Timeline for Has incorrect notation ever led to a mistaken proof?
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| Oct 5, 2019 at 20:39 | comment | added | Toby Bartels | To really remove all ambiguity, I should have written $(\partial T(-,-)/\partial x)_y$ whenever I wrote $\partial T(-,-)/\partial x$ in my previous comment; the subscript indicates what quantity is being held fixed. (This is a point that @Kostya_I was making earlier.) | |
| Oct 5, 2019 at 18:13 | comment | added | Toby Bartels | @LSpice : That's not literally meaningful at all, and I have seen $\frac{\partial T(1.2,1.5)}{\partial x}$, especially in handwriting. In practice, if course, we understand it; in this example, it's $\partial T(x,y)/\partial x\rvert_{x=1.2,y=1.5}=D_1T(1.2,1.5)=2.4k$. But why not $\partial T(y,x)/\partial x\rvert_{y=1.2,x=1.5}=D_2T(1.2,1.5)=3.0k$? Only because we know that it's always $T(x,y)$ and never $T(y,x)$, which is a convention that we can reasonably adopt in this context but not across all of mathematics. | |
| Oct 4, 2019 at 16:41 | comment | added | LSpice | @TobyBartels, I'm not quite sure how to read the penultimate sentence of your comment, but isn't the usual "parenthetical notation" for evaluation of partial derivatives $(\partial T/\partial x)(1.2, 1.5)$ (arguably unambiguous since we already understand a coördinate system just by writing $\partial x$), or probably more usually $\dfrac{\partial T}{\partial x}(1.2, 1.5)$? | |
| Aug 20, 2018 at 7:23 | comment | added | Toby Bartels | … To really use notation like $T = k(x^2 + y^2)$, you need to have notation like $T|_{x=1.2,y=1.5}$, but if $T(1.2,1.5)$ is going to be ambiguous, then it's really not any worse, and better than $T(x,y)|_{x=1.2,y=1.5}$. Note that (in a context where it's clear that $y$ is to be held fixed, and assuming that $k$ has been constant all along), $\partial{T}/\partial{x}|_{x=1.2,y=1.5}$ works the same way; there is nothing like $\partial{T(1.2,1.5)}/\partial{x}$ that, when read literally, is asking for a partial derivative of a constant (in this case, of $3.69k$). [And now my comments are done!] | |
| Aug 20, 2018 at 7:17 | comment | added | Toby Bartels | … to go into that level of abstraction in an elementary course would be as out of place as it would be to go into a construction of the set of real numbers and prove the ordered-field axioms and other basic properties. (Which is to say, maybe for honour students, or for the French, but not for ordinary people, not even math majors at first.) But before the 20th century, Calculus was a theory of variable quantities (as Algebra usually still is, at least at first), as described in the question mathoverflow.net/questions/84221/… | |
| Aug 20, 2018 at 7:11 | comment | added | Toby Bartels | [Note: not a continuation of my previous comment, for once!] Eike is right that we should just write $T = k(x^2 + y^2) = kr^2$ and be done with it. I blame neither the scientists nor the mathematicians for this, but rather the mathematics teachers, who want to teach elementary Calculus (and increasingly elementary Algebra) as a theory of functions like Drey & Manogue's $T$ when it should really be taught (for applied purposes) as a theory of variable quantities like Eike's $T$. In terms of modern mathematics, a variable quantity is formalized as a map on an abstract manifold, but … | |
| Aug 20, 2018 at 7:03 | comment | added | Toby Bartels | … Note that if $T\colon \mathbb{R}^2 \to \mathbb{R}$ is a map (as Michael would say), then there is no corresponding ambiguity in the meaning of $D_1 T$ and $D_2 T$, which you might also write (adopting a common convention) as $\partial{T}/\partial{x}$ and $\partial{T}/\partial{y}$. But this $T$ can't actually be a temperature, but at best a map that, when applied to the coordinates of a point in some particular coordinate system laid on a flat object, yields the temperature at that point. Personally, I never mix the notations and always use $D_1 T$ and $D_2 T$ for maps. | |
