On rotations as continuously-reflected translations

Image of PDF file - single page

rotatn_is_reflected_translatns.pdf is a pdf of a little diagram I created this morning, where I muse with the ideas of transformations and how we can express Rotations (and circularity or curvature) using “not” – the ‘omni-dimensional’ Reflection (how would you describe a zero-yet-infinity-dimensional thing? “multi-“? or “trans-“?).

I don’t know why I called those vectors in the diagram ‘rays’  – they’re vectors which describe a translation transformation.

If this is right, why would we need ‘curved’ non-Euclidean spaces?

Hmmm….

In-between-ness

This zero-D Reflection is a strange bird…

The very existence of a boundary to separate a space in two gives rise to a higher dimension in which both can exist. In zero dimensions, a Thing bounded by a Reflection (“Not”), then (and I know the words are dangerously wrong here) has a kind of ‘inside’ which is not ‘outside’. Strictly speaking, this ‘inside’ is no longer even in the apprehensive realm of whatever is ‘outside’ – precisely because of the boundary which ‘shuts the door’ to the ‘inside’. It is clearer in 2D and then you can roll-back the analogy to 0D later:

Think of a circle – an enclosed section of 2D space. To a 2D observer, that circle is closed and has a boundary. Because it is closed, then the 2D observer might conclude that the circle has an ‘inside’ – but has no way of getting to it, unless one of two actions are taken:

1) Using a special 2D scalpel, the 2D observer ‘cuts’ the circle open. By doing this, it has reduced the circle to  a 1D line – an object in a lower dimension than the observer, and so both ‘inside’ and ‘outside’ can be observed (though arguably both ‘inside’ and ‘outside’ are lost, the moment the circle was ‘cut’)

2) Trying to ‘step over’ the boundary. Like trying to jump a fence – the only way to do this is to briefly traverse the 3D realm, only to land ‘inside’ the circle. But because the observer is 2D, now ‘inside’ the circle, then the observer loses all apprehensive ability of ‘outside’ the circle. It’s either-or.

The Reflection in a dimension, I’ve said elsewhere, has a special property in that it is a Half-Rotation in a higher dimension. So to pass from ‘inside’ to ‘outside’ (and vice-versa) you have to be subjected to a higher-dimensional Half-Rotation. But you’ll start and finish in the original dimension. What if you wanted to stay above 2D? Well, you’d have to ‘not’ complete the Half-Rotation. You’d have to do a Quarter-Rotation of sorts. This is obvious when we consider that the third axis of 3D is 90° perpendicular to the other two, but it’s also beautiful when we consider that some Thing both Is and Is Not in a higher dimension. So back to zero-D, where what can be said of a Thing is that it “Is” or “Is not” – in 1D then you can ‘see’ both.

Consider this line: —–IS—–<NOT>____ISN’T____<NOT>—–IS—–

The ‘region’ bounded by both those ‘not’ Reflections, is invisible (incomprehensible) to a 1D observer. We see it because we’re in a higher dimension.

Look again! Law of Paradox is True…

This is my most risky post. This is one of those posts I know will peg me as mad, but will turn out to be true, if only someone were to come along later and maybe re-word it so that it is more palatable to a wider audience. So here goes…

Much of philosophy is dependent upon three laws – what are called the three laws of formal ontology:

  • The Law of Identity (ID). It states “That which is, is.”
  • The Law of the Excluded Middle (EM). It states “Everything either is or is not.”
  • The Law of Non-contradiction (NC) (also strangely called the Law of Contradiction). It states “Nothing can both be and not be.”

I will show you that the last law (NC) is a contradiction in itself, and by being so, admits the validity of paradox.

