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I would like to ask if anyone could share any specific experiences of discovering nonequivalent definitions in their field of mathematical research.

By that I mean discovering that in different places in the literature, the same name is used for two different mathematical objects. This can happen when the mathematical literature grows quickly and becomes chaotic and of course this could be a source of serious errors. I have heard of such complaints by colleagues, mostly with respect to definitions of various spaces and operators, but I do not recall the specific (and very specialised) examples.

My reason for asking is that I'm currently experimenting with verification using proof assistants and I'd like to test some cases that might be a source of future errors.

EDIT: Obviously, I would be interested to see as many instances of this issue as possible so I would like to ask for more answers. Also, it would be even more helpful if you could provide references.

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    $\begingroup$ A classic example is "compact", which can mean either "Hausdorff and every open cover has a finite subcover", or just "every open cover has a finite subcover". $\endgroup$
    – Arturo Magidin
    Commented Nov 22, 2017 at 19:12
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    $\begingroup$ @ArturoMagidin This leads to the classic dialogue "The space $X$ is quasicompact, so..." "What does quasicompact mean?" "It means compact." $\endgroup$
    – Will Sawin
    Commented Nov 22, 2017 at 19:35
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    $\begingroup$ In my experience at least, the only people who think "compact" includes Hausdorff are algebraic geometers. I've never heard a topologist use "compact" to mean anything other than "every open cover has a finite subcover". $\endgroup$
    – John Pardon
    Commented Nov 23, 2017 at 3:29
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    $\begingroup$ @JohnPardon: there is also a cultural thing here. In France, it is standard to include "Hausdorff" in the definition of a compact space. $\endgroup$
    – Taladris
    Commented Nov 23, 2017 at 14:49
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    $\begingroup$ @Taladris, maybe in france, everyone's an algebraic geometer $\endgroup$
    – Vivek Shende
    Commented Nov 29, 2017 at 0:54

71 Answers 71

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The wide range of choices in the definition of an automorphic form is particularly annoying. Depending on the purposes, it could be a meromorphic function fully invariant by a certain discrete group of transformation, a holomorphic function almost-invariant, a differential form, a subrepresentation of an $L^2$ space, a classical or an adelic object, a solution to a partial differential equation, etc.

These are sometimes related, sometimes definitely different, and the lack of vocabulary consistency in such an active field requires some care when dealing with object (leading authors to usually restate a precise definition of the objects they are dealing with, or at least quoting the literature associated with the paradigm they have chosen.

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Graphs.

Some define them without loops or multiple edges. Some with both. They can also be directed or undirected too. Some define quivers to be directed graphs with multiple edges and loops which is a atart on the naming confusion.

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  • $\begingroup$ This was previously mentioned, twice. $\endgroup$
    – David White
    Commented Sep 12, 2023 at 17:20
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The word conjugate has several different meanings. Wikipedia lists 16 mathematical meanings, some of which are related, but many of which are very different.

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    $\begingroup$ Is this thread supposed to be about particular words that have disparate definitions or about particular concepts with actually conflicting definitions? $\endgroup$
    – Michael Hardy
    Commented Sep 12, 2023 at 2:14
  • $\begingroup$ ${}\qquad\uparrow\qquad\ldots\,$and now I see that I took it as the former in my posted answer to this question concerning the word partition, and as the latter in may posted answer about linear regression. $\endgroup$
    – Michael Hardy
    Commented Sep 12, 2023 at 18:06
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The notion of spectrum in operator theory. Some people assume that if $\lambda$ has a property that $T-\lambda I$ is injective with dense range which is not the whole space and with continuous inverse then $\lambda$ is not an element in spectrum since $(T-\lambda I)^{-1}$ can be extended continuously other authors consider such $\lambda$ as an element of spectrum.

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In set theory a forcing notion can be a pre order with a largest element or lowest element, depending on the style of the author.

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    $\begingroup$ This is a nonexample. Both versions are trivially equivalent. $\endgroup$
    – Andrés E. Caicedo
    Commented May 25, 2019 at 23:37
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Let $X_1,X_2,X_3,\ldots$ be independent random variables.

Sometimes it is said that a stopping time for this random process is a random variable $T$ for which the truth value of $T=n$ (for $n=1,2,3,\ldots$) is determined by the values of $X_1,\ldots,X_n.$

And sometimes it is said that a stopping time for this random process is a random variable $T$ for which the each of the events $\big[T=n\big]$ for $n=1,2,3,\ldots$ is independent of the sequence $X_{n+1}, X_{n+2}, X_{n+3},\ldots\,.$

The second definition is more inclusive than the first.

