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 fom 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, his 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.

It is strange. Zero, 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 rapped 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 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.

Remember, this is after Wisdom, where the psycho-social concept actually revealed itself to be two different systems: the system going on in our heads, and the system going on out there. The out there bit is predicated on our action, and consists of social objects. The in-here bit is predicated on our thought, beliefs, and so on. What we have in here partially determines what happen out there, but it doesn’t really matter what we think is going on.

Here’s a couple of thoughts regarding rates of change.

We are all on a continuum of how much change we think is possible. The more we are engaged with the current system, the more fixed we are about what can and can not happen. The top dog is essentially fixed in position, responsible to maintain it. The further out we are, the more flexibility our thinking, the more we can see the bigger picture perhaps, unbounded by any specific institutional directive. There is a relativity of sorts going on here.

Things get ugly when two people from different systems engage, each embedded in their own. This is just about understandable for two companies competing, what about different companies? What about job system versus a governmental system versus the natural world system? Embedded systems. Messy.

Can’t be bothered writing this in html… gotta wait until I am off the ipad…. jeeez, the hassles…

Zero has several … uses of it or approaches to it.

First, the obvious, zero as a digit, used in place value with the other digits to signify numbers. eg 302 is the number three hundred and two, indicating three hundred and two things. The 3 stands for three hundreds, the 0 for no tens, and 2 for two units. A useful shorthand.

Second, the subtle, zero even as a digit, does not behave like the other digits. In eg 3003, the 3 stands for three thousand and the other 3 for three, and three thousand is quite different from three. You can’t confuse them. Whereas, the 0 stands for no hundreds, the other for no tens, that is the same number, none.

Third, the obvious, zero as a marker for the number zero. And what is the number zero? This gets tricky. No anything. Nothing. Is it even a number? Well, whatever it is, we have a symbol for it.

Fourth, the subtle, zero. Nothing. Not even the symbol to represent it. Beyond thought or thing. Void.

So, to map it to subjective experience. Fourth, the subjectivity exists as null-state, beyond conception, the void. Third, we can call this buddha. We can call it anything, as it happens, because it is just a symbol, “buddha” or “0″ or “void”. Second, subjectivity as defined by context, a socially derived self. There is no meaning to the person beyond the engagement they have with others. And first, subjectivity as I, a specific person, here and now, thinking and reading this and believing they have a personality or perhaps beliefs independent of others, an essential self.

As pure that. What number are you?

I have had some experience like 0 as place value, and I have met a lot of people who know about the marker “0″. They talk a lot, they know a lot, they know how it fits into the number system and all that. But they are a million miles from the actual experience of 0, the void. And I’d rather engage other numbers, than those who talk about and signify 0.

hehehe, that’s actually funny.

Simply, he is absolutely affirmative that there is nothing inside. This is rather nice. It echoes the experience I had when I was high for three years, that state that was self-sustaining. It is all about engagement with the world, and the engagement is the world. And so the individual is freed from the prison-interpretation that the I is trapped inside the body, and the world is out there, and all I can do is respond to sensations, electrical impulses and the like that the body generates. Nope, not at all, says Harding. The world is engagement, the world is the head, not a meaty thing. Nice. Definitely recommend the book.

Right, to math! He talks about the science of the 1st person, appreciation of no-thing inside, works perfectly or not at all, says Harding. The (non)experience of no-thing is the same for everyone, and hence the science of the subjective first person is actually the only objective thing that can be determined. But consider bananas: their DNA might be identical, but each banana is still quite distinct. So, is this not the case with us? I also have the notion of zero, the end of the psycho-social continuum ending in each of us, and yet because of its similar quality, fathoming one’s own nothing is tantamount to fathoming anyone else’s.

It comes down to how we use zero. 0 as in “no beans on the counter”, 0 as “no millions, no thousands, no hundreds, perhaps even no units. That is, zero as a marker for “none” in terms of place value.

Contrast this with, 0 as in “the absence of any number” or even, “the absence of anything and everything”. That is, the void.

As we know, in actuality, there is no such thing as nothing. Or at least, what we thought of as empty space, isn’t. I am not sure what is going on in an atom, the “space” between electrons”, but I suspect we are injecting our sense of “nothing” into these “spaces”.

So, this causes me to reflect on whether we actually use “zero” in anything other than the first way, as a marker for place value. We never come close to the actual (actual?) meaning of void. And here we encounter the buddhist sense of void, and indeed, I have suggested that mind is effectively the absence of matter, the stuff of mind is “negative”. Indeed, we thought of negative numbers as being purely mental, and that -7 meant “not 7″ rather than the additional concept of “opposite” to positive.

