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INTEGRAL WORLD: EXPLORING THEORIES OF EVERYTHING
An independent forum for a critical discussion of the integral philosophy of Ken Wilber
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Publication dates of essays (month/year) can be found under "Essays".
Peter 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 2: Where Mathematical and Spiritual Reality Meet
Peter Collins
Restating the Vision
I concluded the first part by once again referring to the important fact that properly understood, mathematical understanding relates to all bands on the spectrum, with a continual transformation in the nature of its symbols taking place throughout development.
Unfortunately, Mathematics has become ever more rigidly identified with the understanding of just one band (Band 2 in my model) so much so indeed that it is all but impossible to have views considered, which challenge the conventional wisdom in a fundamental manner.
What is perhaps unusual regarding my own development is that I realised (dimly at first) from an early age that a more comprehensive perspective was required. Now 50 years later, I can say with considerable conviction that despite the appearance of great rigour, the standard interpretation of the most important mathematical concepts lacks any true coherence. Put another way, Mathematics is in urgent need of recovering a proper integral dimension.
Though the actual experience of mathematical notions is inherently of a dynamic interactive nature, for centuries  indeed millennia  we have increasingly sought to present its truths in an absolute abstract manner. This approach has indeed greatly facilitated a certain important form of specialised knowledge in quantitative terms. However it has been at the cost of a gross reductionism that has blinded us to the much richer reality that Mathematics should properly represent.
So once again instead of just the one recognised (analytic) aspect to Mathematics, suited to mere quantitative interpretation, in a more balanced approach we require three.
Thus as well as the analytic (Type 1), we include an  as yet unrecognised  holistic aspect suited for true qualitative appreciation of all mathematical symbols (Type 2).
Then when a certain specialisation has taken place with respect to both analytic and holistic (in relative isolation) the final most comprehensive form of understanding is based on the increasing dynamic interpenetration of both aspects (Type 3).
So as these two aspects really comprise an important unity, it is only in the light of this third comprehensive approach (which I refer to as radial understanding) that both the analytic and holistic aspects themselves can attain their full realisation.
Now again in terms of the full spectrum, I would see the first analytic type appreciation of mathematical symbols as largely confined to Bands 1 and 2 (with specialised understanding occurring at Band 2); corresponding holistic type appreciation would then significantly unfold through Bands 3 and 4 (with specialised understanding at Band 4); finally radial understanding of the dynamic interpenetration of both analytic and holistic aspects would mainly occur through Bands 5, 6 and 7 (with specialised appreciation at Band 6).
Now of course, it would be totally unrealistic to expect that we can quickly move from the (recognised) merely quantitative interpretation of mathematical symbols to this third comprehensive stage of radial understanding (entailing the balanced interaction of both quantitative and qualitative aspects) in the near future.
Though, by definition, we already possess a certain degree of access to all bands, I imagine that it will take some considerable time with respect to our human evolution before this will be possible in any sustained manner.
So my true purpose here is to awaken in followers of this forum a new vision of Mathematics (that is incomparably greater than present conventional wisdom).
And because the Euler formula is admitted  even by practicing mathematicians  as transcending in significant ways current comprehension, I am thereby seeking to apply this new vision to the formula so as to unlock more of its potential riches.
Numbers 1 and 0
We will now return to the 5 most important mathematical constants, by looking at the deeper holistic meaning of the numbers 1 and 0.
When used in an analytic sense, 1 refers to an independent unit (in quantitative terms).
Indeed all natural numbers can be expressed in this manner through addition of such independent units.
So 3 (representing the cardinal notion of number) = 1 + 1 + 1.
However, 1 has an equally important holistic meaning, referring to the notion of interdependence (with respect to separate units).
Customarily, we use the word “unity” or “union” or even “oneness” to refer to this latter notion of 1. So when we speak for example of a political union (such as the EU) this implies a certain integration (or interdependence) with respect to its 28 individual (unit) members.
0 (zero) in an analytic sense has a merely static meaning (literally as nothing) in the absence of phenomenal form. In this sense the notion of nothing is clearly independent of something (implying 1).
However 0 also has a much deeper holistic meaning, where it likewise becomes increasingly interdependent with the holistic notion of 1.
In terms of spiritual union, the goal in the various mystical traditions is to outline the path that ultimately leads to complete interdependence with respect to all form (initially envisaged as phenomenal units). However, paradoxically this unity (with respect to all form), in the deepest appreciation of the nature of holistic interdependence, thereby becomes identical with nothingness (or emptiness) in the absence of all form.
