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Perhaps this question overlaps with similar ones, ... but I want to focus on a particular possible cause of confusion. I notice that students are often confused by the concepts of "infinite" and "unbounded". So, when asked if the set of invertible matrices is compact, they reply "no, because there are an infinite number of matrices with non zero determinant, therefore the set is unbounded". Actually this happens in Italian, where the corresponding words ("infinito" and "illimitato") are almost synonyms in everyday language. Does this happen in English too, or other languages?. I wonder: what if we chose another name for the two concepts? Would they make this mistake anyway? One way to check this would be to compare with what happens in other languages, where perhaps the words chosen do not create the confusion. Do you have other examples of this situation? Can you suggest different math concepts which in one language are named with synonyms, but not in another? Do you know if this problem has been studied anywhere?

Charles Matthews
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Diego Matessi
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  • Your actual question seems to be more specific and linguistically oriented than your title question. (I'm sure everyone would agree that the answer to the title question is a resounding yes.) Would you consider editing the title to better match the question? – Pete L. Clark Jun 18 '10 at 12:00
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    Infinite and unbounded probably don't have much of a distinction for nonmathematicians in English either, but they make sense once one gets used to them.

    Most of my fellow students (even those not intending to become mathematicians) find the terminology mostly logical, although Spiro's answer may be an exception...nevertheless, people make mistakes with definitions without regard to other definitions anyway, and it's part of learning the definition in the first place, so I think that for the most part students should just be encouraged to think longer.

     – Jason Polak Jun 18 '10 at 13:25
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    @Pete L. Clark. I don't know how to change the question. The reference to languages is just a suggestion on how one could find out whether it is the names which cause the confusion. Simply because other languages may have chosen better names. The answers that are coming in are in theme. Thank you all... – Diego Matessi Jun 18 '10 at 14:09
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    Suppose, in your language, the word for "multiplication" resembles a nasty word. Perhaps that would be distracting for students. In English, in some 18th century writing, we find a polynomial is called "sexual" if it has degree six. We don't say that any more! – Gerald Edgar Feb 27 '12 at 19:08
  • I feel "adjoint" also falls under this category – IAmNoOne Sep 29 '16 at 06:32

20 Answers20

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"Open" and "closed". Every reasonable human being on the planet, who has not studied topology, will assume that something can either be open or closed, but not both. This often causes students to make statements like "Set A is open, therefore it is not closed, thus ..."

Rune
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  • The majority of open sets are not closed... There is a notion of "closed form"; however "open forms" don't exist. – Wadim Zudilin Jun 18 '10 at 13:52
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    Perhaps open sets should be called exposed sets, and closed sets should be called clothed sets (you can see the interior through a dotted line, but the solid line prevents you from looking inside). This would be consistent with some pictures in $R^2$ and $R^3$ students see early on.

    And then, later, you can talk about sets being clothed and exposed. If a student says that a set cannot be exposed and clothed, you just have to mention Lady Gaga.

     – Marty Jun 18 '10 at 15:18
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    At least concluding a set is not closed because it's open makes sense for a proper, non-empty subset of a connected space. Worse is the reasoning "Since $A$ is not open, it must be closed." – PersonX Jun 18 '10 at 16:03
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    Can't resist quoting a comment from another thread: Munkres is fond of saying: "Sets are not doors!". Also, there are, of course, spaces with clopen sets. – Victor Protsak Jun 19 '10 at 02:22
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    It's a bit clunky, perhaps, but I always thought 'enclosed' for closed and 'edgeless' for open captured the intuition from open/closed intervals without introducing problems that occur by the natural linguistic analogy to doors. 'Enclosed' because you can't get out (with a sequence), 'edgeless' because, well, there's no edges. – Ketil Tveiten Jun 19 '10 at 12:12
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    But it can be ajar! – timur Jan 20 '11 at 02:54
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    This linguistic wrinkle has evidently occurred to a number of people. To follow on Victor Protsak's comment, Kelley's General Topology has a humorous and not-very-serious exercise on "door spaces" -- such a space is a topological space in which every subset is either open or closed. – Carl Offner Feb 29 '12 at 02:10
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My favourite is this: some books use "completely reducible" for semi-simple and "irreducible" for simple. As a result, every irreducible module is completely reducible.

