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Peter CollinsPeter Collins is from Ireland. He retired recently from lecturing in Economics at the Dublin Institute of Technology. Over the past 50 years he has become increasingly convinced that a truly seismic shift in understanding with respect to Mathematics and its related sciences is now urgently required in our culture. In this context, these present articles convey a brief summary of some of his recent findings with respect to the utterly unexpected nature of the number system.

Part I | Part II | Part III

Recognising the Shadow

A Reply to Elliot Benjamin

Peter Collins


Firstly I want to sincerely thank Frank for posting my latest articles (Dynamic Nature of the Number System, Part I, Part II and Part III) on the Integral World website.

I also want to thank Elliot for engaging in the discussion (thereby displaying a generosity of spirit that would not always be associated with his professional colleagues).

One of the advantages of a discussion is that it enables one to talk more informally about issues, providing further context for the articles and thereby perhaps more easily conveying to the general reader the nature of the issues involved.

So what we are talking about here is the fundamental nature of our number system!

Indeed it would be difficult to imagine a more important topic, as it serves as the corner stone of Mathematics and by extension all the sciences.

Though we may not directly advert to this, the very way we tend to view reality (especially in a scientific sense) is fundamentally conditioned by our cultural assumptions regarding number.

Now what may come as surprising news to many is that recent developments within Mathematics and Physics are now strongly suggesting that the number system is significantly different from what we previously imagined.

This really started in 1859 when Bernard Riemann showed that underling the customary natural number system (i.e. the everyday counting numbers 1, 2, 3, 4….) is an amazingly intricate alternative harmonic wave system, which in some mysterious way seemed to provide a perfect synchronicity as between the behaviour of individual prime numbers (2, 3, 5,..) and their overall collective relationship to the natural numbers.

This is all very reminiscent of the nature of quantum physical reality at the sub-atomic level, where matter reveals itself in the form of both particles and waves.

However, it was still too early to make this connection as between the number system and quantum mechanics. It was not till the 1920's that the strange new behaviour of the quantum world emerged! Unfortunately Mathematics had become much more isolated from Physics at the time of the quantum discoveries. So basically the relevance of these radical new physical findings for the world of number was ignored by the mathematical community.

However from the 1970's, dramatic new evidence started to emerge that the famous Riemann zeros (giving rise to the wave nature of number) showed an uncanny resemblance to the physical energy states of certain quantum systems. This was then narrowed more precisely to quantum chaological system (i.e. quantum systems with an underlying classical basis in the theory of chaos).

So it is now undisputedly accepted that the Riemann zeros are spectral in nature (representative of energy states). And the search is on to find the dynamic system (of a physical or quasi-physical nature) whose energy states perfectly match these zeros.

However what I find amazing is how the obvious message that the number system itself is inherently dynamic in nature is being missed. So what the physicists are really searching for is simply the number system!

There is an enormous mental block out there (conditioned through millennia of a fixed quantitative view of the number system) to making this transition to an inherently dynamic appreciation of number.

So the implications are very clear. If the natural number system is indeed fundamentally different from what we imagined - as is now seems inevitably the case - then the implications are truly enormous for our understanding of Mathematics, for all the sciences, our place in the world and our relationship with each other.

As stated in the articles, I believe we will soon face the most radical intellectual revolution in our history which will entail massive social consequences.

And we need urgently to start preparing for this transition!

Reply to Elliot

I will now address some of Elliot's points raised in his reply with a view to further clarification of the issues he raises.

Elliot mentions the simple example I give of multiplying 2 * 3.

He accepts the validity of expressing the result in geometric terms as square units, However he is avoiding the key point made, in that this was used to indicate how a qualitative (i.e. dimensional) change has thereby taken place with respect to the nature of the units.

So therefore we cannot think of even the simplest case of multiplication without a qualitative transformation being involved.

I found it interesting that Elliot never mentions the word “qualitative” in this exchange, though it is really key to appreciating the relevance of the issue.

You see, the difficulty is, that once we squarely (excuse the pun) accept that we cannot multiply two numbers, without a qualitative change being involved, this inevitably raises the question of how this qualitative notion can then be accommodated within mathematical interpretation. And to put in bluntly, the conventional mathematical approach is to simply ignore the issue altogether, by studiously avoiding any reference to qualitative notions with respect to fundamental operations such as multiplication!

