# What are the most overloaded words in mathematics? [closed]

This is community wiki. In each answer, please list one word at the top and below that list as many different meanings of that word in mathematics as you can think of, preferably with links or definitions. ("Adjective" and "adjective noun" count as the same adjective.) People should edit previous answers as appropriate.

(This is mostly just for fun, but I'm also curious if there have been successful attempts to rename concepts that involve overused words.)

Edit: I may have been slightly unclear about the intent of this question.

• When I say "overused" I don't mean "used too often," I mean "used in too many different ways." So I'll change the title of the question to reflect this.
• Different concepts named after the same mathematician, while potentially confusing, are understandable.
• I mostly had in mind adjectives that get recycled in different disciplines of mathematics. Different uses of the same noun tend to be less confusing, e.g. the example of "space" below. I think it's good to be intentionally vague about what we consider a "space."
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## closed as no longer relevant by Harry Gindi, Noah Snyder, Qiaochu Yuan, François G. Dorais♦, Scott Morrison♦Apr 30 '10 at 4:56

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I'm -1'ing this because I think it's a rather uninteresting question. – Kevin H. Lin Dec 1 '09 at 12:47
I'm +1'ing it because the answers are useful. – Greg Kuperberg Dec 1 '09 at 18:54
Here's a harder and possibly even more useful project: grouping the multiple answers into related meanings. Is that worth it? – Elizabeth S. Q. Goodman Dec 3 '09 at 6:13
I don't like the question because it invites too many bad answers. I (-1)ed a bunch of answers for the following reason. Just because an adjective is used with many different nouns or a word is used frequently doesn't mean that it's overloaded; the use may be perfectly consistent. I realize the the question was edited, so this is a slightly unfair criticism. – Anton Geraschenko Dec 14 '09 at 16:55
I'm voting to close; I didn't really see the point of this when it was first posted, and I still don't really see the point now. – Kevin H. Lin Apr 20 '10 at 22:27

Regular. To start off:

The regular representation of a group $G$ over a field $k$ is the action on $k[G]$ given by group multiplication.

A topology is regular if a closed set and a point not in that set can be separated by disjoint open sets.

A point $\zeta_0$ on the boundary of a domain in $\mathbb C$ is called regular if there exists a subharmonic barrier function $b(z)$ defined within $D$ near $\zeta$. This may not be the standard definition but Gamelin's complex Analysis defines it as a subharmonic function $\omega(z)$ on $\{|z-\zeta_0|<\delta\}\cap D$ which is negative everywhere, tends to 0 at $\zeta_0$, but $\limsup(\omega(z))<0$ as $z$ tends to any other boundary point of $D$ within distance $\delta$ of $\zeta_0$.

I've borrowed/paraphrased the following from the Wikipedia disambiguation page but removed a couple that either are not too relevant to pure math or qualify the "regularity" more. Feel free to put them in too.

Regular cardinal, a cardinal number that is equal to its cofinality

Regular category

Regular element, and regular sequence and regular immersion.

Regular code, an algebraic code with a uniform distribution of distances between codewords

Regular graph, a graph such that all the degrees of the vertices are equal

The regularity lemma, which has nothing to do with regular graphs

Regular polygon, and regular polyhedron

Regular prime: a prime $p$ that does not divide the class number of the $p$th cyclotomic field $\mathbb Q[\zeta_p]$.

Regular surface in algebraic geometry

Regularity of an elliptic operator

JS Milne's comment: A regular map is a morphism of algebraic varieties.

Regular value of a differentiable map

Regular ring (Note: this definition can be made noncommutative. A right noetherian ring R is said to be right regular if every finitely generated right R-module has finite global dimension. See Lam's Lectures in Modules and Rings, Section 5G.)

(von Neumann) Regular ring

Regular language, a language that can be accepted by a finite state machine.

Absolutely regular is a synonym for $\beta$-mixing in stochastic processes.

Regular matroid, a matroid which is representable over every field. In this sense, all graphs are regular (their cycle matroids are regular), which has nothing to do with regular graphs.

