Reality as Number

MW Lapidus actually being onto something, because if he’s correct, it turns out that his ‘flow’, this very strange, highly counterintuitive, noncommutative geometrical ‘flow’ projects down into a simpler realm, into the number system, as a flow of ‘generalised prime numbers’ on a line. 

This is very close to some strange speculative ideas I made public back in

1996. Lapidus contacted me a few years ago to say this, as it had come to his attention when I first put it up on the Web.  Now, it’s not that I influenced him, it’s almost as if I caught a glimpse of some future mathematics which will follow from his current work. I don’t know how I can explain what happened… it’s as if I caught a glimpse of something which was coming, but I didn’t have the language to describe it accurately, so I just described it as well as I could in this rather naïve way.  And so in a way I now feel somewhat vindicated concerning my slightly crackpot idea, because of Lapidus’ work.  * In some ways, I think all this intellectualising and mathematics isn’t really that good for me, and isn’t really what I ‘should’ be doing.  But part of me can’t entirely detach myself from it.  The speculation I just mentioned, which now appears to be at least partly vindicated, gripped me in a profound way.  This event had a precedent a few years earlier when I became convinced there was some connection between the Gaussian probability distribution and the prime numbers: that was driven by a sort of compulsion that was, looking back, was quite...not psychotic – it didn’t lead to any sort of negative behaviour – but it did rather take over my psyche.  I was one of those kids who, it was obvious fairly early on, could excel at mathematics, and being a fairly scrawny, unattractive young person, one latches on to anything one is good at — it provides a sense of importance.  It was as simple as that, it wasn’t any sort of noble motive for seeking the truth or anything.  I was living in the States as a teenager, starting to get interested in things like radical art movements and philosophy.  If circumstances had been different I’d have probably studied something else, but I wanted to get out of the States, that was quite a big thing for me then, so to get into a British university my best bet was to apply to do a maths degree, which I did. And then I sailed through that, and got offered a place on a PhD programme, which seemed like a great idea – effectively being paid to explore ideas which I found quite interesting and which I seemed to have an aptitude for exploring.  So that was all fairly accidental, and there was no real motive behind it, if you like.  It was just the way my life unfolded.  But by the time I was doing the PhD I was starting to engage with a lot of other non-mathematical ideas and people, and there was a real sense that, hang on, where is this going, is this really what I want to be doing?  At that point I was more interested in ‘seeking the truth’ – it sounds a bit grandiose, but I wasn’t interested in a stable career, and the idea of deriving some sort of self-esteem from being an accomplished mathematician, that no longer seemed to be of any importance.  So I started thinking, if I’m seeking the truth, is the truth to be found here, is this really what I should be doing?  And then the disillusionment set in. After a year of being on a Royal Society European fellowship, there was a distinct sense that modern mathematics was becoming irreparably fragmented, and I felt like I was being made very comfortable in an ivory tower, in a vast field of other ivory towers, between which there was relatively little communication.  And then there were all sorts of personal factors, just the way my life was going, people I knew, a sense of imminent global catastrophe... This was 1995, so perhaps there was a touch of millenarian hysteria involved! There was a sense that, as a mathematician, I was part of the problem rather than part of the solution.  A lot of my friends were involved in ecological activism and things like that, and I started to formulate a worldview wherein science had become the new, unacknowledged, religion of industrialised society, and mathematics was the inner priesthood of science.  To put it in very simple terms, Western culture runs on science, and science runs on maths.  So I saw myself as being trained up for this priesthood which was unconsciously steering the world to complete destruction and meaninglessness.   And so there was a sense of guilt, almost, that I was involved in this.  So I just broke out and floated around doing all sorts of interesting things for a few years, had a great time — I don’t regret that at all. I never imagined that I’d get involved in mathematics again.  But then certain ideas about prime numbers started to percolate in my mind.  I’d never really looked at number theory in any detail, had just a very basic number theory course as an undergraduate.  But shortly after I ‘dropped out’, I started thinking about prime numbers and the fact that they have a sort of ‘random’ quality…and at the same time thinking about the Gaussian distribution, the bell curve, and the ubiquity of that, the fact that almost anything that you can name, count, measure, and gather data on tends to scatter along this particular ideal exponential curve.  I remember posting a question on an Internet newsgroup back in 1995, trying to get somebody to explain to me why this thing shows up everywhere: not just in the biological realm, but in much more convoluted ‘cultural’ realms – I expect that you could count the number of appearances of a letter of the alphabet on the front page of a newspaper over so many years or months, and you’d find the same thing.  And the purely mathematical explanations put forward made sense to some extent, but I still felt there was some huge mystery lurking behind the Gaussian distribution, the fact that it shows up everywhere.  I scribbled all sorts of half-baked ideas down, some of which seem ridiculous now, some still of great interest.  But I became convinced – and I still don’t know where this came from – I became utterly convinced that the distribution of prime numbers in some sense was very deeply linked to this, to the ubiquity of the Gaussian distribution, that they were two sides of something.  And what’s strange is that almost seven years later, I discovered there was something called the Erdös-Kac theorem, which was proved in 1940, and which I’d never even heard mentioned before.  This was the beginning of probabilistic number theory, and it basically states that the distribution of prime factors of large integers follows a Gaussian distribution.   Obviously, the larger the integer, the more prime factors it is likely to have, but you rescale in a way that takes that into account, so you’re dealing purely with the seeming randomness in the fact that some numbers have got lots of prime factors and some numbers have only got one – and you end up with a bell curve.  And not just an approximate one, this is what really struck me:  if I was to measure the population over time of sparrows in the garden out there, or the way that those sunflower seeds fall on the ground [pointing to bird-feeder hanging in a tree], if I had large enough numbers I may well get very nice approximations of the bell curve.  A high-resolution computer image might even match the ideal mathematical bell curve in every detail. But they’re always approximate; in fact all use of statistical inference in science is based on finite amounts of data, which give rise to approximate bell-curves or other distributions.   With the Erdös-Kac theorem on the other hand, the n, the number of elements in your data set, actually tends to infinity. This is what really struck me about all this: n can tend to infinity only when you have an infinite amount of whatever it is you’re dealing with. And integers are the only thing, effectively, which we have – at least theoretically – access to an infinite amount of.  So, I haven’t fully delved into this, but there’s a problem with the use of infinity in statistics and probability theory.  It’s fine in some sort of abstract Platonic sense, but when you start applying it to the world, there is no infinity. But it does apply absolutely, precisely – and this is the theorem which Erdös and Kac proved – that as n tends to infinity, the distribution of prime factors tends to this distribution.  So, in some sense, that’s the only ‘true’ Gaussian distribution there really is, the ‘oldest’ one, the most primordial.  As soon as you’ve got positive integers, that’s hidden there within them.  Any other instances of the Gaussian distribution, you know, bird populations or currency fluctuations or anything else like that, not only are these approximate, but they require all sorts of complicated categories and definitions.  So, anyway, I still can’t quite explain why I was so gripped by this idea of the prime numbers and the Gaussian distribution being linked, but I was, and it’s as if I was somehow unconsciously aware of something and couldn’t manage to pin it down, you know. I tried endlessly to find some way of relating these things and failed.  Had this been 2005 rather than 1995 I probably would have quickly found out about the Erdös-Kac theorem using websearches. So as a result of this unresolved compulsion, I had a certain amount of prime number-related activity going on in my mind.  Then, in the winter of 1998 I went back to the States to visit my parents who were still out there, and I had a lot of free time.  I found a long thin piece of cardboard and drew a number line, circled all the prime