Once again the simple identity connecting the two holistic notions of unity and nothingness is:
1 – 1 = 0.
However, in holistic terms the very notion of 1 (in its completely interdependent nondual sense) combines both positive and negative aspects through the complementarity of opposite poles.
It must be remembered that in dynamic terms, all mathematical experience is conditioned by the interaction of opposite polarities.
For example, we cannot posit a phenomenal object in mathematical terms in the absence of a corresponding mental construct (which relatively is of a negative nature). So there is always by definition a dynamic relationship as between mathematical truth (as external) and its corresponding (internal) interpretation.
And as we shall see, a key dilemma with respect to the nature of the Euler identity is that because of a reduced absolute treatment with respect to both poles, the standard approach leads to a breakdown as between mathematical truth as objectively verified and its corresponding mental interpretation. In other words, it is not possible to give an intuitively meaningful explanation of the Euler identity in a solely analytic manner.
The very recognition of interdependence from a mathematical perspective (as indeed all experience) is based on this dynamic complementarity of opposites (entailing negation of the static analytic notion of 1).
In my first article on the “Dynamic Nature of the Number System”, I went into considerable detail in explaining how our understanding of a crossroads implicitly entails such a notion of interdependence.
So once again, heading for example in a northerly direction, when one encounters the crossroads, a left or a right turn can be posited individually in an unambiguous manner (represented as + 1).
Equally when heading in a southerly direction, again a left or right turn can be posited in such an unambiguous manner.
However when one attempts to consider both north and south directions simultaneously, a left can equally be a right turn and a right a left turn (depending on context). So left and right only have a relative meaning from this perspective.
So the recognition of interdependence (where each turn can be both left and right) is represented as (+) 1 – 1 = 0.
In other words the dynamic notion of the union of opposites here is inseparable from 0 (in qualitative terms).
Put more simply the notion of interdependence (as opposed to independence) is of a qualitative nature (which is thereby 0  i.e. literally nothing  from a quantitative perspective).
Now, in my discussion of the crossroads, I was at pains to emphasise that the very notion of interdependence requires a 2dimensional logic, where meaning is based on the dynamic interaction of two poles. This contrasts sharply therefore with 1dimensional logic (based unambiguously on a single pole).
Quite remarkably, in formal terms, all accepted mathematical interpretation is 1dimensional.
The implication here is quite stark as it entails that  by definition  such Mathematics has no means of dealing with the key notion of interdependence (in any context) except in a grossly reduced merely quantitative manner.
So I am not engaging in hyperbole when I say that, due to such gross reductionism, the fundamental basis of present mathematical understanding thereby lacks any true coherence.
In the contemplative spiritual life, all experience increasingly entails a dynamic interaction as between dual notions (of phenomenal independence) and nondual notions (relating to the interdependence of such phenomena). For some individuals, the desire to reconcile both fully becomes the driving force of all subsequent existence.
The culmination of this process then entails a highly refined experience, where the unity of all form as pure interdependence is likewise experienced as nothingness in the emptiness of all distinct notions of form. Implicit in this realisation is that nothingness represents the potential for all created form, which is then actualised through the total attainment of unity with respect to such form.
As the Buddhist Sutra states:
“Form is not other than Emptiness
Emptiness is not other than Form”
So in holistic mathematical terms the first line is represented as:
(+) 1 – 1 = 0; and the second as 0 = (+) 1 – 1.
We could equally say with respect to Mathematics:
“the analytic aspect is not other than the holistic
the holistic aspect is not other than the analytic.”
In other words, for truly coherent mathematical understanding that is consistent with the dynamics of experience, analytic appreciation automatically entails underlying holistic understanding; equally holistic understanding entails underlying analytic appreciation.
Again, as there is no formal recognition whatsoever of this vital holistic aspect in Mathematics, the qualitative nature of its symbols is thereby completely ignored and thereby greatly misunderstood.
So to sum up here, the analytic interpretation of mathematical symbols such as 1 and 0 is appropriate with respect to quantitative notions of independence; however the corresponding holistic interpretation of these symbols is then appropriate with respect to qualitative notions of interdependence. And as stated in my previous articles, this notion of qualitative interdependence arises directly in the context of our everyday use of ordinal numbers!
In the dynamics of experience both of these aspects continually interact. However, in all formal mathematical contexts, the qualitative aspect is simply reduced in a merely quantitative manner.
Revisiting the Transcendental
If you have encountered calculus, you will be already familiar with the notions of differentiation and integration in this mathematical context.
