discovery by Madhava of Kerala, Southwest India, who lived around 1400 CE, of infinite power series that are expansions of pi and the trigonometric functions sine, etc. This took place almost three centuries before these same series were re-discovered in Europe by Newton and Leibniz. In Europe, that event led to the infinitesimal calculus, which could not have been expressed without the help of an artificial mathematical language. In India, that revolution in language did not take place: Madhava and his followers continued to write in Sanskrit or Malayalam, the Dravidian language of Kerala. The accompanying illustration depicts, on top, the infinite power series that expresses the circumference of a circle with diameter D  (i.e., two times the radius R) in Sanskrit, followed by a translation into English by A. K. Bag. At the bottom is the series in its modern form which is basically the same as what was written by Newton.

Infinite Series Expansion of the Circumference of a Circle with Diameter D

Newton's laws were not always written in an artificial form. He formulated in cumbrous Latin, later disambiguated and clarified by Euler, the law of motion that is now taught to children as   f = ma.

Absence of artificial notations and especially of the calculus go far towards explaining why modern science did not originate in India or China. Earlier forms of Asian mathematics inspired the algebra of the Arabs but to what extent was that an artificial language? India did develop a formal or artificial language but that was in linguistics and two millennia earlier. Our first workshop began to study these and other remarkable facts.

The second  quoted d'Alembert: "algebra is generous: she often gives more than is asked of her." It means that notations and equations achieve far more than that for which they were originally designed. An example from modern logic started in 1942 with J.C.C. McKinsey coming to study with Alfred Tarski on intuitionistic logic. But Tarski had at that time already seen, that the work would best be reformulated in algebraic terms, and so they wrote together a study that tied three topics together: "The Algebra of Topology." In the 1970's, the computer scientist Edgar F. Codd developed a method for dealing with relational data bases. Later it was shown that that was another notational variant. Such unexpected generosities explain that Dirac declared of his own equation: "it is smarter than I am." Stephen Hawking put it thus: "Even if there is only one possible unified theory, it is just a set of rules and equations."

The ten participants in our second workshop tried to study these questions within the wide historical perspective of the first. They did not make a principled distinction between the earlier sciences of the Eurasian continent and modern science, nor did they ignore what Joseph Needham had written: "to write the history of science we have to take modern science as our yardstick - that is the only thing we can do - but modern change will change, and the end is not yet." The result was a wonderful mix of disciplines, mutual surprises and unexpected recognitions, scintillating discussions and conversations that lasted deep into the night when the official meetings, interspersed by Dutch "broodjes," had given way to sumptuous suppers and spirits.

I cannot do justice to all the contributions, let alone the lively rulings of the chairs who included Kamaleswar Bhattacharya, Henk Barendregt, Dirk van Dalen, Fenrong Liu, Kim Plofker and Bram de Swaan. The meetings began with a discourse on "Generous - sometimes too generous?" by Jens Høyrup. His first example of over-generosity was the extension by a fourteenth century Italian mathematician of the rules that govern positive and negative algebraic powers and that include the less-than-intuitive "less times less makes plus". Another example was Cantor's unrestricted acceptance of sets as members of other sets. It became apparent in the course of our discussions, that such over-generosities correspond to over-generalizations in natural language. If we know the English plural trees we can make the plural blows. Children pick it up soon but may create too much as in mans or sheeps. Philosophers, European as well as Indian, have always done it - claiming, for example, that the world may be explained in terms of substances and qualities because sentences consist of subjects and predicates.

Jeffrey Oaks in "Medieval Algebra as an Artificial Language" presented a missing piece: a historical survey of algebra applicable to Arabic and European languages. Starting in the ninth century with systematic verbal solutions of equations, it reached a symbolic form in the twelfth century in the western part of the Islamic world. In the afternoon, Joachim Kurtz gave an account of the surprising adventures of European Syllogistics - medieval reformulations of Aristotelian logic - in Late Imperial China. Since it involved the introduction of some 800 unintelligible new terms, it relied on Kanji characters found in logic textbooks from Japan.