| Aug 20, 2018 at 6:53 | comment | added | Toby Bartels | … To interpret $\partial{x}/\partial{r}$, you have to know what other variables are being held constant. As you can see at en.wikipedia.org/wiki/Heat_capacity there are two common precise meanings of heat capacity, one where pressure is held constant and one where volume is held constant. In some contexts, it's obvious which is meant, but in others, one had better say! I don't know if confusion between these has ever led to an error in print, but if it has, then that's an answer to Mike's question where the usual notation for partial derivatives is incorrect and at fault. | |
| Aug 20, 2018 at 6:46 | comment | added | Toby Bartels | @Kostya_I : Indeed, you should never use such an expression as $\partial{x}/\partial{r}$! Or rather, you should never use it out of the blue (with only $x$ and $r$ previously introduced) but only as an abbreviation after you've given enough additional context to understand its meaning. This is a real problem in, for example, thermodynamics, where heat capacity is defined as $\partial{Q}/\partial{T}$, the partial derivative of heat with respect to temperature. (I'm abstracting away the issue that $Q$ isn't meaningful, which isn't important since $\mathrm{d}Q$ is reasonably well-defined.) | |
| Aug 17, 2018 at 15:02 | comment | added | Qfwfq | @Carl Offner: it was a way to keep using the effect of the abuse of notation while keeping your conscience clean at the same time: you are writing down something that is essentially identical to the abuse of notation, so if you understand it that's fine, but if you have some doubts then the typography will notify you that a composition with a change of variables is understood. | |
| Aug 14, 2018 at 19:49 | comment | added | Wolfgang | Never seen such a bunch of comments to one answer yielding moreover such a substantial discussion! 😀 | |
| Aug 13, 2018 at 8:04 | comment | added | Eike Schulte | @Kostya_I I guess you’re right: Requiring $r$ to be a “coordinate function” in the sense that it belongs to some “invisible” set of coordinates is just as much abuse of notation as using $r$ to denote different things depending on context. So it comes down to preference in the end. Now of course, everybody would like their preferred way to be the standard way … | |
| Aug 12, 2018 at 21:18 | comment | added | Michael Bächtold | @Kostya_I I suspect we might be talking past each other, and this comment section is not the best place to clarify this. If you want we can continue this in chat. | |
| Aug 12, 2018 at 20:51 | comment | added | Kostya_I | @MichaelBächtold, I don't see how you can claim that $r$ is 'nothing else than a function', and still make sense of $\partial /\partial r$. Coordinate functions carry more structure than 'just functions'. Otherwise, you seem to imply that there's a notion of partial derivative for "a function of something" that is different from a notion of a partial derivative of a map from $\mathbb{R}^n$ to $\mathbb{R}$. I wonder what this notion is and of what use it is. | |
| Aug 12, 2018 at 19:28 | comment | added | Michael Bächtold | The real difficulty here seems to be due to the confusing terminology: to be a function in the modern sense is not the same as to be a function of something in the original sense of the word. (Or actually it is, but that will only cause more confusion. That is why I prefer to use the word map for modern functions.) | |
| Aug 12, 2018 at 19:26 | comment | added | Michael Bächtold | @Kostya_I: all those three meanings are the same and coincide with what Eike suggests: a coordinate in a chart is nothing else than a function $r:U\to \mathbb{R}$ on a manifold $M$ (defined on a subset $U$ of the manifold), and if you make yourself clear what it means to be "a function of something", then such a function $r$ can also be a function of other coordinate charts. So if $(x,y)$ are coordinates on the manifold, then such an $r$ will always be a function of them. | |
| Aug 12, 2018 at 19:20 | comment | added | Kostya_I | ... and $\partial r/\partial x$ is not a partial derivative, but a notation for a vector field $\partial/\partial x$ applied to $r$. This is already strange, since if it's denoted like a duck, walks like a duck, etc., why is it not a duck? But what's worse is that since we cannot just define $\partial/\partial \varphi$ for an arbitrary $\varphi$, we have to insist that $x$ is a special kind of function - a 'coordinate function'. This looks like an ad hoc solution to salvage an unsound notation, and essentially not much different from saying that $x$ also denotes a coordinate label. | |
