If you’ve read along this far, you’ll have noticed how mercurial these matters can be. And, like mercury, madness isn’t that far away…
Well, I feel like I’ve hit something with such resonating truth, such obvious obviousness, that I know this is it! Now, to extract it while avoiding insanity is the hardest part – I don’t want to get high on my own fumes, so to speak. If, at any point, you feel like I’ve strayed into crazy-town, please, please let me know where you think that fork in the path lies – where did I begin to talk nonsense?
OK, let’s get started:
Let us consider what I call Wu – that is, the vast infinite expanse of Everything-that-is, ‘before’ the Big Bang. Wu is effectively Nothing (to our animal brains) because there was ‘nothing’ remarkable, no referential from which to even begin to perceive Wu. And yet, Wu is Everything because it is everything that we can work with to create a Universe, to create Existence. Being ‘Wu’, then, means that there were no dimensions (because a Dimension requires an origin, a point of reference from which to designate a magnitude extending into a Dimension). No Dimensions means No Space and No Time. In particular, No Time implies No Change. No Change means it’s pointless (pardon the pun) to even talk about Wu. We have no idea how long it lasted, nor could we. The essential thing is that there is a ‘we’, which means at some point there was Change. How can this be?
First, consider a transformation (which is just another word for Change) – a Rotation, a Translation, a Magnification or a Reflection. These are the four elementary transformations from which all others can be derived. Movement along a curved arc is but the combination of a Rotation (around a point) and a Translation (of a fixed ‘radius’).
Transformations exist en soi, regardless of your point of reference. The only thing is that you’d describe it differently – depending on your point of reference, but that doesn’t change that particular transformation in any way or form. Transformations are essential – they exist in essence.
Also, of prime importance, is to understand that transformations cannot be destroyed. They certainly can be transformed (a Translation ‘transformed by’ a Rotation becomes a new transformation – movement along an arc, as above). But even a Rotation (say, by 15°) can’t be destroyed by a contra-Rotation (by say -15°). In Matrix mathematics, this is the same as multiplying a matrix by its inverse, which results in the Identity Matrix. But they are still transformations! (arguably not much of a transformation, seeing as any transformation matrix multiplied by the identity matrix always results in the original matrix itself, so a sort of no-change transformation).
Right, let’s look at these transformations:
Translation: It takes something and translates it a certain distance, in a certain direction, along a certain line. These three underlined things are ‘magnitudes’, intensities of sorts. These are actually the three descriptive aspects of a vector: A vector has a given length (the magnitude of ‘distance’ above), a direction (again, as above), and a line of action (again). So, Translation can be described by a vector with three prerequisite bits of information.
Rotation: It takes something and rotates it a certain amount, in a certain direction, along a certain axis of rotation. Once again, the three bits of info needed to describe a vector.
Magnification (a.k.a. Scaling): It takes something and scales it a certain amount, in a certain direction, along a certain line. Again, a vector.
Reflection: It takes something and reflects it
a certain amount, in a certain direction, along a certain line. Hmmm… that first one doesn’t work. Something is either reflected or it isn’t. So there’s no notion of ‘by how much’. That second one also doesn’t work, for the same reasons. And that last one, though I’ll allow it for now, is actually quite tricky.
Consider a reflection of a point in 2D space. We say that the point was reflected relative to a certain line. Who’s to say it wasn’t actually rotated by 180° around another point exactly equidistant between the original point and its reflection? Sure, if the object weren’t a point, but a triangle, then we could distinguish it from a mere rotation because every point of the triangle would be reflected (and so it is impossible that it was a mere rotation). But that 2D triangle in 2D space, reflected relative to a line, is actually rotated by 180° in 3D space along the axis of rotation formed by that line. So a 2D ‘reflection’ is a 3D rotation. And, if you think about it, a 1D ‘reflection’ is a 2D rotation. (Questions for later: By extension, is a 3D reflection a 4D rotation? Is a 1D rotation a zero-D reflection?)
Reflection, when you reflect upon it (oh, c’mon, I had to!), is a bird not of the same feather as the other transformations.
So now back to Wu. When Wu is all you got, and you want to take a bit of it (to get ‘something’ you can work with), how do you separate it? What makes the boundary between the bit-of-Wu and Wu? By definition, the boundary must also be made of the same ‘stuff’ as Wu (because it is Everything, and thus all that you have on hand to ‘make’ a boundary with), and therefore so too must the bit-of-Wu. How on earth can you use the very thing you’re trying to separate from?
Well, let’s look at the boundary. What is it? What does it do? A boundary separates two things (‘bit-of-Wu’ from ‘Wu’). It is an action – it does something to one thing to make it different from another thing. It transforms. A boundary is a transformation.
Wait, what? A boundary is the description of the transformation which makes one thing different from something else. In geometrical optics, we call that boundary the diopter. And the diopter is defined as the boundary between two media where light behaves differently in one relative to the other (light might travel more slowly through glass than through air, for example, and so the diopter is the boundary between air and glass).
