APPENDIX I: SOME THOUGHTS ABOUT DIMENSIONS
Before we take a look at our proposition about 'Time', I'd like to discuss with you the implication of Einstein's theory that what we think of as 'time' is in fact a dimension, just like the space dimensions. In other words, we have four dimensions: length, breadth, height and time.
But first, a light-hearted quotation from Lewis Carroll:
"It's called 'wabe', you know, because it goes a long way
before it, and a long way behind it...
'And a long way beyond it on each side,' Alice added."
Einstein proposed that Space and Time are inseparable components of a 4-dimensional space-time universe.
To see what this means, let's imagine a miniature world which is a cube. Now suppose that one of the faces of the cube - say the bottom face - is a little 2-dimensional world, a Flatland, inhabited by creatures called 'Toodies'(2-D). The Toodies' world then is the plane surface which is the bottom face of our cube. They and their world are not just paper thin, they have no thickness at all. They can move right and left or backwards and forwards, but the 3rd dimension, up and down, simply doesn't exist for them.
Plane ABCD is the 2-dimensional world of the Toodies
Unlike our imaginary Toodies, the Threedies (3-D) are solid citizens like you and me. They inhabit the 3-dimensional cube as their world, so that all parts of the cube are accessible to them. They can move to any part of the Toodies' Flatland and by moving upwards (or downwards) they can vanish from it.
Cube ABCDEFGH is the 3-dimensional world of the Threedies.
Since the Toodies' Flatland is infinitely thin (which simply means that it has no height - an infinitely thin sheet of paper, if you like), then an infinite number of Flatlands could be stacked into the cube. Or we could put it the other way round and say that the cube which is Threedy's world is made up of an infinite number of slices, of an infinite number of Toody paper-thin worlds.
But let us now suppose that a Toody is subjected to some force which can lift him up the 3rd (up and down) dimension of the cube. So he is propelled out of his own paper-thin world, the bottom face of the cube, right up through the cube to its top face. As he does so, he will pass through all the 2-dimensional 'paper' Flatlands which lie in between. Since the whole cube exists, then all of these Flatlands exist, even though they won't exist for Toody until he reaches them. So they lie in Toody's future.
How will Toody feel about this? Well, he will note that each world he passes through is slightly different from its predecessor. So his impression will be that 'things are changing'. But change occurs, and can only occur, in time. So his movement in this 3rd (up/down) space dimension will seem like the passage of time to Toody: it is his time dimension.
Anything that has existence must endure: i.e. it must have a time dimension. And Toody is no different: if he did not endure in Time, he could not exist. So if the 3rd space dimension is Toody's time dimension, then he must endure in it, which means that he must move along it.
And since the 3rd (up-down) dimension of our cube is Toody's time dimension and Toody's progress along it is upwards, then all the space below him lies in his past and all the space above him lies in his future. And since all parts of the cube are accessible to Threedy, this means he can observe the part of the cube which is below Toody and is Toody's past and the part of the cube which is above Toody and is the scene of all his possible futures.
Note that Toody still has choice. He cannot alter what lies in the future as a whole, but he can choose which way to move in his Flatland and this exercise of choice will determine his path through the future, and so will affect his own individual future.
Toody travelling in (his) time
As I shall be arguing in this book, it is quite illogical to suppose that existence occurs only at an instantaneous present, in a 'now' so short as to be non-existent. No, each individual, whether animate or inanimate, must have a time dimension, must have a 'now' which has permanent duration. This means that a Toody must have some finite extension in his time dimension, in his third dimension - a dimension which to Threedy is the third space dimension. So Toody's range of perception of the 3rd dimension could be finite, limited by the length of his 'Now', provided he had some means of perceiving.
Let's take an analogy. Suppose Toody is like an ant climbing up a brick wall. As the ant climbs up the wall, all it will be aware of is the red clay of the brick under its feet. It will observe at intervals that this gives way to something whiteish, a strip of mortar, which after another interval reverts to red brick. That is all the ant will see. The actual wall will be inconceivable to it, let alone the purpose for which it was built. But it still has a measure of choice, of free-will: it can choose which route it takes up the wall.
We have no reason to suppose that these conclusions are unique to a Toody, so let us suppose that there is a general rule that what a lower dimension observer experiences as his duration in time would appear to a higher dimension observer as a 'something' moving in space. Applying this to Threedy, our 3-dimensional material friend, then his Time must be a 4th dimension along which he must necessarily endure (i.e. have extension in it and journey along it). So to a Fourdy, a still higher dimension observer, Threedy would be a 'something' forever moving in a 4th space dimension.
Threedy marching along (his) time
Note that this perception of an extra space dimension as 'time' occurs when the observer has fewer dimensions: it is because Toody's observing instrument is 2-dimensional that he sees a 3-dimensional object (such as Threedy, for example) as comprising an ever changing series of 2-dimensional 'slices' or 'snapshots of events'. A 3-dimensional observer, a Threedy, on the other hand, would observe all the other Threedies 'whole' and though he might concede that they were changing, he would not realise that he was really seeing them as a series of 3-dimensional 'snapshots of events' of a 4-dimensional space-time. Note also that Threedy is 'outside' Toody's 2-dimensional world.
It seems to me that this follows from one of Einstein's great discoveries: that space and time are inseparable. This discovery is a vital ingredient of the seventh proposition of our theory.
What I'm saying is that an extra dimension has two aspects: Looked at subjectively by its recipient, it is experienced as time: looked at objectively by a higher dimension observer, it appears as an extra space dimension.
Moreover, the possession of an extra dimension enables an observer to take an outside view of a lower dimensioned world. Let us suppose, for example, that Threedy has an instrument which enables him to watch Toody as he moves (not impossible, because Toody's 'Now' gives him a little extension in the 3rd dimension). At any moment in Threedy's time, a moment Threedy calls 'Now', he can observe a portion of Toody's progress - how much of a portion will depend on how fast Toody is going and how long Threedy's 'now' is. That portion can be divided up into a series of instants. Each instant will have a past, when Threedy can see what Toody has been up to, and a future, where Threedy can only guess what Toody will do (see diagram above). Obviously, whatever Toody decides to do at any particular instant will change the portion of his path that lies in his future. So any portion of Toody's progress will seem to Threedy to have two parts: a distinct observable part and a constantly changing indistinct part.
This dual aspect of Toody arises from the fact he has no extension at all in the 4th dimension, Threedy's time dimension, a fact which gives Threedy a bird's eye view of him.
Similarly a 4-dimensional Mr Fourdy would feel himself outside Threedy's world, but he would need a 3-dimensional observing instrument to observe Threedy, who would have no extension in a 5th dimension, which would be Fourdy's time dimension. And by the same token, a Fivedy would feel himself outside Fourdy's world.
I hope to show that these concepts may lie at the heart of some of the mysteries of reality.