Check out my review of Ken Wilber's latest book Finding Radical Wholeness

TRANSLATE THIS ARTICLE
Integral World: Exploring Theories of Everything
An independent forum for a critical discussion of the integral philosophy of Ken Wilber
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: www.benjamin-philosopher.com.

SEE MORE ESSAYS WRITTEN BY ELLIOT BENJAMIN

INTEGRAL MATHEMATICS

A FOUR QUADRANTS APPROACH

Elliot Benjamin, Ph.D.

1. Introduction: Integral Mathematics Perspectives

Mathematics is both an art form and a scientific discipline. When philosopher Ken Wilber writes about the differentiation of “The Big Three” (i.e. Art, Morals, Science), we find mathematics in the rather unique position of simultaneously entering territory that is both an extremely subjective art form as well as an extremely objective science form, encompassing both the rational and vision/logic levels of consciousness in Ken Wilber's Integral model (c.f.[11],[12],[13]). As a mathematician, this split has meaning to me as the division of the field into pure mathematics and applied mathematics.

This dual perspective of mathematics is described exceptionally well by Jerry King in The Art Of Mathematics (c.f.[8]). King stresses how these two disciplines of mathematics are often worlds apart; that they may be as far apart as the mystical poet and the objective scientist. However, King also calls for a uniting of these two mathematical disciplines; i.e. an integration of the worlds of pure mathematical thinking and pragmatic mathematics application in objective reality. In other words, King is asking that the realms of art and science be integrated in the world of mathematics, in quite an analogous way to the Integral model's quest to integrate the inner and outer domains in various branches and disciplines of knowledge into the “four quadrants” of Upper Left: Individual, Upper Right: Behavioral, Lower Left: Cultural, and Lower Right: Social, including psychology, spirituality, medicine, law, politics, government, education, business, etc. I would like to propose in this article that we add the discipline of mathematics to the recognized list of emerging integral disciplines.

We can view Integral Mathematics from a variety of different perspectives, including the following:

  1. Ken Wilber's “A Calculus Of Indigenous Perspectives” approach to integral mathematics (c.f. [14],[15]).
  2. The whole range of mathematics in the context of an independent line, as part of an integral four quadrant analysis (c.f. [7],[9],[10]).
  3. An integral four quadrant analysis applied to mathematics as a discipline and specific subject, focusing upon the various branches of mathematical study.
  4. An integral four quadrant analysis applied to a particular age group.
  5. An integral four quadrant analysis applied to mathematical research.
  6. Integral Mathematics in the context of a bona-fide mathematical discipline in itself, applied to all branches of mathematical study.

Each of the above perspectives of Integral Mathematics is a legitimate approach, but my main concentration in this article will be on perspective #3. I will be describing how the pure mathematics disciplines of Number Theory and Group Theory (a Group Theory/Consciousness problem is described in the Appendix) can be extended into an applied mathematics context that involves all four quadrants in Integral theory.

The disciplines of pure and applied mathematics are related to the division of Upper Left (UL) and Upper Right (UR) quadrants in the four quadrants of Integral theory, where we are utilizing a four quadrant perspective on the disciplines of mathematics. In the Upper Left quadrant, we have such things as the “cognitive line” as well as a continuum of mathematical knowledge that can be effectively described as “transcend and include” in levels of true “holarchical” fashion (c.f. [11],[12],[13] for definitions of these terms). For example, when fourth graders learn their multiplication and division facts, they do not forget these facts (hopefully) when they eventually learn high school Algebra, but include their ingrained Arithmetic skills to solve higher-level mathematical problems. Of-course our phenomenal technological advances in recent years make it unnecessary to retain many formerly necessary mathematical skills: Arithmetic, Algebra, and beyond, although I contend that it is intrinsically beneficial for us to retain the mathematical knowledge upon which our technology is based.

However, incorporating these technological developments into our Integral Mathematical model brings us into the Lower Right (LR) quadrant of the AQAL (All Quadrants All Levels in Integral Theory) model. For example, as a pure mathematician researcher in the field of Algebraic Number Theory, I have learned to appreciate the tremendous usefulness of the mathematical computer software program “Pari” in furnishing me with extremely complicated examples of theoretical mathematical results I have proved. There is no way I would have ever been able to come up with these examples without the use of this technology, and I view my pure mathematics research in this context as an example of Integral Mathematics perspective #5. (For the interested reader, I have a paper that has appeared in the Ramanujan journal which weaves my pure mathematics results together with my technologically based examples (c.f. [4])). For me, it has been truly enlightening to gradually assimilate this blend of technology into pure mathematics, and when I gave a talk about my results to the Maine/Quebec Number Theory Conference in 2002, I received feedback that it was very refreshing to see this kind of well-balanced mixture of theory and technology.

