What are the most overloaded words in mathematics? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-21T19:17:20Z http://mathoverflow.net/feeds/question/7389 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics What are the most overloaded words in mathematics? Qiaochu Yuan 2009-12-01T07:43:37Z 2010-06-21T05:12:42Z <p>This is community wiki. In each answer, please list <strong>one word</strong> 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.</p> <p>(This is mostly just for fun, but I'm also curious if there have been successful attempts to rename concepts that involve overused words.)</p> <p><strong>Edit:</strong> I may have been slightly unclear about the intent of this question.</p> <ul> <li>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.</li> <li>Different concepts named after the same mathematician, while potentially confusing, are understandable. </li> <li>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."</li> </ul> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7390#7390 Answer by Qiaochu Yuan for What are the most overloaded words in mathematics? Qiaochu Yuan 2009-12-01T07:48:07Z 2009-12-02T15:30:06Z <p><strong>Separable</strong></p> <ul> <li><a href="http://en.wikipedia.org/wiki/Separation_axiom" rel="nofollow">Separation axioms ($T_0$,$T_1$, etc.)</li> <li><a href="http://en.wikipedia.org/wiki/Separable_space" rel="nofollow">Separable space</a> (countable dense subset)</li> <li>Separable differential equation</a></li> <li>Separable scheme (although analogous at least in spirit to the Hausdorff axiom)</li> <li>Separable field extensions / polynomials</li> <li>Separable subgroup (ie a subgroup that's closed in the profinite topology)</li> <li>Separable quantum state (it means mixed unentangled)</li> </ul> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7391#7391 Answer by Elizabeth S. Q. Goodman for What are the most overloaded words in mathematics? Elizabeth S. Q. Goodman 2009-12-01T07:50:17Z 2010-04-22T17:17:09Z <p>Regular. To start off:</p> <p>The regular representation of a group $G$ over a field $k$ is the action on $k[G]$ given by group multiplication.</p> <p>A topology is regular if a closed set and a point not in that set can be separated by disjoint open sets.</p> <p>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 ${|z-\zeta_0|&lt;\delta}\cap D$ which is negative everywhere, tends to 0 at $\zeta_0$, but $\limsup(\omega(z))&lt;0$ as $z$ tends to any other boundary point of $D$ within distance $\delta$ of $\zeta_0$.</p> <p>I've borrowed/paraphrased the following from the <a href="http://en.wikipedia.org/wiki/Regular" rel="nofollow">Wikipedia disambiguation page</a> 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.</p> <p><a href="http://en.wikipedia.org/wiki/Regular_cardinal" rel="nofollow">Regular cardinal</a>, a cardinal number that is equal to its cofinality</p> <p><a href="http://en.wikipedia.org/wiki/Regular_category" rel="nofollow">Regular category</a></p> <p><a href="http://en.wikipedia.org/wiki/Regular_element" rel="nofollow">Regular element</a>, and regular sequence and regular immersion.</p> <p><a href="http://en.wikipedia.org/wiki/Regular_code" rel="nofollow">Regular code</a>, an algebraic code with a uniform distribution of distances between codewords</p> <p><a href="http://en.wikipedia.org/wiki/Regular_graph" rel="nofollow">Regular graph</a>, a graph such that all the degrees of the vertices are equal</p> <p>The <a href="http://en.wikipedia.org/wiki/Szemer%C3%A9di_regularity_lemma" rel="nofollow">regularity lemma</a>, which has nothing to do with regular graphs</p> <p><a href="http://en.wikipedia.org/wiki/Regular_polygon" rel="nofollow">Regular polygon</a>, and regular polyhedron</p> <p><a href="http://en.wikipedia.org/wiki/Regular_prime" rel="nofollow">Regular prime</a>: a prime $p$ that does not divide the class number of the $p$th cyclotomic field $\mathbb Q[\zeta_p]$.</p> <p><a href="http://en.wikipedia.org/wiki/Irregularity_of_a_surface" rel="nofollow">Regular surface</a> in algebraic geometry</p> <p><a href="http://en.wikipedia.org/wiki/Elliptic_operator#Regularity_properties" rel="nofollow">Regularity of an elliptic operator</a></p> <p>JS Milne's comment: A regular map is a morphism of algebraic varieties.</p> <p><a href="http://en.wikipedia.org/wiki/Critical_point_(mathematics)#Definition_for_maps" rel="nofollow">Regular value of a differentiable map</a></p> <p><a href="http://en.wikipedia.org/wiki/Regular_ring" rel="nofollow">Regular ring</a> (Note: this definition can be made noncommutative. A right noetherian ring <em>R</em> is said to be <em>right regular</em> if every finitely generated right <em>R</em>-module has finite global dimension. See Lam's <em>Lectures in Modules and Rings</em>, Section 5G.)</p> <p><a href="http://en.wikipedia.org/wiki/Von_Neumann_regular_ring" rel="nofollow">(von Neumann) Regular ring</a></p> <p><a href="http://en.wikipedia.org/wiki/Regular_language" rel="nofollow">Regular language</a>, a language that can be accepted by a finite state machine.</p> <p><a href="http://books.google.com/books?id=u1TmA-lm7loC&amp;lpg=PA62&amp;ots=Fc43T5JMXd&amp;dq=absolutely%20regular%20beta%20mixing&amp;pg=PA62#v=onepage&amp;q=absolutely%20regular%20beta%20mixing&amp;f=false" rel="nofollow">Absolutely regular</a> is a synonym for $\beta$-mixing in stochastic processes.</p> <p><a href="http://en.wikipedia.org/wiki/Matroid#Regular_matroids" rel="nofollow">Regular matroid</a>, 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. </p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7392#7392 Answer by Qiaochu Yuan for What are the most overloaded words in mathematics? Qiaochu Yuan 2009-12-01T07:51:12Z 2009-12-01T10:03:15Z <p><strong>Normal</strong></p> <ul> <li><a href="http://en.wikipedia.org/wiki/Normal_distribution" rel="nofollow">Normal distribution</a></li> <li><a href="http://mathworld.wolfram.com/NormalVector.html" rel="nofollow">Normal vector</a></li> <li><a href="http://en.wikipedia.org/wiki/Normal_space" rel="nofollow">Normal space</a></li> <li><a href="http://en.wikipedia.org/wiki/Normal_extension" rel="nofollow">Normal extension</a></li> <li><a href="http://en.wikipedia.org/wiki/Normal_subgroup" rel="nofollow">Normal subgroup</a></li> <li><a href="http://en.wikipedia.org/wiki/Normal_operator" rel="nofollow">Normal operator</a></li> <li><a href="http://en.wikipedia.org/wiki/Normal_convergence" rel="nofollow">Normal convergence</a></li> </ul> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7405#7405 Answer by Philipp Lampe for What are the most overloaded words in mathematics? Philipp Lampe 2009-12-01T09:23:19Z 2010-04-19T22:18:46Z <p><strong>Perfect</strong></p> <ul> <li><a href="http://en.wikipedia.org/wiki/Perfect_number" rel="nofollow">A perfect integer</a> is the sum of its proper divisors.</li> <li>A perfect complex is locally quasi-isomorphic to a bounded complex of finitely generated projective modules.</li> <li>A perfect field is a field whose algebraic extensions are all separable.</li> <li>A perfect square is a natural number of the form $n^2$ for some $n \in \mathbb{N}$.</li> <li>A <a href="http://en.wikipedia.org/wiki/Perfect_group" rel="nofollow">perfect group</a> is equal to its own commutator subgroup.</li> <li>A <a href="http://en.wikipedia.org/wiki/Perfect_set" rel="nofollow">perfect set</a> is a closed set with no isolated points.</li> <li>A <a href="http://en.wikipedia.org/wiki/Perfect_graph" rel="nofollow">perfect graph</a> is a graph such that each induced subgraph's chromatic number is equal to its clique number. </li> </ul> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7406#7406 Answer by Alon Amit for What are the most overloaded words in mathematics? Alon Amit 2009-12-01T09:27:55Z 2009-12-02T02:39:36Z <p><strong>Elliptic</strong>.