Friday, May 7, 2010

Transcendental Numbers

Transcendental numbers especially e and pi play an important role with respect to prime number behaviour.

For example the simplest version of the prime number theorem, providing a general means of calculating the frequency of primes uses the natural log of n (which is based on e).

Also the sum of terms for even number dimensional values in the Euler Function - which again can be shown to have an intimate relationship with the primes - contains neat numerical expressions involving powers of pi.

We have seen that all number types (as quantities) can be given a corresponding holistic meaning (in qualitative terms). In this context we have already looked at the primes, (algebraic) irrational and imaginary numbers.

Though transcendental numbers are also classed as irrational, they differ in an important way from other (algebraic) irrational numbers such as the square root of 2.

All (algebraic) irrational numbers can ultimately be derived as solutions to higher dimensional polynomial expressions (with rational coefficients).

For example, in this context the square root of 2 arises as the solution of,

x^2 - 2 = 0.

So the irrational number here serves - literally - as the reduced expression of a number that is rational with respect to a higher dimension (or combination of dimensions) but then becomes irrational when interpreted in a reduced linear manner.

So the number 2 in this context has a rational meaning when expressed with respect to the 2nd dimension (i.e. where x is raised to the power of 2) which then becomes irrational when given a reduced linear value with respect to the 1st dimension (where x is raised to the power of 1).

Now the corresponding qualitative interpretation here is very revealing.
Interpretation according to 2-dimensional understanding is indeed rational within the same dimensional perspective.

As we have seen, such 2-dimensional understanding represents both/and logic that is based on the complementarity of opposite polarities. However when we attempt to express such interpretation through the medium of standard either/or linear logic (relating to the 1st dimension) it appears as irrational (i.e. paradoxical).

In fact Hegel is very interesting in this context as identifying "true" reason with 2-dimensional (rather than standard 1-dimensional logic). Such reason then appears very circular and paradoxical (irrational) from the conventional linear standpoint.

One of the problems that frequently arises in the spiritual contemplative life is that intuitive type awareness - properly pertaining to higher dimensional understanding - inevitably gets reduced to a degree to the standard 1-dimensional understanding (that governs so much of practical affairs).

St. John of the Cross carefully distinguishes lower-level attachments (with respect to conventional understanding) and higher-level attachments (with respect to intuitively inspired consciousness).

The first - with respect to linear (1-dimensional) understanding he terms active and the second type - with respect to higher dimensional understanding he terms passive.

So the famous dark nights that he depicts are with respect to the need for cleansing the spiritual aspirant of all passive attachments (so that spiritual intuitive illumination is not confused with the conscious rational symbols through which it is mediated).

And the direct relevance of this for Mathematics is that strictly this same contemplative process is necessary to avoid confusing - in any relevant context - infinite with merely (reduced) finite notions. And unfortunately, conventional mathematics is rife with such reductionism (that is not even recognised as such!)

Now all of this sets up the proper context of appreciating the true qualitative significance of what is meant by a transcendental number.

A major clue comes from consideration of the geometrical nature of pi (the best known of all transcendental numbers).
Pi as is well-known expresses the ratio of the circumference of a circle to its line diameter.
In corresponding qualitative terms, a transcendental number expresses the relationship as between circular and linear interpretation (with respect to such a number).

Now the distinction here from the earlier (algebraic) irrational numbers is that transcendental numbers cannot be reduced in the same linear manner.
In quantitative terms this entails that transcendental numbers cannot arise as the reduced 1-dimensional expression of a variable (with respect to polynomial equations with real rational coefficients).

In other words because the emphasis is now - neither on linear or circular interpretation as separate - but rather on what connects both, (qualitative) transcendental understanding cannot be expressed through either aspect (as independent).

Thus there is something more elusive about both the nature of transcendental numbers (as quantities) and corresponding transcendental numbers (as qualitative understanding).

In spiritual terms such appreciation would relate to a more refined contemplative state where finite symbols can mediate the pure spiritual light (with little conscious attachment). Realistically some degree of attachment would remain (at an imaginary unconscious rather than real conscious level).

The work of Cantor was to point to this elusive nature of transcendental numbers. Though they are amazingly dense with respect to the number system, remarkably few are directly known. Indeed Cantor was able to point to their existence without directly identifying any particular example!