| Aug 12, 2018 at 19:03 | comment | added | Kostya_I | I'd say that $r$ can have three different meanings: a function on a manifold, a label for a coordinate in a chart (as in $\partial x/\partial r$), and a function of coordinates in another chart (as in $\partial r/\partial x$.) A traditional way to proceed is to acknowledge this abuse of notation and live with that. A perfectly rigorous way would be to give three different names to these three objects; nobody does it as it would be confusing and impractical. A third option, as @EikeSchulte suggests, is to insist that $r$ is a function on a manifold, so it is never a function of $(x,y)$... | |
| Aug 12, 2018 at 12:01 | comment | added | Eike Schulte | @Kostya_I You can use $\partial f/\partial r$ just fine when you interpret $r$, $\theta$ as coordinate functions. You just have to be aware that it depends on the full set $\{ r, \theta \}$ and not $r$ alone. (Basically, $(\partial/\partial r, \partial/\partial \theta)$ is the dual basis to $(\mathrm{d}r, \mathrm{d}\theta)$ and that depends on both $r$ and $\theta$.) Of course, this dependency is hidden by the notation, but in praxis you use disjoint names for all sets of coordinates most of the time anyway. (And $x$ is just a (maybe local) function on the manifold, so you can use it for $f$.) | |
| Aug 12, 2018 at 10:55 | comment | added | Michael Bächtold | @Kostya_I: "we should never use expressions like $\partial x/\partial r$?" Are you asking this because you read the Jacobi post, or are you also asking about $dx/dr$ notation? I just answered an older question about the latter here. | |
| Aug 12, 2018 at 9:55 | comment | added | Kostya_I | @MichaelBächtold, indeed, it does make perfect sense for me now, thanks. But does it also mean that we should never use expressions like $\partial x/\partial r$? | |
| Aug 12, 2018 at 9:27 | comment | added | Michael Bächtold | @Kostya_I: the question doesn't say "The temperature is $T$ ..." rather it suggest that $T(x,t)$ is the temperature. If you switch notation to $T=k(x^2+y^2)$ you are interpreting $T,x,y$ as maps from a manifold to $\mathbb{R}$ and the equation makes perfect sense. | |
| Aug 12, 2018 at 9:20 | comment | added | Kostya_I | As for 'modern definition of function', $T$ is arguably not defined to be a function of two real variables. It is defined as a physical quantity, and no physical quantity is a function of several variables - they only become such if a coordinate chart is chosen. @TomLeinster, for "what is $T(1.2,1.5,)$", the answer is given above by LSpice. BTW, I really don't see how switching to $T=k(x^2+y^2)=kr^2$ would be of any help. On the left is a function (on a manifold) and on the right are symbolic expressions, how can they be equal? | |
| Aug 12, 2018 at 9:01 | comment | added | Kostya_I | @MichaelBächtold, $(x,y)$-plane and $(r,\theta)$-planes are two different copies of Euclidean plane. As such, they are isomorphic, but not canonically so. When you plugging $(r,\theta)$ into the function specified in $(x,y)$, you are implicitly using this isomorphism. But since it's not canonical, there's no single situation when it's useful. You may view this as a formalisation of what physicists mean by "you cannot add quantities of different dimensions" etc. | |
| Aug 12, 2018 at 0:42 | comment | added | Carl Offner | @Qfwfq: Does this really help? Even after reading what you wrote, I had to read it again before I saw the "little space". We really aren't expecting to notice such typographical niceties, I think. | |
| Aug 11, 2018 at 21:36 | comment | added | Qfwfq | To save the formal correctness of the notation and at the same time not be clunky, one could put a little space before the parentheses when a change of "chart"/variables is left implicit, for example: $T(x,y)=T\;(r,\theta)$. | |
| Aug 11, 2018 at 18:04 | comment | added | Tom Leinster | @Kostya_I: you say "which function we mean is indicated by what we plug in." So what if I ask you to plug in some specific numbers? E.g. what is $T(1.2, 1.5)$? | |