That is, if we are going to conduct appliedXQ, which is to say, if we are to consider XQ seriously with application to our lives, then comprehension of zero is essential. Not just in the guess we have made about infinitesimals as buddhist perform a kind of calculus on their subjective experience, as time tends to zero, but in terms of performing mathematical calculations with the deep appreciation of zero as void.

One of the attempts by mathematicians to rationalise maths was to ground counting in set theory. They do this funny thing, which I can’t quite get my head around. They talk about the empty set, then the empty set as a thing — that’s one. Then the empty set and one — that’s two. Now the empty set, one and two — that’s three, and so on. It’s a bit weird.

But as far as I can understand it, 0, or the empty set, is not a number. 1 is a number, and 2 is a number, they are the same sort of thing. The 2 in 201 is similar to the 1. Two apples are suggesting the apples are the same, ignoring the fact that they are different, slightly different flavours etc; we are just drawing attention to their distinctness, their unique thingness. Just like the 0 in 201, that there are no apples. Not that there is no anything, just no apples. So, 0 can be considered a number, when it is used like a digit like other numbers. But when is it ever used as the absence of everything? Ie, a non-number?

What does it mean when we add the absence of anything? Or we multiply by the absence of anything? Or we multiply to the power of the absense of everything? This gives some sense, ironically, not to the absence, but to the function. With addition, the absence of everything doesn’t change a thing. With multiplication, the absence of everything nullifies whatever value it touches. Multiplication is like the black hole function when associated with the absence of everything. And with indices, the answer is 1, always one, a number, the only number, the whole, perhaps even “everything”.

Does this give us an insight into what multiplication means subjectively speaking? Like when we multiply -6 by -1 we get +6, a state change, something we have suggested earlier is involved in our mental processes: if we master the “multiplication by -1″ it doesn’t matter what value of negative we encounter, we can convert it immediately with the same function, to its positive. That is, we can turn any apparently disadvantageous situation into an advantageous one.

So, what does multiplication by 0 mean? Neutralising whatever is good or bad? Meditation? But meditation we have surmised is calculus, as time tends to zero. Hmmm, when we look at how Newton invented his calculus, he used multiplication by a fluxion, or a number which tended to zero. This was ironed out of the explanation later, and we adopted Leibnitz’s interpretation I think, because Newton’s fluxion was eventually used to divide things, and as we know, we can’t divide by zero. That’s interesting, now… we can’t divide by zero…

Ok, let’s say that the inner state of being is the zero state. We can multiply concepts, insights, beliefs, visions, whatever consists of the substance of the mind by zero. We can perform this function of multiplication, we can apply the zero state to our substance of mind. And by doing so… we realise the empty state of mind? That everything we thing and so, is actually empty, void? That makes some kind of sense. Can this turn out to be “therapeutic”? Can we apply this function, when we are actually performing maths, and it has an equivalent resonance at other levels of our thinking? 7 x 0 is… 37 x 0 is… 6,431,407 x 0 is… Meditate on this. Or rather contemplate this, in your mind, as you sit and have objects of thought. Imagine seven apples, and applying the function of multiplication as an application of the void that is self. Or, the obverse, from these considerations, the effect of multiplication by zero, come to an appreciation of the empty state of self, of zero itself.

Following closely, contemplate an object of thought, a feeling, a sensation, and apply the function of division, of separation, of analysis, of parts to it. Into no parts, into the absence of any parts. Well, clearly, we can’t. We can’t divide a thing into no parts. Well, if we did, it should be just the one part, itself… one? Or, if we apply mind as void to it… what does this mean? What light does this cast on the mental process of division? That is, can not be done? What can not be done in the mind? An object of the mind can not be divided by the empty state of mind. To divide, requires something. Requires discernment, requires categories. And hence the mind that is empty, the core of the being that is void, can not be used to divide, to analyse, to consider the parts of any mental object, or perception, or sense? Does this make sense?

And what might be the practical, appliedXQ, exercise here? It is like a koan. 7/0. 37/0. x/0. Observing the mind’s… neutrality, or stillness.

Another way of thinking about divide is to share. So we share 8 sweets between two people and they get 4 each. We share 7 things between absolutely no people… hmmm, no… There are other ways of thinking about division, and one of them means that numbers divided by zero become infinite. Divide a chocolate block with 8 knobs into two parts; each part will have 4 knobs. Divide it into four parts and each will have 2 knobs. The less number of parts you divide it into, the more knobs. Dividing by a fraction, produces more. The chocolate bar divided into 1/2 knobs can be shared by 16 people.

