Set theory, Mereology, boundaries and continuity

With this notion of wuji, or apeiron, or continuum, or whatever, the whole idea of set theory never sat well with me. The null set (the set which has no members) doesn’t quite make the cut. Consider the continuum – perfect homogeneous Everything – and you realize, like the definition of apeiron, it is limitless, or has no boundaries. Thus it cannot be contained and so cannot be a member of a Set – ‘the set of Everything’ for example. Then I thought about the boundaries. These boundaries are the only way to distinguish something within the apeiron. So instead of thinking of sets, I thought of boundaries within a limitless whole. Being infinite, the first boundary is actually a bisection, splitting everything in the middle. without actually ‘cutting’ anything.

Think of it like a strip of paper with black on the reverse side, white on the other side facing you. If you give that paper a half-twist, you are now differentiating the strip into half-white, half-black. The first thing in a continuum would be a Point which, in 1D, requires two boundaries So a second half-twist (I would opt for an un-twist actually but I’ll get to that) is needed. Sure enough, you now have a ‘segment’ of the strip-continuum which is clearly black in the middle and white on either side. So what we know about a Point’s position in a line is that it is not the point to its left and not the point to its right. The twist and un-twist is so that you have the truth value ‘within’ the boundaries:

---|>0---0<|---   (my crude graphic of two NOT gates facing each other)

This got me thinking further: What is the smallest number of boundaries required to enclose a 2D region? I realized that three line segments joined as an equilateral triangle was the most basic enclosable region. So a 2D region is enclosed by three 1D regions. Then a 3D region is enclosed by four 2D regions. This was a trend that allowed me to reaffirm my earlier concept: a 0D ‘region’ is ‘enclosed’ by one transformation – the zero-D Reflection, Not.

Let nB = the minimal set of boundaries required to enclose a region in the nth dimension. |nB| is the number of boundaries within that set. nBm where m is the index of nB sets

0B={Not}, |0B|=1

1B={0B1, 0B2}, |1B|=2

2B={1B1, 1B2, 1B3}, |2B|=3

3B={2B1, 2B2, 2B3, 2B4}, |3B|=4

nB={n-1B1, n-1B2, n-1B3, …, n-1Bn+1}, |nB|=n+1

So the smallest 3D region is delimited by four 2D regions. This is, of course, the tetrahedron, which can be made by four planar ‘cuts’. A triangle can be ‘cut’ out of paper using three cuts. A segment of string can be made with two ‘cuts’.

So, also, I discovered another field of study in mathematics called Mereology – the study of wholes and their parts. There is mention of Alfred North Whitehead – a strong proponent of Process Philosophy, which is very strongly similar to the rationale that I’m using on this site. Everything is a process, or act of transformation.

Well, once again, this is a short entry, but more to think about… Hope you enjoyed it!

Substance vs. Space

It is important to distinguish the apeiron, or wu, from the space which emerges from it. They are not the same thing – or if they are, I need to be careful.

Just because the Dynamic and the Static define Points and then 1D Rotations etc., that does not mean that every Point is the Dynamic or whatever. Instead it means there is a Point wherever a zero-dimensional Reflection occurs. Wu is the substance, and the Point is the space which emerges from it. Transformations are the substance of Reality, at least I maintain. But transformations do not ‘occupy’ space, they define it.

This is the slipperiness of the notions we’re working with. Wu (apeiron) is not the null set, no matter how tempting it is to define it that way. Yes, the null set has no members, and Wu is the uniform unchanging substance of Everything and consequently has no relations, but it is not a set precisely because it has no boundaries, and so cannot be delimited by the concept of a set (remembering that null is “The set containing no members”). Wu cannot be contained but is the container. So the first ‘element’ of Wu is the null set, i.e. a delimited portion of Everything, which is not Wu. The null set is defined by the zero-dimensional Reflection ‘not.’

Maybe Wu can be defined by x ∉ ∅, without {}, thereby making it not-a-set, i.e. an element – but an element that cannot be an element of any set!