Also the paradoxes of infinity that Cantor was able to demonstrate, whereby different number sets exhibit different degrees of infinity, I would see as an ultimately unsatisfactory way of understanding the relationship as between finite and infinite. From a spiritual perspective, infinite degrees of infinity would refer to the fact that each object (as finite) can uniquely reflect the spiritual light (that ultimately transcends and yet is potentially immanent in every object).
In this context - of dynamic experiential interaction - the notion of one homogeneous infinite concept makes no sense.

Though Mathematics uses the linear rational approach it is quite clear, in the very names we use to describe the main number types, that their inherent nature is far from rational. For example as well as the primes with connotations of primitive, we have the (algebraic) irrationals, the imaginary, the transcendental and ultimately transfinite numbers.

In spiritual contemplation, as one moves away from the rational world, there are various hierarchies or degrees of more intuitively inspired understanding. likewise with respect to number types as we move away from the rationals they exhibit greater and greater degrees of subtlety (not directly amenable to rational interpretation). Ultimately any remaining finite quality ultimately disappears (as with the transfinite numbers).

Again what is missing from conventional appreciation is that these number types require distinctive means of qualitative interpretation (that ultimately transcends all rational understanding).

In this way the Pythagoreans were indeed correct that - properly appreciated - mathematics should offer an extremely important means of attaining ultimate contemplative awareness of reality.

We have already looked at the inherent nature of pi (which in qualitative terms entails the relationship as between circular and linear understanding).

It is fruitful to also look at the inherent nature of e (which plays perhaps an even more significant role with respect to the primes).

As we know differentiation and integration play very important roles in Mathematics; equally they play an extremely significant role with respect to psycho spiritual development.

If we differentiate the simple function y = x^2 we get dy/dx = 2x.

So what has happened here is that we have reduced the higher dimensional expression to a lower dimensional form (where the dimension 2 as qualitative becomes transformed to 2 as quantitative number).

Differentiation in psychological terms is somewhat similar and entails a form of reductionism whereby object phenomena can become identified in quantitative terms.

Now integration (in this context) in both mathematical and psychological terms entails the reverse procedure whereby the lower quantitative understanding becomes transformed in higher dimensional terms. So when we integrate 2x we obtain x^2 (with number as quantity now becoming the corresponding number as qualitative dimension).

In psycho spiritual terms (and indeed in all biological life processes) it is very similar. Integration here essentially entails moving the quantitative to the qualitative perspective (where various quantitative parts can assume a coherent qualitative whole identity).

Now the simple function y = e^x is unique in the sense that when we differentiate (and once again integrate), its value remains unchanged.

Therefore from a qualitative mathematical perspective, the number e plays a unique role in that it represents that very point whereby differentiation becomes inseparable from integration in development.

In psycho spiritual terms, this would entail a highly refined state where the differentiation of phenomena becomes so rapidly interactive that they no longer even appear to arise in experience. Therefore in this state, differentiation (in the generation of discrete phenomena) would become inseparable from continuous integration in maintenance of a stable spiritual equilibrium.

Now we have already seen that the inherent nature of a prime number entails the extremely close relationship as between both linear and circular modes of behaviour.

Not surprisingly therefore prime number behaviour (in its general distribution) is linked very closely with e (which in its qualitative nature entails the reconciliation of both aspects).

The ultimate goal in terms of psycho-spiritual development is the complete unravelling of all primitive impulses. This is achieved when a purely continuous intuitive state, combined with the rapid unattached generation of discrete phenomena in experience, unfolds.
This further entails the final reconciliation of both the linear and circular modes of interpretation.

In this way the prime (i.e. primitive) problem is eventually solved (though in truth human experience can always only approximate to this goal).

In a very similar fashion the prime number problem is likewise eventually solved through obtaining the complete reconciliation of the linear and circular methods of behaviour (and corresponding interpretation).

This reconciliation is already implicit in a general distributional manner in the prime number theorem (that incorporates the use of e). However errors still remain here with respect to the exact prediction of the number of primes.

However, as we shall see the full reconciliation of linear and circular aspects is in fact implied by the Riemann Hypothesis.

Indeed, once again, in this context the Riemann Hypothesis simply operates as the necessary condition to ensure the full reconciliation (or consistency) of both linear and circular aspects of quantitative prime number behaviour (together with reconciliation with respect to the corresponding linear and circular aspects of qualitative interpretation).

Thursday, May 6, 2010

Imaginary Numbers

A key problem that I faced for many years related to the provision of a satisfactory holistic mathematical interpretation of imaginary numbers.

Ultimately familiarity with Jungian psychology provided the key to the breakthrough.

