Holistic Mathematics 1 - Introduction

 Nature of Holistic Mathematics

Q Can you tell us briefly, what is Holistic Mathematics?

PC In conventional mathematics, symbols are given a static absolute type interpretation where opposite poles are clearly separated.
Thus mathematical relationships are assumed to have an independent validity (that is not influenced through interaction with interior mind). So for example its propositions are considered in absolute terms as either true or false.
In fact for many the great appeal of conventional mathematics derives from the seemingly clear-cut objective nature of the truths it represents.

However because in experiential terms all understanding is inherently dynamic, this necessarily applies to the interpretation of mathematical symbols which intimately depends on the corresponding understanding of the stage of development from which they are viewed. Thus when understood appropriately, the interpretation of symbols continually changes with each major stage of development leading to a distinctive mathematical system. Indeed in the most fundamental sense these symbols carry a merely relative and paradoxical meaning that ultimately dissolves in the pure mystery of Spirit.
When approached in this manner, the nature of mathematics is utterly transformed so that the interpretation of its symbols becomes inseparable from the very nature of understanding itself, thereby providing the appropriate tool for scientific encoding of the inherently dynamic structure of that understanding.

I state my position firmly without any desire to exaggerate.

In conventional analytic terms Mathematics - as Gauss said - is "the queen of the sciences". Indeed modern science is unthinkable in the absence of the highly valuable tools provided by such mathematics.

Equally - when appropriately understood in an experiential interactive manner - Holistic Mathematics has an entirely distinctive interpretation with the inbuilt capacity to become "the queen of the development sciences".

In other words in order to interpret (i.e. encode) development - with respect to all its varied stages - in an appropriate scientific manner we need to use the powerful tools deriving from the dynamic interpretation of mathematical symbols.

Q You seem to be proposing something very radical here!

PC Yes ! I really think so. Mathematics - as commonly understood - represents just one possible interpretation of its symbols which is especially suited for analytic science.
And this science - with its supporting mathematical tools - is based on the differentiated meaning deriving from clear separation of the fundamental polar opposites (e.g. exterior and interior).

However there is an alternative interpretation of mathematical symbols that is especially suited for integral science (as applied for example to all transformation processes in development).
This alternative science - again with its supporting mathematical tools - is based on the dynamic interactive meaning deriving from corresponding complementary recognition of the fundamental polar opposites.

As we have seen in our previous discussions the actual dynamics of experience necessarily entail - to a degree - both the separation of opposites (as independent) and their complementarity (as interdependent).
Thus the analytic and integral scientific interpretations necessarily derive their meaning from this overall dynamic context.

So I would broadly distinguish three types of science supported in each case by their corresponding mathematical tools.

1) Analytic Science - supported by Conventional Mathematics

2) Integral Science - supported by Holistic Mathematics

3) Radial Science - supported by both Conventional and Holistic Mathematics.

Q  Just one small point! Why do you customarily use the term Holistic Mathematics (rather than Integral Mathematics)?

PC Integration (and differentiation) have specialised analytic interpretations within Conventional Mathematics (as those familiar with Calculus will recognise).
However in the context of my approach integration has an inherently dynamic interpretation (based on the complementarity of polar opposites). So my use of the term "Integral Mathematics" could possibly lead to unwanted identification with the standard meaning.

The Pythagoreans

Q You seem to be saying that that there are vast new areas of holistic mathematical interpretation that have remained undiscovered. Can this be so? Surely there must be precedents for what you are doing?

PC This is a very interesting question indeed and one that has engaged my mind considerably over the years.
Though the archetypal nature of mathematical symbols has been well recognised through the ages, I have yet to find such understanding properly structured in a precise scientific manner with direct applicability to all development processes.
So in this sense - with its key symbols redefined in an inherently dynamic manner - I believe that I am proposing the truly original form of mathematics that potentially has an immense integrative capacity. And insofar as my limited ability allows I intend to demonstrate how it can be frutfully applied to the interpretation of all the key stages of development.

I do not claim to be a historian of mathematical ideas so I will only relate those which have either influenced or supported in some manner my thinking.

The Pythagorean school is especially interesting in this regard as it promoted an inherently integrated view of mathematics. This could be expressed by saying that they believed in important correspondence as between the quantitative order revealed through mathematics and the qualitative philosophical paradigm revealing this order. Furthermore the balanced appreciation of both aspects was seen as the ideal preparation for spiritual contemplation (thereby serving as the ultimate goal of mathematical activity).

