For many years I have strongly felt the need for new thinking and this has led to the development of Holistic Mathematics. It provides the basis for the new scientific vision.

I do not see Holistic Mathematics as my own creation but rather as the expression of a contemporary need for this truly scientific approach to integration. Therefore I want to share it with anyone who wishes to accept).

Lugi asked sometime ago for short summary of Holistic Mathematics.

The basic point is very simple.

(Conventional) Mathematics is extremely important as an important analytic tool for (differentiated) scientific studies.

However I see Holistic Mathematics as an equally important synthetic tool for (integrated) scientific studies.

There are really two logical systems inherent in understanding.

The first system – which has now been developed to a high level of specialisation - is the either/or logic based on the clear separation of polar opposites in experience.

Thus in this logical system a proposition is either true or false; understanding is either objective or subjective; movement is either forward or backward etc.

This logic corresponds with what we understand as reason and is the direct basis for differentiation in experience and our notions of conventional (analytical) science.

The second system – which has not yet been coherently formalised – is the both/and logic based on the complementarily (and ultimate identity) of opposites in experience.

In this logical system a proposition is both true and false; understanding is both objective and subjective; movement is both forward and backward. Put another way, in this latter systems all statements are understood to have a purely relative validity.

This logic corresponds with spiritual intuition and is the direct basis for integration in experience. It is an inherent component of all authentic mysticism. However what is not properly recognised is that all levels of intuitive understanding have precise indirect rational translations which provide the basis for the holistic (synthetic) approach to science (i.e. integral science).

Perhaps the most common mistake is to try and approach integration through the first logical system (for which it is not suited). Despite his enormous contribution I find that Ken Wilber is very much at fault in this respect. He does not employ what I would consider a truly integrated approach. His work is therefore full of internal inconsistencies (which I have been pointing out for some time on the Forum).

The difference as between these two logical approaches can be characterised in another way.

The first logical system basically lends itself to a static (absolute) treatment of relationships characterised by a one-directional way of looking at reality. In other words logical connections are separated in time and take place a one-way sequential fashion.

The second logical system lends itself to a dynamic (relative) treatment of relationships characterised by a bi-directional way of looking at reality. In other words logical connections are seen to unfold in two directions simultaneously.

Both systems - though equally important - have distinctive uses.

We could say that a differentiated approach is characterised by one-directional understanding (i.e. where logical connections are sequential and unambiguous).

An integrated approach by contrast is characterised by bi-directional understanding (i.e. where logical connections are simultaneous and paradoxical).

(Conventional) mathematics represents a system of thought that is based directly on the specialised use of the either/or system of one-directional understanding.

What is truly remarkable is that all mathematical symbols, operations and relationships can be given an alternative explanation based directly on the both/and system of bi-directional understanding.

I refer to this mathematical understanding as Holistic Mathematics (or Holomatics). It is an inherently dynamic way of looking at relationships and provides the appropriate tool for an integrated scientific appreciation of reality.

The bi-directional approach - which is the basic feature of Holistic Mathematics - in itself represents the dynamic translation of the fundamental mathematical operations of addition and subtraction (i.e. positing and negating).

In static conventional terms these are understood as separate (mutually exclusive) operations. However in dynamic terms they are complementary (and inherent in all processes (physical and psychological).

In psychological terms, to posit is simply to make conscious in experience. All conscious phenomena are thereby - by definition - positive phenomena (in this dynamic sense).

Established notions of science represent a merely positive approach to understanding (as they are based on an objective conscious approach to reality).

However we have also the unconscious which is dynamically neutral. In rational terms this involves the equal balance of both positive and negative poles. Thus the crucial insight here is that in moving from conscious to unconscious understanding we must dynamically negate in experience (what has already been posited).

Though this process necessarily takes place in all experience, it generally remains undeveloped and merely implicit.

Authentic spiritual development involves the specialised development of this negative direction of understanding.

The fusing of this with the already formed positive direction leads to the generation of spiritual intuition. With the negative direction strongly developed a very pure form of intuitive experience is attainable.

