Talk:Abuse of notation

message: Sound power is measured in Watts and sound power level in dB. Also the text states that the A in dB(A) denotes a particular reference level, which is not correct. The A indicates A-weighting, a form of filter/frequency weighting. Both sound pressure level and sound power level can be weighted using A-weighting.
Mikael Ogren (talk) 17:24, 1 March 2011 (UTC)
thread_title: Sound power should be "sound pressure level"

message: I am not sure what you are trying to do with this page, in particular the opening comment is almost content-free.  I always use screwdrivers to open paint tins.  8-)  (and no, this is not vacuously true - I've been painting my house recently).  I would think a page on abuse of notation should also describe why it is a useful thing to do, as well as describing why the examples are actual abuses of notation.  (e.g. always insisting that functions and variables have distinct symbols leads to proliferation of symbols that only the most anal retentive mathematician (or painter) would delight in.) Andrew Kepert 01:45, 6 Apr 2005 (UTC)
thread_title: Screwdriver?

message: sink(x) doesn't simplify exposition, nor suggests any correct intuition.
It's just an arbitrary exception that creates ambiguity, so I propose we designate it a misuse of notation.
< rant >
: It's being widely taught to school kids, roughly together with teaching f-1 as function inverse, and creates unjustified confusion.  It's a shame!
: (To make it worse, f-1 is usually introduced without explaining the general fk notation for composition, nor it's origin in paren-less "f f x" function application notation, and the wide abuse of exponentiation nota).
: Presumed rationale for the exception:
: * Trig functions are among the first functions that are commonly written without parentheses.  This notation is commonly introduced without discussing order of operations w.r.t. exponentiation, making sin x2 ambiguous.
: * Repeated application of trig functions is nearly useless.  However, sin-1 is useful, limiting the exception to positive powers, which is ugly.
< /rant > 79.179.39.171 (talk) 22:24, 14 June 2009 (UTC)
thread_title:  Misuse rather than abuse? 

message: The present version would seem to make a better stub for this topic.  I don't think there is much of a controversial nature left, but there remains enough structure so that it's pretty clear what the intended topic is. --J. W. McLeod 12:48, 10 Apr 2005 (UTC)
thread_title:  Have retreated from language criticized above 

message: Who is John Harrison? --Abdull 08:29, 30 May 2006 (UTC)
thread_title:  John Harrison 

message: I don't think \lim_{x \to \infty}f(x) = \infty qualifies as abuse of notation.
If the domain and codomain under consideration are the extended real line, the limit may very well exist, and have the precise value of \infty, without any notational or conceptual difficulties whatsoever.
If the domain and codomain under consideration are \mathbb{R}, then, as described, the limit does not exist (edit: and neither does the infinity), and, therefore, the sentence is not false but meaningless when considered merely as the sum of its parts, so the idiom (essentially bringing the extended real line into a real context) gives meaning to an otherwise meaningless sentence, rather than giving an additional meaning to a meaningful one.
Dfeuer 04:36, 30 October 2007 (UTC)
thread_title:  Infinite limits 

message: Consider the question "f(x) = 0; Is it true that f'(x) = 0?"  This can either mean "at some particular point x, f(x) = 0, in which case f' evaluated at the same point need not be 0, or it can be taken as a definition, in which for all x, f(x) = 0, and f'(x) is indeed 0 at all points.  Is there a standard way of disambiguating these?  —Preceding unsigned comment added by 76.113.64.59 (talk) 09:25, 17 June 2008 (UTC)
replies: text: The = operator means that something is always true.  Without qualification, f(x) = 0 means that x is 0 for all (valid) x.  So \frac{df}{dx} = 0 for the same domain. Xihr (talk) 10:11, 17 June 2008 (UTC)
replies: text:  The = operator in itself does neither imply always nor true, e.g. you can easily state "x=2" or even "2=3". Therefore, I agree with the poster of the unsigned comment above. As long as x is a free variable the question is incomplete. However, the concept of always meaning "for all x" could also be implied with the equivalence operator like this: f≡0. Schellhammer (talk) 16:38, 5 November 2008 (UTC)
thread_title:  Quantifiers or Definition vs Fact 

message: 1.  The determinant formula for the vector product is just a mnemonic device to help in remembering the definition.  Therefore I can't see how it is an abuse of notation. McKay (talk) 09:39, 3 June 2009 (UTC)
thread_title:  Two comments 