Spiro Karigiannis
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    "Semisimple" is not without its perils: for example, 1-dimensional abelian Lie algebra is simple, but it's not semisimple! (There are good reasons for making this choice: I just want to point out that making sure that naive parsing works isn't always the goal.) – Victor Protsak Jun 19 '10 at 02:20
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If you count "or" as a mathematical concept, the the fact that it is fundamentally inclusive in mathematics but often exclusive in most other uses of English can lead students to mistakes.

Q.Q.J.
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    This reminds me about the French version of "or" and a related puzzle which I learned from Don. He can't find an explanation of why one of the extended letters on French keyboards does not come as the whole word in which it is only used. And this is the word very often used in maths! (As this is a puzzle I am not supposed to provide the answer, but I know that it is not very obvious: ù.) – Wadim Zudilin Jun 18 '10 at 12:13
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    Où in fact translates to where, which is nevertheless common in mathematics. (Ou, without the accent, is French for or.) – Niel de Beaudrap Jun 18 '10 at 13:04
  • @Niel: I said "this reminds me" and "a related puzzle": ou reminded me about où. – Wadim Zudilin Jun 18 '10 at 13:58
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    "Or" is not the same as "exclusive or" in English if there are three or more disjuncts. "A or B or C" in English commonly means "exactly one of A, B, C" whereas "A xor B xor C" is true if and only if an odd number of disjuncts holds. – Timothy Chow Jun 18 '10 at 14:04
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    And it gets even more xorsting as the number of disjuncts increases! – Q.Q.J. Jun 18 '10 at 16:00
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    You just need a little xorcise... – Victor Protsak Jun 19 '10 at 09:08
  • I always thought that "or" in English is inclusive (same as the mathematical meaning). If you want to use xor in English you use "either or". Maybe some grammarphile can chime in here though. – Tony Huynh Feb 29 '12 at 04:44
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I'm surprised nobody has mentioned "one-to-one function" for injection and "one-to-one correspondence" for bijection.

JeffE
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  • I hadn't realized that in teaching (maybe because I teach subjects where those terms rarely arise), but now that you mention it that did confuse me when I was a student... – Michael Lugo Feb 29 '12 at 00:13
  • I didn't even realize this was a thing! For me, the word "correspondence" just means "binary relation between two sets." – goblin GONE Mar 05 '16 at 08:02
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It's not just students who get confused by terminology. I was recently puzzled for quite a while until I realised that finite von Neumann algebras can be infinite-dimensional.

Woett
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Peter Shor
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My favourite example is "complex analysis" (as well as "complex" and "imaginary" numbers). Students, mostly in advance, feel it too complex. There should be probably a better name but it's too late to change...