However differentiation and integration have a much wider meaning from a developmental perspective, with all physical and human developmental processes entailing the dynamic interaction of both aspects.
What is termed the exponential function, i.e. y = e ^ x, contains a very unusual and important property.
Thus when one either differentiates or integrates with respect to this function, its value remains unchanged.
If we attempt to look at the nature of e in dynamic holistic terms, it likewise bears a remarkable property with respect to human growth.
In other words, e deeply symbolises the nature of development, where differentiation (in the form of discrete phenomena of a dualistic nature) becomes ultimately inseparable from their corresponding integration (in a continuous nondual manner).
As we have seen from the analytic context, the Euler identity is based on e.
We then further noted that rather than a linear, it relates to a circular notion of exponential growth.
On reflection, this describes very well the nature of spiritual contemplative development.
Again, one starts out on this journey with linear dualistic notions still intact. Gradually however, significant erosion of such notions takes place with nondual awareness becoming an evermore prominent reality.
Then at the higher stages of such development, one now no longer views reality in terms of dual and nondual notions (as separate) but rather as the highly refined relationship where they become increasingly interdependent with each other.
Thus, whereas the dimensional framework for phenomenal experience initially is strongly linear, with advancing contemplative development it is transformed in a circular (paradoxical) manner.
And with respect to daily activity (within this framework), one thereby attempts to seamlessly reconcile the discrete differentiation of each distinct phenomenon with the continuous integration of all phenomena.
So from the holistic perspective, the Euler identity can be seen as an especially powerful mathematical relationship that scientifically encodes the most advanced nature of spiritual unity. Indeed as I stated at the beginning of this article, such spiritual development is itself necessary, so as to enable the refined holistic ability to then decode these mathematical symbols in the appropriate manner.
Pi of course is likewise a transcendental number with potentially an enormous holistic mathematical relevance.
Again in analytic terms, pi describes a relationship with respect to the (line) diameter and its (circular) circumference. In like manner from a holistic perspective, pi now describes the dynamic relationship as between (unambiguous) linear and (paradoxical) circular type understanding. And as we have seen, the proper appreciation of the notions of independent and interdependence respectively always entails this dynamic relationship.
Indeed we can precisely define the very nature of transcendental understanding in holistic terms. Here, reality is interpreted in neither dual or nondual terms (as separate) but rather as the highly refined relationship of both types of meaning.
This provides the deeper qualitative explanation as to why a number such as pi cannot be the solution to any algebraic equation as this would imply some means of expressing nondual meaning (associated with higher dimensions) in a reduced dualistic manner. However clearly this is not possible when one defines such meaning in terms of the central relationship as between both notions.
However there is an important distinction to be made as between the analytic and holistic appreciation of the circle. In analytic terms, the circle, by definition, must be given linear extension (to have phenomenal meaning). So, both the circular circumference and its line diameter are seen as somewhat separate.
However in holistic terms, both the (line) diameter and its (circular) circumference must ultimately be reconciled as identical. This requires the contraction of the circle to a central point (where, by definition, diameter and circumference fully overlap).
Thus in this sense, purely circular becomes inseparable from purely linear (where both notions imply each other), and becomes just another way of expressing the ultimate identity of form and emptiness.
However the hugely significant relevance here is that when understood appropriately, the interpretation of symbols used in the Euler identity, requires a holistic  rather than analytic  interpretation for their meaningful comprehension.
Thus very much as Benjamin Pierce expressed, the Euler identity leads to a central paradox with respect to mathematical understanding. 1
Though the identity can be framed and indeed proved in an analytic manner, its coherent interpretation requires the corresponding holistic appreciation of these symbols.
In other words, the Euler identity points directly to a reality that greatly transcends the conventional restricted analytic interpretation associated with its symbols.
There is a huge message here with equal relevance for modern physics.
For example string theory has become increasingly mathematical (in the restricted analytic sense) in the attempt to uncover the ultimate Theory of Everything.
However such mathematical progress has become significantly divorced from the physical world in an intuitively meaningful manner. So by definition, these ever more abstract mathematical interpretations of reality will eventually be rendered meaningless from a philosophical perspective (due to the total lack of a holistic dimension).
Thus, when we attempt to push the mere abstract interpretation of mathematical symbols to ever further extremes, ultimately this leads to complete incoherence with respect to overall understanding.
Hidden Importance of the Imaginary
The last of the 5 constants relates to the imaginary unit, which in analytic terms refers to the square root of – 1.
