David Madore's WebLog: Popularizing mathematical concepts

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(Tuesday) · Premier Quartier

Popularizing mathematical concepts

Some time ago I got into my mind that I should try to write a little text (perhaps entries into this 'blog, perhaps a standalone document) explaining something about the meaning of three mathematical objects, in a way understandable to the layman (whose a priori level of mathematical culture remains to specify: it should include a basic understanding of natural numbers and plane geometry but no knowledge of ordinals).

The three mathematical objects, which I chose because they fascinate me, because I think there is genuine beauty in them, and I would like to share with others the vision of that beauty, and because of their tantalizing relation to “infinity” (in a broad sense) are: the Long Line, the Stone-Čech compactification of the natural numbers (also known as beta-omega), and Zariski's Ciel Étoilé (Starry Vault). These objects all convey a geometrical, almost graphical, intuition: yet they defy us (in a way akin to the bewilderment caused by Escher's prints, but much stronger) as they cannot be represented accurately by any pictorial representation, they cannot be “embedded” in our Universe.

For the benefit of my colleagues I can define these three objects (and describe some basic properties of them) assuming a certain knowledge of mathematics:

To define the Long Line, first define the (closed) Long Ray: as a totally ordered set, this is the product of ω1 (the first uncountable ordinal) by the half-closed real interval [0;1[ (closed in 0 and open in 1), totally ordered lexicographically (giving the greater weight to the first component); as a topological space, it is that totally ordered set with the order topology: and this topological space is actually a (non-metrizable) topological manifold (with boundary). The Long Line is simply obtained by glueing two copies of the Long Ray, in opposite directions, at their origin. So it is a one-dimensional topological manifold (without boundary), and it has the remarkable property that any continuous real-valued function upon it is bounded; in fact, any one-dimensional manifold (without boundary) is homeomorphic to either the circle, or the real line, or the open Long Ray (the closed Long Ray minus its origin) or the Long Line, so the latter is the only noncompact boundaryless one-dimensional manifold with this property. There exist an infinite number of pairwise non-diffeomorphic differentiable structures on the Long Line.

The Stone-Čech compactification of the natural numbers is the set of ultrafilters over the set of naturals numbers. (Recall that a filter over a given set is a non-empty collection of subsets of it, not containing the empty set, closed under enlargement and intersection. And an ultrafilter is a filter which is properly contained in no other filter, or which, in other words, for any given subset of the given set contains either that subset or its complement. The set of all subsets containing a given element is an ultrafilter: such ultrafilters are called principal ultrafilters.) It is made into a topological space by declaring that, for any given subset of the naturals, the set of ultrafilters containing that subset is closed (and any closed set is an intersection of such basic closed sets). Alternatively, we can give a general characterization of the Stone-Čech compactification of a (completely regular) topological space: it is the initial object in the category of continuous maps of that space to a (Hausdorff) compact space, in the sense that any continuous map from the given space to a (Hausdorff) compact space factors, in a unique way, through the Stone-Čech compactification; and here we are interested in the Stone-Čech compactification of the discrete space of natural numbers. The naturals themselves embed in this space as the principal ultrafilters; but the cardinal number of the space is the cardinal of the power set of the real line. (Ironically enough, the Long Line, although conceptually much more “difficult” than the set of integers, has a Stone-Čech compactification which is much more simple: just add one point at infinity in each direction.)

To define Zariski's Ciel Étoilé, we first recall what the blowup of a differentiable manifold at a point is: it is a manifold which is locally diffeomorphic to the original manifold everywhere save at the point which has been blown up, the latter being replaced by the set of all possible tangent directions at this point (technically, the projective space of the tangent space at this point, i.e. the set of all lines through the origin in the tangent space). Topology and differentiable structure on the blowup are defined in the more or less straightforward way: for example, neighborhoods of a direction in the blowup contain wedges around that direction, and the latter are claimed to be diffeomorphic with the suitable open subset of real affine space. Now consider the projective plane, and blow up various points on it, then points on these blowups, and so on: this lattice of blowups forms a projective system, and the projective limit, as a (compact) topological space (which is not a manifold) is Zariski's Ciel Étoilé. In other words, a point of the Ciel Étoilé is given by a point of the projective plane, then a point of the blowup of it at that point, then a point of the blowup of that at that point, and so on.

Assuming I have made no error, and if my description is sufficiently accurate, all mathematicians should now agree on the demonstrable properties of the three objects under consideration. So even if we do not believe in the platonic heaven in which these objects should “live”, at least their existence is guaranteed, in a certain way, by a common understanding between mathematicians as to what their properties are. Intuition is another thing altogether. Intuition is what convinces a mathematician that this-or-that property should hold, before he sets out to prove it; now mathematical books and treatises do not endeavor to dictate how mathematical intuition should be formed, and each mathematician is left to build his own, which may or may not agree with his peers' on this or that subject. In other words, mathematical truth (or at least, provability) is a common understanding between mathematicians; intuition is not necessarily so. Making a mathematical concept popular is difficult because it requires to work at the level of intuition (since we are not proposing to give rigorous definitions and properties), which is slippery.

