Dynamic Considerations

MW: Yes, and so something with this kind of quasi-mathematical character is generally regarded with a certain suspicion; it’s neither one thing nor the other.

C: A mystery rather than a problem, then.

MW: Yes, and I suppose I tend to be attracted to the mysteries.

C: Practically speaking, how does the hypothetical positing of a Riemann dynamics change the nature of the search for a proof of RH?

MW: It brings other people in, it brings the physicists in. Before, you had analytic number theorists hammering away at this problem.  And now probability theorists, geometers and physicists are all contenders, and they all have pieces of the puzzle.  It’s broadened the scene, if you like, of people concerned with the problem.  But it also has given a deeper sense of what’s at stake; again, if there is a dynamic system underlying the Riemann zeta function, well then it underlies the number system; if it underlies the number system then it underlies everything, or at least everything that rational scientific thought concerns itself with.  And so, again, we’re force to ask what is it, where does it ‘live’, what does it ‘do’?  And perhaps the most important question is, what is the time parameter? Because a dynamical system always has a time parameter according to which it ‘evolves’ – so what kind of time are we talking about in this case?  So it basically opens a whole new can of philosophical worms. It makes me think of what Hilbert said, when he was asked about RH, he said that it isn’t just the most important problem in mathematics, it’s the most important problem.  And I think a lot of people might just think, yes, that’s because he was a mathematician, he was biased...but I think he knew what he was talking about. He and Pólya first proposed that there might be a ‘Riemann operator’, that the zeros might be a spectrum of something.  They didn’t suggest a dynamical system as such, but they could be said to have laid the groundwork for that.  So I think Hilbert may have sensed something very big going on there, which he was trying to express in that pronouncement.

C: The first steps towards elaborating the nature of the Riemann dynamics comes with Julia’s interpretation of the zeta function as a thermodynamic partition function. What is a partition function, and in what sense can one speak of the primes as a numerical gas – Julia’s ‘free Riemann gas’?  Is it simply a useful metaphor taken from thermodynamics, or is there a more substantial link?

