There are two types of space, not just the one. A mathematician might disagree with this statement of course since in mathematics there are many different types of space. Essentially – however – we can argue that there are only the two types – the type that we can say something about, and the type that we can’t. The type of space we can say something about is Abstract (or Euclidean) Space and the type we can’t measure or define in any way is what we might call Paradoxical Space. Abstract Space is very easy indeed to explain and this is because it accords so perfectly with our expectations. To travel in a specific direction in Euclidean Space is to get further and further away from our point of origin; to reverse this movement is to get closer and this clearly comes as no great surprise to anyone! There is therefore this abstract quantity called distance that accumulates as we continue to move in a particular direction. The axis that we are travelling along can therefore be explained by saying that there is ‘more of’ in one direction and ‘less off’ in the other. That’s what the axis is – it’s a way of marking off, in a continuous way, this continuum of ‘more of’ versus ‘less of’. That’s what makes a continuum into a continuum!
It doesn’t matter in the least what this ‘more of’ is all about – this is a purely abstract space that we talking about here so we don’t have to worry about that. We’re talking about the relationship between the measurements or quantifications we make of things here not the things themselves. The quantity could be ‘distance’ but it could equally well be weight or temperature or density or pressure or any number of different variables. Abstract space isn’t a ‘thing in itself’ therefore, it’s just had a system of relationships of something or other that we have ‘taken for granted’. ‘Taken for granted’ means that it doesn’t matter (as far as our calculations go) what that ‘something’ is, we just ‘assume’ it and then carry on with the business of ‘elaborating the relationships’. The continuum has no real ‘end’ to it in the sense that we can always continue having ‘less of’ at one end and ‘more of’ and the other. Wherever we are we can always still ‘add to it’ or ‘take away from it’, in other words.
There is no need for us to continue explaining Abstract Space because we have such great familiarity with it. It happens to be the way the thinking mind works anyway so of course we have great familiarity with it. We automatically assume the existence of Abstract Space every time we think. The key point that we need to make here is that there doesn’t need to be anything ‘real’ there – ‘more of’ or less of’ don’t actually need to be about anything for the system of relations which we are calling Abstract Space to work. In this AS is just like Jean Baudrillard’s ‘realm of the hyperreal’ in that it doesn’t need anything outside of itself in order to continue propagating – it quite happily ‘feeds off itself’ in other words.
Paradoxical Space is very different in that isn’t ‘about’ anything, it just ‘is’ – only in a negative rather than a positive (or ‘asserted’) sense. Whilst Abstract Space is constructed in relation to something that has been ‘assumed’ Paradoxical Space isn’t a construct and it doesn’t exist in relation to anything else; we can say therefore that even though Paradoxical Space can’t be quantified it has a quality of reality to it that doesn’t exist in its abstract counterpart. This is plainly counterintuitive – the space that we can measure isn’t real (but is only a projection of our assumptions) whilst the domain which we can in no way measure and quantify is a ‘reality in itself’, albeit a negative one.
We can move freely in Paradoxical Space but we can’t ever accumulate anything! We can neither accumulate some quantity of other nor can we lose it. We have no means of obtaining anything as a result of our movement and we can’t say that we are getting closer to something or further away from it, no matter how much movement we engage in! There is no possibility of us ever obtaining positive knowledge about our situation via an action of referring to an external framework.
There is no ‘polarity’ here in other words; there is no asymmetry that gives us directionality. There is no ‘more of’ at the one pole and ‘less of’ at the other. Does this mean that all movement (or change) is ‘hollow’ or ‘meaningless’, therefore? Again this is counterintuitive – it is the measurable type of movement that is ‘hollow’ or ‘only apparently real’. The reason for this lies in the nature of the logical continuum itself – the logical continuum advances in gradations, step-by-step, measure-by-measure, but as we have said, it isn’t a real thing. It’s a ‘formal system’ – it’s an extension of our initial set of assumptions. It’s an artificial framework, the artificial imposition of a type of order that doesn’t actually exist until we say it does. This is Abstract Space in a nutshell.
