Imagine there's a pure, disembodied intellect, and you somehow have the ability to communicate with it. It's a very clever intellect, so it's perfectly receptive to abstract, a priori mathematics: but it has never had any experience of spatial things, and it wants you to explain space to it. How might you go about this?

Well, first you might explain the number line. You invite it to consider an infinite set of objects (we'll call them 'real numbers'), all different from one another, but continuously ordered in two directions from a particular element that we'll call 'zero' (or 'the origin'): ever greater to a positive infinity, and ever less to a negative infinity. And now, with the number line in place, you invite the intellect to take three such lines. That is to say, you invite it to consider an infinity of ordered triples of the form <x, y, z>, where x, y and z are all real numbers from this same set, but are capable of varying independently of one another. Let's call each of these triples a 'point', and call its three individual elements 'Cartesian coordinates'. And then you add some more specific definitions, framed in these terms. For instance, you define a 'sphere' as a set of points such that x2+y2+z2=r2. (Where r is yet another number: we can call that one the 'radius'). And then, once you feel that you've got enough definitions in place, for different kinds of 'shapes' (for that's what we'll call such sets of points), you proceed to offer the intellect a bunch of axioms, still framed in these same terms. Let's opt for Euclid's axioms. (At this point, perhaps the mind asks you: "So am I to take it that these axioms define the *only* way that space could be?" You might reply: "No, other systems do exist, the so-called 'non-Euclidean' geometries: but, for present purposes, let's just keep things nice and simple"). And then, once the intellect has had a while to mull over all this, and to digest all the mathematical consequences of the axioms you've given it, you can proudly declare that you have succeeded in teaching it geometry. It now knows all that there is to know about shapes and the relations between them.

Once you're satisfied that it's comfortable with that, you then carry on. You proceed to explain that certain shapes instantiate things that we call 'sensible qualities'. The intellect is puzzled: "What are those like?" But here, you concede, your powers of communication do unfortunately run out. You might say: "There's one that we call 'red', but I'm afraid I can't convey to you what red is actually like, because you could never imagine what it is from a description of it. Sensual experience is necessary, before you can get the idea, and that's precisely what you're lacking." But at least you can tell the intellect a few things about these qualities. For a start, there are lots of them. And they're all different... only some are more different than others. They're connected amongst themselves by higher-order relations, and in particular a relation that we call 'resemblance'. The quality we call 'red', for instance, resembles the quality we call 'orange' more than it resembles the one we call 'green', and *much* more than it resembles the one we call 'rough'. But the one we call 'rough' kind of resembles the one we call 'prickly'; more so, at any rate, than it resembles the one we call 'smooth', and a lot more than it resembles 'quiet'. And then some pairs among these qualities can combine together, while others can't. For instance, the qualities 'red' and 'smooth' can both be spread over the same set of points, while the qualities 'red' and 'green' cannot be. And so on.

At least, the same shape can't be red and green *at the same time*: but then, we haven't introduced time into the story at all yet. Okay then, let's do that. We'll just add a fourth coordinate, so that our points will now be of the form <x, y, z, t>. And then let's add some more axioms. Perhaps we might make them Newton's laws. Or just pick your own favourite system of mechanics: you can make it Einstein's if you prefer. For, where geometry deals with shapes and their abstract spatial relations at an instant, what mechanics tells you is how one configuration of shapes will evolve into another through time. So we'll plug in some mechanical laws and... we now seem to have a world.

And we've managed to convey all of this to our friendly disembodied intellect, *apart from* the sensible qualities that are supposed to be giving some real character to these shapes, as they interact and move around through time. But those sensible qualities don't constitute space. They do characterise the objects that are *in* space (at least in relation to our own sensory capacities): but the space itself is something distinct from them all. It's an eternal and universal structure that's fit to give a home to them. And that structure, it seems, *has* been adequately communicated to the disembodied intellect: for the structure is a purely mathematical one. Fundamentally, all that it really is is a bunch of numbers, ordered into certain sets, plus a handful of axioms and laws. And, once the intellect has grasped that, what's missing? It still doesn't know what it feels like to experience objects in space: but that's just a matter of how things make us feel, which is independent of the space itself. As far as genuinely spatial properties and relations are concerned, you can take any you like: 'round', 'square', 'thin', 'thinner than', 'half as big as', 'further away from', 'three times as far as', 'in between', or even something as precise as '4.7 metres long' (given that this is really just expressing a comparison between a given object and an agreed standard). Indeed, once we factor in time, we can even add 'growing', 'moving', 'accelerating at 9.8 m/s2', etc. All of these properties and relations can be adequately understood just by taking some Cartesian coordinates and doing a bit of elementary arithmetic on the numbers themselves.

- Log in to post comments