## On dimension

From my experience, the (purely mathematical) concept of “dimension” is frequently misconstrued, by people outside scientific fields, as something inherently physical in nature.

The following is a citation taken from Serge Lang’s 1985 book MATH! Encounters with High School Students, after a transcript of a geometry talk he gave to tenth-grade students in Toronto:

“The ready acceptance of $n$ dimensions by students at an early age is a common experience for me. Given the history of “dimensions”, and all the fuss made around Einstein’s “fourth dimension” as time, this might be surprising to some people. But on the basis of experience, I find that young people will find problems about higher dimension only if grown ups suggest the existence of problems.

The question: “Is time the fourth dimension?” is a very bad question, because it already prejudices what one might mean by “dimension”, and also the use of the definite article “the” implies that if there is a “fourth dimension” there is only one. But today it is generally accepted that whenever you can associate a number with a notion, then you have a dimension. This idea was already seen clearly by d’Alembert, when he wrote the article on “dimension” for Diderot’s enclopoedia:

Cette manière de considérer les quantités de plus de trois dimensions est aussi exacte que l’autre; car les lettres algébriques peuvent toujours être regardées comme représentant des nombers, rationnels ou non. J’ai dit plus haut qu’il n’était pas possible de concevoir plus de trois dimensions. Un homme d’esprit de ma connaissance croit qu’on pourrait cependant regarder la durée comme une quatrième dimension et que le produit du temps par la solidité serait en quelque manière un produit de quatre dimensions. Cette idée peut être contestée mais elle a, ce me semble, quelque mérite, quand ce ne serait que celui de la nouveauté.

When I brought up higher dimensions with the students during my talk, there was absolutely no resistance from anyone. The students saw right away that the important thing is how we operate with numbers in the context of several dimensions. Then they have the correct idea concerning this abstraction, and give the right answers. Sometimes in similar circumstances, I do get the question whether a fourth dimension exists. I answer it as above, pointing out that it all depends on how you wish to use the word. If by “dimension” you want to mean only the spatial dimensions of ordinary space around you, then there are only three dimensions. If you accept to give the word a wider meaning, then you have arbitrarily many dimensions, with which you can work just as easily. The automatic response that in $n$ dimensions under a dilation by a factor of $r$ volume changes by a factor $r^n$ is then both immediate and correct. Of course, when I ask how volume changes under dilations in four dimensions, Serge answers: “It’s $rst$ whatever.” He shows that he has well understood what’s going on. The only thing that would remain to be done is to point out that his “whatever” corresponds to a choice of letters, and that if we continue to use letters for dimensions, we shall run out of choices after the 26 letters of the alphabet have been exhausted. Therefore, one may denote the numbers associated with three dimensions by something like $x_1, x_2, x_3$; and then there is no difficulty to denote the numbers associated with $n$ dimensions by $x_1, x_2, \ldots, x_n$. Then if we dilate by factors of $r_1, r_2, \ldots, r_n$ one sees at once that volume of boxes changes by a factor of $r_1 r_2 \cdots r_n$ (the product). Whatever difficulty exists here (and it is very slight) lies only in an appropriate choice of letters and indices; that is, in the choice of notation to transcribe in symbols what the mind has already grasped.”