The Number Paradigms

                                               SUMMARY OF LINEAR STAGES   I am developing here, what I define as a transrational approach. This is based - in terms of developed consciousness - on the fundamental complementarity of the conscious (rational) and unconscious (intuitive) processes. When understood in such dynamic transrational terms, (rational) mathematics and (intuitive) psychology are themselves complementary. We have therefore what can be called a psycho-mathematical understanding of both areas. When suitably formulated, seen from one direction, every (horizontal) mathematical relationship has a (vertical) psychological counterpart which is complementary. Seen, alternatively from the other direction - which I have been chiefly dealing with - every (vertical) psychological relationship has a (horizontal) mathematical counterpart. ...

                                                              MIRROR VISION   QUALITATIVE NUMBERS: NEGATIVE   Mirror Development   The development of these structures marks the decisive transformation from the linear to the circular level, where the focus gradually switches from differentiation of the conscious to the unconscious structures. Each of the crucial transformations between levels is marked by a key distinguishing characteristic. We have been examining development in terms of process, mode and direction. The choice in terms of process is conscious v unconscious understanding (finite v infinite in psycho-mathematical terms). The choice in terms of mode is cognitive v affective understanding (real v imaginary in psycho-mathematical terms). The choice in terms of direction is extern...

                                                    PYTHAGORAS REVISITED   PSYCHO-MATHEMATICAL APPROACH Special Numbers There is a very interesting example, in the history of mathematics of an explicit psycho-mathematical method. It is associated with the Pythagoreans who flourished in Greece (circa 600 BC). Their many discoveries in number theory help to illustrate this approach. For example there is the special class of numbers (associated with them) called perfect numbers. A perfect number is one that is equal to the sum of its proper factors. 6 is the first of these numbers. Its proper factors are 1, 2 and 3 and the sum of these is 6. The next three perfect numbers are 28, 496 and 8128 respectively. (In all slightly more than 30 are known). Perfect numbers have several unique properties and were frequently used, in earlier ti...

                                                                        A NEW LOGIC   QUALITIVE NUMBERS: IRRATIONAL   (A) Concrete Stages   Positive:  The Square Root of 2 We have looked at the dilemma of the Pythagoreans in discovering that the  Ö 2 is irrational. They rightly saw that there was a lack of correspondence between their qualitative paradigm which was rational and the number system (which now extended to irrational quantities). From a psycho-mathematical approach (which requires complementarity), this was unacceptable. Needless to say, there are much philosophical riches contained in the notion of a square root. This is where the distinction as between the horizontal approach of numbers representing quantities (within a given dimension),...

                                             SUMMARY OF CIRCULAR STAGES   The Pythagoreans were highly significant in that they adopted explicitly a psycho-mathematical approach to reality. Their approach was apparently validated in the belief that all number quantities were rational, corresponding in turn to the use of a qualitative paradigm which was also rational. The order of this universe was thus shattered by the discovery of numbers (e.g.  Ö 2) which were irrational. The approach of mathematics has subsequently developed in a highly reduced fashion. Many important types of mathematical quantities have been discovered which though they are not rational (i.e. irrational, imaginary, transfinite) are interpreted, in a highly reduced fashion, within a non-matching qualitative paradigm.   The rational paradigm is the product of the linear leve...

                                                         VIRTUAL DISCOVERY   QUALITATIVE NUMBERS: IMAGINARY   Remarkable Transformation We have seen earlier how the transition from linear to circular levels involves a remarkable psychological transformation in the differentiation of mirror or negative structures. One then realises that actual experience of reality involves polarised directions of experience in that objective and subjective experience take place in opposite systems of space and time. In other words if movement of phenomena in objective terms (for the world) takes place in a positive or forward direction, then in relative terms, movement of the same phenomena, in subjective terms (for the self), takes place in a negative or backward direction. Thus experience physically and psychologically moves...

                                                                   PI IN THE SKY   QUALITATIVE NUMBERS: TRANSCENDENTAL   Introduction We have seen how the circular level is concerned with the development of superstructures which in mathematical terms are irrational. We have irrational qualities (in the form of vertical stages of psychological growth), that exactly complement irrational quantities (in the form of the horizontal reduced mathematical interpretation of such numbers). Strictly speaking these psychological structures correspond to mathematical quantities that are algebraic irrational. For example  Ö 2 is algebraic irrational as a solution to the equation (i.e. x  2  = 2). This means that though this number is irrational in the 1st dimension, it is in fact...