It is amazing how so often the actual symbols and terms used in mathematics with respect to quantitative interpretation are deeply suggestive of their true holistic nature!

For example rational numbers (as quantities) bear a direct relationship with - what is termed - the rational paradigm (as qualitative).

We also saw that the holistic interpretation of irrational numbers leads to the need for a paradoxical type of circular logic which - quite literally - seems irrational in terms of accepted linear understanding.

We also saw that the very symbols used to represent 1 and 0 are (with small variations) the line and the circle. So the holistic interpretation of the binary digits incorporates both linear (1) and circular (0) interpretation.

Then we found that the holistic nature of prime numbers is deeply rooted in the nature of primitive behaviour (with respect to both physical and psychological reality).

Likewise in common scientific terms what is "real" is generally interpreted as that which conforms to conscious understanding (of a linear rational nature).

Imaginary i.e. what relates to the imagination, has the corresponding connotation of more intuitively inspired understanding that pertains directly to the unconscious.

So this would suggest from a qualitative perspective that the imaginary notion in mathematics in some way incorporates holistic meaning that properly pertains to the unconscious.

However we can do a little better than that in providing a precise interpretation of what an imaginary number entails (from a holistic qualitative perspective).

In quantitative terms the imaginary number i is defined as the square root of - 1 (which is not therefore reducible in a real manner).

Now we have already defined in holistic terms the 2nd dimension
- 1 as the (unconscious) negation of what has been formerly posited in real (conscious) terms.

Such unconscious negation is of a dynamic nature entailing interaction with the former posited direction of experience. So through such interaction we have the fusion of opposite polarities in the generation of spiritual intuitive energy. In this respect the process is remarkably similar to the manner in nature by which matter and anti-matter particles likewise interact to generate physical energy!

However this second intuitive dimension of understanding is strictly of an empty holistic nature (and therefore not directly amenable to rational interpretation).

However indirectly it can be given a reduced linear rational expression through obtaining the square root of this negated direction ( - 1).

In this manner - though again of a necessarily indirect paradoxical manner - holistic intuitive understanding properly pertaining to unconscious meaning can be formally incorporated in an acceptable scientific manner.

It took a long time for Mathematics to recognise the importance of imaginary numbers and then their comprehensive expression (together with the real aspect) of complex numbers.

I think it is only fair to say that the full acceptance of complex numbers in quantitative terms has truly revolutionised mathematical understanding. For example the Riemann Hypothesis which is our immediate concern here, would have been impossible to conceive in their absence!

However what is still completely missing from present mathematical understanding is any corresponding understanding that imaginary numbers need to be equally incorporated with real in a complex qualitative appreciation of mathematical symbols.

So mathematical interpretation has both real (analytic) and imaginary (holistic) aspects.

But remarkably only the real aspect - related to mere quantitative appreciation - is recognised.

So the equally important imaginary aspect - related to scientific qualitative appreciation - is totally ignored.

So Mathematics while recognising the importance of complex numbers (real and imaginary) as quantities, attempts such interpretation from within a qualitative approach (that is solely real).

And, as I will demonstrate in later contributions, this restricted approach breaks down with respect to appropriate interpretation of the Riemann Hypothesis.

Indeed I would now see a far deeper relevance to the Riemann Hypothesis.
Riemann made truly extraordinary strides with respect to the quantitative aspect of prime number behaviour through the sophisticated use of complex techniques of analysis.

However appropriate interpretation of what these results entail, actually points to a corresponding need to adopt a complex approach in qualitative terms (incorporating both conventional and holistic mathematical understanding).

Indeed ultimately the Riemann Hypothesis points to an essential condition that is required to ensure the consistent interaction of both aspects!

So in this context I will once again outline the basic programme for a comprehensive approach to Mathematics

1) Conventional Mathematics; this is the real aspect of mathematical understanding geared directly to the quantitative interpretation of mathematical symbols. Unfortunately Mathematics at present is almost completely limited to this real aspect.

2) Holistic Mathematics; this the imaginary aspect of mathematical understanding geared directly to the qualitative appreciation of mathematical symbols. This has been my own speciality now for more than 40 years!

3) Radial Mathematics; this is the most comprehensive approach entailing both real (conventional) and imaginary (holistic) aspects in a close interactive manner.

I would classify my own recent resolution of the Riemann Hypothesis as a very preliminary version of the radial approach (that still remains closer to holistic rather than conventional appreciation). However as the Riemann Hypothesis in this context relates more to a philosophical rather than strict quantitative truth, such an approach is still adequate to resolve the matter.

