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INTEGRAL WORLD: EXPLORING THEORIES OF EVERYTHING
An independent forum for a critical discussion of the integral philosophy of Ken Wilber
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.
The Art Form of Mathematics
A Response to Peter Collins
Consequently I take personal offense from what I perceive as Peter Collins's condescending depiction of pure mathematics and number theory...
This essay is in response to Peter Collins' latest series of essays about what he perceives as the urgent need to extend the realm of mathematical thinking .
Peter begins by explaining to us that when we multiply numbers, such as 6 = 2 x 3, we are missing a crucial aspect of the problem by not associating the problem with its geometric application of area; i.e. as a rectangle with length 2 units and width 3 units with an area of 6 = 2 x 3 “square” units (cf. ). But of course we all learned in grade school that such is the case, and we were required to include the dimensional notation of “square” units when talking about area. This is certainly a valid and simple example of applied mathematics; i.e. we are applying numerical units to elementary geometry. However, I strongly disagree with Peter that this invalidates us talking about 6 = 2 x 3 without always referring to this particular geometric application. From my understanding of Peter's philosophy of mathematics, he is advocating nothing less than the elimination of the whole field of pure mathematics and number theory as legitimate fields of study in their own rights (cf. ).
What I believe Peter is missing here is the aspect of mathematics as an art form . Yes he makes a reference to “the music of the Riemann Hypothesis” (cf. ), but for him this music is severely lacking unless it has real world applications. I beg to differ. Let me give a few examples. Please bear with me, as my examples will not be difficult to follow mathematically—I promise!
First off, how do mathematicians “prove” that there are infinitely many natural numbers; i.e. ordinary counting numbers 1, 2, 3, ....Well before giving the simple proof, we need to say what we mean by “infinitely many,” as “infinity” is a major stumbling block for Peter (cf. ). But I must confess that I do not see a problem with defining “infinitely many” as “more than any natural number.” There is no “infinity” needed in the concept of “more than,” at least not in the simple way that I am using the term, and I will therefore make the assumption that readers are with me thus far.
O.K. lets get back to the proof. Assume there is a largest natural number and call it n. Well, would you not agree that n + 1 is a larger natural number than n? So if I assume that n is the largest natural number then I have just produced a natural number that is larger. This is what I believe we can all agree is a blatant contradiction—it is impossible to be largest and then have something larger. The only way out of our predicament is to come to the conclusion that our initial assumption must have been mistaken; i.e. there is no largest natural number—end of proof—there are infinitely many natural numbers.
For the life of me I cannot imagine that anyone—other than Peter—would have a problem with this proof. Well perhaps Peter might grant us this one, since I made no use of multiplication, which according to Peter necessitates talking about area and dimension, and “perhaps” we could convince him that in this particular case our definition of “infinitely many” as “more than any natural number” is not too outrageous. However, it is quite likely that Peter would still have a problem because of the fact that “any” could refer to “any” of “infinitely many” natural numbers, but lets give him the benefit of the doubt and proceed to a more interesting number theory proof.
How many prime numbers are there? We'll define prime number essentially as Peter defined it; a prime number is a natural number, other than 1, such that its only factors (i.e. divisors that leave no remainder) are the number itself and 1. The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Let's prove there are infinitely many prime numbers, keeping in mind that we are now in territory that Peter would have some definite problems with. For not only are we talking about “infinitely many” again, but now we are indirectly talking about multiplication, since a factor of a number means precisely that we are multiplying; for example 2 is a factor of 6 because 2 x 3 = 6. Oh well, I'll proceed anyway, as this proof was given a few thousand years ago by Euclid  and has stood the test of time as being a particularly “aesthetic” mathematical proof that I believe illustrates the title of this essay: “the art form of mathematics.”
The proof is based upon the well-known fact that any number can be factored into prime numbers. For example, 120 = 10 x 12 = 2 x 5 x 2 x 6 = 2 x 5 x 2 x 2 x 3 or using the language of exponents, 120 = 23 x 3 x 5. Now the factorization into prime numbers happens to be unique (which is called The Fundamental Theorem of Arithmetic); i.e. no matter how we started we would end up with the same factorization into prime numbers. For example, 120 = 2 x 60 = 2 x 2 x 30 = 2 x 2 x 3 x 2 x 5 = 23 x 3 x 5. However, to prove there are infinitely many prime numbers all we need is the fact that any number can be factored into prime numbers. Is this fact a problem for Peter? Once again it may be because of the word “any” but I will continue.