Wadim Zudilin
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    A different example is "continued fraction", with the Russian literal translation "continuous fraction". But they are quite discrete! Especially classical ones (which many of my Russian colleagues call "chain fraction" as obviously a better name coming from German). – Wadim Zudilin Jun 18 '10 at 11:40
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    I agree that "complex" has unfortunate overtones, but it does allow one to say things like "Let's complexify this to simplify it.". On second thoughts, perhaps that's a good reason to find a new term for it! – Andrew Stacey Jun 18 '10 at 11:46
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    And how did we ever allow ourselves to say "complex simple Lie algebras"!? – John Stillwell Jun 18 '10 at 12:03
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    If the French revolution could produce a new measurement system from whole cloth and manage to convince the world that it's better than its prdecessor, I think it would be a simple task to revise the unfortunate terminology of 'real', 'imaginary', and 'complex' and propagate it forward. The only question is whether mathematicians decide that this particular piece of unfortunate nomenclature is a 'real' problem, or an 'imaginary' one. – Niel de Beaudrap Jun 18 '10 at 13:01
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    Niel: Gauss complained about the real/imaginary terminology. A quote: "That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question." – KConrad Jun 18 '10 at 19:49
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    Wadim, I am sure you've heard about the professor who wrote a very complicated equation on the board and with his characteristic modesty, called it $\mathit{ordinary}$ differential equation (cited by V.I. Arnold). – Victor Protsak Jun 19 '10 at 02:16
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    Excellent point, except for the "too late to change" part. If it helps that much, we should probably just do it. There's only a finite amount of historical work to rephrase, but there's a potentially unbounded number of future students to benefit. – Noah Jun 19 '10 at 22:47
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    And let's not forget the seemingly oxymoronic "simplicial complex." – Jesse Madnick Jul 19 '10 at 02:44
  • We should use Function theory for Complex analysis. – timur Jan 20 '11 at 02:53
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    More stories of this sort: http://golem.ph.utexas.edu/category/2007/11/the_dangers_of_complex_analysi.html – Kevin O'Bryant Feb 27 '12 at 18:23
  • When I was an undergraduate student, I had been puzzled to find a book in the math library entitled "Complex multiplication"... – ACL Feb 28 '12 at 08:56
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    @ACL, my advisor mentioned that a middle-school teacher came into his office, saw Serre's "Introduction to arithmetic", and wondered if it might be useful for her courses …. – LSpice Feb 17 '16 at 21:13
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The article "Surprises from mathematics education research: student (mis)use of mathematical definitions" by Edwards and Ward addresses some of your concerns in the context of U.S. undergraduates.

From the introduction:

...[T]asks involving the definitions of "limit" and "continuity," for example, were problematic for some of the students. Ward's intuitive reaction was that those words were "loaded" with connotations from their nonmathematical use and from their less than completely rigorous use in elementary calculus. He said, "I'll bet students have less difficulty or, at least, different difficulties with definitions in abstract algebra. The words there, like 'group' and 'coset,' are not so loaded."

...

He was surprised to see his algebra students having difficulties very similar to those of Edwards's analysis students. (So he lost his bet.) In particular, he was surprised to see difficulties arising from the students' understanding of the very nature of mathematical definitions, not just from the content of the definitions.

Jonas Meyer
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    Here's a link to the pdf from Ward's website, in case you don't have JSTOR: http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf – Jonas Meyer Jul 19 '10 at 00:47
  • That is an interesting article, but its relevance to the present question seems to be confined to the distinction the authors draw between word definitions that are "extracted" (from use) and "stipulated" (by arbitrary rules). If anything, their "surprising" conclusion, based on limited data, is that even in the absence of an obvious connotation, students presented with a new, entirely abstract definition will try to relate it to mental images of things they have seen and used before rather than proceed literally from the definition. Duh! (They passed the Turing test.) – Victor Protsak Jul 19 '10 at 06:39
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    There are many questions in the present question, but among them is to what extent do the words' ordinary meanings affect students' misunderstandings. The authors of this article claim that the effect didn't show up in their study; "coset" has no ordinary meaning as opposed to "limit", and yet similar problems were observed among undergraduates regarding misuse of both. At least, that is what I took out of it, and why I thought it was relevant. – Jonas Meyer Jul 19 '10 at 17:30
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    The idea that 'group' (or 'ring' or 'field' or 'sheaf') doesn't have everyday meaning is a little strange …. – LSpice Feb 17 '16 at 21:14
  • I agree with L. Spice, looking back at this; why would "group" be less loaded than "continuity"? – Jonas Meyer Mar 01 '17 at 01:38
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Not only is the word "complex" a problem, the word "simple" is too. I mean, how "simple" are the sporadic simple groups? I have to tell my students that "simple" does not mean "not complicated", but rather "cannot be simplified further."

John Stillwell
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I don't know of any research on this question specific to learning about mathematics. But the question opens up a big can of academic worms, outside of mathematics.