| Aug 11, 2018 at 17:15 | comment | added | Peter LeFanu Lumsdaine | Has this notational ambiguity ever led to an actual mistake in published work? It’s a neat shibboleth, and the discussion spawned in comments is great fun; but as far as the original question is concerned, it seems to be just a potential answer so far, not an actual one. But it does appear to have genuine potential for having caused actual mistakes — so does anyone know of any such? | |
| Aug 11, 2018 at 9:23 | comment | added | Michael Bächtold | @Kostya_I: It was you who suggested this was a 'gotcha' question. If you take the modern definition of function serious and consider the notation $T(x,y)$ in that context, then the only correct answer is B. Since for you it's obvious that A is correct, but you don't see that there must be "non-standard" semantics of $T(x,y)$ involved in arriving at A, I conclude that you're not using the modern definition. "What is the domain of exp?" I would never say it's the $x$-line or $y$-line or $\theta$-line. Could you define what that even means, and in which sense these lines differ? | |
| Aug 11, 2018 at 8:51 | comment | added | Kostya_I | As for 'alternative semantics', I am not sure what you mean. Semantics always depend on context. Here the notation T is overloaded - the same letter denotes several functions with different domains (physicists actually stress that domains are different by calling them $(x,y)$ plane and $(r,\theta)$ plane). This is common in math: e. g., what is the domain of exp? It could be numbers, matrices, operators, elements of a Lie group, etc., and easily more than one of them in the same equation. No confusion arises: which function we mean is indicated by what we plug in. Just as in our example. | |
| Aug 11, 2018 at 8:31 | comment | added | Kostya_I | @Michael, 'trick question'? Quite the contrary. Assuming that this is a meaningful problem, the only sensible answer is (A). It is to answer (B) that one has to assume that this is a trick question, having no actual math contents, with phrasing and notation intentionally chosen to confuse and distract. I do agree though that the question should be phrased more clearly if intended for students. But the point is: can you point out a situation other than an artificial teaching one where the intended answer would be (B)? | |
| Aug 11, 2018 at 6:03 | comment | added | Michael Bächtold | @Kostya_I : so what you’re saying is that we go through all the trouble of teaching students the meaning of the notations $T\colon \mathbb{R}^2 \to \mathbb{R}$ and $T(x,y)$, only to tell them later that if they applied it correctly to arrive at B they were posed a trick question? Because as far as I know, the alternative semantics of $T(x,y)$ which people are using to arrive at A are not clearly laid out in any book. Or do you know one? | |
| Aug 11, 2018 at 3:42 | comment | added | nick012000 | As a computer programmer, my initial reaction was to answer "B", personally. I just looked at it and went "Okay, it's a function with an input of two numbers, that's getting two numbers passed into it." | |
| Aug 11, 2018 at 2:45 | comment | added | Daniel R. Collins | @Kostya_I: For what it's worth, the paper I was reading today used $\pi$ as the prime-counting function. Probably the time before that it was used as the parameter for population proportion. The simple fact is, if you don't define it explicitly in the current piece of writing, then it's poorly-defined. | |
| Aug 10, 2018 at 21:14 | comment | added | Mateusz Kwaśnicki | One more story in the same vein: Our chemistry students asked me once for help, as they had been asked the following question at their final exam in probability: "The parameters of the normal distribution are: (a) $(0,1)$; (b) $(\mu, \sigma)$; (c) $(\mu, \sigma^2)$; (d) $(\alpha, \beta)$." I found this question really amusing. | |
| Aug 10, 2018 at 20:53 | comment | added | Kostya_I | I meant "Just like (B)", of course. | |
