INTEGRAL WORLD: EXPLORING THEORIES OF EVERYTHING
An independent forum for a critical discussion of the integral philosophy of Ken Wilber
Publication dates of essays (month/year) can be found under "Essays".
Part I | Part II | Part III
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.
Dynamic Nature of
the Number System
Part I: Mathematics at a Crossroads
I firmly believe we are now approaching the greatest revolution yet in our intellectual history, where Mathematics and all the Sciences will be understood in a completely new light.
This revolution is pointing to the need for a radical new interpretation of the nature of the number system, which ultimately concerns the relationship of quantitative to qualitative type understanding.
For several millennia, we have sought to interpret the number system in a fixed absolute manner, independent of normal experience.
However developments within Mathematics itself are already suggesting that this view is fundamentally in error.
Unfortunately however, due to the reduced nature of present understanding, the profound implications of these changes have not yet been grasped by the mathematical community. More than anything, as we shall see, the situation reflects a profound lack of a coherent integral dimension to the conventional appreciation of Mathematics.
From my early childhood, I have had an abiding deep fascination with the nature of number (especially the relationship as between addition and multiplication). Before hearing about logarithms, which provides a useful means of converting from multiplication to addition, I had developed my own system (using 2 as base).
Then, while attending primary school in Ireland, I became deeply aware of an obvious - though unaddressed philosophical issue - regarding the nature of multiplication.
This arose in the context of simple arithmetical problems relating to the area of (rectangular) fields.
So, for example if the length of the field is given as 3 units and the width as 2 units, the resulting area of the field is 6 square units.
This result represents both quantitative and qualitative aspects of number transformation.
Thus, in quantitative terms 3 * 2 = 6. However equally in qualitative terms, we have changed from linear (1-dimensional) to square (2-dimensional) format with respect to the nature of the units.
This qualitative change is inevitably involved whenever we multiply or obtain the powers of numbers. (Equally it is involved when we divide or obtain roots!)
However in customary mathematical usage, a merely quantitative (1-dimensional) interpretation is given with respect to such numerical results. So the qualitative nature of the number transformation involved, is simply reduced in a quantitative manner.
So for example the result of 3 * 2 i.e. (3^1) * (2^1) is given as 6 i.e. 6^1. Again, though the qualitative dimensional nature of the units has clearly changed, the result is given in a reduced (i.e. 1-dimensional) merely quantitative manner.
This issue, though simple to state, goes to the very heart of the nature of the number system.
When discussing the Riemann Hypothesis (which directly relates to the nature of this system) many prominent mathematicians admit that it reveals an unresolved problem with respect to our understanding of the relationship of addition to multiplication.
So mathematicians look on prime numbers (2, 3, 5, 7…) as the building blocks of all the natural numbers (except 1). And it has long been the quest to probe into the precise nature of this relationship of the primes to the natural numbers.
However the fallacy being maintained is that somehow we can hope to coherently view natural numbers in a merely quantitative manner.
For example 6 is the natural number that results from multiplying the prime numbers 2 and 3 and as we have seen - properly interpreted - this already entails an unrecognised qualitative dimension!
When we add numbers (expressed with respect to the same dimension) no qualitative change is involved, However when we multiply, a qualitative as well as quantitative transformation in the units is inevitably involved.
So we can see right away, that the relationship of addition to multiplication inherently entails quantitative and qualitative aspects of number with respect to each other.
However - by definition - this key issue cannot be satisfactorily resolved in a merely quantitative manner. By its very nature, it thereby transcends the scope of conventional mathematical interpretation!
Type 1 and Type 2 Aspects of Number
We can see this problem more clearly by looking at the simplest possible case involving addition and multiplication.
When we add 1 + 1 (with both defined in a 1-dimensional manner as 1^1) the answer is 2 (i.e. 2^1).
So (1^1) + (1^1) = 2^1
Thus, a merely quantitative change is involved in this case (with no corresponding change in the qualitative nature of the units).
However when we multiply 1 by 1 (with again both defined in a linear manner) we obtain 1^2.
So (1^1) * (1^1) = 1^2.
Therefore, 2 in this case (representing the power of 1) relates directly to a qualitative dimensional notion of number.
Thus in this case - in reverse - while no quantitative change in units has taken place, a qualitative change (in the dimensional nature of the units) is involved.
So the important insight to grasp - using 2 as an example here - is that every number can be defined in both a quantitative and qualitative manner. 
This in fact leads to two distinctive aspects of equal importance in the natural number system.
In the Type 1 aspect, the quantitative number (as base) can change, while the number as dimension remains fixed (with the default value of 1). 
So the natural number system here is represented as:
1^1, 2^1, 3^1, 4^1,.......
In the Type 2 aspect - in a directly inverse manner - the qualitative nature of the number (as dimension) can change, while the number as base remains fixed (with the default value of 1).
So the natural number system here is represented as:
1^1, 1^2, 1^3, 1^4,.....
From the conventional mathematical perspective the second aspect of the number system seems both trivial and irrelevant as the reduced quantitative value of each of its members = 1.
