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Suppose you are writing a mathematical text (say an article) and you want to call an object (for example, a set) by a letter. It would be cool then to have some reference (optimally available on the internet) where you could find some standard letters and notations of mathematical objects and pick one that you like. Does such a "notation dictionary" exist?

ADDED. Thanks everybody for interesting answers! Maybe it is worth to add that I had in mind rather basic things. The question was trigged by my attempt to find a good letter to denote a subset of the segment $[0,1]$. Finally I decided to call it $T$ (in the course of the proof it turns out that $T$ is equal $[0,1]$ :) ).

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    $\begingroup$ @Dmitri: This is not directly relevant, but you might want to check out mathoverflow.net/questions/18723/… $\endgroup$
    – Harry Gindi
    Commented Jul 23, 2010 at 23:40
  • $\begingroup$ Whereas people tend to standardize symbols for particular concerete objects (e.g. the Cantor set is C) there is no need to standardize variables. You can really denote your generic subset of $[0,1]$ by any letter, in lower or capital cases, of the Latin or Greek, or other aplhabets. $\endgroup$
    – Pietro Majer
    Commented Jul 26, 2010 at 7:16

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Really the closest that you can get is Wikipedia or the right kind of search in Google Scholar. The community needs tools to establish or recognize consensus, which of course is an open-ended problem. These tools, while they are certainly far from perfect, are the best tools that exist. If you did embark on a project to document standards, that could be a great thing to do, but it would probably eventually be co-opted by Wikipedia.

When I think about quantum algebra, a topic which is notorious for "notation sprawl", I use Wikipedia and Google Scholar. The more traditional method is follow a few respected papers and textbooks, and this is also still reasonable.

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    $\begingroup$ Greg, since you value Wikipedia so highly, would you like to rewrite the article "Quantum group" so that it contains something besides notation (whether standard or not)? It's one of the weakest advanced math articles I've seen there. By the way, there is no real quality check mechanism on WP: it's all consensus based and depends strongly on the level of expertise of a small set of contributors and degree of voluntary compliance with various policies. $\endgroup$
    – Victor Protsak
    Commented Jul 24, 2010 at 4:29
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    $\begingroup$ Note, however, that notation is not uniformized even across various articles on the same topic on WP! That is the only sensible thing to do, since different people contribute to different articles and editorial coordination involves enormous amount of work and can be difficult to implement on an open wiki. $\endgroup$
    – Victor Protsak
    Commented Jul 24, 2010 at 4:35
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    $\begingroup$ Regardless of my opinion of Wikipedia, it is objectively an extremely useful and influential mathematical reference despite its obvious shortcomings. It would be a great public service for someone to rewrite the article on quantum groups, and various other articles. Unfortunately most people have no professional incentive to do it, and I am no exception. It could be a good issue to discuss with a financial patron. Otherwise, all I can say to people who improve Wikipedia is, "thank you". $\endgroup$
    – Greg Kuperberg
    Commented Jul 24, 2010 at 5:01
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    $\begingroup$ I don't understand the comment about "no professional incentive" to edit Wikipedia. Is there a professional incentive to participate in MathOverflow? $\endgroup$
    – András Salamon
    Commented Jul 25, 2010 at 16:42
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    $\begingroup$ It's true that I don't "need" the visibility. Nonetheless, I enjoy interaction with colleagues, and in this respect MO is more fun than WP. $\endgroup$
    – Greg Kuperberg
    Commented Jul 26, 2010 at 4:15
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The question presupposes the existence of some standard letters and notations of mathematical objects, which I'm doubtful about in many research areas. My experience with subjects that have a long history suggests that notation in mathematics evolves over time in less than logical ways. In some areas there simply isn't any "standard" notation, while in many others some influential sources have tended to establish a de facto standard. But quite a few mathematicians just make up their own symbols as they go along (I won't name any names), forcing readers to translate according to their own taste. It depends a lot on whose earlier work you most rely on. Even Chevalley, in volumes 2 and 3 of his abandoned series of books on the theory of Lie groups, made some really eccentric choices of fonts and letters.

On the other hand, there are some good lists of LaTeX symbols, as already pointed out. But such a list can only reflect overall usage, not tell you what is currently thought to be "standard" in a given subject area.

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  • $\begingroup$ As a slight aside, there are mathematicians out there trying to improve mathematical notation. A good example of this is the book Concrete Mathematics (by Graham, Knuth and Patashnik) en.wikipedia.org/wiki/Concrete_Mathematics. They use (and possibly introduce) a much superior, memorable and obvious notation for rising factorials and falling factorials than the traditional notations (which are also subject to ambiguity between authors). There are many other instances of new notations. I hope that these are adopted widely enough to displace the poorer old notations. $\endgroup$
    – Rhubbarb
    Commented Jul 31, 2012 at 12:35
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I'm not sure this is quite what you have in mind, but there is a "comprehensive" LaTeX symbol list: http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf

Unfortunately, it doesn't make suggestions about what kinds of symbols should be used for what kinds of objects, but that's usually a moving target anyway.

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  • $\begingroup$ Greg no. 2: +1 for a phantastic sense of humour! $\endgroup$
    – Wadim Zudilin
    Commented Jul 24, 2010 at 11:14
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Indeed, when in your research you happen to meet an interesting object, totally unknown to you, first you have to check whether it has already defined, as it's quite likely to be. How to find a reference then? One good thing in modern mathematics is that today we have reached a good level of standardization; very often names are given just following the most reasonable and obvious way, avoiding fancy or weird terms. Therefore, in this situation the first question one has to ask is simply: "how would I name this guy?" and then, just google it. So, yes, the dictionary does exist and is the whole Internet. In my experience, it's not harder than guessing the name of a TeX command: often trying the obvious term is even quicker than go and look at the userguide. Let me tell you an example. Weeks time ago for some reason I become interested in subsets of Banach space that are stable for infinite convex combinations, i.e. with countably many positive coefficients summig to 1. How would you call such a special sort of convex? (I leave it as a riddle; you can check it directly on google). Once one has found a reference, everything is there, included the notation ( of course guessing directly the letter denoting an object would be much more difficult). Oh, and then there is MO: what could you imagine better than a living dictionary?

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The International Organization for Standardization (ISO) offers a standard for mathematical notation as part of the more general ISO 80000 standard for quantities and units. It is widely used in the physical sciences and also adopted by the National Institute of Standards and Technology (NIST) in the USA, for example. The full standard can be found around in the internet.

It is a self-consistent very well-thought standard. It generally agrees with common notation in maths literature. It covers areas from basic set-theory, first-order logic, tensors, functional analysis and special functions, and much more. The specialized notation of some areas such as differential geometry or category theory or higher-level formal logic is not quite covered. Notation for probability theory is given separately in the Guide to the Expression of Uncertainty in Measurement.

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    $\begingroup$ So I can’t see any actual notation on this site without handing over 158 Swiss Francs, or am I doing something wrong? I can’t really see how a standard behind a paywall will get enough traction. $\endgroup$
    – Gordon Royle
    Commented Feb 9, 2021 at 9:40
  • $\begingroup$ @GordonRoyle check here for example. I absolutely agree with you on that. This is a more general problem; a lot of science is behind paywalls today (and then is it still science?). $\endgroup$
    – pglpm
    Commented Feb 9, 2021 at 13:02

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