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Peter CollinsPeter 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.



Peter Collins

I thank Elliot for his generous and gracious reply [to my Riemann Hypotesis essay]. Also I congratulate him on his article “Integral Mathematics” where his obvious love for his subject clearly shows in every paragraph. And quite deservedly it has met with a continued great reaction in Frank's Reading Room.

In responding, I hope to make use of his various comments and observations so as to offer a little more perspective on the rationale adopted in the Riemann Hypothesis article.

Elliot mentions two unproven prime number theorems i.e. the Goldbach Conjecture and Twin Prime Hypothesis.

Just to clarify, I did not in fact claim that these can never be proven within the axiomatic system of Conventional Mathematics. Rather I suggested that following on from my conclusion in the article that the Riemann Hypothesis is indeed not provable within this system that these other outstanding prime number problems need to be re-examined in that light.

Elliot helpfully adds to the list the case of whether the set of perfect numbers is infinite and the also whether odd perfect numbers exist. And because every (even) perfect number is associated with a distinct Mersenne prime then the first of these problems reduces to whether the set of Mersenne primes is infinite.

He is perhaps understandably reluctant to let go of the hope that all these problems will eventually be resolved in a satisfactory manner (within the axiomatic framework of Conventional Mathematics).

However I would respond by saying again that we need to view prime numbers in a similar manner to quantum particles in physics. And an uncertainty principle is at work here that has played havoc with understanding based on the traditional classical approach. So likewise - especially in the context of prime numbers - we should be prepared to experience similar fundamental problems with the conventional mathematical perspective.

I share with Elliot his love of perfect numbers and have indeed spent many a joyful hour exploring their fascinating features.

These numbers have a long history and were certainly well known to the ancient Greeks with Euclid providing the quantitative formula that we still use for all (even) perfects numbers as early as 300 BC.

However what is remarkable regarding perfect numbers (and other related mathematical symbols) is the archetypal or holistic meaning that they conveyed in earlier times. Once again the Pythagoreans were to the fore here in studying perfect numbers for their qualitative mystical significance.

Indeed when these number symbols were adopted by early Christian writers they assumed an even greater archetypal relevance.

Thus for example we find that the Christian Bible is extensively sprinkled with number references that are used in a qualitative rather than strict quantitative sense.

So in the very first Chapter of Genesis we are told that God created the World in six days.

Now as a scientific statement of creation (suited to modern tastes) this clearly is unacceptable. However it is really a story told in the form of a myth designed to convey a holistic qualitative - rather than scientific quantitative - significance.

6 is of course the first perfect number (representing the sum of its three proper factors) i.e. 1 + 2 + 3 = 6. As well as being the easiest of all perfect numbers to appreciate, it is perhaps more perfect than other numbers in the sense that 6 also represents the product of these same proper factors i.e. 1 X 2 X 3 = 6.

To appreciate the profound archetypal significance that this perfect number once conveyed we can do no better than refer to a famous quote of St. Augustine from his “City of God” (written in the 5th century).

“Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect.”

If we quickly move forward now and contrast this with a modern story of creation (as in string theory) we are provided with an abstract mathematical account almost utterly devoid of any qualitative resonance with reality. So rather than enabling a greater sense of collective belonging, such abstract science can easily deepen alienation from our world.

So in attempting to divorce (conscious) quantitative from (unconscious) qualitative appreciation, Conventional Mathematics by its very methods leads to a continual loss in the numinous appreciation of its symbols. Not alone is this potentially very damaging from a psycho-spiritual perspective, but even in conventional terms can lead to a considerable loss with respect to the unconscious inspiration that is so necessary to fuel any truly worthwhile mathematical endeavor.

Elliot is no doubt also well acquainted with amicable (friendly) numbers that operate somewhat as poorer first cousins of the perfect numbers. Once again these were studied by the Pythagoreans with clear appreciation of their qualitative relevance.

The most accessible amicable numbers occur in pairs with the first relating to 220 and 284.

Here the sum of the proper factors of each number is equal to the opposite number suggesting a complementary type relationship (as in a good friendship).

