Future Vision of Mathematics

MW: Yes, and in children as well, with their favourite numbers, and feelings about each of the first few positive integers — ethnomathematics and children that one finds in 'primitive' numerological systems.

C: Something that gets beaten out of people by mathematics: when people start learning mathematics, it’s as if the first task is to extirpate any idea that numbers have quality.  Mathematics is in fact often seen as constitutively opposed to any such intuition.

MW: Yes. Marie-Louise von Franz, one of my favourite writers, who studied under Jung and wrote a lot about number archetypes, she talked about number having both quantitative and qualitative aspects.  The quantitative is obvious, we all use numbers to count.  Cultures who revere certain numbers and have mystical beliefs about them which we might laugh at, they still use them to count with and to trade, they recognise that they have a quantitative aspect.  This is the aspect of number that has given rise to economics and technology; but equally, perhaps even more importantly, there’s the qualitative aspect that only survives in our culture in children having favourite numbers, some adults having lucky numbers, not wanting to sleep on the thirteenth floor of an hotel, the way they might choose lottery numbers, that sort of thing.  But, you know, in ‘serious’ society numbers are supposed to be entirely quantitative.  Von Franz wrote about a traditional Chinese story involving eleven generals who, faced with some very difficult military situation, took a vote as to whether they should attack or retreat. Three voted to attack and eight voted to retreat.  So what did they do? They attacked, because three was a more favourable number — it wasn’t a bigger number, but it was a number associated with unanimity, or some other favorable quality like that.  And the attack was a success.

So it’s interesting that they could build a civilisation that was able to have a functioning economy and military and to govern millions of people — clearly they were intensely aware that number had a quantitative aspect — but there was also a serious engagement with the qualitative aspect which is dismissed in our present culture as entirely superstitious.  Now I’m not encouraging people to engage in completely arbitrary numerology, I mean, I’ve looked at a lot of that new age numerology literature, and the problem is, nothing can be verified: someone can write a book saying a particular number means something, and someone else can write another one saying it means the complete opposite.  It just confuses matters, as there’s never any consensus or certainty in these interpretations.  That’s why professional mathematicians would almost unanimously just react against it and say it’s all rubbish.

C: But is there any way to talk about it which doesn’t get into that morass of mysticism?

MW: There are two approaches: one is the serious attempt by Jung and his followers to catalogue all of the ethnomathematical systems, undertaking a serious study and survey of various cultures and their relationship with number, trying to find common threads, and through psychoanalytical work and dream studies, trying to find extract essential patterns to build up a body of material from which we could possibly deduce something about how number interfaces with the psyche at a fundamental level.  The other approach is to seriously study number theory, because as far as I’m concerned, that is numerology, really — you’re looking at the properties of integers, and if you study it to a certain depth it takes you into the realms of what you could only call the mystical or the uncanny, where cracks seem to open up in your normal understanding of reality.

C: Is that perhaps what characterises number theory as opposed to mathematics, what makes it a very different discipline?

MW: Well, number theory is universally acknowledged as a branch of mathematics.  It can’t really be separated from it like that.  But it arguably has a unique status at the very heart of mathematics.  You’re working at the very root of it all, dealing with the simplest objects, the positive integers.  And yet you come across these counterintuitively complicated structures and results.  You can separate mathematics into branches and disciplines but they all ultimately overlap and interrelate. Gauss (who himself was called the ‘prince of mathematicians’) called mathematics ‘the queen of the sciences’ , and number theory ‘the queen of mathematics’.  The idea is that number theory is generally seen as the pinnacle, in that it contains the most difficult problems; also it’s concerned with the integers, and all of the rest of mathematics ultimately relies on integers.  Hence it’s not surprising that problems of number theory do seep into other areas of mathematics, and even physics.  What is surprising is that physics is beginning to shed light on number theoretical structures like the zeta function, as if it were just one of a class of objects, whereas it’s meant to be this fundamental object underlying everything. What I’m trying to describe with my clusters of bubbles isn’t intended as any sort of serious mathematical proposition, it’s just a picturesque visualisation — trying to look at the number system from another angle, if you like.  But there’s a hidden assumption within the Peano axioms, I think, which needs to be addressed — although I don’t think I’m the one to address it.  It concerns the axiom which allows you to always add one.  Even in the proof of the infinitude of primes, I sense some sort of subtle circularity there — the idea is that, if the number of primes were finite, you could multiply them all together and then add one.  And that rapidly leads to a contradiction concerning primeness and divisibility…hence there must be infinitely many primes.  So that takes you back to the Peano axioms, the idea that you can always add one. But in my visualisation, multiplying them all together would correspond to building one mighty cluster using one of each type of bubble.  And in that visualisation ‘adding 1’ is a far less obvious operation.  This ties in with problems of time, the idea of time, repetition, even basic physical questions: you know, this ‘adding 1’ business presupposes that you’ve got a physical space, something like the space we’re familiar with, in which you can make a sequence of marks, or a time continuum in which you can make a sequence of utterances or beats. And I feel there may be subtle assumptions concerning the homogeneity of time and space involved in this, too.

C: These questions of time and space must fall out from the primes’ intimate connection to the relationship between multiplication and addition.

