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From: Stephen Wolfram, A New Kind of Science
Notes for Chapter 10: Processes of Perception and Analysis
Section: Human Thinking
Page 1103

Languages. There are about 140 human languages and 15 full-fledged computer languages currently in use by a million people or more. Human languages typically have perhaps 50,000 reasonably common words; computer languages usually have a few hundred at most ( Mathematica, however, has at least nominally somewhat over 1000). In expressing general human issues, different human languages tend to be largely equivalent - though they often differ when it comes to matters of special cultural or environmental interest to their users. Computer languages are also mostly equivalent in their handling of general programming issues - and indeed among widespread languages the only substantial exception is Mathematica, which supports symbolic, functional and pattern-based as well as procedural programming. Human languages have mostly evolved quite haphazardly over the course of many centuries, becoming some× simpler, some× more complicated. Computer languages are almost always specifically designed once and for all, usually by a single person. New human languages have some× been developed - a notable example being Esperanto in the 1890s - but for reasons largely of political history none have in practice become widely used.

Human languages always seem to have fairly definite rules for what is grammatically correct. And in a first approximation these rules can usually be thought of as specifying that every sentence must be constructed from various independent nested phrases, much as in a context-free grammar (see above). But in any given language there are always many exceptions, and in the end it has proved essentially impossible to identify specific detailed features - beyond for example the existence of nouns and verbs - that are convincingly universal across more than just languages with clear historical connections (such as the Indo-European ones). (One obvious general deviation from the context-free model is that in practice subordinate clauses can never be nested too deep if a sentence is expected to be understood.)

All the computer languages that are in widespread use today are based quite explicitly on context-free grammars. And even though the original motivation for this was typically ease of specification or implementation, I strongly suspect that it has also been critical in making it readily possible for people to learn such languages. For in my observation, exceptions to the context-free model are often what confuse users of computer languages the most - even when those users have never been exposed to computer languages before. And indeed the same seems to be true for traditional mathematical notation, where occasional deviations from the context-free model in fields like logic seem to make material particularly hard to read. (A notable feature that I was surprised to discover in designing Mathematica 3 is that users of mathematical notation seem to have a remarkably universal view of the precedence of different mathematical operators.)

The idea of describing languages by grammars dates back to antiquity (see page 875). And starting in the 1800s extensive studies were made of the comparative grammars of different languages. But the notion that grammars could be thought of like programs for generating languages did not emerge with clarity until the work of Noam Chomsky beginning in 1956. And following this, there were for a while many efforts to formulate precise models for human languages, and to relate these to properties of the brain. But by the 1980s it became clear - notably through the failure of attempts to automate natural language understanding and translation - that language cannot in most cases (with the possible exception of grammar-checking software) meaningfully be isolated from other aspects of human thinking.

Computer languages emerged in the early 1950s as higher-level alternatives to programming directly in machine code. FORTRAN was developed in 1954 with a syntax intended as a simple idealization of mathematical notation. And in 1958, as part of the ALGOL project, John Backus used the idea of production systems from mathematical logic (see page 1156) to set up a recursive specification equivalent to a context-free grammar. A few deviations from this approach were tried - notably in LISP and APL - but by the 1970s, following the development of automated compiler generators such as yacc, so-called Backus-Naur context-free specifications for computer languages had become quite standard. (A practical enhancement to this was the introduction of two-dimensional grammar in Mathematica 3 in 1996.)

Stephen Wolfram, A New Kind of Science (Wolfram Media, 2002), page 1103.
© 2002, Stephen Wolfram, LLC