So the sum of the proper factors of 220 = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

In turn the sum of the proper factors of 284 = 1 + 2 + 4 + 71 + 142 = 220.

Now we can look at the perfect as a special case of amicable numbers where each number in the pair is equal (and thereby half of the total value of the pair).

This would suggest therefore in qualitative terms a harmonious friendship based on true equality (which indeed would be perfect).

Indeed the qualitative significance of perfect numbers can be made even more explicit.

One could fruitfully describe a perfect personality as the preservation of true harmony as between complementary aspects. The Chinese referred to these complementary aspects as the masculine and feminine principles. Indeed using terminology in keeping with our discussion, the feminine principle is frequently identified with poles such as interrelated and intuitive with the masculine in turn represented by the complementary opposites i.e. autonomous and rational.

Now perfect numbers in their very structure provide the quantitative aspect of this perfect personality.

For example I made reference in the article to the prime number 127 (as a Mersenne prime) which is 1 less than 128 (i.e. 2 raised to the power of 7).

So once again we have here the number 128 as quantitative counterpart of the feminine principle, which is the most interrelated possible for its size (with all the factors = 2).

127 then represents the quantitative counterpart of the masculine principle which is the most independent and autonomous for its size (as a prime with no factors).

And when we multiply both of these numbers and obtain the average i.e. as half of a pair we have (128 X 127)/2 = 8128 which is the 4th perfect number.

Now if we use this model of a perfect number to suggest what should be a “perfect mathematics” then clearly it would represent both quantitative (rational) and qualitative (intuitive) aspects which I refer to in the article as “Radial Mathematics”.

So conventional Mathematics as a formal discipline is based solely on the masculine principle (which I imagine Elliot would accept as unhealthy).

Indeed we can profitably extend this example further to show what is precisely at fault with conventional mathematical interpretation.

Once again in our example the number representing the “feminine” component is 128. As I stated early in the article, though a qualitative dimensional transformation in units is involved when we multiply two numbers together, that this is simply ignored in conventional mathematical terms.

Therefore though 128 = 2 X 2 X 2 X2 X 2 X 2 X 2, represents a significant degree of qualitative transformation, the result 128 is expressed in a merely (reduced) quantitative manner.

Thus, in a very precise way this absence of the feminine principle can be seen to be intimately rooted in the way multiplication is treated (or more correctly mistreated) in conventional mathematical terms.

I think it is very striking that two of the key practitioners (and authorities) speak about the crucial gap, preventing resolution of the Riemann Hypothesis, in precisely the same manner as a failure to properly understand the relationship as between addition and multiplication.

So we can perhaps see now how this ultimately points to the lack of the feminine (intuitive) dimension in formal mathematical understanding.

This is why I was confident that interpretation of the Riemann Hypothesis from the perspective of Holistic Mathematics would quickly open up important doors that would always remain firmly shut when approached using the conventional route.

And the amazing discovery that then unfolded was:

(i) that zeta values in the Riemann Function require both quantitative and qualitative interpretations, with Right Side values in the complex plane (s > 1) corresponding to conventional linear quantitative and Left Hand side values (s < 0) corresponding to an alternative qualitative interpretation (based on a distinctive circular logical approach)

(ii) that both Right-Hand (quantitative) and Left-Hand (qualitative) interpretations are indissolubly linked. (Interestingly though I did not have the space to deal with this in the article, the remaining values for s in the critical region between 0 and 1 can be shown to combine both quantitative and qualitative aspects. And, as .5 lies in this region, the Riemann Hypothesis thereby requires an explanation that simultaneously combines quantitative and qualitative interpretation!)

I have been persistently making this case for many years now that the framework of mathematics needs to be greatly enlarged to include both quantitative and qualitative aspects both as separate specialized areas serving distinct purposes (Conventional and Holistic Mathematics respectively) and finally in simultaneous conjunction with each other (Radial Mathematics).

Though the Riemann Hypothesis is not required to demonstrate this need, my investigation of it provided a truly remarkable confirmation of long-held beliefs. Indeed in the qualitative sense that I have given to perfect numbers it can be validly viewed as the Perfect Hypothesis! I believe therefore that when the true nature of the Hypothesis eventually dawns on the mathematical community that it can act as one extremely important catalyst for a truly revolutionary change in accepted understanding.