There is a massive difference between a number, however small it might be, and zero, in this sense of absolute nothing, absence, void.

What this may mean for powers, I don’t know. It is another country! Mandelbrot land, me thinks.

When we apply the state of void to numbers in an “addition” or “subtraction” way, we get no change. When we apply the state of void to numbers in a “multiplication” way, we get the void. And when we apply the state of void to numbers in a “division” way, we can’t… or I have a vgue recollection that if you do, we get infinity.

Definitely related to the reimann sphere.

Briefly, repetition of a function, multiplication by constant, will make a value get bigger or smaller, stretch towards infinity, or approach zero. But they never reach “infinity” whatever that means, and they never reach “zero” as the absence of everything. The numbers will be finitely large, and finitely small. The leap to “infinite” or “zero” is a step change. They are not numbers. However, when we multiply by zero, bang, we have the step change. And if we divide, by symmetry, bang, we have the step change of infinity, whatever that is.

Oh look, no-state — bang we have 1, a thing separate from void, number. Then we can see we have no-state, and 1, the thing separate, that’s 2 — and like an unzipping we get 3 (no-state, 1 and 2). All because of the mistake of thinking of zero a number. If it is not a number, we can settle on being aware of no-state. 1 and 0. Something being aware of nothing, without it incrementing.

Which really brings out the comparison of addition and multiplication. What are the mental versions of this? Repeated addition is multiplication. 2+2+2+2 is 2×4 = 8. 2 added to itself x times, 2x or for some reason i’d like to write x2, would be infinite too. And x3 would be a larger number than x2. Counting up in tens, x10, normally written n10 (or 10n) as n increases.

2 added to itself 4 times is 2×4 = 8. 2 added to itself no times is 2×0 = 0. 2 added to itself once is 2×1 = 2. Or shouldn’t that be, 2 added to itself once is 2+2 = 4. This is a language mistake. 2 added to itself 4 times is 2 +2+2+2+2 = 10. That is, we have the original 2 to count in. So, multiplication doesn’t count its original. It changes it. Adding 2 doesn’t change the original value. Multiplying does. Or we could say that multiplication implicitly contains a zero to start with in terms of addition. My that’s ugly. But all that means is 2×4 means 0 +2+2+2+2 = 8.

And of course, multiplication means perform the same addition function. Just like when we count two apples, we are saying they are the same. So, we are saying +2 is the same. When of course, we know that 2+2 is quite different than 100+2.

So, continuing the ante-previous paragraph, the reason why multiplication by 0 end up with zero is because that is what is assumed we start with. The function 2×4 is more correctly (+2) done four times. It is not the number 2 that is manipulated, but the function of +2, add 2. That’s pretty smart. So if we multiply a function of +x by zero, if we perform the +x function no times, then clearly, we have nothing.

But we already knew this, of course…. the trick is applying… reflecting on what the processes that are going on in our heads, not so much in the maths…

Multipling by -1 means performing a function of +x negative one times. Or better, performing a function of -x (ie a “bad” thing) negative one times. Performing a function of a bad thing precisely not once. Performing a “bad thought” precisely “not once”. Performing “bad thinking” precisely “not once”. Performing “a thought that is going in the wrong direction” precisely in the “opposite direction”.

Multiplying by negative one is precisely the operation of the mind’s capacity to be oppositional. So, if the direction of a thought is going towards zero, ie has negative alignment, ie is aiming to self, multiplying by -1 means be oppositional, and turn the negative alignment into a positive one, ie aiming out to the world. hmm, not that it should be battered back like a tennis ball, at the zelf, zero, that is the other person. No, one doesn’t aim. One reverses, into the world. It would take the other person to see this as something which is directed at them, negatively. Hmmm, is this why buddhist wisdom is knife-edged? It can be taken “negatively”. Or why, when encountering a person who is “nothing”, all you can see is what is “reflected” back at you.

Hmmm, this is a lot… we veered too much off our course. Making a little progress towards the subjective appreciation of addition as contrasting multiplication. Oh, and counting is just the simple case of multiplication on the +1 function. Or rather, just repeated +1 function. Addition is wider ranging phenomena, eg +5+7 = +12. When we are adding things that are not similar. And another thing: we can count holes, but these are like counting discrete items, eg not7 and not5 is not12. This is quite different that void, the absense of all things.

Wait a mo. The void of things, that’s like the interpretion of negative numbers as not. The void of self is zero.