</p> <ul> <li><a href="http://en.wikipedia.org/wiki/Elliptic%5Fintegral" rel="nofollow">Elliptic Integral</a></li> <li><a href="http://mathworld.wolfram.com/EllipticPartialDifferentialEquation.html" rel="nofollow">Elliptic Equation</a></li> <li><a href="http://en.wikipedia.org/wiki/Elliptic%5Foperator" rel="nofollow">Elliptic Operator</a></li> <li><a href="http://en.wikipedia.org/wiki/Elliptic%5Fcurve" rel="nofollow">Elliptic Curve</a></li> <li><a href="http://en.wikipedia.org/wiki/Elliptic%5Fpoint#Classification%5Fof%5Fpoints%5Fon%5Fa%5Fsurface" rel="nofollow">Elliptic(al?) Point</a></li> <li><a href="http://en.wikipedia.org/wiki/Elliptic%5Ffunction" rel="nofollow">Elliptic Function</a></li> <li><a href="http://en.wikipedia.org/wiki/M%C3%B6bius%5Ftransformation#Elliptic%5Ftransforms" rel="nofollow">Elliptic (Moebius) Transformation</a></li> </ul> <p>Of course, these are not entirely independent, but there are several distinct meanings involved.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7408#7408 Answer by Alon Amit for What are the most overloaded words in mathematics? Alon Amit 2009-12-01T09:35:01Z 2009-12-01T10:47:11Z <p><strong>Abelian</strong>.</p> <ol> <li><a href="http://en.wikipedia.org/wiki/Abelian%5Fgroup" rel="nofollow">Abelian Group</a> (also other commutative algebraic structures, and related structures like Abelian extensions)</li> <li><a href="http://en.wikipedia.org/wiki/Abelian%5Fand%5Ftauberian%5Ftheorems" rel="nofollow">Abelian theorem</a></li> <li><a href="http://en.wikipedia.org/wiki/Abelian%5Fvariety" rel="nofollow">Abelian Variety</a> (as well as surface)</li> <li><a href="http://en.wikipedia.org/wiki/Abelian%5Ffunction" rel="nofollow">Abelian function</a></li> <li><a href="http://en.wikipedia.org/wiki/Abelian%5Fintegral" rel="nofollow">Abelian integral</a></li> <li><a href="http://en.wikipedia.org/wiki/Abelian%5Fcategory" rel="nofollow">Abelian Category</a></li> <li>Abelian equation (used in <a href="http://projecteuclid.org/DPubS?verb=Display&amp;version=1.0&amp;service=UI&amp;handle=euclid.bams/1183548587&amp;page=record" rel="nofollow">web geometry</a>, also appears as Abelian relation)</li> </ol> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7409#7409 Answer by Alon Amit for What are the most overloaded words in mathematics? Alon Amit 2009-12-01T09:40:14Z 2009-12-02T15:33:24Z <p><strong>Primitive</strong>.</p> <ul> <li>Primitive polynomial (in the sense of finite field theory, namely minimal polynomial of field generator)</li> <li>Primitive polynomial (in the sense of ring theory, namely gcd of coefficients is 1)</li> <li>Primitive element (and primitive extension)</li> <li>Primitive function (antiderivative)</li> <li>Primitive permutation group (no non-trivial equivalence relation preserved)</li> <li><a href="http://mathworld.wolfram.com/PrimitivePolytope.html" rel="nofollow">Primitive polytope</a> (rarely used, I think).</li> <li><a href="http://en.wikipedia.org/wiki/Primitive%5Fring" rel="nofollow">(left) Primitive ring</a></li> <li>Primitive recursion (in logic and complexity theory)</li> </ul> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7412#7412 Answer by John D. Cook for What are the most overloaded words in mathematics? John D. Cook 2009-12-01T10:39:50Z 2010-02-11T20:39:27Z <p><strong>Obvious</strong></p> <p>"'Obvious' is the most dangerous word in mathematics." -- E. T. Bell </p> <p>Example: all examples You are using to answer this post are obvious.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7419#7419 Answer by QPeng for What are the most overloaded words in mathematics? QPeng 2009-12-01T11:45:07Z 2009-12-01T20:24:37Z <p><strong>Base/Basis</strong></p> <ul> <li><a href="http://en.wikipedia.org/wiki/Base%5F%28group%5Ftheory%29" rel="nofollow">Group Base</a></li> <li><a href="http://en.wikipedia.org/wiki/Base%5F%28topology%29" rel="nofollow">Topological Base/Basis</a></li> <li><a href="http://en.wikipedia.org/wiki/Basis%5F%28universal%5Falgebra%29" rel="nofollow">Algebra Basis</a></li> <li><a href="http://en.wikipedia.org/wiki/Basis%5F%28linear%5Falgebra%29" rel="nofollow">Vector Space Basis</a></li> <li><a href="http://en.wikipedia.org/wiki/Base%5F%28mathematics%29" rel="nofollow">Logarithm Base</a></li> </ul> <p>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.