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"Strong" and "weak" are also used quite a bit. I would say that these are not overloaded a la programming languages in the same "strong" sense that "regular" is. – Steve Huntsman Dec 1 '09 at 8:06
I suggest the word "denominal" for "regular element" in a ring, because it makes for a nice denominator :) And then the related terms "denominal sequence" instead of "regular sequence", "denominal immersion" instead of "regular immersion"... – Andrew Critch Dec 1 '09 at 8:09
regular map: morphism of algebraic varieties – JS Milne Dec 1 '09 at 8:12
All right, I've tried to add those and am now done editing. Please do add more and organize or paraphrase as you see fit! I am not familiar with all of the terms; that's why it's community wiki. – Elizabeth S. Q. Goodman Dec 1 '09 at 8:21
I've added two notions of "regular ring." I would add a third notion of Artin-Schelter regular ring, if I was familiar enough with the concept. – Manny Reyes Dec 1 '09 at 23:26
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en.wikipedia.org/wiki/Normal_(mathematics) has the whole list! – Harry Gindi Dec 1 '09 at 8:57
What makes this worse is that it often means something different that "regular." If somebody can teach me a way to remember the difference between a regular space and a normal sapce, I'd be extremely grateful. – John Goodrick Dec 1 '09 at 18:56
@John: Sometimes a picture is all it takes: brownsharpie.courtneygibbons.org/?p=395 – Loop Space Dec 1 '09 at 21:37
@Andrew: Now I'm curious, but I can't get that to load (internal server error). Do you know what's wrong? – Elizabeth S. Q. Goodman Dec 3 '09 at 6:22

The most overloaded word in mathematics is the empty word. The one that comes between $a$ and $b$ in $ab$, meaning multiplication. Or does it mean the binary operator in a more general monoid or group? Or one of the two binary operators in a ring? Or the action of a monoid or group on a set, or the action of the base ring on a module? (And if so, is it a left or right action?) Or the application of a function (or functor) on its argument? Or even three or four of these in one expression, or, even worse, two at the same time in the very same place, exploiting associativity to ensure the ambiguity is mostly harmless? Or one of countless other things?

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Perfect

• A perfect integer is the sum of its proper divisors.
• A perfect complex is locally quasi-isomorphic to a bounded complex of finitely generated projective modules.
• A perfect field is a field whose algebraic extensions are all separable.
• A perfect square is a natural number of the form $n^2$ for some $n \in \mathbb{N}$.
• A perfect group is equal to its own commutator subgroup.
• A perfect set is a closed set with no isolated points.
• A perfect graph is a graph such that each induced subgraph's chromatic number is equal to its clique number.
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I never understood why people call it a perfect square, rather than a square. (frustrating) – muad Apr 19 '10 at 22:21
There are also perfect codes, which are those that attain the Hamming bound; see en.wikipedia.org/wiki/Hamming_bound#Perfect_codes – John Stillwell Apr 20 '10 at 3:55
Perfect post.......LOL – The Mathemagician Apr 20 '10 at 4:24

Simple

• Simple field extension
• Simple group
• Simple ring
• Simple module
• Simple algebra
• Simple graph
• Simple polygon
• Simple curve
• Simple zero
• Simple function
• Simple connectivity
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I disagree with this one as being "overuse" because all of the ones that I know about do have some connection to the ordinary meaning of the word "simple". – Loop Space Dec 2 '09 at 8:34
The word "overused" was changed to "overloaded" in the question, but I can see your point. – Jonas Meyer Dec 2 '09 at 9:58

Obvious

"'Obvious' is the most dangerous word in mathematics." -- E. T. Bell

Example: all examples You are using to answer this post are obvious.

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I think it's synonym "clearly" is even more dangerous. (At least, I rarely write "obviously", but I do write "clearly" all the time to justify statements that should be true, but I can't explain why very clearly). – Ilya Grigoriev Dec 19 '09 at 20:41
I love when a famous mathematician says it's obvious and it takes me 5 pages to justify it. That's him just being too lazy to write it out and he makes you feel like an idiot for not using the Force to see it like he does. – The Mathemagician Apr 20 '10 at 4:07

If you happen to work on del Pezzo surfaces, don't make the mistake of standing in an airport security line talking about "blowing up a plane at eight points".

