I remember once becoming absurdly excited in Princes St. Gardens in Edinburgh – that was just where I chanced to be, not the cause of the excitement – when I realised that an interesting thing happens if you number the dimensions in the reverse of the conventional order.

My brother had once explained the concept of the fourth dimension to me by saying that it was ‘at right-angles to the third’ which he elucidated by explaining that, as a point drawn out in one direction gives a line, so that line, moved at right angles to itself, generates a plane surface, which in turn generates a volume when moved at right angles to itself; hence the next dimension, the fourth, must involve performing an identical operation with a cube.

This I found pleasantly bamboozling: I could follow the first three steps of the procedure perfectly, yet with the fourth there seemed nowhere left to go; yet I convinced myself that dwellers in a linear or planar universe would experience the same difficulty at stages two and three. Later, when the same brother suggested that the fourth dimension was Time, I formed some notion of a cube moving through space and leaving a sort of cubic trail behind it – which was the direction I was headed in when I started thinking about Duchamp’s Nude Descending a Staircase, here . There is a similar though less marked suggestion of movement in time in this picture, where Miss Kate Ward has admirably captured the headlong speed of my Dursley-Pedersen:

John Ward at speed on his Dursley Pedersen, photographed by his daughter Kate

But in Princes Street Gardens it struck me that if we begin with volume as the first dimension – and picture a cube, say – we can abstract a second dimension from that by attending to one of its surfaces and ignoring the rest; then by the same method, we can consider the edge of that surface in isolation to give us a line – from where, if we wish, we can go to its end, and arrive at a point – but that is, indeed, the end, and there is literally no point in going beyond it.

I still consider that a neat manoeuvre, though nowadays I am more interested in the point you end up with than in the doing away with the need for any further dimensions.

Duchamp’s nude shows a temporal sequence as a spatial one, and that is consistent with our normal usage, which is to talk about time in spatial terms – we speak of a length of time, a short time, a space of time and so on, and we use ‘before’ and ‘after’ – which denote spatial relations – to speak of temporal ones.

It is clear that the ideas of space, time and movement are bound together: indeed we use movement through space (which takes time) as a way of thinking about time, even where no movement through space is involved – and since we developed film technology, we have reinforced this with time-lapse photography, so that we see flowers opening and closing, buildings being erected, even carcases decaying – though in fact there is movement there, without change of location, though it is normally too slow for us to perceive as such – we only notice that there has been a significant change from an earlier state to a later one, without detecting it from moment to moment.

But could you arrive at the notion of movement (and so of time) in a world of immutable Forms, such as Plato envisages in his Republic? The Forms are timeless and unchanging, but if they are distinct and separate (and extended, for how else could we picture them?) it would seem that they must occupy space and there must be distance between them; and where there is distance, there is surely the possibility of movement, and so of time, even if there is none actually?

However, we have introduced another element here, the one that Berkeley drew to our attention: a perceiver. To picture a world of timeless forms is to do so from some point of view, a particular location within that world, and it would seem to be from there that the notion of movement arises – and indeed, might we go further and suggest that it is location that gives rise to the concept of space?

Does a point imply space? in other words, is even a point – location without extension – sufficient to require the whole of infinite space for it to be located in, or is that just a product of our thinking being tied to a three-dimensional model to start with?

But there is some sleight-of-mind here: we might picture a point in space from which we look outward, and so gain some sense of depth and distance, but in what direction are we looking? more to the point, what are we looking with? It seems we have smuggled in an eye, albeit an invisible one – though you might well ask how you can have an eye without a lens, an iris, an optic nerve – and some sort perceptive apparatus at the other end of that nerve.

This leads me to Kant, and his observation that ‘space, time and causality are the mode and manner of our perception.’ Hume – whom Kant spoke of as having roused him from his dogmatic slumber – had suggested that the only way we could arrive at an idea of causality was through observing invariable succession – we see that A is invariably followed by B and in time conclude that A causes B. This is not a very satisfactory account, and it is thoroughly demolished by Schopenhauer, who points out that the most familiar and invariable succession of night and day does not lead us to suppose that one causes the other.