numbers, and then started drawing arcs between the prime numbers and their multiples.  So every number was connected to all of its prime factors by arcs emanating out from that number to the left.  The number fifteen would have two arcs emerging from it, one going to number three, the other to five.  And a prime number would have no arcs going to the left, only arcs going to the right.  Now obviously you can never draw the complete thing, but I drew enough of it that you could get a sense of there being something, a connectedness, a ‘messy’ connectedness,  like a nervous system, or mycelium, or...I don’t know, I can’t quite describe it, but I just spent a long time looking at this, I had it up on the bedroom wall.  And as a result of internalising that image, I started to think that it was perhaps the gaps between the primes that were most important…but I was somehow naïve enough to think that possibly no-one else had thought of that, whereas in fact quite a lot of work has been done on the gaps between the primes and yes, they are important.  But I started thinking that maybe the gaps, suitably rescaled, are the things which distribute in a Gaussian way.   I tried to run some computer models, to calculate the gaps and analyse their distribution — but not having access to the necessary computational power, that wasn’t really going anywhere.  And then this image of the interconnectedness of the primes, the whole number system as a single connected entity, with each prime as a sort of ‘nexus’ , the whole thing exploded in my mind — it was something very sudden, and the initial impression I got was that the primes themselves were imbued with a sort of ‘charge’...I think I’d read somewhere that average gaps between consecutive primes are logarithmic, that is the average gap between a prime p and the next prime is log p, the natural logarithm of p. Obviously the gaps can vary wildly from this average, but the average is a precise mathematical result, becoming increasingly precise as we allow p to tend to infinity.  I was suddenly gripped by this idea that the primes themselves were imbued with a kind of charge, something like an electrical charge, and that that log p was the clue, that was the charge of the prime p. At the time I was unaware of Julia’s thermodynamic approach which associates with each prime p the energy log p, and also that certain proposed dynamical schemes involve ‘orbits’ with period log p associated with each prime p.

C: So the magnitude of the gap before the prime would be its charge?