MW: Brian Conrey, who’s President of the American Institute of Mathematics, and Alain Connes have both been quoted as saying that RH is ultimately concerned with the basic intertwining of addition and multiplication. And if we haven’t really got a clue how to prove RH — which we don’t — we’re going to have to own up, we don’t even understand how addition and multiplication interrelate. A more succinct, precise way of describing these two possible constructions of the primes that I have outlined — the conventional ‘just add 1’ approach, and my ‘lexicographical’ approach with its equivalent clusters-of-bubbles visualisation — is given by Grald Tenenbaum, who certainly knows what he’s talking about: Addition and multiplication equip the set of positive natural numbers with the double structure of an Abelian semigroup.  The first [addition] is associated with a total order relation as it is generated by the single number one. So if you’ve got addition and you’ve got this single number 1,  you can generate the postive integers just by adding 1 plus 1, 1 plus 1 plus 1, etc.  If you take 1 as your ‘additive generator’, the universe generated is the set of positive integers. The second [multiplication], reflecting the partial order of divisibility,

This probably isn’t the time to get into the subtle issues of ‘order’ in mathematics — you’ve got ‘total order’ and ‘partial order’: addition relates to total order, where something definitively comes before or after something else; and divisibility relates to partial order, a less distinctive type of order, although I won’t get into the details of that… [Multiplication], reflecting the partial order of divisibility, has an infinite number of generators, the prime numbers. So, now, rather than starting with just the number 1 and combining it with itself in every possible way using addition, we start with this infinite set of primes and then take all possible multiplicative combinations.  Defined since antiquity, this key concept has yet to deliver up all of its secrets, and there are plenty of them.

It has the quality of a square peg in a round hole, this tension between addition and multiplication.  It’s almost like, despite the inarguable perfection of the number system, they don’t really fit together very well, and they generate what I feel is something like friction, and this produces the sprawling mass of definitions, theorems, lemmas and conjectures that we call analytic number theory.  There’s a novel by Apostolos Doxiadis called Uncle Petros and Goldbach’s Conjecture — it’s written as fiction, but he gets some key ideas across through an elderly mathematician character.  This is very well put, I feel: Multiplication is unnatural in the same sense that addition is natural.  It’s a contrived second order concept, no more really than a series of additions of equal elements. So that’s the point, that 3x5, you can see that as 0+3+3+3+3+3 — you start with nothing, zero, and add three five times.  So in a sense you can build multiplication out of addition, whereas it doesn’t work the other way around.  So addition is a first order operation, and multiplication is, as he’s saying, unnatural, in that it’s ‘second order’. The thing that struck me about it when I was dwelling on this for a while was that it has to do with time, it has to do with repetition.  And it also relates to the very deep issues concerning the whole idea of where number comes from and how we define number.  As I hinted earlier, I’ve spent a lot of time thinking about how you could ever have two of anything. You know, there are two people sitting here in this room right now, but that relies on the definition of what a ‘person’ is.  We define the category linguistically, and we think we know what a ‘person’ is, but you can imagine some sort of genetically-engineered mutant that may or may not be a ‘person’ depending on how the definition was formulated, and the definition’s made of words and each word is imprecise, is subject to interpretation.  So any type of category you define is going to have a ‘fuzzy’ boundary, so...although it works quite well for day-to-day affairs, counting things works fairly well, you’ve got fifteen sheep in your paddock.  But you can always contrive some convoluted situation where, maybe it’s fourteen sheep or maybe it’s fifteen — is that odd looking creature really a ‘sheep’ or is it something else?  So, it comes down to issues of language and definition.  We consider chunks of spacetime, we recognise patterns and say, yes, that chunk of spacetime falls into suchand-such a category.   As I said, I started to wonder how you can really have two of anything.   Every entity ultimately distinguishes itself from every other, these categories are not mathematically precise, there’s an arbitrary element involved in deciding whether things get included — “where do you draw the line?” as they say.   And yet these categories are the essence of counting, and if there’s aproblem with applying the concept ‘2’ to our experience then there’s going to be a problem with all of the other positive integers. 

Exceptionally, when you get down to the subatomic level you can have two of something, because each individual electron is absolutely indistinguishable from the others.  So that’s interesting, that this concept makes sense at the subatomic level but then  ‘fuzzes out’ at macroscopic scales.   But the thing is, when you say ‘3x7’, you’re effectively saying ‘three sevens’.  So, seven pebbles in a row — you count out seven by adding one plus one plus one, etc. That feels quite ‘natural’.  But then, to make the leap to ‘three lots of seven’…you can have three giraffes or three potatos, the fuzzy boundaries mean that’s a difficult enough issue as it is, but ‘three sevens’ presupposes that a ‘seven’ is something that there can be more than one of in some sense...

C: One would have to say that the multiplier and the multiplicand are somehow of a different order, two different types of numbers are involved in the operation.

