## Thursday, April 28, 2016

### Approaching the Infinite

I mentioned that around the age of 10, I formed serious misgivings regarding the nature of multiplication. To my way of thinking, even at that young age, a comprehensive explanation required a qualitative - as well as quantitative - aspect. However - quite literally - the qualitative dimension was effectively removed from conventional interpretation.

And this proved far from a passing concern. In fact I spend some considerable time thereafter in the attempt to "capture" the qualitative dimension.

Thus to focus clearly on this missing aspect, I considered a square with side of 1 unit (i.e. expressed in 1-dimensional terms). The area of the square is then given as 1 square unit (i.e. now expressed in a 2-dimensional manner). Therefore, though the quantitative result has remained unchanged as 1, clearly a qualitative dimensional change has taken place.

12 thereby represents the quantity of 1 (expressed in 2-dimensional units).

Then as all real number quantities are ultimately expressed with respect to linear (1-dimensional) units, I considered the possibility that the “dimensional” could be thereby “converted” in a quantitative manner, by obtaining the two roots of 12.

This idea of “conversion” in fact occurred quite naturally to me as much earlier, before encountering them in maths class, I had - using 2 as a base - discovered the principle behind logs, whereby multiplication could be “converted” to addition.

And the two roots of 1 are + 1 and – 1 respectively.  So, something very strange seemed to be happening here, whereby two answers resulted that were the direct opposite of each other.

Though I was not able to properly understand this conundrum, it laid the firm foundation for an eventual solution while attending University.

However first year at University studying Mathematics proved a most disillusioning experience whereby my former misgivings, regarding its reduced nature, were to return in a much deeper fashion.

The course involved extensive exposure to the notion of limits. Though apparently great refinement had been brought to the modern discussion of the limiting notion, I quickly realised that at bottom it was in fact a sham so as to enable treatment of the infinite in reduced terms.

In fact this represented just another important example of how a fundamental qualitative notion is again reduced in a merely quantitative manner.

And the relationship as between finite and infinite is completely unavoidable, as it underlies every possible mathematical relationship.

For example the concept of number potentially applies to every number (and in this sense is infinite). However all actual numbers are necessarily of a finite nature. Thus there is a dual sense in which the very notion of number is used (entailing both finite and infinite aspects).

However the infinite notion is then reduced in formal interpretation. So instead of the realisation that this notion correctly applies in a holistic potential fashion, the infinite is misleadingly identified in a finite actual manner.

From this perspective, one therefore approaches the infinite through linear extension, whereby it is considered as greater than any finite number.

However, strictly speaking, this is just nonsense! What is greater than any finite number is always another finite number! So we cannot meaningfully approach the infinite in this manner!

Thus infinity is not greater than any pre-assigned finite number (however large). Also the infinitesimal notion is not less than any finite number (however small).

This simply represents the reduction of holistic (qualitative) notions in a reduced quantitative (analytic) manner.

Then I realised that this has important implications for what we know as mathematical proof.

We might maintain that a proof applies universally to all cases within its class.

So, the Pythagorean Theorem for example (that the square on the hypotenuse of a right angled triangle equals the sum of squares on the other two sides) thereby universally applies to all triangles in this class.

However, properly speaking there is an important distinction as between potential and actual meaning.

So if we say that the theorem potentially applies (infinitely) in all cases, this strictly this does not entail any (finite) cases (in an actual manner).

Then when we try to maintain the reduced conclusion that it applies in all actual situations, we are left with an inevitable problem as the very notion of “all” does not have a determinate meaning in finite terms.

Therefore the very truth of a proposition (such as the Pythagorean Theorem) with respect to a limited set of finite cases necessarily entails an unlimited set of other finite cases which can never be determined.

Now this not of course mean that there is no value in mathematical “proof” (as currently pursued) but rather that such truth is of a relative - rather than absolute - nature.

So an uncertainty principle strictly attaches to the proof of every proposition (which correctly must be understood in a dynamic interactive manner).