MW: Well, for sufficiently large primes p, the gap before and the gap after would both be approximately log p. And I had the idea that these primes were in some sense repelling each other and that the bigger the prime, the greater the charge and the stronger the repulsion, hence the bigger the gap.  This all came tumbling in as a single thought, really — the account I’m giving now is an attempt to reconstruct and coherently describe it.  But rapidly following this initial impression was the idea was that, well, if there’s that kind of repulsion involved then what I’m looking at is a frozen image of something which was previously in motion — this is what I got a very strong inner visual sense of.  I try to describe it to people like this: imagine attaching a wire to a wall and then stretching it away from the wall, effectively off to infinity, and then marking out with tiny white dots equal spaces representing the integers, and then imagine little tiny magnetic beads, mutually repulsive particles, positioned along the wire at positions 2,3,5,7,11, etc., that is, at the positions of what we call the prime numbers.  Now set up a camera, and then subject the whole area to a huge fluctuating magnetic field, causing the beads to move up and down the wire, driven not just by the field, but by their mutual repulsion.  Film that, and then run the film backwards.   What you’d see is all these particles moving around on the wire and repelling each other, responding to each other, and then eventually coming to rest at the positions we associate with the primes.  That’s the image. Now I was well aware of the obvious question: how do we interpret the time parameter here?  This is a huge problem – we’re not talking about time in the familiar clock sense, not in the historical sense.  I certainly wasn’t under any illusion that anything like this had ‘happened’ at any point in the past.  I was suggesting that the system had a ‘past’, but that it wasn’t part of the historical past, rather of some other time-like dimension.  And rather than thinking, that’s ridiculous, I won’t think that, I tried to suspend disbelief and see where it would take me.  So the basic thought then was, okay, if what we’re looking at is a frozen image of something which was previously in motion, the motion must have subsided for some reason – so what we’re looking at must be something in a state of equilibrium.  So, what kind of equilibrium?  Well, I came up with a crude notion of ‘arithmetic equilibrium’: Why have the magnetic beads come to rest where they are?  Well, if we freeze the motion at any moment, so you’ve got an infinite sequence of tiny beads whose positions don’t necessarily correspond to positive integers – they could be any real numbers – and then generate all possible finite multiplicative combinations of those numbers, that would produce something analogous to the positive integers.  The positive integers, recall, can be generated as the set of all finite multiplicative combinations of the primes.  But these new ‘integers’ would not be anything like the familiar integers, they’d generally be all over the place.  They wouldn’t be nicely arranged, equally-spaced.  But if the particles ever happened to reach the point where they collectively inhabited the positions associated with what we now call the primes, the ‘integers’ they’d generate would be equally spaced. So, I thought, it’s equal-spacedness which is a key to this ‘arithmetic equilibrium’ which, according to my scheme, has been achieved in the number system.

C: Something like an entropic sequence, heading towards an attractor.

MW: Something like that, I was thinking in terms of all sorts of ideas I had partial understanding of – my understanding of physics is very piecemeal, it was even more so then.  So many ideas were feeding in.  I started to think, how would it begin?  Maybe something like a big bang, where you’ve got all the particles squeezed together at the wall, at the end of the wire, but with something like an infinite magnetic field produced by the wall, and then you let go, and they all explode outwards.  At any moment you could freeze the image and generate all the finite multiplicative combinations, the set of ‘integers’ that they generate: I called these ‘generalised primes’ and ‘generalised integers’ . Well, it turns out that Arne Beurling, a relatively obscure Norwegian mathematician, had come up with this idea of generalised primes and generalised integers many decades previously.  To better understand the familiar primes he’d started looking at the question, suppose we ‘change’ the primes, what can we then say about the associated integers and their asymptotic distribution?  Martin Huxley (Cardiff University), who’s quite an eminent number theorist, got in touch with me as a result of my original website, to say, oh yes, there is actually a name for those, they’re called ‘Beurling generalized primes’.

C: The distinction being between the primes as we know them and, as it were, a generalised function of ‘priming’ by which a number system is generated.