MW: Yes, one is operating on the other.  If you add, it doesn’t matter...I mean, it’s true to say that 3x7 is the same as 7x3, you’ve got this basic ‘commutative’ property applying to the positive integers.  But when you consider the ‘act’ of 3x7, the three is how many times you’re doing something, whether it’s laying out a row of seven beans or playing seven drumbeats, and the seven is some kind of an extension in space or time.  3+7 or 7+3, both numbers play the same rôle.   So there’s something there, not easy to pin down, which we don’t understand, and I have a very deep sense that we won’t really understand it until we really understand time. It has something to do with time.  Our inability to understand the primes, our inability to prove RH is a symptom of our inability to understand the relationship between addition and multiplication, and that is related to our relationship with time.

C: On your site you quote J.J. Sylvester: I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in regard to time, which like space may be in essence poly-dimensional and that this and other such sort sort of truths would become self-evident to a being whose mode of perception is according to superficially as opposed to our own limitation to linearly extended time.

MW: I think he must have been thinking about the relationship of multiplication and addition in terms of time. This was 1888, so RH had been posed, but mathematicians long before RH understood that the enigma of the prime numbers was rooted in the uneasy relationship of addition and multiplication.  So possibly he had a

sense that the relationship had something to do with time. But he says ‘the profound mystery which envelopes our conceptions relative to prime numbers’ — in other words, the puzzling interface of addition and multiplication — ‘depends upon the limitations of our faculties in regard to time’.  So if there were a higher dimensional, a two-dimensional ‘time surface’ or something like it — the word ‘superficially’ is being used by Sylvester in the original sense meaning ‘relating to surfaces’ — our minds, normally constrained to a ‘timeline’, could perhaps ‘spread out across it’ in some sense.  It’s perhaps a bit like being able to come up off the surface of the earth and look down from a third dimension to get a sense of how things are laid out, whereas when you’re stuck on the ground, certain things are not at all apparent...but these are all very vague and intuitive ideas. * C: In one of the papers you link to in the archive11, Volovich suggests a most extreme and startling explanation for the concurrence of physics and mathematics.

MW: Yes, and you may have noticed that he quotes Pythagoras at the beginning, a slightly amusing Greek-to Russian-to-English compound translation of “all is number” — “the whole thing is a number”.  I got very excited when I first found that paper, because he’s suggesting that number theory is the ultimate physical theory.  That came out in 1987 as a preprint at CERN — he’s an accomplished physicist — but it was never published in a journal.  The fact that it never got published and the fact that he hasn’t responded to my questions about it could suggest that he’s backed away from it somewhat.   I can’t speak for him, but I wonder if he’s slightly embarrassed by its more grandiose claims, in the way I was suggesting earlier that physicists and mathematicians can be. But the thing is, he has done this vast body of work on p-adic physics, which I referred to earlier. And the rise of p-adic physics is a very interesting thing in itself because, you see, even though the universe at the scale of this room is Archimedean — I can lay my ruler end to end and will eventually reach the end of the room — the universe is not Archimedean at all scales.  Below the Planck scale, it’s no longer Archimedean.  Below this 1035m or so — which to some people sounds too small to worry about, but you just take a metre, then a tenth, then a tenth, not that many times, really…It’s not that our instruments aren’t precise enough to measure below that scale, it’s that the whole idea of measurement as we’ve formulated it ceases to make consistent sense.  And effectively, space becomes non-Archimedean below that scale. There’s a similar scale with time and other fundamental quantities, below which they become non-Archimedean. You can theoretically join some unit of measurement end-to-end and never achieve a given, finite extension. This has led people to think that maybe p-adic physics, where you’re dealing with a non-Archimedean number system, would be more appropriate for application at the sub-Planck scale.  And Volovich seems to be suggesting that different non-Archimedean number systems could apply to different regions of space and time at different scales.  Again, I’m not entirely sure: large parts of the paper are beyond the scope of my present understanding.  I’m intrigued by his referenced to ‘fluctuating number systems’, but I don’t know whether he means fluctuating with time, or in some other more generalised sense.  People are now starting to think about applying p-adic mathematics to the physical world.  Each p-adic number system provides a different sense of ‘distance’ between two rational numbers, and that notion of distance then allows you to define all the other numbers which aren’t rational via precise mathematical concepts involving ‘limits’.  I mentioned this earlier. This distance or ‘metric’ is defined in terms of divisibility of primes.  It has to do with highest powers: for instance, in a 7-adic metric, finding the distance between two rationals involves basically looking for the highest power of 7 that divides into the numerator of their difference — that difference of course is also a rational number — when it is expressed as a fraction in lowest terms.   As a result of that, number theory comes flooding into your p-adic physics: if you start looking at p-adic or adelic space and time, issues associated with the prime numbers become directly relevant.  Of all of this number theory/physics material I’m archiving this is the area I’m least familiar with.

C: Saying that the means of measurement, that the possibility of measurement has changed is one thing, but saying that numbers are actually the ‘atoms’ themselves, so to speak, is something else: that means that there is no longer some thing you’re measuring.  The measurement itself takes on a sort of substantiality.

MW: Yes, these are very difficult notions to grasp, in so far as I understand what’s being proposed.  I think, perhaps like myself, Volovich caught a glimpse of something, got quite excited about it and wrote it down; he’s quoted Pythagoras — it’s as if there’s some mystical quality to his insight.