It is, however, the best you can do if you want to insist – as Hume did – on the empirical principle, i.e. that all knowledge is derived from experience, via the senses. What Hume failed to grasp, as Schopenhauer pointed out, is that perception is an intellectual act, not the passive process Hume supposed it to be – the mind works on and arranges the data supplied by the senses – in fact it makes sense of them.

The stepping stone between Hume and Schopenhauer is Kant, who realised that causality was not something we derived from experience, but rather a pre-requisite for making sense of it – it is, if you like, part of our intellectual apparatus; and it is not alone – he couples it with Time and Space, grouping all three together as ‘the mode and manner of our perception’ and putting forward the notion that, far from being derived from experience, these three are the means by which we make sense of experience itself. In this, they have been likened to a set of spectacles that we cannot remove, even when we realise that they  condition all that we see. We are therefore in the frustrating position of knowing that the world as we know it exists only for us (or those similarly equipped) which is the truth that Berkeley realised with his esse est percipi ; what the world is actually like in itself (i.e. unconditioned by the apparatus of space, time and causality) we cannot imagine.

This is what started, like a hare from its form, the ding-an-sich or thing-in-itself which Schopenhauer eventually ran to ground (with the aid of oriental philosophy). Schopenhauer’s brilliant move is to point out that, while the world as it is known to us via the senses is indeed much as Kant suggests, a world of representation, of objects-for-the-subject, conditioned by our intellectual apparatus, that is not the sole aspect we have access to – if we turn our eyes inward, as it were, we become aware that there is, in ourselves, a sort of privileged glimpse of the inner nature of the world, the very thing-in-itself – namely, the Will.

For Schopenhauer, we as individuals are self-conscious outposts of a single and otherwise blind and unconscious Will which is manifested everywhere and in everything and whose sole aim is to exist. This has echoes in our own day in the ideas of the evolutionary biologists, that we are in effect the mere by-product of our genes’ determination to reproduce ad infinitum. It is not of course original to Schopenhauer, who derived it from his reading of ancient Eastern, particularly Indian, philosophy.

Schopenhauer draws pessimistic conclusions from this: effectively, all that self-consciousness has done is make us helpless spectators, aware that we are embodiments of a will that we cannot control but which rather drives us: all we can do, at best, is to quiet the will, to turn away from existence (though Arthur himself was happy to keep going to what was then the ripe old age of 72). Nietzsche, who saw himself as a disciple of Schopenhauer, takes the idea in a more sinister direction: we can embrace our situation, and recognise that the will takes us beyond good and evil – if we are strong, we should follow its promptings wherever it leads us and glory in it, rejecting the ‘slave mentality’ of judaeo-christian thought, which he saw as essentially a conspiracy of the weak to keep power from the strong. You do not need to go far beyond Nietzsche (he died, insane, in 1900) to find the horrors to which that ultimately led.

Yet this interpretation seems to turn on a needlessly pessimistic interpretation of the nature of the Will coupled to an erroneous aggrandisement of our own status – both Schopenhauer and Nietzsche assume that, in coming to consciousness in Man, the Will has attained its highest form of existence (though Nietzsche might qualify that by adding ‘so far’). However, if we look more closely at the judaeo-christian tradition which Nietzsche dismisses and at the Eastern thought that Schopenhauer derived his ideas from, we find a different way of looking at it – having looked into ourselves, and seen that we are embodiments of a single will (and might that will not be higher, rather than lower, and ourselves just waking into it, not yet fully comprehending it?) we can then look outwards again and see others not as different from ourselves but essentially the same – which is the foundation of compassion and the Golden Rule which characterises all the great belief systems – ‘treat others as you would have them treat you’ – or, if you prefer, love your neighbour as yourself  – (because that is what, in effect, he is). (For an excellent and inspiring 10’ talk on the Golden rule, click here. )