Certainly some intuition is shared. Presumably all mathematicians will agree that the Long Line somehow “looks like” a line, only longer, much longer, and qualitatively longer: somehow it stretches beyond infinity in the usual sense (assuming there is a usual sense for infinity). But this is very vague: saying that a line is long is essentially meaningless. Here's another attempt to convey the main properties of the Long Line in a way that mostly anyone should be able to understand:

First of all, the Long Line is “locally like a line”: this means that if you stand at a given point on the Long Line, what you see around you, in your neighborhood, is exactly what you would see while standing on an ordinary line.

Next, the Long Line, like the ordinary line, has an order: some points are to the left of others and some are to the right (or you may prefer to call them up and down, or front and back, or whatever: this is of no importance, but let's stick to left and right). Like in the ordinary line, there is no leftmost point and no rightmost point: if you pick any point on the Long Line, there are more points to the left, and more points to the right.

Furthermore, just as for the ordinary line, all points on the Long Line are essentially worth the same: none of them is distinguished in any way, they all look the same, if you will. There is no “midpoint” or anything like that (though nothing prevents you from choosing a point and calling it the midpoint if you will).

Given two points on the Long Line, there is only one path leading from one to the other, just as for the ordinary line: to get from A to B, assuming, say, that A is to the left of B, you have to go through all points which are between A and B, that is, to the right of A and to the left of B. Furthermore, the part of the Long Line which is between A and B (whatever A and B are!) looks exactly like the part of any ordinary line between any two of its points. So the Long Line really looks a lot like an ordinary line.

There is no “distance” or “uniformity” on the Long Line. Let us be a little more precise than this. It makes no sense to say that A is closer to B than C is to D, because distance can be stretched or shrinked as one will. The only case in which it does makes sense is to say that A is closer to B than A is to C, when B and C are on the same side of A: then it means that B is between A and C, in other words that one must go through B to get from A to C (so certainly B is closer to A than C is). This may appear to be a difference with the ordinary line, but that is simply because we conceive the ordinary line with a distance on it; it need not be so: we can forget about this distance, and allow to stretch and shrink arbitrary parts of the line, as if it were made of (infinitely stretchable) rubber; the Long Line is not different (except that it cannot be given a distance, it is simply too long for that).

So far the Long Line looks exactly like the ordinary line. Now let us try to describe in what way it differs from it.

The ordinary line is “infinitely long” in both directions. However, it can be “exhausted” in the following way: pick a point on the line, then pick a point one foot to the right (assuming we have a distance on the line, and a “foot” of some kind), then a point one foot further to the right, and so on to infinity. Then in infinitely many steps we exhaust the line: there is no point which stands to the right of all the chosen points. Now this simple-minded attempt fails for the Long Line: it cannot be exhausted in this way. Specifically: take an infinite number of points on the Long Line, one for each integer (take a point 0, any one you will, then a point 1, then a point 2, then a point 3, and so on to infinity, exactly as you like them, as far or as close to the previous ones as you will), then there are still points on the Long Line which are to the right of all the chosen points, and points which are to the left of all the chosen points. Let's make this a little more picturesque and a little less precise: assume you can travel instantaneously from any point to any point on the line; then on the ordinary line you can move out of bounds (“out to infinity”) by making such hops, but on the Long Line you cannot—no matter what large hops you make and no matter if you make them until the end of time, there are still points which are further than all the distance you traveled (and you could choose to get there immediately, but there would still be points which are even further).

In short: no matter how long you travel, and no matter how fast you travel, you cannot come to the end of the Long Line, you cannot speed out to infinity, you must remain within bounds.

Lastly, here's another remarkable property of the Long Line. Assume we are given a continuously varying quantity on it (that is, at every point of the Long Line we have a value for this quantity, and it varies steadily in neighboring points). Then, no matter how this quantity is defined, it is eventually constant: in other words, there is a point on the Long Line such that, to the right of that point, the quantity always takes exactly the same value, and there is a point such that to the left of that point the quantity always takes the same value. Certainly this shows that there cannot be a definable distance on the Long Line, because if there were, the distance to a certain point O would be a continuously varying quantity that could not be eventually constant, an absurdity.

I am afraid I have failed miserably in making the intuition behind the Long Line accessible to non-mathematicians (and I'm not even sure that mathematicians would agree on all these “intuitive” properties, although of course the theorems behind them are unquestionable). But my attempt fails in an even more basic way, really: it fails to explain why and in what way this mysterious object, the Long Line, should exist (even assuming the above properties are understood); or even, really, what it means for it to exist. Certainly it doesn't exist in any physical sense: our Universe is far too small to contain a copy of the Long Line in any sense whatsoever (assuming even that it should contain a copy of the ordinary line, which is far from obvious). Mathematically, we have a precise theorem:

There exists one, and only one up to (non-unique) homeomorphism, topological manifold which is a (non-empty) totally ordered set with the order topology, in which any countable subset is bounded. (This manifold is exactly the Long Line.)

Only a layman need not know what a topological manifold is, and, even if he knew, need not agree that it coincides with the intuition afforded by the above description.

So have I failed utterly in trying to make non-mathematicians grasp this elusive “Long Line”? Probably. Can it be done? I don't know.

I'm afraid the situation is similar for the other two objects. Certainly I have not given myself an easy task.

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