However, I must fully admit that I am essentially a pure mathematician, and my interest in mathematics is primarily for its inner artistic beauty, as described so eloquently by Jerry King in The Art Of Mathematics (c.f.[8]). But upon reading Ken Wilber's “A Calculus Of Indigenous Perspectives” (c.f. [14], [15]), I felt the inclination to write an article in which mathematical group theory, the basic logical structure of the pure mathematics discipline of Abstract Algebra, could be applied to a theory of shifts into higher levels of consciousness addressed in Integral theory. I call this theory “A Group Theoretical Mathematical Model Of Shifts Into Higher Levels Of Consciousness” (c.f.[5]). I view the ideas in this theory as a step towards a unification of pure mathematics (UL) and applied mathematics (UR), i.e. a merger of the Upper Left and Upper Right quadrants in the Integral model for the cognitive mathematical stream. The basics of this mathematical model for my Group Theory/Consciousness example is included in the Appendix for interested readers, though the mathematics does require a high degree of concentration to follow.

In regard to the Lower Left (LL) quadrant for Integral Mathematics, which can be described as the cultural “We” realm, this is the realm of sharing the integral view of mathematics with others. A prime (excuse the mathematical pun) example of my own experience in all four quadrants of Integral Mathematics has been the work that I have done in both elementary schools and liberal arts college courses teaching the ideas in my Recreational Number Theory book Numberama: Recreational Number Theory In The School System (c.f.[2]). In the context of Recreational Number Theory, which involves exploring the intriguing patterns in our number system as an enjoyable recreational pastime, I promote the exploration and discovery of a variety of interesting and stimulating number patterns as a unique learning experience that children and liberal arts college students may have in the realm of pure mathematics (UL). At the same time, I stimulate elementary school children to diligently practice their multiplication and division skills while all my students are learning how to make use of their age appropriate technology tools of Arithmetic or scientific calculators, in order to do the necessary trial and error work of discovering these patterns (Upper Right (UR) and Lower Right (LR) quadrants).

However, it is important to keep in mind that in this context of Integral Mathematics I am addressing Integral Mathematics perspective #3, i.e. mathematics as a particular subject and discipline, when I view the calculation skills as an UR quadrant activity. In integral mathematics perspective #2, where the focus is upon the entire range of mathematics as a developmental line within people, all mathematical thinking including calculation skills would be considered an UL activity, while the UR activities would consist of such things as the observable behaviors and brain wave states of students. The LR aspects would include mathematical symbols and written language, the classroom settings where mathematical communication takes place, the actual external communications of language utilized to discuss the mathematics, etc.

For older children and college students, their exploration and discovery of mathematical patterns eventually result in concrete algebraic formulas (c.f. [1],[2],[3])). There is much collaboration amongst students in working together to explore my problems, and my book includes twenty games that teachers, children, and parents can reproduce and play together to further practice the skills they are working on (c.f. [2]). For me, it is a way to bridge the gap between my own rather ivory tower mathematical interests and the extremely pragmatic view of mathematics that the great majority of people in the world have. Another example of the Lower Left quadrant being utilized in my own Integral Mathematics work has been my offering of “Family Math” workshops for parents and children working together on these Recreational Number Theory problems. In this Family Math context we have a strong collaborative LL quadrant activity of families working with each other in wonderful collaboration and mutual understanding to explore the UL quadrant Recreational Number Theory problems that I use, with much arithmetical calculation practice in the UR quadrant, in conjunction with classroom settings, verbal exchanges of the mathematics involved, and technology in the form of calculators and occasionally the internet, all of which comprise LR quadrant activity.

We thus have all four quadrants well represented in the Integral model: the Upper Left for the intrinsic artistic pure mathematical experience of exploring and discovering patterns of numbers; the Upper Right for the objective disciplined arithmetic skills practice with eventual concrete algebraic formulas; the Lower Left for the collaboration of liberal arts college students or children, parents, and teachers working together to discover these intrinsic Number Theory patterns via objective arithmetic skills practice; and the Lower Right for the outward forms of communication and physical resources, and for the use of technology in the form of calculators and eventually computers as the numbers become increasingly larger and reach the point where it is no longer feasible to explore the patterns without the use of this more advanced technology.

To give a concrete illustration of how I utilize Recreational Number Theory in the context of a four quadrant Integral model in teaching the joys of mathematics to others, I will focus upon the example of Perfect Numbers, which is described in more detail in my Numberama book (c.f.[2]). Although I have taught Perfect Numbers in an Integral Mathematics context to all age groups, including college students as well as children, the description I will be giving in Section 4 is particularly well suited for the age group of upper elementary school children, which is an illustration of Integral Mathematics perspective #4. As I have indicated in my above six Integral Mathematics perspectives, the Integral Mathematics discipline of study approach (Integral Mathematics perspective #3), of which Perfect Numbers is a specific example taken from Recreational Number theory, is a very different perspective from Wilber's Integral Calculus Of Indigenous Perspectives (Integral Mathematics perspective #1).

Wilber's approach is essentially a mathematical symbolic language to describe various first-person, second-person, and third-person perspectives, with various layers of further perspectives on the horizon. This approach is at the cornerstone of a more refined grid to the four quadrants, where each of the four quadrants is further divided to include both an inside and outside perspective. Wilber has come up with an interesting display of mathematical symbolic language to describe these perspectives (c.f. [14], [15]), but this is in a very different context from my main goal of demonstrating how the four quadrant Integral model can be applied to mathematics as a subject and discipline of study, focusing upon pure mathematics in the context of UL intrinsic mathematical thinking. I would also like to add that Integral Mathematics perspective #6, which views Integral Mathematics as its own mathematical discipline, applied to all branches of mathematical study, actually includes both my Perfect Numbers example in Section 4 as well as my Group Theory/Consciousness example in the Appendix. To give two well known examples from an applied mathematics context of Integral Mathematics perspective #6, we will take a brief look at the Pythagorean Theorem from high school Trigonometry, and the Fundamental Theorem Of Calculus from first year Calculus for math and science majors.

