tag:blogger.com,1999:blog-264226589944705290.post2593247283079826729..comments2023-06-03T04:58:32.301-07:00Comments on God Plays Dice: Mathematics ≠ notation Mathematics is not notationMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-264226589944705290.post-60817973819609646632008-12-25T15:51:00.000-08:002008-12-25T15:51:00.000-08:00So for Riemann, the symbol = meant "equal to" with...<I>So for Riemann, the symbol = meant "equal to" without the verb, and he wrote the equivalent of "A is = B" as a text sentence.</I><BR/><BR/>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).<BR/><BR/><I>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><BR/><BR/>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.<BR/><BR/>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).<BR/><BR/>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.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-83690870174524552642008-12-11T21:22:00.000-08:002008-12-11T21:22:00.000-08:00I was always struck by Riemann's grammatical usage...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<BR/><BR/>"Das Integral um den Werth n2pii aber ist = (-n2pii)^s-1(-2pii), man erhaelt daher..."<BR/>and<BR/>"Die Anzahl der Wurzeln von xi(t) = 0, deren reeller Theil zwischen 0 und T liegt, ist etwa <BR/>= (T/2pi)log(T/2pi)-T/2pi; ..."<BR/><BR/>So for Riemann, the symbol = meant "equal to" without the verb, and he wrote the equivalent of "A is = B" as a text sentence.rokirovkahttps://www.blogger.com/profile/09819224673158288550noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-48024342679940879562007-11-02T06:04:00.000-07:002007-11-02T06:04:00.000-07:00I've faced this issue a lot while doing editorial ...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."Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-33852764423089869332007-10-22T17:38:00.000-07:002007-10-22T17:38:00.000-07:00Concerning the main issue, I beg to differ. While ...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:<BR/><BR/>Since x = y, it follows that ...<BR/><BR/>or similar constructions, as the sentence reads perfectly well for various sensible replacements (equals, is equal to, ...) for the = sign.Michael Alberthttps://www.blogger.com/profile/06406323435945694109noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-56196122099810245412007-10-22T11:28:00.000-07:002007-10-22T11:28:00.000-07:00Aaron,I pronounce g(f(x)) as "g of f of x", and (g...Aaron,<BR/><BR/>I pronounce g(f(x)) as "g of f of x", and (g o f)(x) is <I>probably</I> "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.<BR/><BR/>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 Lugohttps://www.blogger.com/profile/15671307315028242949noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-59041924599529682132007-10-22T11:23:00.000-07:002007-10-22T11:23:00.000-07:00Most people I know pronouce it "g of f."Incidental...Most people I know pronouce it "g of f."<BR/><BR/>Incidentally, I find myself wishing maybe 100 times a week that it were acceptable to write things like, "x not necessarily = y." :PAaronhttps://www.blogger.com/profile/18281785407407667986noreply@blogger.com