Perhaps I can illustrate this with reference to the most famous of their "discoveries" i.e. the Pythagorean triangle which is one of the best known theorems in Mathematics.
This states that in any right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides (adjacent and opposite).
Thus for example if the two sides are measured as 3 and 4 units respectively then the hypotenuse will be 5 i.e. 32 + 42 = 52.

But the simplest possible case where the two sides (adjacent and opposite) are both 1 raised an intractable problem.
In this case the hypotenuse is the square root of 2.
Now the Pythagoreans believed that all numbers were rational (i.e. with unambiguous values that could be expressed as fractions).

However the square root of 2 is an irrational number that cannot be expressed in terms of a rational fraction.

To appreciate why this discovery was so disturbing we need to consider again the nature of their integral approach which assumed a direct correspondence as between the quantitative order of numbers and the qualitative (i.e. philosophical) worldview through which they were interpreted.

So in qualitative (philosophical) terms the Pythagoreans adopted a rational approach which was then balanced by their belief in a corresponding quantitative order (i.e. where all quantities were rational).
Thus the real dilemma regarding the discovery of an irrational number was that it challenged the very basis of their integral assumptions.
In other words in the context of their philosophical approach they had no means of explaining why a number could be irrational.

Following the Pythagoreans a significantly reduced interpretation of mathematics has emerged where quantitative has become largely divorced from philosophical interpretation.

We have in fact two types of irrational numbers (algebraic and transcendental).
(The square root of 2 is the best known example of the former and pi the most famous of the latter type)!

Numbers can also be transfinite. And - as we have seen - we have imaginary as well as real numbers (and complex numbers which comprise both aspects).

However all of these number types are given a merely reduced interpretation within conventional mathematics so that their - potentially - great philosophic riches have been lost.

For example, though mathematicians define and successfully use imaginary numbers in a (solely) reduced analytic context they can offer no insight as to the extremely important philosophical significance of an imaginary number.

Q I know we have dealt with this somewhat in previous discussions but can you briefly elaborate again on this significance?

PC. When appropriately understood - in a dynamic context - the notion of an imaginary number has a vital role in the scientific appreciation of the fundamental interaction between wholes and parts (and parts and wholes) in all development processes.
For example it explains the subtle interplay as between perceptions and their corresponding concepts which necessarily entail all phenomenal experience.

Q And presumably you are implying that the other number types such as irrational, transfinite and complex - when correctly understood in dynamic terms - can potentially play an equally important role in the scientific interpretation of development!

PC That is true. When one reflects on it number is synonymous with quantitative notions of order. We need number to structure experience in quantitative terms.

However what is not readily appreciated is that number is - potentially - equally synonymous with qualitative notions of order. Thus to structure the dynamics of experience properly in a scientific manner we need to use the alternative holistic (i.e. integral) interpretation of number.

Just as the rational paradigm, which defines analytic science, can be associated with the rational numbers, equally - in qualitative terms - we have important alternative number paradigms which can be associated with the other main number quantities.
As we shall see later associated with each stage is a specific qualitative number interpretation which precisely defines the very nature of the stage. It is the understanding of these respective stages that gives rise to a spectrum of scientific number paradigms (or rather meta-paradigms).

Q I think I am beginning to grasp what you are saying. You are implying that the limitation with conventional science is that it is largely confined to the rational understanding of just one major stage of the Spectrum where interpretation is of an absolute unambiguous nature.

However there are several "higher" stages in the Spectrum with which are associated uniquely distinctive cognitive interpretations which directly correspond with the appropriate qualitative holistic interpretation of these other number types.

So instead of just one valid type of scientific enquiry - which is associated with the understanding of the middle stage - we have a range of potentially valid types of scientific enquiry that are associated with the middle "higher" and radial levels.

PC Yes! And the great problem is that so far we have largely failed to properly identify and clarify these alternative scientific approaches (which are equally rooted in the corresponding types of mathematical understanding associated with the "higher" and radial levels).

Q Why do you think that this is the case? Surely you accept that the nature of the more advanced stages have been clarified to a considerably level of detail by the major mystical traditions.

