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Tallis in Wonderland

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On Points

Raymond Tallis pinpoints the mathematics/reality divide.

Readers of this column will have had a hint of my views on the limitations of the ability of maths and physics to capture lived experience – in particular in ‘Time, Tense and Physics: The Theory of Everything But…’ in Issue 81. Here, I want to focus on something absolutely central to the interpretation of the world in which we live in terms of mathematics: the notion of a point – in particular, a point in space. Points, which lie at the heart of the mathematisation of space (and, subsequently, of space-time), achieved this elevated status via the discipline of geometry.

Those of you who remember your early encounters with Euclid may recall, amidst memories of fear, humiliation, boredom and the other accompaniments of the pedagogic experience, that Euclid begins with a series of definitions and axioms. Some of them seem so obvious that you may have wondered, “Why is this guy telling me this? Did I really need to be informed that ‘Things equal to the same thing are equal to each other’ or that ‘A whole is greater than any of its parts’?” You stop wondering when, as if by magic, he conjures from these ideas a series of theorems that are far from obvious. What’s more, they were accepted as the last word on space for two thousand years.

Euclid’s first definition is of a point: ‘A point is that which has no part’. It can’t be divided because it is entirely without extension: without length, surface area or volume. It does not occupy any space. This is equally true of a line (which is ‘breadthless length’) and also of a plane, which seems to enclose space but does not fill any because it has no depth. But it is most obviously true of a point, which is zero-dimensional. This provokes the naïve inquirer (which all philosophers should be) to ask a (pointed) question: If a point does not occupy space, how can it be located in space? How can a spatial nothing designate one bit of space rather than another bit of space? How can we say that one point in space is such and such a distance from another? Surely something can have a particular location in space only in virtue of occupying some of it. My computer mouse could not be next to my computer if it did not occupy space next to that occupied by my computer.

Marked Up

You might not see this as a real question if you confuse mathematical (geometrical) points with the material means we use to mark them out, which are not dimensionless, and so do occupy space, otherwise we couldn’t see them. But we must distinguish the dots, such as blobs of ink, by which we designate points on a page, from the things they designate. Being designated by such a mark is the most obvious way in which a point borrows space from something else in order to be located in space. (Lines and planes do the same thing, of course: they, too, are squatters; but points are the most bare-faced hitch-hikers.) The space-occupying mark has an obvious spatial location, but it is not a Euclidean geometrical point because it has parts. It has a spatial extension. We can cut it in half, by cutting through the paper marked by it. So the biro mark is not itself a point, but a sign of a point. So we need to look a little harder to see how we can locate a geometrical point in space.

There is another, deeper, mode of spatial parasitism that enables Euclidean points to seem to inhabit locations in the world (and consequently, to be located at one place rather than another) without taking up any space. I am referring to the use of mathematical co-ordinates to designate points. So I can refer to a point as being at x=2, y=3, and z=4, where x, y and z are three axes that designate the up-down, side-to-side and back-front dimensions. By this means, it seems possible to designate a point in space without using anything that occupies space. Cartesian co-ordinates rescue Euclid’s dimensionless entities, enabling them to do useful work. End of problem.

Not so fast. The Euclidean point is here rehabilitated into real space only by means of an intermediary: a co-ordinate system, whose three axes, taken together, establish a frame of reference. But this simply moves the question on. Does the frame of reference mark off a volume of space? Surely, you might say, the volume marked out is located in space: real space.

Not so. Typically, the three axes are located on a page as the elements of a diagram. The space they putatively enclose is not the space where the page, or I who am writing on it, is located. Co-ordinate ‘2,3,4’ in a diagram in a book I am reading in Stockport railway station is not itself a place in Stockport, even though the book in which is it drawn is in my hand at Stockport station. Consequently, the point on the page isn’t a point in real space – the kind of space you and I walk through and dance in and bodily occupy. The mathematical point so designated still remains within the thought-space of mathematics.

Points of Paradox

What, you may ask, has this got to do with philosophy? Well, it has everything to do with rescuing lived space (and lived time) from a reduction to mathematics and mathematized physics. To see this, let us look at something a bit more familiar: those famous paradoxes that have engaged philosophers since Zeno first introduced them to support his teacher Parmenides’ claim that motion is logically impossible, and hence illusory.

