Thursday, April 15, 2010

Early Disillusionment

In these blogs I am attempting to accurately show the precise nature of my disillusionment with Conventional Mathematics and how this has led to the gradual development of a complementary qualitative mathematics that I refer to as Holistic Mathematics. This alternative mathematics then proved invaluable in terms of formulating an integral scientific approach for my personal study of many areas e.g. psychological development, physics and economics.

More recently I have used it in the attempt to unravel key issues within Mathematics.

My first serious investigation was with respect to the mysterious Euler Identity which led to a novel and unexpected understanding of its true significance.

I then used the insights acquired here to tackle the Riemann Hypothesis. This enabled me ultimately to see the problem in a completely new light with its resolution - at least to my own satisfaction - unfolding in a very simple manner.

The preparation for this resolution has entailed however more than 40 years in the painful exploration of new ground (that would be deemed totally irrelevant by the vast majority of practising mathematicians).

Both at Primary and Secondary School in Ireland I loved Mathematics often attempting to come up with my own solutions to problems. Indeed before I heard about logs as a child I had invented a working system (based on using powers of 2) and later had stumbled on Calculus ideas before becoming aware of the subject.

Not surprisingly therefore when I went to University I opted to pursue a Honours Course in Mathematics. However the first year was to prove a very chastening experience as I gradually lost confidence in my ability for the discipline. This coincided with deep philosophical questioning with respect to fundamental mathematical notions.

The first year course included the subject of Real Analysis which entailed considerable use of the mathematical concept of a limit.

Though apparently sophisticated procedures had been developed to handle limits, from my perspective I considered them something of a sham and unacceptable in philosophical terms.

The deeper issue here concerns the true relationship of finite to infinite in Mathematics.
I began then to strongly suspect that Conventional Mathematics was employing but a reduced notion of the infinite concept (where effectively it was reduced to finite meaning).

For example as a finite series such as 1, 2, 3, 4,.... gets larger, we are misleadingly led to the conclusion that in the limit it becomes infinite.

Now strictly speaking this is just nonsense. While it is true that the terms in this series will get progressively larger, however they always remain finite and at no stage become infinite!

Also in Calculus limiting notions, are very important. This time as a finite quantity becomes progressively smaller and approached zero, we are again presented with the mistaken conclusion that in the limit it becomes infinitesmal (infinitely small). Indeed Bishop Berkeley had already skillfully exposed the philosophical weakness of "infinitesmals" centuries earlier referring to them wittily as "ghosts of departed quantities".

So what we have in both cases is the attempt to reduce what is properly infinite to merely finite notions (as befits a linear rational interpretation of mathematical concepts).

And we cannot simply avoid this issue by attempting to reframe limit concepts so that we do not apparently need to introduce infinite notions. For the relationship between the finite and infinite is everywhere in Mathematics!

For example when we speak about number in fact two related notions are involved. Firstly we have the concept of number that potentially applies to all numbers in a general sense; secondly we have actual number perceptions that are specifically located within the number system. So the number 2 for example represents an actual finite number extracted from the overall number system (that is potentially infinite!)

Now the point is that the potential number concept relates to what is properly infinite whereas actual number perceptions relate to what is finite.

Thus we cannot deal with the very notion of number without accepting the interaction of finite and infinite aspects.

What we continually get therefore in Conventional Mathematics is the attempted reduction (in any relevant context) of what is strictly potential and infinite to merely finite rational interpretation.

Indeed such reductionism lies at the every heart of mathematical proof.

For example the proof of the Pythagorean Theorem entails that in every right handed triangle the square on the hypotenuse equals the sum of squares on the other two sides.

However "every" or "all" in this context has a merely potential meaning that is strictly infinite; when we empirically apply the theorem however, we are in the realm of the finite (where "every" has no strict meaning). In other words we can only determine the truth in a finite number of cases through leaving other finite cases always indeterminate.

So strictly speaking there is no absolute correspondence as between the truth of any proposition (in general) and its actual application in specific terms, as the former relates to the infinite and the latter to the finite domain. And once again finite and infinite represent notions that are qualitatively distinct from each other!

So the belief in absolute proof with respect to mathematical theories depends on this basic reductionism whereby infinite (whole) are reduced to finite part notions.

In more correct terms all mathematical proof is of a relative approximate nature reflecting a particular form of social consensus that expresses the orthodox beliefs of the mathematical community.

However a key indication that mathematical proof (such as the Pythagorean Theorem) is not absolute is given by the very fact that I am here genuinely questioning the nature of its validity. In this respect I do not share the particular kind of consensus that would accept its absolute validity. Of course those who accept the current mathematical orthodoxy could simply attempt to dismiss my position as irrelevant and carry on regardless. However, this would still confirm the existing orthodox position as representing a majority consensus (that excludes outside criticism). In other words mathematical truth is of a strictly relative nature. It only appears absolute within a linear (1-dimensional) interpretation.