PC I would only partially accept this assertion. It certainly is true that the spiritual contemplative nature of these stages has been clarified but - I would maintain - largely in terms of secondary expressions that are unduly associated with the religious symbols of the cultures in which they emanate. So we have for example Buddhist, Christian, Hindu and Islamic mystical expressions all pointing to the same basic experience but which are not fully comparable in terms of each other.

Also I would maintain that undue attention has been placed on the directly spiritual aspects of these stages with insufficient attention devoted to the precise nature of associated cognitive, affective and volitional structures. As spiritual development necessarily interpenetrates with these structures in an increasingly refined manner at the more advanced levels, it is vital therefore to properly clarify their nature.
In particular I see a great lack of appropriate recognition of the very subtle bi-directional cognitive understanding that is associated with the "higher" levels which provides the appropriate basis for integral science.

Thus - from the cognitive perspective - there remains a great absence of any universal type language to unravel the dynamics of mystical development, which traditionally have been unduly wedded to the interpretations of particular cultural traditions.

Now the great value of scientific (and mathematical) symbols is that they attempt to provide this universal type interpretation.
Therefore - in a certain qualified sense - even though scientists may come from widely differing backgrounds and cultural experiences they share the same common interpretation of scientific truth.
Furthermore the establishment of the universal type language and procedures of science has greatly accelerated its progress.
So what is true in terms of the analytic approach to science can equally apply in a holistic context.

Q Are you therefore saying that there is great need for a universal language for interpreting the understanding traditionally associated with the more advanced mystical stages of development and that this constitutes radically new notions of integral - as opposed to analytic - science?

PC Yes! I would add however that the firm basis for this integral science are corresponding notions of integral (i.e. holistic) mathematics. Now such understanding in no way replaces the need for existing secondary expressions of mystical development. Indeed properly understood it can provide a more general framework with the capacity to enhance the value of such expressions.
Also scientific expression cannot substitute for directly spiritual or artistic type understanding.
However equally we should not underestimate the great value of developing a dynamic (and subsequently radial) interpretation of symbols that is properly scientific (and in a qualified sense therefore of universal applicability).

So I am proposing an approach to the more advanced stages of the Spectrum that does not specifically require the traditional religious terminology of the mystical traditions.
Of course the secondary expressions of these traditions would still remain vitally important though necessarily in some respects of an arbitrary nature.

Thus it should be possible for example for a Buddhist and Christian to share a wide level of agreement regarding the universal nature of the more advanced stages of development i.e. through adopting a common scientific language, while choosing very distinctive secondary means (dictated by their respective religious traditions) for expressing this development.

The Pythagorean Dilemma

Q Can the Pythagorean Dilemma - as you call it - be satisfactorily resolved?

PC Indeed, it can! One of my earlier little "successes" with Holistic Mathematics came from providing - what I considered - a satisfactory solution.
Though the Pythagoreans ultimately viewed mathematics as an aid to spiritual contemplation, their cognitive perspective was limited due an adherence to somewhat unambiguous notions of scientific truth (i.e. the rational paradigm).

However the refined cognitive understanding properly associated with the "higher" spiritual levels is of an increasingly paradoxical nature and therefore not rational in this limited sense.

So what is required to solve the "Pythagorean Dilemma" is to provide an appropriate holistic mathematical interpretation of the cognitive understanding that typifies H1 i.e. the subtle realm which exactly matches in dynamic terms the analytic interpretation of the square root of 2.

Q How briefly is this done?

PC As we have seen the very nature of H1 is that it is bi-directional with opposite horizontal polarities of form (positive and negative) coinciding as empty Spirit.

Now we can only express this relationship in reduced linear format through identifying the opposite polarities involved in a somewhat dualistic fashion.

In this way the spiritual nature of the "higher" - ultimately ineffable - two-dimensional reality where opposite poles coincide, is expressed in the (reduced) dualistic language of form by two separate poles (that are positive and negative with respect to each other).

Likewise the square root of 2 (i.e. the solution for x in the equation x2 = 2) equally entails the attempt to express a "higher" two-dimensional relationship in reduced linear (one-dimensional) terms. Again the result splits into two separate poles (which are positive and negative).

So x (i.e. the square root of 2) has equally positive and negative expressions.

As we know this square root is irrational (cannot be expressed as a rational fraction).
This means that its value can only be approximated in rational terms.

Thus, correct to four decimal places the square root of 2 is either + 1.4142 or - 1.4142.
However its true value continues indefinitely and cannot be precisely determined.
So irrational numbers (such as the square root of 2) always contain both finite (discrete) and infinite (continuous) aspects that are interrelated.

