the function of i

30 04 2011

Interested development of thought, tying together multiplication by i on the complex plane, and the standard model as represented below:

If we take the dimensions to be unconscious to conscious to be our x axis and incompetence to competence to be our y axis, we get a similar mapping from bottom left quadrant to bottom right and so on, anti-clockwise, which is equivalent to multiplying a complex number by powers of i.

We can abstract this system further, if we consider any two dimensions of “opposites”. The establishment of zero on the number line as a continuous and contiguous part, gives rise to the extension of the number line into the negative direction. It allows the notion of reflection, how 6 can be converted into -6 by the multiplication of -1. That is, the negative identity in multiplication, -1, transforms any number x to -x, and vice versa. I mentally refer to this as reflection, whereas it can be conceived as being rotation.

When we superimpose one dimension with another at right angles centreed at the mutual zero, the usual x and y axes, the multiplication by i (ie the root of -1) rotates a point through 90˚. Note that multiplication by i^2 is the same as multiplying it by -1, the negative identity of multiplication, and hence performs the same function: rotation by 180˚ (the notion of reflection breaks down in two dimensions in this case).

How can this function occur if we do not have zero in our number line? When we consider simply A and not A as “opposites”?





zero not on number line

30 04 2011

If we consider -x as being “not x”, then the natural conclusion is to say that absolute zero is not on the number line.

Zero was adopted in western maths relatively recently as a means of ascribing a symbol for the lack of something. Namely in dark and middle ages when using counting boards, there were no beads on a particular power of ten space. Hence, “0” represented no units, or tens, or hundreds. A digit to represent no beads.

Seems natural, therefore, to have it starting counting in a number line since 0 precedes 1, and the distance between 6 and 7 say is the same as the distance between 0 and 1. Yes, for sure, if we are measuring things. But, if we are not measuring and counting things, how does this effect our use of zero? Again, the direction of XQ.

If we have -7 as meaning not 7 specifically, and -3 as being not 3 specifically, then the absence of all number, not x of any kind, can be indicated by the number zero, the absence of all numbers.

(The reason things get tricky, is because we are using iteration — the ten symbols to represent digits, combined with place value; thus 7 may mean 7 units or 7,000. This folding in of itself, gives the power of modern mathematical notation. The playing around with zero in this context might confuse us. To keep things straight, not twenty has no “zero” in it; there are twenty things, there are precisely not twenty things. The zero used in the number-glyph is “20” is not significant wrt the number, but our notation system. There is no “not” in twenty, as a number of things.)

There are certain mathematical progressions that do not have zero in them, zero as meaning not any and every number. It is not reached, it can not be, it is not part of the same substance as number. x^y. For every number x>0, there is no value of y that brings the total x^y to be equal to zero. On this path lies the significance of the mathematical constant e.

This relates to something Leon came up with a long time ago. I am a person who approaches zero; zero is part of what I do, deeply, experientially. Whereas, Leon had as a basic metaphor, the Egyptian binary number system. He did not appear to have a zero in his being vocabulary. His closest association was, and I bet still is, with one. In the old context of our conversation, I think I related this to identity, of the individual, as well as tot he mathematical sense of identity: 0 is the identity for addition, and 1 is the identity of the function of multiplication. That is, a number is not altered if we add 0 to it; a number is not altered if we multiply it by 1. In this context, we have completely different number systems,: one where zero is part of the number line, and one where it is not.

That is, the notion of zero included in our number line gives rise to the potential of negative as we commonly understand it. It is the pivot around which negative may come into being; it is the means by which opposition enters into maths; the gateway for duality of mind to find itself on the page. Without zero as part of our contiguous number line, we can not have negative.

What, then, comes of a mathematical system that does not have zero as part of the number line? And what of a person who has this… metaphor… identifying idiom… as a part of their character? Their being? A person without negative…?





counting cycles

15 04 2011

The experience of counting lends itself to thinking of a line. We count successive markings in a line. This appears natural to us. It appears true. There appears to be no other way. It is clearly useful. It has furthered our understanding and allowed us to bind our understanding to the nature of the universe. And this is true for the physical universe, the objective, or at least the crude or obvious aspects of the objective, physcialised universe.

But there is another way of thinking about counting, equally natural. It is to count in cycles, through the medium of time. (An observation: as a few cm in a line are bound within a greater measure of length, and yet greater, so we may consider cycles of time may be contained within greater and greater measures of time. And: the application of fractals in space, the koch curve etc, may have similar application in time… that way lies the unfurling of consciousness, its secrets revealed by the methodology of patient meditation; not the peeling or cutting of an onion, mind, but the gentle opening and revealing of a flower… the lotus blossom, no less.)

So, in one of my first ever jaxed tracks, before I even knew what jaxing was, I counted the beats in a piece of music. I counted through eight, then repeated. Thus, I was counting the micro cycles that constituted a beat, and one of the longest, most easily recognisable patterns came in chunks of eight. A pattern of melody. I did not count the number of patterns of eight, but there was definitely movement in them, and crescendo, and shift to a completely different set of patterns. And so we have patterns within patterns in music, that is, in time. I can comprehend, that is recognise, some of Beethoven’s Ninth. The final movement, so… iconoclastic, so… self-defeating, a magnificent release of human ego, an ultimate submission to the source of creativity, and humble gratitude — and let us name it God! for convenience, for glory, for ease, for that which begets the music in the first place!

So, the ninth inspired, and inspires. My mind, like any other, is taken up with these patterns in time, and its pressing edge against the inconceivable. It was performed only twice around Beethoven’s end of life, clearly it was not recognised at the time whatever the retrospective scholars and enthusiasts say. He managed a leap into the unknown, and it is praised in the act of part of it being chosen for the european union anthem. A wonder. And I wonder how many appreciate it…. deeply, alone, in one’s listening….?

We do not need to concern ourselves with such scale, but simply conduct a thought experiment of any music we appreciate, and count. Observe the counting in one’s mind. You need not apply numbers. Listen again, and hear the combination of patterns, one mounting on another. Listen again, and again, which is easy since it this music is something one loves, it catches the mind and the heart, it moves the body or lifts the spirit, it has magical effect on what it is to be conscious. This is what we are observing, by our counting in time — forget the numbering! This is what the direction of XQ is.

And it is the careful examination of the effect of sense, and feeling, and thought, that is subjective science, and precisely our nature in the infinitesimals of time, is what the buddhists intimately contemplate.








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