message: I believe the direct sum "operator" is abuse of notation. If V, W are vector spaces, V+W gives you a new vector space. However, V\oplus W does not: it's either true or false. Writing U=V\oplus W is even worse, because the notation suggests you're comparing a boolean to a vector space. Also, usually operators allow you to define new spaces (or whatnot) but \oplus does not. You can define U \stackrel{\rm def}= V+W and then ask whether V\oplus W. Writing U \stackrel{\rm def}= V\oplus W would be invalid. Do you agree? Itayperl (talk) 16:09, 10 August 2009 (UTC)
replies: text: I'm not sure what your complaint is. According to the linked page, V\oplus W refers to the vector space where the underlying set is the cartesian product of V and W, and the addition and scaling operators are defined component-wise (this is standard notation). This is clearly a vector space. On the other hand, V+W in general means nothing. The only time it has meaning is when V and W are subspaces of some larger subspace X, in which case V+W refers to the set {v+w : v in V, w in W} which happens to also be a vector subspace. I don't see any abuse of notation here; all I see are two different notations which mean different things. skeptical scientist (talk) 08:13, 20 November 2009 (UTC)
replies: text: skeptical scientist is correct.  \oplus does not assert anything.  It's a binary operator on vector spaces. Zaslav (talk) 18:53, 3 January 2010 (UTC)
replies: text: Sorry about that. My textbook defined \oplus differently. The definitions on Wikipedia make a lot more sense. Thank you!  —Preceding unsigned comment added by 132.66.234.217 (talk) 18:17, 9 August 2010 (UTC)
thread_title:  Direct sum operator 

message: seems to be rather dry. Sorry for complaining when I can't offer up a superior alternative, but I'm hoping that someone finds a more colourful quote :D 118.90.20.3 (talk) 11:48, 8 October 2009 (UTC)
thread_title:  The Quotation 

message: I propose adding to the article a mention of the ubiquitous misuse of "dB" in sound level measurements. A dB (decibel) is only a numerical ratio of two quantities. Common sound level measurements are in dB(a) where the suffix "a" defines a reference level. Cuddlyable3 (talk) 17:21, 3 January 2010 (UTC)
replies: text: "Abuse of notation" is a specifically mathematical concept referring to a certain way of sloughing over certain technicalities of notation.  I don't believe it's used outside math, as for physical concepts like dB.  But maybe you want to start a fashion of abusing the language of math by applying it to physics?  (You'd be in good company.) Zaslav (talk) 19:01, 3 January 2010 (UTC)
replies: text: There you go abusing dB yourself. "dB" is not a physical concept. I don't wish to start any fashion. Cuddlyable3 (talk) 22:47, 3 January 2010 (UTC)
replies: text: "The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity ..." (Wikipedia, Decibel). Zaslav (talk) 05:09, 4 January 2010 (UTC)
replies: text: That's a fair quotation Zaslav but expression is not the same as identification. I may tell you that a signal magnitude increases by 3 dB and that we understand. If I tell you a signal magnitude is 3 dB I make no sense. It is routine to use, not abuse, dB expressions of physical ratios in radio communications. Cuddlyable3 (talk) 15:45, 23 May 2010 (UTC)
thread_title:  Sound dB 

message: I've made a subheading to avoid messing around trying to indent lists etc. correctly.
Well, Zaslav, you've been waiting long enough for further comments - over three years!  And I heartily concur in your opinion: that none of the following:
#"partial function"
#"generalized function"
#"law of composition not everywhere defined" (or its allegedly abusive precursor)
is in fact either:
*an abuse of notation (since they are all terms, not notation), or
*an abuse of (the English) language.
Another example of a non-restrictive adjective used often in maths?: "approximate'" or "approximately".  Technically speaking, the approximate includes the exact; two is certainly "approximately two", even if it is also "exactly two". 
The writing is, as you wrote, POV.  It reminds me of one of those oh-so-amusing newspaper pieces that regularly appears exposing the supposed illogicality of English and, metaphorically shaking its head and clucking its tongue, suggests that perhaps we really should go back to learning Latin in schools in order to impart the virtues of clear thinking.  These three examples will only confuse the reader seeking a clear understanding of how mathematicians systematically abuse notation and use language to avoid pedantically complete specification within certain contexts.  The following comment is also POV, but I believe has support from decades of research in linguistics:
It is a correct and sophisticated use of any language to qualify its terms no more than is absolutely necessary to understand them unambiguously from their context.
We should therefore remove these incorrect examples forthwith.
yoyo (talk) 17:33, 12 April 2013 (UTC)
thread_title: Three examples are wrong and should be removed.