In like manner both finite and infinite aspects are inherent to the phenomenal understanding of H1.
So from one perspective we identify phenomenal form in finite terms. However by its very nature, such phenomenal form (finite) interacts with spiritual emptiness (infinite) in an ultimately indeterminate mysterious fashion.

Therefore - in a precise holistic mathematical fashion - the understanding of H1 is qualitatively irrational in a manner that directly corresponds to the quantitative behaviour of (algebraic) irrational numbers.

So just as the rational paradigm - which defines analytic science - corresponds to the quantitative behaviour of rational numbers, equally the "irrational" paradigm - which potentially defines the integral science of H1 - corresponds to the quantitative behaviour of irrational numbers.

Q This is very interesting! I gather that what you are suggesting here is that a proper understanding of irrational numbers - that properly integrates quantitative and qualitative type appreciation - requires the spiritually inspired "higher" cognitive understanding that goes beyond the rational scientific understanding that typifies the middle levels of the Spectrum. Presumably this would apply to all the other "strange" number types such as transcendental, transfinite, imaginary and complex.
Thus, conventional Mathematics attempts to deal with "higher" order quantities through reducing their interpretation to the understanding of the middle levels (that is properly suited solely for interpretation of rational quantities).
Is this correct?

PC Very much so! Remember again how vitally important number is as a means of quantitative ordering!
So when we look at the various number types in holistic mathematical terms, we realise that potentially they have an equally important role as a means of qualitative ordering that define a whole range of distinctive scientific meta-paradigms (analytic, holistic and radial).

These meta-paradigms in turn directly correspond to the fundamental understanding that defines the intrinsic structure of all the main stages in development.

Q Are you therefore implying that as number in holistic terms provides the appropriate means for the qualitative ordering of phenomena that all the stages of development correspond (in their basic structure) to number types (as defined in a dynamic qualitative sense)?

PC That's it! Just as we have a Spectrum of numbers (as analytically defined) enabling the quantitative ordering of phenomena, equally we have a Spectrum of numbers (as holistically defined) enabling the qualitative ordering of phenomena (i.e. the scientific understanding of all the major stages of the Spectrum).

There is in fact - though not yet appreciated - a direct correspondence as between the analytic and holistic interpretations of the Number Spectrum. This would mean for example that the discovery of a major new (quantitative) number type in analytic terms would thereby imply the definition of a corresponding (qualitative) type in a holistic manner. In other words it would lead to the definition of the appropriate dynamic structure for a major new stage of development (with its associated scientific meta-paradigm).

Equally the discovery of a major new stage in development (as qualitative number type) would entail the definition of a new (quantitative) number type in analytic terms.

Q I can appreciate the logic of what you are saying. However it puzzles me greatly as to why such understanding - which seems indeed to have truly fundamental implications - is largely "missing" from the culture. Why do you think this is so!

PC Unfortunately, major divisions still exist as between mystical spirituality and rational science.
Very little interpenetration of these two realms has taken place with representatives of both camps speaking languages that are largely incompatible in terms of each other.

Now it is certainly true that when viewed from the differentiated conventional perspective, science and mystical spirituality have little in common.
Science - and especially mathematics - represent the rational analytic extreme; mystical spirituality represents the intuitive contemplative extreme.

However when viewed from the corresponding integral viewpoint, mathematics and (mystical) spirituality are dynamically interdependent and have the capacity to mutually serve each other in a truly remarkable fashion.
However this appreciation can only flow from an understanding which creatively combines both in a balanced experiential manner.

There are no shortcuts here.
The proper interpretation of holistic mathematical symbols requires authentic development of pure mystical intuition (as spiritual emptiness). Likewise it requires the precise dynamic mathematical appreciation associated with very refined bi-directional interpretation of cognitive forms.

With few exceptions, mystical development has been traditionally associated with pure spiritual intuition (without the corresponding capacity to express such understanding through integral mathematical symbols).
Likewise mathematical development has been associated with a special type of rational cognitive understanding (without the corresponding capacity to realise a pure contemplative level of spiritual awareness).

So I see Holistic Mathematics as the integration of twin aspects of understanding (intuitive and rational) that have heretofore developed in a largely separate manner. So through combining the two in a uniquely original manner, the potential power associated with both aspects can thereby be greatly enhanced.