As Jung portrayed so well, the unconscious is always present in conscious understanding (in an unrecognised manner). What then happens is that the unrecognised aspect projects itself involuntarily on to conscious understanding.

Ultimately the desire for meaning and fulfilment is of an unconscious holistic nature. However this then becomes projected on to conscious phenomena (that are misleadingly seen to contain this meaning).

In this sense mathematical truth is never strictly of an absolute nature with unconscious blindness all the stronger for the fact that it is not formally recognised.

Indeed it is such blindness that has successfully blotted out recognition of the complementary holistic element of mathematical understanding which - quite literally - is still not seen as having any relevance.

From a quantitative perspective, the imaginary number i results from obtaining the 4th root of unity and is therefore lies on the circle of unit radius (in the complex plane).

In corresponding reciprocal fashion the imaginary number i (as qualitative) results from raising 1 to 4 (i.e. the 4th dimensional expression of 1). It likewise is of a circular nature (i.e. entailing a circular logical explanation).

The four roots of 1 in quantitative terms are 1, - 1, i and - i respectively. The four dimensions in qualitative terms therefore entail both the positing and negating of phenomena with respect to both real and imaginary polarities.

The two real polarities relate to conscious internal and external aspects which in dynamic interactive terms are always positive and negative with respect to each other.

It has to be remembered that even at its most abstract level, mathematics represents experiential activity which necessarily entails a dynamic relationship as between the knower and what is known. Though in linear terms these polarities are understood as corresponding with each other in absolute terms, strictly this is not so with an experiential dialectic (that continually changes) taking place.

The two imaginary polarities then essentially relate to the dynamic manner in which whole and part notions interact in experience.

Again in conventional mathematical understanding (which is qualitatively 1-dimensional) whole and part notions are reduced in terms of each other.

However what is properly whole - relatively - is of an infinite nature whereas the part is finite.
And as we have seen the whole pertains directly to intuitive and the part to rational understanding respectively.
When appreciated in this light the whole - while necessarily related - is not confused with the parts; likewise the parts - again while necessarily related - are not confused with the whole. Using spiritual language the whole maintains a distinct quality that transcends (in an infinite manner) the parts; equally the parts maintain a distinct identity through which the whole is made immanent (again in an infinite manner).

Thus switching from (transcendent) whole to (immanent) parts and in reverse manner (immanent) parts to (transcendent) whole entails a corresponding switch as between positive and negative with respect to imaginary polarities.

This distinction as between real and imaginary is of the first magnitude intimately affecting all understanding at its deepest levels. For example - properly understood - any concept and its related perceptions are real and imaginary with respect to each other. So for example if specific numbers (as parts of the number system) are real, then the corresponding (whole) number concept is imaginary. Once again whereas specific numbers possess an actual finite identity, the number concept to which they relate is potentially of an infinite nature. Also in reverse fashion if we now take the number concept as real, the number perceptions to which they relate are potentially imaginary (i.e. through which the infinite concept is made immanent).

When viewed in this manner, just as experience keeps switching as between conscious and unconscious aspects of understanding, in qualitative mathematical terms it likewise continually switches as between real and imaginary aspects.

Seen in this new light, scientific phenomena are necessarily of a complex nature (with real and imaginary aspects). This is even evident in normal speech. For example a person may speak of buying a "dream" house. So here the house is given a real identity as a finite specific conscious object (that can be identified locally); however equally it possesses an infinite aspect pertaining to unconscious desire as the embodiment of holistic meaning.

And strictly speaking this is true of all phenomena (including mathematical symbols). Whereas they have a real identity (as conscious) they equally possess an imaginary identity as in some way fulfilling a quest for meaning. Indeed without this imaginary aspect (pertaining to the unconscious) it would not even be possible to pursue mathematical truth!

Though expressed in a rational linear manner, the remarkable feature of imaginary numbers (as quantities) is that they are actually expressive of the alternative circular logical system (based on the complementarity of opposites).

So even at a quantitative level two logical systems are at work. Indeed it is this latter circular aspect of imaginary numbers that gives them amazing holistic properties in many quantitative contexts. Once again this is why Roger Penrose keeps referring to the magic of complex numbers.

However once again what is greatly missing is true appreciation of why complex numbers possess such holistic properties! Thus as we have seen, recognition is likewise needed in qualitative terms of the complex nature of mathematical activity.