(Yes, this really happened, and ended happily, or at least not in Guantanamo.)

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This answers belongs to the mathematical jokes question not here. (A good joke though.) – Gil Kalai Apr 30 '10 at 7:16

Natural

Very often I read things like "Now, it is natural to ask...", ""X is a natural generalization of Y" or "A natural question is..." when the problems are by no means natural, and the feeling of "naturalness" is only achieved post factum.

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To complicate matters, "natural" has a technical meaning in category theory that may or may not coincide with the colloquial use of the word. – John D. Cook Dec 1 '09 at 16:54

Hedgehog.

Just one use of this word in mathematics is "overuse".

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I got 143 hits on MathSciNet (that was in 'anywhere') and 127 on the arXiv. My favourite is "Hedgehogs of Hausdorff dimension one". – Loop Space Dec 1 '09 at 15:19
Really? I appreciate the use of funny words in mathematics. – Qiaochu Yuan Dec 1 '09 at 16:16
Really? It should be used more often. "Wait, by 'hedgehog," do you mean a topological space, a Lie superalgebra, or a cohomology theory?" – John Goodrick Dec 1 '09 at 18:53
Vocabulary is a barrier in any subject, though. All I'm saying is math would be more fun for me if I could use words like "strange" and "charm" like them physicists. – Qiaochu Yuan Dec 2 '09 at 17:42
I was thinking something similar about the word "coconut." No one wants to study a coconut or it's dual, the cococonut. – Bill Kronholm Apr 20 '10 at 15:10

Separable

• Separation axioms ($T_0$,$T_1$, etc.)
• Separable space (countable dense subset)
• Separable differential equation
• Separable scheme (although analogous at least in spirit to the Hausdorff axiom)
• Separable field extensions / polynomials
• Separable subgroup (ie a subgroup that's closed in the profinite topology)
• Separable quantum state (it means mixed unentangled)
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Although these are all used in completely different ways, because they do all (at least, those that I know what they mean) actually link to the common notion of separation then I don't find them at all confusing. – Loop Space Dec 1 '09 at 8:46
I've never understood what having a countable dense subset has to do with the common notion of separation. – Alison Miller Dec 2 '09 at 2:48
Well, for the real line (say) it means that every pair of distinct real numbers is separated by a rational number. – Qiaochu Yuan Dec 2 '09 at 3:01
Also, there's separated subsets of a topological space: A and B are separated if they are respectively contained in disjoint open sets. A space X is disconnected iff it is the union of two separated sets, which is sometimes called a "separation" of X. (Munkres, for instance, uses this terminology.) – Nate Eldredge Apr 20 '10 at 14:26
@Qiaochu: that shows why one wants to say something like “the reals are separated by the rationals”. But a space could just as well be separated in that sense by some uncountable set — so this would justify a terminology like countably separable,κ-separable, etc., but not why separable is used just for the countable case. – Peter LeFanu Lumsdaine Dec 19 '10 at 18:01

Elliptic.

Of course, these are not entirely independent, but there are several distinct meanings involved.

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I'll let someone else do "hyperbolic" :-) – Alon Amit Dec 1 '09 at 9:28
Parabolics are more mysterious! ;-) E.g. consider parabolic subroup/subalgebra. – mathreader Dec 2 '09 at 2:05
Go for it... . – Alon Amit Dec 2 '09 at 3:34

Deep

I'm not sure whether it has one meaning or zero. Either way, I think that it is deeply overused.

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It has at least one meaning, as a term of art in model theory: some of us study things called "deep theories." (Sounds impressive, right?) wilfridhodges.co.uk/mathlogic03.pdf – John Goodrick Dec 1 '09 at 18:45
Deep hole: eom.springer.de/D/d120080.htm. And if something with great depth is deep, some modules may be deep: eom.springer.de/D/d031180.htm Nonetheless I agree that it is often used without meaning. – Jonas Meyer Dec 2 '09 at 9:43
deep learning: en.wikipedia.org/wiki/Deep_learning – mkatkov Jun 13 '12 at 5:20

trivial

Besides being a synonym to 'obvious', like in 'the proof is trivial', it has the meaning of 'shallow' ('the question is trivial') and moreover denotes a bunch of mathematical notions:

trivial group

trivial representation

trivial topology

trivial solution (in ODE/PDE)

etc.

