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I've personally tripped up on a few concepts that came down to an abuse of notation, and I've read of plenty more on stack exchange. It seems to all be forgiven with a wave of the hand. Why do we tolerate it at all?

I understand if later on in one's studies if things are assumed to be in place, but there are plenty of textbooks out there assuming certain things are known before teaching them. This is a very soft question, but I think it ought to be asked.

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No, it is perfectly fine here! :-) – Mariano Suárez-Alvarez Dec 24 '12 at 15:11
Perhaps you can elaborate on how you define "abuse of notation": is it when notation is introduced, but not explained or defined (i.e., assumed to be understood)? or do you mean when unconventional notation is used in place of what is standard? Or both. Examples would help. – amWhy Dec 24 '12 at 15:11
Sometimes, good notation doesn't exist; I've even heard it said that in some cases, simply coming up with good notation for something can be an important mathematical advance. Alas I can't find a reference. – Hurkyl Dec 24 '12 at 15:56
@Hurkyl Maybe this: "The invention of the symbol $\equiv$ by Gauss affords a striking example of the advantages which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic."? (G. B. Matthews in "Theory of Numbers", 1892) – Old John Dec 24 '12 at 16:33
One abuse that obfuscates, serves no one and should be eradicated immediately is the awful use of ${\cal L}\{f(t)\}$ for the Laplace transform. Just write ${\cal L}f$. – copper.hat Dec 25 '12 at 5:13

I doubt I could put it better than this:

"The student of mathematics has to develop a tolerance for ambiguity. Pedantry can be the enemy of insight." - Gila Hanna

I also highly recommend Terence Tao's article describing the "pre-rigorous", "rigorous", and "post-rigorous" stages of a mathematician's development.

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+1 for reminding me of that article! – Hurkyl Dec 24 '12 at 15:38
Interesting. Is that how you are taught in the USA? I can't recall the "pre-rigorous" era in my undergrad studies. I could suggest high-school, but that doesn't count (especially when you have three years of army, and roughly a year or two extra before you return to the academy). We just zap the kids into the rigorous era. It's fun, and painful to most. I remember liking it very much as a freshman. The problem is that often the post-rigorous era is introduced (here) way too early... so that is another trauma for most students. – Asaf Karagila Dec 24 '12 at 19:15
@AsafKaragila:Virtually all forms of "calculation-oriented" math are actually pre-rigorous forms of the actual mathematical discipline. The confusion comes because a) we tend to give them different names and b) far more students learn the pre-rigorous versions than ever learn the true discipline. For instance, all school children learn arithmetic, but only a few will ever learn Number Theory. Likewise, most science undergrads learn advanced calculus, but only the Math students are likely to take Real and Complex Analysis. And the "proofs" in these earlier forms are actually only derivations. – RBarryYoung Dec 24 '12 at 23:29
@RBarry: Obviously most people learn calculations and pre-rigorous mathematics. That wasn't my point. I pointed that as a math undergrad I never met that pre-rigorous mathematics, and that the post-rigorous stage came in too soon for my taste (and I am not a fan of rigor!). I was asking whether math undergrads in the USA have a pre-rigorous part in their academic journey. One could make the same comparison with cooking - "heat-it-and-eat-it stage" vs. "recipe stage" vs. "creative cooking stage". You missed my point by a lot. – Asaf Karagila Dec 24 '12 at 23:34
@AsafKaragila: That is the question that I was answering. If you had Calculus in college then you had the pre-rigorous course. In the U.S. virtually all math undergrads do (math students are a subset of science students). It is only after they have Calculus that they would normally be eligible for Real and Complex Analysis. Post-rigor is not generally encouraged until late in graduate school. – RBarryYoung Dec 25 '12 at 1:30

When one writes/talks mathematics, in 99.99% of the cases the intended recipient of what one writes is a human, and humans are amazing machines: they are capable of using context, guessing, and all sorts of other information when decoding what we write/say. It is generally immensely more efficient to take advantage of this.