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7420#7420 Answer by Marcin Kotowski for What are the most overloaded words in mathematics? Marcin Kotowski 2009-12-01T11:55:22Z 2009-12-01T11:55:22Z <p><strong>Natural</strong></p> <p>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. </p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7426#7426 Answer by lhf for What are the most overloaded words in mathematics? lhf 2009-12-01T12:51:53Z 2009-12-02T07:03:19Z <p><strong>Simple</strong></p> <ul> <li>Simple field extension</li> <li>Simple group</li> <li>Simple ring</li> <li>Simple module</li> <li>Simple algebra</li> <li>Simple graph</li> <li>Simple polygon</li> <li>Simple curve</li> <li>Simple zero</li> <li>Simple function</li> <li>Simple connectivity</li> </ul> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7428#7428 Answer by Deane Yang for What are the most overloaded words in mathematics? Deane Yang 2009-12-01T13:23:41Z 2009-12-01T13:23:41Z <p>"Let" (which does not meet the 15 character minimum)</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7431#7431 Answer by Scott Carter for What are the most overloaded words in mathematics? Scott Carter 2009-12-01T13:53:38Z 2009-12-01T13:53:38Z <p>Closed:</p> <p>Closed set Closed surface Closed geodesic Closed function This question is closed :-)</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7437#7437 Answer by Andrew Stacey for What are the most overloaded words in mathematics? Andrew Stacey 2009-12-01T14:55:19Z 2009-12-01T14:55:19Z <p>Hedgehog.</p> <p>Just <strong>one</strong> use of this word in mathematics is "overuse".</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7440#7440 Answer by Greg Kuperberg for What are the most overloaded words in mathematics? Greg Kuperberg 2009-12-01T15:02:30Z 2009-12-01T15:02:30Z <p><strong>Deep</strong></p> <p>I'm not sure whether it has one meaning or zero. Either way, I think that it is deeply overused.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7453#7453 Answer by Ady for What are the most overloaded words in mathematics? Ady 2009-12-01T16:22:18Z 2009-12-01T16:22:18Z <p><strong>Space</strong></p> <p>Affine space</p> <p>Banach space</p> <p>Cauchy space</p> <p>Euclidean space</p> <p>Function space</p> <p>Hardy space</p> <p>Hilbert space</p> <p>Inner product space</p> <p>Kolmogorov space</p> <p>Krein space</p> <p>Klein space</p> <p>Pontrjagin space</p> <p>Lp space</p> <p>Measure space</p> <p>Metric space</p> <p>Minkowski space</p> <p>Normed vector space</p> <p>Locally convex space</p> <p>Linear topological space </p> <p>F-space</p> <p>Frechet space</p> <p>Nuclear space</p> <p>Operator space</p> <p>Affine space</p> <p>Projective space</p> <p>Polish space</p> <p>Quotient space</p> <p>Sobolev space</p> <p>Topological space</p> <p>Uniform space</p> <p>Vector space</p> <p>Harmonic spaces </p> <p>Conformal space </p> <p>Complex analytic space</p> <p>Affinely connected space</p> <p>Algebraic space</p> <p>Symplectic space</p> <p>Measurable space</p> <p>Measure space</p> <p>Probability space</p> <p>Riemann space</p> <p>Lorentzian space</p> <p><em>And so on.</em></p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7464#7464 Answer by Theo Johnson-Freyd for What are the most overloaded words in mathematics? Theo Johnson-Freyd 2009-12-01T17:45:15Z 2009-12-01T17:45:15Z <p>Well-defined. Overused not because it has too many definitions but because it has too few.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7495#7495 Answer by mathreader for What are the most overloaded words in mathematics? mathreader 2009-12-01T20:03:31Z 2009-12-02T01:01:02Z <p><strong>trivial</strong></p> <p>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:</p> <p>trivial group</p> <p>trivial representation</p> <p>trivial topology</p> <p>trivial solution (in ODE/PDE)</p> <p>etc.</p> <p>Sometimes it produces confusion as it is not quite clear which sort of triviality is meant.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7515#7515 Answer by Ady for What are the most overloaded words in mathematics? Ady 2009-12-01T22:23:40Z 2009-12-01T22:23:40Z <p><strong>Reflexive</strong> (relation, locally convex (Banach) space, operator algebra, module, a.s.o.)</p> <p>It is an adjective.</p> <p><strong>Proposition</strong> <em>Every infinite dimensional von Neumann algebra is reflexive, and also it is not reflexive.