2. Mathematics As a Discipline: UL & UR

If one were to take an informal survey of the range of present day mathematics in regard to where they might fit in the four quadrants of study (technically known as “quadrivia” c.f. [15], although I will take the liberty of referring to “quadrants” for ease of presentation), I believe that virtually all mathematics subjects have components in the inter-subjective (LL) and inter-objective (LR) quadrants.. Through cultural collaboration and communications amongst mathematicians and scientists, and the tremendously widespread use of the internet and computer programs in addition to textbooks, research papers, seminars, classroom settings, etc., it seems quite evident that the cultural and social quadrants (LL and LR) are well represented in virtually every field of mathematics.

However, when we look at the individual and behavioral quadrants (UL and UR) in the various branches of mathematics in the context of the pure and applied mathematics divisions that I outline below in relation to the UL and UR quadrants for Integral Mathematics perspective #3, then the picture is not quite as simplistic or as universal. In terms of where particular mathematical disciplines belong in this UL and UR classification, I will make the following distinctions, keeping in mind that this is a generic classification and is not meant to be airtight or complete. The particular classification scheme that I have devised can also be described in Ken Wilber's symbolic language of indigenous perspectives (Integral Mathematics perspective #1) where pure mathematics would have an interior perception first person description and applied mathematics would have an exterior perception concrete world focus third person perspective (c.f. [14], [15]).

Figure 1. Disciplines of Mathematics: UL & UR
PURE MATHEMATICS (UL)
Number Theory
Abstract Algebra
Topology
Trigonometry,
APPLIED MATHEMATICS (UR)
Statistics
Differential Equations
Pre-Calculus (Analytic Geometry,
high school Algebra,
Arithmetic)
PURE AND APPLIED MATHEMATICS (UL & UR)
Calculus Analysis (Real & Complex)
Probability
Geometry (Euclidean & Projective)
Set Theory

To illustrate Integral Mathematics perspective #6, i.e. Integral Mathematics itself as a mathematics discipline, I will take a look at two well known applied mathematics problems from Trigonometry and Calculus. To begin with, we can measure the width of a river without crossing it by applying the Pythagorean Theorem from Trigonometry, which says that in a right triangle (a triangle with a perpendicular corner), the square of the side opposite the right angle (the hypotenuse) is equal to the sum of the squares of the two other sides, which is generally described algebraically as c^2 = a^2 + b^2, where the symbol ^ denotes an exponent; thus 5^2 = 5 X 5 = 25. The crossing the river problem starts out as a highly pragmatic example from Trigonometry in the UR quadrant, but one can study how to prove the Pythagorean Theorem using logical mathematical thinking, which involves the UL quadrant. One can have all kinds of cultural connections with others thru interactive learning communities via the context of classroom settings as well as field experiences in surveying the land or shores of the river, which involves the LL quadrant.. Finally, one can employ various technologies in terms of surveying equipment as well as classroom calculators, utilized in the external formats of classroom settings with verbal mathematical exchanges, which enters into the LR quadrant.

A similar argument can be made using Calculus for the problem of finding the area under the parabolic bell shaped curve y = x^2, say for the range from x = 2 to x = 7. A parabola can always be described by a quadratic (highest exponent of 2) equation in Algebra, which is a common representation for many kinds of scientific problems, ranging from measuring the height of an object thrown from the ground, to the probability distribution of so-called “normal” distributions from Statistics; once again we are starting out with an UR quadrant problem. As it turns out, we can get an estimate of this area by measuring the area of a number of small rectangles inside the parabola. As the sum of the areas of our rectangles approach the whole space of the parabola by our rectangles increasing in number and reducing in size, the better our area approximation will be. However, we can get the exact area of the parabola by using what is known as The Fundamental Theorem Of Calculus, which relates the theory of anti-derivatives to finding the area of various geometric curves. But for our purposes right now, what is important is that the Fundamental Theorem Of Calculus can be proven, and this is generally done for first year math and science major college students, and is certainly in the context of an UL quadrant activity. The LL and LR quadrants once again can be utilized in various teaching/learning scenarios with a rich array of educational and technical resources.

From this brief glimpse into the world of well known applied mathematics, together with the examples I will be describing from pure mathematics in Section 4 and the Appendix, it appears that there is much potential to view Integral Mathematics from perspective #6, in the context of a separate field of mathematical study itself.

3. The Developmental Line of Mathematics

One way of approaching the developmental line of mathematical thinking is to use Piaget's levels of cognitive development: sensori-motor, pre-operational, concrete operations, and formal operations (c.f. [10]).

