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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 230 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:


Mathematics or Philosophy or Science?

Collins/Benjamin Debate Continued

Elliot Benjamin

In my previous Integral World article The Art Form of Mathematics [1], my intent was to offer my perspective on the legitimacy and aesthetic value of pure mathematics and number theory in its own right, in response to the recent series of Integral World articles by Peter Collins, under the generic title Dynamic Nature of the Number System [2]. In Peter's reply to my "Art Form of Mathematics" response article [3], in which I very much appreciated his professional and respectful tone in spite of our enormous differences of perspective on this matter, it has struck me that what I believe is going on here is primarily a matter of where does one draw the line between mathematics and philosophy and science?

Everything that Peter talks about in regard to the “qualitative” realm of mathematics, the “psycho-spiritual” realm of mathematics, intuition and geometric visualization in mathematics, etc. is not anything I would have a problem with if it were under the heading of philosophy or exploratory science. However, what Peter wants to do is to transform the whole realm of mathematics to include all these realms and to refer to it as a “higher” or “wider” realm of mathematics. For Peter, mathematics needs to “transcend” the use of logic and rationality and to always (at least “in principle”) be associated with real world phenomena. And he is certainly entitled to his perspective.

I don't want to go on and on in this debate with Peter, as we would just be going around in circles saying the same things over and over. Frank Visser has already indulged us with much patience and kindness, and I think we have essentially both said our pieces.

But I do want to point out a very simple example of what I perceive as Peter's attempting to force mathematics into a “straight jacket one size fits all” kind of thinking.

Lets go back to 2 x 3 = 6. Yes it is correct, as Peter has stated, that I agreed that it makes perfect sense to think of this as units of length, width, and area of a rectangle in a geometric application. But what I said in my Art Form of Mathematics article is “I strongly disagree with Peter that this invalidates us talking about 6 = 2 x 3 without always referring to this particular geometric application” (cf. [1]). The area of rectangle application is an important and significant way to visualize multiplication—I have no disagreement when Peter says this. But in my opinion, counting how many candy bars there are in two packages each consisting of three candy bars is also a legitimate way of thinking of 6 = 2 x 3, as simply 3 + 3 = 6, without any need of this dimension interpretation of multiplication that Peter appears to me to be obsessed with. The difference is between seeing the geometric interpretation of area as an “important application” of the concept of number vs. seeing the whole concept of number as invalid without this geometric interpretation.

If I may offer one more mathematical reflection on Peter's latest response, let's take another look at the concept of infinity. Peter claims that when we talk about infinity we are entering the “qualitative” realm and we need to shift gears in our whole nature of mathematical thinking. In actuality this is not very different from what historically took place in the realm of mathematics that Peter refers to as “quantitative.” It is certainly true, as Peter says, that infinity is not a number; rather it is an indication that numbers increase without bound, and it was a revolution in mathematics when it became widely used in the development of Calculus [4]. There are some very bizarre paradoxes that one can prove by using mathematical definitions of infinite sets utilizing the idea of one-to-one correspondence, such as the result that there are as many even counting numbers as “all” counting numbers (both even and odd). The idea is that for every counting number we can associate an even counting number to it (associate 1 to 2, 2 to 4, 3 to 6, and in general n to 2n); therefore there cannot be more counting numbers than even counting numbers. Yes—the concept of infinity is strange. But from a “quantitative” mathematics perspective, the proofs I gave in my last article that there is no largest natural number and no largest prime number immediately result in there being “infinitely many” natural numbers and prime numbers, using my hands-on definition of “infinitely many” as “more than any natural number.”

Thus what is going on here in these widely different perspectives on the nature of pure mathematics and number theory between myself and Peter Collins is essentially the whole question of are we talking about mathematics, philosophy, or science? If we are talking about philosophy or science then I have no quarrel about what Peter is advocating for, at least in regard to philosophical and scientific explorations. Indeed much of what he writes about in regard to psychological and spiritual development, the applied mathematics and number theory results related to quantum physics, inclusive of the Riemann hypothesis, and various other related matters, I find interesting, intriguing, and valuable.

However, where Peter and I part company is on the validity of abstract mathematical thinking without the necessity to find a philosophical or scientific basis in the “real” world to justify this thinking. Once again I am arguing that pure mathematics, and in particular abstract number theory, is an art form, and there may or may not be any applications to much of this thinking in the physical world—even “in principle.” I find it rather humorous that Peter thinks that “in principle” there are physical and psychological applications of my abstract algebraic number theory results that I referred to in my last article [5], and for all I know he may be right. But my main point is that whether or not there are some kinds of philosophical and scientific applications lurking in the depths of my abstract mathematical thinking is to me superfluous to the legitimacy of my mathematical thinking.

I understand that to Peter I am “conditioned” by my upbringing as a mathematician, and that I am not able to view mathematics in the wider scope that he has dedicated his life to advocating for, which includes the non-rational “qualitative” component. But I think at this point we need to “agree to disagree” and not continue to say the same things over and over again. I think Peter is completely immersed in a life mission to transform mathematics beyond the quantitative logical realm in which it has been defined as a subject of study for the past few thousand years. My suggestion to Peter is that he continue with his ideas to extend mathematics, but to accept that there is a realm of pure mathematics, a realm of applied mathematics, and a realm of mathematics philosophy, without invalidating the realm of pure mathematics as an art form—for the joy of mathematical thinking in its own right. In the realm of applied mathematics, I see Peter discussing some cutting edge scientific explorations involving quantum physics applications to various aspects of number theory. In the realm of mathematics philosophy, I see Peter discussing some interesting philosophical speculations about applications of number theory in the realms of psychology and spirituality. And this is my perspective on Peter Collins' Dynamic Nature of the Number System.


1) See Elliot Benjamin (2013), The Art Form of Mathematics: A Response to Peter Collins (

2) See Peter Collins (2013), Dynamic Nature of the Number System (I): Mathematics at a Crossroads; (2): Holistic Role of Zeta Zeros; Dynamic Nature of the Number System, (3): The Bigger Picture (

3) See Peter Collins (2013), Recognizing the Shadow: A Reply to Elliot Benjamin (www.integralworld.net0

4) See Carl B. Boyer (1968), A History of Mathematics. John Wiley & Sons: New York.

5) See Elliot Benjamin (2001), On Imaginary Quadratic Number Fields with 2-Class Group of Rank 4 and Infinite 2-Class Field Tower. Pacific Journal of Mathematics, Vol. 201, No. 2, pp. 257-266.

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