discrete numbers

9 10 2011

Had more of an idea about this a few days ago, but never got down to writing it. So, here is the vestiges of it.

The very act of numbering cuts up the world. Two as distinct from three. It is less a distinction, between this and that, and more of a boundary. Drawing a boundary around a thing, much in the same way a word might be used to package a thing perceived.

The structuralist might necessitate a distinction, of opposites. And there is, of course, a difference between two and three apples.

I’ve already remarked on sameness as being important for counting, or multiplying. There was something in here about multiplication…

3×7       7×3

Three times seven. Three groups of seven. Seven times three. Seven groups of three. Quite different descriptions of reality, or indeed different situations in reality. When calculated to the “answer”, 21, there is a further loss of information. Three boxes of matches with 7 matches in each, versus seven boxes of matches with 3 matches in each… same number of matches, different numbers of boxes.

It wasn’t this… it was a different tack. I just can’t remember… it was something very very simple about this notion of discrete numbers. What this means in terms of our mind’s processing. What the last observation seems to be about, is our mapping of number to “things”, discrete mental objects that might match some material situation, eg matches and boxes. This is happening at a slightly… later, or higher… aspect of consciousness; what comes first, is the discrete mapping of number like a word to a thing. One, two, three…

Wait a mo — another thought. Counting is one, two, three… the pointing at a new thing and including the old… two includes one, three includes two and one. Whereas, there is the whole-image form of counting (can’t remember the correct name for this… it’s not “counting”), where eg 7 things are immediately recognised as seven things even if the perceiver doesn’t have a word for “seven”. Hmmm three as in third…

For some reason, this makes me come round to music. Thinking about counting in time. Counting objects, I think, comes later. It’s a combination of this pattern matching, immediately taking in a pattern, and noticing a pattern in time. Combine those, and you get counting things.

But this is miles away from my initial thought about discrete numbers.

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positive and negative infinity

5 10 2011

Just noticed, perhaps for the first time, I can’t remember, that Reimann’s trick was to conflate positive and negative infinity to the same point. This is quite remarkable. This actually connects in my head to the higher-dimensional twist, the mobius loop in time, that I intuit is going on with consciousness. Ho hum, but there you have it.

It is strange. Start with zero, then the number line going off to infinity to the right along the positive axis, and going off to infinity to the left along the negative axis. And strangely, these two opposite directions meet at the same point.

OMG, as I write this, I am amazed I haven’t seen this. No I am amazed that Reimann saw this. I think this is what he saw.

It’s actually pretty simple. In terms of fairly standard human, enlightenment thinking, we are standing on the planet, let’s say at the equator. You point off east with your right hand, and west with your left hand, and sure enough, if you follow your pointing around the globe, you are actually pointing to a thing at the opposite side of the planet. And indeed, you could point through this and continue until you are pointing back along the opposite direction: your right finger pointing eventually meets the end of your left finger pointing in the opposite direction. In a way, you are pointing at yourself.

But this is predicated on the curvature of the earth. Now imagine this pointing is not bending. So you are standing in space. You point off in one direction, and you point in the other. You point off to infinity. The only way this could possibly make sense, is if you think there is a similar thing going on with the universe. That your right pointing eventually ends up coming back to meet your left pointing. This may or may not be the case with the universe. But what Reimann does, is suggest that it is, in effect. That is, be bends infinity. That is, infinity is not this endless thing, but converges. That is, he captures our ability to conceive of infinity. Or, more prosaically, our ability to label it, “infinity”.

We have taken a step from received understanding, I suspect. We are definitely performing within XQ space here now.

It is not so much that the infinity used in the reimann sphere has anything to do with existence, with the physical universe. It is to do with our human immersion in it. But not so much our physical embodiment, but our subjective orientation within it. Eg, left and right.

Returning to the math of Reimann, the strange thing is, positive and negative infinity meet at the same point. If you visualise this, it is a simple circle, as we have circumscribing the equator as in our initial thought experiment. This is ok if we are thinking of planets which are curved. This is a bit stranger if we are talking about subjective qualities, eg good things and bad things. A super extreme bad thing ends up meeting a super extreme good thing. The more they are apart, the more they converge on the same point.

It may be a mistake to think of this as a point, but it is definitely done with the Reimann sphere. Anyhoo, however you think about it, the reimann sphere is useful because it allowed mathematicians to map arithmetical functions to geometric functions. Multiplying by 1 and -1 and i and -i result in rotations of the Reimann sphere.