So in holistic mathematical terms, (linear) reason, which is one-directional, is based solely on the positive direction of experience.

(Circular) intuition, which is bi-directional, incorporates both positive and negative directions.

So, we are here dealing briefly with the holistic equivalent of the most basic operations in (conventional) mathematics (i.e. addition and subtraction).

However this leads to extremely important insights. For example it means that all phenomena have mirror image equivalents which are necessarily involved in the dynamics of experience.

So to understand - for example - stages of development, in dynamic terms, we must recognise that every stage has a complementary mirror stage (with which it ceaselessly interacts).

Indeed my fundamental critique of Ken Wilber's methodology is largely based on this holistic mathematical interpretation of addition and subtraction.

Ken's method is largely a positive (one-directional) approach, which is suited to differentiation.

However a true integrated approach must be both positive and negative (i.e. bi-directional).

This is just a very brief introduction to one idea in Holistic Mathematics (which yet has enormous implications). However we may get some insight into its great potential value by accepting that ever mathematical symbol has a corresponding holistic interpretation.

Addition and subtraction are operations that apply to numbers. Most of my research this far has centred around a translation of number in holistic mathematical terms.

As Jung realised so well, number is the best archetype we have for order. My key purpose has been to redefine the various number types in dynamic terms as a precise scientific ordering for integral studies. This has led to many highly significant insights, which I will briefly try and explain.

There are many different number types. The most fundamental are the binary digits (1 and 0).

Then we have prime numbers (like 7 which have no factors). Then we have the natural or counting numbers (I, 2, 3, 4 etc which are uniquely related to prime numbers. We have negative numbers, and the other rational numbers (which include fractions).

We also have irrational numbers (which cannot be expressed as fractions). These are either algebraic (such as the square root of 2) or transcendental (such as pi).

We also have imaginary numbers (based on the square root of –1), and complex numbers (which comprise real and imaginary parts). Finally we have transfinite numbers (which transcend finite definition).

Now in dynamic terms the fundamental structure of reality can be precisely defined in terms of the holistic interpretation of these various number types. Furthermore this applies to both the physical and psychological aspects of reality, (now understood in complementary fashion).

Thus in physical terms we have various levels (and stages) comprising a Spectrum of Nature. In this approach each level has both a linear and circular interpretation. In other words though levels – from one perspective - are relatively distinct, they yet show various complementary features with other levels.

In psychological terms we equally have various levels (and stages) comprising a Spectrum of Consciousness.

Again the value of the holistic mathematical approach is that each level is interpreted as being both relatively distinct (linear) yet also relatively interdependent (circular) with other levels.

What is truly extraordinary is that all the various levels (and stages) in both spectra can be precisely defined in terms of holistic mathematical number types.

In other words the structure of reality in principle - even at an advanced degree of detailed investigation - can be given a precise mathematical interpretation (based on the three fundamental polarities of "The Theory of Everything").

In fact – and this is something that Jung would have appreciated – these spectra (of nature and of consciousness) have their most precise scientific translation as number types (holistically defined).

I would consider this a most fundamental breakthrough.

For example it paves the way for major developments in terms of the creation of new scientific paradigms.

The conventional scientific paradigm is often referred to as the rational paradigm. Indeed it is aptly named for it indeed accurately represent the holistic equivalent of the rational number system.

However rational numbers represent only one of many different classes of numbers. Associated with each of the other number classes (e.g. irrational, imaginary, complex) are distinct paradigms (with equal scientific value). Holistic Mathematics provides a means of precisely distinguishing the features of these paradigms.

I stated that the most fundamental numbers are 1 and 0.

Now as we know the use of these two binary digits through computerisation and information technology is revolutionising modern reality (which is increasingly referred to as the digital age).

So all information can (potentially) be encoded in binary digital format.

However there is an equally valid dynamic interpretation of digits possible using holistic mathematical notions. This insight opens up enormous vistas.

For just as the (linear) interpretation of digits is the basis for encoding all information processes, equally the (circular) interpretation of digits is the basis for encoding all transformation processes.