MW: Well, firstly, Julia’s work doesn’t directly address the issue of the Riemann dynamics, although there may well be a deep connection there.  Your last question is difficult to answer, but it would be hard to deny that there’s a sort of a metaphor here, in that there’s a strong resemblance between certain aspects of the zeta function and the theory of thermodynamic partition functions.  But it goes deeper than a superficial resemblance. There are enough corresponding elements, that Julia included what he called a ‘dictionary’ in the paper he first published about this. It consists of two columns, with number theoretical structures on one side and corresponding thermodynamic structures on the other.   And the correspondences are such that, if you’re sufficiently familiar with number theory and statistical mechanics, you can’t deny there’s something...there’s a very strong link there.  So you could call this a metaphor, but I would maintain that it’s more than just a metaphor in the familiar sense, i.e. a useful way of explaining what something is by means of something else which isn’t directly related to it.  Now what is a partition function, in statistical physics, or statistical mechanics?  Well, in classical mechanics, a billiard table is often used as an example: you’ve got a finite number of billiard balls bouncing off each other, bouncing off the sides, they’re colliding, energy is being transferred between them, there are various angles, positions and momenta involved.  And the idea is that you’ve got a sufficiently simple system that you can keep track of each individual object and what it’s doing.  But a problem arises when you’ve got something like a box of gas: that’s effectively like a giant three-dimensional billiard table, but there are too many components to keep track of what each one is doing.  You’re not actually going to be able to do anything in that way, so you’re going to have to study it in the sort of way sociologists study society — they can’t possibly consider all the specifics of each individual person, so they must look at overall statistical trends in the population.  Suppose you had a quantity of gas particles in this room, and they were all roaming freely.  It would be very surprising to find them all clustered up in one corner. One expects a more uniform spread.  But, there’s no real reason they can’t do that. It’s like if I toss a coin fifty times, I’d be very surprised if I got fifty heads or fifty tails, but there’s no reason why that can’t happen.  That would be no more unreasonable than any other outcome of fifty coin-tosses, it’s just that it’s extremely improbable because, unlike any other outcome, there’s only one way of arriving at it. Similarly, there are proportionally few possible configurations of those gas particles where they’re all squashed in one corner, compared to the vast proportion of configurations where they’re more-or-less uniformly distributed.  Now suppose you have a box of gas, and the gas consists of particles which can jump between different energy levels in an effectively random way.  This time you’re concerned, rather than with the spatial distribution, with the total energy of the system – that’s simply what you get when you add together all the individual particle energies.  You can ask about the probability of  the system having a particular total energy, and it turns out to be rather like the situation with the spatial distribution.  That is, the system tends towards a mid-range total energy on the whole, while the highest and lowest ranges of possible total energy are much more improbable – because their occurrence requires something akin to a huge number of coin-tosses producing almost all heads or almost all tails. So what you’re looking at with thermodynamics is the probability that you’ll find a box of gas or some similarly complex system in one state or another.  And the partition function takes a unit of probability and ‘partitions’ or subdivides it, so that you end up with a curve describing in precise terms the relative probability of finding the total energy of the system at any particular level.  So the partition function will basically return probabilities that a system is in one of any number of possible states.  The partition functions Julia refers to are functions of temperature — as the temperature of the system varies, the probabilities also vary, and the partition function is able to provide a precise probabilistic distribution of possible total energies at any given temperature. Now partition functions, it turns out, are the key to understanding statistical mechanical systems; they ‘encapsulate’ such systems.  The partition function in this context is a function of temperature, and temperature would naturally be seen as a variable which varied on the real line — on the positive real line, if you’re working with absolute temperature.  Well, nineteenth century mathematics suddenly allowed the possibility of extending such a variable to the complex plane, regardless of what a complex-valued temperature might actually refer to. You take your partition function which is supposed to be returning probabilities of a system being in some energy state or other based on a real-valued temperature variable, extend it to the complex plane, and you find there are singularities hidden out there which tell you about the possible existence of phase transitions in your system.  These are very important for understanding the system, but, as I said earlier, you wouldn’t see them if you didn’t have access to the complex plane.  Now Julia wasn’t the first — George Mackey got there first, although it wasn’t widely noticed.  Julia discovered it independently and then Donald Spector, a couple of years later — they all noticed that if you treat the primes as your basic particles, and each prime p is thought of as having as its ‘energy’ the natural logarithm of p – that logarithm turns out to be very important, logarithms show up everywhere in analytic number theory – then the Riemann zeta function very naturally falls into the role of being the partition function of an abstract numerical ‘gas’ which is made of this set of particles – what Julia calls the ‘free Riemann gas’.  Imagine a fluctuating integer, where prime factors are coming and going all the time, joining and leaving, so the energy of that integer is going up and down, the more prime factors there are the higher the energy, and the less prime factors the lower the energy.  The zeta function naturally becomes the partition function of such a system.  The ‘pole’ of the zeta function – this unique singularity of the zeta function at the point 1 in the complex plane where

the zeta function isn’t defined, where it effectively becomes infinite – corresponds very naturally to something in thermodynamics called a Hagedorn catastrophe, a phenomenon involving the energy levels crowding together so the system hits a critical state and shifts into an altogether different mode.  So the pole of the zeta function is associated with this ‘catastrophe’, and based on what I was just saying, the Riemann zeros also become linked to phase transitions, in a way that no-one entirely understands.  And there’s more...those are just the basic points, there are further subtleties which suggest that, in some sense, thinking of the zeta function as a partition function goes beyond mere metaphor.  It’s a metaphor, but it’s a metaphor that goes deep enough to suggest to me that the number system has some sort of quasiphysical quality.