I also dealt with several related perspectives, where qualitative notions are inevitably involved with respect to all mathematical operations. For example, the notion of “interdependence” is qualitative in nature. Therefore the very fact of bringing any two numbers into a relationship with each other implies a qualitative aspect. Also, the infinite notion (as we shall discuss) and the related use of intuition in mathematical understanding are properly qualitative. Finally the ordinal notion of number is also qualitative.

And again, let us be totally clear about this! One cannot deal satisfactorily with the qualitative aspect of number in a merely quantitative manner. The mistaken belief that it is indeed possible represents the strongly reduced approach that pervades all conventional mathematical relationships!

Now, I show in my article how the quantitative and qualitative aspects in fact relate to two distinct notions of number (which I refer to as Type 1 and Type 2 respectively).

So incorporating both these aspects entails a dynamic relative context where both Type 1 (cardinal) and Type 2 (ordinal) interact. This happens naturally in experience. But this experience is then severely reduced in conventional mathematical interpretation.

Of course I have no objection to a Type 1 approach to number (with respect to mere quantitative understanding). It is perfectly appropriate in its rightful context. So I am certainly not trying to eliminate this as a valid field of enquiry as Elliot suggests. Indeed, I believe that it could ultimately become greatly enriched through being understood as representing just one component of a more comprehensive and balanced approach to mathematical enquiry.

However just as Newtonian Physics would now be seen as an approximate approach to reality that works very well at the macro level of reality, but then breaks down at the sub-atomic, equally the reduced (Type 1) quantitative approach to Mathematics should be seen in the same light.

So, when we look at the basic assumptions of this approach at a more refined level of investigation, they do not hold up to scrutiny. This then becomes especially important when one examines more fundamental problems such as the Riemann Hypothesis (which entails the basic relationship of addition to multiplication).

Admittedly you won't hear many saying this!

Those inside the mathematical community, by definition accept the basic assumptions. And the comments of someone like me, who would not be considered a valid member, are in the main just ignored and treated as totally irrelevant.

Effectively therefore, the mathematics profession does not enable criticism of its rationale from either inside or outside its ranks. This is an extremely unhealthy situation for both the profession itself and for society in general and needs to be strongly challenged as I am attempting in this article.

So again I thank Frank for allowing me the space to make such comments. And also I give great credit to Elliot in this regard. He might not like my comments (which is fully understandable); however he has not ignored them.

With reference to “the music of the primes” I thing that Elliot is perhaps misreading my intentions (though I would not blame him for this as it is just a footnote to a long article). The point here is this! Clearly there is a legitimate quantitative role in the interpretation of music with respect to the analysis of structure, chords, notes, instrumentation etc. However the direct experience entails a qualitative aspect, in appreciation of the way all these separate aspects are holistically combined.

Even though mathematicians frequently refer to the “music of the primes”, they insist on an interpretation that is merely quantitative in nature. A qualitative holistic aspect of mathematical appreciation is likewise required. However again because of quantitative conditioning this obvious inference is repeatedly overlooked!

Elliot then goes on to defend the conventional treatment of the infinite notion in Mathematics. But once again appropriate interpretation is akin to scaling with respect to a map.

If once again we are content with a reduced quantitative interpretation which defines the Type 1 approach, then Elliot's views are acceptable.

However with a finer level of scaling, we need to recognise the dynamic interplay of two distinct notions that are quantitative and qualitative with respect to each other.

Elliot says “but I must confess that I do not see a problem with defining “infinitely many” as “more than any natural number”

The problem here is that this is strictly a meaningless statement in actual terms.

For a number to be more than any natural number (that is finite), then such a number must be identified in actual terms. So if Elliot wants to suggest any natural number, I will provide a larger number that is also finite in nature. So to avoid this situation, the proposition of any natural number implicitly requires reference to a potential i.e. infinite notion of number which cannot be actually identified in finite terms (and thereby entails circularity with respect to his definition).