C: There might be thousands of these papers hidden everywhere that people haven’t published.

MW: I’m not sure it would be in the thousands, but who knows…There’s a general hesitance to stick one’s neck out.  If I’m helping to encourage that sort of thing, then I suppose that’s a useful contribution.

C: Exeter University has granted you an honorary fellowship and hosts the web- archive, but there is no funding available for your work.   Apart from your own fascination with the subject, what drives you to continue this labour of archiving and making your own speculative connections public?

MW: Over the years after I’d dropped out of formal academia, I spent a lot of time thinking through and honing these ideas about mathematics being some sort of inner priesthood of our scientistic culture that’s in the process of destroying the ecosystem, and wondering what could be done about it, how do we change this, you know? I felt that campaigning to stop the destruction of this or that rainforest isn’t going to be enough, you’ve got to go right to the core, to the root of the problem, the fulcrum.  And, reading von Franz, with her ideas about ethnomathematics, and quantity and quality, and reading René Guenon, who — although I don’t embrace his traditionalist fundamentalism — wrote a fascinating book called The Reign of Quantity and the Signs of the Times, I started having this idea that only when Western Culture re-evaluates its relationship with number can there be any real change in the way we relate to the world, because we’ve got stuck in a ‘quantocentric’ view of the world.  And so I have felt at times that what I was trying to bring forth — whether it was in my strange 1998 ‘evolutionary’ notion or just in my networking of various people’s work via my web archive—was an acceleration towards an imminent transformation in our relationship with the number system.  I was quite driven for a while,  but I’ve become considerably more cautious and sober in my approach to this since.  I saw what I perceived to be clues...felt that it had to be coming, and only through that sort of transformation will the Western project ever be able to steer itself in a less destructive direction.  At times I’ve felt that I had an important rôle to play — not that I was ‘chosen’ to do it or anything, but that my work was cut out for me, and it was an important mission.  Other times, I’ve been much less certain, and wondered, you know, why am I sitting in front of this computer editing HTML, when I could be spending the same time and effort campaigning for, say,  the rights of an indigenous tribe having its land ravaged by a multinational corporation.  I had to justify this to myself when people I knew were involved in things like that, by telling myself, well actually they’re just dealing with the symptoms, whereas I’m trying to deal with the root of the problem.  So it verged on an idealism, almost an activism. 

C: The point being that rather than lamenting the destructive rôle of number and of science, one tries to recognise that there’s something else within number, and as you said, to re-evaluate our relationship to it, which is not to say to reject it, but to become more numerate...

