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Peter Collins is from Ireland. He retired recently from lecturing in Economics at the Dublin Institute of Technology. Over the past 50 years he has become increasingly convinced that a truly seismic shift in understanding with respect to Mathematics and its related sciences is now urgently required in our culture. In this context, these present articles convey a brief summary of some of his recent findings with respect to the utterly unexpected nature of the number system.

The Spectrum of Mathematics

Reply to Elliot Benjamin

Peter Collins

Once again I wish to thank Elliot for both reading my "The Problem with Mathematical Proof" and taking the time to graciously reply with "Mathematical Proof and Logic." And I very much admire his willingness to actively engage with important issues relating to Mathematics that I fear many of his fellow practitioners would disregard as simply irrelevant.

Elliot repeats the response with which he concluded his reply to the Riemann article. So I will briefly address his comments here with a view to further clarification of my position.

“Adding a qualitative dimension to mathematics as a philosophical perspective is certainly a legitimate and useful means of enabling people to appreciate the beauty and mystery involved in mathematical thinking. But this is as far as I am able to go with Peter's ideas and article, as for me the logic and inherent value of mathematics is all based upon its logical structure and foundations.”

Certainly, I would agree that the inclusion of a qualitative dimension can be useful in enabling people to appreciate the beauty and mystery involved in mathematical thinking.

However I would go considerably further than this! Basically what I contend is that all mathematical symbols can be interpreted in accordance with two distinct logical systems that are linear and circular respectively.

So a properly balanced mathematical understanding requires that equal emphasis be given to both quantitative and qualitative aspects. However, because of the almost total domination in Mathematics of the merely quantitative aspect of appreciation, we have gradually lost any clear conception of what the qualitative might entail!

However it is precisely because of this great lack of balance with respect to mathematical understanding that the issue needs to be urgently addressed.

It might help to place my remarks in perspective through reference to the notion of the Spectrum of Consciousness which Ken Wilber introduced in his first book.

As with the physical bands of electromagnetic energy, there are likewise several psychological bands with respect to the full range of potential development.

Now Mathematics as we know it represents an extremely specialised form of rational interpretation that characterises the Middle Band. From a formal perspective, it is defined merely in terms of the rational dualistic aspect of understanding.

However the key characteristic of higher bands on the psychological spectrum is that they entail a considerable growth in nondual intuitive modes of awareness. So, cognitive understanding at these levels becomes increasingly refined through interpenetration with intuitive appreciation leading to the more circular use of logic.

It is not that one thereby disregards the value of conventional mathematical appreciation. Rather one realises that this represents but a convenient approximation that indeed works well for one band of the spectrum but starts to break down badly at the higher levels.

And conventional appreciation disintegrates at the higher bands precisely because it has no means within its own set of interpretations of reflecting the qualitative type understanding of mathematical symbols that is consistent with the more contemplative vision acquired.

In other words with qualitative understanding, mathematical symbols are reflected primarily through an intuitive lens (expressed with increasingly refined circular forms of reason). By contrast with quantitative understanding, mathematical symbols are reflected primarily though a linear rational lens (though even here intuition necessarily plays an unrecognised supporting role).

Quite simply, if we are to deny the qualitative as opposed to the quantitative interpretation of mathematical symbols, then equally we must deny any coherent concept of an integral science.

True integration implies holistic qualitative notions of a nondual kind; differentiation by contrast implies analytic quantitative notions that are dualistic in nature. So integral science clearly requires underpinning with mathematical appreciation that is based primarily on qualitative rather than quantitative mathematical notions.

I gave a few important illustrations in my article to illustrate this qualitative approach. So for example, when appropriately appreciated in this new light, the structure of all stages of development on the psychological spectrum can be seen to have a holistic mathematical interpretation.

Also, just as the (quantitative) binary system based on 1 and 0 is invaluable as a means of encoding all information, a corresponding (qualitative) binary system based on both linear (1) and circular (0) understanding exists with the power to potentially encode all transformation processes!

And yes! This is certainly Mathematics though admittedly of a distinctive new variety with an unlimited potential to enrich present knowledge.