As I have stated the true relationship as between a number quantity (and its corresponding dimension) is as real to imaginary. As real pertains directly to what is quantitative and imaginary to what is qualitative respectively, this implies the real to imaginary connection!

Now the holistic (i.e. qualitative) number system is defined as 1 raised to a dimensional number which varies.

Remarkably when we raise 1 to a real number e.g. a fraction, a transformation takes place so that the result is circular (lying on the unit circle in the complex plane).
(Though not recognised in conventional terms this equally applies in a complementary qualitative manner to all whole number dimensions!)

Then in reverse fashion, when we raise 1 to an imaginary number dimension, again a transformation takes place so that the result is now real (in a linear manner).

So once again the appropriate relationship as between a number quantity and its related dimension is as linear to circular (or alternatively in more scientific terms real to imaginary).

Again all of this is deeply relevant to understanding the true nature of the Riemann Hypothesis.

The Riemann Zeta Function is defined with respect to dimensional numbers that are complex. The implication therefore is that both quantitative and qualitative transformations pertain to the resulting numbers that materialise.
And - as we shall see - a wide range of these quantitative results cannot be given a conventional linear interpretation. However they can be given a coherent holistic mathematical interpretation!

The Riemann Zeta Function clearly demonstrates that mathematical operations can generate numerical results whose meaning is qualitative rather than quantitative.
The clear implication therefore is that we cannot properly understand such behaviour without incorporating holistic mathematical interpretation.

We can also perhaps now see why complex numbers have proven so valuable in uncovering some of the mysteries of the primes.

Prime numbers embody in their inherent nature patterns of behaviour that are linear and circular with respect to each other. Likewise when converted into more scientific amenable language, this implies that prime numbers embody behaviour that is both real and imaginary (in qualitative terms).

It is therefore by no means surprising that complex methods of analysis would be needed to uncover many of the quantitative secrets of prime numbers.

However, what is not recognised is that it then requires complex methods of qualitative interpretation to ultimately make sense of this quantitative behaviour.

Wednesday, May 5, 2010

Holistic Nature of Prime Numbers

In the previous contribution, I indicated that each major stage on the spectrum of possible psychological development is closely linked with the qualitative interpretation of a corresponding number type.

We start with the two original numbers 1 and 0. 1 is directly linked in holistic number terms with conscious reality and the rational interpretation of (unitary) form.
0 - by contrast - is directly linked with unconscious reality and the intuitive awareness of spiritual emptiness (i.e. that is empty of phenomenal form).

In turn, linear interpretation - based on the either/or logic of separate polarities, is directly related with the former number (as qualitatively understood); circular interpretation - based on the both/and logic of complementary polarities - serves as the indirect expression of the latter holistic number.

At the commencement of development, both conscious and unconscious remain in an undifferentiated state; equally this represents an - as yet - unintegrated state (as successful integration depends on prior differentiation).

This equally implies that neither linear nor circular understanding can yet emerge in development.

In holistic terms, the emergence of conscious understanding (where phenomena can be differentiated) coincides with the appreciation of duality (in the first separation of opposite polarities). In this way the infant in some measure is able to obtain some initial awareness of a self as distinct from the collective environment.

So in such duality here we have the birth in holistic terms of the first prime number 2 which then serves as the basis for the holistic emergence of the further primes.

Now the very word that is commonly used to suggest the nature of such development i.e. primitive is highly suggestive of its true holistic mathematical nature.

The very essence of primitive development is that both conscious and unconscious still remain entangled with each other to a considerable extent.
Therefore in a primitive state, conscious phenomena cannot be properly separated from an overall undeveloped unconscious state. In other words both conscious and unconscious remain confused with each other.

The nature therefore of primitive instinctive behaviour is that the holistic desire for meaning (properly pertaining to the unconscious) remains embedded with specific phenomena. Initially with infant experience this confusion is so strong that phenomena remain extremely transient (disappearing as soon as they arise). Gradually however as conscious life becomes more differentiated, phenomena can assume a more stable independent existence (free of unconscious confusion).

This holistic nature of prime numbers has equally important implications for understanding of the physical universe. Initially with highly primitive matter i.e. subatomic particles, the existence of distinct phenomena cannot be properly distinguished from their holistic dimensional background. Indeed this is even recognised to a degree in string theory with dimensions in a sense understood to be contained in the strings.

However with the greater differentiation of matter, phenomena gradually attain a more stable distinct existence whereby they can be successfully placed in a dimensional context of space and time.

The key implication of all this for appreciation of the nature of prime numbers is that their inherent nature closely incorporates the operation of two logical systems that are linear and circular with respect to each other.