Even the ‘finesse’ available in set-theory which is the proper subset, where every element of a subset is identical to the set (from which it is ‘subbed’) yet this is not the same set, doesn’t fit the bill. This is why the null set is its own subset but not its own proper subset. Apeiron still doesn’t fit, because you still can’t say it is a proper subset of the null set, since it’s not a set!

There is another tack available to my reasoning though – because I said initially that Wu is transformations, it doesn’t mean that all transformations are identical. Yes, they are all inter-dependent (transformations transforming other transformations), but the first Static is sufficient for the emergence of Space, and the first Dynamic is sufficient for the emergence of Time, but is that to say that all of space is an infiny of points? I don’t think so. Yes, we use Points to refer to the position of something, so yes, this creates the possibility of reference.

Unfortunately I know far too little of Topology to begin to understand the various configurations available for a collection of Points. Yes, they certainly could be spherically arranged. They can also be flat (Euclidean). But an infinite expanse of an infinity of Points renders the ‘curvature’ of space a moot point, since an infinity of points allows for any shape imaginable – the concept of curvature ceases to hold any meaning.

Bah! I’m losing steam here. Running out of ways to move forward. Could my reasoning have gone down a dead-end path? Could I have thought myself into a corner? It looks like it, but I’m not giving up. I may have to start from Nothing again.

 

Back to Physics

While it is fun to philosophize and make stuff up about “Wu” and zero-dimensional reflections, at some point, this has to tie-back to ‘real’ reality – the actual, physical world of physics and matter.

I mean, seriously, what is this ‘zero-dimensional Reflection’ I keep talking about? How can a reflection be ‘of itself’?

Firstly, starting with absolute, conceptual Nothing, in order to ‘cordon-off’ a part of Nothing, it needs a boundary of some kind. This boundary is ‘made of’ the same ‘stuff’ as Nothing, which at this point we realize is Everything.

Supposing that this Nothing/Everything is stuff (I’m not yet talking of Matter though), then this boundary we need must do something to this Nothing/Everything to make it different from itself somehow (so that we may consider whatever is inside that boundary to be different from Everything ‘else’). But by the definition of Everything, this boundary is also part of Everything, so it must be of the same stuff. So a boundary, which does something, and yet is something, can only be a Transformation (a ‘mathematical function’ or operation). Transformations are the only ‘thing’ which satisfy the above three criteria, namely: A) are immaterial, B) can be something, C) can do something – that is to say, transformations can act on other transformations, producing new transformations (transformed transformations, so to speak), and yet all-the-while being immaterial. This means we have an immaterial Nothing-yet-Everything.

My first postulate: Everything is Energy, and that Energy is Transformations-operating-on-Transformations, infinitely.

But a Transformation isn’t a thing – it’s a description of a thing, right? Yes and no. Yes, of course it’s a description. When two different things are being moved in exactly the same way, how do you describe what’s happening to them? The action taking place is independent of what it’s acting upon, and is therefore a distinct thing.

There’s a first problem though: Just because something can change, doesn’t mean it does. There is no implication or proof of actual transformation – Action. Since Transformations are operations, they have an input and an output. The input, being other Transformations, infinitely recursively, as well as the output being another Transformation, means that Everything is inter-dependent and poised in a pregnant moment of indeterminacy. But you have to think about it a little further. There’s no Space, and no Time, so what kind of Transformation can possibly exist in a dimensionless space? I know only of four ‘fundamental’ transformations, as enumerated in Geometry:

  • Translation
  • Rotation
  • Magnification (a.k.a. Scaling)
  • Reflection (inversion)

The first three all require at least one dimension, since all three require three bits of information – that is to say, they are all vectors requiring a line of action – a dimension. That last one, however, and interestingly enough, is possible in a zero-dimension space. Reflection, in zero-d space, is simply an existential ‘inversion’ – Something either is or it isn’t. It is the Boolean NOT operation of sorts. Being an operation, it has an input and an output. Most mathematicians sorely dislike paradox, so would instinctively reject the possibility of a single NOT where its input was tied to its output. The equivalent of “This statement is False”. But, even if I were to accept the rejection of paradox, and say ‘No NOT operation may have its output as its own input’, what of three NOTs in a loop? Two NOTs effectively cancel each other out, they are semantically equivalent to ‘is’, so the third NOT’s output is it’s input. This happens for all odd-numbered NOT-loops. It seems paradox is unavoidable. I personally think this particular paradox is not a problem but, better yet, an answer to the problem ‘Just because it can, doesn’t mean it does’ as we will see with my second postulate.