MW: Yes, it’s a bit like that, taking the idea of the primes not as indivisible integers, but as a set of generators.  But the idea of them flowing or moving, no-one as far as I knew had ever put that idea forward.  And so I came up with what I decided was almost a ‘creation story’, some sort of strange mythological mathematics – the creation story behind the number system.  Whether there was this ‘big bang’ thing at the beginning or not, I wasn’t sure...but the idea was that, okay, these generalised primes were somehow set in motion.  Remember, there are these generalised prime particles, and then there’s a kind of invisible set of generalised integers that they’re embedded in, that they’re generating, which are also in motion.  And, at any moment, the ‘heterogeneity’ of these generalised integers, their lack of equal-spacedness, is creating some kind of ‘tension’ which is affecting the particles’ charges. The idea of fixed log p charges gave way to the idea of fluctuating charges, governed by the spacing within the generalised integers at any given moment.  So you can almost think of the distribution of these generalised integers trying to space itself out by ‘influencing’ the generalised primes and their charges so that their mutual repulsion eventually leads them to a stable configuration, an attractor point – that would be the arithmetic equilibrium.    Having reached that – the familiar configuration of primes – the generalised integers would be nothing but the familiar positive integers 1,2,3,... The perfect equal-spacedness of these would result in all forces on the generalised primes dropping away, and the number system has then ‘come into being’. That was the ‘story’ I came up with, that all came a bit later, trying to make sense of this image that I originally had of the primes being charged, mutually repulsive, and in motion — or having been in motion. At the time, it had felt like, this is profoundly important and I have to act on it, I was being somehow compelled to act on it.  It felt like the most important...certainly the strangest idea ever to enter my mind.  And, insofar as I can grasp what is meant by ‘numinous’, it was charged with a numinous quality. I was hoping to be able to actually describe the scheme in serious mathematical terms, to reveal that there was some mathematical integrity behind it, but that never happened...So all I had was this nebulous idea about an evolutionary dynamical system underlying the primes.  And it was an idea which seemed very strange, I can’t emphasise that enough — I couldn’t really justify it using any sort of logical or mathematical reasoning, and yet it gripped me psychologically with such force that I couldn’t let go of it, I was driven to try and make sense of it.  And that led me to create a website...you know, this is what you do in 1998, you create a website, and then you start emailing various eminent mathematicians and physicists to try and get them to look at what you’re doing.  And as a result of that, a few people were quite

helpful and responsive, I was sent some relevant literature, and I started to realise that actually, there are a lot of strange, unexplained connections between number theory and physics. These things seemed to me to be circumstantial evidence supporting my strange insight, whatever it might have been, or whatever value it might have had.  They too suggested the number system had some mysterious ‘quasi-physical’ character.   This may have been wishful thinking on my part, but the material was undeniably fascinating in its own right, so I started compiling it into a web-archive, intended to, at least indirectly, back up my idea.  Eventually, though, my original idea began to become a bit of an embarrassment to me – it seemed quite nave and ill-informed.  So, as the archiving took on a life of its own, and I became fascinated with all this serious maths and physics that I had become aware of, I gradually buried the original idea inside a vast web-archive.  But I never entirely removed it, somehow still sensing, or hoping, that there was something of value there. All my attempts to come up with a mathematical model, a dynamical system that would correspond to that image, had failed.  I had struggled because I didn’t have anything like the mathematical abilities that would be required for that.  And in fact, I now feel vindicated in that it’s not that I wasn’t capable enough to do it; in order to describe anything like a flow in this space of Beurling prime configurations wherein what’s called the classical prime configuration, the usual primes, constitutes some

kind of dynamical equilibrium – in order to describe anything like that you need to do what Michel Lapidus has done, and introduce a noncommutative flow on a moduli space of fractal membranes.  And there was no way in 1998-9 that I could have had access to those ideas.  So – and again, this isn’t a serious proposition, but the only way I can make sense of this for myself – it was as if I’d caught a little precognitive glimpse of some future mathematics, sensed the importance of it, tried to get it down, but didn’t have the language to get it down, did the best I could, and put it out on the Web.  This then led on to me putting a lot of time and effort into what was effectively public service web-archiving for a few years, which has been quite fulfilling, but it was initially just a consequence of the original ‘flash’, and the compulsion it induced in me.  Now I’m feeling somewhat vindicated that someone appropriately qualified has shown that there does appear to be something like this underlying the number system. 

C: Is there an analogy between what you’re describing and what happened historically with non-Euclidean space — could it be seen as an arithmetical version of that, with the unknown time parameter as something as unanticipated as the curvature of space?

MW: Yeah, in the sense that you’re breaking out of what is considered to be the only possible version of something, into a whole range of possible versions, and that initially seems ‘mad’ to many onlookers.

C: At the time, the idea that space could be folded or that space could be curved seemed insane.  Nevertheless, such new generalisations are arguably the very movement of science itself.

MW: I think it was Gauss, Boylai and Lobachevsky who simultaneously came up with the same basic idea of parabolic geometry, and at least one of them was afraid to even mention it to anyone.  If I had still been involved in serious mathematical research in 1998-9, if there had been a career at stake, my guess is that, having had the same experience, I may well have thought twice about going public with these ideas.  Whereas as it was, it didn’t really matter.

C: An interesting example of how being embedded in a discipline, having a reputation, and no doubt having funding depending on it, would actually stop you from saying something — there wouldn’t be any channel through which to get it out.

MW: In a way, I was in a perfect position to just have a go, to push it out there. I’ve read accounts of mathematicians trying to describe how they made certain great fconceptual leaps. The big difference is that the leaps they made were into something that could actually be mathematically described, and ultimately, you know, were incorporated into legitimate mathematics.  Whereas I just had a sort of mad flash, a glimpse of something which, as yet, is not legitimate mathematics, it’s just a vague impression.

C: Yet the structural detail in which you described it makes it something more than simply a vague idea.