The whole point about Abstract Space (obviously enough!) is that it’s an abstraction, that it doesn’t actually relate to anything real. It doesn’t actually matter what ‘more of’ or ‘less of’ refer to, as we keep saying. Ultimately there is nothing that we are having ‘more of’ or ‘less of’ – there’s no actual concrete entity or phenomenon there, just a scale, just ‘a world that is made out of numbers’. We are relating in a very literal-minded way to the reflection of our own assumed measuring stick, but this does not make the world that we are relating to so literally real.
What we are saying here is simply that there is no such thing as ‘continuous change’. This sounds wrong to us straightaway because we are so used to thinking of change in terms of scales, in terms of measurement, in terms of the increments that we assume to have actual existence in the real world, but actually the structure of the physical world (as opposed to the formal realm of our measurements) isn’t continuous at all, as we know very well now. The physical world exists in a type of ‘punctuated’ space – it is made up of units of measurable space which are separated by ‘a discontinuity’ – something that doesn’t (and can’t) make sense in terms of the scale that is being used. Time, space, energy – all exist in ‘packets’, in ‘quanta’. There is no ‘overall continuity’ to the world – that’s just in our heads. Who said there has to be some ‘overall continuity’? In maths, Gödel has shown that there isn’t – that’s the whole point of the incompleteness theorem.
No change can occur in the continuity; no change can occur in the continuity because change means ‘jumping’ and there is no jumping in a continuum. Change means that we have to leapfrog across the uncharted abyss of the discontinuity! We have to ‘get real’ in other words and leave ‘the safety of the map’. The discontinuity equals ungrounded change and ‘ungrounded change’ equals reality, as David Bohm says. ‘All is flux; Nothing stays still’, as Heraclitus put’ it. Our fundamental mistake is to assume that reality is, in its essence, capable of being wholly or exhaustively defined, and this is why we find it so very hard to factor the discontinuity into our equations. We can’t for the life of us understand why reality can’t just be one happy logical continuity – it annoys us to be told that it can’t…
And yet when we ‘leave out the discontinuity from our equations’ we end up with ‘the Dehydrated Land’; we end up with a world with no actual change in it, only the apparent change that is associated with our linear scales. We can’t have change without jumping out of ‘the known’ – to insist that we can be is utterly absurd! All we have to do in order to see how utterly absurd an assumption this is is to reflect for a moment on what it is that we are actually saying here. How can there be change without us changing our viewpoint, our outlook, our set of static assumptions about what reality is? What sort of ‘change’ doesn’t involve us radically re-evaluating our way of seeing things? Do we imagine we can see change happening if we never change our basic assumptions? This just goes to show how deeply confused we are in our everyday thinking – change means change. Change means ‘movement into the unknown’. The movement from one known to another isn’t movement, as Krishnamurti points out.
There is no need to argue this point any further – it is so very clearly true. No one who is actually awake is ever going to argue against this! Change means change. How can we have change and yet at the same time not have change at the same time? How can we have change and yet to keep the framework within that change is supposedly taking place? We can’t have our cake and eat it – we can’t have change yet not have it the same time.
Paradoxical Space is ‘space within which real change can take place’ therefore. It’s the only place within which real change can take place. What’s more, as we have already pointed out, real change is synonymous with reality – there is no other type of reality than the reality of ‘change that can’t be measured’, ‘change that can’t be scaled’, ‘change that can’t be neatly fitted into a nice, safe map’. From a psychological perspective, we might start wondering what exactly it is that we have against Paradoxical Space. What do we have against the type of space within which there is no possibility of gain and no possibility of loss, the type of space within which there is no ‘more of’ and no ‘less of’? The answer isn’t long coming – there is no possibility of existence for the ‘fixed point of reference’ which is the self or ego in Paradoxical Space. The self or ego IS loss and gain, success and failure, ‘more of’ and ‘less of’ – there’s nothing to the self other than this and that is why we have to live out the course of our lives in the ‘dehydrated land’ that is Abstract Space …