The important point to remember is that just as it is now accepted that non-Euclidean Geometries with important applications exist, at an even deeper level non-Conventional Mathematics also exist with even more significant applications.

In time it will be accepted that what is currently regarded as Mathematics more correctly represents understanding (as qualitatively interpreted in 1-dimensional terms). Admittedly this represents an especially important case (where qualitative can be reduced to quantitative interpretation). However this should not exclude other dimensional interpretations (where qualitative and quantitative aspects remain distinct).

From the important psychological perspective, mathematical activity entails both rational and intuitive aspects in dynamic interaction with each other.

Now properly understood the rational aspect relates to the finite and the intuitive to the infinite domains respectively.

However in the interpretation that characterises Conventional Mathematics the intuitive is reduced to the rational aspect so that it is formally characterised in merely rational terms.

The true holistic aspect of understanding is thereby of an intuitive nature. Though it is commonly reduced to the rational aspect it remains of a qualitatively distinct nature.

We have already mentioned the Pythagorean Theorem. Though the proof of such a theorem is presented as being merely rational, this is not strictly true. Without corresponding intuition, the penny as it were would never drop and one would therefore remain unable to - literally - see what the rational linkages imply.

Indeed this constitutes a major problem with mathematical truth. So often it is presented as a merely abstract body of rational connections. This entails that few - apart from specialised practitioners who can uniquely "see" such truth - are able to qualitatively confirm that the conclusions are correct. In other words the more abstract Mathematics becomes in rational terms, the less amenable it becomes to holistic interpretation of a qualitative kind.

Coming back to my earlier College career I left the Mathematics Course after a year nursing major misgivings. I still wanted to study Mathematics; in fact I was now interested in the discipline at a much deeper level. However I no longer wished to pursue the conventional route (built as I saw it on reduced - and ultimately flawed - assumptions).

Over the next couple of years I became immersed in philosophical issues developing in particular a great interest in the Hegelian system. Indeed it was this association with Hegel that led to a decisive breakthrough in my holistic mathematical thinking.

I had reached the conclusion that mathematical activity entailed both quantitative (rational) and qualitative (intuitive) aspects in dynamic interaction with each other. However the big problem that remained was how to give this qualitative aspect a coherent mathematical basis (going beyond mere philosophical supposition).

Now Hegel in his philosophy used dialectical reason where very thesis gave rise to a corresponding antithesis with the fusion of both leading to a new dynamic synthesis.

I began to see that thesis and antithesis could be represented in mathematical terms as opposite unit polarities (that were positive and negative with respect to each other).

It then occurred to me that this bore a close association with the two roots of 1 (+ 1 and - 1).

So a key insight was the recognition that 2-dimensional understanding (qualitatively) was in fact identical with the structural nature of the square root of 1 (in quantitative terms).

More generally the structure of D (as qualitative dimension) is identical with the corresponding structure of 1/D i.e. the corresponding root of 1 (expressed in quantitative terms).

This ultimately led to the defining of a new circular number system (designed for qualitative rather than quantitative interpretation).

In the conventional system each number is defined with respect to a (default) qualitative dimension of 1.

Thus the natural number system,

1, 2, 3, 4,..... more correctly can be represented as

1^1, 2^1, 3^1, 4^1,......

So in the conventional approach, when we raise a number to a dimension (other than 1) its resulting value is expressed with respect to the (default) 1st dimension. So
2^2 (for example) = 4^1.

This gives a very precise demonstration as to why the conventional system is in fact qualitatively linear (i.e. 1-dimensional).

However an alternative system can be defined where the dimension as qualitative number changes with the base quantity remaining unchanged as 1.

i.e. 1^1, 1^2, 1^3, 1^4,......

Now the secret to interpreting this second qualitative system is the recognition that the structural form of the dimension D is identical with 1/D (as interpreted in quantitative terms).

So again to obtain the structural form of the 2nd dimension, i.e. 1^2, we obtain 1^(1/2) in quantitative terms = - 1.

- 1 can then be given a distinctive holistic mathematical meaning (as the qualitative interpretation of the 2nd dimension).

Likewise just as - 1 is a number lying on the circle of unit radius (in the complex plane) likewise all such roots are circular in quantitative terms.

Therefore the qualitative interpretation of dimensions entails a circular - rather than a linear - use of logic.

Now one might be tempted to question the relevance of all this for the Riemann Hypothesis. However there is in fact a fundamental connection!

As is well known the Riemann Hypothesis proposes that all the non-trivial zeros of the zeta function lie on the line with real part = 1/2.