message: (The article mentions this to support that certain manipulations of differentials are notational abuses.) Could someone add an example (y[x] to demonstrate this)? Cesiumfrog (talk) 00:42, 23 May 2010 (UTC)
replies: text: Consider a circle with equation x^2 + y^2 = R^2.  Then x dx + y dy = 0.  Hence dy/dx equals -x/y if y is nonzero but is undefined for y = 0.  Similarly, dx/dy equals -y/x but is undefined for x = 0. Hence dx/dy = 1/(dy/dx) holds if and only if both x and y are nonzero.  This example depends on being strict about 1/0.  If you want to be less strict and replace "undefined" by "infinite", together with some rules you decide to accept, you may still accept that dx/dy = 1/(dy/dx) holds everywhere and want a different example. Boute (talk) 09:51, 7 August 2010 (UTC)
replies: text: Well, yeah, I would like a different example. But I'm starting to doubt whether a more persuasive example (e.g., finite dx/dy) exists? Anyway, it seems almost like a circular argument (or rather assuming the conclusion): the article faults the lack of strictness (in manipulating derivatives like fractions) purely on extremely strict grounds (distinguishing 1/0 from infinity). Would it not be better to simply point out the standard way that the derivative notation is defined (such that the notation represents in shorthand a single complex entity rather than a simple ratio of two independent entities), without making any stronger claim? (If anything, it would seem justifiable to me if the article separately added 1/(1/0)=0 to its list of examples of notational abuses, since it seems to be a separate case of something disallowed in strict contexts but nonetheless tending toward correct answers.) Cesiumfrog (talk) 04:16, 9 August 2010 (UTC)
thread_title:  dx/dy is not always equal to 1/(dy/dx) 

message: Is the quotation right? "We will occasionally use this arrow notation unless there is no danger of confusion." Shouldn´t the "no" be removed, for example?--190.188.2.122 (talk) 15:33, 24 December 2010 (UTC)
thread_title: Quotation

message: Generalized function can be a non-function.--刻意(Kèyì) 00:11, 31 December 2010 (UTC)
thread_title: Some adjectives, such as "generalized", can only be used in this way

message: I always used \mathrm{sin}(x)^n notation to raise trigonometric functions to powers. 
I'm a programmer and thought that () is the function invocation operator. 
So the sine comes first then the power.
Now I think a mathematician is confused by this and thinks I meant \mathrm{sin}\,x^n.
Now I have quite a few papers to fix...
Calmarius (talk) 14:17, 28 June 2013 (UTC)
thread_title:  Some thoughts about the trigonometric function thing 

message: Right now, the derivative section reads like an opinion piece or argument. It cites no sources, and makes unverifiable claims. E.g. "fully justified", "completely rigorous". If there is a source for these claims, then the content should be attributed to that source as an opinion. If it is absolutely not an 'abuse of notation', then it should say so in an objective tone, or be removed from the article. If there is some sort of debate on the matter, all sides should be represented.  
I tagged it with original research. 
GregRos (talk) 23:16, 10 August 2013 (UTC)
replies: text: Why do you object to this, and what on earth makes you think it is "original research"? It is perfectly standard that dx and dy can be taken as real non-zero numbers in the ratio \frac{dy}{dx} : 1, and this fully justifies manipulating them as though they are numbers, because in that case they are numbers. (It is probably true that very few mathematicians habitually think of them as such, but that is not the point: the point is that they can be so regarded, and that doing so justifies the techniques. Far from being original research, it was presented to me as perfectly standard and accepted in my time at university, over 40 years ago, and it was there in the text books. No doubt it would be possible to find such a text book and cite it to satisfy your demands, but the trouble of doing so would be disproportionate. JamesBWatson (talk) 20:45, 12 August 2013 (UTC)
replies: text:  I agree with GregRos and disagree with JamesBWatson. I believe the statement "dx and dy can be taken as real non-zero numbers in the ratio \frac{dy}{dx} : 1" is simply false and as a mathematician I would never allow it. The thing about "separating dx and dy" is that in certain circumstances we have theorems that tell us that the result of such manipulation leads to the correct answer; it is certainly not because dx and dy are numbers. 09:09, 14 August 2013 (UTC)
replies: text:  Good heavens, I wasn't suggesting that "the result of such manipulation leads to the correct answer" is "because dx and dy are numbers", or that the notation is somehow a substitute for "theorems that tell us that the result of such manipulation leads to the correct answer". Obviously, the fact that the method works requires analytical proof. However, granted that it can be proved that it does work, there is the quite separate question of whether it possible to produce a sound, logical interpretation of such notation, rather than regarding them either as some sort of fictitious device or else as infinitesimals. Indeed, it turns out that there is such a sound interpretation. We can take, for example, f'(x) {dx} = {dy} as referring only to real numbers, and this means that the manipulation of such notation is perfectly sound and logical, not some sort of "abuse of notation". In graphical terms, dx and dy refer to the lengths of lines parallel to the axes making a triangle with a segment of a tangent, just as δx and δy refer to the lengths of lines parallel to the axes making a triangle with a chord. If the expressions dx and dy are interpreted that way, then there is no "abuse of notation" involved, as it is literally true that the product of the real number f'(x) and the real number dx is equal to the real number dy. The point is not that this interpretation is "the reason" why the method works, or that it somehow avoids the need for proof that it does, but simply that this is a perfectly logical interpretation of the symbols, without any "it doesn't really mean anything, but I will do it because it works" nonsense. In my experience the place where this interpretation of the notation is most useful is in connection with partial differentiation, because uses with ordinary differentiation can always be rewritten without differential notation without significant increase in complexity, whereas such notation as  \frac{\partial f}{\partial x} {dx} + \frac{\partial f}{\partial y} {dy} cannot so easily be replaced. However, the point at issue as far as this article is concerned is not whether or under what circumstances the notation is useful, but simply whether or not a perfectly logical interpretation of the notation is possible. And the answer to that is that it certainly is possible, whether or not most mathematicians are unaware that it is, and think they are "abusing" the notation every time they use it. JamesBWatson (talk) 14:12, 14 August 2013 (UTC)
thread_title:  Derivative section 