Therefore, we have the inevitable interaction of two aspects, which are - relatively - analytic (quantitative) and holistic (qualitative) with respect to each other.

Thus if we attempt to concentrate solely on the analytic aspect - which represents the present mathematical approach – we thereby reduce the infinite notion in a finite manner.

If on the other hand, we concentrate solely on the holistic aspect (as what is potentially infinite), the finite then is elevated to the infinite notion, so that we can no longer apply mathematical propositions in actual terms.

This inevitable dynamic interaction as between analytic and holistic aspects of understanding corresponds in psychological terms with the interaction of conscious and unconscious aspects through reason and intuition respectively.

So reason by its very nature tends to be analytic, whereas intuition is of a holistic
nature.

So the reduction of qualitative to quantitative type understanding in present Mathematics directly complements the corresponding formal reduction of intuitive to rational type interpretation.

Therefore, while at University for me the bottom completely fell out of the mathematical world (as conventionally understood).

I realised then that a completely new approach was required, which - even if remaining in a minority of one for the rest of my life - I must pursue.

And this was to have profound implications for the integration of mathematical with spiritual type reality, for in the correct appreciation of the infinite notion, Mathematics intrinsically possesses an inescapable spiritual dimension.

## Friday, April 22, 2016

### Holistic Science

As stated in the last blog entry, I see a comprehensive science as consisting of 3 distinct aspects, which for convenience can be named Type 1, Type 2 and Type 3.

Type 1 would correspond to the conventional scientific approach representing the analytic (quantitative) extreme to the interpretation of reality.

Its fundamental feature is that it is of a linear (1-dimensional) nature. This means that all relationships are rationally interpreted within single isolated polar reference frames.
In particular the (external) objective is abstracted from the (internal) subjective aspect; and the (individual) parts are abstracted from the (collective) whole.
Thus with Type 1 science, we have the reduction of subjective interpretation to objective explanation; equally we have the reduction of the whole (in any context) to parts.

By contrast Type 2 represents the holistic (qualitative) extreme to interpretation, which is all but unrecognised in our present world.

Whereas the Type 1 (analytic) approach corresponds with the linear use of reason, the Type 2 (holistic) approach is primarily based directly on intuitive insight, indirectly expressed in a circular rational manner. This implies that holistic type appreciation always appears paradoxical with respect to the conventional (i.e. linear) use of reason.

Linear reason is essentially geared to the quantitative analytic investigation of the parts of a system, where logical connections are made in a sequential manner. However intuitive awareness by contrast is directly geared to the qualitative awareness of the whole (where the relation as between parts are viewed simultaneously in a synchronous fashion). Indirectly, this awareness is then translated in a circular rational manner (that appears paradoxical in linear terms).

Though the importance of intuition might indeed be accepted by scientists, especially where creative work is involved, it is always in effect subsequently formally reduced in a merely rational manner.

So just as Type 1 science continually misrepresents the relation between analytic and holistic aspects in physical terms, equally from a psychological perspective, it fundamentally misrepresents the relationship as between (linear) reason and (circular) intuitive awareness.

Type 2 science starts where Type 1 science ends.

There are two fundamental polar sets that necessarily underlie all phenomena.
These are 1) the external/internal and 2) the whole/part aspects of experience.

The basic requirement with Type 1 science is that these poles be absolutely separated from each other, thereby eliminating consideration of the two-way interaction that necessarily takes place.

So for example as I am presently typing, I am aware externally of the computer screen (relative to my internal self). However through the dynamics of experience this keeps switching so that I also become aware of my internal self  (in relation to the external screen). So a dynamic two-way relative interaction ceaselessly takes place (leading to transformation with respect to both external and internal aspects). However conventional science absolutely freezes this interaction, thereby creating the illusion that the external world has a coherent meaning (independent of the interpreting self).

Likewise there is a ceaseless two-way interaction in every context as between the whole (that provides a qualitative relationship for parts) and the constituent parts (that provide the quantitative data to be related).

However conventional science once more absolutely freezes this two-way relative interaction so that the (qualitative) whole aspect is inevitably reduced to its (quantitative) parts.