Therefore, in conventional terms, the natural number system is represented merely in terms of its Type 1 aspect as:
1^1, 2^1, 3^1, 4^1,....
And, as in this system, number values are always ultimately expressed in a linear (1-dimensional) manner, 1 (as dimensional power) is likewise omitted.
So the natural number system is thereby, from this reduced perspective, more simply represented as:
1, 2, 3, 4,....
This in turn tends to consolidate the mistaken impression, that somehow these numbers represent abstract unchanging entities (independent of experience).
However, nothing could be further from the truth!
In fact, our actual experience of number, entails a continual dynamic interchange between base and dimensional aspects, with both continually switching as between quantitative and qualitative type meaning.
This also concurs in psychological terms with the important fact that understanding of number necessarily involves the interaction of both reason and intuition.
Thus in dynamic interactive terms, which defines actual experience, when we identify the quantitative nature of number with distinct number perceptions, the corresponding qualitative nature of number relates to the general concept (to which the individual perceptions are related).
Again in conventional mathematical interpretation, this crucial dynamic relationship as between number perceptions and concepts is completely missed.
Thus, our experiential understanding of number entails the continual interaction of perceptions and concepts which are quantitative and qualitative with respect to each other. The implicit recognition of this complementary distinction generates intuition.
This is vitally necessary in enabling the dynamic switching as between perceptions and concepts on the one hand, and then in reverse manner, concepts and perceptions to take place.
So we could validly say that in experience, the qualitative aspect of number interpretation is directly related to intuitive appreciation (of a holistic kind). Then the recognised quantitative aspect, directly relates in turn to rational appreciation (in an analytic manner).
However, just as in conventional mathematical interpretation, the qualitative aspect of number is grossly reduced in a merely quantitative fashion, the intuitive aspect of mathematical understanding is likewise grossly reduced in a merely rational manner.
So, from a conventional perspective, we identify both general concepts and individual perceptions of number in rational terms. We thereby fail to realise that intuition (of a qualitative holistic nature) is vitally necessary in enabling the dynamic switch as between these concepts and perceptions (and perceptions and concepts) to take place.
Therefore, when we misrepresent experience in this manner, the intuitive energy, so necessary to fuel these number interactions, itself becomes greatly depleted. Then number concepts and perceptions become increasingly rigid in experience, seemingly confirming each other in a merely quantitative manner.
And this mistaken static view comprises the essence of existing mathematical interpretation.
So Conventional Mathematics is formally interpreted solely in rational terms.
Thus the deeper - and truly fundamental - issues with respect to the number system of how qualitative is related to quantitative meaning (in an external objective sense) and how intuitive is related to rational interpretation (in an internal manner) remain completely unaddressed by the mathematical community.
This is why I realised from an early age that these key issues for Mathematics could only be properly addressed by someone working outside the recognised mathematical community.
Unfortunately the peer pressure to conform without question to accepted reduced procedures is so strong, that eventually it becomes exceedingly difficult for practicing mathematicians to even recognise the relevance of these vitally important issues!
Finite and Infinite
The confusion as between quantitative and qualitative is especially evident in the accepted mathematical treatment of finite and infinite notions.
Indeed such confusion is inherent in all mathematical proof.
I have illustrated before in an earlier article on this site the reduced nature of mathematical proof. Thus, when a proposition is proved (e.g. the Pythagorean Theorem) in a conventional mathematical manner, it is thereby assumed to apply to all cases (within its class) in an infinite manner. It is then likewise assumed that what is true for all (i.e. in infinite terms) can thereby be applied to any specific finite example.
In other words a direct quantitative relationship is assumed to connect both the finite and infinite realms.
Put more bluntly, it is assumed that the infinite notion can be successfully reduced in a merely finite quantitative manner.
So the conventional treatment of the infinite in Mathematics is to misleadingly view it as a linear extension of finite notions, in the mistaken impression that if we could somehow extend the number line indefinitely that it would eventually become infinite!
However, strictly speaking, this is utter nonsense. No matter how far the number line is extended, we still remain in the realm of the finite.
In fact the infinite represents a qualitatively distinct notion altogether!
Rational understanding corresponds with the quantitative notion of number in an actual finite manner.
However intuitive appreciation corresponds with the qualitative notion of number in a potential infinite manner.
In the dynamics of understanding, both finite (actual) and infinite (potential) notions continually interact (through reason and intuition) leading to ceaseless transformation in our number experience.
However in formal mathematical interpretation, these dynamics are then completely short-circuited.
The intuitive aspect of understanding, directly relating to the qualitative (infinite) aspect, is edited out of the picture altogether.
What remains is but a convoluted quantitative explanation (corresponding to mere rational interpretation) that is greatly lacking in coherence.
So here the general proof (say of the Pythagorean Theorem) is assumed to apply quantitatively to all actual cases within its class (i.e. in an infinite manner). And as “all” cases are now seemingly identified in actual terms, there then appears to be no problem in applying the proof to any specific example.
However the problem here is that “all” is strictly meaningless in an actual finite context because it is impossible to clearly define what it is supposed to represent!