MW: Well I’m not sure that the detail of what I’ve described adds any validity.  Had it not been for Lapidus’ work coming along, I probably would have entirely disowned it by now.  But at the time, there was a conviction that there was something in it, but it was hard to know what to call it.  There was an awkwardness because, it falls between the usual categories...I suppose it could be called phenomenology or something, there’s probably a legitimate-sounding name that someone could come up with.  But when I put it out on the Web I was quite careful, because I was well aware of all sorts of cranks on the Web ranting about how they’ve discovered this or that revolutionary idea, or proved Einstein wrong, or whatever. And I so I tried to be very understated in how I presented it – you know, I’ve had this idea, and I don’t know what it means, it may well be meaningless, but I invite people to either show me why it’s meaningless, or else indicate what it might lead to.  And gradually it began to happen.  But I don’t know if it really contributed to anything.  I think Michel Lapidus would probably have reached the same conclusions regardless.  Perhaps it did influence him, I don’t know, but I don’t think so.  So in a way, if I did catch a glimpse of some sort of future mathematical discovery, it would have occurred anyway, so what’s the value of what I did?

C: At least, it does lead one to think about mathematics not in terms of the points at which people draw everything together, make it into a formal system, but rather these discontinuous moments when, inexplicably, things move, things split apart and something new opens up?

MW: A crack opens up and something doesn’t quite make sense.

C: From what you know of the mathematical community, is it the case that the sort of research you are pursuing is not accepted, that they’re not interested in it?

MW: There’s a small enclave of perhaps slightly more open-minded, more unusual mathematicians, who are prepared to discuss these sorts of things privately.  The vast majority are slightly bemused or just not interested, they’re too busy with their own work to stop and think about what it all might mean.  Mathematicians aren’t generally encouraged to think about ‘meaning’.  They don’t really need to, they’ve got a very exact discipline, they’ve got theorems to prove and things like that. Basically, what I’m doing, I couldn’t call it mathematical research.  You’ve called it fundamental research...you could call it that, I know what you mean.  The way I see it I’m just trying to raise certain questions and generate discussion, and I’d say the vast majority of the mathematical community just isn’t going to engage with that, which is okay.  Because I’m not actually doing mathematics, I’m not engaged in mathematics research in the way they are; I’m playing a different game, asking questions about what mathematics means, what it is, how we relate to it.  But at the same time I’m not part of the philosophy-of-mathematics community either, which is involved in something much more rigorous and disciplined than what I’m doing. I suppose because I’ve got more time I’m in a better position to just stop and think: what’s the point, why are we looking at this stuff anyway, what does it mean? Professional mathematicians these days tend to be extremely busy, they’ve got to theorems to prove, papers to publish, conferences to attend.  They need to keep their careers afloat, and so they’ve got a lot less time to think about what this stuff might mean. But the thing about the Web – and this is quite an important factor in what I’m doing – is that it’s possible for me to say what I think and to discuss it with large numbers of other people in the academic world, without having any formal academic status and without having to get anything published.  And I can change it as I go along –there’s no final document, that’s the other thing.  I don’t publish articles, I can just put together vague rambling webpages and then keep changing them as my ideas change.

PC: I was very interested to read about your ‘strange speculative ideas’ which clearly occurred as the result of an important peak insight regarding the true dynamic nature of the primes.

And I would see such an insight as representing a natural progression with respect to the well constructed criticism regarding academic and cultural issues in relation to mathematics that you make in both the present discussion and the “Secrets of Creation” trilogy.

However it is your very position as a professional mathematician, which has been holding you back from taking the important next step, as it seems to me that despite breaking much new ground through your personal efforts, that you still value the more conservative feedback of your fellow mathematicians too highly.

Though in some significant ways you have detached yourself from identification with the mathematical club, you still to a degree require its professional approval to properly value what you have already discovered. And in particular this seems to be true of your relationship with Michel Lapidus.

We have contrasting experience in this regard. I would not be considered a valid member of the club by professional mathematicians. However my talents, such as they are, relate mainly to the vitally important holistic area of interpretation, which is not even recognised by the profession. So I remain fully confident of the basic criticisms I make regarding the nature of mathematics, irrespective of whether any club member chooses to take notice.

This all dates from an initial eureka moment at the age of 10, when engaged in measuring areas during arithmetic class. Though the area for example of a rectangular field obtained from multiplying the length by the width relates to square (2-dimensional) units, in the customary treatment of multiplication, no account is taken of this fact, with all answers expressed in a linear (1-dimensional) manner. 

So even at this age I recognised that there was something seriously wrong with the standard treatment of multiplication and suspected that there was an important hidden dimension to mathematics that I would eventually uncover.

This mathematical awakening occurred shortly after a key existential experience, which probably then triggered the intervention of the unconscious in a profound manner.

Because such misgivings started at such a young age, I have been in the habit of continually questioning the accepted assumptions underlying the present approach. And these in truth are of a highly reduced nature. So I have long reached the position that nothing short of a major revolution with respect to the very nature of mathematics is now vitally necessary if we are to successfully adapt to the increasing demands of our world.

