Why do we think of space as three-dimensional? Faced with that question, the Dr Samuel Johnson within us will emerge and say firmly: 'We just do, and there's an end on't'. Well, I will suggest that Dr Johnson may be correct in his conclusion. But I shall explore a more interesting route to that conclusion. So here's a start on't.
One route to the conclusion might involve an appeal to Euclid (fl 300 BC). I am not sure whether Euclidean geometry is still taught in schools but most people have a rough grasp of what it involves: definitions, axioms, proofs and so on. You may remember that the sum of the angles of a triangle equals 180 degrees, or perhaps you remember the theorem of Pythagoras. Anyway, Euclid clearly thought he was offering a geometry of space, and his account of space had three dimensions. A first answer to the question then is: We think of space as having three dimensions because Euclid has proved by geometry that it has three dimensions.
But that answer will not do. Whatever Euclid thought he was doing, he was in fact working out by deduction the geometrical implications of his starting point, which consisted of his definitions, axioms and so on. The point is that most mathematicians and philosophers nowadays would say that the propositions of pure mathematics, such as the theorems of Euclid, are neither true nor false in any empirical sense. They are simply validly or invalidly deduced from the axioms and definitions of the mathematical system in question.
While this is true of pure mathematics, it is also true that Newton accepted the three-dimensional geometry of Euclid in working out his natural philosophy (physics). (In fact, he also devised the calculus and used it in his calculations, but that is another story.) The abstract world of Euclid is given empirical content by Newtonian physics. This suggests another answer to my question: space is three-dimensional because Euclidean geometry has been given content by Newtonian physics. Together they demonstrate that space is three-dimensional. I shall park that as a first shot at the answer, an answer that would have been accepted for several centuries.
But, millennia after Euclid, a non-Euclidean system of geometry was developed by Bernhardt Riemann (1826-1866). Again, as with Euclid, his system was developed as pure mathematics with no bearing on physical reality. It turned out, however, that his geometry fitted in well with the relativity physics of Einstein. In this combination of the geometry of Riemann and relativity physics we find conceptions of curved space and of a space of more than three dimensions.
Einstein's own teacher, Hermann Minkowski, took the ideas even further and treated time as a fourth dimension added to the three dimensions of space. His view can be summed up in his own words that space and time will 'sink into mere shadows, and only a kind of union of them shall survive'. I could not possibly query Minkowski's mathematics, but I hope it is not arrogant to ask what on earth that is meant to mean?
Readers might reasonably think that I should return to my starting point – the world of ordinary experience. Can an explanation be given of why, or what it means to say, that the world of our ordinary experience is three-dimensional? The word 'dimension' in ordinary life generally means 'direction', and – despite Riemann, Minkowski and many others – it seems to be a fundamental feature of the way we perceive our world that there are only three dimensions or directions: height, breadth and length. That was the view of Aristotle. He writes that if something can be divided one way it is a line; if two, a plane; if three, a body. Like much in Aristotle, that seems to be the view of common sense – these three magnitudes are all there are, and that is how we in fact experience space. Dr Johnson would concur.
Yet that view of space, if it is intended to be an account of how we in fact experience space, is not quite right. Psychologists tell us that our visual perception of near and far is less precise than left and right or up and down, and our perception of auditory space differs in respect of left and right, up and down, and near and far. More significantly, our perception of tactile space is radically different from that of visual or auditory space in being largely the perception of the relationships among objects.
Indeed, the philosopher/mathematician Leibniz argued with great sophistication that space is not a matter of 'dimensions' at all but is rather the relationship among objects. In everyday life, we follow Leibniz rather than Newton. Wittgenstein notes: 'It is anything but a matter of course that we see
three-dimensionally with two eyes. If the two visual images are amalgamated, we might expect a blurred one as a result' (
Investigations IIxi).
It seems then that there is not an exact correspondence between Newtonian/Euclidean space and everyday perceptual space. And there is no correlation at all between everyday perceptual space and the geometry of Riemann or the space-time conceptions of Einstein or Minkowski. Moreover, the phenomenology of our perceptions of space – how we in fact perceive it – does not fit with our assumption that space is three-dimensional. Grant this disturbing lack of correlation on all fronts, the question arises as to why our language persists in a three-dimensional picturing of space, rather than reflecting more closely on our actual visual, auditory or tactile perceptions of it.
Well, our actual perceptions of space might permit or indeed be better suited to a two-dimensional language, but that would be inconvenient. For example, we would need to abandon words such as 'near' or 'far' or 'behind'. When an object was approaching, we would need to say it was getting larger, and when it was receding, we would say it was getting smaller. When an object went behind another, we would need to say it had vanished, and then re-appeared. Such a two-dimensional language might be possible and enable us to describe the world in a rigorously phenomenological manner. I guess that some modernist painters have depicted space in this two-dimensional way. But a two-dimensional language is too impoverished for many things we might want to say.
Having noted all the foregoing bumps and bends on the route, I can now return to my starting point: Why do we think of space as three-dimensional? I suggest the following answer.
In pure abstract geometry, space is three-dimensional if our axioms make it so, as in Euclid. But equally, in pure abstract geometry, space can have many dimensions – Riemann actually suggested it could have infinite dimensions. But again, that is the case if the axioms of the system make it so. Space in itself, apart from what we call it, how we divide it and how we fix the positions of objects in it, is without any identity of its own. Newton gave it one sort of identity and Einstein gave it a different sort, and nowadays cosmology suggests other possibilities. Space is given these identities because it suits the purposes of the science of the time. But space itself has no identity of its own.
People sometimes say: 'Einstein has shown that space has many dimensions or that time does not move uniformly'. But scientific theory never provides grounds for denying a belief held in a context outside the range of its scientific method; there is nothing in scientific theory that either denies or affirms the validity of other methods of making sense of experience or the attainability of objective truth.
Hence, my suggested answer to the question – why do we think of space as three-dimensional? – is that our language both follows from and reinforces our three-dimensional thought patterns. In other words, it is the hard-wiring of our thought patterns as reflected in our language which gives our perceptions of everyday space in three dimensions. Einstein, Rieman and many other mathematicians, geometry and physics don't come into it.
Robin Downie is Emeritus Professor of Moral Philosophy at the University of Glasgow