In linguistics, the Whorf hypothesis (sometimes called the Sapir-Whorf hypothesis) can be summarized as the notion that different peoples have different languages (syntax, lexicon, etc..), and these differences influence how they think about things. For example, different languages have different tenses available for use -- does this affect how speakers perceive time?

So, I'd say to start by looking up the Whorf hypothesis -- maybe it's been considered by some applied linguists studying education.

The other linguistic can of worms is the use of metaphor in mathematical language. Some words we use are directly visual, like "smooth" and "compact", some are strange (to me) analogies like "sheaf" and "flabby", and others are part of larger metaphorical systems like "consider a variety over a finite field" (the use of the positional word "over", to express dependence like a building resting on its foundation).

If you want to read up on these aspects of mathematical language, I'd recommend books by the Berkeley linguist Lakoff -- the classic "Metaphors we live by", and the application to mathematics in "Where mathematics comes from". Not that I agree with everything in the latter book, but it's an interesting read. I don't think you can address your question seriously without reviewing the linguistics literature.

Marty
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    I do not think that the Whorf hypothesis is taken very seriously these days. – Chris Godsil Jun 18 '10 at 22:22
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    I don't think it's taken seriously by those who follow Steven Pinker's criticisms -- certainly it's been taken down on a number of occasions. But I think it still rears its head in some contexts. The question definitely suggests Whorf to me... – Marty Jun 18 '10 at 23:26
  • I agree with Marty, this Whorf hypothesis seems related. Also, it seems like one of the take-away messages I got from reading 1984, though I never knew it had a name. Thanks for writing this answer – David White Feb 27 '12 at 23:02
  • Actually these days it seems linguists (patterning after Noam Chomsky) think that "language affects cognition in /no/ way". So yes, a can of worms. – Elizabeth S. Q. Goodman Feb 29 '12 at 00:06
  • I don't think this question has anything to do with what Whorf was suggesting. (And for that matter, I'm under the impression that what people refer to as the "Whorf hypothesis" is arguably rather stronger than Whorf himself would have argued for.) In any case, what's at issue here is simple confusion of terminology. That happens in any language, in any field, and has nothing to do with the idea (quite likely wrong, or at least vastly overstated) that the inherent structure of a language has some determining effect on how its speakers view the world at large. – Carl Offner Feb 29 '12 at 02:02
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This is very elementary, but I find it surprisingly common: many students talk about "infinite" numbers. You know, like

$$0.3333333\ldots$$

Victor Protsak
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  • The problem here is not terminology, but that while such elementary concepts such as this are taught, teachers stay as far away as possible from talking about infinity. There is no opportunity to discuss the difference between an infinitely long representation of a number, and a number which is 'semantically' infinite. Not that I advocate teaching infinite cardinals in grade-school, but if we did (or taught some other notion of infinity with some amount of seriousness), an immediate remedy would be to point out that 0.333333... is bounded between 0.3 and 0.4. – Niel de Beaudrap Jun 19 '10 at 09:35
  • Right. I suspect that teachers themselves are far from comfortable with infinity, and not just grade school teachers, either. – Victor Protsak Jun 19 '10 at 10:29
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    I am far from comfortable with infinity as well... Fortunately understanding infinity is not a necessary condition to work in mathematics. – darij grinberg Jul 18 '10 at 21:11
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    I think this is a symptom of the teaching of most students wherein Calculus is the pinnacle of their education in mathematics. If someone takes a calculus class, doesn't get an A on the sequences and series chapter and never thinks about sequences and series again, what can we expect them to pass on to others besides "funny things happen at infinity" ? – stankewicz Jan 20 '11 at 14:59
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Oh, and “convex”. As far as convex sets go, the mathematical usage accords well with everyday language. Not so with convex (and concave) functions. Educationists (?) have tried to remedy this by using terms “concave up” and “concave down” in calculus textbooks, a usage that I detest. (I have a hard time remembering which is which of those two.)