I have already satisfied myself that this is the case. I have found that it is possible to encode all the various levels (and stages) of the psychological spectrum in this holistic binary digital fashion.

Another key notion is what I call "The Theory of Everything".

This relates to the fact that reality (at all levels) can be precisely defined in terms of three sets of polar relationships (which I refer to as horizontal vertical and diagonal respectively).

These sets of relationships in turn have a precise holistic mathematical interpretation (relating to the eight roots of unity). What is fascinating is that the geometrical representation of such relationships (in the complex plane) leads the well known Jungian mandala patterns. So underlying these mandalas is a fundamental mathematical structure.

Horizontal polarity relates to the fundamental inner–outer division of reality (exterior- interior, object-subject etc).

Vertical polarity relates to the fundamental whole-part division (individual-collective, quantitative-qualitative).

Diagonal polarity relates to the fundamental form-emptiness division (actual-potential, manifest-unmanifest).

Now these three fundamental relationships (underlying reality at all levels) can be defined in accordance with two logical systems (which ceaselessly interact).

We can view them symmetrically (in terms of the circular system of complementary opposites) or asymmetrically (in terms of the linear system of separate opposites).

Pure contemplative awareness would approach perfect symmetry in terms of these poles. In other words in this spiritually intuitive experience, separate opposites (at horizontal, vertical or diagonal levels) would no longer exist.

The Holistic Mathematics research project involves a number of interelate stages (all of which have engaged my attention).

The first involves a detailed experiential map of the various levels (and stages) of the Spectrum that is performed in a coherent scientific manner (detailing cognitive and affective as well as spiritual stages).

This particular mapping is then used to provide an explicit translation in holistic mathematical format. So all key levels (and stages) of the two complementary spectra (of nature and of consciousness) are now seen to have a coherent (holistic) mathematical interpretation (which involves the dynamic understanding of all the main number types).

Associated with each major level (there are 7 in my approach) is a distinct major paradigm. Again the key features of these paradigms (their relation to each other) can be mapped out in holistic mathematical terms.

These (major) paradigms then form the basis for a range of differing scientific worldviews.

(Conventional) science is based on just one such worldview.

However fascinating alternative worldviews (which are based on complementary relationships) are also possible. I refer to the science based on these other worldviews as Holoscience.

So whereas the basis of the (analytic) differential approach is (conventional) Science, the basis of the (synthetic) integral worldview is Holoscience.

All established disciplines - such as Mathematics, Physics, Psychology, Philosophy, Biology, Economics etc. can be translated in a Holoscientific manner.

This provides the true basis for an integral (rather than a merely differentiated) approach.

Holoscience (with its integrated perspective) represents just one of three levels of scientific investigation.

Conventional Science represents the first level. This is the differentiated (analytic) approach (based on the either/or logical system).

Holoscience (Holistic Science) represents the second level. This is the integrated (synthetic) approach (based on the both/and logical system).

Radial Science represents the third (and most comprehensive level). This combines the differentiated (analytic) and integrated (synthetic) approaches (based on the interaction of both logical ssytems).

So the ultimate goal of scientific understanding should be Radial Science that would combine both analytic and holistic aspects.

However before this can be achieved the holistic aspect must first be developed.

In conclusion I would like to take the opportunity of clearing up some difficulties (and confusions) regarding my approach.

I appreciate that many find it very difficult to follow its rationale.

I believe that this is largely due to the fact that Holistic Mathematics represents a truly original approach (with only very tenuous historical precedents). It requires therefore the willingness to look at reality in a very different manner.

Secondly by its nature it represents a highly condensed (and abstract) cognitive type understanding (generally not associated with spiritual development).

Its formulations in no way are meant to replace he need for authentic spiritual progress. Indeed it is not be possible to properly appreciate these without such development.

I greatly welcome more informal discussion as a means of unpacking the compressed insights of Holistic Mathematics.

In no way do I want to see it as a replacement for other valuable means of enquiry.

However if we are serious about a truly scientific approach to integrated studies, then I believe that Holistic Mathematics cannot be ignored.