Sometimes it produces confusion as it is not quite clear which sort of triviality is meant.

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I'm with you. And worse, for trivial topology, I've heard people mean both "indiscrete topology" and "contractible" because contractible spaces, of course, are "topologically trivial" – Charles Siegel Dec 1 '09 at 23:30
I think this should be split. I agree on the casual use (first sentence) but disagree on the others because they do connect to the ordinary meaning of the word "trivial" and so I don't regard them as "overuse". Charles' point about "trivial topology" is more a question of ambiguous use than overuse. – Loop Space Dec 2 '09 at 8:37

Primitive.

• Primitive polynomial (in the sense of finite field theory, namely minimal polynomial of field generator)
• Primitive polynomial (in the sense of ring theory, namely gcd of coefficients is 1)
• Primitive element (and primitive extension)
• Primitive function (antiderivative)
• Primitive permutation group (no non-trivial equivalence relation preserved)
• Primitive polytope (rarely used, I think).
• (left) Primitive ring
• Primitive recursion (in logic and complexity theory)
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Admissible, a colorless synonym for "which lies in the class of objects we're studying".

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But, like "nice" mentioned above, a good term to use if you eventually want this class of objects to be named after you :) Do you have some more colorful alternatives that you prefer? – Nate Eldredge Apr 20 '10 at 14:31
Teleman classes. :) – userN Apr 20 '10 at 22:32
This is so true. Admissible is the least useful adjective ever. – Harry Gindi Apr 30 '10 at 4:59

"Cauchy theorem"

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The word "stable" is used in many different contexts. Also "elementary" has many usages. The word "lattice" has two entirely different meanings which are ar time confusing. So is the word "field".

I have two more: "deterministic" refers sometimes as "not random" and it is also a central concept in computational complexity where "non deterministic" has another meaning (very different from random). The word "classical" is used in various confusing ways.

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Can you comment about where "stable" is used confusingly? The only applications I know of are differential equations and homotopy theory, and I never thought these two notions could be confused somehow ... – Konrad Voelkel Dec 15 '09 at 13:05
Interestingly, both pairs of meanings you mention are in fact distinguished in German: Verband / Gitter and Körper / Feld. Since I assume the origin is German, I wonder why this got lost in translation. "Bond" could've been used for lattices in logic, and, obviously, "body" for fields in algebra. Too late now. – Tilman Apr 20 '10 at 18:09

Nice

Mostly because it is such a local word - 'such and such is called 'nice' if...'

Segal once formally defined a 'nice simplicial space' - these days called simplicial spaces satisfying the Segal condition, a very sensible move.

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So, if you want to have something named after you ... ;-) – Kevin H. Lin Dec 22 '09 at 20:11
Good is used in the same way... – Adam Gal Apr 21 '10 at 10:54

Complete/Completion

complete metric spaces,

complete measure spaces,

completing a ring at an ideal,

complete graph

complete category

complete lattice

and many more uses (a lot in computation theory/logic) at

http://en.wikipedia.org/wiki/Completeness

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I disagree that these are "overuse" as they do all link to the idea of "nothing missing" (at least, the ones that I know about do). – Loop Space Dec 2 '09 at 8:38
@AndrewStacey, it seems that that logic would rule out most of the answers; for example, one could argue that (to pick one that's relevant to me) the word 'good' is not overused because, in all contexts, it refers to some desireable quality. – L Spice Feb 13 '14 at 14:54

Spectrum.