I’ve come to call these two forms of zero-d reflections The Static and The Dynamic. The dual-not-loop is The Static. It is stable. The single-not-loop is The Dynamic, and it is ultimately unstable because it is a paradox in the sense that it is indeterminable. If the value or state of these three Reflections were to be evaluated, then at each ‘attempt’ to evaluate the Dynamic, the Static would be either/or (like ‘either 1 or 0’).

My second postulate: There is Action so long as The Dynamic cannot be evaluated.

This is like some kind of Reality-Computer, or a scratched-record. At each encounter with the ‘glitch’, it starts again, oblivious of its previous attempts. “Ok, the Static is one. The Dynamic is –glitch-. Ok, the Static is zero; the Dynamic is –glitch-. Ok, the Static is one…” Another analogy I feel is suitable is that of the programmer’s “infinite While loop”:

while(True){       Do stuff;}

 

But consider why I claim that the Static alternates. Why wouldn’t it just stay ‘one’ each time around? Well, if it did, then we wouldn’t be ‘different’ from the previous evaluation attempt. This violates the very possibility of ex-sistence. Further, ‘staying’ at a value is only half way around the loop. You cannot go backwards around the loop (to do so is to go anti-NOT-wise, i.e. not-not, which is The Static). And finally, the interdependence of all these transformations (you will see when we get into higher dimensions) makes it impossible to ‘stay the same’ at each evaluation.

It’s a pretty tall claim, to say that the paradox of the single-not-loop creates a sort of evaluatory context in which the dual-not-loop actually happens, but that is precisely my point – A transformation must have an outcome, it must happen. If it does not happen at all, then there is no Transformation to speak of. It ceases to be a Transformation at all.

My third postulate: Entropy is anti-Energy, where Energy is ‘having the quality of being different to itself’ and Entropy is ‘having the quality of being not different to itself’.

The Dynamic is pure Energy – the degree to which something is different to itself at each evaluation, and The Static is pure Entropy – the degree to which something is not different to itself at each evaluation. By ‘the degree to which’, one can consider ‘having the quality of’. By ‘different to itself’, I mean ‘of a knowable value’. The Dynamic, at any moment is neither True nor False, and so is the slipperiest of fish – it cannot be pinned down, cannot be determined or identified, so is ultimately different to itself instantly and eternally. The Static, in contrast, is the very opposite – it is either True or False. It cannot help but be something at some point – at any instant you look at it, it is something.

I believe (naïvely?) that this definition of Entropy isn’t too-far-removed from the present ‘accepted’ definition, and can still be used; only that Entropy now becomes the conjugate of Energy. A hot body has Energy because its internal state (its displacement on the axis of Time) is changing vigorously. A cold body has Entropy because its internal state is quite stable (lesser displacement on the axis of Time). As the two bodies meet, Energy and Entropy are exchanged and Equilibrium is attained. I use the word Entropy to describe a sort of anti-Energy. Potential Energy is pure Entropy – pure Static – the transformation hasn’t happened yet and it is highly self-similar, highly stable. Kinetic Energy Is Dynamic Energy, where a body in motion has a high degree of self-difference.

The notion of Rotation now seems plausible. Imagine a cylinder with one half (lengthwise) painted Black, and the other White. If you were to spin the cylinder around its length axis, but watch it from above, you’d see it alternate White-Black-White-Black, etc. This ‘rotation’ of existence, of ‘is-ness’ is what I’m talking about. Further, the closed-loop transformation in zero dimensions seems to be the smallest possible ‘entity’ – the Point.