message: Just passing by and the tone in several sections reads too much like editors sending passive-aggressive digs at each other.  It's important to summarize the opposing rationales for the conflicting attitudes but please keep the tone encyclopedic rather than evangelical.  — Preceding unsigned comment added by 67.87.17.222 (talk) 07:41, 2 December 2013 (UTC)
replies: text: Can you give specific examples of what you mean? Without them, it is difficult to take any steps to improve matters. Also, are you sure that an "Original Research" tag is appropriate? "Editors sending passive-aggressive digs at each other" sounds as though it means something very different from "Original Research", and I am at a loss to see how the tag applies in this case. JamesBWatson (talk) 15:56, 3 December 2013 (UTC)
thread_title: Added OR template

message: I put two templates on the "Big O notation" section: a POV template and an "unreferenced section" template. This section appears to me to be a personal essay, rather clumsily constructed, and to convey the particular point of view of its author(s). It states that the notation "f(x)=O(g(x)" is abusive, which is not at all universally admitted in the mathematical community, and particularly not amongst number theorists. The claim that it is abusive follows from the very rigid (and impossible to keep) position that a symbol should have one unique meaning in mathematics. But the use of alternate meaning of a symbol or group of symbols is common in mathematics and does not necessarily indicate an abuse of notation. (In this particular instance one can perfectly well consider that "=O" is one single operator, in which the part "=" has no individual meaning, and in particular is not an equivalence relation): with this acception there is no abuse of notation in the formal definition. This other point of view is not even addressed in the section.Sapphorain (talk) 19:46, 10 July 2015 (UTC)
thread_title:  "f(x)=O(g(x))" is not considered abusive by all mathematicians 

message: ...but not quite.  After all, a few things in here are reasonable, and it's a phrase that does come up in mathematical writing often enough.  Rather than a full delete, I'd propose eliminating all but the sections on Structured mathematical objects and and Functional notation (and maybe associativity of the cartesian product also), and then maybe merging into mathematical notation.  
Various problems (and they're pretty fundamental) include, but are not limited to:
* The subsections on various kinds of numbers under "Equality vs. isomorphism" are all poorly written, very muddled, and very dubious/controversial as to what the exact abuses are.
* The Del operator isn't so much a notational abuse as a mnemonic device for remembering formulas (this one's borderline, and I suspect that it could actually be made rigorous, but I've never looked at the issue).
* Trig functions: This isn't an abuse of notation – it's just bad notation (that we're all unfortunately stuck with).
* Values of a RV: This section should be removed anyway as it's more like a personal essay
* Bourbaki: This whole section is just a confused mess of pedantry
As an aside, a better example of an abuse under the equality-vs-isomorphism type would be something like writing  \pi_1(S^1) = \mathbb{Z}  for the fundamental group of a circle.  Deacon Vorbis (talk) 18:54, 1 February 2017 (UTC)
thread_title:  This article should almost be deleted... 

