Talk:Abuse of notation

From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated Start-class, Mid-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
Start Class
Mid Importance
 Field: General

Sound power should be "sound pressure level"[edit]

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)

Screwdriver?[edit]

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)

Hi Andrew, Sounds like you could have made a better start at this than me. I used an abuse of notation recently in Combinadic so thought there should be such a page, but am still at a loss what should go into it. --J. W. McLeod 09:28, 6 Apr 2005 (UTC)

Very common abuse?[edit]

"A very common abuse of notation is using sin2(x) instead of (sin(x))2."

It's not an abuse at all according to other things I've read - fn(x) = [f(x)]n for n not equal to -1 (for n=-1 it refers to the inverse function). The article Function (mathematics) seems to make no reference to this. Brianjd | Why restrict HTML? | 04:56, 2005 Apr 8 (UTC)

  • Especially in the abstract, fn(x) = f[f(x)] However, sin2(x) is an exception to the rule. Bluap 16:48, 3 May 2005 (UTC)
    • Seems like a double standard to me. So sin² means the square of the sine—i.e. the sine times itself—but sin-1 is the inverse function? OneWeirdDude (talk) 23:00, 16 October 2008 (UTC)
sin2x is an unusual notation but probably not abuse. For most functions, f^2(x)=f(f(x)), rather than [f(x)]^2. Still, all trig functions represent a common and systematic exception. This is true of all exponents, not just 2. —Preceding unsigned comment added by Eebster the Great (talkcontribs) 21:35, 2 February 2009 (UTC)

Misuse rather than abuse?[edit]

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)

Have retreated from language criticized above[edit]

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)

John Harrison[edit]

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

Infinite limits[edit]

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)

Yes, \lim_{x \to \infty}f(x) = \infty has a precisely defined meaning, as our own article on limits shows. I'm getting rid of that example. -- 75.162.71.236 04:15, 8 November 2007 (UTC)

Quantifiers or Definition vs Fact[edit]

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)

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)
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)

two comments[edit]

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)

2. The section on O(.) reads like a personal essay. The way "=" is used in this context is imo an abuse of notation, but this needs to be cited from a suitable source. The other claim, that f(n) is just a value rather than a function, doesn't belong here as it is not specific to this notation. Also, ambiguity of notation is not at all the same as abuse of notation, so the example O(nm) doesn't belong either. McKay (talk) 09:39, 3 June 2009 (UTC)

Inner product vs. v^T w[edit]

Many people seem to write the inner product <v,w> between two vectors as v^T w, although strictly speaking the result of the latter operation should be a 1 by 1 matrix rather than a scalar. This is not the same thing: consider an m by n matrix A with m and n both > 1. Then, like any matrix, A can be multiplied by a scalar, but not by a 1x1 matrix. I'm not aware of any mathematical operator that will take a one by one matrix and extract its element as a scalar or vice versa.

Can anyone knowledgable either confirm or disconfirm this as a case of abuse of notation? —Preceding unsigned comment added by 131.111.20.201 (talk) 13:28, 8 June 2009 (UTC)

The inner product is always defined as a map V\times V \to \mathbb{F} where \mathbb{F} is the underlying field of V. You can define the standard inner product in a Euclidean space as \langle v,w\rangle = v^T w if you look at it as a scalar rather than a 1x1 matrix. You don't need a special operator in order to use isomorphic spaces interchangeably (it's trivial to prove that R – as a vector space over R – is isomorphic to M^\mathbb{R}_{1\times 1}, i.e., the space of all 1x1 real matrices). This is exactly the same issue as writing 1\in \mathbb{C} instead of (1,0)\in \mathbb{C}: \{(x,0)\mid x\in \mathbb{R}\}\subseteq \mathbb{C} and \mathbb{R} are isomorphic.
For the not too mathematically inclined I'll sum up: I don't think this qualifies as abuse of notation. —Preceding unsigned comment added by 132.66.234.217 (talk) 15:53, 10 August 2009 (UTC)
I agree. This is an example of a different but similar phenomenon from abuse of notation, which is the tendency for mathematicians to identify objects which are in some sense interchangeable. Here we see the identification of the 1x1 matrix [s] with the scalar s; another common example is the identification of a set and its characteristic function. skeptical scientist (talk) 08:07, 20 November 2009 (UTC)
There actually is an operator that will map a 1x1 matrix to a scalar. This operator is the det() determinant operator, which in the case of a 1x1 matrix is simply the number inside the matrix. 168.156.170.146 (talk) 18:00, 6 January 2012 (UTC)
To me, au contraire, it seems an obvious but usually harmless case of abuse of notation. (Reason: The 1x1 matrix is treated as a scalar multiplier of another matrix. Technically this is impossible. That's what defines abuse of notation.) But this abuse is so simple that it would be distressingly pedantic to flag it as "abuse of notation", and therefore in a practical sense I agree with skeptical scientist. If we draw attention to an "abuse of notation", it should be a more substantial kind of abuse than this. (Personal disclosure: I was put off from certain matrix calculations for years by distress over exactly this "abuse". That's one reason I say it is abusive.) Zaslav (talk) 18:50, 3 January 2010 (UTC)

Direct sum operator[edit]

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)

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)
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)
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)

The Quotation[edit]

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)

I disagree, the commonly accepted interchangability of vector notations from bold letters, to underlined with tildes, to the quoted arrows make it very applicable from my experience. 204.52.215.3 (talk) 16:57, 21 October 2009 (UTC)

Sound dB[edit]

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)

"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)
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)
"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)
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)

Pedantry[edit]

I regret to say that I find some of the remarks in the Bourbaki section to be pedantic in the extreme and reflecting a lack of understanding of the normal flexibility of the English language, as well as being POV. Specifically, the claim that the term "partial function" is abusive because a "partial function" on A is not a "function" on A is just a mistake of English. In language, the addition of an adjective is not required or expected to be always a narrowing of the meaning. I could give many examples in addition to "generalized X" if I could only remember them. (I'm sure they'll come after I close this comment.)