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I agree, but when teaching a new concept to a student, where there is no context in the mind of the student, then it's absurd to hold to this habit. – Korgan Rivera Dec 24 '12 at 19:07
Create the context and then abuse it. It is subestimating students to suppose they will not be able to deal with small abuses of notation and language. Of course, one has to be explicit about what one is going to abuse. But imagine a course on linear algebra which uses notation to distinguish the zeroes of different vector spaces! – Mariano Suárez-Alvarez Dec 24 '12 at 19:53
@Mariano: You just gave me a flashback to the first linear algebra course I took, in which the text did make these distinctions. One of the identities listed on the inside cover of the text was $L(0_V) = 0_W$. ("Love equals ow.") – Pete L. Clark Dec 25 '12 at 0:29
@Mariano: Tastes vary. In an introductory linear algebra course I routinely subscript the zeroes to indicate their respective spaces until we’re well into the course, and I’m similarly careful in intro. abstract algebra courses. I find that it reduces confusion significantly. – Brian M. Scott Dec 25 '12 at 10:40
+1 Nicely put. Extreme formality makes me mostly suspicious. – WimC Dec 25 '12 at 18:27

Since Bourbaki is rather busy and is not (yet) a member of this site, I'm posting His answer (which He preemptively wrote about 70 years ago) on His behalf:

As far as possible we have drawn attention in the text to abuse of language, without which any mathematical text runs the risk of pedantry not to say unreadability.

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Why the capital "He"/"His"? – Joe Z. Dec 25 '12 at 16:23
As far as I know, that's usually only used for God. – Joe Z. Dec 25 '12 at 20:13
@Joe: ah, the notation is abused here, but we'll tolerate it. – Konerak Dec 25 '12 at 20:34
@Jared: That was unnecessary. – Rahul Jul 9 '13 at 16:42
@Jared: I have very delibarately capitalized the pronouns and adjectives pertaining to Bourbaki, out of respect for Him. Please do not edit my text. – Georges Elencwajg Jul 9 '13 at 21:12

Abuse of notation is tolerated when the alternative is worse!

In some cases, abuse of notation isn't really abuse at all, but simply a lack of fleshing things out. For example, I'm sure many would consider

$$\arctan(+\infty) = \pi/2$$

an abuse of notation that is meant as shorthand for

$$ \lim_{x \to +\infty} \arctan(x) = \pi / 2 $$

But if you take a short trip into theory of the extended real line, the identity is seen to be a literally true fact about the $\arctan$ function on the extended real line (which is the continuous extension of the $\arctan$ function on the reals).

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I would say rather that whether $\arctan(+\infty)=\pi/2$ is an abuse of notation depends on the context. In freshman calculus, for instance, it’s at best an abuse of notation and at worst simply wrong. – Brian M. Scott Dec 25 '12 at 10:35
This reminds me of Orwell's six elementary rules ("Politics and the English Language"), of which the sixth is "Break any of these rules sooner than say anything outright barbarous". (From the Economist style guide.) – copper.hat Dec 26 '12 at 23:27
@Brian: Maybe differential forms would have been a better example, since students are taught to use them heuristically (e.g. integration by parts, substitution in integrals) long before they are formally introduced. – Hurkyl Dec 26 '12 at 23:53

As I stated in my comment/question below your question, it seems that you are "abusing" (mis-using) the phrase "abuse of notation."

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion). Abuse of notation should be contrasted with misuse of notation, which should be avoided. A related concept is abuse of language or abuse of terminology, when not notation but a term is misused.

In particular, I'm referring to your observation:

I understand if later on in one's studies if things are assumed to be in place, but there are plenty of textbooks out there assuming certain things are known before teaching them.

Here, it seems to me that you are complaining that you are encountering the use of notation that you do not understand and have not yet encountered, and for which the author/instructor has not explicitly defined. This is NOT an abuse of notation. This is where you "speak up" and ASK what is meant (if in class). Alternatively, in such a situation, you need to take the initiative to understand the notation, to look to see if the text in question has an appendix or index defining the notation it uses, or you can appeal to some reference to better understand the symbols/notation and its various uses, which are usually context dependent.

That said, with respect what actually is meant by "abuse of notation": we are all human, and mathematical notation, like any language, is subject to ambiguity, perhaps less so than natural language, but nonetheless, it is still subject to ambiguity.

Notation also provides a means to communicate, compactly, what would be laborious to try to communicate otherwise, even if at the cost of "abusing notation."

In any case, being human also means it's usually a good thing to avoid pedantry and to learn to tolerate the use//abuse/misuse of any language (mathematical or otherwise) by others. Certainly, you may want to it out when you take something to be an erroneous use of notation/language (and doing so in a helpful way), but deciding not to tolerate it is perhaps going too far.