In a similar way as we did before, lets assume there is a largest prime number and we'll call it p. Let's make a larger number than p by multiplying all the prime numbers from 2 to p and then adding 1 to our product of numbers. For example, if I thought that my largest prime number was 11, my new number would be (2 x 3 x 5 x 7 x 11) + 1 = 2311. But notice that 2311 cannot be divisible (i.e. without a remainder) by 2, 3, 5, 7, or 11. How do I know that? Well just from how I defined 2311 = (2 x 3 x 5 x 7 x 11) + 1. If I divide 2311 by 2, I will get a quotient of 3 x 5 x 7 x 11 = 1155 and a remainder of 1. Similarly for the primes 3, 5, 7, and 11. However, there are plenty of other prime numbers that conceivably could divide (“divide” will always mean for us “divide without remainder”) into 2311 for all that we know (although 2311 does turn out to be a prime number).
But we now need to get a bit more abstract—and Peter is not going to like this. So we have by assumption our largest prime number p and we create this new number by multiplying all the prime numbers from 2 to p and adding on 1. I'll call this new number n, and certainly n is larger than p. Thus n = (2 x 3 x 5 x 7 x 11 x ... p) + 1. But just as we saw before in the case when p = 11, our number n cannot be divisible by any of the prime numbers in the product 2 x 3 x 5 x 7 x 11 x ... p. But by our assumption these are all the prime numbers there are! So we have somehow managed to create a number larger than our largest prime number p that is not divisible by any prime number. But this contradicts the fact that any number can be factored into prime numbers—end of proof—there are infinitely many prime numbers.
Well I'm on a roll here and I can't resist giving you one more aesthetic art of mathematics well-known proof of another number theory result. Lets talk about fractions—once again with the stipulation that Peter is going to have a problem with this. For fractions imply multiplication; for example 5/6 can be interpreted as 5 x (1/6), and 1/6 means a number “multiplied” by 6 that gives you 1. And to make matters worse for Peter, I'm going to now talk about “irrational” numbers. An irrational number is a number that cannot be written as a fraction (a fraction is called a “rational number”). Thus a fraction is a quotient of integers, a/b, where b is not 0; by an integer I mean any counting number, positive or negative, other than 0; thus I mean 0, 1, 2, 3, ..., -1, -2, -3.... What is the aesthetic number theory proof I am referring to? It is one that was also established a few thousand years ago by the ancient Greeks. It is the proof that the square root of 2 is an irrational number (cf. ).
By a square root of a number n, I mean a number multiplied by itself to equal the number n; for example, we can say that the square root of 9 is 3 since 3 x 3 = 9. As Peter wrote about at great length, square roots can be positive or negative (for example, we could also say that the square root of 9 is -3 since (-3) x (-3) = 9), which is a major problem for him and leads to all of his “bands” of mathematics (cf. ). However, I will simplify matters for our purposes and just consider a natural number (i.e. a positive integer) as a square root; therefore the square root of 2 for us will be unique. But how did the Greeks prove that the square root of 2 is irrational?
Well here is the argument—see what you think. Assume the square root of 2 is rational, and therefore the square root of 2 (which I will now denote as √2) = a/b where a and b are integers and b is not equal to 0. Now any fraction can be reduced to lowest terms; i.e. it can be reduced to an equivalent fraction with no common divisors. I don't think that even Peter would have a problem with this, as it can be demonstrated quite geometrically; thus 18/24 = 3/4 and 51/68 = 3/4. So let's assume that √2 = a/b is reduced to lowest terms, which means that a and b cannot both be even numbers, since even numbers are divisible by 2. Now lets square both sides of the equation (I know—I am “multiplying” again—oh well) and from basic high school algebra (I will assume for our purposes right now that high school algebra is a “legitimate” subject of study) we obtain 2 = a2/b2 (note that √2 squared is 2 because √2 is defined as the number whose square equals 2). Now multiplying both sides of the equation by b2 we obtain a2 = 2b2, which means that a2 is an even number. But by using basic high school algebra it is not at all difficult to show that the square of a number is even if and only if the number itself is even (try out some examples for yourself, such as 36 = 62 and 36 and 6 are both even, while 25 = 52 and 25 and 5 are both odd). Thus our number “a” is an even number, and therefore we'll let a = 2m for some number m. But then we have a2 = (2m)2 = 4m2 (again some basic high school algebra) and we obtain that 2b2 = a2 = 4m2. Now divide both sides of the equation 2b2 = 4m2 by 2 to obtain b2 = 2m2. But this means that b2 is an even number and consequently from what we saw before, b is also an even number. But now we have √2 = a/b where a and b are both even numbers; i.e. a and b are both divisible by 2. Do you see the problem? We assumed that the quotient a/b was reduced to lowest terms. Contradiction! End of proof—the square root of 2 must be irrational.