Thus, the extraction of any finite set of cases (in a determinate manner) always implies another finite set of cases (that will always - by definition - remain indeterminate). So a proof cannot therefore actually apply to these - necessarily indeterminate - cases?
Now, I am not suggesting here that mathematical proof has no value.
What I am suggesting however, that it is necessarily of a merely relative nature.
The deeper issue, once we face up to the confusion implied by mere quantitative interpretation, is that a satisfactory defence of the notion of mathematical proof requires that an initial consistency be established as between the quantitative and qualitative aspects of number interpretation.
Thus properly understood in a dynamic interactive manner, the holistic intuitive appreciation associated with a mathematical proof potentially applies in all cases.
However the application of the proof in specific circumstances actually relates to a limited possible set of finite examples.
The huge issue therefore - preceding all conventional proof - relates to the very appropriateness of relating the distinct notions of finite and infinite, which are quantitative and qualitative (actual and potential) with respect to each other.
And of course, from the complementary psychological (internal) perspective, this relates to the appropriateness of combining both reason and intuition in mathematical interpretation, which again are quantitative and qualitative with respect to each other.
Thus, the most vital issue in Mathematics, prior to all conventional proof is the requirement to establish a consistent correspondence as between finite and infinite notions in objective terms and also reason and intuition from the complementary aspect of internal interpretation.
Indeed the Riemann Hypothesis, as we shall see, relates directly to this key issue.
Therefore the Riemann Hypothesis - when appropriately interpreted - establishes this initial condition for the consistent interaction of both quantitative (rational) and qualitative (intuitive) type appreciation with respect to number.
This of course implies that it is futile attempting to prove (or disprove) the Riemann Hypothesis in a conventional mathematical manner.
Again it transcends - by its very nature - conventional mathematical interpretation. More importantly, it cannot be even properly understood in this manner!
So the truth of the Riemann Hypothesis is already necessarily assumed in the conventional mathematical procedures used to establish proof.
And this is the true reason why the Riemann Hypothesis is so vitally important, as it points to the stark limitations in current mathematical understanding.
Cardinal and Ordinal Numbers
There is another deceptively simple way of illustrating the equal importance, in dynamic terms, of both the quantitative and qualitative aspects of the number system.
Again in conventional terms, the natural numbers are viewed as independent entities in an absolute manner.
However if natural numbers are indeed independent in this manner, it begs the question as to how such numbers can then be related to each other (as interdependent)!
So the very fact that mathematical operations assume that interdependent relationships between numbers are possible, once again betrays the fundamental reductionism in conventional interpretation.
Thus when the quantitative basis of number relates to number as independent, the corresponding qualitative basis of number relates to number as interdependent (with other numbers). 
Thus remarkably - and this is central to the issue of interpreting the nature of prime numbers - Conventional Mathematics is completely lacking any genuine notion of interdependence (which is of a qualitative nature). It thereby can only attempt to deal with such an inherently qualitative notion in a reduced - ultimately unsatisfactory - quantitative manner!
So in truth, the natural numbers are only independent in a relative manner; this equally implies that their interdependence is also of a merely relative nature.
And both of these aspects of number only have proper meaning in a dynamic interactive context.
Now, the quantitative notion of a number can initially be directly identified with its cardinal aspect; by contrast the qualitative notion is then - relatively - directly identified with its ordinal aspect. 3
So let us first look carefully at the cardinal notion of number!
A cardinal number e.g. 2, is defined quantitatively in collective whole terms (i.e. as an integer).
Thus, when we attempt to define a cardinal number by its individual units, they thereby lack any qualitative distinction.
So 2 = 1 + 1 and as these units are homogenous (exactly similar) we have no means of making a qualitative distinction as between each unit.
Thus the cardinal is an independent notion of number based on its quantitative whole identity (i.e. as an integer).
And this view is represented by the Type 1 aspect (of the number system) that we mentioned earlier.
However the ordinal notion of number properly relates to an inverse complementary interpretation.
Just as the cardinal notion relates to number independence, the ordinal relates to number interdependence.
Thus the ordinal notion of number relates necessarily to a relationship within a collection (or group) of numbers.
For example in the simple case of 2 members, the ordinal notions of 1st and 2nd only acquire meaning through their relationship with each other. Thus if we fix one member as the 1st, the other member - in this relative context - can then be defined as 2nd.
Thus whereas in Type 1 terms, 2 is defined by its general collective nature (as quantitatively independent), in Type 2 terms, 2 is defined by its unique individual members (as qualitatively interdependent).
So 2 in this context = 1st + 2nd.
And just as the individual units of the cardinal notion of a number have no distinct qualitative identity, in an inverse complementary manner, the collection (sum) of the individual ordinal notions of number has no distinct quantitative identity. (We will show how this is demonstrated presently).
Perhaps there is no area in Mathematics that is more deficient than its treatment of the ordinal nature of number.
Once again, because of reduced quantitative interpretation (in a cardinal manner), ordinal relationships are assumed to directly correspond with cardinal notions.