This is perhaps the most important and blatant form of reductionism in conventional science, that is so widespread that it is rarely ever even noticed!

The central concept in Type 2 (holistic) science is that of complementarity.
Thus what are understood in linear rational terms as direct opposites are now seen in Type 2 terms as constituting an essential unity.

To make this a little more accessible, I will deal firstly with a simple example that should resonate with everyone.

Imagine one is travelling along a straight road approaching a crossroads. Now once we define the direction from which it is approached in 1-dimensional terms (i.e. with respect to a single isolated pole of reference) Type 1 (linear) rational interpretation applies.

So if one is heading N and reaches the crossroads, then both left and right turns can be unambiguously defined. Then if having passed through, one then reverses direction heading S again both left and right turns can be unambiguously defined with respect to the crossroads.

However if we now simultaneously attempt to view the situation where the crossroads can be approached from both N and S directions, then an inevitable paradox applies to directions.

Thus what is defined as a left turn when approaching the crossroads in the N direction is a right turn when approaches from the opposite side heading S. And what is a right turn heading N is by the same token a left turn heading S.

Thus in this second context (where two reference frames are viewed simultaneously) both left and right have a contingent arbitrary meaning (depending on relative context).

Now  the very ability to appreciate to the merely relative nature of the two turns at the crossroads requires holistic intuition (where two reference frames can be viewed simultaneously).

And as we have seen, this leads to a (circular) paradoxical outcome from the conventional rational perspective.

Now one might initially query why this example should have important implications for science. However the key here is the recognition that all actual experience is based on mutually switching polar reference frames. Thus on the one hand we have the necessary interaction of external and internal aspects of experience; likewise we have the mutual interaction of whole and part aspects.

Thus once we move to consideration of this dynamic interactive nature of experience, we must necessarily move on to a new holistic type of scientific appreciation (i.e. the Type 2 approach).

As we saw in our crossroads example, there is an inevitable complementarity as between left and right turns (when viewed simultaneously from 2 reference frames).

However, this complementarity universally applies to all experience (which is dynamically conditioned by interacting polar opposite reference frames).

Perhaps most critically, though it is not yet even been considered by the profession, it applies to all mathematical relationships.

My first realisation that there was something seriously wrong with accepted mathematical interpretation came at the age of 10, when I found myself in class carrying out empirical estimates of the area of various rectangular fields.

Thus if for example we have a field that is 80 yards long and 60 yards wide the area will be 4800 square (i.e. 2-dimensional) yards.

Thus as well as the quantitative change area with respect to the area there is an inescapable qualitative change also in that we move from linear (1-dimensional) to square (2-dimensional) units.

However in conventional multiplication, the qualitative aspect is simply ignored altogether.

Therefore 40 * 80 = 4800 (in a merely quantitative i.e. 1-dimensional manner).

Even at that tender age, I knew that there was something seriously wrong here! And this subsequently led me on a life-long quest to properly incorporate the neglected qualitative aspect into mathematical understanding.

This problem becomes acute, when we come to consider the relationship of the primes to the natural numbers (which is the most fundamental area of mathematical research).

The primes are considered as the "building blocks" of  the natural numbers. So each composite number - say 6 - is uniquely expressed as the product of prime factors (i.e. 2 * 3).

However in conventional terms, only the quantitative aspect of the primes is considered.

So if we take the first prime i.e. 2 and attempt to break it into its constituents it would be given as

2 = 1 + 1. So each of the units here are interpreted in a homogeneous quantitative manner (i.e. without qualitative distinction).

This represents the cardinal notion of number.

However there is a corresponding ordinal approach (where the qualitative aspect is now emphasised).

So from this perspective 2 = 1st + 2nd. However 1st and 2nd only have meaning in the context of a circular type relationship. Thus their sum has no quantitative meaning.
I then realised that both the cardinal and ordinal approaches to number are themselves directly complementary.

In the cardinal approach, we deal with the quantitative collective aspect of number (where individual units have no qualitative distinction); in the ordinal approach by contrast, we deal with the individual qualitative aspect of number (where the collective sum has no quantitative distinction).

