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Elliot BenjaminElliot 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.

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MATHEMATICAL PROOF
AND LOGIC

A Response to Peter Collins

Elliot Benjamin

What Peter advocates for as an extension of mathematics, I see as a perspective other than mathematics

About two years ago I responded to Peter Collins' original Integral World article advocating the use of “qualitative” mathematics to resolve The Riemann Hypotheses [1]. I concluded my response article as follows:

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. [1]

In Peter's present Integral World article on the same topic [2], "The problem with mathematical proof", he extends his analysis of “conventional” mathematics to the extent of finding problems with mathematical proof in general. To support his ideas, Peter makes much use of the Pythagorean triangle, both in the Pythagorean Theorem and in the proof that the square root of 2 is irrational. But the conclusions that Peter draws from these examples once again do not work for me. I do not have a problem with for example representing any quantity by the variable “x” when there may be infinite quantities to choose from. If you want to prove the Pythagorean Theorem and you take generic sides of a right triangle to be x, y, and z, this is representative of any possible real numbers you can choose, and of course there are infinitely many to choose from. This is bona-fide logic in mathematical proof and to me it is the essence of pure and sound mathematical thinking.

In regard to proving that the square root of 2 is irrational, yes it is true that rational numbers (i.e. fractions) and irrational numbers (i.e. non-repeating and non-terminating decimals) are quite different entities. But the kind of qualitative context that Peter is talking about is to me something “other than” mathematics. What Peter eloquently argues for is in my opinion a combination of philosophy, physics, and mysticism, and I by no means discount his interesting examples and arguments. Where I part company with Peter is in the inclusion of his ideas into his advocacy for some kind of “qualitative”, “holistic”, or “radial” mathematics.

In Peter's previous response to my comment on his original article, "Reply to Elliot Benjamin on my Riemann Article" [2], he discussed the dual nature of subatomic particles as waves, representing the foundations of quantum physics in two very different contexts—discrete and continuous, which he argued demonstrated the need for this kind of qualitative perspective on mathematics. But it happens to be the case that the way these two perspectives were mathematically resolved as a mathematical foundation of quantum physics was done in an ultra-logical “conventional” mathematics tour-de-force by the great mathematician John von Neumann in the 1950s with his publication of Mathematical Foundations of Quantum Mechanics [3]. Essentailly von Neumann demonstrated with complete mathematical rigor how particular infinite sums (representing the discrete particle functions) and infinite integrals (representing the wave functions) can be put in one-to-one correspondence as they both satisfy all the axioms and properties of Hilbert space [3]. Consequently I do not agree that the different contexts of being discrete and continuous means that a problem cannot be tackled through what Peter refers to as “conventional” mathematical thinking.

It is certainly the case, as Peter points out, that there are problems that cannot be solved based upon the axioms of the system, which is the important contribution that Godel has made, and there are also paradoxes in mathematical systems, as conveyed to us in particular by Bertrand Russell, especially when the words “all,” “some,” and “none,” are used, such as the phrase “the set of all sets.” However, within the given axioms of a mathematical system, I see mathematics as the branch of science that utilizes bona-fide logical thinking to prove results in the context that it is able to work in. What Peter advocates for as an extension of mathematics, I see as a perspective other than mathematics, which has relationships to what can be thought of as the “art of mathematics.” [4]

Thus to conclude, I do not have a problem with anything that Peter describes in his articles pertaining to viewing mathematics in an extended context that involves philosophy, physics, and mysticism. But I do object to calling this extended context “mathematics.”

REFERENCES

1) See Collins, P. (2009). A deeper significance: Resolving The Riemann Hypothesis. Retrieved June 11, 2011, from www.integralworld.net and Benjamin, E. (2009). Commentary on Peter Collins' Riemann Hypothesis article. Retrieved June 11, 2011, from www.integralworld.net

2) See Collins, P. (2011). The problem with mathematical proof. Retrieved June 11, 2011, from www.integralworld.net

3) See von Neumann, J. (1955). Mathematical foundations of Quantum Mechanics. Princeton University Press: Princeton, NJ.

4) See King, J. P. (1992). The art of mathematics. Fawcett Columbine: New York.




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