I could go on and on here, but I think you see my point. Mathematics can be an art form. Certainly there are times we can transform our mathematical number theory problems into various kinds of applications. For example, the notion of the square root of 2 can be thought of geometrically as the hypotenuse of an isosceles right triangle with its two legs each of length 1. Perhaps for this reason the proof I just gave for the irrationality of the square root of 2 would even be considered as “legitimate” by Peter; perhaps he might even consider it to be an example of one of his higher integrative levels of mathematics, such as “band 6” (cf. ). My ex-Ph.D. mathematics advisor and current colleague Chip Snyder and I have recently proved a significant result in constructive geometry using a combination of Euclidean geometry, abstract algebra, and number theory that perhaps Peter would also consider to be acceptable . For anyone interested, we proved (based on Chip's ingenious algebraic number theory formulation) that one can construct a regular 11-sided polygon (i.e. a hendecagon with all sides of the same length) by using a compass and a straightedge with exactly two marks on it, but I'll spare you the details (cf. ). However, for most of my career as a pure mathematician my research involved highly abstract ideas from algebraic number theory, which is a combination of abstract algebra and number theory, that to the best of my knowledge have no apparent application in Peter's “real” world . I'm talking about things like “proving that the Hilbert 2-class field is infinite for imaginary algebraic quadratic number fields with 2-class group of rank 4 with 4-class rank 2 in certain cases,” but once again I'll spare you the details (cf. ).
As you can see, I love mathematics—I'm a pure mathematician and number theorist, and I think I made my love of mathematics quite clear to Integral World readers from one of my first Integral World essays, entitled Integral Mathematics: A Four Quadrants Approach . Consequently I take personal offense from what I perceive as Peter Collins's condescending depiction of pure mathematics and number theory that he describes in his recent articles (cf. ). His perspective conveys to me that my career and my life as a pure mathematician is—for lack of a better word—“bogus.” Well perhaps Peter would use a kinder word—something like “partial” or “incomplete.” But whatever word Peter would use to describe my life as a pure mathematician, clearly he feels that I have missed the boat along these lines. But then he would most likely reassure me that I should not feel too bad, as so did every other mathematician who ever lived—including Euclid and the ancient Greeks.
I won't get into Peter's issue of the pure mathematics legitimacy of the Riemann Hypothesis here, as I know you have already indulged me with my mathematical expositions far more than you need to, as this is a philosophy site and not a mathematics site. My reason for once again putting on my mathematical hat on Integral World is simply to offer another perspective on the legitimacy of pure mathematics and number theory, a perspective that is strikingly different from the exposition that Peter Collins has given. When Peter talks about the urgency of incorporating experiential spirituality and ethical inner values in the world, I don't have any disagreement with him (cf. ). But I think he has absolutely no understanding of the intrinsic value of mathematics as an art form for its own right or of the intrinsic enjoyment one can obtain from logical thinking about mathematical ideas that may or may not exist in the real concrete world. I described my experience in this context as a “natural dimension of mathematics” in another of my early Integral World essays, entitled My Conception of Integral . And with this I shall end my defense of myself, Euclid, the ancient Greeks, and all the pure mathematicians and number theorists who have dared to think about mathematics as an aesthetic art form in its own right.
1) See Peter Collins (2013), Dynamics Nature of the Number System, (I): Mathematics at a Crossroads; (2): Holistic Role of Zeta Zeros; (3): The Bigger Picture. www.integralworld.net
2) See Jerry P. King (1992), The Art of Mathematics. Fawcett Columbine: New York.
3) See Carl B. Boyer (1968), A History of Mathematics. John Wiley & Sons: New York.
4) See Elliot Benjamin & Chip Snyder (2014), On the Construction of the Regular Hendecagon by Marked Ruler and Compass. Mathematical Proceedings of the Cambridge Philosophical Society (to appear).
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.
6) See Elliot Benjamin (2006), Integral Mathematics: A Four Quadrants Approach at www.integral.world.net; see also my (1993) book Numberama: Recreational Number Theory in the School System. Natural Dimension Publications: Swanville, Maine (available by contacting me at firstname.lastname@example.org).
7) See Elliot Benjamin (2006), My Conception of Integral at www.integralworld.net