Notation of points with coordinates

Question

At least in Germany, nearly all teachers and textbooks use the notation $$P(x,y)$$ for the point $P$ with coordinates $x$ and $y$.

My own math professors at university always cried about this, as the notation should be $$P=(x,y)$$ and I've come to understand the truth of this. Alas, it's difficult to convince other teachers of this rationally.

Where can I find ressources to prove my point, that at least at university level, the second notation is used and the first notation is at least deemed formally incorrect?

Answer 1

First, notation sometimes evolves, sometimes not. I don't think $P(x,y)$ is bad, but it is somewhat old, I believe, although still in use but not quite à la mode, somewhat like French these days. :) It's not as bad, say, as the notation $\sin^2 x$ or worse (for me), $\ln^2 x$, which I first learned meant $\ln \ln x$ but my students use for $(\ln x)^2$. But what I have to say should not be taken as promoting the old notation. As $P(x,y)$ is used less and less, fewer and fewer people will be familiar with it. That in itself is argument against it.

Let $E^2$ be a Euclidean plane. A coordinate system on $E^2$ is a bijection $E^2 \leftrightarrow {\bf R}^2$ (with certain properties). The point $P(x,y)$ is a notation represents a point $P \in E^2$ corresponding to the element $(x,y) \in {\bf R}^2$. The point $Q(-1,3)$ is similarly understood. When this convention is well-known, it is not particularly hard to understand.

There are a couple of competing impulses in current mathematics. One is that notation should clearly indicate a single object that depends only on the symbols out of which the notation is constructed. I can trace this back to Frege, but it seemed to take hold in the 20th century. The other is that notation should be efficient. The desire for efficiency quite often leads to a violation of the first principle. The notation $P(x,y)$ is certainly efficient, the parentheses being arguably extraneous; but since they are used to denote a coordinate pair, they might also be defended. The implicit coordinate mapping is rarely important. Usually when the notation $P(x,y)$ is used, it is in a simple geometric problem and there is only one coordinate mapping. The context can be made clear by words, which is often a better way, in my opinion, than with abstruse and superfluous notation.

When I was young and impatient, I did not like $P(x,y)$. The $P$ seemed extraneous. In many cases the letter $P$ was not needed for the problem, but the author seemed to feel every point needed a letter attached to it. Part of the problem was that I thought ${\bf R}^2$ was the same as the Euclidean plane. If they are considered distinct, and they often are in modern synthetic geometry, then $P = (x,y)$ is certainly wrong, unless $(\_,\_)$ denotes a function ${\bf R}^2 \rightarrow E^2$, which normally it does not. If the plane and ordered pairs are identified, as I was led to think in my early years, then the notation is okay.

The normal way is to define or posit a mapping $\phi \colon E^2 \rightarrow {\bf R}^2$ (by introducing coordinate axes), and let $P = \phi^{-1}(x,y)$. But this seems cumbersome. So it seems to me one can chose which is the lesser of two evils: the notation $P(x,y)$, which does not conform with current functional notation; and the notation $P = (x,y)$, which treats the plane as identical with ordered pairs of real numbers. The current trend in usage is for the latter, of course.

Answer 2

As mentioned in the comments already, there isn't one universally accepted correct notation. And the prevalence of both notations throughout the literature would make it quite a Herculean task to correct this notation in the writing of others.

I find aspects of both notations bothersome. So in my own writing, I tend to avoid using either. Instead, I write let P have coordinates (1,2) or P has coordinates (1,2) or P is a point with coordinates (1,2).

The problem I have with P=(1,2) is, perhaps, a philosophical one (but being logician by training, I like to think philosophically). To me, it seems that the plane exists prior to the coordinate system. We are perfectly capable of talking about points and their relationships amongst themselves and other objects without taking recourse to any coordinate system. (Read Euclide to see the extent to which he avoides ever even talking about numbers. Indeed, coordinate systems are a rather modern development.) After all, there are other coordinate systems, why identify a point with one particular choice of coordinate system. Consider how, if we decide our plane is inbedded in a larger 3 dimensional space (or even higher dimension), the point is still P, but it's coordinates must change: let's say surreptitiously P has coordinates (1,0,2) in this higher dimension. Given the transitive nature of equality, would we really want to say (1,2)=(1,0,2)? I think not. But all notations have their short-comings: so perhaps in this context we're willing to disclaim that the transitivity of equality can be so cavalierly applied.

A coordinate system helps us locate the point and differentiate one point from another via functions: that is, a coordinate system allows us to analyze what's happening between points by reducing the analysis to that of what's happening between numbers and vectors etc. And a great deal of mathematical thinking has gone in to showing that thinking about numbers and vectors will not lead us astray in regards to points and their ilk. But I would caution that there's nothing obvious about how coordinates relate to each other being equivalent to how points relate to each other. And in that sense P=(1,2) suggests that this equivalence properties obvious where, in fact, it took a long time to be discovered and verified.

I would say it's best to look at the context in which a particular notation is being used as respect that within a particular field one style of notation may be preferred over another. Perhaps I overgeneralize, but I don't believe many mathematicians put too much thought into why they use a particular notation until they find the notation creates more problems then it solves. But also, mathematicians are generally not worried about the deeper philosophy a particular notation suggests.