Brendan Gillons's "Panini's Ashtadhyayi and Linguistic Theory" started with the outline of a simple formal language from which he derived the structure of Panini's Sanskrit grammar by a series of steps. He then showed that Panini's problems concerning compositionality, implicit arguments and anaphoric dependence are at the core of current formal thinking in linguistic theory.

Frits Staal's "Creativity and Generosity in Language" started with an explanation for the surprisingly early development of an artificial meta-language for linguistics by pointing to Vedic ideas about a hierarchy of languages of which the lowest is our common spoken language. He wondered whether innate faculties of language and number can be dissociated from each other and features of civilization. Chinese and Latin written traditions (highlighted by Karine Chemla and Charles Burnett during

the first workshop) do not necessarily lead to greater clarity but always to greater separation than an oral tradition like the Vedic. Such facts suggest that artificial languages result from a fusion of the two faculties.

Roddam Narasimha's paper "Observe, Numerize, Theorize" started from the contrast between Greek theorizing, as in the derivation from axioms, and Babylonian and Indic numerizing, which starts from observation and computation. Newton had a foot in both camps, but the advent of the large-scale integrated circuit is beginning to cause another shift that may lead to new and different artificial languages. Narasimha referred to Brouwer's intuitionist critique of language, logic and axioms which appeared to exemplify the innate faculty of number. Chair van Dalen clarified the concerns of intuitionists and a lively discussion ensued.

John Kadvany's "Positional Notation and Linguistics Recursion" related the decimal positional number system of Indian mathematics to the structure of Sanskrit numerals and showed to what extent mathematical recursion is based on linguistic recursion. Martin Stokhof's "Hand or Hammer?" discussed ‘grammatical form' and ‘logical form' in early twentieth century Euro-American analytical philosophy. Adding linguistics and the philosophy of language, he wondered whether the distinction between natural and formal languages can be maintained.

On the final morning, P.P. Divakaran discussed "Natural and Artificial in the Malayalam text of the Yuktibhasa." This text of around 1560 CE contains no formulae, there are no diagrams, the language is subtle but devoid of abstractions. Divakaran took his audience step for step through a literal translation, leading up to a calculation of the surface area of a sphere. Though the detailed expressions seem with hindsight to point in the direction of the calculus, they stopped just short of that.

In the final talk, entitled: "Can the world be captured in an equation?" Robbert Dijkgraaf  discusses a variety of examples, some of them suggesting  that physics benefits from the generosity of mathematics, others (especially in the quantum and string theories) that they develop simultaneously, others again that reductionism plays a role or that a sense of playfulness or beauty is decisive.

This report is not long enough to explain the meanderings and delights of the general discussion that was led by Fenrong Liu and Bram de Swaan. Some of the basic issues were barely touched. Do our equations show how the universe hangs together, or do they prevent precisely that because our species with its innate faculties could not have been selected for any such lofty reason?

The meeting was not widely announced in Amsterdam and attended mainly by specialists and a few distinguished visitors. In the end, all participants declared that it had been one of the most eye-opening and enjoyable events they had attended. Writing later from the Great Hall of the People in Beijing, where he had just met the vice-premier responsible for Science and Technology together with Stephen Hawking and other physicists who had addressed an audience of 6,000 enthusiastic students, Robbert Dijkgraaf referred to our workshop as "the great little meeting in Amsterdam. It was a gem."

Thirty years ago, comparing China and India, Joseph Needham emphasized the riches of China's "organic philosophy of nature" which made up to some extent for its "mathematical and theoretical backwardness" (quoted in Concepts of Science, IIAS 1993, ‘94, page 21). Much more is now known, especially of Chinese mathematics, but if there still is any truth to any such evaluation, the Chinese Government is obviously set to change it -- because technology depends on science and/or for the sake of truth, generosity, "advancedness" (xianjinxing) or some other reason. It gives us something to ponder not only for the history of science but for the future.

Robbert Dijkgraaf, writing later, referred to the "little Amsterdam meeting" as "a gem."