Mathematics as a Natural Language

6 06 2007

My friend Luke Biewald pointed me to an interesting post suggesting that the language of mathematics is actually best viewed as a natural language rather than a formal language. I think some of the points the author makes about math don’t separate natural and formal languages (recursion, self-reference, alphabet and rules for combination, and so on). And I think that since he works in natural language processing, he may be thinking of being natural as a sort of simplicity rather than complexity (as I would think of it). But it’s an interesting point.

Mathematicians basically never write fully formal proofs in the sense that logicians like to talk about. They regularly “abuse notation” and overload symbols in order to simplify their way of speaking. Many of these changes are in fact quite historically contingent – if we hadn’t originally started abbreviating things one way, further developments that way would have looked incomprehensible to a community that had abbreviated things differently.

Given the fact that it is actually used by a relatively large community for certain essential (to those people) types of communication, it has most likely picked up a lot of the irregularities and “irrationalities” that plague natural languages – probably much more so than constructed languages like Esperanto and Klingon. I don’t know what all the relevant differences are between “real” natural languages, Klingon, and math, but they may help reveal something interesting about at least one of these languages.




One response

14 06 2007

I think you are right that the list of things the author gave do not separate natural and formal languages. The post seems a little unclear on an important point. From the post, his point seems to be that mathematics tends to be done in natural language supplemented with some more formal symbols. Some of the symbols have a conventional use and meaning, and if one doesn’t use them one has to look them up to make sure one gets them right (just like the Japanese phrases one used to know but doesn’t any more). Some of them even have standard pronunciations. The important point is how this makes math a natural language as opposed to what Vendler (I believe) would call a scientific sublanguage, a fragment of the larger language with a more specialized vocabulary and maybe even additional syntactic forms that are allowed cause, while otherwise awkward, they have been used in the smaller community. Since I like a math background, I might be missing something important in his post. It seems like if math is in any sense a natural language, then it would be a dialect or sublanguage of a real natural language. Or, why even view it as a language at all? Didn’t Brouwer say math wasn’t a matter of language but of mental construction? (Just throwing that out there.)

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