So just as primitive behaviour - either in psychological or physical terms - entails the entanglement of distinct phenomena with a qualitatively distinct collective context, likewise, properly understood it is the same with prime numbers.

In other words we cannot just attempt to understand prime numbers as the independent building blocks of the natural number system (that befits a linear method of interpretation).

From the overall holistic perspective, prime numbers display a remarkable degree of interdependence with the natural numbers so that their precise location intimately depends on such numbers.

And once again these two aspects i.e. quantitative independence and qualitative interdependence are properly distinct from each other (pertaining to different logical systems).

Now let us see more precisely what actually happens with the treatment of prime numbers in Conventional Mathematics.

Though the inherent nature of prime numbers corresponds to a much earlier stage of development (where qualitative and quantitative characteristics remain embedded with each other) the actual interpretation conventionally used derives from the much later middle stages of development (where linear rational understanding has achieved its specialised expression).

Therefore Conventional Mathematics attempts to understand prime numbers within an exclusively linear (rational) context even though this misrepresents their inherent nature (as properly combining both linear and circular aspects of understanding).

This linear aspect is then clearly manifest in the interpretation of prime numbers as the independent building blocks of the natural numbers.

Now in fairness considerable attention has been given also to the distribution of the prime numbers. However this has essentially been done through further extension of merely linear methods of investigation.

This is a huge problem the implications of which are yet adequately understood. For example in a static context if I walk up a road I can unambiguously identify for example a left turn. Now in a different context (where I now walk down the road) I can again unambiguously identify a left turn.

However though with respect to each independent frame of reference, both turns are unambiguous as left, in relation they each other they are necessarily both left and right.

It is exactly the same with the study of prime numbers. In a static context one can attempt to interpret the individual identity of prime numbers or their overall collective nature from within a linear rational context. However in dynamic interactive terms these two contexts are complementary requiring both linear (quantitative) and circular (qualitative) understanding.

In other words we cannot properly understood the nature of prime numbers in the absence of a holistic mathematical context. Furthermore, as I will show later, coherent appreciation of the Riemann Zeta Function incorporates results (pertaining to both conventional and holistic mathematical interpretation).

From my current perspective, I would find it patently absurd to view prime numbers - as conventionally understood - as the basic building blocks of the natural number system as equally their very location intimately depends on these same natural numbers.

Once one recognises the dual nature of prime numbers i.e. with respect to both their linear and circular characteristics (in turn requiring two distinctive methods of logical interpretation), then a key question arises with respect to maintaining consistent interpretation according to both aspects.

And in is in this reformulation of the inherent nature of a prime number that the Riemann Hypothesis obtains its true context i.e. as the fundamental requirement for guaranteeing such consistency.

Tuesday, May 4, 2010

Number Types and Stages of Development

The are various number types (as quantities) recognised within Mathematics.

For example we can start with the original - and most fundamental - numbers 0 and 1.

Then we have 2 and the other odd prime numbers e.g. 2, 3, 5, 7,.....

These then lead on through multiplication to the natural number system 1, 2, 3, 4,...

Then we have the integers allowing - including 0 - which allow for for both positive and negative natural numbers.

Next come the rational numbers which can be expressed as fractions i.e. the ratios of the integers

Then we have the (algebraic) irrationals such as the square root of 2 and phi which arise as solutions to polynomial equations using integer coefficients.

We also have (transcendental) irrationals such as pi and e (which do not arise as solutions to the aforementioned polynomial equations.

We also have imaginary as well as real numbers and the combination of both as complex numbers.

And finally we can define transfinite numbers representing infinite classes (as derived in a linear manner).

However a key realisation for me (following on the attempt to resolve the Pythagorean Dilemma) was that all these number types could equally be defined in a coherent qualitative manner.

Furthermore the qualitative interpretation of each number type defines a unique scientific paradigm (or more properly metaparadigm) with which to interpret reality.

Right away this suggests how limited is the accepted rational paradigm which defines the conventional scientific - and indeed mathematical - approach to reality.

Though clearly very important in their own right, the rational numbers (as quantities) represent just one subset with respect to the overall set of numbers.

In like manner the rational paradigm likewise represents just one important form of scientific interpretation (from a varied range of possible interpretations).

Looked at from another perspective each qualitative number type closely corresponds with the understanding that unfolds at a particular stage of development.