Therefore though I greatly admire at one level the remarkable analytic ability of its specialised practitioners, at another level I can clearly see that an enormous shadow characterises the whole profession in a total refusal to formally recognise the qualitative holistic aspect (which intimately applies to all mathematical understanding).

Then in my late teens when I pursued a university degree in mathematics, this disillusionment with accepted assumptions resurfaced in a much more intense fashion now mainly relating to the interpretation of the infinite notion in real analysis.

So after a year I gave up mathematics in favour of a more liberal arts degree that embraced English literature, economics and politics.

However privately while studying these other subjects I was already developing the bones of a new holistic approach to mathematics.

This started with a dynamic appreciation of the psychological manner in which the nature of number is actually experienced.

I realised that as all experience is necessarily conditioned by fundamental poles such as external and internal and whole and part, that this equally applies to the mathematical treatment of number.

So rather than number representing some unchanging absolute notion, a continual two-way interaction necessarily takes place as between number objects (perceived as external) and number perceptions (perceived relatively as internal). Likewise a continual two-way interaction takes place as between the number concept (as whole) and specific number perceptions (as parts).

Then indirectly as a result of studying Hegelian philosophy at the time around 1970, I was able to translate this psychological appreciation through recognised mathematical symbols (which now acquired a holistic rather than analytic meaning).

So positing (as the holistic notion of addition) could be directly identified with conscious understanding. Negation (as the corresponding holistic notion of subtraction) could then be identified with the unconscious.

Thus the incorporation of both positive and negative signs with respect to number required explicit recognition of both the conscious and unconscious aspects of understanding from a holistic mathematical perspective. Clearly therefore a comprehensive interpretation of mathematics would require that both aspects be explicitly recognised.

This then led to a profound investigation of the holistic nature of number, initially focusing on 2, which as I have mentioned previously, relates directly to the qualitative notion of “twoness”. This in turn served as a basic blueprint for the holistic understanding of all other numbers.   

Thus every number possesses a shared (qualitative) as well as separate (quantitative) identity.

Thus, when I say that there are two cars in a driveway (as an independent group) this relates to the quantitative notion of 2. However if I compare each car as 1^st and 2^nd units respectively, this now entails the shared qualitative notion of 2 (as “twoness”).

We could also say, as indeed you have mentioned, that in terms of its qualitative shared aspect that 2 acquires an archetypal meaning as a universal class to which all groups of 2 belong. So we have again two distinct notions of number as separate (in a quantitative sense) and shared (in a qualitative manner) respectively.

Unfortunately, these two meanings (quantitative and qualitative) are never properly distinguished in conventional mathematical interpretation. This therefore represents the most basic form of reductionism, which equally implies the most basic form of confusion.

However as growth in holistic appreciation relates directly to ever more refined intuitive states, traditionally associated with spiritual contemplative development, further mathematical progress was initially slow.

Then some 10 years later when now immersed in Jungian thought, I found the means to define the important imaginary notion, thus enabling complex numbers to be handled in an appropriate holistic mathematical manner.

I had been especially interested in developmental psychology for some time. So by the early 1990’s, I felt it was time to apply my holistic mathematical ideas to the scientific appreciation of the full spectrum of human understanding (including emotional, rational and spiritual type aspects). So I wrote a book and placed it on the internet. The basis contention was that the various number types such as prime, natural, rational (positive and negative), irrational (algebraic and irrational), imaginary and transfinite could be fruitfully matched in holistic terms with the major bands of development on the spectrum. In fact my basic finding was that all possible stages of development are thereby configured in a holistic mathematical fashion.

For example the conventional mathematical approach can be directly related to the rational numbers (in holistic qualitative terms). However this then immediately suggests that distinctive forms of mathematical understanding exist that are qualitatively related to the other number types, which are not presently recognised by the profession!

So we have here not only the holistic scientific basis for the study of psychology but for all disciplines such as physics, biology, economics, philosophy etc. where a truly integrated appreciation is required. 

Of all the various number types, the holistic interpretation of the primes initially caused me most difficulty.

Then I recognised that there were in fact close connections as between the psychological notion of primitive with respect to childhood experience and the corresponding holistic notion of prime numbers.

With instinctive behaviour as in infant development, the conscious aspect remains largely embedded with its unconscious counterpart. So the child, as in magical thinking, directly associates holistic notions with specific objects.

The physical counterpart here to such primitive behaviour can be found at the sub-atomic level of particles, thereby giving an important appreciation of the increasingly transient existence of such particles. So particles that lie close to the ground from which they originate cannot yet achieve a stable independent identity. Thus we could correctly refer to such particles as existing in a confused holistic dimensional framework of prime space and time, just as the young infant exists in a corresponding confused psychological framework of prime space and time.

This likewise implies for example that p-adic notions of number would be of special relevance at the micro scale of quantum measurement!