Finally, the qualitative approach to perfect numbers can throw interesting light on Elliot's desire to know if any odd perfect numbers exist.

Though I would not argue the point in absolute terms, I would say that true qualitative appreciation of the nature of perfect numbers considerably strengthens the probability that odd cases do not in fact exist.

This is due to the fact that the kind of complementarity that typifies the known (even) cases cannot - by definition - be present where the number is odd. Thus if by some chance odd numbered cases existed they would be of a significantly different character from their even counterparts where complementary aspects (already identified for psychological perfection) could not be properly matched.

I have dealt with this topic of perfect numbers at some length to demonstrate the value of complementary (qualitative) appreciation of mathematic symbols.

Elliot in his response argues that Conventional Mathematics has in fact managed to successfully deal with several important paradoxical issues thereby incorporating them within an ever expanding framework of understanding.

He makes special reference here to the tackling of limits in calculus and to the work of Cantor on infinite sets and also on developments with respect to imaginary and complex numbers.

Though I would acknowledge the great work that pioneers such as Cantor contributed to their chosen field, I would interpret this somewhat differently, seeing it as ultimately pointing to the severe limitations of what is mathematically possible within the current framework of accepted understanding.

The case of Cantor is especially fascinating. My interest in his work stems – perhaps surprisingly - from reading an account by Etienne Gilson on “The Philosophy of St. Thomas Aquinas”.

St. Thomas adopted a hierarchical system of angels as an essential component of his overall approach. Though this system seems extremely quaint to the modern mind, through appropriate reinterpretation it can be accurately translated into a very interesting account of the higher stages of human spiritual development. So from this new perspective, angels are seen as projections of the unconscious potential of personality with the various grades of angels representing the stages through which spiritual development can take place.

You might readily ask what has this to do with Cantor and his infinite sets! Well, a great deal in fact as Cantor was deeply influenced by medieval theology and essentially was attempting to deal with the same problems that occupied theologians with respect to the infinite, in the mathematical context of numbers! So, for example the - often ridiculed - theological question of how many angels can dance on a pinhead, is no different in principle from the mathematical question of how many numbers exist within a small interval on the number line.

Just consider for a moment this quote from Chapter V111 on “The Angels”

“The angels are creatures that can be proved and, in certain cases observed; their suppression would render the universe, taken as a whole unintelligible;”

Now consider this amended quote:

“The transcendental numbers can be proved and, in certain cases observed (e.g. pi and e); their suppression would render the set of numbers taken as a whole unintelligible;

You perhaps may now appreciate the close connection between both areas!

Thus numbers possess a qualitative (unconscious) aspect with various degrees of higher infinite potential. So just as psychological development starts with primitive instinctive behavior, the number system starts with the primes. At the middle stages specialized rational development takes over with its correspondent in rational numbers. Then when development moves on through the higher spiritual stages we have the equivalent transition in number terms through (algebraic) irrational, transcendental and ultimately transfinite numbers.

However just as I would view Angelology as an ultimately unsatisfactory means of attempting to classify the higher stages of human infinite potential, I would see Cantor's theory of infinite sets somewhat in the same light.

In other words, Cantor was attempting to grapple with issues, both philosophical and mathematical, considerably transcending the confines of rational linear logic (while attempting to adhere to that restricted framework).

And this of course is the source of all the paradoxes that thereby materialized.

When paradox arises it is a clear sign that what can really only be grasped through intuitive and ultimately nondual awareness is being conveyed in a reduced rational (linear) manner.

However, such paradox cannot be itself resolved at the level of rational linear appreciation.

This is precisely why alternative circular appreciation which is based on the dynamic complementarity of opposite poles - that create paradox at the linear level - is required.

So rather than displaying once again the triumph of the conventional mathematical approach, Cantor's work, perhaps more than any other single factor, has in fact greatly undermined any absolute basis for this system.

Elliot correctly mentions the opposition that Cantor met from the mathematical community at the time which affected his mental health. However there was another factor present which greatly contributed to the problem, which was his failure to resolve the Continuum Hypothesis.