The problem is that finite and infinite notions properly are quantitative and qualitative (or alternatively actual and potential) with respect to each other.

This likewise concurs with experience that combines both rational (finite) and intuitive (infinite) aspects.

Put another way, from a psychological perspective, the relationship of finite and infinite (in any mathematical context) entails the interaction of both conscious (rational) and unconscious (intuitive) aspects of understanding.

So the bigger issue for mathematical interpretation is this prior relationship of the consistency of two distinct domains of understanding (that again are quantitative and qualitative in terms of each other). From a psychological perspective it entails the consistency of both conscious and unconscious notions with respect to all mathematical understanding.

This is a vitally important point to which the Riemann Hypothesis (when appropriately interpreted) directly relates.

Since it precedes mathematical proof, the Riemann Hypothesis cannot be thereby proved (or disproved) in a conventional manner. In other words, by its very nature, it thereby transcends the domain of recognised mathematical interpretation.

And all of this can only be properly recognised when one subjects finite and infinite notions to proper scrutiny.

Now at the Type 1 level of interpretation, we can of course blur these distinctions in the way that Elliot outlines. But again the analogy of Newtonian Physics and Quantum Mechanics is relevant here.

Thus an approximation that works well in conventional terms, can be shown to be quite inadequate at a more refined level of interpretation.

Elliot is anxious to convey to me that mathematical proofs can represent an aesthetic art form (and that he himself derives considerable aesthetic enjoyment from his pursuit of mathematical problems).

I do of course recognise this! However in a sense, it once again begs the question of considering Mathematics as a rational mode of enquiry. Clearly there are considerable non-rational aspects also involved which should be formally recognised in a more comprehensive mathematical appreciation.

However this is not the case at present where it is viewed in a merely rational manner.

So my key point again is that ultimately the proper appreciation of Mathematics is inseparable from the full process of human development, entailing rational (cognitive), affective (aesthetic) and spiritual (intuitive) elements.

In respect to Elliot's comments on the square root of a number, they perhaps manage to trivialise my position (and the importance I attach to it).

I could sense from a very early age that there was something gravely inadequate in the conventional mathematical interpretation of a square root. Then eventually - as I describe in the first article - I found myself able to precisely provide - at least to my own satisfaction - a coherent explanation.

I discovered in fact that the square root (of any number) cannot be properly explained in conventional mathematical terms. As I put it in my articles such Mathematics is based solely on 1-dimensional interpretation (where in any context one pole is solely allowed as a valid reference point).

However explaining the - apparently - simple operation of a square root requires 2-dimensional interpretation.

In this context, Elliot goes on to give the famous quantitative proof (by contradiction) of the square root of 2 (as again demonstrative of the art form of Mathematics).

The irony of the situation here is that I referred to this very proof at some length in my last mathematical article on this web-site (which was reviewed by Elliot).

However there is a much deeper issue here which perhaps he is overlooking.

As we know the finding that the square root of 2 is irrational had a devastating effect on the Pythagorean School. This arguably was a major factor in subsequently leading to an unbalanced obsession in Western Mathematics with mere quantitative type enquiry (which has now reached a very unhealthy extreme).

What is interesting is that the very fact that the Pythagoreans accepted that the square root of 2 is irrational, implies that they had already formulated an acceptable quantitative proof (possibly along the lines of the famous proof in Euclid).

But clearly they did not take much consolation from this discovery. In other words the key problem remained that they could not provide a coherent qualitative explanation on philosophical grounds of why the square root of 2 is irrational!

So at this stage of the development of Mathematics, it was accepted that a correspondence should be maintained as between the quantitative (analytic) and qualitative (holistic) aspects of enquiry. And this balance was fatally undermined by the failure to understand the qualitative nature of the simplest of irrational numbers.

Now if we fast track to the present time, this qualitative aspect of appreciation - which the Pythagoreans saw as so important - has all but been eradicated (certainly in formal mathematical terms).

Once again, I say that this situation is extremely unbalanced and unhealthy both for Mathematics as a discipline and for society as a whole (which is so influenced by mathematical conventions).

Interestingly, Elliot refers to my Band 6 level of enquiry and what it might entail in terms of such a proof.

Well, I would suggest that there are three key elements involved here.

First of all, we must provide the standard (Type 1) analytic proof of a quantitative nature. And as he has demonstrated, an ingenious proof of this nature has long been available.

Secondly we must provide an additional (Type 2) holistic “proof” of a qualitative nature. Just as the Type 1 depends greatly on reason, the Type 2 is more heavily dependent on refined intuition (expressed in an appropriate circular rational manner).

In a society which valued intuition, a consensus would soon emerge as to what constituted an acceptable “proof” in this regard. Now the Pythagoreans searched in vain for such a qualitative proof.

When I embarked on my holistic mathematical journey, one of the first tasks I set myself was to provide this “missing” qualitative explanation of why the square root of 2 is irrational. I outlined this search in that article on mathematical proof already mentioned. Anyone interested might also like to look at this blog entry “The Pythagorean Dilemma”.

Thirdly, we most establish a two-way correspondence in Type 3 terms as between the Type 1 (quantitative) and the Type 2 (qualitative) aspects of proof.

Now I could perhaps go part of the way with respect to this final requirement (as it is a problem I have considered at length).

However, needless to say with respect to the notion of mathematical proof generally, we are still light years away from Band 6 requirements.

If we could imagine however, some future golden age of comprehensive mathematical enquiry, the implications would be enormous.

It would entail for example that one who had formulated an acceptable quantitative type proof (in any area of Mathematics) would then be enabled, through the nature of understanding thus attained, to clarify its holistic experiential implications (in both physical and psychological terms). So at this level there would be no such thing as pure abstract Mathematics, as the consequences of every proposition in principle could be translated in an experiential intuitively meaningful fashion.

It would also entail from the corresponding side that the holistic appreciation of mathematical propositions would then lend themselves readily to the formulation of analytic type proofs.

So Mathematics at this level would be both amazingly productive and creative in a balanced dynamically interactive manner.

What Elliot perhaps does not sufficiently realise is that the very belief among mathematicians, that highly abstract work can have a meaning independent of applications to the real world, itself stems from the limited absolute type assumptions on which such work is based.

From a more balanced perspective, where all mathematical relationships are defined in dynamic experiential terms, this belief in pure abstraction would quickly fade, with every mathematical proposition potentially possessing applications in both physical and - most importantly - psychological terms. Indeed my own route to understanding of the holistic nature of the Riemann zeta zeros, initially owed more to an internal psychological - rather than external physical appreciation.

It is true that it would take an extremely advanced holistic type awareness to provide the experiential meaning (in physical and psychological terms) of Elliot's stated example of “proving that the Hilbert 2-class field is infinite for imaginary quadratic fields with 2-class group of rank of rank 4-class rank 2 in certain cases” but mark my words, in principle it can be done and one day given sufficient evolution in our mathematical development will be achieved.

In fact in the 3rd article I outline a new appreciation of number where all phenomenal processes (physical and psychological) are inherently encoded in number (as their root nature). This would imply that any mathematical manipulation of symbols (though apparently abstract in nature) would be associated with a corresponding dynamic configuration of phenomenal reality (again in both physical and psychological terms).

Now it might indeed be exceptionally difficult in many cases to successfully decode abstract mathematical notions with respect to such reality, but again in principle it can be done.

We sometimes forget that we are still, in all probability, at a very early stage in our human evolution. So what will be possible in mathematical terms in future eras is currently unimaginable (though this hasn't deterred me from pushing the boat out somewhat further than present limited conceptions).

Elliot expresses his love again for Mathematics, which I accept is totally genuine and sincere.

And I am sure if his many excellent contributions (on a wide variety of topics) on this site are anything to go by, I am sure that his work as a mathematician is indeed top rank..

However, we should not lose sight of the deeper issue with respect to the overall nature of Mathematics i.e. that it has indeed become - especially in recent times - hugely unbalanced.

I think that Elliot would agree that we would now find it totally unacceptable to attempt to explain all psychological behaviour merely from a manifest conscious perspective. Since Freud there has been a growing recognition of the importance of the unconscious in influencing all human behaviour.

Now, when one attempts to view Mathematics in a balanced manner, giving equal recognition to both conscious (quantitative) and unconscious (qualitative) aspects, it then equally becomes totally unacceptable to attempt to explain all mathematical relationships in a solely rational manner.

So using Jungian language, there is an enormous unhealthy shadow collectively hanging over the Mathematics profession at present in the complete failure to formally recognise its equally important holistic dimension (which is directly of an unconscious nature).

What makes this even more damaging is that Mathematics as a discipline insists on maintaining itself within a hermetically sealed chamber, safely insulated from viewpoints such as expressed in this article.

The basic attitude that prevails is that anything not in keeping with unquestioned assumptions is thereby irrelevant from a mathematical point of view.

Once again this is an extremely unhealthy position both in terms of the development of Mathematics and with respect to its influence on society generally.

I believe that Integral Studies could be very important in providing an overall spectrum (from which a more comprehensive approach might emerge (leading to the recognition of distinctive types of Mathematics at different stages).

So from this standpoint, a great limitation of accepted Mathematics - with which I imagine Elliot might reluctantly agree - is that it is totally lacking such a development perspective.

Elliot surely recognises that conventional mathematical interpretation effectively is limited to the 2nd Band (in terms of my Spectrum)!

So if he does indeed recognise the existence of many further possible bands on the Spectrum, is he seriously trying to suggest that valid understanding should remain at Band 2? Can he not see that this is quite incompatible with a comprehensive all-band approach?

Or in terms of the close analogy with the electromagnetic spectrum, is he intent on insisting that natural light alone is relevant and that all the other bands e.g. gamma, x-ray, infra red, microwave and radio wave radiation can thereby be ignored?

Again in fairness I would see Elliot as considerably more open-minded and intellectually rounded than many of his professional colleagues.

However, even he, with a demonstrated commitment to Integral Studies, still apparently finds it difficult to recognise that the Spectrum of Development applies as much to Mathematics as spirituality. This therefore says a great deal about the way we are so strongly conditioned to treat this discipline as a special case, exempt from any fundamental criticism of its limited rationale.


I would like to return once more to this new vision of the number system by summarising some of its key features.

Firstly it is inherently of a dynamic nature, entailing the complementary interaction of both quantitative (cardinal) and qualitative (ordinal) aspects. It equally entails the complementary interaction of an external aspect to number (as objective) and a corresponding internal aspect (as mental interpretation).

There are two extreme poles, which we can identify with respect to this number system.

At one extreme, we have the long accepted analytic interpretation of the natural number system in an unchanging absolute manner.

From this perspective, pure number theory is seen as a pursuit that can be abstracted from human experience (though implicitly it must always remain an unrecognised part of such experience).

The key feature of such analytic interpretation is that the qualitative aspect of number is reduced in mere quantitative terms.

There are of course two aspects to the natural number system i.e. cardinal and ordinal (though in analytic terms, the ordinal is again simply reduced in a cardinal manner).

At the other extreme we have the - as yet - unrecognised holistic interpretation of this system as represented by two sets of zeta zeros (Zeta 1 and Zeta 2) in an incredibly dynamic manner. The Zeta 1 (Riemann) zeros are admittedly recognised in analytic terms. However the key to appreciation of their true nature is holistic in nature where they are seen as complementary with the (unrecognised) Zeta 2 zeros!

So whereas analytic interpretation entails the rigid extreme of number as absolute type form, the zeta zeros represent the other extreme of number as representing pure energy states (with both physical and psychological manifestations) that lie on the very borderline of formless ineffable reality.

The key aspect of these zeta zeros is that they closely approximate - as far as is possible in the relative finite realm - the perfect harmony in the number system of both its quantitative (cardinal) and qualitative (ordinal) aspects.

In this new perspective, the primes and natural numbers are ultimately seen as perfect mirrors of each other. From the Type 1 (cardinal) perspective the natural numbers appear to depend on the primes; then from the Type 2 (ordinal) perspective, the primes seem to depend in reverse on the natural numbers!

So the key significance of the primes and natural numbers with respect to the number system is that they enable the two-way transmission of both its quantitative and qualitative aspects.

Now the role of the zeta zeros is truly critical and can in fact be explained quite easily.

Imagine we are at an EU summit meeting where two important politicians meet from - say - Italy and Germany! Imagine also that each politician only can understand in his/her native language!

For effective communication to take place, once the Italian politician speaks, the language will need to be converted accurately by a translator into German for the other politician to understand. Equally when the German speaks, this time the language will need to be converted into Italian for the second politician to understand.

Now this brings us back to the fundamental relationship as between multiplication and addition. As we have seen, we can identify pure addition with the Type 1 aspect of the number system and pure multiplication with the Type 2 aspect respectively.

The trouble is that these two aspects of the number system are initially incompatible, so that they - literally - cannot speak with one another. Therefore in a truly wonderful, and ultimately fully mysterious, manner, the zeta zeros solve this problem with respect to such incompatibility.

So the Zeta 2 zeros (which are easier to intuitively grasp) provide the means of translating the Type 2 aspect of the number system in a Type 1 manner. Then in a reverse fashion, the Zeta 1 (Riemann) zeros provide the corresponding means of translating the Type 1 aspect of the number system in a Type 2 manner.

So the analogy with the problem of communication using different languages is very apt.

However the significant difference is that language translation is a somewhat inexact process. How often for example do we hear of an important intended meaning being “lost in translation” possibly leading to considerable misunderstanding?

However for communication to take place with respect to the number system (in relation to its two aspects) the translation provided by both the Zeta 1 and Zeta 2 zeros must be completely precise. If this was not the case, then we could never guarantee consistency with respect to the conventional use of number (in either cardinal or ordinal terms).

Thus, put more simply, the zeta zeros play a vital direct integral role with respect to the operation of the number system.

Using Jungian language, from a psychological perspective, the zeta zeros represent the perfect (fully recognised) shadow of mere analytic type understanding of the number system, whereby the (unconscious) holistic nature of number is brought fully to (conscious) light.

So once again an enormous collective shadow hangs over Mathematics as a recognised discipline, in its complete refusal to formally recognise its (unconscious) feminine side in the holistic appreciation of its symbols.

It has to be said again that this situation is hugely unbalanced and thereby hugely unhealthy. And it is a very sad reflection that it requires someone standing outside the profession to recognise this problem clearly.

As the zeta zeros, require specialisation with respect to holistic - rather than analytic - appreciation, their true relevance therefore cannot be properly understood in conventional mathematical terms.

The fundamental importance of the number system is greatly enhanced in this new dynamic appreciation. Not alone is number now viewed with respect to quantitative notions of order and measurement but equally with respect to all qualitative phenomenal notions.

Put quite simply, number is now seen as the encoded nature of all phenomenal reality (in both physical and psychological terms). Phenomenal reality in turn is simply seen as the decoded nature of the number system.

So through evolution, we have continual transformation both with respect to number and phenomenal reality.

I sometimes imagine that out there in the Universe where more highly developed intelligent beings have in all probability evolved that they will have long since discovered this fundamental relationship between the number system and reality.

And then if communication was to effectively take place with other evolved intelligent beings, the common currency would inevitably be this number system. In that sense there is nothing more fundamental, nothing more important in phenomenal existence.

And you might ask can we go beyond the zeta zeros? Where did they come from?

Well! Here we are on the borderline of complete mystery.

We need to remember however that ultimate knowing does not culminate in some clear cut rational explanation of the nature of reality. Rather it leads us to the point where we readily abandon all remaining vestiges of thought and simply surrender to the mystery.

I think that there is indeed a further layer of possible enquiry - of necessarily a highly elusive transparent nature - where the relationship between the natural number system and the primes itself is embedded in a prior relationship entailing the binary numbers 1 and 0. But that itself will inevitably be shrouded in even deeper mystery.

Finally, I want to sincerely thank Elliot again for engaging in the discussion in such a passionate manner showing that he cares deeply about what is at stake.

I hope that the sharp contrast with respect to our positions will help to better clarify issues raised for the general audience.


Elliot Benjamin (2013) The Art of Mathematics,

Peter Collins (2013) Dynamics Nature of the Number System, (I): Mathematics at a Crossroads; (2): Holistic Role of Zeta Zeros; (3): The Bigger Picture.

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