Other Influences

Q Let us move on! Can you briefly outline some other influences?

PC Leibniz would certainly be included. As is well known he independently discovered  calculus (along with Newton).
He also had a fascinating philosophical system of "monads" which lends itself to the holistic mathematical interpretation of the dynamic relationship as between whole and part (and part and whole).
He developed the binary number system (based on 1 and 0) which has played such a leading role in the digital revolution.
Remarkably he also appreciated the metaphysical significance of 1 and 0 which is the important first step towards recognition of the complementary holistic significance of these same digits.

Nicholas of Cusa is another interesting figure. Though living in the 15th century he had a surprisingly modern view of the relative nature of space and time (in keeping with Einstein's Theory of Relativity).
He also saw very clearly into the fundamental nature of polar opposites and was one of the few to properly combine mystical insight with mathematical understanding.
I imagine him as the type of person - who if around today - would readily appreciate the nature of Holistic Mathematics.

I also owe a debt to Hegel who greatly influenced my early philosophical appreciation of the dynamic nature of developmental processes.
Indeed Hegel's emphasis on "dynamic negation" proved the catalyst in enabling me to make the vitally important link as between the analytic interpretation of the fundamental operations of addition and subtraction (i.e. positing and negating) and their corresponding holistic mathematical equivalents.

Carl Jung is an important influence. His approach - though it never adequately deals with advanced spiritual stages of development - is inherently dynamic. I think that there is little doubt that he implicitly thought in a holistic mathematical fashion. Not surprisingly therefore many of his important ideas readily lend themselves to subsequent explicit interpretation in this manner.

I find his treatment of personality types especially valuable in this regard. The manner in which he views the complementary nature of conscious and unconscious aspects with respect to the various types helped me greatly to make a vital connection with the dynamic mathematical interpretation of the complex number system (with its "real" and "imaginary" aspects).

Jung also greatly appreciated the importance of mandalas as symbols of psychological integration. These mandalas which often entail ornately designed circular figures symmetrically divided into four quadrants (or alternatively eight sectors) lend themselves to profound holistic mathematical interpretation.

I would see important holistic mathematical notions deeply implicit in the mystical traditions.

For example Taoism lays significant emphasis on the complementarity of polar opposites so that the primal indivisible unity (which is purely spiritual) splits at a phenomenal level into opposite polarities (that are positive and negative with respect to each other).

There are important connections here (in dynamic holistic terms) with the analytic notion of the square root of 1, and the very manner of expression of Taoism proved a valuable catalyst in helping me to see this unexpected connection.

In the Hindu tradition, yantra yoga - based on symmetricaly designed geometrical shapes - is used in a manner similar to the mandalas as a deliberate mathematical focus for spiritual meditation. Because Holistic Mathematics is based on the integral interpretation of symbols it can clarify why such patterns potentially possess great spiritual power.

In Christian mysticism the ultimate goal of spiritual unity is often sharply contrasted with the nothingness of self (and of all phenomenal created reality). We have here in fact the dynamic holistic equivalent of the binary digits (1 and 0) which are so important at a mathematically analytic level of interpretation.

Likewise the most famous Buddhist sutra

"Form is not other than Emptiness
Emptiness is not other than Form"

could be alternatively expressed as

"Unity is not other than Nothingness
Nothingness is not other than Unity"

This thereby provides a holistic dynamic expression of the ultimate interdependence of unity (1) and nothingness (0).

So the binary digits which are fully independent at the (static) analytic level become fully interdependent at the corresponding (dynamic) holistic level.

I remember being especially struck by a Chapter in "Blackfoot Physics" by David E. Peat entitled "Sacred Mathematics". The culture of the Indian tribes he was describing had an inherently dynamic understanding of mathematical symbols which resonated strongly with many of my own insights.

Now one could be unfairly critical and dismissive by saying that in many respects this understanding was conveyed through magical and mythical representations and thereby of a "prepersonal" nature.
However in dynamic terms prepersonal and transpersonal are structurally complementary. Therefore the dynamic understanding conveyed through prerational symbols points directly to a "higher" understanding that is properly transrational.

The problem about our rational culture is that the dynamic meaning, that is inherent in all mathematical interpretation, has been all but lost. So we are cut off both from the prerational roots and thereby also the enormous transrational potential of its symbols.

