Extended Boundaries for Mathematical Language?March 9, 2001
Philip J. Davis
Language Death. By David Crystal, Cambridge University Press, Cambridge, 2000, x + 198 pages, $19.95.
Occasionally a book crosses my desk that, although itself having little to do with mathematics, suggests the possibility of a mathematical analogy. This is the case with David Crystal's book.
The substance of the book is easily described. Something between 5000 and 7000 languages are currently spoken on earth. In some parts of the world---including, not so long ago, a certain Danish island---contiguity is no assurance of mutual comprehensibility. Many of these languages are fast disappearing, and it has been estimated that half of them will have disappeared by the end of the 21st century. Half of the world now speaks one of the top twenty languages.
Describing in some detail a number of the languages that are in extremis, Crystal, who is a student of linguistics and a prolific author, details the various reasons for the death of a language, what is lost when one of them dies, and what, if anything, can be done to preserve them.
Natural catastrophes, diminishing populations, outside exploitation of indigenous raw materials, cultural hegemony and subsequent assimilation, economics, wars, politics, are some of the many reasons for the death of a language.
What is lost when a language dies? A language encapsulates a culture and its history, an identity, a particular way of looking at the world. A language finds unique structural ways of saying things, e.g., verb tenses, and its varieties consequently fascinate the abstract-minded.
But languages are also born, and some have been revived. Consider how many individual languages Latin spawned even as it decayed. Crystal suggests new possibilities:
"Although at present, Singaporean, Ghanian, Caribbean, . . . continue to be seen as varieties of English, it is certainly possible for local sociopolitical movements to emerge to upgrade them to language status."
Under the influence of nationalisms and war, Serbo-Croatian (which fifty years ago was promoted as one language) is even now splitting into three languages.
There are numerous books on the death of languages, and there is a great deal of related literature. There are foundations for the preservation of endangered languages, and there are societies for the preservation of individual ones. In contrast to this concern, there is also a considerable degree of indifference to language death on the part of both the general public and the professionals. One of my university colleagues in linguistics told me point blank that the death of languages is not a mainstream concern and, using terms that I didn't understand, indicated where the real action is: the abstract structure of languages.
Is mathematics a language? Well, it is certainly not a "natural language" like Swahili or Gaelic, in which we can communicate a wide variety of everyday things. On the other hand, mathematics does have certain language-like features, and it is profitable to explore several of them. Mathematics is written down (for the most part linearly) in symbols, and its sentences, symbols and all, can be read aloud. Years ago, when I was writing papers with Joseph L. Walsh, he gave me the following advice: When you write a technical paper, check that your mathematical statements are embedded properly within complete English sentences. And use appropriate punctuation marks to ensure this.
From this point of view, mathematics is a subset of English, German, or any other natural language in which its specialized symbols are embedded. This means that readers, whatever their familiarity with mathematical symbols, remain mathematical illiterates unless they know the carrier language. To alleviate this problem for anglophones, we have available specialized mathematical dictionaries from a number of languages into English.
Does mathematical language die? Well, parts of it certainly do. I have on my shelf a standard American elementary school arithmetic book, dated 1874, by one Daniel Fish. It devotes six pages to alligation. Reader: See if you can find the definition of "alligation" in the dictionary you have at hand. And do you know what the arithmetic rules of simple fellowship are? If not, consult Carleton's Practical Arithmetic (Boston, 1810). And this is grammar school stuff. We may wonder then, without prejudice, how long the topics deemed essential in the latest suggested national curriculum will maintain their importance.
A more significant example: If you read James Gregory: Tercentenary Memorial Volume (edited by H.W. Turnbull, 1939), you will find that it is not easy to interpret what Gregory (1638-75), a much underrated Scottish mathematician of the early days of calculus, has written. Turnbull has done a marvelous and painstaking job of working from primary manuscript material-figuring out the author's handwriting; translating from a mixture of Latin and English; making sense of long-departed mathematical terms, such as "chiliad," "cycloform," "adfacted"; interpreting different mathematical symbols. We learn what things were then deemed interesting, what methods were employed, what difficulties were seen, what connections were made to people and to the cosmos. In supplying translations and copious interpretations, in supplying historical notes and references to the 17th-century scene and to "what came later," Turnbull has reconstructed for us a departed mathematical culture and sited it with respect to his own.
Except for people with historical or antiquarian interests, there seems to be little regret or interest in the decay of mathematics that occurs even as mathematics grows. One of the cliches of our business is that mathematics is cumulative, that the mathematical future always incorporates and greatly improves on its past. But this is true only because the present ascribes significance to precisely those parts of the past that are consonant with what the present now considers significant. The universal time-invariant comprehensibility and validity of mathematics are among the great myths of the subject.