According to Piaget, young children before the age of 6 or 7 are in the pre-operational stage and have not yet developed “number conservation,” meaning that their sense of number is not intact. For example, if one increases the distance between a given number of objects, effectively spreading them out, this may cause children to believe that there are more objects present when they are spread out than when they are bunched close together, despite the fact that the number of objects remains the same (c.f. [10]). However, it should be pointed out that there is also disagreement with Piaget's conclusion that young children do not have a real number sense (although Piaget's actual levels of cognitive development are universally accepted), from recent research in neuropsychology and brain physiology that focuses upon how young children may not be understanding the instructions of the experimenter, and may have a different interpretation of what is meant by “more,“ “less”, etc. (c.f. [6]). At any rate, let us assume for the moment that children between the ages of 7 and 11 generally enter Piaget's concrete operations stage, and are quite capable of engaging in arithmetical calculations with a true sense of what a number actually represents. Their ability to engage in more symbolic mathematics involving the manipulation of algebraic quantities representing whole sets of numbers, does not come into prominence until age 11 or so, when they have entered the formal operations cognitive level. This ability to manipulate formal mathematical symbols, representing various sets of mathematical objects, continues to grow and expand thru adolescence and young adulthood.

However, the higher levels of mathematical ability and the sublime creative productions of mathematicians appears to move beyond Piaget's highest stage of formal operations into what we may correspond to Integral Theory's vision-logic level of cognition, where complex inter-relationships are processed symbolically and metaphorically in highly creative ways (c.f. [9]). This vision-logic level of mathematical cognition is the essential vehicle that allows for the discovery of new mathematical ideas, and in particular for the highly abstract mathematical disciplines involving combinations of various fields such as Number Theory, Topology, Abstract Algebra, Real and Complex Analysis, Projective Geometry, etc. (see Figure 1). At the same time, this vision-logic level of mathematical cognition allows for the highly theoretical logical proofs of some of the key theorems of applied mathematical disciplines, such as Calculus and Analysis and Statistics (see Figure 1). The actual application of mathematics to the world-at-large is a combination of the vision-logic and formal operations cognitive levels with concrete real world applications, represented in applications of Statistics, Calculus, Differential Equations, etc. and most especially in the combined mathematics/science fields such as Mathematical Physics and Mathematical Biology, etc.

An even more focused perspective on the mathematical line of Integral theory can be seen from the work of Howard Gardner on Multiple Intelligences (c.f. [7]). For Gardner, the logical-mathematical line of “multiple intelligences” is one type of intelligence in addition to the intelligences which he characterizes as linguistic, musical, spatial, bodily-kinesthetic, and personal (inner and outer-directed awareness). Although there are various relationships amongst these diverse intelligences, the main features of Gardner's logical-mathematical intelligence include Piaget's descriptions as follows (c.f. [7], parenthesis my inclusion):

“Its origins are in the child's action upon the physical world; the crucial importance of the discovery of number; the gradual transition from physical manipulation of objects to interiorized transformation of action (UR to UL); the significance of relations among actions themselves; and “higher tiers of development, ” where the individual begins to explore the relationships and implications obtained from hypothetical statements.”

These higher tiers of development appear to have significant connections to the vision-logic level of consciousness, involving a beyond logic realm of intuition as well as long and complicated chains of abstract reasoning. This can be described in more specific mathematical terms as the hierarchical development from the concept of “number” to the creation of “Algebra,” where numbers are regarded as a system and variables are introduced to represent numbers, to the more general concept of “functions,” where one variable has a systematic relation to another variable. Functions may involve real values such as length, width, time, etc., but may also involve non-real quantities such as imaginary numbers, functions of functions, and significantly more complicated abstractions as well (c.f. [7]). The two examples I have given from Trigonometry and Calculus are examples of functions of real values. The example of Perfect Numbers in Section 4 is an example of the concept of number represented in Algebra. And the Group Theory/Consciousness example in the Appendix is an example of a more abstract formulation of functions, though applied to the real world in the context of shifts into higher levels of consciousness thru the practice of meditation.

The above discussion of Piaget and Gardner for the cognitive mathematical line can be put into the context of Integral Mathematics perspective #2. From this perspective, essentially any kind of mathematical thinking, whether it is computational or symbolic, would fall in the UL quadrant. The production of written mathematical language and symbols to describe this mathematical thinking would be placed in the UR quadrant. The teaching and learning (thru social interaction) of mathematical ideas and skills would be the crux of the LL quadrant, encompassing all our interpersonal and interactive educational settings. And the use of textbooks, calculators, computers, external classroom settings and verbal mathematical communications, etc. would be the nuts and bolts of the LR quadrant. We thus see from the work of Piaget and Gardner how Integral Mathematics perspective #2 deals with the cognitive mathematical line in the context of a four quadrant analysis.

4. The Four Quadrant Mathematics of Perfect Numbers

We come now to our primary example of Integral Mathematics perspective #3, mathematics as a discipline and particular subject of study, which will be taken from the area of Recreational Number Theory and will involve the topic of Perfect Numbers. This topic is also an excellent illustration of how the world of pure mathematics can be introduced to upper elementary school children, illustrating Integral Mathematics perspective #4 (an integral four quadrant analysis applied to a particular age group).