Put another way our present culture represents an extremely specialised differentiated interpretation of mathematical symbols (suited for analytic science).
However equally it represents the great loss of any meaningful holistic interpretation of these same symbols (which is properly required for integral science).

Q Are there any other modern influences?

PC Surprisingly few have attempted to combine mathematical understanding with genuine transpersonal awareness. However there are always exceptions.

In this context I would refer to Franklin-Merrell Wolff. In some ways he represents a modern version of the Pythagorean ideal in his manner of appreciation of how mathematical symbols e.g. transcendental numbers can provide the most refined means of spiritual meditation.

However I would especially like to mention the Russian polymath Vassily Namilov who combines both wisdom and learning in an unusually original manner.

For example - using a probability approach derived from his mathematical background - he shows how stages of development have both discrete and continuous interpretations (with consciousness in its ultimate integral state fully continuous). Not surprisingly therefore he is critical of the (discrete) hierarchical perspective as an approach to integration.
We would certainly share much common ground on this point!

He also expresses the view that the inherent processes of consciousness conform to a  geometric pattern. I would agree here but go further to extend this to an overall mathematical structure (which can be equally represented in arithmetical, algebraic and geometric format).

Also I would emphasise that that holistic mathematical understanding is very distinct from its analytic counterpart. I think this understanding is deeply implicit in what Nalimov says. However I believe much greater clarity of expression can arise from making this understanding fully explicit.

Reduced Mathematical Notion of the Infinite

Q Was there any particular event in your life that precipitated the development of Holistic Mathematics!

PC Though I studied Mathematics at University I became somewhat disillusioned with - what I could see - was a reduced manner of intellectual interpretation. In particular my disenchantment centred on the manner in which Mathematics attempts to deal with the notion of the infinite.

From a spiritual perspective the infinite is qualitatively distinct from finite notions. However within Mathematics this distinction is lost with the infinite being treated - in effect - as a linear extension of the finite.

For example we can represent the finite number system by a straight line stretching in both directions (representing positive and negative values respectively).
Now the customary interpretation of the infinite is given as an "indefinite" extension of these two linear directions. So we are led to the utterly misleading view that as numbers become larger and larger in finite terms they eventually approach infinity.
So technically in the limit if a number is greater than any arbitrary quantity we may assign, it is thereby infinite.
However strictly this makes no sense. By definition any number that we can assign is finite and any number greater than this assigned quantity thereby is also finite.

We also have the same confusion in relation to "infinitesimals".
In calculus the notion of an "infinitesimal" is vitally important in studying rates of change. So if y is defined - as is customary - as a function of x, we can define dy/dx i.e. the rate of change of y with respect to a change in x as we make this change smaller and smaller. Now again in the conventional manner of interpretation the change in x approaches zero by a limiting process whereby it becomes "indefinitely" small and infinitesimal (with no finite meaning).
Technically this limiting value is expressed as a number that is less than any value - however small - that we may assign.

However once again, strictly speaking, this interpretation is utter nonsense and simply reflects the reduction of the infinite to finite notions of understanding.
If we assign a finite quantity - however small - it remains a finite quantity. Likewise the "limiting" value which is less than this remains a finite quantity!

Now when properly appreciated this confusion of the infinite with finite notions of interpretation, undermines the very concept of mathematical "proof".

Let us take the well-known case we have already discussed i.e. "The Pythagorean Theorem". The general theorem states that in any right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

This theorem potentially applies to "all" right-angled triangles.
However "all" in this case is potentially infinite.
Thus the reduced assumption - that lies at the heart of all mathematical proof - is that what is true in the potentially infinite case applies to every specific actual case (that is finite).

In other words we have here the direct confusion of potential (infinite) and actual (finite) interpretation.

So strictly speaking a general proof which potentially applies for "all" cases does not logically apply in any actual particular case.
The problem is that - dynamically speaking - "all" cannot be absolutely defined in finite terms.
In other words with respect to any general class, what we determine as finite is always against the background of other finite cases that remain indeterminate.
So uncertainty necessarily applies to all finite determination, which can only take place in the context of what also remains indeterminate (in finite terms).
Thus to pick any actual finite number (or group of numbers) from the general set of number (which potentially applies in "all" cases) requires that other finite numbers cannot be selected.
In other words we can never exhaust the general set (that potentially applies to all numbers) through the finite selection of actual numbers.