This then implies the crucial finding that the ordinal nature of number (which relates to the holistic notion of number interdependence) cannot be properly defined within the conventional (Type 1) mathematical approach.

And as both cardinal and ordinal recognition are necessarily intertwined in the very manner we understand numbers in experience, this means that we cannot properly define the cardinal notion either (in the absence of Type 2 understanding).

In other words, Mathematics is inherently of a dynamic interactive nature, where all relationships entail the interaction of complementary poles (in quantitative and qualitative terms).

However Conventional Mathematics is misleadingly represented in reduced absolute terms as the relationship between fixed static entities.

Thus the greatest revolution yet in our intellectual history will eventually begin to unfold when the inherent dynamic interactive nature of Mathematics is eventually embraced. And this will entail the recognition of both analytic (quantitative) and holistic (qualitative) aspects as equally important partners.

This of course equally applies to all the sciences (that are rooted in mathematical understanding).

Once again I find it striking how two developments with very opposite implications occurred at the same time in 1859. Darwin's theory of evolution really represents the last big application of the wholly reductionist approach to science. Riemann's findings regarding the prime numbers are however pointing directly to the severe cracks in the very foundations of that reductionist approach.

## Thursday, April 21, 2016

### Richard Dawkins: An Appetite for Reductionism

I completed recently the first part of Richard Dawkins' Biography "An Appetite for Wonder: The Making of a Scientist: a memoir". In fact - due to its ready availability in my local library - I had read the 2nd part "Brief Candle in the Dark" earlier.

In many way I found the first part more interesting as it provided insight into how Dawkins later came to adopt his particular view of science.

Though some might describe his earlier life in Africa as idyllic, I would not see it that way. Certainly it provided a range of interesting experiences, but it seems to me have been a somewhat unsettled and lonely existence. This was compounded by the fact that Dawkins comes across as an unusually sensitive child with a very trusting nature.

And this trust was severely tested as he tried to adapt to the many uncertainties of his world.

It is very revealing in this context that Dawkins frequently admonishes his younger self for his "childhood gullibility" (though to many readers this may indeed appear as his most attractive characteristic). He is thereby suggesting that if he could have reasoned then in an adult manner, that he could have avoided much of the hurt and confusion of his early life!

Now it is in the very nature of development that a child will go through magical and mythical stages of growth where the holistic unconscious remains embedded to a degree with emerging conscious rational ability.

Then - at least in modern Western culture - by early adulthood the more specialised development of conscious ability takes place. This then tends to dictate our normal waking activity, though - as Dawkins is so well aware - significant mythical elements generally remain with respect to conventional religious understanding.

There are in fact further important stages of development that can occur beyond the specialised rational level. These have been documented in great detail in all the great spiritual traditions, where highly refined intuitive type capacities, culminating in a pure contemplative vision of reality, can unfold. Then at most advanced levels, both specialised reason and holistic intuition can be merged with each other in an extraordinarily creative, yet immensely productive, manner.

So in psychological terms, this most advanced stage represents the mature interpenetration of both conscious (analytic) and unconscious (holistic) understanding.

Thus like the electromagnetic spectrum in physics, we have a full spectrum with respect to the possible stages of psychological development.

However Dawkins shows little appreciation of this psychological spectrum. While reluctantly conceding the existence of "lower" stages where both conscious and unconscious still intermingle in a somewhat confused fashion, Dawkins believes that we should encourage children from an early age to dispel all myths. In other words, we should train children to think just like adults.

However this displays a remarkable lack of appreciation of the role of the unconscious mind.

So the true task in life is not to throw the baby out with the bathwater (in discarding the unconscious aspect), but rather to seek to develop both aspects in a mature manner, so that they can mutually serve each other.

So proper development does not end with the specialised development of the conscious mind. Rather it should then ideally proceed to ultimately attain corresponding specialised development likewise with respect to its unconscious aspect i.e. in a refined holistic intuitive form of awareness.