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"Linear regression"

Statisticians know better than to think that the term "linear regression" is so called because one is fitting a straight line. Often in linear regression one fits a parabola or a sinusoid or some other sort of functions via ordinary least squares, and the word "linear" is just as apt as when one fits a straight line, and is more appropriate than it would be if "linear regression" meant fitting a straight line, since, if "linear" referred to the function one is fitting rather than to how it is done, then the word "affine" would be more appropriate.

So where is the "other" definition --- the one according to which it does mean fitting a straight line? One example is when an astronomer taught a course in a statistics department on applications of statistics to astronomy, and reserved the word "linear" for fitting a straight line. Respectable scientists from outside of statistics do that.

Consider the model $$ Y_i = a + bx_i + cx_i^2 + \text{error}_i, $$ and the model $$ Y_i = a + b \cos x_i + c\sin x_i + \text{error}_i, $$ where the ordinary least-squares estimates are $\hat a, \hat b, \hat c.$

For either of those two models, the mapping $$ \left[ \begin{array}{c} Y_1 \\ \vdots \\ Y_n \end{array} \right] \mapsto \left[ \begin{array}{c} \hat a \\ \hat b \\ \hat c \end{array} \right] $$ (with the $x$s fixed) is linear.

Contrast this with the model \begin{align} & \operatorname{logit} \Pr(Y_i=1) = \operatorname{logit}( 1- \Pr(Y_i = 0) ) = a + bx_i, \\[8pt] & \text{where } \operatorname{logit} p = \log \frac p {1-p} \text{ for } 0<p<1. \end{align} (long "o", soft "g")

Here the usual estimates $\hat a, \hat b$ are not found by least squares, but rather \begin{align} \left(\hat a, \hat b \right) = {} & \operatorname*{argmax}_{a,b} \Pr( Y_1=y_1\ \& \cdots \&\ Y_n=y_n\mid a,b) \\[8pt] = {} & \operatorname*{argmax}_{a,b} \prod_{i\,:\,y_i\,=\,1} \operatorname{logit}^{-1}(a+bx_i) \\ & \qquad \times \prod_{i\,:\,y_i\,=\,0} \left( 1-\operatorname{logit}^{-1} (a+bx_i) \right). \end{align} This is one of many instances of nonlinear regression.

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  • $\begingroup$ Can the person who down-voted this explain? $\endgroup$
    – Michael Hardy
    Commented Oct 1, 2023 at 14:31
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sample

Thoroughly stir the population of jabberwockies and pick one at random, then stir again and pick another independently of the first, and so on until you have $50$ jabberwockies.

Do you have $50$ samples, or one sample consisting of $50$ independent observations?

It seems that in signal processing and data science and various other disciplines, this is called $50$ samples, and in statistics it is one sample.

Notice that the term "sample size" is commonplace: the size of this one sample is $50.$ And the two-sample t-test is called the two-sample t-test. A two-sample t-test may use one sample of $50$ northern jabberwockies and another sample of $40$ southern jabberwockies.

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Does entropy decrease or increase? What mathematicians call "entropy" is actually what physicists would call "free energy." Free energy includes a term proportional to entropy, but with a minus sign.

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When is a function concave? When is it convex? Do you determine this by looking at the graph "from above", or "from below"?

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  • $\begingroup$ I can never remember which way this goes without looking it up, but are there really varying conventions in upper-level mathematics (as opposed to calculus textbooks)? I thought one spoke exclusively about convex functions in, say, measure theory. $\endgroup$
    – LSpice
    Commented Nov 29, 2017 at 18:59
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    $\begingroup$ @LSpice Instead of looking it up, you could just look up (i.e., look at the graph from below) and you'd have the standard convention. $\endgroup$
    – Andreas Blass
    Commented Dec 1, 2017 at 23:55
  • $\begingroup$ @AndreasBlass, hah, very nice! Indeed, once I know the answer I can justify it ex post facto, but unfortunately that doesn't help me personally remember, as @‍RodrigoA.Pérez puts it, whether to look "from above" or "from below". Maybe thinking of it as looking up will help. $\endgroup$
    – LSpice
    Commented Dec 2, 2017 at 1:39
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    $\begingroup$ The definition of "convex function" is universally agreed and standard, luckily $\endgroup$
    – Pietro Majer
    Commented Jun 12, 2019 at 12:11
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I think the concept of simple Lie groups has some non equivalent definitions.

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    $\begingroup$ What are they?. $\endgroup$
    – LSpice
    Commented Sep 10, 2023 at 0:34
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    $\begingroup$ @LSpice I copy past the wikipedia text "Unfortunately, there is no universally accepted definition of a simple Lie group" for existence of several definitions. The same link contains definitions, references, etc $\endgroup$
    – Ali Taghavi
    Commented Sep 10, 2023 at 8:17
  • $\begingroup$ en.wikipedia.org/wiki/Simple_Lie_group#Definition $\endgroup$
    – Ali Taghavi
    Commented Sep 10, 2023 at 8:20
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