Interpreting multiplication as repeated addition, repeated function +x or -x, zero times, ie by not repeating it. Leave the flower to do its flower thing. No need to think about it. Name it. Colour it. Event appreciate it. Because, the body does that anyway. No need for mind to re-present, or repeat. No need to count. Multiplication by zero negates addition or subtraction. Nullifies.

That’s pretty simple. That’s rather neat. The judicial application of “not”.

Applying repetition oppositionally in mind, is to create a version of a thing in mind. Seeing 7 and thinking -7. Or Seeing a thing, and thinking the bad version of it. Or thinking the opposite of it. Not so good. Misapplication of x-1. The correct application of mind, of x-1, to repeat once in opposition, or precisely not once, is on the mind itself, turning a -7 into a 7. Seeing a bad thing, and thinking the better version of it.

I really need to get in touch with this zero, subjectively speaking, for me to be able to apply it as zero repetitions, or the extreme form of apply precisely never. To eg itch, so that it is nullified, no action follows. I have been using the force of my mind. And after a few days of negating itch, I eventually break, and I end up scratching terribly. The trick is not negation, but nullifying. In this case. And this is effortless if it is “employing” the absolute zero.

Phefw.

The symbology for iterated functions happens to be the same super-position we use for indices. That is

f^3(x |-> x+1) when x=2 is

f((((2)+1)+1)+1) = 5

This is neat when we consider the tentative thoughts we have made about indices and numbers of people, link this to versions of other people’s thought, and even link it to Bateson’s insight that ratio was a first order function, shouldn’t really be considered to be part of the number line in some way, and should be mapped on the i axis. Ok, I am bodging things, but I am merely making a note for myself. What is important is that this method of symbolic representation links up algebra, function, geometry, and a few thoughts we have traced through complex mapping of functions between eg tango and social organisation, or the versions of ideas we carry around in our head. That is, the models we carry around within the psycho-ended bit of the psycho-social dynamic, and how our different versions of eg “democracy” as they related to the complex social dynamic which is in existence supported by our actions.

This is inspired by this chris post. Incredible! I’ve always wondered how julia sets link to mandelbrot set visually, and here is this kid just exploring. Marvellous.

First, mapping actuality. I’ve been approaching this a few times over the last two years. Representing an actual object, and mind object. My first exploration while writing XQ Conditional in 2008 was that mind was in the negative space. It seems like a sensible… convention? But it did derive from thinking about subtraction and the mind’s capacity to filter, as well as imagine things not there. Earlier, as a teacher, holding up an empty hand and saying to the students, “I am holding up two oranges, exactly two oranges,” as a way to teach negative numbers, and asking where these “not oranges” were. So, here is the current mapping, or at least the mapping of last night:

positive numbers — discrete aspects of Actuality (actually undifferentiated from everything else), eg this cat here (3), that table there (eg 7)

negative numbers — the sense-object of mind (sense, married to an internally comprehensible meaning unit, possibly perceptual node, found in animal mind, me thinks), eg this cat-shaped perception (-3), that table-shaped perceptual object (eg-7)

imaginary number i — the linguistic label attached to the mental object or thing out there, eg the word “cat” (3i), the word “table” (-3i)

imaginary number -i — the linguistic label attached to a mental object that does not exist, eg the word “darkness” (-1i), or “peace” (-9i)

To construct this, I see I have bodged meanings of negative as being not, as well as opposite, especially in the interpretation of negative i. That is, darkness as an absence of light, as opposed to a thing in itself. And yet we find ourselves thinking, the darkness spreading. That is, purely imaginary objects of the mind. So, in this way, it is the same relationship between a thing out there and the mind (positive and negative 1), and a word and the concept it applies to (positive and negative i).

Interesting consequences of thinking this. Multiplication by -1 has the effect of mapping actuality to the sense-object, of word to concept, and the converse. Multiplication. The number 4, and it’s two roots +2 and -2,

(An aside: roots, as nudged by reading Arthur Young, can be found in the complex plane too. But just sticking to positive and negative roots of eg 9, we get +3 and -3. Notice, not +3 OR -3. They are both roots. And then it came to me. +3 x +3 is +9. -3 x -3 is +9. Same time? Two branches. But in terms of a specific case, most physics, they look only at the positive solutions, because these numbers refer to things… And… +3 x -3 is -9… which suggest to me that the root of -1 is +1 and -1, not i. Or rather, it gives weight to the notion that i is both positive and negative, the elevated station, the both-and of buddhist logic, that which is above happiness and sorrow, or true happiness. I know… this sounds odd: I am mixing too much up, but we are mapping not only objects, but subjective entities, so we are entitled to metaphor to whatever field in our heads… the buddhist space is at least minimal.)