However we can never hope to attain to this understanding if we insist on identifying Mathematics, as at present, exclusively with Middle Band appreciation.

It might help to clarify further what is involved here by drawing an analogy with respect to geometrical developments.

As we know for over 2000 years in our Western history, Geometry was identified exclusively with the Euclidean variety. Then early in the 19th century through the investigations especially of Lobachevsky, Gauss, Bolyai and Riemann, strange alternative geometries arose through the questioning of one key assumption that informed Euclidean understanding.

And Einstein as we know in his General Relativity was to make considerable use especially of the curved geometrical notions of Riemann!

It is not that Euclidean Geometry is now seen as in error! Rather through an enriched perspective, it can be understood as an important special case of a more general phenomenon.

At a deeper level the questioning of a key - and ultimately unwarranted - assumption of the exclusively (linear) rational basis of mathematical appreciation, opens the way for a whole new range of non-conventional mathematical interpretations.

And in a manner similar to Riemannian geometry, these find their proper role in the context of curved interpretation of reality where both dual (rational) and nondual (intuitive) notions interpenetrate.

And once again from this perspective, Conventional Mathematics is not seen as in error. Rather it represents but an important special case of a much more comprehensive mathematical understanding.

“This logical structure and foundations may be extended into quite mysterious ranges, such as has been done with infinite arithmetic, complex numbers, and ultimately with The Riemann Hypothesis itself, but I believe we are still in the world of logical mathematics when this extension is being made, perhaps paradoxical logic, but the solution of mathematical problems requires bona-fide mathematics, in my opinion.”

I will seek to be brief here.

It is certainly true as Elliot states that - what I term Conventional Mathematics - has indeed been extended into the areas that he has mentioned.

However in all cases key problems have emerged. For example the extension of linear logic into the area of infinite (or transfinite) arithmetic has led to a whole series of paradoxes that remain unresolved.

As we discussed in our last exchange, this ultimately led to the famed continuum hypothesis which has dramatically demonstrated Godel's claim that important mathematical problems would always exist that could not be resolved within the accepted axiomatic system

Now as Ken Wilber has often stated, paradox arises from attempting to appropriate nondual notions in a dualistic manner. The clear implication therefore is that the infinite notion is essentially of a nondual nature (and cannot be therefore properly understood through dualistic reason).

The area of complex numbers also raises many important qualitative issues that are not addressed in present Mathematics.

Roger Penrose for example repeatedly refers to the magic of complex numbers. By this he means that in many important contexts, they possess totally unexpected holistic properties. This strongly suggests that the very notion of the imaginary as used in Mathematics is directly of a qualitative nature (that is then given an indirect quantitative expression).

For example this would help to explain why complex numbers (with both real and imaginary aspects) are so useful in probing the nature of prime numbers as these inherently combine both quantitative and qualitative aspects in their very nature.

It is also interesting to note that imaginary numbers behave so differently from real numbers when used as dimensional powers.

For example if we raise the number 1 to a real power such as 2, in conventional terms we get just one unambiguous answer.

However if we now raise 1 to i (where i is the imaginary number representing the square root of – 1) we generate potentially an unlimited number of possible correct answers.

Elliot finally mentions the Riemann Hypothesis where despite the incredibly specialised mathematical techniques brought to bear on issues surrounding the problem, Conventional Mathematics is still unable to provide convincing answers to some very obvious anomalies.

Fro example if we square each of the natural numbers and add up the series 1 + 4 + 9 + 16 + … etc in conventional terms the sum is infinite.

However according to the Riemann Zeta Function, in what represents the first of the trivial zeros, the sum of the same series = 0.

So Conventional Mathematics generates yet another paradox (which it is powerless to resolve within its own methods). Indeed this suggested to me some years ago that the very reason why the Riemann Hypothesis appears so impenetrable is directly due to the lack of a qualitative dimension that - among other things - could successfully explain why such a paradox arises.

And this ultimately resulted in obtaining - what I consider - a wonderfully coherent and richly satisfying interpretation of the true nature of the Riemann Hypothesis (which would have been impossible using conventional understanding).