Spectrum (homotopy theory)

Spectrum of a matrix, in linear algebra

Spectrum of an operator, in functional analysis (a generalisation of the spectrum of a matrix)

Spectrum of a ring, in commutative algebra

Spectrum of a C*-algebra

Spectrum of a theory, in mathematical logic

Stone space of Boolean algebra

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I disagree! Most of these are closely related, and it's right that they should be denoted by the same word. It's only the homotopy usage that's really different. E.g. let T be a linear operator on a fin-dim vector space V. Then T has a spectrum, its set of eigenvalues. But T also gives rise to a ring, the subring of End(V) generated by T. The set of prime ideals of this ring - its spectrum - is in one-to-one correspondence with the set of eigenvalues. (The prime ideals are (T-lambda), where lambda is an eigenvalue.) So, the two uses of "spectrum" are really the same. – Tom Leinster Dec 2 '09 at 15:54
How "spectral sequences" fit it? – Gil Kalai Dec 22 '09 at 19:58

Abelian.

1. Abelian Group (also other commutative algebraic structures, and related structures like Abelian extensions)
2. Abelian theorem
3. Abelian Variety (as well as surface)
4. Abelian function
5. Abelian integral
6. Abelian Category
7. Abelian equation (used in web geometry, also appears as Abelian relation)
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They are all connected to the mathematician Abel! I think if you were going to say something like this the word Riemann or Euler should come to mind first. – Steven Gubkin Dec 1 '09 at 15:01
I'm willing to let things named after mathematicians slide. – Qiaochu Yuan Dec 1 '09 at 16:13
To be fair, abelian groups, abelian varieties, and abelian categories are all pretty closely related. – Harrison Brown Dec 2 '09 at 0:53
Well... I guess, although I don't know exactly how "abelian variety" is related to "abelian group", other than it's a group (not necessarily abelian!) If I knew what "abelian group" was and had to guess what "abelian variety" means, I would surely have failed miserably. – Alon Amit Dec 2 '09 at 1:57
According the the wikipedia page, en.wikipedia.org/wiki/Abelian_variety , the group law of an abelian variety is always commutative. – Chris Schommer-Pries Feb 11 '10 at 21:55

Closed:

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I'm not convinced that "closed" is overused. It is certainly used a lot, but is it really too much? – Loop Space Dec 1 '09 at 14:56
"Closure" certainly has many meanings. – lhf Dec 1 '09 at 15:46
"Closed" is certainly overused in Mathoverflow ;-) – vonjd Apr 21 '10 at 7:32

Uniform. Most of these do have the intuitive sense of "being locally the same everywhere," but by no means all of them, and their sheer number gets pretty confusing.

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Base/Basis

Edit: It's been clarified that we're really more interested in adjectives but I think the use of base in these examples are quite substantially different.

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Reflexive (relation, locally convex (Banach) space, operator algebra, module, a.s.o.)

Proposition Every infinite dimensional von Neumann algebra is reflexive, and also it is not reflexive.

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P.S. The above (intentionally confusing) Proposition is both true and non-trivial. – Ady Dec 3 '09 at 19:40

"Let" (which does not meet the 15 character minimum)

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I agree with Deane. It's too easy to write prose of the form "Let n be a natural number. Let m = n^2 + 1." The two uses of "let" mean quite different things. While there's no danger of misinterpretation, it has a slightly jarring effect, and it's one of those things that makes the text just a little bit less transparent, just a little bit less easy to read. It would be better to change the second "let" to something like "put" or "write". – Tom Leinster Dec 1 '09 at 15:02
Fix, of course! – Harry Gindi Dec 15 '09 at 3:50

Canonical... would be a canonical example. I guess.

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No, that would be a bad example. – Harry Gindi Dec 15 '09 at 3:49

Index, Order, and Rank certainly qualify

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Well-defined. Overused not because it has too many definitions but because it has too few.

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– Qiaochu Yuan Dec 1 '09 at 17:49
I think Gowers has effectively ended this long-suffering confusion with this blog post-glad you found it,Qiachu. – The Mathemagician Apr 20 '10 at 4:04

Generically

A word used a lot when you don't want to precisely specify under which conditions something is true, but its true in most cases. An example would be that generically all square matrices are invertible.

The precise meaning of this - at least in algebraic geometry - is that whatever property we are talking about is true on a dense open subset. Another example would be given a function between two smooth manifolds then a generic point in the target manifold is not a critical value of the function.

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