But first, some ‘ground rules’ or tools with which to break down the three laws:

  • Truth:
    • Take to be True that which is permanent, which ‘hold still’. Truth forever is True.
  • Falsity:
    • Take to be False that which is not permanent, that which does not ‘hold still’. Falsity is never (not forever) True.
  • Not:
    • The inverse of ‘is’, where ‘is’ is ‘not-not’. So, ‘not’, on its own, is of similar quality to False, because it is unstable, undeterminable, and can never ‘hold’. All odd numbers of ‘nots’ are identical to a single ‘not’.
  • Is:
    • The inverse of ‘not’, or not ‘not’. ‘Not-not’, is of similar quality to True, because it is, stable and determinable, and will forever ‘hold’. All even numbers of ‘not’s are identical to a double ‘not’ –> ‘not-not’.
  • Or:
    • This is ‘not’, in the sense of ‘one, not the other’, this is a part, a disjunction, as per above False.
  • And:
    • This is not ‘Or’, or ‘not-not’, in the sense ‘Not(one not the other)’, this is the rejection of the part, acceptance of the whole, a conjunction, as per above True.
  • All:
    • That which exists and that from which is existed, both Thing and not-Thing. “Is and is not” which is ‘not-not not-not not-not not’, which is False, forever changing and unpredictable and cannot hold still.
  • Thing:
    • A ‘thing’ is what is, that which stands-out from the All, i.e. that which is not-All. Thing evaluates to “not-All”, so one more ‘not’, and so evaluates to ‘not-not’. True.
  • Nothing:
    • That which is not-Thing, i.e. “not not-All” – where ‘not-not’ is, so “Nothing is All”. Nothing == All. Nothing has the same qualities as All and as Falsity, it is forever changing and unpredictable and cannot hold still. False.

 

Now let us look at these laws:

The Law of Identity: “That which is, is.”

  • “That which is”, is “Thing” as per above. Thing is ‘not-not’.
  • “Is”, we’ve seen, is ‘not-not’.

Rebuilding this statement, we have “Thing is”: ‘not-not not-not not-not’: ID is True.

 

The Law of the Excluded Middle: “Everything either is or is not.”

  • “Everything” is ‘every (single) thing’. We can thus consider just one ‘thing’ and consider its truth to be applicable to every one of them. Thing is ‘not-not’.
  • “Either” is superfluous to “or”, so we can rephrase it to “Everything is or is not.”
  • “Is”, we’ve seen, is ‘not-not’.
  • “Or”, above, is ‘not’.
  • “Is not”, is ‘not-not not’.

Rebuilding this statement, we have “Thing is or is not”: ‘not-not not-not not not-not not’ – eight ‘nots’, which is True. EM is True.

 

The Law of Non-Contradiction: “Nothing can both be and not be.”

  • “Nothing” is ‘No Thing’, is not Thing, is ‘not not-All’, is ‘is All’, evaluates to ‘not-not-not’.
  • “Can”, is permission, so is equivalent to “is”, ‘not-not’.
  • “Both” is superfluous to ‘and’, just as ‘either’ was superfluous to ‘or’.
  • “Be” is “is”, ‘not-not’.
  • “And”, as above, is “is”, ‘not-not’
  • “Not be” is not “Be”, is not “is”, is “is not”, ‘not-not-not’.

Rebuilding the statement, we have “Nothing is and is not”: ‘not-not-not not-not not-not not-not-not’ – ten ‘nots’, which is True.

 

But let’s look at that last one (NC) again: We’ve seen that Nothing is All. So “All can both be and not be” This is a permissive statement indicating that A=¬A, a paradox! How can the Law of Non-Contradiction contradict itself?!? Because it admits paradox! At every moment, we remain coherent, even though it sounds so strange: “Nothing is and is not” <–> ”All is and is not”<–> ”All is True and False” (which, finally, is of vital importance if we’re to even have a concept of Truth, because there can be no Truth without what is not-True). This only admits that our Reality allows for the existence of Paradox (which, when you think about it, makes sense A) because we’ve got a name for it: Paradox, and B) we’re a part of this Reality, and we can conceive of Paradox, and C) it is a truly pesky thing which keeps cropping-up whenever mathematicians try to formalize the logic of the fundamentals of mathematics – paradox doesn’t go away!).

All, some may argue is not Nothing. But when you take Everything, and remove all boundaries between every Thing, you are left with a masse whole with no Thing – which is the apeiron – the All with out limit.

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!