Rational understanding - that defines conventional scientific interpretation - conforms to a narrow band towards the centre of the psychological spectrum. However there are several "higher" and "lower" bands that potentially define equally important modes of scientific interpretation (that are all but unrecognised).

My own interest over the years has largely been devoted to the "higher" stages of development.

In the spiritual traditions these generally are associated with contemplative type development requiring an ever more refined type of intuitive awareness.

However what has been greatly missing from such accounts is any proper clarification as to to the implications of such awareness for scientific (and mathematical) appreciation.

So recognising this deficit I decided to try and fill in the gaps to best of my own limited abilities. and it is through these endeavours that Holistic Mathematics (and associated Holistic Sciences) emerged.

Indeed nearly 20 years ago I wrote two online books "Transforming Voyage" and "The Number Paradigms" summarising my progress to that point.
So I basically was at pains to trace the nature of each of the major stages of development before then showing how a distinctive number paradigm was closely associated with each main stage.

In the present context i.e. resolution of the Riemann Hypothesis, two number types of special importance are the prime numbers and the imaginary numbers.

Once again though these are conventionally understood merely as quantities, they equally possess an important qualitative aspect.

So in the next contribution we will look firstly at the important holistic (i.e. qualitative) nature of prime numbers which will help to illuminate their true inherent nature (which in turn is vital in terms of obtaining proper appreciation of the fundamental significance of the Riemann Hypothesis).

Sunday, May 2, 2010

The Pythagorean Dilemma

The Pythagorean School is very interesting in that it pursued a much more comprehensive notion of Mathematics than what now conventionally exists.

Basically for the Pythagoreans, mathematical symbols possessed both an (intuitive) qualitative as well as (rational) quantitative significance.

Put another way their appreciation combined holistic as well as analytic aspects. Through this holistic aspect, certain mathematical symbols were seen to possess universal archetypal properties. Thus the proper nature of mathematical activity greatly transcended mere rational knowledge opening up the way to authentic spiritual contemplative awareness.

For the Pythagoreans, numbers (i.e. the integers) were especially important mathematical symbols, through which the secrets of reality, as they believed, were encoded.

They also believed that all numbers were rational (i.e. could be written as ratios of the integers).

So there was an assumed correspondence as between rational numbers (as quantities) and - what might be called - the rational paradigm (in qualitative terms).

However the application of the famed Pythagorean triangle was to shatter this belief. In the simplest case of the right angled triangle, where both adjacent and opposite sides = 1, the hypotenuse = the square root of 2.

Now the Pythagoreans knew that this did not represent a rational number but rather an irrational quantity (i.e. that could not be expressed as the ratio of two integers).

The reason therefore why this discovery was so significant was that it undermined the vital correspondence they believed to exist as between both the quantitative and qualitative aspects of Mathematics. In order words, the Pythagoreans in their enquiry of nature adopted then - as now - the rational paradigm. So they believed that all scientific investigation would conform in qualitative terms to the rational approach; however the right angled triangle clearly demonstrated the existence of irrational quantities. So the Pythagorean Dilemma - as I refer to it - relates to the fact that they lacked the qualitative means of explaining why irrational number quantities could arise.

Subsequently in Western Mathematics the Pythagorean Dilemma has simply been ignored rather than resolved through a basic form of reductionism whereby its activity is now identified solely with its quantitative aspect. From this perspective there is no need to explain the deeper why with respect to the nature of irrational numbers. Rather they are treated in reduced fashion where their quantitative value can be approximated in finite terms to any required degree of accuracy.

However, though it must be readily admitted that Western Mathematics has indeed made enormous advances with respect to its specialised quantitative development that equally, from an overall comprehensive perspective, it has become hugely unbalanced (with its important holistic aspect no longer formally recognised).

So an important task in redressing this imbalance is to show now how the Pythagorean Dilemma can be properly resolved.

The key to answering this problem is to demonstrate that just as we can have irrational as well as rational quantities, likewise in qualitative scientific terms we can define an irrational as well as rational paradigm with which to interpret phenomenal reality.

The very essence of an irrational number (such as the square root of 2) is that it combines both (discrete) finite and (continuous) infinite aspects in its very identity. Thus for example we can approximate the square root of 2 (thereby giving it a merely reduced discrete rational identity) to any given level of accuracy. So correct to 4 decimal places its value is 1.4142 (which is rational).

However in truth an irrational number possesses an irreducible qualitative aspect in that its decimal sequence continues indefinitely (and can never be represented as a fraction).