So for example, 1st, 2nd, 3rd, 4th,... are assumed to correspond with the cardinal notions of 1, 2, 3, 4.
In the mathematical textbooks on ordinal numbers, a total conspiracy of silence is maintained regarding their inherently qualitative nature.
Though much mention is made of number rankings in this context, it will never be formally admitted that these strictly relate to a qualitative (relational) - rather than quantitative (independent) - notion of number.
So once again, the ordinal (qualitative) notion of number is simply reduced to cardinal (quantitative) type interpretation.
However properly understood - as I have been illustrating - cardinal and ordinal notions bear an inherently complementary relationship to each other (as polar opposites) which can only be properly understood in a dynamic interactive manner.
When one recognises this true dynamic nature of the number system, combining both cardinal (quantitative) and ordinal (qualitative) aspects, the key issue then relates to the ultimate reconciliation of these two distinctive notions in terms of each other.
Once again, if we are unable to establish the prior consistency of both of these aspects, then the whole subsequent mathematical edifice is built on sand (as it simply assumes such consistency as its starting basis).
A Crossroads Illustration
Shortly after my initial disquiet in primary school (arising from the qualitative issue that arises with multiplication) I soon became aware of a related problem, which subsequently was to have a profound influence on my mathematical development.
I had isolated the qualitative aspect relating to multiplication through 1 * 1 = 1^2.
However, when one now attempts to obtain the square root of 1, two distinct answers are possible i.e. + 1 and – 1.
This seemed distinctly odd, as from my perspective, I believed that a symmetry should exist as between the squaring of a number and the subsequent obtaining of its square root.
When we square a number in conventional mathematical terms, only one answer results; however when we then reverse the operation, two answers then apparently emerge which are the quantitative opposites of each other.
So, just as it would not be feasible from the conventional mathematical perspective to consider that a theorem could be proved true one day and false the next, equally it didn't appear feasible to me that we could accept the truth of two opposite answers in quantitative terms.
Even though I could not properly articulate my reservations at the time, I suspected that Conventional Mathematics was restricted to just one logical system (which could not give a coherent explanation of the square root of 1). So even then, I began to consider that other logical systems, associated with these dimensional numbers, to which I already attached so much importance, might exist.
It was some 10 years later following intensive study of Hegelian philosophy that I finally attained full clarity regarding the answer.
Though the issues here are indeed quite subtle, a basic explanation can be given in terms of the - apparently - simple task of assigning directions at a crossroads.
Imagine a vertical road heading from South (S) to North (N) and then a crossroads drawn horizontally East (E) to West (W) at the halfway mark.
Now if starting at the bottom of this vertical road heading North (N), one encounters the crossroads, it would be easy, for example, to unambiguously assign the direction West (W) as a left turn.
Then, if later starting at the top of the vertical road heading South (S), again it would be easy to unambiguously assign the direction East (E) also as a left turn.
The point here is that the notion of direction is unambiguous when we use just one isolated frame of polar reference.
So in the first case, the left turn was assigned with respect to the North (N) direction of movement; in the 2nd case, it was assigned with respect to the South (S) direction.
Thus, both these single polar frames of reference, when treated as independent of each other, give unambiguous answers.
This precisely defines what is meant by 1-dimensional interpretation (i.e. interpretation based on one unambiguous pole of reference).
However when we simultaneously try to combine both North (N) and South (S) directions, deep paradox applies with respect to assigning left and right turns.
So what is left from one polar frame of reference is right from the other; likewise what is right from one reference frame is left from the other.
Thus, 2-dimensional interpretation entails using both poles of reference simultaneously, which thereby defines the nature of interdependence in this case.
Therefore, what is unambiguous in a linear (1-dimensional) context, appears deeply paradoxical from a circular (2-dimensional) perspective.
The great relevance of this example is that all experience of phenomena is necessarily conditioned by fundamental reference poles, which keep switching dynamically in experience.
Thus analogous to East (E) and West (W) directions in our crossroads example, we have internal (subjective) and external (objective) poles.
Then, likewise analogous to North (N) and South (S) we have individual (part) and collective (whole) poles. Put another way, we have quantitative and qualitative frames of reference.
Therefore, all mathematical knowledge is necessarily conditioned by these same fundamental poles, which dynamically interact in experience.
And since our mathematical knowledge (including most importantly the number system) necessarily springs from human experience, it likewise is inherently of a dynamic interactive nature.
However due to centuries - indeed millennia - of reduced interpretation, we have all but lost sight of this simple conclusion.
We can in fact use our crossroads example to graphically illustrate the problem with conventional mathematical interpretation of the prime numbers.
Because of its 1-dimensional nature, when mathematicians attempt to interpret the individual nature of primes, they do this within a (merely) quantitative frame of reference.
Then, when they now attempt to interpret the collective nature of the primes, with respect to the frequency of prime number distribution, they again do this within an unambiguous quantitative frame of reference.
However as we have seen, individual and collective are polar opposites with respect to each other. Therefore when we simultaneously attempt to combine both the individual and collective nature of the primes, deep paradox results, as the relationship is now quantitative as to qualitative (and qualitative as to quantitative) with respect to each other!