</em></p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7536#7536 Answer by Harrison Brown for What are the most overloaded words in mathematics? Harrison Brown 2009-12-02T01:12:57Z 2010-06-21T05:12:42Z <p><strong>Uniform</strong>. 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.</p> <ul> <li><a href="http://en.wikipedia.org/wiki/Uniform_polytope" rel="nofollow">uniform polytope</a></li> <li><a href="http://en.wikipedia.org/wiki/Uniform_convergence_%28combinatorics%29" rel="nofollow">uniform convergence in machine learning</a> (related but not the same)</li> <li><a href="http://en.wikipedia.org/wiki/Uniform_distribution" rel="nofollow">uniform distribution</a></li> <li><a href="http://en.wikipedia.org/wiki/Uniform_convergence" rel="nofollow">uniform convergence</a></li> <li><a href="http://en.wikipedia.org/wiki/Uniform_continuity" rel="nofollow">uniform continuity</a></li> <li><a href="http://en.wikipedia.org/wiki/Uniform_integrability" rel="nofollow">uniform integrability</a></li> <li><a href="http://en.wikipedia.org/wiki/Uniformly_bounded" rel="nofollow">uniform boundedness</a></li> <li><a href="http://en.wikipedia.org/wiki/Equicontinuity" rel="nofollow">uniform equicontinuity</a></li> <li><a href="http://ncatlab.org/nlab/show/uniform+space" rel="nofollow">uniform space</a> (from uniform continuity)</li> <li>The Riemann <a href="http://en.wikipedia.org/wiki/Uniformization_theorem" rel="nofollow">uniformization theorem</a></li> <li>uniform circuit family (complexity theory)</li> <li>Gowers uniformity norms</li> <li><a href="http://planetmath.org/encyclopedia/UniformModule.html" rel="nofollow">uniform modules</a></li> <li><a href="http://en.wikipedia.org/wiki/Matroid#Uniform_matroids" rel="nofollow">uniform matroid</a></li> </ul> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7562#7562 Answer by Alison Miller for What are the most overloaded words in mathematics? Alison Miller 2009-12-02T06:00:59Z 2009-12-02T06:00:59Z <p>Spectrum.</p> <p>From <a href="http://en.wikipedia.org/wiki/Spectrum#Mathematics" rel="nofollow">http://en.wikipedia.org/wiki/Spectrum#Mathematics</a>:</p> <p>Spectrum (homotopy theory)</p> <p>Spectrum of a matrix, in linear algebra</p> <p>Spectrum of an operator, in functional analysis (a generalisation of the spectrum of a matrix)</p> <p>Spectrum of a ring, in commutative algebra</p> <p>Spectrum of a C*-algebra</p> <p>Spectrum of a theory, in mathematical logic</p> <p>Stone space of Boolean algebra</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7563#7563 Answer by Zev Chonoles for What are the most overloaded words in mathematics? Zev Chonoles 2009-12-02T06:01:55Z 2009-12-02T06:01:55Z <p><strong>Complete/Completion</strong></p> <p>complete metric spaces,</p> <p>complete measure spaces,</p> <p>completing a ring at an ideal,</p> <p>complete graph</p> <p>complete category</p> <p>complete lattice</p> <p>and many more uses (a lot in computation theory/logic) at</p> <p><a href="http://en.wikipedia.org/wiki/Completeness" rel="nofollow">http://en.wikipedia.org/wiki/Completeness</a></p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7565#7565 Answer by moby for What are the most overloaded words in mathematics? moby 2009-12-02T06:38:43Z 2009-12-02T06:38:43Z <p>Canonical... would be a canonical example. I guess.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7574#7574 Answer by Jakob Blaavand for What are the most overloaded words in mathematics? Jakob Blaavand 2009-12-02T12:05:04Z 2010-04-20T14:29:06Z <p><strong>Generically</strong></p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/7575#7575 Answer by Gil Kalai for What are the most overloaded words in mathematics? Gil Kalai 2009-12-02T12:15:12Z 2009-12-22T19:55:39Z <p>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".</p> <p>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.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/8499#8499 Answer by David Roberts for What are the most overloaded words in mathematics? David Roberts 2009-12-10T23:26:07Z 2009-12-10T23:26:07Z <p><strong>Nice</strong></p> <p>Mostly because it is such a local word - 'such and such is called 'nice' if...'