So the very nature of an irrational number therefore is that it combines both finite and infinite aspects (which cannot be reduced in terms of each other).

The corollary of this in qualitative terms is that irrational understanding likewise combines both finite and infinite aspects. And as we have seen this entails the combination of both rational and intuitive modes of interpretation.

And as discussed in the previous blog, whereas the rational is conveyed through the standard linear logical mode, the intuitive aspect is indirectly conveyed through the circular mode (based on the complementarity of opposite polarities).

Now to put this in perspective we need to consider what actually happens in experience when authentic contemplative development unfolds!

Initially in the spiritual life this entails considerable detachment (wich in holistic mathematical terms entails the dynamic negation of linear rational forms). This negation of conscious phenomena then causes a decisive switch in experience whereby a new type of meaning incubates in the unconscious. Then when the time is ripe it bursts forth as authentic intuitive awareness in a brilliant spiritual illumination.

Quite literally experience now becomes 2-dimensional and transformed in an irrational qualitative manner. Once again the 1st dimension relates to the conscious (posited) direction as linear understanding and the 2nd to the unconscious (negated) direction (as intuitive awareness); Whereas previously a phenomenon e.g. a flower was given a discrete local identity in experience, now it enjoys a two-fold identity. At one level one can again identify it (in more refined fashion) as a local phenomenon; however it now equally possesses an archetypal holistic identity (as mediator of a divine spiritual light). And this holistic aspect in turn relates to the dynamic complementarity of opposites in the fusion of both external and internal aspects of understanding.

In other words, all phenomena are understood to possess both finite and infinite aspects (that cannot be reduced in terms of each other).

The implication therefore for science is that the irrational (2-dimensional) paradigm itself must necessarily combine understanding of phenomena according to both linear logic (where opposites are clearly separated) and circular logic (where they are considered as complementary).

In this way the Pythagorean Dilemma is thereby solved.

In quantitative terms, obtaining the square root of a number entails raising that number to a dimension of 1/2 (i.e. power of 1/2).

And the holistic interpretation of what is involved here equally requires raising understanding in inverse qualitative terms to the dimension 2 (or power of 2).

So 1/D (relating to the dimension or power of a number) in quantitative terms entails the corresponding interpretation of D (in qualitative terms) with the nature of both structurally similar.

The key limitation of the Pythagorean worldview is that the relationship as between both quantitative and qualitative aspects of understanding is viewed in merely 1-dimensional terms.

So just as from a 2-dimensional perspective we see reality in qualitative holistic terms as representing both positive and negative polarities (that are dynamically interdependent) in corresponding quantitative analytic terms we see the square root of a number, such as 1, as representing both positive and negative polarities i.e. + 1 and - 1 (that are statically independent).

And again just as 2-dimensional understanding qualitatively combines both local discrete understanding of a finite and holistic continuous awareness of an infinite nature (where mathematical symbols operate as archetypes) respectively, likewise with an irrational square root number quantity such as 2 (obtained by raising to the inverse dimension of 1/2) again both discrete finite and continuous infinite aspects are combined in its nature.

Of course when - as in Conventional Mathematics - the holistic aspect is disregarded, mathematical symbols greatly lose their qualitative numinous properties and become treated as merely reduced quantities.

Again one may be tempted to question what relevance any of this can have for understanding the Riemann Hypothesis!

Well you see it is all a matter of perspective and from the holistic mathematical perspective it is indeed of paramount significance.

Once again the Riemann Hypothesis relates to the fact that all the non-trivial zeros of the zeta function lie on the real line = 1/2.

Now 1/2 in this context relates to a dimensional number. And of course a square root relates equally to the same dimensional number.

So the whole point about this exercise is that actual understanding of the Riemann Hypothesis requires corresponding qualitative interpretation relating to the 2nd dimension. This in turn entails that it cannot be properly interpreted using 1-dimensional linear logic but must also incorporate circular understanding (corresponding to the 2nd dimension).

In fact as we shall see later the very nature of a prime number (when properly interpreted) requires the incorporation of both linear and circular aspects.
For example from the linear aspect it is customary to view prime numbers as independent building blocks from which all the natural numbers are obtained. However from the equally important circular perspective prime numbers are fully interdependent with the natural numbers (with their general distribution intimately depending on these same numbers).

Indeed ultimately the Riemann Hypothesis represents the fundamental condition required for the consistent use of both logical systems (which is inherent in the very nature of prime numbers).

And this simple truth cannot be appreciated from within the standard 1-dimensional interpretation of Conventional Mathematics!