OK, but if the boundary is a transformation, then Wu must be (a) transformation, and then the bit-of-Wu must also be a transformation… Everything is change, and change cannot be created or destroyed (Energy, anyone?). So, enormously, we’ve also realized that our first foray into ‘pre-Big-Bang’ was totally wrong. There never was any kind of unchanging Wu. Wu has and always will be Change.
Remember that Wu, at this point, is still Zero Dimensions. So how can you have a zero dimension transformation?
And here’s where I came upon the clincher. Reflection is the only transformation that can be described in Zero Dimensions! Because, in essence, what is Reflection if not the fact of saying ‘This is not That’? Reflection is the existential provider of Difference! “Not” is the zero-dimension reflection ‘operator’. So now you have:
bit-of-Wu is not Wu.
And here we have Douglas Hofstadter to ‘save the day’. This is a beautiful example of a Strange Loop. How can something, which is made of the same stuff as Wu not be Wu? Well, strangely, it all works out:
Lets look at Strange Loops and ‘Not’s for a second: Firstly, if ‘Not’ is a transformation, then it is a function with an input and an output. Let’s represent ‘Not’ by a picture, crudely: >o–
I imagine this as a kind of Pac-Man guy with his mouth open, facing to the left, and has a little tail on the right. The reason I chose this representation is to allude to the Boolean logic gate symbol for Not which is the triangle with a circle on one vertex: -|>0-, but because I’m lazy and my European keyboard makes the vertical bar a pain to type, I decided to cut it. Besides, little dude’s gotta eat, so why stick a bar in its mouth and and jam it open?
So, imagine an infinite bag of these little dudes, all jumbled-up. Give it a good shake, and see what kind of things you can pull out of the bag. There are no rules except that dudes gotta eat, so they bite a tail, any tail.
Well, the first thing you might pull out of that bag is an infinitely long chain of dudes-biting-tails: >0–>0–>0–>0–>0–…..>0–>0–>0–>0–….
This poses a problem: What’s the first dude biting? Well, the only answer has got to be a tail, so the first dude is biting the last dude’s tail. Ok, so now we see that the only thing we can pull out of the bag is loops of ‘nots’.
And another problem: What about a dude biting his own tail? Technically, it’s allowed. What does it mean? What is a ‘not’ biting its own tail? Well, it’s the Strange Loop:
This sentence is false (not True)
Ok, so we have a huge assortment of loops, ranging from ‘one’ (above) to ‘infinity’.
Let’s look at these: If you don’t like the first one (because it’s paradoxical and self-contradicting and messes with your head), then we can start with two ‘nots’:
.->o-. | | '-o<-'
If you can pardon the crude ‘graphics’, that’s supposed to represent two ‘nots’ each biting the other’s tail. Well, that’s:
The next sentence is not True
The previous sentence is not True
And we see, that quickly works out OK. It makes sense and doesn’t cause a paradox. As a matter of fact, in Electronics, that kind of double-not-loop is called a ‘bistable’ and is used to store one binary digit in an array of memory.
Ok, what about three ‘nots’? At this point, I suggest you draw it on paper, because my next illustration will really be an eye-sore:
.->o->o-. | | '---o<--'
The next sentence is not not True
The previous sentence is not True
And we see that we’re back with the Strange Loop:
The next sentence is True
The previous sentence is not True
So which is it? True or Not True? Well, if you remember that we’re talking about transformations (0-D reflections), then the answer is both.
This is actually the most essential ‘thing’ to ex-sist. This is the ‘bit-of-Wu’. I have called it a Dynamic Truth, or just The Dynamic. This is because we’re talking about mathematical transformations which have no time, so this kind of transformation is perpetual, instantaneous, and simultaneous.
This strange loop is essential to our existence. If you remember that, as I wrote previously in the blog, for something to ex-sist it must be different from everything else, and the instant it is the same as it was before then it ceases to ex-sist, from which we established a sort of ‘life expectancy’ of a ‘bit of Wu’ was very short, unless it was permanently and instantly different from itself, then you’ll realize that this Dynamic is the answer to that dilemma – at all times, the Strange Loop is indeterminable, and so is inherently different to itself always.
While these strange loops (one for every odd number of ‘not’s) remain ‘stuck’ in indeterminacy, the normal loops remain stable (even numbers of ‘not’s, forming bi-stable loops). The whole system – Wu – is in a state of tension, ‘dogged’ by the attempts to determine a value for the Strange Loops. Wu is an infinite ocean of Static loops (bi-stables) and Dynamic Loops (Strange Loops). Wu is yin and yang.
And one final tidbit: programmers will be familiar with this state of affairs because they exploit it all the time. They use the ‘while loop’ to trap the program’s execution into an infinite loop which allows a program to ‘listen’ to user input. These loops are so ‘tight’ that the user thinks that the program responds in ‘real-time’, but in actual fact, it’s all just sequential operations. The same thing is happening here – ‘while’ Dynamic Loops remain indeterminable, Static Loops are are operated on as stable entities, thus giving rise to the first Dimension, through Translation transformations… The Dynamic Loops give rise to (are, even) the Dimension of Time, and permit Change to happen…
I have to stop here for now, but mull this over, and I’ll be back to push this further…
Please let me know what you think in the comments below! Thank you!