C: How are we to interpret this?  There’s a perplexing quality to these propositions, one is never sure whether what’s being revealed is a progression, or simply a restatement of the same problem in different terms.

MW: Possibly, but you see, the mathematics that’s come out of studying things like boxes of gas, that that should be applicable at all to studying something as fundamental as the positive integers, to me comes across as sort of uncanny. I think that’s a good word to capture how a lot of people have reacted to these discoveries.  It’s hard to see how it’s simply a reformulation of the problem.

You’d never have got there if you hadn’t studied the boxes of gas in the first place.  When you ask how we can best interpret this, the only answer I can come up with is I honestly don’t know.  To me it points to something fascinating which we haven’t yet entirely understood or taken into account.  Now, interestingly, Alain Connes’ (College de France, IHES, Vanderbilt) model involving what’s called a C*-dynamical system – his attempt to try and describe the Riemann dynamics, which hasn’t yet fully succeeded, although it’s certainly opened up some new vistas – was inspired by Julia’s paper, but Connes uses the partition function in a somewhat different sense. The partition functions I’ve been describing, the ones associated with boxes of gas, etc., could be called ‘classical partition functions’ as they belong to ‘classical statistical mechanics’.  But there are also partition functions used in quantum statistical mechanics, which take some of the same concepts down to the quantum level.  Connes takes certain elements of quantum statistical mechanics and applies them to the zeta function, treating it as a partition function, and this reveals certain things which again push the metaphor, in my mind, so far that it can’t be regarded as just a metaphor.

C: So there is a direct link between the quantum-mechanical interpretation and the thermodynamic?

MW: I think there must be, although it’s not yet entirely clear what it would be.  Of the two most extensive pages in my web-archive, one deals with the spectral interpretation — Hilbert and Pólya’s suggestion that the Riemann zeros might be vibrational frequencies of something and Michael Berry’s (Bristol University) physics-inspired work concerning what that ‘something’ might be.  Berry and his colleague Jon Keating have outlined a whole set of dynamical properties characterising this hypothetical Riemann dynamics.  And the other page deals with the thermodynamic or statistical mechanics side of things – you’ve got Julia, Spector, Mackey, who all put forward the idea that the zeta function is a partition function, which would suggest that the zeros are in fact phase transitions of something.  So these two currents of research are seemingly different approaches, not obviously compatible.  Alain Connes has begun to bridge the gap, though.  He has taken Julia’s suggestion about zeta as a partition function, shifted it into the realm of quantum statistical mechanics, and then brought in p-adic and adelic number systems, and a lot of other very deep mathematics including something called noncommutative geometry, which is about as difficult as current mathematics gets.  He’s managed to describe a dynamical system, or at least sketch out the beginnings of one, which produces the Riemann zeros as vibrational frequencies, but where the zeta function is also playing the role of a partition function, so there is a link there. 

C: Connes’ adele is an infinite-dimensional space in which each dimension is folded, so to speak, with the frequency of each prime.

MW: Yes, that’s almost it.  An adele is a generalised kind of number which contains an infinite number of coordinates, one associated with each prime number, effectively, and then an extra one, which corresponds to the continuum of real numbers.  The adelic number system embraces all of the different p-adic number systems — 2-adic, 3-adic, 5-adic, 7adic, etc.  p-adics and adeles constitute yet another aspect of number theory finding its way into physics, thereby suggesting that things aren’t the way we thought they were. The Archimedean principle, the basic principle of all measurement, is based on rational numbers, on ratios.  If you have a line segment and a longer line segment, by taking the shorter line segment and joining it end to end a finite number of times, you will always be able to exceed the longer line segment.   That seems obvious – it’s the basis on which I can take a ruler and measure this room.  If I kept joining it end to end and I never got to the end of the room, then measurement wouldn’t work very well!  So, the universe at the macroscopic scale is Archimedean: the Archimedean principle applies.  And the number system we generally use, the continuum of real numbers, is an Archimedean system.