MW: Yeah, which is what I saw around me, people being very suspicious towards mathematics, hating it, seeing it as controlling and evil, and I thought, no, we need to get inside it, try to understand where it comes from and how it works.  But then I started to question whether I was just  creating a whole set of complex and noble motivations for myself when in fact it was just my ego or desire to be acknowledged for what I’d achieved, or, you know, just wanting some sort of recognition or status.  I was continually wondering what it was that was motivating me, and trying to rein myself in and consider the worst possible motivations as well as the best. I had a kind of motivational collapse in early 2005, when I was struck by a very deep sense of there being insufficient time; you know,  I had this grandiose hope of helping to effect some sort of long term change in culture and the way in which we deal with the number system. I started to think, maybe what I’m contributing to would have that effect if there were a few more centuries left of relatively leisurely culture and well-funded academia to take these ideas on and develop them, but, you know, we’re facing multiple global crises, and this sort of thing is never really going to have time to take root.  I’ve since drifted in and out of this activity periodically, found what I think is a healthy level of interest in these matters.  But I don’t strongly believe that I’m part of some current of cultural change anymore, I’m just...I suppose you just can’t know what effect you’re having, particularly with the Web, when you’re pushing ideas out. You don’t know who’s reading them and what they’re going to do with them — a bright teenager who reads my website might be inspired to study mathematics and, influenced by some of the hints, clues, suggestions, etc. I’ve assembled, go on to make amazing discoveries...who knows? There’s also the whole relationship between psyche and matter which seems to have been at the centre of all my interests over the years. I got involved in parapsychology for a while, online psychokinesis research in 1996, wondering whether there really was something in that, and what it would imply concerning the psyche-matter interface.  There’s also a very exciting interdisciplinary field of ‘consciousness studies’ emerging, and which I’ve been following, people trying to understand the physics of consciousness, looking at microtubules in brain cells and how quantum mechanical phenomena at that scale might help to explain the origins of consciousness — physicists, neurologists, philosophers, psychologists, anthropologists, psychopharmacologists, etc. are all contributing to this field.  Then there’s all the Jungian  theory concerning myth, archetype, synchronicity and the ‘psychoid’ level of reality — a kind of psycho-physical interface.  The simple fact that mathematics is able to describe the world at all, that’s a mystery involving mental constructs being mapped mapping onto material reality.  There’s the ‘mind-brain problem’ which philosophers debate.  And then dreaming, shamanism, schizophrenia, quantum-mechanical paradoxes, these are all things I’ve spent a lot of time thinking about, reading about — generally wondering how it all fits together.  And it had occurred to me that these topic cluster around the central mystery of how matter and psyche interface. But I’d been thinking about prime numbers, etc. for a few years before it occurred to me that this is very much part of the same picture.  I’d been exploring the interface of physics — which concerns matter, obviously — and number theory, which, as that Tenenbaum quote suggested, is really an exploration of  ‘the mind itself’.   And the research I’ve been interested in archiving displays a two-way traffic: Number theorists have been providing concepts and structures which physicists have used to better understand the world of matter.  Physicists have been able to, using their understanding of matter, shed light on the internal workings of the number system. Even number theory without the physics is implicated: although number is widely considered as a mental construct, at the same time it manifests directly in the world of matter: when you consider a quartz crystal or a five-petalled wildflower, it’s hard to deny there’s an essential ‘sixness’ or ‘fiveness’ there.  So, number itself is a bridge of sorts between psyche and matter. This last idea, that number is a bridge between psyche and matter, comes quite close to something Jung was exploring in his later career.  He left a lot of incomplete work when he died, and I believe he left von Franz to look at number archetypes.  He’d looked at individual integers, the first few integers and their various associations.  But later, more importantly, he’d come up with the idea that, not individual numbers with their associations, but the set of positive integers as a single entity is in itself an archetype, the archetype of order.  Now what has distinguished Western culture from the rest of humanity, what characterises the Sumerian-toBabylonian-to-Greek-to-Roman-to-Western-European cultural current that dominates the planet with its measurement and science and so on, is the way we’ve dealt with this archetype which normally inhabits the collective unconscious. I picture it as a sort of mysterious sea creature — we’ve hooked it and we’ve hauled it out from the dark depths into the daylight of consciousness. We’ve taken something that was primarily unconscious, and which would naturally manifest primarily via the number archetypes and number associations in other cultures. We’ve dragged this thing out of the sea and onto the land, cut it up and studied it, studied its anatomy in great detail in order to obtain a new kind of magic, if you like, and that, I came to believe, was the root of all the world’s problems.  But then we have this emergence into consciousness of the set of the prime numbers buried within the set of positive integers, a hidden archetype within an archetype, a kind of chaos within order, the black dot in the yin half of the yin-yang symbol; the emergence of that archetype — the prime numbers, the zeta function and everything they entail — into mass consciousness, is just starting now, really.   The first four ‘popular’ books on RH have all come out in the last couple of years...it’s strange that this should all be happening so suddenly.  Thinking along the quasi-Jungian lines I’ve sketched out, the integration of these ideas into consciousness, the idea of the Riemann zeros having their origins in some ‘older’ or ‘deeper’ numerical reality, something more ‘primordial’, etc. may turn out to be of profound historical significance. According to the insanely optimistic wishful thinking which I’ve since distanced myself from, this could be the event that would start to alleviate the effects of rampant ‘quantocentrism’ and put things back into balance.

C: I wonder whether the growth of ‘popular science’ could play a rôle here — thinking in particular of the many books which have been published on RH.

MW: The fact that you’ve got four books on RH out suddenly — why is this, why hadn’t this happened before?  I’m sure a few years ago most people involved would have said that it’s impossible to explain RH to laypeople.  But four authors have done their best, with varying degrees of success.  The books have all been well received, have sold fairly well.  So why is this happening? The mystically inclined might invoke an unseen force that’s trying to bring these ideas into consciousness. Jungians might talk about ‘compensation’ and the collective unconscious. But more simplistically, more materialistically, it’s market forces, it’s capitalism, and it’s because people are looking for meaning.  Many are turning to New Age cultism, some are turning to born-again Christianity, Scientology, fundamentalist Islam, whatever. But there are a lot of people who are aware that the real ‘guardians of truth’ these days are not priests and monks, but scientists and mathematicians, and yet, they find themselves in a position where they don’t know anything about the essential subject matter.  So they want someone to explain, say, the mysteries of quantum physics to them. I get this all the time, people really wanting me to explain quantum physics, fractals, relativity, the golden mean, chaos theory, p; there’s a handful of things that people get really excited and obsessed about, you know.  And of course the market system rises to meet a demand, a growing demand for meaning.  The problem is that capitalism doesn’t care whether a book is accurate or well-written, it just cares about sales figures.  So as a result you get gross oversimplifications hitting the market and sometimes selling quite well.  Because the market has expanded, there is more competition, and ideally, if you believe in the effectiveness of capitalism, then the ‘best’ stuff will float to the top — but ‘best’ in this sense doesn’t necessarily correlate with truthfulness or accuracy, rather with how successfully the book quenches readers’ thirst for meaning.  There does seem to have been a certain amount of progress, though.   I don’t really watch much TV, but it does now appear that with the computer graphics available, it’s possible to make some things a lot more visually accessible, so viewers can at least get a flavour of the problem, or of what’s at stake.  But the really deep stuff, the major philosophical problems underlying maths and physics…it’s hard to imagine that there really is a shortcut to years and years of disciplined study. I mean, you might be able to get the basics of something across to a few, a small section of the population who are already interested and whose minds are structured in a certain way — it’s not to do with levels of intelligence, just a certain kind of intelligence.   You’ve got committees for the popular understanding of science and things of that nature, but they’re very marginal. Unless there were a major cultural shift, unless you had major government funding, and the top layer of mathematicians and scientists committing themselves full-time to bringing this stuff through into popular culture...but there’s no motivation for that to happen — governments aren’t interested in educating their populations except in ways which will further economic growth.  They want a certain proportion of young people to be trained up to be economists, accountants, engineers, etc. ‘Truth’ doesn’t really come into it.  So I doubt it…but, again, you never know, some major cultural shift could occur where the demand for this sort of knowledge reaches the point where the best people would feel obliged to provide it. Or, possibly, there could be some sci-fi type breakthrough involving direct brain-to-brain knowledge transfers, you know, you can’t rule these things out, but I’m not holding my breath! You’ve probably noticed, part of my website is very formal-academic, the web-archive aspect; and part of it is just about getting fundamental ideas across to people who are open to them and just want to understand their reality a bit better.  I have felt in the past, with my ‘activist’ hat on, that it’s important to bring some of these issues to widespread public attention — the basic issues of the number system.  At this stage I don’t know if it is ‘important’ or not, but I’d be very interested to know what the overall effect of that kind of exposure would be. Again, I suppose I am still gripped by the idea that, if we transform humanity’s relation with number, that could have a positive transformative effect.  I suspect I’m still partially motivated by that belief at an almost subconscious level. The only thing I can really say with any confidence at all is that I think we’re on the verge — and again, the timescale is very indefinite here — but Western Civilisation is on the verge of collectively realising that the number system is something very different from what it had previously thought it to be.  I haven’t got a