As Elliot points out in relation to the real number system, Cantor distinguished two types of infinities. Firstly there was the set of all the rational numbers. Then there was the set of all real numbers in this system (including the irrational). Again as Elliot ably describes, Cantor proved that the infinity of this second set was of a greater magnitude than the first.

Now he dearly wished to prove that there was no other type of infinite set (lurking around as an unrecognized guest) between the rationals and the real.

However he failed in this quest.

When Hilbert presented his famous list of 23 important unsolved mathematical problems in 1900, he did so confident in the belief that all could be eventually resolved within the accepted axiomatic system.

However this dream was to be shattered in the 1930's when Godel proved to the mathematical community that there would always be valid problems that could never be solved within the accepted – and even any enlarged – axiomatic framework of mathematics. He then demonstrated this conclusion with respect to the prime numbers. Godel also apparently held the belief that the existing axiomatic system of Mathematics was insufficient to solve the Riemann Hypothesis!

The first of the important unsolved mathematical problems on Hilbert's list was the Continuum Hypotheses (with which Cantor had struggled in vain).

Then following separate results provided by Godel earlier and Paul Cohen in 1964, it was concluded that the Hypothesis was undecidable i.e. could neither be proved nor disproved within the accepted axiomatic system of mathematics.

So at last here was clear evidence with respect to a fundamental mathematical problem that its accepted axioms were quite inadequate for the task of seeking a proof.

And if we look at this problem now from a qualitative philosophical perspective the reason should appear quite obvious.

The Continuum Hypothesis deals with numbers, discrete and continuous, that actually correspond to two distinct logical systems. So one cannot solve a fundamental problem relating to the relationship between two distinct logical systems with reference to just one! However this elephant in the room is continually ignored by conventional mathematicians through insisting on a merely reduced linear interpretation of number quantities.

In this respect the Riemann is very similar to the Continuum Hypothesis in that once again it requires, for appropriate interpretation, the same two logical systems (i.e. linear and circular).

You know there is a delicious irony in the fact that the real key to resolving the Riemann Hypothesis comes from interpretation of the simplest of the so-called trivial zeros (s = - 2) which I believe can be explained in a manner which everyone can appreciate.

If one takes the sum of squares of the natural numbers then clearly the series in conventional linear interpretation diverges i.e. is infinite. Thus,

1 (squared) + 2 (squared) + 3 (squared) + 4 (squared) +…… = infinity.

However according to the Riemann Zeta Function (where this series corresponds to the value of the Function when s = - 2),

1 (squared) + 2 (squared) + 3 (squared) + 4 (squared) +.…… = 0.

So we have here a clear example of paradox where two diametrically opposite results seemingly can exist for the same series.

Thus the obvious question then arises as to how to reconcile the two results!

Now bearing in mind my earlier point that such paradox always points to a limitation in the linear manner of understanding, clearly this suggests than an alternative type of interpretation is required.

And as I explain - in some detail in the article – this alternative explanation is provided by the circular logical system (corresponding directly to qualitative rather than quantitative meaning).

So the reason why the series can have two apparently contradictory results is that this same series can be interpreted according to two distinct logical systems!

And the startling fact that then emerges is that in the Riemann Zeta function it is the qualitative – rather than quantitative – interpretation that is required to interpret this numerical result for the series.

When one properly appreciates what is involved here then it is no longer possible to associate numerical data - even in the strictest mathematical context - with a merely conventional linear interpretation.

And this is precisely why the implications of the Riemann Hypothesis are so revolutionary. Whereas the Continuum Hypothesis exposes a fundamental limitation regarding the structure of Conventional Mathematics in general terms, the Riemann Hypothesis - or more accurately Riemann Zeta Function - does this in a very specific mathematical context where an entire region of numerical results (for s < 0) cannot be satisfactorily interpreted in a quantitative manner.

I thoroughly agree with Elliot that Andrew Wiles solution to Fermat's Last Theorem was a truly outstanding achievement leading to important new (conventional) mathematical developments in the process.

Up to the time of its proof in 1995, it was perhaps the best known of the unresolved mathematical problems, though this owed a great deal to the simplicity of its formulation (making it easy to communicate to the ordinary layperson).