Now, the real number continuum is based on a particular arbitrary choice of how we ‘close’ the system of rational numbers.  The rational numbers are fairly simple, well-determined, or given, if you like – canonical. You’ve got your integers, and then you start taking ordinary fractions and that fills in the gaps – it doesn’t fill in all the gaps, but it densely fills in the number line.  The ‘holes’ that remain are the irrational numbers, which can’t be expressed as ratios of integers, √2 being the one that, it’s widely believed, was first discovered, and π being undoubtedly the most famous.  But there’s not just a handful of exceptions, these irrational numbers are in some sense more common than the rational numbers. The question is, given the system of rational numbers, how do you fill in the holes, how do you seal the whole thing up?  Well, the method we’ve ended up adopting produces the system of real numbers, which is a system in which the Archimedean principle applies. And that’s based on defining the holes, the irrational numbers, as the ‘limits’ of sequences of rational numbers. But to define the limits, you have to have a sense of distance; put simply, a sequence converges when its elements get closer and closer to something, and the notion of ‘closer’ requires some sense of distance.  The sense of distance we use to define the real numbers is the obvious one: the distance between any two rational numbers on the real number line is what you get when you subtract the smaller from the larger. But that’s an arbitrary way of defining distance.  It turns out that, within the logical constraints which apply, there are an infinite number of other meaningful, consistent ways you can define what distance is, and each leads to a different notion of ‘closure’ and hence to a different number system.  So you’re still starting with the rationals, but the way you ‘fill in the holes’ is completely different, and you end up with a different kind of mathematics.  Now this was discovered by Hensel in the late 1890’s, and very quickly the possible ways of closing the rationals were classified.  It turns out that there are infinitely many of them, and that they correspond to prime numbers: there’s the 2-adic system, the 3-adic system, the 5-adic system, the 7-adic system, all the way up, and then finally there’s the ∞-adic system, which corresponds to the usual system of real numbers, and which suggests the existence of what’s called the ‘prime at infinity’, a deeply mysterious thing, which an Israeli mathematician called Shai Haran has written a whole book about5. But the point is, in a 2-adic, 3-adic or 5-adic number system, the distance between two rational numbers has nothing to do with the traditional distance between two points on a ruler anymore, rather it’s about arithmetic relationships involving divisibility of numerators and denominators by the prime p which characterises the p-adic system in question.  So things that would look very close together on a ruler could be huge distances apart, and vice versa, things that are vast distances apart in a normal Euclidean sense could be very close together in a p-adic sense.

C: And the adelic system is built up of all these?

MW: An adele is a generalised number which has an infinite number of co-ordinates.  One’s a 2-adic number, one’s a 3-adic number, one’s a 5-adic number: one for each prime.  They’re usually written as:

(2-adic number, 3-adic number, 5-adic number...; real number)

so you get one of each.  When, at the end of the nineteenth century, these p-adic number systems were discovered, it was realised that we’ve been doing all our physics on the basis that time and space are like the real number continuum.  That’s the assumption; all the Einsteinian, Riemannian, Minkowskian manifolds, spacetime manifolds, were based on real numbers extending in different dimensions.  But why should we assume the universe is ‘real’, in that sense?  You could formulate a 17-adic manifold and do space-time physics in it, or a 37-adic manifold; but then, why pick one prime rather than another?   Hence the idea arose, why not chuck them all in, create a system which involves all of them at once — this is the adelic approach, described in very crude terms. Hence p-adic and adelic physics — there are people developing models of p-adic physics where the p is just left as an arbitrary p, where it would work for any prime, basically re-building physics according to these new number systems.  So you’ve got p-adic models of time, p-adic models of probability.  A lot of it really turns your ideas of the world on their head. Now Connes has come up with a dynamical system on a space of adeles, which generates the spectrum of Riemann zeros.  The problem is that the system he’s starting with has already got the prime numbers built in to it, so some people would say, well, he’s really only reformulated the problem.  But I suspect there’s a lot more to it than that.  It’s not quite the dynamical system that is being sought in connection with RH, but it is widely seen as a valuable step in the right direction. Even more interesting than Connes’ work, from my point-of-view, is that of the lesser-known Michel Lapidus (University of California-Riverside), another Frenchman with a staggeringly broad view of mathematics and physics.  I recently had the privilege of proofreading his latest book – I hope it will come out this year, it’s been a long time in the pipeline.  It’s called In Search of the Riemann Zeros and it brings all of these ideas together.  And he’s taken Connes’ idea even further. He’s got a set of ideas involving quantum statistical mechanics, p-adics and adeles, dynamical systems, vibrational frequencies, partition functions, it’s all in there, but also fractals, string theory...