As already stated, I would say, that perhaps more than anything else, the recognition of the true holistic significance of the imaginary notion has the power to greatly transform our customary appreciation of reality.
In holistic terms, the operation of subtraction (i.e. –) relates to the negation of dualistic type interpretation (associated with the positive direction). So whereas the positive relates to conscious type appreciation, the negative leads by contrast to unconscious type awareness.
In dynamic terms, we can only unconsciously negate what has been already consciously posited. So properly speaking, the unconscious relates thereby to twodimensional understanding. So the very notion of interdependence, which we have already dealt with, is properly of an unconscious nature.
The imaginary unit thereby arises in attempting to express  what is inherently  a (circular) 2dimensional holistic notion, in a 1dimensional (linear) conscious manner.
So here in holistic terms, i^2 = – 1; therefore i = the square root of – 1.
Put another way, the great importance of this holistic interpretation of the imaginary notion in Mathematics is that it represents a hidden way of dealing with qualitative notions of interdependence in a reduced analytic fashion.
Not surprisingly therefore, imaginary numbers often play an unexpected holistic role in Conventional Mathematics.
Roger Penrose in particular continually refers to the “magic of complex numbers” in recognition of their amazing holistic properties!
Though the complex nature of number quantities (with both real and imaginary aspects) is now fully accepted within Mathematics and Science, the overall paradigm is solely real from a qualitative perspective.
So the attempt is made to interpret reality in a solely real (i.e. conscious) manner.
However in experiential terms, all experience of reality (including mathematical) is complex from a holistic perspective including both conscious (real) and unconscious (imaginary) aspects. This then has important implications for the present scientific paradigm which is based solely on real (conscious) modes of interpretation.
When the unconscious aspect of experience is not properly recognised, it tends to project itself into consciousness in an involuntary manner (creating a blind prejudice with respect to conscious judgement). With however appropriate recognition, it can then flow freely into consciousness as a refined form of holistic intuitive awareness.
Because Mathematics in formal terms remains in complete denial of the unconscious, this has created an enormous shadow blindly protecting it from fundamental criticism of its rationale (such as detailed in this article).
So, quite simply, just as in quantitative terms, Mathematics has ultimately been led to the acceptance of an imaginary aspect to all numbers, likewise it will ultimately be led to incorporation of an imaginary (holistic) aspect with respect to comprehensive understanding.
This in fact is very relevant to the interpretation problem that we have mentioned with respect to the Euler formula. The crucial reason for this is the presence of the imaginary unit (i.e. the square root of – 1) in the dimensional power  or exponent  of e.
Though this can indeed be expressed in reduced quantitative terms, because of its inherent qualitative meaning, it switches appropriate interpretation from an analytic to a holistic perspective.
Normally, when we raise a number such as e to a dimensional power (representing another number) we expect an intuitively meaningful answer in analytic terms (i.e. a number that can be placed on the number line).
For example consider the expression e ^ (pi).
If you check on a calculator you will find it gives the result 23.14 (correct to two decimal places). This corresponds to the Type 1 aspect of number (lying on the real number line). So fully expressed, this Type 1 expression of number is 23.14 ^ 1.
However, if we now multiply the dimensional power by i, to obtain e ^ (i *pi), it introduces a dramatic shift in number behaviour.
So e ^ (i * pi) = – 1. This in fact represents a number that now belongs to the Type 2 aspect of number (indirectly expressed in a circular manner).
In terms of the Type 2 aspect of number, the result, i.e. – 1 = 1 ^ (1/2)
Thus fully expressed the second result conforms to 1/2 with respect to the Type 2 aspect.
So we change here from the Type 1 to the Type 2 aspect of number, through the behaviour of the dimensional power. And the reason for this is that i itself properly entails the indirect expression of the Type 2 number corresponding to 1 ^ (1/4).
Thus the key point here is that we cannot meaningfully understand the Euler identity in the absence of the Type 2 aspect of number (which corresponds directly to the holistic interpretation of its symbols).
In fact, put even more simply the fundamental Euler identity,
i.e. e ^ (2i * pi) = 1, simply represents the holistic mathematical expression of 1.
So, the great mystery attached to the expression, is that we have now literally come full circle with respect to the interpretation of 1. So we started with the standard analytic Type 1 appreciation to eventually fully arrive through the Euler identity at its corresponding holistic expression.
This is really the true great secret of the Euler identity, which cannot be appreciated in the conventional mathematical manner.
Recognition of the holistic notion of the imaginary likewise has huge implications for understanding of the physical world.
It is long been recognised that the study of subatomic reality requires the use of complex numbers in quantitative terms.