</p> <p>Segal once formally defined a 'nice simplicial space' - these days called simplicial spaces satisfying the Segal condition, a very sensible move.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/9565#9565 Answer by Ian Weiner for What are the most overloaded words in mathematics? Ian Weiner 2009-12-22T22:28:07Z 2009-12-22T22:28:07Z <p>Index, Order, and Rank certainly qualify</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/9567#9567 Answer by Dan Piponi for What are the most overloaded words in mathematics? Dan Piponi 2009-12-22T22:36:57Z 2009-12-22T22:36:57Z <p>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?</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/9580#9580 Answer by Jon Awbrey for What are the most overloaded words in mathematics? Jon Awbrey 2009-12-23T02:28:03Z 2009-12-23T02:42:11Z <p><strong>-ary</strong></p> <p><strong>-ary</strong>, as in $k$-ary numeral $s$, refers to the number $k$ of values in the domain $K = \lbrace 0, 1, \ldots, k-1 \rbrace$ that affords the basis of numeration.</p> <p><strong>-ary</strong>, as in $k$-ary relation $L$, refers to the number of domains $X_1, \ldots, X_k$ for which $L \subseteq X_1 \times \ldots \times X_k$.</p> <p><strong>-ary</strong>, as in $k$-ary operation $f$, refers to the number of domains in the domain of the function $f : X_1 \times \ldots \times X_k \to Y$, the rubric being, "a $k$-ary operation is a $(k+1)$-ary relation".</p> <p>Some writers use Greek roots and the Greek suffix "-adic" for the number of domains in a relation, hence <em>medadic</em>, <em>monadic</em>, <em>dyadic</em>, <em>triadic</em> for relations of 0, 1, 2, 3 places, respectively. This usage actually has a degree of historical precedence and it can serve to sidestep conflicts with the domainance of binary numerals in our modern world, but of course the wrinkle but moves to other domains where writers are adicted to other habits.</p> <p><strong>NB.</strong> All puns are intended.</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/15030#15030 Answer by kakaz for What are the most overloaded words in mathematics? kakaz 2010-02-11T20:45:08Z 2010-04-20T22:08:10Z <h2>Elementary</h2> <p>Sense 1 = basic, simple, concerning the <em>elements</em> ("first steps") of a subject.</p> <ul> <li>elementary fact</li> <li>elementary introduction</li> <li>elementary problem</li> <li>elementary proof</li> </ul> <p>Sense 2 = treating mathematical objects as elements of a collection.</p> <ul> <li>elementary number theory</li> </ul> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/15035#15035 Answer by userN for What are the most overloaded words in mathematics? userN 2010-02-11T21:45:19Z 2010-02-11T21:45:19Z <p>Admissible, a colorless synonym for "which lies in the class of objects we're studying".</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/21926#21926 Answer by akopyan for What are the most overloaded words in mathematics? akopyan 2010-04-20T03:49:48Z 2010-04-20T03:49:48Z <p>"Cauchy theorem"</p> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/22024#22024 Answer by vonjd for What are the most overloaded words in mathematics? vonjd 2010-04-21T07:33:46Z 2010-04-21T07:33:46Z <p>Fundamental theorem:</p> <pre><code>* Fundamental theorem of algebra * Fundamental theorem of arithmetic * Fundamental theorem of calculus * Fundamental theorem of curves * Fundamental theorem of cyclic groups * Fundamental theorem of surfaces * Fundamental theorem of finitely generated abelian groups * Fundamental theorem of Galois theory * Fundamental theorem on homomorphisms * Fundamental theorem of linear algebra * Fundamental theorem of projective geometry * Fundamental theorem of Riemannian geometry * Fundamental theorem of stochastic calculus * Fundamental theorem of vector analysis * Fundamental theorem of linear programming </code></pre> http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/23074#23074 Answer by Scott Morrison for What are the most overloaded words in mathematics? Scott Morrison 2010-04-30T04:54:24Z 2010-04-30T04:54:24Z <p>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".</p> <p>(Yes, this really happened, and ended happily, or at least not in Guantanamo.)</p>