My own childhood experience in many ways is the opposite of Dawkins in that I displayed unusual scepticism for myths from a very early age. I had already discarded the notion of Santa Claus at about the age of 5 or 6. I even then used "a scientific experiment" to prove my point by suggesting to my older brothers the most likely places in the house where the Christmas presents would be stored by my parents. Then on Christmas Eve I duly carried out a search to quickly locate them in the preferred location.

Then about a year later when being instructed on the story of Adam and Eve in the Bible, I stood up in class to declare that I believed in evolution (which, as one might imagine in the Ireland of the 50's went down like a lead balloon with my teacher).

However strangely, this early uncovering of religious myth did not reduce - but rather increase - my desire for authentic spiritual meaning.

So whereas I would now readily agree with Dawkins that the major religions support the literal adoption of myths in their attempt to convey spiritual truths, I do not equate spiritual meaning with the acceptance of such myths.

So again, one can remain totally sceptical regarding the literal meaning of religious myths while maintaining a sincere commitment to spiritual type meaning!

Likewise, while remaining personally sceptical, I would not dispel entirely the present need for the preservation of myths in popular religious terms. For the role of myth here is to convey, however imperfectly, holistic type meaning that resonates with the unconscious - rather than the conscious - aspect of personality. And this type of meaning, by its very nature, cannot be conveyed in a rational scientific manner!

So the real problem is that the unconscious aspect of personality still operates at an immature level in modern society necessitating the preservation of myths to convey spiritual type truth!

And paradoxically the very adoption of the rational scientific model adopted by Dawkins would only serve to delay further the realisation that such unconscious development requires the recognition of a distinct type of meaning that is not catered for in conventional scientific terms.

In fact there are clear paradoxes evident in relation to Dawkins' own adoption of "rational science".

The very title of his memoir "An Appetite for Wonder" draws attention to this!

Now the capacity for wonder is a indeed a marvellous human gift. However it pertains directly to the unconscious - rather than the conscious - aspect of personality.

The very nature of conventional scientific interpretation requires that a clear dichotomy be drawn as between the "knower" and "what is known". Therefore in conventional science - certainly in the kind of science that Dawkins advocates - an unwarranted supremacy is given to the merely objective data of experience (as if they can somehow exist independent of the enquiring mind).

So such science attempts to deal - misleadingly - with what can be objectively known. However the real issue in science points directly to the relationship of the knower to what is known (i.e. to the manner in which the subjective aspect of experience through interpretation, interacts with  its corresponding objective aspect).

By its very nature therefore, the relationship as between the "knower" and "what is known" cannot be satisfactorily addressed in a detached conscious manner.

However it can be addressed in terms of the unconscious where both aspects are holistically understood as complementary (and ultimately identical with each other).

Thus the capacity for wonder, enticing rapt attention, already presupposes a certain mergence of the "knower" with "what is known". So through wonder, we can become momentarily lost in the objects of our investigation by - literally - to an extent becoming united with them.

And of course this capacity for wonder is initially associated with childhood (where the unconscious is still immaturely embedded with conscious understanding).

Thus it would seem clear to me that Dawkins' own capacity for wonder is rooted very much in his childhood and later teen experience (where magical and mythical type thinking still abounded).

And - though I am sure that Dawkins would not choose this form of expression - wonder really serves as the innate expression of a spiritual instinct

Thus if he had been successful in eradicating this "childhood gullibility" (through the premature adoption of adult reason) this very capacity for wonder would likely have been its severest casualty.

It is also interesting to find that Dawkins was - and remains - a lover of poetic verse, especially in its most romantic expression.

For example he quotes - among others - lines of W.B. Yeats which made on him a lasting impact.
And this poetic sensibility is clearly evident in the titles he has chosen for some of his recent books "Unweaving the Rainbow", "An Appetite for Wonder" and "Brief Candle in the Dark".

And he demonstrates how this affective dimension is intimately involved in the experience of a rainbow.

"And it doesn’t matter how many rainbows you see throughout your life. The glory is reinvented afresh, and the heart leaps up every time".