Now properly understood, this leads to the important realisation that reality at this level cannot be understood in merely analytic terms, but entails the dynamic interplay of both analytic and holistic aspects.
In other words, the subatomic level inherently entails recognition of the underlying synchronous holistic nature of the universe. Experimental verification of the nonlocal effect, where subatomic particles can communicate with each other (even at a great distance), has now taken place. However most physicists are still in considerable denial regarding the obvious implication, that this indeed implies an inescapable holistic dimension underlying what we know as physical reality.
The Main Operations
I will deal briefly with the holistic understanding of the operations of addition, multiplication and exponentiation concentrating on just one aspect, that when properly appreciated, is of enormous potential relevance.
Once again mathematicians attempt to view these operations from a merely quantitative perspective.
This then leads to failure to recognise the inherent manner in which multiplication for example is significantly different from addition.
Mathematicians often attempt to portray addition as a longhand version of multiplication.
So, in terms of addition, for example, 2 + 2 + 2 = 6.
It will then be stated that this in fact implies multiplication. So because 2 occurs three times, we could represent this operation more succinctly in multiplication terms as 2 * 3 = 6.
There is a crucial point that is thereby missed. With addition we treat the three instances of 2 in an independent manner. However the very recognition of their common identity or “twoness”, thereby enabling representation in terms of multiplication, implies the qualitative notion of interdependence.
Thus in contrast to addition, there is an inherent qualitative (as well as quantitative) aspect to multiplication, which is the fundamental reason why the operations are incompatible with each other (in merely quantitative terms). [2]
The same problem is inherent in the relationship of multiplication and exponentiation.
We could represent 2 * 2 * 2 in multiplicative terms = 8. So again, 2 in each case is treated as independent.
However because 2 repeatedly occurs, we could represent this “twoness”, through exponentiation as 2 ^ 3 = 8.
Once again, the recognition of 2 as a repeating factor implies the qualitative notion of interdependent (rather than independent) identity.
Finally here we will deal briefly with the equality sign connecting both sides of an equation.
Thus once more, the fundamental Euler equation is e^ (2i * pi) = 1.
Indeed the inclusion of 2 adds another important aspect in the holistic appreciation of 2 as duality. As the very process of dealing with any phenomenon implies duality (in the separation of opposite poles), it is appropriate therefore that 2 should be holistically included along with the other constants.
In conventional mathematical terms, the equality sign with respect to the fundamental Euler identity has a merely static interpretation (where it is given a fixed absolute meaning).
However from the corresponding holistic perspective, the equality sign is interpreted in an inherently dynamic manner.
Then in the most comprehensive type of radial understanding, both the analytic and holistic interpretations of equality interpenetrate in an increasingly balanced fashion.
So, in the next section we will elaborate further on the experiential context for the true holistic appreciation of the Euler identity.
Contemplative Spiritual Union
When viewed from the appropriate perspective, the fundamental Euler identity, i.e.
e^ (2i * pi) = 1, can be seen as a truly remarkable holistic mathematical expression of the nature of mystical union.
Again in holistic terms, e implies activity so refined that the discrete (dual) nature of the individual differentiated phenomena in experience approaches identity with their (nondual) continuous integration.
The exponent 2i * pi then literally relates to the dimensional framework in which such activity takes place. As we have seen pi is a transcendental number and i the imaginary unit. So in holistic terms, we have experience here of an imaginary transcendental nature which dynamically approaches a central point as the pure activity of will.
So what could this mean?
Well once again, the transcendental meaning of pi implies the realisation that reality does not represent the linear (conscious) or circular (unconscious) aspects as separate but rather the central relationship between both (associated directly with the will in a pure form of volitional intent). Imaginary then implies that this relationship can now be applied intimately to the deepest shortlived projections emanating from the unconscious.
So when one can closely harmonise both the highly transitory (indirect) conscious manifestation of such projections with their inherent unconscious nature, then by definition, their involuntary nature ceases. In this way, full union in psycho spiritual terms (through the holistic notion of 1) of conscious and unconscious can thereby take place.
The Oxherding Pictures
The 10 oxherding pictures and commentaries (originally based on an old Taoist story) provide a very attractive account of the various stages leading to spiritual enlightenment.
The oxherd symbolises the untamed self, whereas the ox refers to the true enlightened self.
