Here is another little chapterette from my book. It follows a chapter based on discussions that followed a post by David Corfield from n-Category Cafe. There, the following thought was raised: Is there something analogous to Chomsy’s theory of language’s structure and language acquisition when it comes to mathematics. One interesting aspects is trying to understand “dyscalculia” which is a term describing children’s learning disabilities in mathematics.
I remember from my youth a book in Hebrew called “Logic, language and method” by Yehoshua Bar-Hillel. Yehoshua Bar-Hellel was a philosopher at the Hebrew University of Jerusalem and among other things wrote together with Micha A. Perles and Eli Shamir some basic papers in the study of automata. Ten years ago I collaborated with his daughter Maya Bar-Hillel (who is a Psychology professor at HU) in studying the “bible code“.
One fact I remember from Bar-Hillel’s book (used there to explain some basic notion of transformations) is the difference between English and Hebrew regarding anti missile missile. In English you say “anti missile missile” and “anti anti missile missile missile” and “anti anti anti missile missile missile missle” while in Hebrew it will be “missile anti missile” “missile anti missile anti missile”, “missile anti missile anti missile anti missile” etc.
Apropos differences between languages, Rodica Simion (see her poem “Immigrant complex,”) once told me that in English and most other languages she knew, you say “more or less” but in Hebrew you say “less or more”. (When I asked around I was told that Greek is like Hebrew; anyway, i will be happy to learn how this saying goes in languages you know.)
These three paragraphs on Chomskian’s linguistics represent a subject where my knowledge is “second hand.” I wonder if it shows.
The Chomskian revolution in linguistics
The Chomskian revolution in linguistics is comprised of three elements. The first is finding common structures and formulating common rules that apply to all human languages (to a much greater extent than before). The second is relating linguistics to studying and making hypotheses about the way children acquire languages. And the third is studying mathematically very abstract forms of languages. Chomsky’s theory of generative grammar is important in all three aspects.
Chomsky’s perception and demonstration of the unifying concepts behind different languages have impacted the way languages are perceived by linguists and by philosophers, and dramatically changed the way linguistics is practiced. Chomsky saw a direct link between the way children acquire language and the internal structure and logic of languages. His works in this direction are regarded as part of the cognitive revolution in psychology. While emphasizing the universal rules behind different grammars, Chomsky also made a strong point regarding the uniqueness of the cognitive aspects of language as compared with other cognitive abilities. He had a famous debate with psychologist Jean Piaget on this subject. Chomsky’s mathematical works on formal languages and the related concept of “automaton” are now fundamental in theoretical computer science.
Chomsky is criticized for being too dominant in the area of linguistics and for leading to unmotivated sharp turns in his own theory. The decline of individual language studies is regarded by some as a negative side-effect of the Chomskian revolution. Others argue that without a major additional statistical ingredient, formal mathematical structures á la Chomsky’s generative grammar and “transformation rules” are insufficient for understanding the structure and acquisition of languages.
A video about dyscalculia