It appears to me that the article is becoming a sounding board for opinions about good writing. Obviously, some editor and I differ, so I have my POV and s/he has his/hers. That's why I call it POV. It's no longer about "abuse of notation/language".

The term "abuse of notation/language" is really not so broad; it means writing something that is technically incorrect without giving it a special definition, in the belief that it will be easily understood by everyone. Thus, for instance, "partial function", which is precisely defined, is not abusive. Using the term without a definition might well be called abuse, but that is not the example provided.

Similarly, to call "law of composition not everywhere defined" an abuse of language is to confuse quality of writing with the concept of "abuse of language". There may or may not be something ungrammatical or confusing or distressing about this construction; but this style of qualification is not "abuse" in the mathematical sense. (Personally, I think it needs two commas and will then be perfectly correct. Others may disagree. This is a stylistic or grammatical question.)

I am not surprised to see examples from Bourbaki. Bourbaki can be very pedantic. And was writing in French, for which the rules might not be the same – my French is not strong enough to justify an opinion.

I propose to erase some of the more extreme complaints about "abuse". But I await further comments. I'm sure they'll be interesting! Zaslav (talk) 06:43, 4 February 2010 (UTC)

The more I read of this article, the more I think "abuse of notation" has been misunderstood by some contributors to mean anything someone who is extraordinarily pedantic could find objectionable. The term as used (or abused?) in this article is losing its meaning and its usefulness. I suggest that a sizable contraction of the article is in order. Zaslav (talk) 06:39, 23 February 2010 (UTC)
An example of pedantry is the section "Misc" [Miscellaneous], which I have deleted. It said,
The so-called reflection through the origin is an involution, but not a reflection.
This is mistaken. There are many kinds of reflection. Reflection through a point is not reflection through a line, but it is a kind of reflection. Perhaps contributors should "reflect" longer before listing examples of abuse (sorry, I just couldn't help it). Zaslav (talk) 00:18, 15 March 2010 (UTC)
Actually, in geometry the conventional meaning of "reflection" is orthogonal symmetry with respect to a hyperplane. So symmetry with respect to a point is not a reflection, except in dimension 1. The reflection article does not give a clear definition, but it does say that exactly one eigenvalue is -1. Marc van Leeuwen (talk) 09:24, 23 March 2010 (UTC)
I'm sorry, but I believe you are mistaken about the correct definition. Perhaps by "conventional meaning" you mean that people who are not extremely knowledgeable about classical geometry -- this includes most expert mathematicians -- think there is only one kind of reflection, namely, in a hyperplane. This limitation appears to be very common among those who study "groups generated by reflections". If you read thorough books on geometry I think you'll find that there are other kinds of reflection. They simply are not as widely known.
Wikipedia articles ("Reflection") cannot be considered authoritative in deciding a question like this. One must go to the source. I regret that I don't have any sources available to me at the present time. I suggest that any book by Branko Grünbaum is authoritative, though perhaps not definitive. Zaslav (talk) 08:30, 30 March 2010 (UTC)

Three examples are wrong and should be removed.[edit]

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:

  1. "partial function"
  2. "generalized function"
  3. "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)

dx/dy is not always equal to 1/(dy/dx)[edit]

(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)

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)
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)

I don't think that this is an abuse of notation at all. When I was at school I formed the impression that separating dy and dx as though dy/dx were a fraction was an abuse of notation, but then at university I found that there is a perfectly simple interpretation which fully justifies the notation: dy and dx are simply real numbers in the appropriate ratio. This interpretation is, in fact, mentioned in the article (though the issue is somewhat muddied by expressing it in terms of the geometry of a graph). The case dy/dx=0 does not justify the remark in the article that "the derivative does not always behave exactly like a fraction (e.g. dx/dy is not always equal to 1/(dy/dx))" at all, because it is is in fact an example of how derivatives do behave exactly like fractions, not an example of how they don't, since if a/b=0 then b/a is no more and no less meaningful than dx/dy when dy/dx=0. I considered removing this section of the article, but on reflection it will probably be better to rewrite it. JamesBWatson (talk) 19:58, 19 August 2011 (UTC)

Quotation[edit]

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)

Some adjectives, such as "generalized", can only be used in this way[edit]

Generalized function can be a non-function.--刻意(Kèyì) 00:11, 31 December 2010 (UTC)

Some thoughts about the trigonometric function thing[edit]

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)

Derivative section[edit]

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)

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)
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) — Preceding unsigned comment added by McKay (talkcontribs) 09:09, 14 August 2013‎
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)

Added OR template[edit]

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)

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)