message: The article has been substantially revised - mostly improved - since I last saw it over five years ago.  IMO, it currently represents the reality for working mathematicians quite well, since it: 
* is informative
* uses fairly accessible examples
* provides balance by mentioning the importance of context and history
Though I wouldn't yet call it "wonderful", the writing is on the way to becoming a quite good article.
Yet it is entirely unsourced, now having the  template warning.  Is this a problem?  According to WP principles, yes it is.  Two possible solutions occur to me:
# Find sources - the illogical (but very human) "appeal to authority" - that WP demands;  OR
# Convince WP to change its principles to defer to a consensus of the cognoscenti - er, editors, I mean … good luck with that approach!
So come on guys, surely some of you have seen some writing in textbooks and published papers that addresses and exemplifies this issue?  I'm sure I've read something along those lines in the last three years or so.  (Some older text on algebra, I think; and something on Terry Tao's blog.)  Well, I'll try to find those sources (if I can first find the time), but please feel free to beat me to it!  yoyo (talk) 15:46, 25 November 2018 (UTC)
replies: text: I agree that it would be much better to find sources if there are any. But, in the case of a consensus of the cognoscenti is it is not necessary to convince WP to anything, as WP policy WP:IAR is  D.Lazard (talk) 16:03, 25 November 2018 (UTC)
thread_title:  This article is informative and balanced - but remains unsourced 

message: This page cites, as Reference 4, a blog post titled "Abuse of Math Notation" which is simply an expletive-filled rant about things that a random person on the internet has decided are "abuses of notation". In particular, I disagree about f(x) being an abuse of notation when referring to a function. So, here are a couple of references I grabbed at  random off my book-shelf:
Studies in Real and Complex Analysis edited by Hirschman and published by the Mathematical Association of America (1965). Functions are defined on page 9 of that book, and  then just 9 lines later the author states "A function f(z) is holomorphic in a domain..." This is the first reference in the book of a function being called "f" and the name is explicitly given as f(z), not just f. 
Methods of the Theory of Functions of Many Complex Variables by Vladimirov and published by Dover (1966). As best I can tell, the first use of the word "function" comes on page 5 in a sentence beginning "We shall say that a function f(x) belongs to a class..."
 
Apologies that my examples are from the 1960s; I only collect older math books. My degree is from the 1990s, and although I agree that it is standard to define a function with respect to its domain and range (or co-domain, or whatever), I have never heard of the idea that using f(x) is considered anything other than equally standard, let alone "abuse of notation". If anything, I would say that insisting that a function be called f and not f(z) (or f(x), or "a machine that takes in inputs and spits out widgets") is an example of math pedantry. Ackshooerry (talk) 19:16, 16 May 2023 (UTC)
replies: text: I don't have a good reference to support that it's an abuse of notation. However, the references you provide don't contradict the claim. Neither does the fact that it's prevalent among mathematicians to use this notation. Yes, the distinction might be a form of pedantry as you call it; but pedantry is what formalism is about. While it may be obvious whether f(x) denotes a function or a number in calculus, the distinction becomes important in set theory (for example), where saying z∈f or z∈f(x) mean two completely different things. bungalo (talk) 01:33, 17 May 2023 (UTC)
thread_title:  A random, expletive-filled, blog post about Wikipedia shouldn't be used as a source on Wikipedia 

message: > This abuse of notation is widely used, as it simplifies the formulation, and the systematic use of a correct notation quickly becomes pedantic.
I fail to see how the correct formulation "Let f be a function ..." is in any way less simple or more "pedantic" than the technically incorrect (unless indeed f(x) is a function!) formulation "Let f(x) be a function ...". 2A02:168:FE38:0:FCB6:AAED:7EDF:9D9 (talk) 11:32, 4 November 2023 (UTC)
replies: text: Used alone, "Let f be a function ..." is correct and non-pedantic, but when you need to name the variable(s), the correct formulation may become cumbersome. Compare "let f be a function; if its variable is denoted x, its derivative is commonly denoted \frac {\partial f}{\partial x}" with "the derivative of the function f(x) is commonly denoted \frac {\partial f}{\partial x}". This is a very simple example. However, when one does not need to name the argument of the function, the correct formulation is often much more simpler. D.Lazard (talk) 12:28, 4 November 2023 (UTC)
thread_title:  f(x) vs f 

message: One of the redirects for this article is Abuse of terminology. Thus the article is on point and has a reliable source. Please re-instate the contribution as the reversion was unjustified. Alternatively, engage in some talk here as is appropriate. Rgdboer (talk) 01:13, 24 December 2023 (UTC)
replies: text: Point conceded about the article topic, but that's almost beside the point.  The author makes no claim of it being an abuse; that's your own interpretation, and so it violates WP:NOR.  And even if the source did mention "abuse" specifically, it's one author's brief opinion about one particular piece of terminology, and its inclusion would be WP:UNDUE.  The way you've added it also runs counter to MOS:QUOTE.  All together, this doesn't belong in the article.  35.139.154.158 (talk) 05:27, 24 December 2023 (UTC)
thread_title:  Hello, Anonymous