particular theory about what it is, I just know it isn’t what we think it is.

PC I applaud the efforts that you have made to draw attention to many existing problems that relate in various ways to our “quantocentric” culture.

And though interest in this number related field because of its unique intellectual requirements remains a small niche area, I believe that you have been more successful than you imagine in becoming an influential figure in this regard.

Your “Number Theory and Physics Archive” and the “Secrets of Creation” trilogy are an excellent contribution to the field. Though there may never be a mass readership for the books in the trilogy, they can still exercise an important influence on the limited number of people who can recognise the supreme importance of the topic.

And I am delighted to see that your  “Prime Evolution” is now available again as it is a very fine contribution. I remember accessing it some years ago on the web and then it seemed to vanish only to reappear again more recently. So I hope many more people will come to discover this superb discussion. 

In terms of the damaging effects present understanding of the number system is having on our culture we would both share a great deal of common ground. I do however hold a much stronger conviction that mathematics is itself in need of radical reform. And this relates to the urgent requirement to discover its unrecognised holistic aspect, which apart from potentially opening up a vast new qualitative field of enquiry has profound implications for the interpretation of existing quantitative relationships. 

This crucial holistic aspect has remained greatly repressed throughout its history and especially with respect to the increasingly specialised development of mathematics in the past two centuries. From a psychological perspective, it therefore entails that the hidden unconscious aspect of understanding, which affects every single mathematical issue must now be slowly brought into the conscious light, before then becoming properly integrated with its conscious counterpart.

This will entail the most radical revolution yet in our intellectual history, which I believe will coincide with a great new spiritual awakening. However apart from a few lone voices crying in the wilderness, it will only properly emerge after a protracted series of crises that gradually awaken the world to the great existing lack of a true holistic appreciation of reality.

So I will conclude these comments now with some personal reflections on a future “golden age” of Mathematics.

Here, I would see three distinct domains (where only one currently exists).

1) The first - which for convenience - can be referred to as Type 1 Mathematics, relates to the traditional analytic approach based on the reduced quantitative interpretation of mathematical relationships.

At present, in terms of formal recognition, mathematics is exclusively identified with this type.

As its methods have become increasingly specialised in an abstract rational fashion, admittedly enormous progress has been made. And this will continue into the long distance future with many new significant analytic findings for example regarding the Riemann zeta function (and associated L-functions).

However an important present limitation, as we have seen, is the manner in which exclusive identification with the analytic, blots out the holistic aspect (with which creative intuition is more directly associated).

While not wishing to prevent further progress in analytic type developments, eventually I believe it will be accepted that the Type 1 represents just one highly important aspect, which should not be exclusively identified as mathematics.

So a strictly relative - rather than absolute - interpretation will thereby eventually emerge for all its relationships.

2) The second - which I customarily refer to as Holistic Mathematics - represents the Type 2 approach.

There have been some precursors in this regard notably the Pythagoreans. However beyond a number of suggestive ideas, they never really succeeded in the explicit development of the holistic aspect, which represents an urgent requirement now for our present age. 

My personal development has been somewhat unusual in this regard. Though displaying a marked ability for mathematics as a child, serious reservations with the standard treatment of multiplication arose at a very early age. Therefore, from that moment I was already reaching out for a new holistic dimension, not catered for in formal terms.

So mathematics for me has very much represented a solo voyage of unique personal discovery.

And in adult life, I have largely concentrated on elaborating the hidden holistic dimension of mathematics, which is truly enormous in scope. Then in the last 15 years or so, using these holistic insights, I have turned my attention to topics such as the Riemann Hypothesis with a view to providing a radical new perspective as to their inherent nature.