C: The adele already intuitively brings to mind string theory, because of the way everything seems to be bound up with the nature of these peculiarly convoluted spaces.

MW: There’s been a lot of work done on p-adic and adelic string theory, but that’s not quite what you mean. Lapidus has actually come up with a fascinating connection.  He was working on something he called ‘fractal strings’, but these didn’t have anything to do with the ‘string theory’ physicists study, it was just the name that he had given to these particular mathematical objects.  And then he generalised them to something called ‘fractal membranes’.  But since he came up with that, oddly enough, he’s found that aspects of string theory relate directly and unexpectedly to the mathematics.   His model involves a dynamical system, a noncommutative flow of fractal membranes in a moduli space...

PC I found the above passages instructive. I was not really familiar with thermodynamic partition functions in physics so found your explanation of Julia’s model using the “free Riemann gas” very interesting and could quickly resonate with many of the ideas expressed.

Again because I have long accepted the holistic aspect of number behaviour, I would have no difficulty at all in treating the primes as akin to physical particles possessing energy.

In fact the relationship is more intimate than this as I would see physical (and psychological) phenomena as ultimately encoded in number. So therefore from this perspective, the energy that applies to physical phenomena such as gas particles equally applies to prime numbers (as the encoded nature of these particles).

So I have already explained how when we combine prime factors they attain an energy (representing a unique qualitative resonance) and this would also imply with numbers that contain more factors a corresponding increase in energy. Though I expressed this directly in psychological terms in the manner that spiritually intuitive energy increases through the ability to “see” the interdependence of ever greater numbers of factors, this equally applies in a complementary physical manner. So the energy levels manifest in physical systems relate to interdependent shared components of the system (such as the interaction of gas particles) which ultimately are encoded in dynamic prime number configurations.

Given that that the gas particles have a charge of log p it is not surprising that the partition function would be represented by the Riemann zeta function. And as we have seen, the distribution, representing both the frequency of primes and of zeta zeros, is intimately connected to this simple log function.

In fact it is interesting to observe how the log function is used in two complementary ways, which again points directly to the true dynamic nature of the number system.

For example we can use log n to approximate the average spacing or gap as between successive prime numbers. And as we know this gap increases when we ascend the natural number scale.

So in the region of 1000 this gap would approximate 7. Then in the region of 1,000,000 it would have effectively doubled to 14.

However log n can equally be used to approximate the average total of factors i.e. divisors, contained by a number (where all factors are included).

Thus from this perspective, a highly composite number such as 12 would contain 6 factors i.e. 1, 2, 3, 4, 6 and 12.

When one studies these two uses of log n, one can better appreciate their complementary nature.

The former use does not refer to number as such but rather the spatial dimensional aspect of number. The latter use then directly refers to numbers (as factors) without reference to any spatial intervals.

And this is a key dilemma with respect to number which you have explained very well both in this discussion and in “Secrets of Creation”. When we express natural numbers as the multiplication of prime factors they appear as awkward clusters that do not relate well to the ordered notion of the number line. So from this perspective we attempt to express the natural numbers without reference to the space intervals that exist between primes. Therefore even though the multiplication of numbers necessarily involves a qualitative dimensional aspect, as we have seen, this is ignored in the reduced quantitative interpretation of number, thereby enabling the composite natural numbers to be placed on the real number line.   