I would dispute the last part of Elliot's statement that (Conventional) Mathematics leads to the use in certain circumstances of paradoxical logic.

The use of the standard techniques does indeed - as with infinite numbers - generate paradoxes. However these paradoxes then remain unresolved within the system precisely because of the lack of paradoxical rational notions. Once again circular or paradoxical logic (based on the complementarity of opposites) serves as the indirect expression of holistic intuitive type awareness. And there is no formal role for intuition recognised within Conventional Mathematics!

Elliot then goes on to defend the status quo with respect to mathematical proof.

But in doing this he does not address the key point that the finite and infinite are qualitatively different concepts. Ultimately this lack of present recognition in mathematics reflects the fact that in formal terms interpretation is conducted solely in a rational manner.

However when one accepts that such interpretation necessarily entails both intuition as well as reason, then we can no longer remain content to reduce the infinite to finite understanding.

As I have stated the intuition is identified with the potential aspect (to which the notion of the infinite properly relates). However any actual example is necessarily of a finite nature. This is recognised well for example in the Myers-Briggs typology for personality types.

Here the S (sense) type is happiest addressing actual concrete situations. The N type (intuitive) by contrast is better recognising the potential inherent in such situations.

So in relating any general proposition that potentially applies to “all” situations to actual circumstances we have the interpenetration of the infinite with finite aspects of meaning (which are qualitatively distinct).

Now of course we can avoid this subtlety by simply collapsing the infinite so that it can then be directly identified with finite understanding. And this corresponds in turn with giving a solely rational interpretation that defines the nature of conventional mathematical proof.

So what Elliot repeatedly refers to as bona-fide logic, correctly represents linear logic (where the role of intuition is formally ignored in interpretation).

This does indeed correspond to an important special case of Mathematics (where qualitative is reduced to quantitative interpretation). But it is only an approximation that breaks down badly when appraised from higher qualitative bands on the psychological spectrum.

Elliot mentions the important role that (Conventional) Mathematics has played in the development of quantum mechanics. I have no wish to question this role (of which Elliot is much better prepared to speak than I am). However his assertions do not directly deal with the nature of quantum reality which in many ways is strongly paradoxical in terms of conventional appreciation.

In particular when we recognise both the wave and particle aspects of sub-atomic processes, an inevitable Uncertainty Principle applies.

In like manner when we realise that mathematical understanding necessarily implies both analytic (rational) and holistic (intuitive) aspects, then a corresponding Uncertainty Principle necessarily applies to all mathematical processes, which includes mathematical proof.

Now once again the very reason why this is not apparent in conventional interpretation is precisely because the holistic (intuitive) aspect is not formally recognised.

Furthermore the Uncertainty Principle in quantum mechanics asserts that the more precisely we try to fix knowledge with respect to one aspect e.g. the position of a particle, the more imprecise and fuzzy becomes one's knowledge of the alternative aspect of momentum.

In like manner because Conventional Mathematics identifies meaning so strongly with the merely quantitative aspect of interpretation, this in effect completely blots out recognition of the alternative qualitative aspect!

So in this context we can use the Uncertainty Principle to predict that the more comprehensive perspective of Mathematics advocated here, is therefore certain to meet with considerable resistance from within the mathematical community.

Finally Elliot refers to the “art of mathematics”. I would agree that there is a valid artistic aspect to Mathematics which can indeed help to enhance appreciation of the subject.

So if we could see for example the famed Egyptian pyramids in their original glory, I am sure that this wonderful artistic expression would indeed enhance mathematical appreciation. Also the fact that an elegant is clearly of much greater aesthetic appeal than an “ugly” proof, generally leads to a clear preference for its utilisation.

However what I have in mind is very definitely of a scientific rather than artistic nature, in using mathematical symbols in a manner that is informed by a distinct form of circular logic (rather than the linear form as used in conventional interpretation).

And crucially, I would see this qualitative aspect of mathematics as the appropriate basis for an integral scientific approach.

Finally may I thank Elliot once again for his engagement and express the wish that our differing perspectives will be helpful in further elaborating the issues raised for readers of the site.