This means in effect that when we attempt to combine both the independence of each individual prime with the collective notion of the overall interdependence of primes (with respect to the natural number system), this must be interpreted as the dynamic relationship of the quantitative (cardinal) and qualitative (ordinal) aspects of number respectively.
However, Conventional Mathematics attempts this combination solely in terms of cardinal notions!
So the position of Conventional Mathematics with respect to the ultimate nature of the number system is like one, at a crossroads who can only identify left turns.
It is simply not possible to successfully identify turns at a crossroads intersection in this manner, with just one notion of direction!
Much more importantly - though not yet recognised within Mathematics - the true nature of the number system equally cannot be identified in a solely quantitative manner! 
So we have demonstrated that the 1st and 2nd dimensions (of 2) holistically apply to two distinct modes of logical interpretation.
The former (1st) is suited to the conscious positing (+ 1) of one pole in an analytic quantitative manner. The latter (2nd) is suited to the (unconscious) negation of this pole (– 1) in a holistic qualitative manner.
Thus in the very dynamics of understanding, when for example we consciously posit a turn as left, we must thereby likewise negate the opposite possibility (i.e. that it is right) in an unconscious manner.
Now, this is especially true in terms of the fundamental poles that we have mentioned (that necessarily condition all phenomenal experience of reality).
So, for example, to (consciously) posit a mathematical object in an external manner, we must thereby (unconsciously) negate its opposite internal mental direction (and vice versa).
Likewise to (consciously) posit a mathematical object in quantitative terms, we must thereby (unconsciously) negate its opposite qualitative aspect (and vice versa).
Thus, once again, our mathematical experience is of a dynamic interactive nature, entailing the ceaseless interplay of conscious and unconscious aspects.
However formal mathematical interpretation completely misrepresents this experience.
So for example it maintains the fiction that mathematical objects can be posited in a merely conscious absolute manner.
It thereby remains - literally - unconscious with respect to the true dynamics involved.
And it is no use just shrugging off such observations as merely philosophical in nature (with no direct relationship to Mathematics), for they expose current interpretation as representing a great distortion of true mathematical reality
And nothing should be of more relevance than this!
So returning to our crossroads, properly understood, for the positive (conscious) recognition of a left turn (travelling in one direction), we have the (unrecognised) shadow recognition in negative terms (that it is not a right turn).
Likewise for the positive (conscious) recognition of a right turn, we have the (unrecognised) shadow recognition in negative terms (that it is not a left turn).
So positive (conscious) recognition (+ 1) is associated with the 1st dimension and negative (unconscious) recognition with the 2nd dimension (– 1).
Then when we combine both 1st and 2nd dimensions, we thereby generate the realisation of the true interdependence of both left and right (when viewed simultaneously from both North and South reference poles).
Now this is thereby represented as + 1 – 1 = 0.
So the important point here is that the notion of pure interdependence strictly has no quantitative meaning.
And this is demonstrated by the use of mathematical symbols which now are given both an analytic and holistic interpretation.
So in analytic (quantitative) terms, as every child will recognise + 1 – 1 = 0.
But equally in holistic terms, the simultaneous positing and negating of opposite poles, as interdependent in experience, results in a purely intuitive type realisation of meaning (which cannot thereby be interpreted in a rational analytic manner).
Using physical type language, the realisation of interdependence in experience, results from the fusion of psychic matter and anti-matter respectively. This then results in an intuitive generation of psycho spiritual energy.
Thus, though we commonly represent numbers as having distinct form (in quantitative terms), properly understood from the extreme qualitative holistic perspective, they represent pure energy states!
Again, all of this is intimately important for mathematical experience, where external phenomena (as mathematical objects) and internal mental constructs (as means of interpretation) continually interact in a dynamic manner.
Equally, this is intimately important as quantitative (cardinal) and qualitative (ordinal) notions of mathematical meaning likewise continually interact.
And Conventional Mathematics is completely unsuited as it stands to obtain any genuine appreciation of such interaction!
All of its symbols possess an alternative holistic meaning which arises directly from the dynamics of such interaction. However as this dynamic interaction is ignored altogether (even though it is the essence of mathematical experience), appreciation of the corresponding holistic meaning of such symbols is completely missing from formal understanding.
So the 1-dimensional interpretation - which defines Conventional Mathematics - is associated solely with analytic type quantitative appreciation, in a static absolute manner.
2-dimensional interpretation, by contrast, entails both analytic (quantitative) and holistic (qualitative) meaning in a dynamic interactive manner.
Though we have used the relatively simple case of 2 to illustrate higher dimensional interpretation, remarkably every number (other than 1) is associated with a unique dynamic configuration of this relationship of analytic to holistic meaning.
So, there is not just one valid interpretation of Mathematics (which is assumed in conventional terms as 1-dimensional).
In fact, potentially an unlimited number of such interpretations holistically exist, all possessing a partial validity in relative terms.
