24
$\begingroup$

The title of this question may make you want to close it immediately, but bear with me a moment. In several older mathematics papers (early 20th century) I have seen statements such as

The motions of three-dimensional space are $\infty^6$.

I am curious what this means. From context I guess that "being $\infty^6$" means something roughly like what we would nowadays call "being a 6-dimensional manifold". But did it have a precise meaning? Who defined it, where and when? When did it fall out of use?

Improve this question
$\endgroup$
8
  • 10
    $\begingroup$ Can you give a reference to some of these papers? $\endgroup$
    – Joonas Ilmavirta
    Commented Jun 5, 2015 at 15:44
  • 5
    $\begingroup$ It ptobably means that you need 6 real variables to describe motion in space, each of which may attain infinitely many values. I'm thinking of coordinates $(x,y,z)$ and the velocity vector $(v_x,v_y,v_z)$. $\endgroup$
    – Franz Lemmermeyer
    Commented Jun 5, 2015 at 15:45
  • 5
    $\begingroup$ I agree with @FranzLemmermeyer that it refers to the number of (independent) real variables needed to specify a motion. But I think the "motions" mentioned here are the members of the Euclidean group (rotations, translations, and maybe reflections). I also think that, in the good old days, people were not so careful about whether those 6 parameters are available globally or only locally. So in the end, it comes down to your guess about a 6-dimensional manifold. $\endgroup$
    – Andreas Blass
    Commented Jun 5, 2015 at 15:48
  • 8
    $\begingroup$ Doesn't it just mean the same (imprecise) thing as "There are six degrees of freedom"? $\endgroup$
    – Noah Snyder
    Commented Jun 5, 2015 at 17:14
  • 15
    $\begingroup$ The notation $\infty^d$ for something parameterized by a $d$-dimensional variety is completely standard in Italian algebraic geometry (Segre, Castelnuovo, Enriques...). I don't know where it comes from. $\endgroup$
    – abx
    Commented Jun 5, 2015 at 18:19

1 Answer 1

Reset to default
19
$\begingroup$

Yes, indeed, this notation was used to state the dimension of the manifold. The idea of dimenson is very intuitive but it took long time and a lot of labor to formalize. Before the modern definitions of "dimension", "manifold" and "homeomorphism" were spread, people, especially geometers, expressed this fact by saying that something depends on $n$ parameters, or is an "n"-parametric family, and wrote $\infty^n$. This notation is out of date now, but they still speak of $n$-parametric families. When I was an undergraduate student (early 1970s) $\infty^n$ was still used in the lectures on projective geometry, for example. I was puzzled because I knew that Cantor proved that all these things have the same cardinaity, until I read about Brouwer's theory of dimension and "domain preservation property". (About a century before, Cantor and Poincare were similarly puzzled until Brouwer clarified the thing:-) I think Cantor was the first who tried to prove that manifolds of different dimensions are not homeomorphic, but he failed.

(Before Cantor it seemed evident to everyone that plane contains more points than the line, I do not know whether anyone cared to prove this, but those who did try, could not).

Improve this answer
$\endgroup$
9
  • $\begingroup$ So are you saying it didn't actually have what we would nowadays call a precise meaning? $\endgroup$
    – Mike Shulman
    Commented Jun 6, 2015 at 18:57
  • $\begingroup$ Not in the continuous setting. It did have a meaning in the smooth or analytic setting due to the inverse function theorem. $\endgroup$
    – Alexandre Eremenko
    Commented Jun 6, 2015 at 19:15
  • 3
    $\begingroup$ Who defined what? Dimension of a manifold was defined by Brouwer, as I said. Whether anyone before him rigorously defined the dimension of a smooth manifold (which is much simpler), I do not know, but people were using it. Notice that the "manifold" itself was rigorously defined later, by Weyl. But of course people were using them before Weyl. $\endgroup$
    – Alexandre Eremenko
    Commented Jun 7, 2015 at 4:40
  • 1
    $\begingroup$ It is like with real numbers. The rigorous modern definition is due to Dedekind and Cantor (late 19 century) but people were freely using them since antiquity. $\endgroup$
    – Alexandre Eremenko
    Commented Jun 7, 2015 at 4:44
  • 1
    $\begingroup$ Such questions are usually hard to answer, and I supposed nobody (rigorously) defined it, the notation was intuitively clear. Who used it first? is difficult to answer, and I conjecture that some projective geometer in early 19 century. $\endgroup$
    – Alexandre Eremenko
    Commented Jun 8, 2015 at 5:08

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .