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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

The Remarkable
Euler Formula

Part 3. Concluding Remarks

Peter Collins

I must say that I feel especially privileged to have been able to write about both the Euler formula and the Riemann zeros in recent months.

One distinction we can make between them is that the former can be treated as an equation, while the latter more commonly is viewed as a remarkable object (representing a very special set of complex numbers).

Whereas the Euler formula would be likely to win the mathematical Oscars for “best equation”, likewise the set of Riemann zeros would be equally likely to win for “best object” (with the Riemann Hypothesis additionally winning the “best unsolved problem” category). These relate to the fundamental nature of the number system, which ultimately represents the foundation for the rest of Mathematics.

However a considerable amount of mystery has always been associated with these topics with many prominent mathematicians indicating through informal remarks that they contain a much deeper meaning (than can be interpreted in strict mathematical terms).

I already mentioned the great German mathematician Hilbert who considered the zeta (i.e. Riemann) zeros not only the most important problem in Mathematics but absolutely (the most important).

Another telling quote associated with Hilbert relates to the importance of the Riemann hypothesis (with which the zeros are intimately connected).

“If I were to awake having slept for a thousand years, my first question would be? Has the Riemann hypothesis been proven?

One final quote of Hilbert which I think is very relevant in the context of the discussion between Andy and Elliot on Mike Hockney's “The Mathematical Universe” is the following:

“The art of doing Mathematics consists in finding that special case which contains all the germs of generality”. (See quotations of David Hilbert)

And the importance of the Euler identity in this context is that it certainly can lay claims as the most special of all “special cases”.

As I have long been developing my own holistic approach to Mathematics - directly based on the contemplative vision of reality - some years ago, I began to consider that it might throw new light on these two key areas (i.e. the Euler formula and the Riemann zeros). What ultimately transpired has surpassed my wildest expectations leading to - what I now consider - a truly coherent dynamic understanding of the number system (radically different from what is presently accepted).

Clearly it is not possible - or even appropriate - to fully communicate the details of such a vision on a forum such as this. However, as it intimately concerns the integral approach to Mathematics, I believe that it is important to attempt to convey at least the general nature of my findings. Unfortunately it is usually considered taboo to attempt to discuss such important issues (outside the recognised channels).

However these for the most part are not really concerned with the deeper matters surrounding the discipline e.g. its unquestioned quantitative assumptions and impact on science and society generally.

So with respect to “Integral Studies” we already have had much discussion on topics such as psychology, philosophy, spirituality, politics, and to a degree economics on this forum. And a considerable amount of recent discussion is now taking place with respect to physics and evolution (which I welcome). Even Mathematics has been getting a look-in as evidenced by Andy's and Eliot's recent contributions.

On a personal note, I am deeply grateful for being granted the opportunity to raise on several occasions the notion of a distinctive “integral mathematics” and what this might entail with respect to future understanding of Mathematics and the scientific disciplines. 1 And once again I would like to pay special tribute to Frank for the wonderful service uniquely provided through his site that facilitates the discussion in an open manner of such integral insights (with potentially a key relevance).

What is not generally known - even among the bulk of professional mathematicians - is that remarkable new findings with respect to the number system have emerged in recent years. In a manner somewhat similar to physics last century, these are destined to eventually lead to an even more profound “quantum type” revolution in Mathematics.

What is now accepted with certainty is that underlying our customary natural number system (of discrete integers) is another amazingly intricate harmonic wave system, with our everyday use of number intimately depending on this previously unrecognised system.

One other point (relevant to the recent discussion) is that these intricate waves, represented by complex numbers that comprise the zeros, can all be decomposed precisely in terms of their various constituent “notes” using Fourier analysis.

There is no real issue among the mathematicians familiar with this area regarding the significance of these developments.

Where however I would part company with their approach, is in the continued attempt to interpret all these findings within the accepted static analytic framework that has defined number theory now for millennia. I would see this as equivalent to a physicist still attempting to interpret new findings relating to the quantum world through Newtonian concepts!

My own conclusions on the matter, which have been crystallising now over many years, point to a radical new interpretation of the number system. A brief outline of this was given in the concluding remarks in my last reply to Elliot.

One important consequence of this new vision is the need for acceptance that implicit in our everyday conscious interpretation of number, is a deep holistic unconscious basis (of which we are all but totally unaware).

Just as maturity in psychological terms entails slowly bringing the shadow to conscious light (through gradually revealing all hidden phenomena in the unconscious), mature mathematical understanding will likewise require bringing its own hidden unconscious basis to light.

What is vital to appreciate is that this will not be at all possible within the restricted confines of the present mathematical paradigm (which is formally based on mere conscious analytic notions).

Rather it will require incorporation with an utterly distinctive type of holistic mathematical understanding. This in turn will require from a psychological perspective, a much more comprehensive approach entailing the full integration of both conscious (rational) and unconscious (intuitive) aspects of interpretation. I can say this with considerable confidence, as I have been in the process of developing such holistic appreciation for some 50 years.

What I eventually discovered is that when properly interpreted, the Riemann zeros relate directly to holistic (rather than analytic) mathematical notions! Likewise the simple result of the Euler identity relates to a holistic - rather than analytic - notion of number. So a comprehensive understanding of number - as indeed of all mathematical and scientific notions - requires an inherently dynamic approach (where both analytic and holistic aspects of appreciation are combined).

So the helplessness that many mathematicians feel when attempting to understand both the Euler identity and the Riemann Hypothesis (with its associated zeros) at bottom derives from an analytic approach that is quite unsuited to the task of proper comprehension.

The Euler identity and the Riemann zeros are in fact intimately connected.

Once again the Riemann zeros arise in the context of attempting to understand the relationship of the primes (such as to 2, 3, 5, 7,…) to the natural numbers. So all natural numbers (other than 0 and 1) can be expressed as the product of prime numbers!

Though the relationship as between primes and natural numbers is highly elusive, it is now possible in principle, using the zeta zeros, to exactly predict the number of primes (and indeed position of these primes) up to any natural number.

In other words the zeros serve as an essential interface through which the consistent relationship as between primes and natural numbers can take place. And the importance of this relationship in holistic terms is that the mysterious interpenetration of quantitative with qualitative meaning is ultimately communicated through the two-way relationship of the primes and the natural numbers. Put another way, through these zeros, perfect synchronicity can be maintained regarding the overall behaviour of the primes with respect to the natural number system. This, then holistically serves as the ultimate requirement for the coherence of all phenomenal communication (in quantitative and qualitative terms)!

And such synchronicity, which has no causal explanation, once again represents a holistic - rather than analytic - attribute of number.

However, by definition, as the original numbers 1 and 0 precede the primes, they cannot be explained in this manner.

So the fundamental Euler identity, as I sought to demonstrate in the last article, can be seen from this light as providing the corresponding holistic (unconscious) basis for our everyday understanding of 0 and 1.

In fact without covering once again ground covered in my previous articles, the more general Euler formula is likewise intimately associated with - what I refer to as - the Zeta 2 zeros. 2

The Euler formula cannot of itself directly explain the all-important relationship as between the primes and the natural numbers. However it still can be seen to comprise an integral component with respect to the deeper underlying holistic basis of the number system.

And why should this be so important?

Well consider again digital technology and all the wonderful inventions that it has made possible in recent years!

Personally, I still marvel each day at how potentially all information can be successfully encoded through the binary digits 1 and 0. Now such technology is based on the analytic interpretation of such digits! If one therefore allows for the corresponding holistic interpretation, then this implies that all transformation processes, can likewise be potentially encoded using the same two digits.

And then if we combine both aspects, this implies that information and transformation could be dynamically encoded through both analytic and holistic aspects with respect to the digits.

And if further one accepts that all evolution relates to a dynamic interaction in terms of both information and transformation processes, one can then perhaps appreciate the potential importance of number.

However again the binary numbers alone cannot properly explain the manner through which information and transformation interact, thereby leading to all the incredible variety manifested with respect to created phenomena (both quantitative and qualitative).

So the code for this is contained in the zeta zeros (Zeta 1 and Zeta 2). In this way number - with respect to the dynamic interaction of both its analytic and holistic aspects - can be seen as the hidden encoded basis of all phenomena. However the full decoding of its inherent meaning is inseparable from the course of all evolution. And from a psychological perspective, this entails all possible bands on the spectrum of development.

The central mystery of existence relates to how both dual and nondual aspects of reality, though corresponding to utterly distinctive types of truth, are yet so intimately related (and ultimately identical). So the direct experience of this union which can be attained in varying degrees through authentic spiritual development is one of complete mystery.

Now the great importance of number (and by extension Mathematics) is that it approximates most closely to the two extremes of both dual and nondual understanding respectively.

Thus Mathematics, as we know, offers the greatest capacity for the exercise of pure rational interpretation (where constructs are used in a highly abstract manner). This once again represents its analytic (quantitative) aspect which is assumed - so misleadingly in our culture - to be synonymous with all valid Mathematics.

However at the other extreme, Mathematics approaches most closely to pure nondual understanding. As mathematical notions such as number are already deeply implicit in all phenomenal experience, therefore when appropriately understood, they operate as the final bridge to the experience of pure ineffable reality. This represents the holistic (qualitative) aspect to Mathematics which - quite remarkably - in our culture is not formally recognised as possessing any validity!

The considerable dilemma however is that both aspects lead to utterly distinctive truth systems.

The great attraction of the accepted analytic approach to Mathematics is that it seemingly offers unambiguous rational truth of an absolute nature.

However at the other extreme of the holistic approach, all rational truth is experienced as deeply paradoxical and ultimately eroded altogether in an intuitive spiritual awareness that represents pure mystery.

So a great problem, which we have not yet addressed in society, is the task of meaningfully incorporating the validity of both these approaches in mathematical and scientific terms.

While the value of rational dualistic understanding will always remain important, a radical shift of perspective is required, whereby its role becomes appreciated in an increasingly relative (rather than absolute) manner.

With this will come a more refined understanding of how all rational interpretation (in Mathematics and elsewhere) has but a limited validity depending on context. In partial circumstances, dualistic validity will still remain very high. However in more universal contexts, dualistic truth will be increasingly seen to lose its usefulness.

Then with the ready acceptance of the - ultimately - relative nature of all phenomenal truth, it will become easier to relate to the world without undue rigid attachment.

This in turn will facilitate the holistic task of continually experiencing the underlying nondual nature of reality as the absolute present moment. Thus both dual (relative events in phenomenal spacetime) and nondual (experience of the absolute present moment) can then go properly hand in hand mutually enhancing each other.

At some future date in our evolution, I have no doubt that both of these aspects (analytic and holistic) will play a more equal role in a balanced comprehensive approach to Mathematics. However this will require considerable specialisation with respect to both rational and contemplative intuitive type appreciation.

I conclusion I will offer some general comments on Mike Hockney's “The Mathematical Universe”.

Overall I believe that Andy has done a fine job in fairly representing both the strength of the book in its interesting and provocative insights and also its chief failing in that many of its dogmatically asserted conclusions are not supported by convincing argument.

For his part, Elliot believes that Hockney makes far too much of the Euler equation (which viewpoint I would support to a certain degree). However in emphatically making this point, he perhaps fails to sufficiently recognise the accepted great importance of the equation in both Mathematics and Physics.

The main problem I would see with Hockney's approach is its reduced nature with little expressed recognition of how dual and nondual are radically distinct notions.

Therefore by appealing to a higher reason in Mathematics, he attempts to deal with reality (notwithstanding many excellent insights) in a rational dualistic type manner. And he keeps jumping quickly between points without explaining them sufficiently.

The notion of “infinite” points (monads) is extremely important in his approach as the basis for the emergence of subsequent phenomenal reality. The very notion of an “infinite” point however entails a qualitative type distinction. And as all the mystical traditions attest, the true notion of the nondual is radically empty (of all such constructs). So in effect - though I certainly would not dismiss the notion of monads outright - he conveniently uses it in a pseudo rational manner to bridge the crucial divide between dual and nondual.

So this all important connection is cheaply explained away as “a mathematical trick” whereby nothing thereby becomes something!

Also I would question the motivation for his secrecy. Mike Hockney seemingly represents a pseudonym for the member of a Neo-Pythagorean cult called “the Illuminists”. Two other prominent writers in this group are “Adam Weishaupt” and “Michael Faust.”

This begs several questions. Who is “Mike Hockney” and what is his background? Who in fact are “the Illuminists”? What is their membership? How do they operate? Are they serious in openly engaging with others holding different viewpoints? Unconfirmed accounts unfortunately would suggest perhaps questionable connections with the occult!