Second, map this onto the Riemann sphere. What does this do to your head? It does funny things to mine. Honestly, if are getting this far, you might as well enjoy the sights from your own mind. (Who am I fooling, no-one is on this path. Not no-one. I am enjoying the vista of a new land unseen by human eyes. Or if it has been seen, such individuals have never returned to humanity… they’ve gone feral, mentally. I wonder how many loons populate our university math departments…? I suspect most do not step off the well-worn tracks at the regular stations, after all, who questions addition, or even counting?). For some reason I think of the riemann sphere as being a thing, not just a surface, a single point manifold, but a substance. If the numbers are the surface, what on earth — or in mind — is mapped inside the surface? Some odd answers pop out, like words. That is, the thin film that is the number sphere is the minimal language set, and so what exists outside and inside that layer are words; perhaps only outside… words lose integrity at a certain level of internal, subjective, depth.

Also, what constitutes the central axis of the riemann sphere, in maths, and in our mapped version? It is like taking zero and pulling it apart into positive and negative numbers, thus creating the space between 1 and -1. Same goes for i and -i, thus creating a 3d space within the equator. Interesting. And then considering the spindle around which this spins — and that is why the riemann sphere is useful, because it relates the arithmetical operations to be related to geometric transformations. Multiplying by i rotates the sphere through π/2 I think. Astounding. Rotation around what? And for some reason, another answer pops into my head, and alternative — time. Because nowhere in this, do we have time, yet.

(Another aside: riemann sphere is a trick to “positivise” all the numbers. Look at the sphere. The bottom is zero. That means that the entire thing above it suggests positivity. In the same way we often represent all our graphs of economics and the world falling apart in the positive quadrant. The point is, it jars with how I usually conceive of a sphere, with zero in the middle. Is this why it is valuable? This contortion gets maths back in line with what I consider to be a mistake of considering negative as opposite, that we should derive a maths which considers -5 as being “not 5″? Who knows… another thread trails its way into the dark. I feel like an old man wandering a vast hall with only a candle to light the way.).

Third, somehow relate the riemann sphere to a torus. This involved quite a few mental mappings. From the second mapping, if we consider the central dimension to be time, then another part of our brain comes up with time cones, which is something I related to being the centre of a torus (with the hole of the torus being a point).

How do you go about relating the time cone shape to the riemann sphere shape? And what comes to mind is that the riemann sphere is rotating. The rotation is the funny thing that makes consciousness stable. We are out on a limb here, but remember I was thinking this while in bed in the early hours and the mind has more freedom to think the absurd. So, consider the riemann sphere — hmm, can’t manage that, just the equator bit, -1, i, 1, -i, as the cross-section of a torus, the centre of which is a point. Which makes one wonder what the dimensionality represents. The upper part of the torus is a circle of i, the outer part a circle of +1, the inner bit the circle of -1, and the lower bit a circle of -i. But this is rotating, like a standing wave, in time.

Ho hum. I am not sure. None of it. But when it gets this far… It’s to do with simulating a moving thing. Most of maths is about capturing a dynamic thing in a fixed thing. Hence, mapping an object’s trajectory on graphing paper, deriving change of change of position, ie acceleration. I’d love to get my hands on modelling software — or better, a person who can actually manipulate, has facility, manipulating computer images.

(Further aside: computers are entirely positive. They have bits which represent negative, etc. They have to simulate “negative”. What does negative mean in terms of computer engineering? Again, pointing at purely positive computation. With negative just being a type of number. Then again, number as computing thing??)

Fourth, mapping the mandelbrot set onto the riemann sphere. This should be possible, but I haven’t been able to find a map of it yet. Looks interesting, in my mind’s eye. And I have always thought of the mandelbrot set to be a bit thin. I mean, 2-d…? And as far as I can guess, the magnification we usually apply to “zooming in” on the mandelbrot set, is actually a magnification of the universe outside of the riemann sphere, from an internal point of projection. That is, being inside the riemann sphere, looking at the mandelbrot set on the surface of the sphere, and then looking more closely at the point at -1, for example, and magnifying the space where the first bulb meets the core base of the mandelbrot set which happens to run along the equator, at least on the negative side. And why doesn’t the mandelbrot set colour as a complex number tends to zero? It usually only shows maps where it colours how fast a point escapes as it tends to infinity.

All of this, these snapshots, somehow come together in a nice way. They do. Just now, they are snapshots. It is all my feeble brain can come up with. I will never be able to pull it together. I don’t have the mathematical precision, nor fine awareness of mind. But a team of people might.

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 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…?

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.