| Aug 10, 2018 at 20:44 | comment | added | Kostya_I | @MichaelBächtold, I don't agree that the notation $T(x,y)$ has only one established meaning. How about the answer "$T(r,\theta)$ is the field of fractions in formal variables $r,\theta$ over T"? Just like (A), it answers the question in a formally correct way, conforming to established notational convention, while completely ignoring the context. It is clear from the statement that $T$ is not a field or ring, but it equally clear that it is not a function of two real variables. So, the answer (B), which assumes that it is, is incorrect. | |
| Aug 10, 2018 at 20:14 | comment | added | Kostya_I | @DanielR.Collins, I think this quote has little to do with reality. While mathematicians 'may' use $i$ and $\pi$ as placeholders for arbitrary complex numbers, or $(r,\theta)$ for Cartesian coordinates of a point, in practice that never happens. Maybe apart from teaching students that it is formally possible. | |
| Aug 10, 2018 at 20:10 | comment | added | Mateusz Kwaśnicki | @LSpice: Well, of course I do not know what they think. I generally prefer readability to utmost formal rigour, so instead of "a function $f : \mathbb{R} \ni x \mapsto x^2$" or "a function $f$ defined by $f(x) = x^2$", I would write "a function $f(x) = x^2$". In any case, this is off-topic here. | |
| Aug 10, 2018 at 19:57 | comment | added | LSpice | @MateuszKwaśnicki, do you actually know that your referees are OK with "the function $\sin x$"? I'm not! (I also often badly want to make this complaint when refereeing, but I don't: I do two passes over a paper when I referee it, one in which I gripe to myself about everything that bothers me to get it out of my system, and another in which I come back and decide what's important enough to put in my report.) | |
| Aug 10, 2018 at 19:46 | comment | added | Michael Bächtold | @TimothyChow: I agree with that. I would only add that in principle there are ways of avoiding this abuse of notation, without making the notation more cumbersome. One was suggested by Eike. The other one would be to introduce a new notation like $T[x,y]$ instead of $T(x,y)$, to denote that the “observable” T is henceforth to be expressed in terms of $x,y$. As I wrote in the answer I linked to in my first comment, computer scientist have something quite similar called ascription. | |
| Aug 10, 2018 at 19:32 | comment | added | Timothy Chow | @MichaelBächtold : If this sort of thing showed up in an actual scientific paper, which it very well might, then 99 times out of 100 the intended meaning would be A, so interpretation B would certainly be a mistaken interpretation. In such a circumstance, I would not say that the notation is mistaken; at most, I would say that it's an "abuse" of notation, and mathematicians are not above abusing notation either. | |
| Aug 10, 2018 at 18:19 | comment | added | Mateusz Kwaśnicki | Huh, this reminds me how annoying I find it when a referee asks me to change "stochastic process $X_t$" to "process $X$" or "process $(X_t)$", but at the same time they find it perfectly OK to write "a function $\sin x$". Coming back to Corinne's Shibolleth, why there is no answer "D: both A and B"? :-) | |
| Aug 10, 2018 at 18:16 | comment | added | Michael Bächtold | @TimothyChow: I think this interpretation amounts to what Eike suggested in his last line, and I agree with it. But since the notation $f(x)$ has only one established meaning in modern mathematics, I’d argue that people choosing A are making the notational mistake. | |
| Aug 10, 2018 at 17:53 | comment | added | Timothy Chow | @MichaelBächtold : They're interpreting $T$ to be a function with two arguments, whereas $T$ is intended to be a function with a single argument (namely, a point in the plane), with $T(x,y)$ referring to the value of $T$ at the point whose Cartesian coordinates are $x$ and $y$ and $T(r,\theta)$ referring to the value of $T$ at the point whose polar coordinates are $r$ and $\theta$. Fans of answer B might be happier if $T$ were defined as $T(p) = x(p)^2 + y(p)^2$ and then we were asked for an expression for $T(p)$ in terms of $r(p)$ and $\theta(p)$, but that notation would be clunky. | |
| Aug 10, 2018 at 17:00 | comment | added | Michael Bächtold | @TimothyChow: Could you expand? Which notational mistake are people who chose B making? | |
