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Publication dates of essays (month/year) can be found under "Essays". Elliot Benjamin is a philosopher, mathematician, musician, counselor, writer, with Ph.Ds in mathematics and psychology and the author of over 150 published articles in the fields of humanistic and transpersonal psychology, pure mathematics, mathematics education, spirituality & the awareness of cult dangers, art & mental disturbance, and progressive politics. He has also written a number of self-published books, such as: The Creative Artist, Mental Disturbance, and Mental Health. See also: www.benjamin-philosopher.com.

COMMENTARY ON PETER COLLINS' RIEMANN HYPOTHESIS ARTICLE

Elliot Benjamin

For me these philosophical and spiritual overtones are quite distinct from bona-fide mathematics.

As a mathematician and philosopher, I thoroughly enjoyed reading Peter Collins' recent Integral World article: A Deeper Significance: Resolving The Riemann Hypothesis. Peter is obviously a capable and experienced mathematician with an ingenious capacity of relating deep mathematical ideas to philosophical and spiritual contexts. His radial mathematics theories that include both quantitative and qualitative dimensions to mathematics, utilizing both logical and circular thinking frameworks, is certainly fascinating reading, at least for anyone like me who has serious interests in both mathematics and philosophy. However, the mathematician in me does take issue with some of Peter's recommendations and claims regarding the solution of the Riemann Hypothesis, as well as his suggestion that other famous historical mathematical problems involving prime numbers are unsolvable in the logical mathematical framework in which mathematics has always been done.

Peter gives two examples of this in footnote 14 of his article: namely the Goldbach Conjecture, which states that every even number, other than 2, can be written as the sum of exactly two primes (such as 10 = 7 + 3, 46 = 41 + 5, etc.); and the Twin Prime Conjecture, which states that there are infinitely many consecutive prime number pairs (such as 5 and 7, 11 and 13, 41 and 43, etc.).

I would guess that Peter would include two of my own favorite unsolved mathematical problems involving primes to this list, which are problems related to perfect numbers, as I have described in my Integral World article entitled Integral Mathematics: A Four Quadrant Approach. For a brief description of these problems, a Perfect Number is a number such that the sum of its proper divisors add up to the number (such as 6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14). There have been something like 43 perfect numbers that have been so far discovered, and all of them are even. The two unsolved problems I am referring to are 1) does there exist an odd perfect number and 2) are there infinitely many perfect numbers? To get a bit of appreciation for these problems, I will remark that perfect numbers become incomprehensibly large, as the fifth perfect number is 33,550,336 and the sixth perfect number is already in the billions. We also know a neat formula involving prime numbers that works for all even perfect numbers. But as Peter so eloquently describes in his article, prime numbers are a great mathematical mystery, and consequently perfect numbers are a corresponding mathematical mystery.

But getting back to the gist of my commentary on Peter's article, I do not agree with him that solutions to these problems are not likely to ever be attained using logical mathematics, and I must confess that although I find Peter's article to be fun and fascinating reading, I see his radial mathematics ideas as little more than philosophical and mystical correspondences to bona-fide mathematics.

I think that Peter's emphasis upon the separation of discrete and continuous perspectives in his example of the square root of 2 is certainly legitimate and useful. In fact, I would expand this to the whole subject of limits in Calculus, as what it means to say that the derivative is the slope of the tangent line, which is the limit of the successive approximations of the slopes of secant lines, or equivalently that instantaneous velocity is the limit of successively approaching average velocities, is of this same challenging intellectual and mathematical nature. It is this kind of paradoxical logic that is at the heart of what can be thought of as infinite arithmetic, formulated by Cantor over a hundred years ago, which enables us to make statements like there are as many even counting numbers as counting numbers (i.e. including the odd counting numbers does not increase the magnitude in regard to the level of infinity), based upon a one-to-one correspondence between every even counting number and every counting number.

However, it can be proven that there is a greater level of infinity for all real numbers, i.e. including the irrational numbers such as the square root of 2 (which Peter has nicely described in his article) then for all the rational numbers (i.e. bona-fide fractions), which are included in the real numbers. These ideas certainly challenged the standard logical mathematical thinking of the time, and Cantor himself suffered through much psychological anguish over this.

The acceptance of imaginary and complex numbers is another case in point that illustrates this kind of expanded mathematical thinking, and once again it took a number of years before this was accepted as part of mathematics. The relationship of prime numbers to complex numbers and ultimately to the Riemann Hypothesis certainly is an amazing and fascinating relationship, and Peter describes this relationship with all of his philosophical and spiritual overtones in a wonderfully refreshing and captivating way. But once again, I am afraid that for me these philosophical and spiritual overtones are quite distinct from bona-fide mathematics. I suppose I am of the old school of logical quantitative mathematics, and perhaps this has more than a little to do with the fact that I am a pure mathematician who loves number theory for its own intrinsic logical, intellectually stimulating, and playful thinking possibilities.

An example of what I believe is a magnificent demonstration of the possibilities of using tremendously expansive mathematical ideas to solve mathematical problems that have been unsolved for hundreds of years, involves Andrew Wiles' solution of Fermat's Last Theorem, which he finally solved in the 1990s after working on the problem for many years.

For those of you who are not familiar with the problem, Fermat's Last Theorem says that there are no solutions in integers (i.e. zero and positive or negative counting numbers), other than x = y = z = 0, of any equation to higher integer powers that is analogous to the Pythagorean equation x2 + y2 = z2. The lore is that Fermat in the 17th century had a proof but did not have room in the margin to describe it, and for the next 300 years mathematician came and went with trying unsuccessfully to find a proof. What it took for Andrew Wiles to come up with his proof, which is something close to 200 pages long, involved deep extensions into various mathematical fields that are very related to the mysteries of prime and complex numbers that Peter Collins describes with such wonderful relish in his article. But my main point is that this proof was finally accomplished, and I have little doubt that Peter would have included Fermat's Last Theorem in his list of deep unproved mathematical results in number theory that quite likely cannot be solved in a logical mathematical framework.

Thus in conclusion I must say that although I find Peter Collins' article to be wonderful mathematical and philosophical reading that portrays the eloquence, beauty, and mystery of some of the most mysterious concepts in mathematics, I find his proposal that results in a solution of the Riemann Hypothesis to be a poetic non-mathematical philosophical portrayal that has little relationship to what I consider to be bona-fide mathematics.

His portrayal of the twofold nature of prime numbers, regarding their distinct quantities as building blocks of all counting numbers and our ability to make accurate statements about their group identity, so to speak, without knowing anything about their individual occurrence as numbers become increasingly large, I do not take issue with. In fact I think this portrayal is accurate and an excellent depiction of the paradoxical nature of much of higher mathematics, inclusive of the infinite arithmetic that I have mentioned above.

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. 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...