Edited based on comments: It seems that in Russian, "convex" can refer a surface curved outward, where in English usually "convex" refers to a solid whose surface curves outward. Perhaps that's why in complex geometry, one might consider a "convex domain" or its boundary, a "convex hypersurface".

Elizabeth S. Q. Goodman
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Harald Hanche-Olsen
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  • At least in America I hear the terms "concave up" and "concave down." The justification, I think, is that one thinks of a little cave whose mouth is pointed up or down. – Qiaochu Yuan Jun 19 '10 at 03:28
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    Qiaochu, this is a classic situation where a picture is worth a thousand words. When I teach it, positive derivative $\iff \cup$ and negative derivative $\iff\cap.$ Any language-based metaphor, no matter how clever, is bound to fail. – Victor Protsak Jun 19 '10 at 04:07
  • (derivative $\to$ second derivative) – Victor Protsak Jun 19 '10 at 09:11
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    Sometimes I just throw up my hands and say "has positive second derivative"; the time it takes to say the few extra words is repaid in the larger number of students who understand. – Michael Lugo Jun 19 '10 at 13:46
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    Ah, right. It's concave up/down, not convex up/down. I'll edit the answer accordingly. Now if we insist on language-based metaphors, wouldn't “smiling” and “frowning” work better? (At least in 1 dimension.) – Harald Hanche-Olsen Jun 19 '10 at 14:05
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    Curiously, it is convex up/down in Russian. And which is which is clear, due to the ordinary (non-mathematical) meaning of the word. – Sergei Ivanov Jun 19 '10 at 18:59
  • So a function which is “convex up” is convex? Because the graph looks convex when you look up at it, from below? Or is it concave, because the graph looks convex when you see it from its up side? I still find it confusing. But it could be caused by a slightly different usage of words like up and down in Russian vs English. – Harald Hanche-Olsen Jun 19 '10 at 19:55
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    No, "convex up" is concave (has negative second derivative). The word "convex" (adjective) in Russian means "bent outwards", and "convexity" (noun) typically means a fragment of a surface resulting from such bending. For example, a low quality table surface may have "convexities" (small hill-shaped defects), and they are clearly convex up (i.e., a layer of the surface is bent up from its normal place). – Sergei Ivanov Jun 19 '10 at 20:50
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    "Smiling" and "frowning", suggested by Harald a couple of comments above, suggest a great mnemonic: a function is happy if its second derivative is positive, and sad if it is negative. – Federico Poloni Feb 27 '12 at 18:53
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    Why not "hill" and "valley"? – Gejza Jenča Feb 28 '12 at 22:29
  • @Sergei Ivanov: Wow, I bet that explains a lot of the trouble I have with the use of "convex" in symplectic/contact and complex geometry. – Elizabeth S. Q. Goodman Feb 29 '12 at 00:11
  • Convex/concave in Chinese is 凸/凹. For functions it is unfortunately exactly the opposite of what the characters look like... – Holden Lee Jun 06 '20 at 01:02
  • Actually according to Chinese Wikipedia (https://zh.wikipedia.org/wiki/%E5%87%B9%E5%87%BD%E6%95%B0) some people use the opposite convention, which accords more with everyday usage. The opposite convention would mean that the "convex" in "convex" set is the "concave" in "concave" function... It seems a large part of the problem is that mathematically, "convex" sets and "convex" functions belong together (ex. in convex optimization), but visually, "convex" sets look more like "concave" functions (looking from the bottom). – Holden Lee Jun 06 '20 at 01:18
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I nominate “trivial”. It can be rather confusing that something can be trivial, but not trivially so.