Though retired from work as a college lecturer in Dublin for nearly six years, it feels only now that I have come full circle in being able to finally resolve (at least to my own satisfaction) those childhood questions regarding multiplication.

However it is very difficult indeed attempting to convey holistic mathematical insights to a professional audience that does not formally recognise their existence.

So my most successful communication has been with talented generalists from diverse mathematical backgrounds, seeking to understand various intellectual disciplines in a more integrated manner.

However even here, there has been considerable resistance to the belief - reflecting the deep-rooted nature of conventional assumptions - that mathematics itself is in need of radical reform.

Bearing the above comments in mind, I will now try to convey some of the flavour of my holistic mathematical pursuits.

As it is directly concerned with the qualitative nature of symbols, much of this work has related to the clarification of the various stages of psychological development through the holistic use of these symbols.

In terms of the physical spectrum, natural light forms just one small band on the overall spectrum of energy.

In like manner, the mental structures (based on accepted common sense intuitions underpinning linear logic) represent just one small band on the overall potential spectrum of psychological development.

Conventional mathematics represents specialised understanding based on this linear band.

However just as there are further forms of physical energy (besides natural light) equally there are further forms of psychological energy (besides the accepted intuitions of conventional mathematics).

These further bands on the psychological spectrum have been extensively investigated by the major esoteric religious traditions East and West, where they are identified with advancing levels of spiritual contemplation of an increasingly formless nature.

However what has not yet been realised - except in the most perfunctory manner - is the important fact that these bands are likewise associated with new forms of holistic and radial mathematical interpretation (utterly distinct from conventional type notions).

In analytic terms, the study of higher mathematical dimensions entails rational understanding of an increasingly abstract nature (where the object aspect is increasingly separated from its subjective counterpart).

However in holistic terms, the study of higher dimensions by contrast entails appreciation of an increasingly refined intuitive nature, indirectly transmitted in paradoxical rational terms, where both objective and subjective aspects are seen as ever more interdependent with each other.

Now one of the extraordinary findings arising from these investigations is that all psychological and physical structures, which are of a dynamic complementary nature, can be given a distinctive holistic mathematical rationale.

And the holistic notion of number is intimately tied to these structures.

So from the holistic perspective, accepted formal mathematical understanding is 1-dimensional (in qualitative terms). This simply means that the qualitative aspect is formally reduced in a quantitative manner.

However as I have already mentioned, associated with every other number (≠ 1) is a distinctive means of interpreting mathematical symbols with a partial relative validity. So the absolute type understanding that we accept as synonymous with valid mathematical interpretation represents just one special limiting case of a potentially infinite set.

And this insight was later to prove of inestimable value in relation to a true dynamic appreciation of the Riemann zeta function.

Also, the holistic relative notion of number is intimately connected to a corresponding new holistic appreciation of the nature of space and time with a direct relevance in physical and psychological terms.

This has immense implications for physics where space and time notions have been reduced in a distorted manner corresponding to a merely quantitative interpretation of number.

So we apply general impersonal notions of space and time in scientific terms. Not surprisingly, physical reality then appears to us as lacking any personal aspect.

However properly understood, all reality contains both personal and impersonal aspects. So the qualitative personal features of nature relate in holistic manner to unique configurations of space and time (which find no place in the conventional approach).

Indeed one could validly argue that the root of the present ecological crisis, which is so rapidly unfolding on our planet, lies in the merely quantitative appreciation of scientific reality (which has greatly distorted its true nature).

For example, as we have seen, the holistic notion of “4” relates to a dynamic appreciation of the corresponding 4-dimensional nature of space and time (with positive and negative directions in real and imaginary terms).

In my own work I have come to appreciate how important the holistic understanding associated with 2, 4 and 8 dimensions respectively are for an integral appreciation of reality.

And once again there are strong links with Jung, who came to realise the significance of certain pictorial diagrams termed mandalas – widely employed in Eastern religions as meditative tools – for psychic integration.

The most common mandalas generally entail highly ornate images based on the division of a circle with four or alternatively eight equidistant points.

From a mathematical perspective, these therefore serve as geometrical representations of the 4-dimensional and 8-dimensional holistic interpretations of reality.

However, just as there is a key distinction with respect to the behaviour of the Riemann zeta function for even and odd values of s, likewise there is a key distinction in holistic terms as between the even and odd dimensional values. Basically, the even values relate in more passive terms with the achievement of a certain attainment of integration with respect to experience (entailed by the corresponding dimensional number). The odd values then relate to a more active state of asymmetrical involvement, associated with a degree of linear understanding (which leads to a departure from the previous equilibrium state).

So in ideal terms, human development should continue moving in and out of progressively more advanced stages of psychic integration.    

In brief, akin to directions on a compass, all the various holistic dimensions can then be looked on as providing unique configurations of the dynamic relationship as between wholes and parts (in objective and subjective terms).

Just one final example I will offer, though all this represents but the tiniest glimpse into a potentially vast new field of investigation, is the holistic counterpart to the accepted binary system!

So the two binary digits 1 and 0 - given an independent interpretation in the standard analytic manner - can be potentially used to encode all information processes.