By contrast the Riemann Hypothesis is quite inaccessible requiring more than a passing acquaintance with Mathematics for proper appreciation of its very nature.

However it is generally considered to be much more fundamental in importance than Fermat's Last Theorem.

In this respect once again it resembles the Continuum Hypothesis and I would suggest that it is with such problems that the possibility – even probability – of no proof being attainable (within the conventional axiomatic system) is most likely to arise.

Actually I was not surprised that a satisfactory conventional solution was found for Fermat's Last Theorem as it was becoming generally accepted that mathematicians were gradually closing in on a proof which was likely to emerge sooner (rather than later) along existing lines of research.

However the Riemann Hypothesis is quite different in this respect. Apart from its more fundamental nature, many important practitioners admit that a crucial key insight is necessary to make further progress!

However Elliot is right insofar as I would attach an important qualitative dimension to Fermat's Last Theorem. This in turn could suggest a shorter (and ultimately more efficient) alternative route for solving the problem in quantitative terms. Though I would not accept that Fermat had already come up with an ingenious proof, I think it perhaps likely that he did indeed have a key insight into its nature which - if revealed - could perhaps facilitate an alternative approach to a solution.

Wiles' achievement also raises interesting questions regarding the nature of mathematical proof.

Elliot will know that after his first announcement of a proof in 1993, that a crucial error in his reasoning was subsequently discovered. Indeed it seemed doubtful for a time as to whether this problem could ever be resolved. However happily for Wiles and the mathematical community eventually he was able with the help of another mathematician to fix the problem.

However this event really highlights the true experiential nature of mathematical proof which represents but a special form of social consensus. Only a handful of specialised mathematicians have the necessary skills and patience to carefully check through a lengthy complicated proof and ensure that no fatal errors in reasoning have occurred. So when a proof is announced it represents a provisional agreement among a small group of elite mathematicians that in their considered opinion the task has been successfully accomplished. As for the rest of us our assent to the proof is based simply on faith that those checking the solution have carried out the job correctly.

Elliot mentions that Wiles' proof is nearly 200 pages long. In fact this is very short compared to the proof relating to the classification of all finite simple groups which is about 15,000 pages in length. So it is quite unlikely that any one individual could ever check such a proof in its entirety. Indeed it is freely admitted here that many problems still have not been fully resolved, though they are not considered of sufficient importance to threaten the overall project. And then there is the added new dimension, as with the first proof of the Four Colour Theorem in 1976, where much of the work was carried out by computer.

This is in no way meant to devalue the on-going need for mathematical “proof” even when understood in a dynamic experiential context. Rather it is to suggest that the very nature of such proof continually evolves. Likewise our understanding is in continual transition. So we should always be open to the possibility of unexpected surprises and even dramatic new developments.

In conclusion, I would like to stress that my real concern in writing relates to many disturbing aspects in our world due in many respects to what I would see as an unquestioning adherence to a limited scientific framework.

One aspect of this is our failure to adequately respond to a considerable global threat to the environment (mentioned briefly in the article).

Another is provided by the inadequate response to the present international financial crisis. In many countries this is exposing a glaring mismatch as between the economic measures actually being adopted and true ethical responsibility. Quite rightly this is evoking outrage among affected citizens around the world.

Just one final example is provided by the increasingly specialised and indeed elitist nature of many branches of both science and mathematics. Here, recognised practitioners tend to comprise exclusive clubs that can become largely removed from the understanding and concerns of the wider population.

To truly move to a more integral perspective will require radical changes in what we understand as science (and most of all what we understand by mathematics). Though I am indeed aware of a considerable reluctance out there to yet address this issue, a great urgency exists. So my contributions over the years have been prompted by a keen desire to awaken such recognition.

The present explosion in information technology in society will soon call out for a corresponding need for radical transformation. If we are not willing to question long cherished patterns of understanding, many will become swamped by great tsunamis of change in society that are inevitable.

Again I thank Elliot for his speedy response. I have greatly enjoyed replying to his various comments and hope that this further contribution helps to clarify at least some of the issues raised.

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