However the clear implication of this is that we cannot hope to successfully reduce the actual experience of a rainbow to its cognitive scientific explanation.

So in actual experience - and this by extension applies to all phenomena - we have the complementary dynamic interaction of both cognitive (scientific) and affective (artistic) aspects.

The very essence of the cognitive aspect is that it is of a detached impersonal nature, whereby a collective universality can be applied to the definition of phenomena. Thus at the level of the scientific description of a rainbow it is quite irrelevant as to how one emotionally reacts to the phenomenon. Rather it suffices that a collective cognitive agreement exists as to this explanation.

However this is all inverted at the affective level, whereby one reacts to  a rainbow through a sensible form of personal response. So from this perspective, each person's experience is unique. And the clear distinction as between the scientific and artistic experience is that in the first case the object is considered in a detached manner (as separate to the observer) whereas in the second case the observer is necessarily viewed in a shared participative relationship with the object of observation.

And without this capacity for sensible response (of an affective nature) it is impossible to see how the observation of any scientific data could properly take place!

But in conventional science the affective aspect is quickly forgotten with the relationships between phenomena reduced to mere rational type explanation (of an impersonal kind).

Thus a deeper issue for science - which Dawkins never really addresses - is to explain how both the affective and cognitive type aspects of experience can be successfully integrated within a more refined scientific type explanation. This requires I believe a complex rational approach entailing both real (conscious) and imaginary (unconscious) aspects. To be more precise the imaginary aspect entails the indirect rational expression of what directly relates to holistic (unconscious) meaning. So it is through recognition of this imaginary aspect that the artistic response to phenomena (which is an indispensable aspect of the experience of phenomena)  can be indirectly interpreted in a scientific rational manner.

So Dawkins’ version of "real" science represents a very reduced version of what true scientific experience properly entails.

Though he readily admits the importance of wonder and affective sensibility, these are not actually integrated into his scientific methodology (which remains at the linear rational level of explanation).

Dawkins considers Charles Darwin his true intellectual hero. As we know Darwin's magnum opus "The Origin of Species" was published in November, 1859. What is very interesting that in the same year - indeed the same month of November - a short mathematical paper was also published by Bernhard Riemann, the long-term implications of which, I believe, will prove more important than Darwin's seminal contribution.

Riemann's paper dealt with the nature of prime numbers, which are considered the basic building blocks of the number system.

In it, he showed how a certain set of solutions to an algebraic equation - now known as the Riemann zeta function - provide the means to perfectly predict the number of primes up to a given number. Now, whereas the individual nature of primes appears highly random, yet a remarkable regularity attaches to their collective relationship with the natural numbers!

And Riemann in effect showed how these solutions, known as the "zeta zeros" could be used to exactly reconcile the individual random behaviour of the primes with their collective ordered regularity (with respect to the natural numbers).

Now this issue might initially appear as of a somewhat limited technical nature.

However the famous German mathematician Hilbert was to later refer to the problem of the "zeta zeros" not only as the most important in Mathematics but absolutely the most important (i.e. of all problems).

Now I happen to agree with Hilbert on this (though for reasons that he would have been loath to consider).

What Riemann in fact discovered, way back in 1859, was that underlying the conventional number system - which we take so much for granted - is an alternative, highly intricate wave-like series of numbers, which is essential for the very operation of the conventional system.

Now something similar was to be discovered much later in the 1920's in physics when in Quantum Mechanics, it was found that all subatomic phenomena have dual particle and wave manifestations

And as we know Quantum Mechanics has created havoc with respect to the traditional classical notions of physical processes (based on Newtonian Mechanics)!

This should have then suggested, due to Riemann's earlier discovery, that a similar problem exists at the very heart of Mathematics. However because the prevailing mathematical orthodoxy is much more entrenched than in physics, this did not happen

However new empirical evidence emerging from the 1970's has lent further weight to the fact that there seems to be indisputable close connections as between the "zeta zeros" and the observed energy states of certain quantum mechanical systems.