The language of mathematics changes within an individual's lifetime. New concepts are added by the score, each with its myriad of special terms and interrelations. In its greening (I dare not say rapid overdevelopment), mathematics has now split so badly that I don't know what colleagues down the hall are talking about when they speak professionally. In this experience, we have a refutation of the vaunted and mythic "unity of mathematics" as a universal language. This unity was already threatened in Poincaré's day by the sheer quantity of the material available. The riches of mathematics, without contemplative judgments, would, Poincaré wrote, "soon become an encumbrance and their increase will produce an accumulation as incomprehensible as all the unknown truths are to those who are ignorant." The unity is now further threatened by self-contained, self-publishing chat groups.
While natural languages are popping off at a rate that alarms those who care, the same is true of computer languages and systems. Where are PL/1, Algol 60, Algol 68, Pascal, Snobol, and APL? All moribund or dead as a doornail. And when conversion software is abandoned, ten years later it will be extremely difficult, if not impossible, to find programs that can make sense of the legion of documents that have been produced and stored.
Now for the upbeat news: There is growth, and not just along traditional lines. Natural language semioticist J.L. Lemke has made the point that the word "literacy" must now be expanded from its narrow, logocentric interpretation to mean
"a set of social practices which link people, media objects, and strategies for meaning making. . . . All literacy has [always] been multi-media literacy: You can never make meaning with language alone."
The non-logocentric aspects of literacy include pictures, diagrams, icons, animations, graphs, tables, statistical charts, maps, models, multimedia and virtual assemblages, together with the construction, manipulation, transmission, integration, interpretation, and validation of these objects.
By way of parallel, I suspect that over the next twenty-five years, mathematics will lose some of its logocentrism. Of necessity, this will result in an enlargement of the notion of what mathematics is and of what it means to be mathematically literate. Even at elementary levels, mere numeracy will not suffice. The enlargement must in some manner incorporate the non-logocentric aspects of general literacy just mentioned. There are a variety of straws in the wind.
In 1974, advocating a concept I called "visual theorems," I urged that the classical notions of what constitutes a mathematical theorem or truth should be enlarged to include a variety of objects that are systematically generated, perceived by the senses, and interpreted by the brain (Proceedings of Symposia in Applied Mathematics, Vol. XX, pages 113-127). This was in contrast to the boast of the Ecole Bourbaki that, unlike earlier mathematical works, its volumes did not include a single diagram.
My colleague David Mumford anticipates a great increase in the invention of new mathematical structures. But a brand new mathematical structure wouldn't necessarily have to be defined axiomatically à la set theory, as is the case for a group or a ring. The new structure might, for example, be a picture or a manner of integrating virtual objects within a multi-media mode.
Keith Devlin's popular book Goodbye Descartes describes the difficulties of the relationship between natural language, logic, and rationality. These difficulties, Devlin asserts, cannot be overcome by traditional mathematics of the Cartesian variety, and he hopes for the development of a "soft mathematics"---not yet in existence---that "will involve a mixture of mathematical reasoning, and the less mathematically formal kinds of reasoning used in the social sciences." He adds that most of today's mathematicians would find it hard to accept soft mathematics as mathematics at all.
Linguists, though hard put to find a satisfactory definition of a language and of literacy, nonetheless see the need for boundary expansion. The late Gian-Carlo Rota came to a similar conclusion through his phenomenological (Husserl, Heidegger) orientation. After listing seven phenomenological properties that he believed to be shared by mathematics, Rota went on to say:
"Is it true that mathematics is at present the only existing discipline that meets these requirements? Is it not conceivable that some day, other new, altogether different theoretical sciences might come into being that will share the same properties while being distinct from mathematics?"
Rota shared Husserl's belief that a new Galilean revolution will come about to create an alternative, soft mathematics, which will adopt idealizations that run counter to "common sense."
From mathematician-semioticist Brian Rotman we have:
"By giving mathematicians access to results they would never have achieved on their own, computers call into question the idea of a transcendental mathematical realm. . . . If that is the case, mathematicians will have to change the way they think about what they do. They will have to change the way they justify it, formulate it and do it."
The topic of mathematics as a language has been treated in recent years mainly by semioticists. Kay O'Halloran, for example, looks at mathematical discourse as multisemiotic---that is, as a discourse that utilizes three semiotic resources: language, mathematical symbolism, and visual display. I feel certain that the actual practice of mathematicians will shortly provide language theorists with much more than that to chew on.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode
Island, and can be reached at email@example.com.