Early child development is largely concerned with the gradual differentiation of conscious from unconscious understanding. Then in adult development, as with mathematics, conscious rational differentiation can reach an extreme degree of specialisation.

The important point here is that prime numbers equally require both conscious and unconscious aspects of interpretation.

However because of rational specialisation, the conventional understanding of the prime numbers remains solely confined to their conscious (quantitative) aspect, where they are given a rigid absolute identity.

Therefore I realised that a comprehensive interpretation of primes would require that the unconscious aspect of personality be equally developed through progressively refined intuitive stages (where the holistic aspect is made manifest). In other words an integrated appreciation of prime numbers would require the marriage of rational analysis with a truly refined form of contemplative awareness.

Though the development of intuitive states is indeed well documented in the spiritual traditions both East and West, the mathematical implications of such understanding, implying increasingly refined dynamic structures of a paradoxical nature have rarely been investigated. So I see such investigation now as an extremely important task for our present age.

Then with sustained development of conscious and unconscious – initially in a somewhat separate manner – finally the relationship with each other in a mature consistent fashion can take place. I identify this 3^rd stage with radial mathematics and as I have stated earlier in the discussion, I would maintain that a preliminary version of this stage is necessary to properly comprehend the true nature of the Riemann zeta function (with its accompanying Riemann Hypothesis and non-trivial zeros).

Interestingly, though I was not yet properly acquainted with the Riemann Hypothesis at the time I had already concluded that the primes and the imaginary transcendental numbers were in dynamic terms related to each other.

So the true nature of the primes could only be resolved through the mature appreciation associated with the holistic nature of the imaginary transcendental numbers.

The strong holistic connection here would equally suggest that the zeta zeros are also of an imaginary transcendental nature (though this has not been proven yet in quantitative terms).

When I look back now at what I had written at this time (some 30 years ago) I am surprised at how well it fits in with my present thinking on the holistic nature of the zeta zeros.

 “When this stage (i.e. imaginary transcendental) is successfully negotiated, the immediate instinctive response to objects directly coincides with an appreciation of their eternal universal significance”.

The key point here is that resolution of the prime number problem (in a proper holistic appreciation of the nature of the zeta zeros) cannot be rightly divorced from the corresponding psychological need to resolve the primitive problem in obtaining mastery of the instinctive self. Only then can one’s human physical response to reality be enabled to properly sustain continual spiritual awareness of an eternal now that underlies all phenomenal sense experience.

As the Pythagoreans realised the true purpose of mathematics should culminate in a direct appreciation of eternal mystery that transcends all our rational notions (while being also immanent in them as their very source).

So in the end the mystery of the primes cannot be solved through rational thought but rather through direct experience of the eternal home from which they originate.

In this sense the primes (and natural numbers) serve as the crucial archetypes bridging the tangible world of phenomenal form with the ineffable nature of spiritual emptiness.

The famous Buddhist sutra states:

“Form is not other than Void

Void is not other than Form”

We could perhaps add that with respect to the phenomenal temporal world, the primes and natural numbers play the truly vital role in mediating the relationship between form and emptiness (in both quantitative and qualitative terms).

This ties in to a degree with your own interesting speculation on the evolution of the primes.

However speaking from my own perspective, I would say that the fundamental relationship of the primes to the natural numbers is set in eternity rather than time.

Thus the very capacity of the primes to maintain a dual identity of a random individual identity with an impeccably ordered relationship at a collective level (which is enabled through the zeta zeros and vice versa) is inherent in matter whenever phenomenal form appears. So if one accepts the beginning of the universe with the Big Bang, then this capacity – which points at its deepest level to the remarkable complementarity of both the quantitative and qualitative aspects of creation – is already present in physical creation from its beginning.

You mention in Volume 3 of your trilogy that there’s some physical-like dynamic system within or “at the root of” the number system.

I would immediately add that there’s equally some psychological-like dynamic system within or “at the root of” the number system (with physical and psychological aspects ultimately identical).

I would also add that this only appears so strange because we have been long accustomed to looking at the number system in the wrong way i.e. in a rigid absolute manner.

However, I would see this dynamic system (both physical and psychological aspects) as simply representing the holistic extreme with respect to the nature of number, which initially remains inherent (as potential for existence) prior to phenomenal reality coming into being.   

However the very process by which this merely blind implicit capacity is gradually made more explicit through the formation of matter, reaching eventually into the self-reflection of nature in a conscious manner as manifested by human beings, does then indeed imply continual evolution with respect to the primes (and natural numbers).

So, as I would see it, all phenomenal forms represent the dynamic configuration of number with respect to both its analytic and holistic aspects. In fact, space and time is directly related to the holistic aspect of number. So in this sense nature undergoes continual transformation in space and time precisely because number itself undergoes a similar underlying transformation. Number is this sense can be looked on as a hidden dynamic software code that determines the very nature of phenomena (with respect to both their quantitative and qualitative features).