What I was trying to get around my head is how this is related to my notion of o being the centre of the individual and 1 representing the individual. I sometimes play with 1 as being the human, and in the Reimann sphere, that would be the equator delineated by 1, i, -1, and -i. I’d like to square this (might have to be careful with my language here;) with notions of internal states, which would be represented by fractions as aspects of being approach zero, or negative numbers as the mind’s reflection of what is, which takes zero as the centre point of the individual.

There’s something like a torus floating around here somewhere, but I can’t find it. Three dimensions of circularity, versus the two of a circle. Not sure if this is correct. Torus can be created from rotations of two circles of different scale. A sphere can be created by a single circle rotated around one axis. That is, a two dimensional shape rotate through the third dimension. A torus can be created by created by rotating a circle through a disconnected axis through the third dimension. So they are the same, except for the connection/intersection with the axis. That is, whether the zero is inside (circle, sphere) or outside (circle, torus). Interestingly, as far as I can conduct the transformations in my head, the first transformation can be conducted with a 180˚ turn, while the second requires a complete 360˚ rotation to complete the torus. But these are artificial means of producing torus, whereas I am more interested in how they form in smoke, for example, or dolphins producing air rings in water.

This was all sparked when I started to thinking of the mathematics of emergence. First spread consisted of systems, circularity, fractal in time, simultaneously iterative, Second spread has one equation/function/expression that simultaneously operates on several different processes which have different periods in time; which also may map to presence, mental, emotional, physical as basic levels of being. Third spread jumped to 0 = 1 – 1, which could be translated as

0 as the centre of the equation/function/expression, if it is to represent consciousness

0 or 1 and -1, if we wish to explode 0 into duality of mind

0 or 1 and not 1, which is kind of a description of the duality of mind

which leads to

e^iπ -1 = 0, euler’s equation

which means that

e^iπ = 1

and this has something to do with period, if i remember correctly, and the Reimann sphere.

There’s a lot in here. A lot. The revelation for today remains, that the positive and negative infinities meet at the same point, at least can in terms of maths. And if we are to take an XQ interpretation, this means the Reimann sphere is more a description of how we bend subjective thought space, eg the notion of infinity, to a well behaved point, eg the word “infinity”.

Somewhere along here is the math of emergent systems. It is do with nested systems in time.

And interestingly, I noted as I started out that my mind approached this from consciousness outwards. I started with zero, and then attempted to derive the other numbers, and found myself with euler’s rule, and thereby to Reimann’s sphere. That is, it is not about trying to work it out from the details, and integrating some kind of theoretical sense, but it is deriving details from some simple starting point. Kinda like the buddhist methodology, and like einsteinian physicists working it out from insight, first principles, and so on. This is quite promising. It suggests, that it is about appreciating simplicity, and from this derive all kinds of complexity. Can we intuit the field equations for consciousness?

Another thing to note, is that I am making progress, albeit slowly, on the notion that we may need a different form of maths. I am not sure about this at all, but it is simply a hunch when appreciating the invention of calculus by newton. A new maths had to be invented to capture the mapping of physical objects, that of functions. A new maths may be necessary to capture the mapping of mental objects. My mind might also have met with some encouragement when I read somewhere recently that the whole path of functions, the entire realm of maths, is limited in some way. Sadly, I can’t remember where I read this, heh. I shall have to wait until it pops up on the radar again, and again almost by random. Still, it will give me plenty of time to explore/prepare other areas.

ADDENDUM: after going back to include the image of the  Reimann sphere

What if the infinity point was the asymptote I keep thinking that mind is, not zero? And zero is the touching point with existence, the flat plane? I know this doesn’t make any sense, but all the numbers are contained in the sphere, and the plane below the sphere is just an illusion. Well, it is all an illusion. No, that’s not quite it. I keep getting fooled into thinking the  Reimann sphere is an object in space, heh, and I have thought about what is inside the Reimann sphere, for example, or outside it. Nope, the extra thought wasn’t this.

It was something about rotating the sphere through one of the axes, eg the real axis. Not the axis that is wrapped around the Reimann sphere, but the axis of the plane surface below. This should create a torus, which touches at zero, or rather does not touch at zero. But all mind is capable of is not really rotating the sphere but the great circle, marked by o, -i, infinity, i, 0, if we are rotating around the real axis. To rotate a whole sphere through space leaves what exactly? And an empty sphere at that?

Ho hum.








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