The topic of Perfect numbers is a magnificent example of an enticing unsolved problem in mathematics that can be easily understood by children, the formulation of which involves very large prime numbers that can shed light on an application to Government security codes. Let us first define a perfect number to be a number such that all the numbers that divide into it evenly – not including the number itself – add up to the original number (i.e. perfect numbers are the sum of its proper divisors). For example, all the numbers that divide into 8 evenly are 1, 2, and 4. The proper divisors of 8 add up to 7 and therefore 8 is not a perfect number. However, 6 has proper divisors 1, 2, and 3, they add up to 6 and therefore 6 is the first perfect number. With a little bit of diligence it can be determined without too much trouble that the second perfect number is 28, as the proper divisors of 28 are 1, 2, 4, 7, and 14, which indeed add up to 28. But now the fun starts, as it turns out that the third perfect number is somewhat larger, but there is an interesting pattern for perfect numbers that can be explored in the context of Recreational Number Theory, and the discovery of this pattern is a good example of the inner creativity of the UL quadrant.

Notice how the first two perfect numbers, 6 and 28, can be written as 6 = 2 X 3 and 28 = 4 X 7. A possible pattern for the third perfect number might therefore be 2 X 3, 4 X 7, and 8 X 11 = 88, where 8 is obtained by doubling 4 and 11 is obtained by adding 4 to 7. Another possible pattern could be 2 X 3, 4 X 7, 8 X 21 = 168 where 21 = 3 X 7, or 2 X 3, 4 X 7, 16 X 11 = 176 where 16 is 4 X 4 (after observing that 4 = 2 X 2), etc. Eventually, with some helpful hints, the pattern 2 x 3, 4 x 7, 16 X 31 = 496 will be arrived at, where the first factor is obtained by squares: 2 X 2 = 4, 4 X 4 = 16, and the second factor is obtained by doubling the first factor and subtracting 1; i.e. 3 = 2 X 2 – 1, 7 = 2 X 4 – 1, 31 = 2 X 16 – 1. It can be readily checked that 496 is truly the third perfect number, as the proper divisors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, and 248, these proper divisors do add up to 496, and there are no perfect numbers between 28 and 496 (wonderful multiplication and division skills practice in the UR quadrant for upper elementary school children: see figure 2).

Figure 2. First Three Perfect Numbers and Their Proper Divisors
First Perfect Number:
6 = 2 X 3;
proper divisors are 1,2,3 which add up to 6.
Second Perfect Number: 28 = 4 X 7;
proper divisors are 1,2,4,7,14 which add up to 28.
Third Perfect Number: 496 = 16 X 31;
proper divisors are 1,2,4,8,16,31,62,124,248 which add up to 496.

However, if this pattern is continued to try to obtain the fourth perfect number, one obtains 256 X 511 = 130,816 since 16 X 16 = 256 and 511 = 2 X 256 – 1. With the use of an ordinary arithmetic calculator (LR quadrant activity) and/or some knowledge of factor trees and prime factorization, it is quite reasonable for liberal arts college students and children in grades 5 and higher to determine that 130,816 is not a perfect number (see [2] for more particular information and mathematical techniques).

What is the correct pattern to find the fourth perfect number? Try doubling the first factor and doubling once again and subtracting 1 to get the second factor. For example, since the second perfect number is 28 = 4 X 7, we would have 8 X 15 as a candidate for the third perfect number, but it can be easily checked that 8 X 15 = 120 is not a perfect number. However, by doing it once more we obtain 16 X 31 = 496, which is indeed the third perfect number. The crucial observation is that the second factors of the first three perfect numbers are 3, 7, and 31, all of which are prime numbers (recall that a prime number is a number that has no proper divisors other than 1) and the second factor of the false candidate for the third perfect number is 15, which is not a prime number. Continuing this pattern once more results in 32 X 63, which would be rejected since 63 is not a prime number, but the next candidate is 64 X 127 = 8128, and it is easy to see that 127 is a prime number. It is quite feasible to determine that 8128 is a perfect number, and it happens to be the fourth perfect number (see figure 3)

Figure 3. First Five Perfect Numbers and Their Correct Patterns
First Perfect Number:
6 = 2 X 3;
3 = 2 X 2 – 1 and 3 is prime.
Second Perfect Number:
28 = 4 X 7;
7 = 2 X 4 – 1 and 7 is prime.
Third Perfect Number:
496 = 16 X 31;
31 = 2 X 16 – 1 and 31 is prime.
Fourth Perfect Number:
8128 = 64 X 127;
127 = 2 X 64 – 1 and 127 is prime
Fifth Perfect Number:
33,550,336 = 4096 X 8191;
8191 = 2 X 4096 – 1 and 8191 is prime.

Continuing this process with much diligence, factor tree and prime factorization use to determine proper divisors (c.f. [2]), serious calculator use, and small group collaborative efforts (productive and fun loving cultural LL activity), students will discover that the fifth perfect number is 33,550,336 (see figure 3). Does this pattern always work to find perfect numbers? How many perfect numbers are there? As it turns out, the topic of perfect numbers has by no means been completely solved by mathematicians, as we do not know how many perfect numbers there are – i.e. if there are infinitely many or not, as only 41 perfect numbers have been found using supercomputers at this time (the exponent itself of the prime number involved in the largest known perfect number has been found thru quite the intensive LR quadrant activity, and it would take 1400 to 1500 pages to write out!). Although we know that all even perfect numbers do follow the particular pattern we have described, and can be readily made into an algebraic formula for college students and middle school children who are learning algebra, we do not know whether or not there exists an odd perfect number (wonderful UL quadrant stimulation). The use of extremely large prime numbers in the formula for even perfect numbers is directly related to how Government security codes are devised (LR), where tremendously large composite (non-prime) numbers would need to be factored into two very large prime numbers in order to unlock the code.