The 1st picture shows the oxherd dissatisfied with life and searching everywhere for the lost ox. In the 2nd, the oxherd discovers the footprints of the ox and the first glimpse of a spiritual path to follow. In the 3rd the ox is glimpsed, but the vision not yet sustained. In the 4th the oxherd last gets hold of the ox though disciplined spiritual practice. In the 5th, through sustained practice the ox is tamed and in the 6th with such practice now automatic, the ox is successfully ridden home.
The 7th, 8th and 9th pictures are especially relevant for our purposes, describing the subtle nature of the transition towards complete spiritual union. In the 7th the ox is transcended. So the oxherd has now successfully attained the spiritual reality beyond all form.
This is often viewed in somewhat impersonal terms. In Christian mysticism (e.g. Meister Eckhart) it is referred to as the Godhead or  even more starkly  naked Godhead.
However there are still some remnants of the old self in the will operating at the personal level. So it is important now to equally attain the immanent aspect of union through realisation of the spirit that is equally prior to all (physical) form.
This is achieved in the 8th picture where both the ox and the self are transcended. Interestingly, this is often depicted by a purely blank circle illustrating a radical awareness of the nature of emptiness (i.e. nothingness). Though the 7th also implies a nothingness, it would be more accurately depicted as a circle containing a point (indicating that to a degree the central point of self has not been fully integrated with nondual awareness).
In Christian terms we now have the balancing of immanent and transcendent aspects in the union of the personal loving God with the impersonal naked Godhead (i.e. spiritual Marriage).
With this double union as it were, the world is reborn in a new light, where from one perspective (through transcendence) the ordinary is made extraordinary and then (through immanence) the extraordinary made ordinary. So both the dual and nondual aspects of phenomena are now substantially harmonised.
In mathematical and scientific terms, this would relate to the commencement of the 3rd comprehensive stage of radial understanding, where both analytic and holistic appreciation of symbols increasingly interpenetrates with each other.
The 10th and last oxherding picture entails the return to the marketplace. Though fully engaged with ordinary life, the truly spiritual person now possesses a unique power in helping others discover their own unique path to enlightenment.
Unfortunately there is a touch of the “happy ever after” regarding the final picture.
Realistically, the final stage does not entail the end of dualistic conflict and existential suffering. Rather it entails an intensification of such experience. However because of the depth of spiritual awareness attained, all difficulties can now be faced with much greater equanimity.
In an interesting article, “What Does Mysticism Have to Teach Us About Consciousness”, Robert Forman, giving several fruitful illustrations, likewise distinguishes between these three stages of nondual meditation.
He refers to the first as the pure consciousness event (which is largely devoid of sensory or mental input). This would equate very well with the transcendent aspect of spirit as beyond all form and would equate well with the 7th of the oxherding pictures.
He refers to the second as the dualistic mystical state. Here there is the continual absorption in a spiritual presence together with active engagement in dualistic activities.
However, a certain divide remains in terms of the integration of both realms. Again this would suggest at a deep level remaining remnants of an unreformed personal self. This continues until union with immanent reality likewise takes place. So this would largely equate with the 8th picture.
The third stage he terms the unitive mystical state. One now can be aware of all the ordinary phenomena, but in a fully reborn manner where their relative distinct identity is experienced as inseparable from the spiritual interdependence of all creation. Again this would equate with the 9th picture.
Relevance of the Euler Identity
When appropriately interpreted, the fundamental Euler identity can be seen to correspond in precise holistic mathematical terms to these 3 stages.
Indeed close consideration of this issue, eventually resolved a problem regarding interpretation of the identity that had puzzled me for some considerable time.
There are in fact numerous differing expressions based on the fundamental identity that apparently give the same result. 3 Thus,
1) e ^ (2i * pi) = 1.
2) e ^ (– 2i * pi) = 1.
3) e ^ 0 = {e ^ (2i * pi)} * {e ^ (– 2i * pi)} = 1.
In conventional mathematical terms, no distinction is made regarding the three results. So the answer is given simply as 1 in each case.
However this would then seem to apply that 2i * pi = – 2i * pi = 0, which clearly makes no sense from the conventional mathematical perspective.
Only when I began to clearly realise that the R.H.S. of the expression corresponds to three distinct notions of 1 that the problem began to resolve itself.
Thus to distinguish clearly the nature of 1 in each expression we must use the Type 2 aspect of the number system.
So now e ^ (2i * pi) = 1 ^ 1;
e ^ (– 2i * pi) = 1 ^ (– 1) and
e ^ 0 = 1 ^ 1 * 1 ^ (– 1) = 1 ^ (1 – 1) = 1 ^ 0.