My fourth postulate: Two N-dimensional Reflections are one (N+1)-dimensional Rotation.

So a Point in 0D is in fact also a Point in 1D. This is because it is a 1D Rotation. However, a line (being the first dimension) is defined to be the infinite collection of Points. So how does one Point differentiate itself from its two neighbouring points on a line? Simply by its boundary: the Reflection. Each Point on a Line is a Reflection of its neighbour. This re-affirms the ‘alternation’ of the Static, because at each evaluation, one Point must be NOT its neighbour. This is of course a 1D Reflection. According to my fourth postulate, two of these 1D Reflections are in fact a single 2D Rotation. So in fact, The Static is 0D, 1D and 2D. Continuing the reasoning, each 2D Point distinguishes itself from its neighbours by being 2D Reflections of them. Once again, using the fourth postulate, 2D Reflections are 3D Rotations. The fourth postulate is all-the-more striking when you consider the 2D Reflection/3D Rotation example:

Draw a triangle on a piece of paper, each vertex numbered. Then draw a vertical line (to represent the line of reflection) a little further away. If the paper is folded along the vertical line, and a new triangle is drawn where the two halves of the paper meet, with each numbered vertex of this new triangle corresponding to each original vertex, then you’ve successfully created its exact Reflection on a 2D piece of paper. But in Folding, you effectively did a half-Rotation in 3D space.

So The Static is a 3D Point also. This reasoning can be extended indefinitely (until proven otherwise), using the fourth postulate. But we’ve said that Rotations are vectors. So Points are Vectors also? Every Point in 3D space can be represented by a vector of norm 0 – that is to say the Zero Vector. This seems to contradict my separation of the first three ‘fundamental’ transformations from the fourth one. But in fact, it doesn’t. Ultimately, when down at the null dimension, the notion of magnitude or intensity is meaningless (the norm of a zero vector, being zero). Essentially, this means that, at zero dimensions, all four types of transformation become one and the same – the ‘pure’ transformation: “NOT”.

My fifth postulate: The Point is the Zero Vector in all spatial dimensions, and all spatial dimensions emerge simultaneously because of the Point.

So where does Matter come in? Matter has Mass, and Mass, it is currently thought, is but the distortion of space-time by Gravity. Space-time is the ensemble of Space and Time as a collection of axes – Three of Space, One of Time.

The following need to be determined:

  • What, really, is Time? Where is it?
  • What is Gravity in this context of Transformations?
  • Can dimensions be bent/distorted and if so, How?

What is Time? I adhere to the reasoning that Time is just the observation of change. There is the further, more human sense of Time which is irreversible Time. But, consider change happening. Take an oscillation, for example. The notion of periodicity requires a notion of comparison (because we need to know what happened before, to see it happen again). When considering ex-sistence, what exists must be different to everything around it, and constantly. If we were to take the periodicity of a Point as the prime frequency – the highest clock-speed that this ‘Reality Computer’ could attain, then every transformation must happen at a slower speed, but there must be change that happens at irregular intervals for something to ex-sist, otherwise there would be no ‘t=zero’ marker for our Universe to begin from. Any periodicity that is a multiple of the base frequency is nothing more than a different clock – and no notion of ‘when’ or ‘how long’ can be obtained. The slower clock-speeds just become the next level of comparison. What is needed is both aspects of Time: periodicity and irregularity. The Static and The Dynamic.

My sixth postulate: Time emerges alongside all spatial dimensions, and exists in all dimensions.

So we now have the three (and plausibly many more) dimensions of Space as well as Time. However, these Points, regardless of how Static or Dynamic they are, remain Points – not Things, not Matter. I’m still struggling to understand how a dimension itself could be bent or curved, but I suspect that it is this kind of transformation of a dimension (which, as we’ve just established, is just more transformations) that results in the ‘appearance of Mass’ through (sub-Plankian?) gravitational distortions of space-time, and therefore Matter. More to come later…