Wikipedia:Reference desk/Archives/Mathematics/2021 March 15

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March 15

Is left-to-right preference in European languages a POV?

Is left-to-right preference in European languages a POV?

A matrix operator works on a column vector to the right or a row vector on the left. References assembled at Row and column vectors show that several geometers have opted for row vector transformations to keep symbols moving left to right. An editor has contended that these references should be discounted as "specially selected". Part of the contention is that a transformation is a function, like f(x) in calculus or f(z) in complex analysis, so it may be written T(v), suggesting a column vector v. But transformations open the world to modern geometry and group theory. Should the geometers' convention be so degraded? Rgdboer (talk) 02:01, 15 March 2021 (UTC)

I'm not quite sure what you're getting at here but... . In my opinion putting the function symbol to the left of the argument is backwards. When you compute sin 72° on a calculator you enter 72 then press sign, so in effect you read sin 72° right to left to compute it. When functions are composed then this also means you read from right to left to get the correct order; f∘g is preform g, then perform f. But I've accepted that this has been the convention for several hundred years, it's not likely to change soon, and it's not Wikipedia's job to change it. I have occasionally seen exponential notation used for functions, which does fix these issues, especially in algebraic contexts. For example the conjugation in a group is usually written x^g = g^-1xg, and this works better because you get x^gh = (x^g)^h as expected. Although there are perhaps a few authors who are seeing sense on this issue and adopting it, I've never had the impression that it's the "preferred" notation in general; I doubt we're going to see 72°^sin any time soon. --RDBury (talk) 11:47, 15 March 2021 (UTC)

(ec) The prevalence of the left-to-right pattern in the writing systems for European languages is a fact. It is not clear that the order in the notation $f(x)$, due to Euler, has much to do with it; I do not know why he did not choose the order $(x)f$, as some later authors have unsuccessfully promoted. Most mathematical notation is purely conventional, based on whoever was the first to introduce a practical notation in an accessible publication, and attempts to replace less felicitous choices by more rational notations have generally been spectacularly unsuccessful. In some cases incompatible conventions coexist, but as to the multiplication of matrices and vectors, the only convention I am familiar with is the one used in our article Matrix multiplication, in which $Mv=w$ and $v^{\mathsf {T}}M=w^{\mathsf {T}}$, where $v$ and $w$ are column vectors. Can you be more explicit what is contentious about this?  --Lambiam 12:03, 15 March 2021 (UTC)

Perhaps Euler used that order because, although he wrote in Latin, he thought in German. In German he would have said eine Funktion von $x$. Had he been Japanese or a Turk, where you say this in the opposite order, $x$ no kansū or $x$'in bir fonksiyonu, the notation might have been the other way around.  --Lambiam 00:12, 16 March 2021 (UTC)

The denigration of the geometric references is wrong. No effort to reform convention is made here, just the right of these geometers' view to be noted, with the reason for their convention. Now an editor has removed the references and stated preference. The tagging as POV was bad; deletion of reliable sources is worse. As noted, there is no controversy as the transpose relates both expressions. The true POV here is that there is something off about using row vectors in English. Rgdboer (talk) 04:35, 17 March 2021 (UTC)

I still do not understand the issue. Whether a vector is a row vector or a column vector is not a property of the vector, but of how it is used in the context. Formally, row vectors are matrices of dimension 1 × n, while column vectors are matrices of dimension n × 1, for some positive integer n. Given n, there is an obvious one-to-one-to-one correspondence between the sets of row vectors, column vectors, and "ordinary" vectors, the latter being members of $\mathbb {R} ^{n}$ that have not been assigned an orientation (row or column). The order "matrix times vector" is more conventional, but any mathematical author may at any time find it more convenient, for a variety of reasons, to express a formula involving a matrix-vector operation in the order "vector times matrix". I don't think the choice is a big deal and not much more can be said about it than that either one is possible, but that some authors in some of their publications have preferred the less conventional approach. Digging up instances of such less conventional choices is, however, OR, unless the choice and its rationale are specifically discussed in some depth. With some more digging, you'll probably also find works in which both orders coexist because now one is more convenient, now the other.  --Lambiam 08:15, 17 March 2021 (UTC)

You got me! "Let's remove those references because they are OR." Introductory texts may use the column vector input, but people using multiple transformations like Hirschfeld, Artzy, Coxeter, Silberstein, and Kemeny & Snell use the row vector input. Since functions and transformations are special types of binary relations, the topic is studied there, especially since transpose is an essential operation in the calculus of relations. Special deference to function composition is given, but generally the composition of relations is uncomplicated by any other tendency to backwardness. The suggestion of "digging instances of such unconventional choices" is unfair since use of row vectors in not unconventional. Why is contributing these sources "digging"? Your thoughtful exposition of how things are going is appreciated.Rgdboer (talk) 04:24, 18 March 2021 (UTC)

Writing "$p^{\mathrm {T} }=Mv^{\mathrm {T} },t^{\mathrm {T} }=Qp^{\mathrm {T} }$" instead of "$p=Mv,t=Qp$" strongly suggest the point of view that row vectors are the norm and that column vectors are transposed "normal" vectors. It is not hard to find texts in which a sequence of transformations defined by transformation matrices is represented by matrix products where the first transformation corresponds to the rightmost factor, so that the matrix-vector product order is $Mv$ – in other words, where a point subjected to these transformations is represented by a column vector.^[1] We can observe from your references that some mathematicians in some circumstances, for whatever reason, have used the product order $vM$ instead of $Mv$. Perhaps we can discover, on further inspection, that a relatively large number of these mathematicians were Swiss, or pacifists, or geometers. That may be interesting enough to publish such a discovery in a book or a respected journal. Wikipedia editors can then use this as a reliable source to add information to the encyclopedia. I do not see the mere fact that some mathematicians in some circumstances, for whatever reason, used the product order $vM$ as encyclopedic information worth dwelling on without falling afoul of paying undue attention to a triviality. Rather in general, as Wikipedia editors we do not report on what we have observed (which is OR), but we report what others have published about it in reliable sources. So if you can find non-cherry-picked reliable sources that discuss the issue of who preferred which when, it may be worth a paragraph or two. If it is based on your own observations, it is OR.  --Lambiam 06:30, 18 March 2021 (UTC)

Integer-only math

Is there a name for math that only uses integers. Obviously, many things don't work if you can only use integers. For example, you can't divide 1 by 2 because the result is not an integer. 97.82.165.112 (talk) 19:33, 15 March 2021 (UTC)

It's the Number theory. (There is also Numerology, but that's not math.) --CiaPan (talk) 22:01, 15 March 2021 (UTC)

In some contexts, you can say "arithmetic" to means specifically what happens on the natural numbers (with integers being a minor extension). For example, in mathematical logic, if you talk about something in the "language of arithmetic", it means that all the variables range over natural numbers. --Trovatore (talk) 22:52, 15 March 2021 (UTC)

Algebraic equations with the constraint that the solutions are in integers are called "Diophantine equations".  --Lambiam 23:58, 15 March 2021 (UTC)

Discrete mathematics is relevant and seems to include what you're looking for; one could also say that the relevant arithmetic is "in $\mathbb {Z}$", the set of integers. Xnft (talk) 17:12, 16 March 2021 (UTC)