| Aug 10, 2018 at 16:01 | comment | added | Timothy Chow | As far as the original question is concerned, I do kind of think that A is the "right answer" and that people choosing B are making a mistake because of the notation. | |
| Aug 10, 2018 at 15:53 | comment | added | Timothy Chow | Here's another problem one could pose: "Solve $ax+b=0$." The answer could be "$a = -b/x$". After all, who is to say that "$x$" is the unknown? | |
| Aug 10, 2018 at 15:53 | comment | added | Daniel R. Collins | @Kostya_I: Interesting perspective. "After all, the author meant something when they chose (r,θ) for their notation, right?" Apparently physicist says "yes", mathematician says "no". Per Redish/Kuo: "In other words, physicists assign meaning to the variables x, y, r, and θ—the geometry of the physical situation relating the variables to one another. Mathematicians, on the other hand, may regard x, y, r, and θ as dummy variables denoting two arbitrary independent variables." | |
| Aug 10, 2018 at 13:50 | comment | added | Kostya_I | So the reason why scientist 'are taught in such a strange way' is that all physically meaning quantities are functions (or vector/tensor fields, etc.) on manifolds, usually specified by picking a chart. To be pedantic, one could introduce coordinate maps and phrase the question as: $T(\varphi(x,y))=k(x^2+y^2)$. What is $T(\psi(r,\theta))$? This is not done, as it is impractical: it blows up formulae while carrying no useful information. The composition with coordinate maps is already implied by the choice of letters. | |
| Aug 10, 2018 at 13:30 | comment | added | Kostya_I | I think this is just plain wrong. I cannot imagine a realistic situation where a mathematician (other than a freshman answering to a professor who is known for silly 'gotcha' questions) would really think of (B). After all, the author meant something when they chose $(r,\theta)$ for their notation, right? Moreover, I find little objectionable about this notation: mathematically, $T$ is a function on a manifold, which has several standard (and standardly denoted) coordinate charts. It is defined in one chart and then calculated in the other. | |
| Aug 10, 2018 at 13:28 | comment | added | Eike Schulte | In my experience (and the linked article has a footnote alluding to this) mathematicians may use the physicists notation as well when working with manifolds: Given a manifold $X$ and a function $T$ defined on $X$, one might choose coordinates $(x, y)$ on some open set $U$ and then write $T(x, y) = k(x^2 + y^2)$ to describe $T$ on $U$. But choosing different coordinates with $x = r\cos\theta$, $y = r \sin \theta$ one then has to accept $T(r, \theta) = kr^2$. (I would argue that writing $T(x, y)$ and $T(r, \theta)$ is not good notation to begin with; simply $T = k(x^2+y^2) = kr^2$ seems better.) | |
| Aug 10, 2018 at 12:08 | comment | added | LSpice | I'm not sure if this is a mistake so much as a different convention. To a mathematician, $T$ is the name of the function, and it doesn't matter what arguments I feed it. To a scientist, $T(x, y)$ is the name of the function, and $T(r, \theta)$ is a different function, so that there is no way to know what $T(1, \pi)$ means without further context. (An example of context: $T(r, \theta)\bigr|_{(r, \theta) = (1, \pi)}$.) It's not clear to me that either of these is objectively wrong. (If you say "always be more explicit", then I think mathematicians fall down in many other cases.) | |
| Aug 10, 2018 at 10:12 | comment | added | Michael Bächtold | @onurcanbektas I won't directly object to your point of view. But if you start to wonder why science students are taught in such a strange way, you'll have to dig deeper. Personally I suspect that the problem goes back to the abuse of notation $y=y(x)$, which startet with Jacobi (as far as I can tell). And if you read Jacobi, you'll see that he was very much thinking about notation. I've also written about this here. | |
| Aug 10, 2018 at 9:56 | comment | added | Our | Well I would argue that this mistake is not due to the notation, but rather due to how science student are taught mathematics. | |
| S Aug 10, 2018 at 8:32 | history | answered | Michael Bächtold | CC BY-SA 4.0 | |
| S Aug 10, 2018 at 8:32 | history | made wiki | Post Made Community Wiki by Michael Bächtold |