Furthermore, Hockney offers his account as the accepted position of “the Illuminists”. However it appears more likely that he is simply using this anonymous group as convenient cover for his own strongly held views.

In conclusion, though I would be naturally supportive of anyone seeking to promote a more comprehensive vision of Mathematics, I would expect - as with all genuine communication - that it would be done in an open manner inviting feedback from other interested parties.


1. Ken Wilber briefly introduced his own version of Integral Mathematics initially back in 2003. This arose in the context of his work on perspectives, which presumably was intended as a major topic in the 2nd volume of his proposed trilogy (still not published)! At the time, in a series of articles “Clarifying Perspectives” on this forum, I offered my own considered position on perspectives and used this as a basis for evaluation of Ken's approach in the final article “Higher Order Perspectives and Integral Mathematics”.

Without repeating what was already said more fully in those articles, it is perhaps fitting in the context of the present discussion to summarise my conclusions at the time.

It would only be fair to say that Ken had indeed come up with an ingenious new development. However my major criticism related to the fragmented nature of his treatment, with little attention given to the overall integration of his various perspectives.

With respect to Ken's “Integral Mathematics”, I certainly support his intention to define Mathematics in the context of all four quadrants (though I have always preferred the corresponding notion of complementary polarities, which inherently is more dynamic than quadrants!) However apart from being an all-quadrant affair, I see the comprehensive approach to Mathematics as likewise embracing all levels on the spectrum (which Ken does not seriously address). Ken's all-quadrant extension of the general context for mathematical enquiry properly represents a vision-logic interpretation, which certainly goes well beyond conventional thinking. However it still remains confined to just one small region of the spectrum (that I personally would not consider suited to true integration).

Ken's “Integral Mathematics” then essentially consisted of a convenient notation system (again ingenious in its own right) for the representation of his various perspectives. However because of the lack of proper integration of these perspectives, I found severe short-comings with his “unpacked” equations.

Indeed one key equation, designed as a representation of mutual understanding (and referred to in my article at the time), was especially lacking from an integral perspective!

Considerable modifications of the preliminary draft may since have taken place. So his “Integral Mathematics” certainly maintains potential as an interesting and valuable way of bringing greater clarification to his important work on perspectives.

However it is still very far indeed from what I would consider a proper integral approach to Mathematics. This would require seriously addressing a wide range of existing mathematical symbols and relationships in a dynamic manner, with a view to seeing how their interpretation would be modified when viewed through “higher” spiritual contemplative stages on the spectrum.

Obviously, I intended all along that my own “Holistic Mathematics” might represent one such integral approach. (However due to the narrower specialised meaning of “integral” within Mathematics, I generally refer to it as “Holistic Mathematics” rather than “Integral Mathematics”)! And though still at a very preliminary stage, I have been turning my attention recently to a more comprehensive form, “Radial Mathematics” where both standard and integral mathematical notions are coherently combined with each other (as attempted in the Euler articles). 2. Once again the Zeta 2 zeros arise as solutions to the finite equation:

1 + s^1 + s^2 +…. + s^(t – 1) = 0.

So for example when t = 3, the relevant equation is

1 + s^1 + s^2 = 0.

The solutions to this equation provide the 2 non-trivial roots (of the 3 roots of 1) which are in fact the Zeta 2 zeros in this case. (Because 1 is always a root, in this sense it is trivial and thereby excluded.)

Now the roots are actually calculated from the Euler formula,

e ^ ix = cos x + i sin x (where x = number of radians)

Also remember that pi radians = 180 degrees.

The two roots in question conform to (2pi/3) and (4pi/3) radians i.e. 120 and 240 degrees respectively.

So the 1st Zeta 2 zero = cos 120 + i sin 120 = – .5 + .866i (correct to 3 decimal places).

The 2nd Zeta zero = cos 240 + i sin 240 = – .5 – .866i.

What is fascinating about these results is that they are indirectly involved in everyday experience (as the unrecognised holistic unconscious basis of ordinal rankings).

So, if we wish to express the qualitative notion of the 1st of 3 members, indirectly in a quantitative manner the 1st zero is required. Then in expressing the 2nd of 3 members, indirectly the 2nd zero arises. So these represent the unconscious quantitative appreciation that is deeply implicit in customary ordinal notions (which strictly involve qualitative type relationships).

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