Now, because the reduced quantitative value of these three values (belonging to the Type 2 aspect of number) = 1, no distinction is made between them in conventional mathematical terms, which strictly renders the expressions meaningless. However from the Type 2 perspective, we now have three numbers 1, – 1 and 0 in this system, which are properly distinct from each other in a qualitative holistic manner.
In other words the truly important point to again grasp is that the Euler identity generates values with respect to the Type 2 aspect of number (and not the conventional Type 1).
This once again clearly implies that a holistic (qualitative) interpretation of the results is necessary. And the requirement for full appreciation at this level is the experience of the various stages of contemplative spiritual union (corresponding to Band 4 on the spectrum) that I outlined at the beginning of the article.
So the initial expression represents the first type of union (with respect to its transcendent aspect) i.e. the union that is experienced beyond all form. Though indeed empty in a quantitative sense, volitional attachment with respect to the central point of the will still remains.
So in this context 2i * pi can itself be viewed as a bindu or point (with a directly qualitative rather than quantitative meaning).
The second expression then represents the negation of this point (strictly unbalanced identification with the point). This requires reducing attachment to the transcendent notion of union (as beyond all form) so as to properly balance it with the corresponding immanent aspect (as prior to all form).
The third aspect then entails the full double union, as it were, with respect to both the transcendent and immanent aspects.
This entails that the very notion of a point (which implies a restriction with respect to the focus of attention) is itself now eroded.
So we have the full union of form that is now inseparable from the corresponding experience of emptiness (i.e. nothingness).
And this is beautifully expressed in the Type 2 system (i.e. the exponent of 1) as 1 – 1 = 0.
So we have now come full circle with respect to the interpretation of the relationship where 1 – 1 = 0.
Thus we started with the conventional (Type 1) analytic dualistic understanding (which would be familiar to a small child).
However full appreciation with respect to corresponding nondual holistic awareness is ultimately inseparable from the experience, where the unity of all form is identical with the corresponding experience of emptiness. This is the Type 2 aspect.
And as nothing is more fundamental in Mathematics than 1 and 0, this thereby represents the deepest most original degree of mathematical understanding that is humanly possible.
Thus perhaps one can perhaps appreciate why the quote of Keith Devlin is so appropriate when he states “that the Euler equation reaches down into the very depths of human existence”.
I will just briefly comment here on the relevance of the above formulations in the context of the origins of the physical universe. Indeed properly understood, true spiritual union equally represents the coincidence of mere potential (prior to phenomenal manifestation of physical reality) and true actualisation (in its psycho spiritual fulfilment).
So thereby the Euler identity is especially relevant in terms of the holistic interpretation of the origins of the universe.
The first two formulations of the identity above would now be relevant (in a reverse manner). So we must initially consider the origins of the universe in terms of a radical emptiness that equally serves as the potential for the emergence of all subsequent form.
The next step towards phenomenal reality could now entail the recognition of qualitative points (or monads) as the infinite basis for emergence of reality in a phenomenal manner.
However there is no short cut rational explanation of how reality as form emerges. This always remains a central mystery (regardless of the sophistication of subsequent scientific accounts).
So properly understood, science as we know it (i.e. analytic science) can only provide explanations of a relative nature. The underlying absolute present moment, which is continuous with phenomenal reality in space and time, always remains truly mysterious.
Properly appreciated, both spiritual (nondual) and scientific (dual) understanding should mutually enhance each other. In this way we can hope to achieve ever more refined explanations of the phenomenal nature of reality (in an increasingly relative manner) while deepening in the nondual awareness of its underlying mystery.
To return then to Mathematics, with attainment of an appropriate degree of specialisation in terms of both analytic and holistic type appreciation, the final comprehensive stage of understanding can commence, where both types of meaning continually interpenetrate in experience.
One day in the future evolution of the human species, we will naturally understand Mathematics in a much more balanced and refined manner. However, even at this stage, the commencing vision of this possibility, can act as a powerful source of inspiration. This then can help to free us from the highly artificial constraints on creative imagination imposed through conventional mathematical training.
As we have seen at the end of the first article, the Euler identity leads to a circular type of exponential growth which makes little sense from the analytic perspective.
However it makes eminent sense from the corresponding holistic perspective in terms of human spiritual development.
Thus we start at 1. (See the diagram in the 1st article). This would correspond initially with the dualistic experience of phenomena (i.e. form as separate units). A quarter revolution with respect to the circle brings us to i. This corresponds again to dualistic experience, this time through imaginary (i.e. projected) phenomena of an unconscious nature. Another quarter revolution and we are at – 1. This represents the dynamic negation of conscious phenomena (referred to as a purgation or “dark night” in Christian literature). Another quarter revolution and we arrive at – i, representing the even deeper negation of involuntary phenomena. Then finally with the last quarter revolution we are back at 1.