However, equally the two binary digits 1 and 0 - now given a holistic interdependent meaning as linear (1) and circular (0) type understanding respectively - can be likewise used potentially to encode all transformation processes.

So for example, in this contribution, I have argued that the number system - and indeed all mathematics - should be interpreted as representing a dynamic interactive transformation process, entailing both quantitative and qualitative aspects.

And these two aspects relate directly to 1 and 0 respectively (in a holistic manner).

In fact, at the most general level all differentiated and integrated processes both in physical nature and psychologically in human terms are encoded in a holistic binary digital manner.

We are now living in the digital age where so much information is already converted in a binary fashion (entailing the analytic interpretation of 1 and 0).

However what greatly concerns me is that because of the dominance of mere quantitative analytic type understanding with respect to such technology, that no adequate means for corresponding transformation exists in our society (which requires the corresponding development of holistic type appreciation).

So at present, a number of related crises are developing in environmental, economic, political and social terms all of which have at their root an inability to tackle global issues in a truly integrated fashion (which demands a radical holistic approach).

And this in turn is deeply related to a distorted view of science which unfortunately is treated as the new world religion. However, at an even deeper level (though not properly recognised) it is due to a distorted view of mathematics which underpins the scientific approach.

3) This, which I commonly refer to as radial mathematics or more simply the Type 3 approach, represents potentially the most comprehensive form of mathematical understanding, entailing the coherent integration of both analytic (Type 1) and holistic (Type 2) aspects.

When appropriately understood, all mathematics is intrinsically of a Type 3 nature, though not yet recognised because of the deep-rooted acceptance of reduced assumptions.

Indeed it is only in the context of radial mathematics (Type 3) that the other two aspects, conventional mathematics (Type 1) and holistic mathematics (Type 2) can reach their fullest expression. So perhaps some day in the distant future, Type 3 will become synonymous with all mathematics.

However even within this category, I would distinguish three important sub-types, (a), (b) and (c) respectively.

Though rooted to a certain degree in the holistic aspect of mathematics, sub-type (a) is mainly geared to the derivation of exciting new analytic type discoveries (with creative insight playing a key role).

There is no doubt that implicitly, Ramanujan represents a truly extraordinary example of such holistically inspired mathematical discovery. At a deeper level, I believe Riemann also perhaps belongs, leading to highly original discoveries that relied on a strong intuitive dimension. However, unfortunately, in neither case was the holistic dimension of mathematics explicitly recognised.

So in the future, even greater creative analytic discoveries in various fields will be possible, when mathematical talent is backed up with a truly mature holistic appreciation of symbols.

I should perhaps stress that with respect to mathematical understanding that analytic and holistic type abilities are utterly distinct. Thus for example a mathematician with an obvious recognised mathematical talent in analytic terms, may have little capacity for corresponding holistic appreciation.

Likewise it works in reverse so that a person who in conventional terms may display little or no mathematical ability could in fact be potentially gifted at the holistic level. Jung for example would readily fall into this category.

However, it would not be possible for someone to successfully operate with respect to Type 3 mathematics without being able to combine in varying degrees both the Type 1 (analytic) and Type 2 (holistic) aspects.  

The second sub-type - while requiring appropriate analytic appreciation (the degree of which depends on the precise context of investigation) - is mainly geared towards the holistic interpretation of mathematical objects.

I would classify my own recent efforts as a most preliminary version of sub-type (b), operating necessarily at a very rudimentary level.

However this is still adequate, for example, to outline the bones of a distinctive dynamic appreciation of the number system. This then provides the basis to radically re-interpret the nature of the Riemann zeta function (with accompanying zeta zeros and Riemann Hypothesis).

Thus, subtype (b) is not geared directly to analytic discovery, but rather a coherent dynamic interpretation of mathematical relationships. However because it is most creative at a deep level of enquiry, indirectly it can facilitate exciting new directions for analytic discoveries.

The final subtype (c) entails the most balanced version of both (a) and (b), opening up possibilities for the finest form of mathematical understanding, that is at once immensely productive and highly creative and readily capable for example of synthesising various fields of mathematical study.

One of its great benefits is that it can also provide the important capacity to appreciate the potential practical applications of new mathematical discoveries.

The reason now why so much abstract mathematical analysis seems irrelevant in practical terms is precisely because it is understood in a manner that completely lacks a holistic dimension.

However, with both aspects (analytic and holistic) properly recognised, the practical applicability of abstract mathematical findings would be more readily intuited.

And from an enhanced dynamic perspective, all mathematical findings which, when properly understood, are experientially based, have practical applications!

In this regard as a general principle, I would imagine that what is considered most important in abstract terms, is likewise potentially of greatest significance both from a holistic and also applied mathematical perspective.

Another great advantage of subtype (c) is that by its very nature, mathematics can now become readily integrated with the rest of human experience, allowing for the fullest expression of personality development.

Thus if we want a vision of what mathematics might look like at its very best, we would choose subtype (c) with respect to the Type 3 aspect.

However, we are still a very long way indeed from realising this wonderful reality, with the great lack yet of an established holistic dimension to mathematics, serving as the chief impediment.