From this brief illustration of how I teach the topic of Perfect Numbers as a mathematics enrichment activity, we can see how all four quadrants of Integral Mathematics perspective #3 have been utilized, perspective # 3 being an integral four quadrant analysis applied to mathematics as a discipline and specific subject, focusing upon the various branches of mathematical study. The intrinsic exploration of number patterns thru generating ideas and hypotheses is an UL quadrant activity focused upon the interior individual realm of cognitive creativity. The actual testing of these ideas to see if these patterns result in successful outcomes involves much practice in the concrete skills of multiplication and division (and algebra for older students), and can be viewed as an UR individual exterior behavioral activity. I generally have students working in small groups collaboratively to both explore ideas for their patterns as well as test them out, which is a LL quadrant activity, sharing in the communion of creative ideas and procedures to test out these ideas. Finally, as the possible candidates for number patterns quickly become extremely large, students utilize technology in the form of calculators (older children and/or college students may utilize computers as well), to obtain results regarding the testing of their ideas for patterns, which is a LR quadrant occasion of inter-objective activity, utilizing technology currently available in our social institutions, in classroom settings..

5. Concluding Statement

From this brief glimpse into the possible ramifications of including Integral Mathematics as one of the disciplines seeking to take an integral perspective, we see that there are rich and enticing possibilities to consider and a number of different approaches that can be taken.

A four quadrant analysis can be applied to the whole range of mathematics as a cognitive line thru the research and theories of Piaget and Gardner (Integral Mathematics perspective #2). A four quadrant analysis can also be applied to mathematics as a particular discipline and subject of study, as seen thru well known applied mathematics examples from Trigonometry and Calculus, a pure mathematics example from Recreational Number Theory, and a combined pure and applied abstract mathematics example joining mathematical Group Theory and levels of consciousness (see Appendix). These two major perspectives of Integral Mathematics have immediate relationships to viewing integral mathematics toward a specific age group (perspective #4) as well as to engaging in mathematical research (perspective #5) and the establishment of a discipline of Integral Mathematics in its own right (perspective #6). And in a context of mathematical perspectives we can study Ken Wilber's Calculus Of Indigenous Perspectives as a symbolic mathematical language to describe the multi-dimensional perspectives inherent in human interaction (Integral Mathematics perspective #1).

We thus see that Integral Mathematics has a great deal of potential to take its place along the other areas of study that are entering the integral AQAL domain. It is my hope that this article may serve as a calling forth to other mathematicians to furnish their own examples of how a Four Quadrants approach to Integral Mathematics may be a rich and valuable inclusion in the development of Integral theory that is currently taking place. I would like to form a network of mathematicians who are excited about this endeavor and who want to share their ideas with one another. I look forward to the first Integral Mathematics conference, and I believe that we may be seeing this in the not too distant future. In the meantime, I welcome hearing about your own ideas and interests concerning what Integral Mathematics means to you and how to get it off the ground.

APPENDIX: Mathematical Group Theory and Consciousness

Introduction

I will now turn my attention to the higher level mathematical world of Abstract Algebra and Group Theory to see another example of how Integral Mathematics perspective #3 can range across all four quadrants as we apply pure mathematics theory to study shifts into higher levels of consciousness in Integral Theory.

To give a concrete illustration of how Mathematical Group Theory, a pure mathematics discipline, (a “mathematical group” is defined below) can be applied to the world of Integral Theory in the context of shifts into higher levels of consciousness, I will present some original ideas from Mathematical Group Theory applied to shifts into higher levels of consciousness. In Ken Wilber's Integral theory, levels of consciousness may vary greatly across various “streams,” for example, a person may be on a rational level in cognition, a conventional level in morals, and an illumined mind level in spirituality, etc. (c.f. [11],[12],[13]). Keep in mind that Wilber uses the terminology waves and structures and levels interchangeably, as well as streams and lines interchangeably. I will apply a mathematical group model to describe the shift from the rational to the vision-logic level of consciousness in the cognitive stream. Specifically we will show how a group theoretical mathematical model can explain how a particular state of consciousness may have a significant impact on a person developing into a higher level of consciousness. I will make the assumption that for a person who is on a continuum between two levels of consciousness, such as rational and vision-logic, vision-logic and illumined mind, etc., repeated experiences of altered and non-ordinary states of consciousness may help a person evolve into higher levels of consciousness in a permanent fashion.

For example, if I am in the middle of a continuum between the rational and vision-logic levels of consciousness, prolonged periods of meditation over a certain time period may be a significant factor in enabling me to move closer toward the vision-logic level of consciousness. We shall refer to this type of meditation experience in accordance with the essence of Ken Wilber's model of “A Calculus Of Indigenous Perspectives” (c.f.14]. However, since Wilber's mathematical notation is quite cumbersome, we shall more simply refer to our meditation experience as M mod x, which means that person x is experiencing meditation M within him/herself via him/herself. It is understood that we are focusing upon some particular type of meditation represented by M, and a particular individual represented by x, as our theory is attempting to capture the subjective individualized framework of both the person and the type of meditation experienced.