And I suspect that we all take "short-cuts", when handy and when we can safely assume the notation we may be "abusing" will be understood. Certainly, there is a "fine-line" between taking advantage of notational short-cuts, and full-fledged "abuse" of notation that fails to convey what was intended by its use.

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@CookieMonster "In any case, being human also means it's usually a good thing to avoid pedantry and to learn to tolerate the use//abuse/misuse of any language (mathematical or otherwise) by others... deciding not to tolerate it is perhaps going too far. And I suspect that we all take "short-cuts", when handy and when we can safely assume the notation we may be "abusing" will be understood. Certainly, there is a "fine-line" between taking advantage of notational short-cuts, and full-fledged "abuse" of notation that fails to convey what was intended by its use." – amWhy Dec 29 '13 at 16:00
Without further research you are saying that I were using non-standard notation. May I show you the following reference: ISO 31-11: ⇒ p ⇒ q implication sign if p then q; p implies q Can also be written as q ⇐ p. Sometimes → is used. . Seems you are suffering from some severe self overestimates, the only thing that is distracting here. – j4n bur53 Dec 29 '13 at 18:32

Personally (I can speak only for myself), I tolerate abuse when it helps keep things clear and simple (with regard to my subjective perspective). Sometimes it might be tolerated when there is not enough resources (e.g. time, space, etc.) available and the details are not that important.

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Done properly abusing notation makes things clearer. Suppose that $f \colon X \rightarrow Y$ and $A \subseteq X$. We can write

  • $f(A)$
  • $\{ f(x) \colon x \in A \} $
  • Define $g \colon X \cup \mathcal{P}(X) \rightarrow Y \cup \mathcal{P}(Y)$ by $g(x) = f(x)$ if $x \in X$ and $g(A) = \{ f(x) \colon x \in A \} $ if $A \subseteq X$. Here we can use both $g(x)$ and $g(A)$. There are difficulties with this method if $X \cap \mathcal{P}(X) \neq \varnothing$.
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Note that $\{f(x): x\in A\}$ is itself an abuse of notation! It is an abbreviation for $\{y: \exists x\in A. y=f(x)\}$. – MJD May 14 '13 at 15:48
@MJD Abuse of notation is a recursive process. We can abuse abuses of notation! – Jay May 14 '13 at 17:55
@MJD: Where is the potential ambiguity that would make that an abuse of notation rather than an extension of it? Some care is required to define variable binding and such properly, but I don't see an inherent problem. – dfeuer Jul 9 '13 at 20:51
@dfeuer What does $\{f(x):x\ne y\}$ mean (where $V$ is the universe of discourse)? Is it $\{z:\exists x\ne y\ z=f(x)\}=f(V\setminus\{y\})$ or $\{z:\exists y\ne x\ z=f(x)\}=\{f(x)\}$ (where $x$ is bound in the first expression and $y$ is bound in the second)? In general, we can only define that expression in certain limited forms, like $\{A:x\in B\}$ where $A$ and $B$ are class expressions and $x$ is a bound set variable. – Mario Carneiro Jul 16 '13 at 1:29
@MarioCarneiro: I think I see what you're saying. It's unfortunate, though, since a bit of modification to the notation would allow that to be unambiguous. Compare the Haskell programming language's list comprehension syntax, which essentially uses an alternative $\in$ symbol (written $\gets$) to indicate that a variable name is being bound there. – dfeuer Jul 16 '13 at 2:58

I came to realize that mathematics (in general all sciences) is a collection of strands of ideas; each strand is no more than a few inches long, i.e., incomplete on its own, but they connect with one another like neurons, and altogether they form a gigantic bridge several miles long. Not abusing notation is like trying to build a bridge using only a single strand -- a perfect, self-contained, single piece of idea. That is very counter-productive, and often hinders progress.

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I think that as new ideas and new ways of thinking about things come to light, any language, lingo or symbolism needs to evolve also. Languages would not be as sophisticated as they are today without some form of creativity element. People who come up with ideas don't always set the conventions (e.g. Latin squares in modern day combinatorial mathematics normally use letters or numbers to represent solutions, not the Latin symbols Euler used). Would he be turning in his grave? Or would the underlying concepts of the mathematics, which are both symbolic and a-symbolic be the overriding factors. I believe that what is needed, is a complicated compromise of tradition mixed with innovation and help for people who don't get one, the other or both.

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