With the circle of development now completed, the desired marriage of analytic and holistic meaning can finally take place.
Notes
1. Strictly speaking, Benjamin Pierce was referring to a mysterious relationship associated with the Euler formula, which has intrigued mathematicians now for centuries.
It can be easily shown from the Euler formula that i ^ i is a real number (i.e. belonging to the Type 1 definition of number). Its value in fact is .207879576…
However though mathematicians can easily demonstrate in analytic terms how this value arises, they cannot explain the deeper philosophical reason as to why such a nonintuitive result might occur.
In fact it is easily explained in holistic mathematical terms.
Remember a Type 2 number arises from raising 1 (which is itself Type 1) to another Type 1 number. So 1 and 1/4 in isolation are both Type 1 numbers.
However 1 ^ 1/4 is a Type 2 number (i.e. i) which is represented on a circular rather than linear number scale. This is so, even though both base and dimensional components, in isolation, constitute Type 1 numbers (with a linear quantitative interpretation). The reason for the switch is that in dynamic interactive terms (just like the turns at a crossroads) they now constitute polar opposites as quantitative as to qualitative, thereby causing this switch in the result to a Type 2 number. The deeper implication is that actual number behaviour can be seen to directly conform here to dynamic interactive notions!
Now i ^ i = {1 ^ 1/4} ^ {1 ^ 1/4}. So this represents a Type 2 number raised to another Type 2 number (viewed in isolation). However, once again when both base and dimensional numbers are viewed as complementary, this again causes a switch, this time to a Type 1 result (i.e. a real number that can be placed on the number line).
This represents just one fascinating example of how holistic mathematical understanding can clarify the deeper reason as to why such apparently strange number behaviour arises.
Indeed it can clarify much more! In conventional mathematical terms, i ^ i does not have a single valued solution but rather can be given an unlimited number of finite results.
However, from a holistic mathematical perspective, this represents very confused thinking! When one equally recognises how numbers have both a Type 1 and Type 2 status, it can be easily explained how these various results properly relate to different base/dimensional number combinations with respect to the Type 2 system (which is not formally recognised in conventional mathematical terms).
So for example in conventional mathematical terms a second value for i ^ i = .0432139…
However, this properly represents the value of i ^ 2i. The root of the problem here is that the first value is obtained from e ^ 2i * pi = 1, whereas the second is derived from
e ^ 4i * pi = 1. However, both these values of 1 correspond in fact to different numbers in the Type 2 system (i.e. 1 and 2 respectively), again not recognised in conventional mathematical terms. (See Note 3)
This then leads directly to the confusion of considering i ^ i = i ^ 2i (which is just as nonsensical as maintaining that 2 ^ 1 = 2 ^ 2). This is a massively important problem with respect to number interpretation simply glossed over in conventional terms.
2. There are two ways of representing numbers one involving addition and the other multiplication which are not easily reconciled with each other.
In the first case a number such as 4 is defined as 1 + 1 + 1 + 1, whereas in the second it is expressed as the unique product of primes i.e. 2 * 2.
One of the greatest mathematicians, e.g. Alain Connes, intimately involved in the search for a solution to the Riemann Hypothesis, sees the problem as relating to a certain incompatibility as between the two modes. In fact the basic reason is easy enough to state as pure addition and pure multiplication (involving repeating both operations with respect to 1) are quantitative and qualitative in terms of each other.
So 1 + 1 = 2 entails both units treated as independent, whereas 1 * 1 entails treating them as interdependent. Thus in the first case we recognise the notion of 1 as an independent unit; in the second we recognise the common quality of “oneness” with respect to both numbers. And this in a nutshell is the crucial distinction with respect to addition and multiplication respectively. However it cannot be dealt with in conventional mathematical terms as no formal recognition is given to qualitative notions!
3. In fact, in conventional mathematical terms when we raise e to an even number multiple of i * pi, the answer = 1.
So, for example, e ^ (2i * pi) = e ^ (4i * pi) = e ^ (6i * pi) = 1.
However when correctly understood, through the Type 2 aspect of the number system, these all represent different members of that system (i.e. 1, 2 and 3 respectively).
So e ^ (2i * pi) = 1 ^ 1; e ^ (4i * pi) = 1 ^ 2; e ^ (6i * pi) = 1 ^ 3.