However, it is important to keep in mind that my theory is at the beginning stages, and I will therefore be making rather simplistic assumptions, such as in increase in the number of hours of meditation results in a corresponding increase in moving from the rational to vision-logic level of consciousness, in order to illustrate the basic mathematical ideas without overdoing the amount of complexity involved in the mathematics. Clearly this mathematical simplification does not realistically describe the actual phenomenon of how people develop into higher levels of consciousness, as many other factors come into the picture to complicate the situation, not the least of which is that simply increasing the number of hours meditating may not have a corresponding effect of shifting into a higher level of consciousness, the personal intention of the person meditating may be a significant variable that needs to be taken into account, etc. We now utilize some mathematical group theory, particularly the theory of cyclic groups.

Mathematical Group Theory

A mathematical group is defined to be a set of elements S with an operation * such that the following properties are satisfied.

  1. If x is an element of S and y is an element of S then x*y is also an element of S.
  2. There is an identity element E in S such that for all elements x in S, x*E = E*x = x.
  3. For all elements x, y, and z in S we have x*(y*z) = (x*y)*z (associative law).
  4. For each element x in S there exists an element y in S such that x*y = y*x = E; we refer to this element y as x^(-1) and call it the inverse element of x.

If for all elements x and y in S we also have the property that x*y = y*x then our group is referred to as a commutative (or abelian) group. A simple example of a commutative group is the infinite set of integers S = {…-3, -2, -1, 0, 1, 2, 3,…) under addition. We see that the sum of two integers is always an integer, zero is the identity element, the associative law holds, for any integer x we have x^(-1) = -x, and for any integers x and y we have x + y = y + x; thus the set of integers is a commutative group under addition. If there is an element x in our group S such that every element y in S can be written as x^n for some integer n where x^n refers to (x*x*x…*x) n times if n > 0, x^(-n) = (x^(-1))^n, and we define x^0 = E, the identity element of the group S, then our group S is referred to as a cyclic group with the generator x. It is an easy mathematical exercise to prove that all cyclic groups are commutative. We see that our infinite commutative group of integers is actually a cyclic group generated by 1. For an example of a finite commutative cyclic group under addition consisting of 12 elements, think of the hour hand clock numbers of an ordinary 12 hour clock (see figure 6). We see that 1 is a generator of the group, 12 is the identity element (12 + 5 = 5, 12 + 7 = 7, 12 + (-4) = 12 + 8 = 8, etc.), and for any hour hand clock number x we have x^12 = (x + x + x + …x) (12 times) = 12. Given any clock number x, we see that 12 – x is the inverse of x since x + (12 – x) = 12 = the identity element E (5 + (12 – 5) = 5 + 7 = 12 = E, etc.

Biquasi-groups and Consciousness Shifts

To formulate a group theoretical model of shifts into higher levels of consciousness, we shall define the transition into the next higher level of consciousness to be a “biquasi-identity element” I of a “biquasi-group” (these “biquasi” terms will soon be defined). Thus in our above meditation example, the transition from the rational to the vision-logic level of consciousness is what we define as the biquasi-identity element I in what turns out to be a cyclic biquasi-group S generated by M mod x. We shall interpret the equation (M mod x)^6 = I to mean that continued practice of our meditation over a certain time period will be a significant factor in enabling person x to move from the rational to the vision-logic level of consciousness. According to Integral theory, repeated contact with altered states of consciousness, such as meditation, may help a person disidentify from their current level of consciousness, thereby enabling a person to take as object that which has been a subject for them. Or, as Wilber put it, transformation involves disidentification with the current stage, identification with the next higher stage, and integration of aspects of previous stages into the higher stage (c.f. [11],[12].[13]). Mathematically, we can think of our meditation example as resembling a finite cyclic group S consisting of 6 elements generated by M mod x; we say “resembling” as opposed to “actual” due to the biquasi nature of our group, which we shall now describe.

For simplistic illustrative purposes, lets make the assumption that M mod x denotes 30 hours of meditation over the time period of one month. According to our group theoretical model (in its preliminary and simplistic form), person x would make significant progress toward moving into the vision-logic level of consciousness if he/she were to diligently continue this meditation practice for 180 hours over a 6 month time period, which is the meaning of the equation (M mod x)^6 = I. We have the equations M mod x + M mod x = (M mod x)^2 (meaning simply that 30 hours + 30 hours equals 60 hours of meditation and is twice as impactful as 30 hours of meditation on person x's potential consciousness shift, once again given our extreme mathematical simplification of assumptions), (M mod x)^2 + (M mod x)^3 = (M mod x)^5, etc.; in general we can say (M mod x)^m + (M mod x)^n = (M mod x)^(m+n) for positive integers m and n. However, once person x reaches the vision-logic level of consciousness, the impact of his/her meditation practice is no longer in the same way relevant to progressing to the next higher (illumined mind in this case) level of consciousness, as we now acknowledge mathematically that whole other factors may come into the picture. We therefore define I*I = I and I*(M mod x)^n = (M_0 mod x)^n, where M_0 mod x means that person x is now meditating while in a vision-logic level of consciousness. Note that by using this scheme, we would have I*(M mod x)^2 = I*I*(M mod x)^2 = (M_0 mod x)^2 and I*(M mod x)^62 = I*(M mod x)^2 = (M_0 mod x)^2. We therefore make no further interpretation of the impact of the meditation experience on person x once the vision-logic level of consciousness is reached. Certainly another mathematical scheme could be devised, but at this point we are merely illustrating the essential ideas in its simplest formulation.