One could validly say that the goal of conscious type transformation is to eventually appreciate in a fully explicit manner, this underlying nature of the prime and natural numbers, which ultimately leads to a pure spiritual awareness of the complete mystery lying at the very heart of creation.

In this sense there is equally both a physical and psychological system, ultimately formless, that co-exists with number, where their very nature – and thereby of all phenomenal reality – is realised in pure eternal mystery. So this realisation properly takes place in experience of an eternal now which once more can not be directly identified with the changing world of space and time.

You later frequently quote Sylvester who suggests that knowledge of the true nature of primes would lead to a poly-dimensional view of time.

I agree with Sylvester here. However clarification of the very nature of such dimensions is not possible through the existing scientific and mathematical framework.

In fact I believe that the true nature of time (and space) relates directly to the holistic notion of number (that is intimately based on the primes). And this finding has massive implications, especially for physics, which is based on reduced linear notions.

So associated with each prime in this holistic sense are unique dimensions (of time and space) applying in both complementary physical and psychological terms. And when correctly appreciated these dimensions intimately relate to the very manner in which we experience reality.

I did on a previous occasion attempt to communicate what the holistic notion of two dimensions entails. And again because this involves a relative notion of both number and of time (and space), it serves as the simplest prototype for appreciation of other prime dimensions.

Once again all experience is necessarily conditioned by external and internal polarities. So as a personal self (that is internal) one is in relationship with a world (which relatively is external).

The illusion of linear (1-dimensional) time is that time moves forward with respect to both self and the world giving thereby just one direction.

However if one experiences time as moving forward with respect to the physical world, then, relatively, time is thereby moving backwards with respect to the psychological self; likewise in reverse if one now experiences time as moving forwards with respect to the psychological self, then it is moving backwards with respect to the physical world.

In other words, time, relatively, possesses both positive and negative directions; and when reference frames switch, as continually occurs in actual experience, what is positive likewise appears as negative and what is negative likewise appears as positive.

And this constitutes 2-dimensional appreciation of time from the holistic perspective.

Appreciation of this paradox of the two directions of time (which requires explicit recognition of the unconscious) leads to a purely spiritual realisation of an eternal now which continually exists.

And from this perspective the dimensions of time (and space) are seen to clearly emanate in terms of phenomenal reality from an underlying present moment (which always is).

This is why spiritually advanced contemplatives come to experience reality as deeply rooted in the present moment and though they may not express it in the manner I have chosen, they then experience temporal events in space and time as having a secondary relative significance. 

However, every prime number is associated with a corresponding unique relative appreciation of the nature of time (and space).

Remarkably all these dimensions of time (and space), which are intimately connected with the holistic interpretation of prime numbers, implicitly occur in actual experience.

So, for example one cannot understand the analytic (quantitative) notion of 17 without the corresponding holistic 17-dimensional appreciation of time (and space) being implicitly involved. And this relates, at its simplest, to the manner in which conscious and unconscious are dynamically configured, thereby providing the unique qualitative aspect with respect to this holistic experience.

There is then a reduced linear sense in which 17 dimensions can be analytically interpreted in a quantitative manner. However this completely lacks resonance with the manner we customarily understand the physical dimensions of space and time.

Accepted scientific notions are tied up with viewing space and time merely with respect to quantitative type measurement. And such science is rooted in conventional mathematical understanding that is 1-dimensional in nature. This means in effect, especially in relation to the flow of time, that linear notions have become hard-wired into our brains. Thus the mathematical notion of a 17-dimensional hypersphere can be given no strict meaning in standard physical terms.

However what is not properly recognised is that space and time intrinsically relate to the holistic qualitative notion of number, where every numerical dimension can be defined in a manner that accords with actual experience (when one has the ability to intuitively “see” what is implied by such a dimension). Space and time therefore relate here to a unique qualitative manner of appreciating phenomenal reality. And this is rooted in the realisation that each prime possesses a special qualitative resonance in holistic terms.

Thus a truly comprehensive mathematical appreciation, which I term the Type 3 radial approach, would require the integration of both the holistic and analytic notions of 17 dimensions.

However because the unconscious holistic aspect of number appreciation is totally overlooked in our mathematical training, we remain completely blind as to its true nature.

So Sylvester was right. For it is the holistic poly-dimensional appreciation of the primes (and of the natural numbers) that enables the Riemann zeta function to be interpreted in an entirely new coherent matter. This then enables one to appreciate the manner in which the zeta zeros fully reconcile the primes with the natural numbers, and thereby, by extension, the quantitative and qualitative aspects of all phenomena.

Though this in no way lessens the true mystery of the primes – in fact it is thereby greatly increased – one is better enabled to directly experience this mystery.

And if we have eyes to see, we can find this mystery deeply reflected in all the phenomena of everyday life, not alone relating to their multivaried quantitative features, but equally with respect to all their remarkable qualitative attributes.