Over the last 10 years or so, I have devoted an enormous amount of time to unravelling this mystery regarding the "zeta zeros" for strange as it might seem, I strongly consider that professional mathematicians are in no position to do so (due to the unquestioned nature of their existing rigid assumptions).

In fact, at the deepest level, the "zeta zeros" relate to the ultimate connection as between quantitative and qualitative type meaning!

For several millennia now, we have tried to understand Mathematics as the mere study of quantitative type relationships. However in truth, the quantitative has no strict meaning in the absence of its counterpart qualitative aspect.

In other words the basic problem in Mathematics - and by extension all conventional science - is the confusion of the nature of an overall relationship in any context (which is qualitative) with the constituents to be related (that are - relatively - quantitative).

For example in present Mathematics, the individual primes are considered in a merely reduced quantitative manner as the random " building blocks" of the natural number system. However the collective nature of the primes displays a remarkably ordered relationship with the natural numbers.

In fact the term "the music of the primes" - with its obvious qualitative connotations - has been used to refer to this ordered relationship!

So the big mistake that is made in present Mathematics is the attempt to understand both the "building blocks" (as parts) and their collective relationship to the natural numbers (as the whole) in a merely reduced quantitative manner.

Rather, what we have here is the two-way relationship of quantitative to qualitative (and in turn qualitative to quantitative) meaning.

And crucially we cannot attempt to meaningfully reduce the qualitative aspect in a merely quantitative manner! In fact we need two distinctive modes of understanding i.e. analytic and holistic for every mathematical symbol and relationship, which are then considered in a dynamic interactive manner.

So to put it bluntly, our present understanding of the number system - and by extension all mathematical and scientific relationships - is, strictly speaking, not fit for purpose as it represents the fundamental reduction of the collective whole (in any context) to its individual parts (which are then interpreted in a merely quantitative manner).

And I am fully confident that future generations will come to accept this observation.

In truth the number system - and the other mathematical relationships - can only be properly understood in a dynamic interactive manner, entailing the complementarity of both quantitative and qualitative aspects (which are analytic and holistic with respect to each other).

In fact, as we approach its origins, the number system is revealed as amazingly dynamic structure with a holistic synchronicity characterising its very nature. So ultimately the primes and natural numbers are revealed as perfect mirrors that mutually interact with each other, both quantitatively and qualitatively, approaching a state of pure ineffability. So rather than a timeless abstract entity, in truth number, with respect to its analytic and holistic aspects, represents the fundamental encoded nature of all phenomena (with natural physical phenomena in turn representing the evolving decoded nature of number)!

So we are light years here from present accepted notions!

I have been outlining for many years now the 3 important stages that I would see in a future "golden age" of both mathematics and science.

First, we have analytic type specialisation of a quantitative kind. This is the form of science that has dominated the last 300 years or so of enquiry. Though the proper differentiation of quantitative from qualitative type meaning has indeed brought remarkable benefits, it is crucially limited in important respects. Because it lacks any true holistic dimension, complete reliance on such science will lead to increasing fragmentation in scientific, economic, social and political terms.

The signs are already there - though not yet sufficiently recognised - of the start of a serious  breakdown with respect to both mathematics and science (where they are unable to adequately explain their most fundamental features)

The second requirement is holistic specialisation of a qualitative kind. This form of science is practically non-existent (except for some unrecognised fringe activity at the margins).

Ultimately it will be required to bring greater integration to our world.
However its rationale is hard to meaningfully discuss here, as it is not yet recognised as having any validity by the scientific profession.

The third requirement is to eventually bring both quantitative and qualitative strands together in a new dynamic synthesis that can be both immensely productive and highly creative.

I refer to these simply as the Type 1, Type 2 and Type 3 approaches that apply equally to both Mathematics and Science.

We are still largely confined to the Type 1 approach - of which Richard Dawkins is such a strong advocate - that formally is of a merely reduced quantitative nature.

Thus if we wish among other things, to properly unweave the rainbow, we will need ultimately to also recognise the pressing need for both the Type 2 and Type 3 scientific approaches.