Then when we alternatively express the natural numbers through the Peano-based approach, where each new natural number is derived from the successive adding of 1, we are now attempting to derive all such numbers with reference to ordered space intervals (of 1 unit). However this ignores the key issue of how prime numbers can be quantitatively related with each other.

So as we have seen, we cannot explain the quantitative aspect of number independence (without including the qualitative aspect of interdependence).

And we cannot explain the qualitative aspect of number interdependence (without the quantitative aspect of independence).

Thus addition and multiplication constitute dynamic complementary activities with respect to the number system, which can only be properly appreciated through a corresponding dynamic mode of interpretation.

Thus once again the two uses of log n with respect to the fundamental nature of number point directly to this complementary relationship.

With reference to the partition function of Julia’s thermodynamic model, you refer to the singularities as phase transition points.

From the psychological perspective we have seen how each singularity (as a successive zeta zero) represents a heightened ability enabling a greater degree of sustained spiritual energy. So from this perspective, one can literally embrace a new phase of harmonious activity, whereby one is enabled to instinctively engage with sense phenomena in a more intimate manner.

Thus from the physical perspective each transition point can be seen in complementary fashion as representing a heightened degree of energy of the system that can be properly sustained with respect to the increasing degree of activity of the gas particles.

The one point at s = 1, in holistic terms represents the standard mathematical approach, which by its very nature is totally unsuited to the dynamic appreciation of number. So not surprisingly, its very method of interpretation unravels when faced with such dynamic considerations, just as the physical system with the Hagedorn catastrophe equally races out of equilibrium in these circumstances.

You mention the approach of Alain Connes and how in some respects it extends Julia’s classical approach to partition functions.

Connes is clearly a very brilliant mathematician who has developed an exciting new framework in his bid to solve the Riemann Hypothesis.

However though it purports to be a dynamic approach, I would see this as true only in indirect terms i.e. where dynamic considerations are investigated in a quantitative manner through accepted mathematical notions (that inherently lack any true dynamic quality).

Though of course I am not qualified to comment on the analytic nature of his highly specialised abstract approach, I can indeed see some interesting parallels with the holistic approach that I have been adopting in recent decades.

So Connes in a certain sense recognises the limitations of the assumptions we make regarding the real number line, where, as you have explained in the discussion, involves just one possible interpretation regarding the notion of distance as between different rational points.

So he shows how an unlimited number of distinct interpretations of distance can be given through the p-adic number systems, i.e. 2-adic, 3-adic, 5-adic, 7-adic and so on.

Thus all these systems, including one final member relating to the real continuum i.e. 1-adic, comprise the adelic approach to number that is necessary for his new interpretation of the zeta function.

Now some time ago, I had come to a somewhat similar notion directly based on a holistic rationale.

I recognised clearly that the current interpretation of the real line and consequently the number system entails 1-dimensional interpretation (which is absolute in nature). Again this is due to the fact that here the qualitative shared aspect of number is completely reduced in quantitative terms.

So, just as the primes in quantitative terms can be used as building blocks of the natural numbers, likewise in holistic terms we can build an entire new set using the primes for interpreting number relationships (all of which possess a partial dynamic relative validity).

So for example the 2-adic system from this holistic perspective relates to the simplest form of dynamic relationship based on the interaction of positive and negative polarities.

What is important to then realise is that all the key assumptions we currently make regarding the number line, no longer apply when interpreted in a 2-dimensional fashion.

So already 2-dimensional appreciation of the number line is inherently dynamic entailing both the analytic (quantitative) and holistic (qualitative) aspects of number.

All the higher p-adic systems can then be seen to represent ever more refined configurations of the manner in which number dynamically interacts.