It is only at the 1-dimensional level that interpretation appears absolute. Once again, this is because - by definition - qualitative is reduced to quantitative interpretation at this level. With all other number dimensions (other than 1) interpretation is necessarily relative with interacting analytic (quantitative) and holistic (qualitative) aspects.
Anybody reading this should appreciate that these are not minor points, but absolutely fundamental going to the very core of what is meant by Mathematics.
Indeed, to be quite blunt about it, current Mathematics is simply not fit for purpose.
Certainly it has attained enormous development with respect to one highly specialised quantitative area of understanding. But the overall philosophical framework in which it operates is hopelessly confused. This then acts to completely blot out recognition of the much greater role that a comprehensive integrated Mathematics can play in our world.
Fortunately for our purposes, in a certain important sense all “higher” dimensional interpretations can be validly seen as an extension of what is involved in the 2-dimensional case.
So “higher” dimensional interpretation (associated with the configuration of the relationship between 3, 4, 5 … reference frames), becomes increasingly more dynamic requiring considerable refinement with respect to both rational and intuitive type appreciation. However, the basic situation remains the same in that they all entail a dynamic interactive context of meaning, entailing both quantitative (analytic) and qualitative (holistic) aspects of interpretation.
2-dimensional interpretation can be referred to as the complementarity of opposites, or coincidence of opposites (as used by Nicholas of Cusa).
An early Western expression is due to Heraclitus:
“The way up is the way down
the way down is the way up”
It is present in all the mystical traditions, especially Eastern. Taoism in particular with its emphasis on the yin and the yang is explicitly framed in terms of such 2-dimensional interpretation.
From a Western mystical perspective, one excellent example of the dynamic nature of such understanding in psycho spiritual development, is provided by Evelyn Underhill's classic work “Mysticism”.
It is found in philosophy, principally Hegel (as I have mentioned).
It is also prominent in the psychology of Carl Jung.
Indeed it is very much part of Quantum Mechanics for example in the complementary nature of wave and light particles of light.
However, its deeper implications with respect to the fundamental paradigm still employed in Physics, have not yet been recognised.
However my key point is that it is intimately required in terms of coherent mathematical interpretation in a dynamic interactive manner.
Indeed since Riemann, it is clear that the number system, very much akin to Quantum Mechanics, also possesses wave (i.e. through the zeta zeros) as well as recognised particle aspects (i.e. standard analytic notions).
So I was finally now able to answer my childhood problem.
In fact, the very notion that the number 1 has 2 roots itself requires clarification.
In fact, the two roots of 1 relate to two distinctive numbers in the Type 2 system.
Thus, 2 in ordinal terms is composed of its two uniquely distinctive members (1st and 2nd) represented through the Type 2 aspect of the number system as 1^1 and 1^2 respectively.
So the 1st root of 1 corresponds to the square root of 1^2 i.e. 1^1 = + 1.
The 2nd root corresponds to the square root of 1^1 i.e. 1^(1/2) = – 1.
Now to (indirectly) demonstrate in quantitative terms the interdependence of the qualitative notions of 1st and 2nd, we combine (i.e. add) both roots.
Thus + 1 – 1 = 0.
And of course the sum of the roots of 1 will always equal zero.
So for example, when we obtain the 3 roots of 1 (1^3, 1^2 and 1^1) by multiplying the dimensional power i.e. exponent by 1/3, we obtain 1^1, 1^(2/3) and 1^(1/3) respectively i.e. – 1/2 + .866i, – 1/2 – .866i and 1.
And the sum of these 3 roots = 0
So this in fact provides the explanation of the complementary behaviour of cardinal and ordinal numbers mentioned earlier.
Thus once again a cardinal number such as 2 has a whole collective identity in quantitative terms. However, when expressed in terms of its individual units i.e. 1 + 1, no qualitative distinction can be made as between these units.
In an inverse complementary manner, the ordinal notion of 2 (representing 2-dimensional) is uniquely expressed in terms of its individual units i.e. 1st and 2nd in a qualitative manner.
However it has no collective whole identity in a quantitative fashion.
This is demonstrated through the sum of its roots (expressed indirectly in a circular quantitative manner).
So 1st + 2nd in this context = + 1 – 1 = 0.
Now the clear complementary connections as between cardinal (quantitative) and ordinal (qualitative) aspects of number, emphatically demonstrate how they are polar opposites with respect to each other. Thus the inter-relationship of both aspects can thereby only take place in a dynamic interactive context.
So 2-dimensional interpretation, entails combining both 1st and 2nd dimensions
(quantitatively represented by + 1 and – 1) as interdependent.
What this means is, that as two directions of understanding (external and internal) are dynamically combined in experience, proof is thereby of a merely relative nature. So + 1 and – 1 are interpreted in 2-dimensional terms through a circular both/and logic (i.e. the logic of interdependence).
When we separate these as + 1 and – 1 (in 1-dimensional terms) we switch to an absolute linear either/or logic (i.e. the logic of independence).
Now, we will see exactly what this entails in terms of mathematical proof.
If we try to maintain the absolute validity of a proof, there are two ways of attempting such an interpretation.