In deeper terms this implies that understanding always entails the dynamic interaction of reason and intuition. The very transformation by which we move from specific quantitative (finite) to general qualitative (infinite) appreciation and in reverse fashion from qualitative to quantitative recognition is always implicitly spiritual and intuitive. However in reduced terms we try to interpret this understanding in merely rational terms (whereby infinite are thereby directly confused with finite notions).

I am not questioning the great value of the conventional interpretation of mathematical "proof" within its own limited domain (i.e. of reduced analytic interpretation).

What I am saying however is that a richer understanding - which avoids gross reductionism - is inherently dynamic where spiritual (intuitive) and cognitive (rational) aspects of understanding interact.
And this inherently dynamic interaction of reason and intuition is the basis for Holistic Mathematics.

Q Briefly, can you explain how you preserve the qualitatively distinct nature of the finite and infinite within holistic mathematical interpretation.

PC If we go back to the conventional interpretation of numbers, positive and negative aspects are clearly separated. So in linear terms the positive number system would be represented by the straight line drawn from centre (representing 0) in a Right-Hand direction whereas the negative number system is represented by the straight line in the opposite Left-Hand direction.

So infinity is represented as the indefinite extension of the line in both directions leading to + infinity and - infinity.

However in dynamic terms, positive and negative aspects are directly complementary.
So whereas finite notions results from the differentiation and separation of these polar opposites, infinite appreciation results from their corresponding complementarity (and ultimate spiritual identity).

Therefore infinite (intuitive) appreciation always results from the dynamic process of negation (of what has been phenomenally posited) in experience.

In static analytic terms, when we negate what has been already posited we get 0.
e.g. 1 - 1 = 0.
Likewise in dynamic holistic terms, when we negate phenomenal form we get nothingness (i.e. spiritual emptiness).
So in holistic terms, 1 represents phenomenal form so that again 1 - 1 = 0.
However, though in analytic terms, 0 and infinity are at opposite extremes (and clearly separated from each other) dynamically speaking they are fully complementary and ultimately identical.
Thus in this holistic context 0 and infinity are complementary expressions of the same ultimate spiritual identity i.e. as the plenum-void (or the nothing that is potentially everything).

(We can even see the "logic" of this dynamic interpretation from the conventional mathematical identity 1/0 = infinity).

Q I see what you are getting at and will attempt to summarise. You are saying that both finite and infinite notions properly arise from the dynamic context of experience where polar opposites are understood in both a separate (differentiated) and complementary (integral) fashion. So the finite relates directly to differentiated and the infinite to integral interpretation respectively (though again in the dynamics of experience these aspects are necessarily interdependent).
Now in turn the rational interpretation of mathematical symbols results directly from the finite (differentiated) understanding whereas the intuitive appreciation comes from infinite (integral) understanding.
However conventional mathematics is necessarily limited in the sense that it tries to reduce the dynamic interaction of both reason and intuition to merely rational interpretation. So one obvious consequence of this is the corresponding confusion of the infinite with merely finite notions.

Did you make much progress with this holistic mathematical understanding while at University?

PC Looking back at some notes from that time I realise that I had made greater progress than I subsequently imagined.
For example I had come up with a dynamic interpretation of the number system (that owed a great deal to the Hegelian influences on my thinking at the time).

Basically this was based on the notion that - in dynamic terms - the positing of a number - always entails corresponding negation. (Remember this is the dynamic holistic application of the operations of addition and subtraction)!
So we can only posit a specific number (as an actual number perception) through the corresponding negation of the general notion of "number" (i.e. the number concept that potentially applies to "all" numbers as infinite).
Likewise we can only posit the general notion of "number" (as number concept) by negating the specify notion of an actual number (as finite number perception).
So the balanced rational and intuitive appreciation of numbers - whereby the qualitative distinct nature of finite and infinite notions is preserved - arises from this dynamic interactive context.

However there is a great distinction as between embryonic insights deeply suggestive of a radical new approach and its fully fledged comprehensive appreciation.
So it took me the best part of three decades before I could properly see what was required to make this early vision a reality (i.e. with the power to comprehensively encode reality in an integral scientific manner).

What I did not fully appreciate at the time, was that my intuitive insight was not sufficiently refined to make the very subtle connections required for the more comprehensive worldview.
Thus the proper development in cognitive terms of holistic mathematical understanding is inseparable from the corresponding spiritual attainment of purer contemplative awareness.