Take Zeno’s Dichotomy Paradox. I am running for a bus. The kind-hearted driver has signalled that he will wait until I arrive. His kindness is ill-rewarded, or will be, if like Zeno, you think that in order to cover the distance between myself and the bus, I have first to cover half the distance; and that in order to cover half the distance, I have to cover half of half, or a quarter, of the distance; that in order to cover this… well, you can see the way it’s going. Given that the process of halving (or ‘dichotomising’) can go on forever, it would appear that I would have to complete an infinite number of tasks to move. I will never reach the bus.

Something has obviously gone wrong. After all, I usually catch the buses that wait for me, and there also must be some explanation as to how I got to the place where I started my dash for the bus.

The commonest way of dealing with this paradox is to suggest that Zeno made a mathematical error: he failed to notice that, as the distances to be covered are divided into smaller portions, the time taken to cover them falls proportionately. As a consequence, although the number of portions of time into which my race for the bus can be divided is infinite, they belong to converging series that adds up to a finite quantity. If, for example, you add 1/2 second to 1/4 second to 1/8 second, etc, you gradually cone down on 1 second, even though the number of steps involved is infinite.

This approach, although valid so far as it goes, misses Zeno’s fundamental error, and this is an error that is still alive and kicking. It lies at the heart of the mistaken notion that the mathematical truth of the world is the truth, period; such that, for example, space is a mathematical entity – a geometrical medium whose components are those zero-dimensional points, one-dimensional lines and two-dimensional surfaces that Euclid played with. Zeno’s error is to imagine that the mathematical description of motion (or indeed of anything else) trumps the experienced or lived reality of it.

We can illuminate this error another way. Zeno essentially argues that any journey requires us to move through an infinite number of infinitesimally small points. No-one knocking off these points at a finite speed could therefore accomplish any kind of movement. In reality, far from being obliged to move from point to point, we could not take steps that small. Long before we reach the end of the halving process we arrive at increments of movement so small that we cannot accomplish them separately. Furthermore, even if it were possible to pass through each point separately, given that each point is non-extended, the passage would not amount to motion. Trying to build up real movement by moving ‘through’ extensionless points is impossible.

We could put it this way: when I walk, I walk through lived, not mathematical space, taking steps, not fractions (1/2, 1/4, 1/8 of a step, etc). This non-mathematicalness is demonstrated by the fact that my journey could be described mathematically in an infinite number of ways – for example, by successive ‘trichotomisations’ (dividing into thirds). If the journey was identical with the mathematical description, I would seem to be doing two incompatible things at once. So what Zeno tells us is not that maths proves that movement through a finite space in a finite time is not possible, but that mathematics does not capture movement, because the mathematical description of space is not the story of lived space and our journeys through it.

Specifying Spaces

The distinction between mathematical points and places in the space in which we live – a mathematical point in space is not a place in space – shows a fundamental rift between the space of mathematics and the space of the living person. The gap is concealed when we use space-occupying signs such as biro marks to represent points, or capture them in values generated in a frame of reference. Except insofar as it is being used for abstract mathematical demonstrations, the frame of reference itself has to be established non-mathematically – as when I use my lived surroundings as a frame of reference. In other words, x=2, y=3, and z=4 does not generate a point in real space unless I have located the axes in a place defined non-mathematically.

Zeno’s paradoxes are the result of taking mathematics rather too literally as the actual material of the world (as if we walked in fractions rather than steps), or taking the space in which we live, move and have our being to be composed of points that somehow manage to have spatial locations without actually occupying space. Exposing the distance between mathematical points and real spatial locations is a way of reminding ourselves of the distance between maths and the world.

Confusion over the nature of points is not a thing of the past. Physicists once thought that space is ‘Euclidean’, and now they think it is ‘Non-Euclidean’. Cosmologists earnestly assure us that the universe began and will end as a ‘singularity’ – a dimensionless (i.e. mathematical) point of infinite density. That’s what happens when you take mathematical points for physical entities located in the world; and when you confusingly take mathematical entities and notions to be constitutive of the world.

There is much more to be said on this. Watch this space.

© Prof. Raymond Tallis 2011

Raymond Tallis is a physician, philosopher, poet and novelist. His latest book Aping Mankind: Neuromania, Darwinitis and the Misrepresentation of Humanity is out now.


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