1) we can associate the absolute truth positively with the external pole as positive. So we now maintain that a proof is absolutely true in an external sense relating to objective reality.
2) we can associate the absolute truth positively with the internal pole as positive. So we now maintain that a proof is absolutely true in an internal sense relating to mental interpretation.
The very nature of 1-dimensional understanding is to assume a direct correspondence as between both aspects (where qualitative and quantitative are reduced in solely quantitative terms), so that the proof (irrespective of what pole is emphasised) is still considered absolutely true.
However from a 2-dimensional perspective, both external and internal are now seen as complementary (i.e. opposites of each other).
Therefore when we reflect 2-dimensional interpretation (through a 1-dimensional lens), if a proposition is absolutely true in an external sense, it is thereby absolutely false (from the internal perspective).
Likewise if the proposition is absolutely true in an internal interpretative sense, then it is thereby absolutely false from the corresponding external objective perspective.
So, the fact that the two roots of 1 are diametrically opposite in quantitative terms (in an absolute manner) reflects the fact that the very process of taking such roots thereby reduces 2-dimensional understanding in a 1-dimensional manner.
Thus, I had finally mastered my childhood problem relating to the two roots of 1 to my satisfaction, establishing (what I had already suspected) that 2 as a dimensional number is indeed associated with a distinctive type of logical interpretation in holistic terms.
Put more starkly, I had convincingly established (from my perspective) that the seemingly simple operation of obtaining the square root of 1 cannot be properly explained in the conventional mathematical manner. So again, the paradoxical outcome i.e. two valid quantitative results, which are the opposite of each other, results from the reduction of 2-dimensional logic in a 1-dimensional manner.
Two-way Switching in Experience
The hardest insights to grasp are the most simple and obvious. The paradox is that because these emanate directly from the unconscious, it requires considerable stillness of thought before they can be properly realised.
So it was only comparatively recently that it clearly dawned on me that implicit in our very recognition of ordinal distinctions at the base number level of interpretation is this alternative holistic qualitative notion of number.
Let me clarify briefly. Initially I identified the cardinal notion with the base number and the corresponding ordinal with the dimensional notion respectively.
So again in the expression 2 ^ 3, 2 represents the base and 3 the dimensional number respectively.
Therefore the base number was treated in a quantitative and the dimensional number – relatively - in a qualitative manner respectively.
However, whereas clearly we can distinguish as between 1st and 2nd dimensions in this general sense, equally we can distinguish as between 1st and 2nd with respect to specific base quantities.
Thus, what actually happens in experience is that both base and dimensional numbers keep switching as between both their quantitative and qualitative meanings.
So in dynamic relative terms, when we choose the base number as quantitative, then the dimensional number is – relatively - of a qualitative nature; however when the reference frame switches, the base number is now qualitative and the dimensional number - relatively - of a quantitative nature.
Put another way, base and dimensional numbers have both quantitative (cardinal) and qualitative (ordinal) aspects (depending on context).
Once again however Conventional Mathematics short-circuits this dynamic interaction completely, by reducing (in both possible cases) qualitative to quantitative interpretation.
Therefore in Conventional Mathematics, both the base and dimensional numbers are treated merely with respect to their quantitative aspects.
Though it is initially valid in Type 2 terms to identify ordinal with qualitative meaning, in Type 3 terms, where we allow for the simultaneous interaction of both reference frames, then all quantitative (cardinal) numbers are seen as possessing a qualitative (ordinal) aspect, and all qualitative (ordinal) numbers likewise a quantitative (cardinal) aspect.
This is exactly analogous with Quantum Mechanics where all particles (such as light) possess wave attributes and all waves possess particle attributes.
In fact, the true (unrecognised) reason why this is so with respect to physical understanding is that waves and particles - properly understood - represent the interaction of both quantitative (analytic) and qualitative (holistic) aspects of meaning.
Now again because the reduced quantitative paradigm of Mathematics is so deeply embedded in Physics, this obvious connection is not made. So the reason why Quantum Mechanics appears so non-intuitive is that its findings are being filtered through a reduced (1-dimensional) mode of interpretation, which is quite unsuited for the task of coherent interpretation.
However, the deeper point that I wish to make is that the behaviour of the number system is equally strange and non-intuitive in terms of the conventional paradigm.
So in this respect, Conventional Mathematics is still firmly locked in a Newtonian type mind-set and not yet ready to embrace the truly extraordinary dynamic nature of its number system.
Indeed - and this might come as a big surprise - the fundamental basis of such strange physical behaviour at the quantum level (i.e. interaction of wave and particle aspects) is rooted in the prior nature of number (when appropriately interpreted in dynamic interactive terms).
Thus again from a relative perspective, where cardinal distinctions are quantitative, ordinal distinctions correspond with (unrecognised) holistic type interpretation in a qualitative manner.
In other words the former relates direct to the conscious aspect of understanding with the latter corresponding to the (unrecognised) unconscious aspect.
So quite literally, we still remain totally unconscious of what implicitly is required in the very simplest task of number recognition.
We believe therefore that ordinal recognition corresponds with conscious appreciation, simply because we have successfully reduced such ordinal understanding in a merely quantitative manner.
