Philosophical Sophistication as Mathematical Sophistication

21 06 2007

How much philosophical sophistication does someone need to count as being “mathematically sophisticated” enough to follow a graduate course in algebra?

When reading through a draft paper by Colin McLarty (addressing different issues entirely), I came upon the following passage from Serge Lang’s canonical text:
I think it illustrates a lot of issues that often arise in understanding mathematical writing.

In actuality, mathematicians almost never write the statement that Lang wrote, except in the same sort of definitional statement. In particular, in place of “f(x)” they would write some expression in terms of “x” that one might use, such as “x3+2x-1″ or the like. Because this expression is just a placeholder, we might expect some neutral term, like “t“. But instead he uses a term that gives the reader the idea of what the overall expression is supposed to mean, at the expense of some abuse of notation.

Another issue of use and mention at work here is what the term “x” to the left of the arrow is doing. He doesn’t say whether “x” is a placeholder for a term denoting a specific element of A, or whether it is a sort of meta-placeholder, representing a variable that itself takes values in A.

In practice, I believe that both options are allowed. By a minor abuse of notation, one can write either “Under function f, 3 \mapsto 9″, or “Under function f, x \mapsto x2“. (I’m using “\mapsto” to stand for the arrow used in Lang’s statement.) In particular, the latter type of statement derives from the former by the standard practice of ignoring certain types of use-mention distinction, and allowing variables to stand either for elements of A or the names of elements of A. This abuse is allowed just about everywhere except in some parts of model theory, where it’s important to distinguish objects and their names.

So getting back to my original point, I think that an ability to know when a term is being used or mentioned, and whether it’s standing for itself, an expression that is partly composed in the way that it’s composed (this might relate to Lang’s famous statement that “notation should be functorial over the language”), or something totally different is important. I suspect that a non-sophisticated math student (or a sophisticated philosopher) would read the statement Lang wrote and suspect that the arrow would never be useful, because we’d always have to specify in some other place what f(x) was (that is, what expression “f(x)” refers to).

One aspect of mathematical sophistication seems to rely being aware of these different levels on some subconscious level, so that you can always jump to the right one, even through multiple abuses of notation.


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.