At any rate, our equation I*(M mod x)^n = (M_0 mod x)^n resembles the requirements for I to be an identity element of a group, but there is a major problem in that M becomes M_0 (signifying that we are now in the vision-logic level of consciousness). This change from M to M_0 does necessitate us calling our identity element something resembling but not quite exactly an identity element; we shall call it a “biquasi-identity element;” the term “biquasi” refers to the fact that we are using a second set to refer to our shift into the vision-logic level of consciousness. Of-course this same general idea could be applied to shifts into any of the various levels of consciousness, in addition to vision-logic. In a similar manner, we don't quite have a bona-fide group, but if we use this biquasi-identity element in the required properties of a group, we find that our group properties are essentially satisfied and we now have what we shall refer to as a “biquasi-group;” more specifically we have a “cyclic biquasi-group” generated by M mod x (see the Appendix in [4] for a more formal mathematical definition of a biquasi-group, along with a few basic lemmas (results that have been mathematically proven but are not as significant as theorems) that describe some of its properties).

Biquasi-group Model Applied to The Four Quadrants

In the rest of my aforementioned paper I describe and compare various kinds of biquasi-groups representing various degrees of meditation and yoga practices that can occur across various streams, both combined and separately, with their resulting effectiveness in regard to shifting from the rational into the vision-logic levels of consciousness (c.f. [4]). But for our present purposes, what is most relevant is that the Mathematical Group Theory used in my paper is at the heart of pure mathematics, an UL quadrant mathematics thinking that comprises the individual and creative mathematical realm.

On the other hand, the application of this pure mathematical group theory to AQAL Integral theory represents the world of applied mathematics, an UR quadrant activity in the Integral model, focusing upon the external behavioral equations derived from the intrinsic UL mathematical model. The application of this mathematical model could involve a variety of communities of meditators and yoga practitioners engaged in a discipline of spiritual activity, thereby entering the LL cultural quadrant. Finally, although the Group Theory equations that I work out in my paper are relatively simple, it is not difficult to see how one could significantly increase both the number and relationship of variables utilized to make the actual phenomenon being investigated more realistic, plus increase the combinations of spiritual disciplines and various streams investigated, to the point where computer software programs would be needed to work effectively with the mathematical equations, thereby engaging in a LR quadrant activity that utilizes technology available mainly within our institutions of higher education. Of course the particular consciousness shift from rational to vision-logic that I have described in my paper is only one illustration, and the same Mathematical Group Theory model can be utilized to describe shifts across various streams and levels in the Integral model; i.e. it applies to the entire Integral AQAL model.

June, 2006

References

1) Benjamin, Elliot (1992). “Number Theory, Developmental Mathematics, and The Discovery Approach.” New England Mathematics Association of Two Year Colleges: Annual Convention.

2) Benjamin, Elliot (1993). Numberama: Recreational number theory in the school system. Swanville, ME: Natural Dimension Publications.(available by contacting the author at [email protected]

3) Benjamin, Elliot (2000). “Number theory—An igniter for mathematical sparks.” Atomim Newsletter.

4) Benjamin, Elliot (2003). “A group theoretical mathematical model of shifts into higher levels of consciousness in Ken Wilber's integral theory.” www.integralscience.org

5) Benjamin, Elliot (2006). “On the 2-class field tower of some imaginary biquadratic number fields.” Ramanujan Journal.

6) Dehaene, Stanilas (1997). The number sense: How the mind creates mathematics. New York, NY: Oxford University Press.

7) Gardner, Howard (1983). Frames of mind: The theory of multiple intelligences. New York, NY: Basic Books.

8) King, Jerry P. (1992). The art of mathematics. New York, NY: Ballantine Books.

9) Lakoff, George & Nunez, Rafael E. (2000). Where mathematics comes from. New York, NY: Basic Books.

10) Piaget, Jean (1952). The child's conception of number. New York, NY: Norton.

11) Wilber, Ken (1995). Sex, ecology, spirituality. Boston, MA: Shambhala Publications.

12) Wilber, Ken (2000a). Integral psychology. Boston, MA: Shambhala Publications.

13) Wilber, Ken (2000b). A brief history of everything. Boston, MA: Shambhala Publications.

14) Wilber, Ken (n.d.). “Appendix B. An integral mathematics of primordial perspectives.” http://wilber.shambhala.com/html/books/kosmos/excerptC/appendix-B.cfm

15) Wilber, Ken (2006). Integral Spirituality. Boston, MA: Shambhala Publications.






Comment Form is loading comments...