Harald Hanche-Olsen
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    "Trivial" as an adjective to a result is indeed subjective. But "trivial" as math concept is not and is what you expect: trivial group, trivial subgroup, etc. – lhf Jun 18 '10 at 13:12
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    @LHF: First of all, in your examples of trivial being used as a math concept, it is also being used as an adjective (and indeed, this is always the case). Grammar aside, I think I disagree with your premise: the adjective "trivial" applied mathematically is context dependent and open to interpretation. For instance is a "trivial torsor" a torsor under a trivial group? what does it mean to say that a group or a space has "trivial cohomology"? Also we had a MO question about Bombieri's intended meaning of "trivial solution" to a certain Diophantine equation. – Pete L. Clark Jun 18 '10 at 17:00
  • ...and is the 'trivial map' a constant map, or the identity? – Ketil Tveiten Feb 28 '12 at 08:30
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Sequence vs series? Particularly if the two notions are introduced one right after the other in a calculus course, students are doomed to mix them up.

Alfonso Gracia-Saz
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    But kids are doomed to mix left and right (which are in most cases introduced to them more or less consecutively) and at that point the two words probably do not mean anything to them. Maybe the problem is that when one is introduced to too many things (i.e. two) essentially at once confusion is to be expected? – Mariano Suárez-Álvarez Jul 19 '10 at 02:30
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    It's a difficult distinction if it isn't stressed: a sequence is a mathematical object. A series is a type of expression. Equality and identity coincide in the former, not in the latter. – AndrewLMarshall Dec 21 '10 at 05:40
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In italian, the words bound and limit sound the same, "limite". This often causes confusion, like in the limit points and the boundary points of a set or a bounded function and its limit.

alberto.bosia
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3

An example is French module monogène for what is "cyclic module" in English. I can't prove that the possibility that a module monogène for a given ring may not be a groupe cyclique would be a stumbling block for a student; but it shows the phenomenon (avoid overloading). (I learned this when Serre pulled up an anglophone lecturer speaking in French on one occasion.)

Charles Matthews
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3

In french, a "monic polynomial" is a "polynôme unitaire", and a "unit vector" is a "vecteur unitaire". What happens to the students when they consider a dot product on a space of polynoms?

Julien Puydt
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    Similar situation in German (with 'normiert' instead of 'unitaire'). –  Feb 29 '12 at 00:52
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In french, a "unitary endomorphism" is an "endomorphisme orthogonal"... and an "orthogonal projection" is a "projection orthogonale" (the 'e' is for female, it's pronounced the same). You can hardly imagine how bad I feel when I have to tell my students that orthogonal projections aren't unitary...

Julien Puydt
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    French has thousands of homonymic pairs of words: most French people seem to understand pretty well the difference between aimait, aimais, aimé, aimée, aimés, aimées, say. Why would university students have any trouble dealing with a few words that have two meanings?! – Mariano Suárez-Álvarez Feb 28 '12 at 23:34
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    Oh, another one where the same happens in German. –  Feb 29 '12 at 00:54
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    @Mariano S-A: the words you cite are not all pronounced the same for one, and second, they're just variants of the same word depending on grammar, so there's definitely no trouble with them. – Julien Puydt Feb 29 '12 at 09:55
  • For matrices, there is also the positive entry / positive semi-definite problem. – reuns Aug 28 '16 at 17:02
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I nominate the word "kernel" as the most abused term in mathematics. The null space of a linear operator, a symmetric positive definite function on a set, a smooth probability density peaked -- am I missing any? I know these are all vaguely related, but the polysemy definitely causes much confusion for students.

Aryeh Kontorovich
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I think it takes sometime for starters to realize the tensor product between representations and tensor product between modules, despite the similarity on the surface that we can treat $g$-representations as $g$-modules. Usually a confusion appears when a concept which makes perfect sense in one area was re-defined or used in a more subtle way in the other area, which might be counter-intuitive in some sense. A concrete example comes up in my mind is Segal's paper on representations of compact Lie groups, where a lot of definitions are rather ad-hoc in modern literature but makes perfect sense when one read his paper careful enough.

Kerry
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Parallel to most people is a precise and useful term, describing for instance railway tracks, even when curved. But when we learn "Euclid's parallel postulate", it merely means "never meeting even if extended indefinitely". This causes difficulties for students introduced to non-Euclidean geometries.

maproom
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