Now in fact this 1/2 relates to a dimension or power (as understood in quantitative terms).

The immediate implication therefore is that a correct qualitative interpretation of what is involved requires corresponding circular understanding pertaining to the 2nd dimension!

It was around the same time that I made the connection clearly as to the holistic interpretation of the two key digits 1 and 0.

In spiritual mystical terms, ultimate truth is often portrayed in Western terms as a qualitative notion of oneness (in the ultimate realisation of form).

In Eastern traditions hoverer such truth is more often portrayed as a void or nothingness (empty of all form) which represents the holistic meaning of 0.

Just as in standard linear terms 1 - 1 = 0,

likewise in holistic terms 1 - 1 = 0.

In other words the experience of Spirit as emptiness requires the negation of all distinct notions of phenomenal (unitary) form.

This also implies that the generation of intuition in experience comes from the dynamic negation of (posited) form.

This further implies that the indirect means of representing the holistic (intuitive) aspect of mathematical understanding comes from the circular use of logic (that in a purely integral sense is based on the complementarity of opposites).

So I now realised that a comprehensive mathematical approach required the use of two logical systems that are linear (1) and circular (0) with respect to each other. And this latter system - as exemplified here by 2-dimensional understanding - has an indirect rational expression in terms of the dynamic complementarity of opposite poles (+ 1 - 1).

And as my own speciality for many years was destined to involve the holistic mathematical approach this would entail considerable further immersion in the circular use of logic.

Thursday, April 1, 2010


The Riemann Hypothesis has fascinated and intrigued me for many years and over the past two years I am confident that I have resolved it to my own satisfaction.

However I am well aware that I will face great difficulties in communicating this fact for appreciation of my very approach requires acceptance of a new - and largely - unrecognised interpretation of Mathematics.

From my perspective, it is now quite clear why attempts to prove the Riemann Hypothesis (within the standard accepted understanding of mathematics) have met with continued failure, as quite simply these are not adequate for the task in question.

So the first great requirement in coming to appreciation of the true nature of the Riemann Hypothesis, is the realisation - and corresponding acceptance - that what is commonly referred to as Mathematics represents in fact just one important aspect of an overall comprehensive mathematical system.

It might help to appreciate this point with reference to the analogy of a scissors.

Clearly the scissors has two blades of equal importance. In like manner Mathematics - rightly understood - contains two aspects of equal importance. However, quite remarkably only one of these aspects is presently recognised!

In other words what is commonly referred to as Mathematics refers simply to the quantitative interpretation of mathematical symbols in what might therefore be referred to as Conventional Mathematics. We will see as we proceed how such mathematics is related philosophically to a linear use of logic!

However there is a corresponding qualitative interpretation of all mathematical symbols that is of equal importance. And it is this latter interpretation that has engaged my attention now for more than 40 years in what I refer to as Holistic Mathematics. By contrast this aspect of Mathematics is based on an alternative circular use of logic relating to the dynamic interaction of dimensions (or directions) in experience.

Of course for true functioning of a scissors, both blades must operate in conjunction with each other. Likewise for a truly comprehensive functioning mathematical system both the quantitative and qualitative aspects must be closely associated with each other in all mathematical interpretation. This combined third aspect I refer to as Radial Mathematics.

Now unless one is willing to considerably widen perspectives as to what is meant by Mathematics, there is little hope of appreciating the sublime secrets enfolded in the Riemann Hypothesis.

Indeed ultimately the Riemann Hypothesis serves as the fundamental axiom necessary for the consistent interaction of both the quantitative and qualitative aspects of Mathematics.

Needless to say therefore, the Hypothesis (which properly relates to the necessary relationship between both aspects) cannot thereby be proved with respect to just one (i.e. the standard quantitative aspect).

So the Riemann Hypothesis has no solution in conventional mathematical terms. However the real message of what follows is how extraordinarily limited in fact is the present conception of mathematics.

As we know the binary system which potentially can be used in computers to encode all information is based on the quantitative use of the two digits 1 and 0.

Likewise a comprehensive mathematical system is based on a holistic binary system that incorporates both linear (1) and circular (0) logic. And such a dynamic system has the power potentially to scientifically encode all transformation processes.

Just imagine how limited digital technology would be if we confined ourselves to the use of just one quantitative digit (i.e. 1)!

Well! that in fact should indicate how limited is in fact current mathematical understanding that is solely based on just the qualitative digit of the linear logical system (1).

So ultimately, what is even more important than the attempt to resolve the Riemann Hypothesis is the realisation of the true nature of mathematical activity (that combines distinctive quantitative and qualitative interpretations).

And paradoxically with this realisation (i.e. of the true nature of mathematics) comes a remarkably simple solution to the Riemann Hypothesis!