However, to make coherent sense of the very simplest example of number recognition, we must formally include both conscious and unconscious aspects of understanding in interpretation.
And once again, as stated so often here, Conventional Mathematics is formally defined in a merely conscious rational manner.
This is why without any exaggeration, I strongly suggest that nothing but a total revolution in our mathematical understanding is now long overdue.
I, for one, have steadily grown over the past 50 years or so to realise how incoherent present interpretation is in explaining the very basics of our number system (and by extension, the basics of everything in Mathematics).
Thus by raising my voice for anyone who wishes to hear, others may likewise be encouraged to look at mathematical reality in a completely new light.
1. As we are referring here to the dynamic relationship as between the base and dimensional numbers respectively, we have to be very careful in interpretation.
Thus when we initially fix the base number as quantitative, the dimensional number is thereby - relatively - qualitative.
However, when we fix the base number as qualitative, the dimensional number is now - relatively - of a quantitative nature.
So symbols representing base and dimensional numbers, possess both quantitative and qualitative aspects depending on context.
In this sense it is exactly similar to identifying the two turns at a crossroads which can be each left or right depending on the direction (N or S) from which they have been accessed.
The important thing to grasp is that actual experience of number is of a dynamic interactive nature, with meaning continually switching as between quantitative and qualitative (and qualitative and quantitative) aspects.
So, the way that the nature of number is formally interpreted within Mathematics i.e. in a static quantitative manner represents a complete misrepresentation of the true experiential position.
2. The notion of “base” with respect to numbers can have different connotations.
So it is important to be clear regarding the manner in which I am using the term in the context of this article.
In general terms, when a is raised to the power of b, i.e. a ^ b, a is thereby the base and b the dimensional number respectively.
3. Again it is important to recognise that we are speaking in a dynamic relative context.
So, because cardinal numbers are now independent and quantitative in a merely relative sense, this implies that when we switch reference frame, they now display interdependent features of a qualitative nature (which in fact is key to appreciation of the true nature of the Riemann zeta zeros).
Equally when we switch reference frames, ordinal numbers display independent features of a quantitative nature (which is likewise key to recognition of the true nature of the unrecognised Zeta 2 zeros that I deal with in the next article).
So like the sub-atomic quantum level of matter, all numbers possess both particle and wave aspects that can interchange depending on context.
Again, though not at all yet recognised by either the Mathematics of Physics community, the behaviour of sub-atomic particles has its deeper roots in the true dynamic nature of the number system.
4. We can see here how there is necessarily a coincidence of both cardinal and ordinal meaning with respect to the designated 1st item. So one member of the group (i.e. with a cardinal identity) must be initially singled out and then designated as the 1st item in ordinal terms.
So once again in a linear (1-dimensional) interpretation of mathematical symbols (which defines Conventional Mathematics) ordinal will necessarily coincide with cardinal interpretation which effectively means they are necessarily reduced in a merely quantitative manner.
5. The qualitative nature of the primes (or more correctly the relationship of the primes to the natural number system) is indirectly recognised by mathematicians.
For example the title of Marcus du Sautoy's eminently readable book on the Riemann Hypothesis is entitled “The Music of the Primes”.
Indeed, exactly the same title “The Music of the Primes” is used by Keith Devlin for his article on the Riemann Hypothesis in his book “The Millennium Problems”.
And Michael Berry, who is to the forefront in terms of elaborating on the connections between the (Riemann) zeta zeros and quantum chaos sums it up in this manner.
“Loosely speaking, the Riemann Hypothesis states that the prime numbers have music in them”.
Quite remarkably however, nobody seems to be able to make the obvious inference, that this clearly points to the fact that the number itself possesses a crucial qualitative holistic aspect that cannot be interpreted in the quantitative manner of Conventional Mathematics.
So this is akin (within Mathematics and Physics) to the situation where many highly gifted people seek to analyse the structure of Beethoven's music at ever greater levels of quantitative depth.
However no one ever considers listening to his music to directly appreciate its quality. Worse still, when requested to listen, the proposal is met with a resolute refusal.
I keep repeating. An enormous cultural blindness has now developed within Mathematics though centuries - indeed millennia - of false conditioning that the number system can be successfully understood in a merely quantitative manner.
Such thinking is very much misplaced and needs to change rapidly! So much depends on it!
Secrets of Creation, Vol 1, Vol 2 and Vol 3: Words: Matthew Watkins, Pictures: Matt Tweed; Immorata Press (2010, 2011 and 2013).
This represents a truly excellent (if rare) attempt by a professional mathematician to intelligibly convey the intricate mathematical issues associated with the Riemann Hypothesis in a most accessible manner for the (interested) general reader.
The skilful manner of exposition (helped by some ingenious illustrations) then leads naturally to the author's own fascinating exploration of deeper issues regarding the nature of the number system.
Riemann Hypothesis - This is my personal blog on the topic, complied over the past 4 years. It traces the evolution in my thinking since “A Deeper Significance - Resolving the Riemann Hypothesis” appeared on this web-site.