22 October 2007

Mathematics ≠ notation Mathematics is not notation

A rule of mathematical style: it is acceptable to write "Therefore, foo is at most bar.", but it is not acceptable to write "Therefore, F ≤ B." (This is due to J. Michael Steele, who has a lot of good rants.)

Steele's logic is that mathematical symbols cannot function as the verb in a natural-language sentence, which makes sense; although mathematical expressions have their own internal logic, an entire expression seems to function as a single noun when embedded in an English-language expression. Thus we write something like "Therefore, we have F ≤ B." or "Therefore, the inequality F ≤ B holds." ifwe want to write things out in symbolic form.

One thing that bothers me about a lot of mathematicians (myself included) is the propensity for their spoken mathematics to be a mere symbol-by-symbol reading of the written mathematics; thus one hears things like, say, "pi one of X" instead of "the fundamental group of X". This isn't too much of a problem in that case, but then one gets to things like "H two of X" -- is that H2(X) (homology) or H2(X) (cohomology)? (People don't seem to go to the trouble of distinguishing superscripts from subscripts when they're talking, which is probably a good thing, lest speech be filled with "uppers" and "lowers".)

Sure, you can usually tell from context, if you're sufficiently experienced in a certain area. But people have a tendency to think that mathematics is the formulas, at least to the point where they speak the notations instead of the notions. (This is one of those fortunate puns that translates well from the original Latin: "non notationes, sed notiones".) Spoken mathematics is not written mathematics. Mathematics is not its notation.

This is probably a corollary of the often-stated fact that a page of mathematical text contains about as many ideas as, say, ten pages of "ordinary" text; thus on a per-page basis, mathematical reading should be ten times slower than literary reading. Unfortunately, a minute of mathematical speech contains many more ideas than a minute of "ordinary" text, but one cannot slow the speaker down.

Incidentally, I used to pronounce ≤ and ≥ as "less than or equal to" and "greater than or equal to"; now I pronounce them "at most" or "at least", because these are shorter. I also pronounce g o f as "g after f", which seems both shorter and less ambiguous than "g composed with f" -- in "g composed with f" the order of composition is conventional. "f before g" or "f, then g" is too confusing, because you read the g before you read the f.

6 comments:

Aaron said...

Most people I know pronouce it "g of f."

Incidentally, I find myself wishing maybe 100 times a week that it were acceptable to write things like, "x not necessarily = y." :P

Isabel said...

Aaron,

I pronounce g(f(x)) as "g of f of x", and (g o f)(x) is probably "g of f of x" in my dialect ("g after f of x" sounds awkward), but using "g of f" to refer to the function sounds wrong to me. I'm not saying it's bad, just that I wouldn't say it.

And I wouldn't mind "x not necessarily = y" on, say, a chalkboard during a lecture, or in very informal notes. But that sort of thing shouldn't creep into "formal" mathematical writing.

Michael Albert said...

Concerning the main issue, I beg to differ. While I believe that most mathematical papers overuse notation, particularly in line, I have no problem with:

Since x = y, it follows that ...

or similar constructions, as the sentence reads perfectly well for various sensible replacements (equals, is equal to, ...) for the = sign.

gowers said...

I've faced this issue a lot while doing editorial work on The Princeton Companion to Mathematics. I've had authors write things like, "Every prime >2 is odd," (not an actual example) and have of course changed ">" to "greater than". (That raises a second interesting point: the symbol ">" should be "is greater than" rather than "greater than".) I would also get rid of quantifiers from running text (this one a stylistic error made more often by students than Princeton Companion authors). But I don't feel nearly so strongly about a sentence like, "Since x=y, the lemma is proved." Putting "equals" there instead would look a bit strange, I think. I think my semi-conscious rule of thumb there is that it's OK because you can pick out an entire sentence "x=y" that's written in a consistent notation. I'd certainly object to, "Since x is the unique positive root of this equation, it =y."

rokirovka said...

I was always struck by Riemann's grammatical usage of the symbol = in his famous 1859 paper. I've always mentally translated = as "equals" or "is equal to", but Riemann instead writes things like

"Das Integral um den Werth n2pii aber ist = (-n2pii)^s-1(-2pii), man erhaelt daher..."
and
"Die Anzahl der Wurzeln von xi(t) = 0, deren reeller Theil zwischen 0 und T liegt, ist etwa
= (T/2pi)log(T/2pi)-T/2pi; ..."

So for Riemann, the symbol = meant "equal to" without the verb, and he wrote the equivalent of "A is = B" as a text sentence.

Anonymous said...

So for Riemann, the symbol = meant "equal to" without the verb, and he wrote the equivalent of "A is = B" as a text sentence.

When I was a kid I spent some time in Germany, and we read "=" as "gleich" ("equal" or "same"). As a verb it would be "ist gleich", but equations were typically read as "a gleich b" for "a=b". This seems to be compatible with Riemann's usage, although I've got no idea whether this has persisted over time (or indeed whether adult Germans say this).

Steele's logic is that mathematical symbols cannot function as the verb in a natural-language sentence, which makes sense; although mathematical expressions have their own internal logic, an entire expression seems to function as a single noun when embedded in an English-language expression.

I disagree. While Steele's idea is internally consistent, I don't think it captures how people do or should write mathematics. (Instead, I'd say that by default an equation doesn't function as a single noun. In the cases where it does, there are essentially implicit quotation marks.) It's an arbitrary prescriptive rule, which leads to clumsy writing in an attempt to avoid natural constructions.

My sympathies are with Gowers: you can put well-formed mathematical expressions into text and treat them as part of the sentence, but you can't mix the two haphazardly. I view a sentence like "Every prime > 2 is odd" the same way I'd view a displayed equation like "a + b + small error term = the answer". They are both perfectly clear and suitable for writing on the blackboard, but they aren't suitable for formal writing. This prejudice may not be any better justified than Steele's, but it seems to be far more widespread (while his seems to be a personal quirk).

My own quirk is that I don't like "we have". I use it sometimes to avoid worse problems; for example, it can be important to insert some words so that two unrelated mathematical expressions aren't separated by just a space or a comma. However, I find "we have" a little stilted, and I'd prefer to avoid it whenever feasible.