A more advanced Type 3 mathematical understanding, which I will not even attempt here, would then show how the holistic and conventional interpretations of p-adic systems can be successively reconciled with each other!

However the key weakness of Connes’ approach from my standpoint is that despite constituting a brilliant analytic attempt to capture the dynamic nature of the zeros, it is crucially flawed in remaining restricted to a mathematical approach, which by its very nature is devoid of such dynamism. Thus whereas I would draw a crucial dynamic distinction as between the 1-dimensional and other p-adic number systems that are analytic and holistic (and holistic and analytic) with respect to each other, Connes attempts to treat all in a reduced analytic manner.

Thus I would see Connes as essentially offering a restatement of the Riemann Hypothesis. Admittedly however it is now dressed up in new clothing that perhaps is suggestive of the need for Mathematics itself to discover its true dynamic nature (through formally recognising its long neglected holistic aspect).

And as I have already suggested, once the holistic aspect is properly recognised, it should become quickly apparent why the Riemann Hypothesis cannot in fact be proved (or disproved) in the conventional mathematical manner.

I would make the same general criticism regarding the dynamic approach of Michel Lapidus.

Like Connes he is undoubtedly a brilliant mathematician in the accepted analytic sense with a wide interest in physics related issues.

You say that his model involves ‘a dynamical system, a noncommutative flow of fractal membranes in a moduli space...’ and also say that he is working on ‘fractal strings’ but that these have nothing to do with the conventional conception of strings.

These very descriptions of his work indicate a key difficulty with the field and indeed with the increasingly specialised abstract findings of mathematics generally.

At the conventional Newtonian level of reality, scientific and mathematical results are accessible to general understanding, precisely because they conform to our everyday intuitive assumptions as to how reality behaves.

However because mathematics and by extension the sciences completely lack a holistic aspect, modern findings have become largely inaccessible to even the highly educated layperson. Indeed outside knowledge of the specialised analytic techniques used to generate their results, the deeper meaning I believe likewise remains largely inaccessible to the mathematicians and scientists concerned.  

For example this is very true with respect to physical developments such as quantum mechanics and string theory.

So, modern physics has been in search of the “Theory of Everything”. However, even if such a theory could be found, it would not resonate with our understanding of reality in any meaningful sense. And the reason for this is the complete lack of any holistic dimension to interpretation.

Therefore this fundamental problem cannot be resolved through acquiring an increasingly specialised knowledge of string theory.

For the real issue relates to the fact that though there are many differing levels of reality, the mathematical and scientific approach remains confined in its methods and interpretations to just one of these levels.

To give a useful analogy, just as there exists a spectrum of electromagnetic energy in physical terms with many different bands (of varying wavelengths and frequencies), equally there exists a psychological energy spectrum with many bands, where the nature of intuition varies considerably.

Conventional mathematics has been built up on just one of these bands (akin to that of natural light) requiring by default the common sense form of intuition to fit its rational explanations.

However in modern times we have been attempting to deal with many important problems that fundamentally relate to differing levels of the spectrum.

Thus when mathematical interpretation stubbornly remains confined to just one level, its specialised methods then yield results that become increasingly non-intuitive in holistic terms.

You even directly refer to this problem later with respect to Lapidus ‘this very strange highly counterintuitive, noncommutative geometrical ‘flow’’.

So I would counter in the same way as with Connes that the key problem here relates to the restricted nature of conventional mathematical interpretation that is formally confined to its analytic aspect.

Thus there is little difficulty in appreciating the dynamic flow of the primes once one recognises the holistic aspect of mathematical interpretation. This holistic understanding is already deeply implicit in the very manner we experience number. However for millennia we have attempted to abstract number in an absolute quantitative manner (which fundamentally distorts its true nature).

Now again, Lapidus may help to indirectly facilitate eventual acceptance of the key fact that the true nature of number is inherently dynamic. However this will not come easily as it will require a radical conversion of the entire mathematical community.