After the Gold Rush

Although string theorists and fans of J. W. Dunne may disagree, most people are convinced that we live in a universe which has three dimensions of space and one of time.

In the early days of topology, one of the tasks that mathematicians took on was to find a topological way to distinguish between spaces of different dimension. The mathematician Pavel Samuilovich Urysohn died at the tragically early age of 26 (drowned swimming in the Bay of Biscay); before his death, one of his very many achievements was a workable dimension theory for topological spaces.

For what I am talking about here, it is not necessary to delve deeply into dimension theory; I only need to distinguish lines from higher-dimensional spaces.

Cantor showed the paradoxical result that the number of points on a line is the same as the number in a plane, by establishing a bijection between them. It is most convenient to consider the unit interval and the unit square. Now each number in the unit interval can be represented as an infinite decimal; taking alternate terms in the decimal expansion we obtain the x and y coordinates of two points in the unit square. (The construction fails at countably many points because of the non-uniqueness of decimal expansions, e.g. 0.257999… = 0.258000…; but this defect is easily patched up.)

A much stronger result is true. The map just given fails to be continuous; but Peano showed the existence of a continuous map from the unit interval onto the unit square, otherwise known as a space-filling curve. A description of his construction and subsequent work is here.

But there is a further step which we cannot take: the unit interval is not homeomorphic to the unit square (or, what amounts to the same thing, the line is not homeomorphic to the plane). To prove non-homeomorphism of two spaces, we have to find some topological property which distinguishes them. (Indeed, it was just such problems that Poincaré was tackling when he came up with his eponymous conjecture.) Now the space obtained by removing a single point from the plane (or higher-dimensional Euclidean space) is connected; but the space obtained by removing a point from the line has two components.

This is not just a topological curiosity; in my opinion, it colours every single person’s experience of the Universe. We have a very strong sense of the three-dimensional space we live in (and the connectedness of the part external to us), but also a very strong sense of the partition of time into past, present and future. Indeed, I am tempted to say that we think of time as 3-dimensional and space as 1-dimensional (though these words are not appropriate here).

First, let us consider space. Of our five senses, touch and taste are local; smell and hearing tell us about nearby space, but are not very directional. Only sight informs our conception of space. Each sight-line from our eye is potentially a semi-infinite ray (a point of the real projective plane), but (except in the open on a dark night) the ray meets a physical object which reflects light to our eye nearer or further. So what we perceive is the real projective plane “coloured” by the objects which have signalled their presence to us. Helped by the sense of depth for nearby objects provided by our binocular vision, we “lift” the projective plane to the 3-dimensional Euclidean space.

So space has two components, myself, and everything outside. I contend that the distinction is not very clear. I can move out of the space I am currently occupying, and then back into it; I notice that space is homogeneous. Try a small experiment: Replay something you did yesterday in your mind… Did you find you were looking out of your eyes while you did it, or looking down on yourself from above, or were you not in the picture at all? I find that the last possibility applies when I do this. So I see space as a single undivided entity.

Time is different, and the distinction into past, present and future is undeniable. But there is another difference. Our senses bring us evidence from different parts of space, some very far away; but we live in the present, and no senses bring us evidence from either the past or the future (apart from perhaps the undocumented “sixth sense”).

But we find it very difficult to live in the present. Percy Bysshe Shelley, in his poem To a Skylark, says “We look before and after, and pine for what is not”. He contrasts this with the unseen skylark which, like a disembodied spirit, seems to speak just of the present; in the last stanza he suggests that if he could adopt this attitude himself he would write poems to which “the world would listen then, as I am listening now” (so even in his approach to enlightenment he is worrying about his reputation).

The song After the Gold Rush by Neil Young captures perfectly the tripartite nature of time. The dreamlike first and third stanzas, one mediaeval, the other science-fiction, are the past and future. The present moment described in the second stanza is just an awakening between two dreams, but is still freighted by all the “looking before and after” that humans are susceptible to: planning for the future (“I felt like getting high”), worrying about the past (“thinking about what a friend had said … hoping it was a lie”).

About Peter Cameron

I count all the things that need to be counted.
This entry was posted in exposition, mathematics and ... and tagged , , , . Bookmark the permalink.

4 Responses to After the Gold Rush

  1. Pingback: Music for Zenna « Log24

  2. Santo says:

    What a beautiful, poetic, thought-provoking post! (Particularly the idea that our visual perceptions form a “coloured” projective plane.)

    Neil Young was honoured at the Juno Awards yesterday with a couple of awards, one for music and the other for his humanitarian work.–indie-rockers-rule-junos-while-host-drake-is-shut-out?bn=1

  3. pkrautz says:

    Thank you for this beautiful post. The best I’ve read all week.

  4. Pingback: Weekly Picks « — the Blog

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