Math as a Natural Language

Mathematics (with a capital "M") is typically considered a formal language. While I would agree that it tends to be more formal than, say, English, I often wonder whether it's truly formal (finding a good definition of formal would be useful, but is apparently somewhat difficult). In other words, are there properties that the usual natural languages have that math does not. I would say there are only a few, and they're mostly unimportant. Secondly, note that I bolded the "a" above. This is because I also feel that Mathematics is more like a collection of separate languages (or, dialects if you prefer since none of them has an army -- except perhaps category theory) that a single language.

First regarding the formality. Typically when I think of formal languages, I think of something lacking ambiguity. This is certainly the case for the sort of math that one would type into matlab or a theorem prover or mathematica. But what one types in here is what I would call "formal math" precisely because it is not the language that mathematicians actually speak. This is perhaps one reason why these tools are not more widely used: translating between the math that we speak and the math that these tools speak is somewhat non-trivial. The ambiguity issue is perhaps the most forceful argument that math is not completely formal: operators get overloaded, subscripts and conditionings get dropped, whole subexpressions get elided (typically with a "..."), etc. And this is not just an issue of venue: it happens in formal math journals as well.

It is often also the case that even what gets publishes is not really the mathematics that people speak. Back when I actually interacted with real mathematicians, there was inevitably this delay for "formally" writing up results, essentially breaking apart developed shorthand and presenting things cleanly. But the mathematics that appears in math papers is not really in its natural form -- at least, it's often not the language that mathematicians actually work with.

To enumerate a few points: Math...

• has a recursive structure (obviously)
• is ambiguous
• is self-referential ("see Eq 5" or "the LHS" or simply by defining something and giving it a name)
• has an alphabet and rules for combining symbols

To some degree, it even has a phonology. One of the most painful classes I ever took (the only class I ever dropped) was an "intro to logic for grad CS students who don't know math" class. This class pained me because the pedagogical tool used was for the professor to hand out typed notes and have the students go around and each read a definition/lemma/theorem. After twenty minutes of "alpha superscript tee subscript open parenthesis eye plus one close parethesis" I was ready to kill myself. It has no morphology that I can think of, but neither really does Chinese.

Moving on to the "a" part, I propose a challenge. Go find a statistician (maybe you are one!). Have him/her interpret the following expression: . Next, go find a logician and have him/her interpret . For people in the respective fields, these expressions have a very obvious meaning (I'm guessing that most of the readers here know what the second is). I'm sure that if I had enough background to drag up examples from other fields, we could continue to perplex people. In fact, even though about 8 years ago I was intimately familiar with the first sort of expression, I actually had to look it up to ensure that I got the form right (and to ensure that I didn't make a false statement). This reminded me somewhat of having to look up even very simple words and expressions in Japanese long after having used it (somewhat) regularly. I think that the diversity of meaning of symbols and subexpressions is a large part of the reason why most computer algebra systems handle only a subset of possible fields (some do algebra, some calculus, some logic, etc.). I believe in my heart that it would be virtually impossible to pin down a single grammar (much less a semantics!) for all of math.

So what does this have to do with an NLP blog? Well, IMO Math is a natural language, at least in all the ways that matter. So why don't we study it more? In particular, when I download a paper, what I typically do is first read the abstract, then flip to the back to skim to results, and then hunt for the "main" equation that will explain to me what they're doing. For me, at least, this is much faster than trying to read all of the text, provided that I can somewhat parse the expression (which is only a problem when people define too much notation). So much information, even in ACL-style papers, but more-so in ICML/NIPS-style papers, is contained in the math. I think we should try to exploit it.