Yesterday we considered how number (as qualitative dimension) can be given an imaginary (as well as real) meaning and that this thereby also applies to time (and space) in both physical and psychological terms.
Basically what this entails is that development can take two complementary directions that are transcendent and immanent with respect to each other. Therefore if we associate real numbers (as dimensions) with the transcendent aspect, then the corresponding imaginary numbers are then - relatively - associated with the immanent aspect.
Though all these numbers (as dimensions) are implicit in actual human experience, remarkably little progress has yet been made with respect to any coherent explicit appreciation. And as I have stated repeatedly the conventional paradigm of Science and Mathematics as we know is merely of a 1-dimensional nature (in qualitative terms).
Now, when appropriately interpreted, the other dimensional numbers do unfold in varying degrees through the process of (authentic) contemplative development.
However as practitioners in the past were rarely directly concerned with the mathematical implications of their newly acquired spiritual vision, the important scientific consequences were never made (except in the most general terms).
So my own special concern from the start has been to marry the contemplative vision with rational understanding through exploration of the amazing new mathematical (and associated scientific) landscapes that thereby emerge.
One significant clue as to the nature of the imaginary number (as dimension) can be given through raising 1 to the power of i.
Now as we have seen when we raise 1 to a rational number (such as 1/3) we generate a new number in the circular number system (i.e. on the circle of unit radius in the complex plane).
However when we now raise 1 to the imaginary number i, we generate a new number in the linear number system. So we can see here how real and imaginary numbers (as qualitative dimensions) are associated with circular and linear quantitative results respectively.
So a key task with healthy contemplative development is the successful balancing of both transcendent and immanent directions. This implies likewise the successful balancing of appreciation of number (as qualitative dimension) in a - relatively - real and imaginary manner.
As we saw yesterday transcendental type understanding (in qualitative terms) is of the most refined manner possible in the phenomenal realm.
So before moving directly into the subject of today's entry, I will briefly summarise on the various types of transcendental dimensions.
Real transcendental dimensions are of the most refined type whereby one understands all phenomenal relations - not in terms of (separate) linear or (circular) holistic notions - but rather as the relationship between both.
The positive refer to the subtle rational appreciation of this relationship; the negative then relate to direct intuitive realisation.
However the final step in the phenomenal realm is making the understanding associated with corresponding imaginary directions explicit.
So the imaginary transcendental dimensions relate to understanding of projections (in the indirect conscious expression of unconscious meaning) as the relationship between both (separate) linear and (holistic) circular understanding.
Indeed it is precisely in successfully being able to understand projections in terms of this necessary relationship of conscious and unconscious that the involuntary nature of such projection ceases. So involuntary projections always arises due to a certain failure in properly relating the unconscious desire for meaning (embodied in such projections) with the conscious phenomenal circumstances through which they are expressed.
Once again the positive expression of such imaginary transcendental dimensions relates to a highly refined form of rational understanding of their nature; the negative expression again relates to the more direct intuitive realisation of this nature.
Now, I have already likened the contemplative journey to a steep mountain climb. The real transcendent aspect of this journey - notice the close association here with the qualitative mathematical meaning of transcendental - relates to the ascent (that ultimately leads to a spiritual experience beyond all phenomena of form).
The corresponding descent then relates to the immanent aspect of the journey resulting in a spiritual experience that is inherent within all phenomena.
Just as the Riemann Hypothesis is generally considered the most important unsolved problem in Mathematics, the Euler Identity is likewise considered its most remarkable equation (formula, relationship).
Now because every quantitative relationship equally has a qualitative significance (that is formally unrecognised in conventional mathematical terms), this suggests that an extremely important qualitative meaning is associated with the Euler Identity (that can only be decoded in the appropriate manner).
Now conventionally the Euler Identity is expressed by the equation,
e^(πi) + 1 = 0;
Therefore e^(πi) = - 1.
Then by squaring both sides
e^(2πi) = 1.
We now have e (which is itself a transcendental number) raised to a dimensional expression (that is of an imaginary transcendental nature).
Notice how when we raised a rational number to a rational number the result was irrational; then when we raised an irrational number to an irrational number the result was transcendental. So we have continued to move in the direction of increasing transcendence (from a qualitative perspective).
However now in this special case where the base transcendental number is e, we raise it to a special case of a transcendental number where the dimension is 2πi, we obtain the simplest of all rational numbers (which in qualitative terms is thereby of the most immanent nature).
So putting it simply, the Euler Identity, when understood in an appropriate qualitative manner, points to the mysterious transformation in contemplative development where both form and emptiness are perfectly reconciled.
We saw earlier how development entails both differentiation and integration (through the odd and even numbered dimensions respectively. So the ultimate task of transformation is to reach a state where differentiation and integration both coincide.
Now e is the perfect numerical symbol of such transformation as both the differential and integral of e^x are uniquely the same!
Now in the real unit circle, the circular circumference is 2π. However if we now consider the radius as imaginary - rather than real - then the circular circumference is 2πi. However imaginary in this qualitative sense combines both positive and negative directions. So the imaginary circle is better represented as a non-dimensional point. Here both line and circle are perfectly reconciled from a quantitative perspective; likewise both linear and circular understanding are perfectly reconciled in qualitative terms.
So e^(2πi) in qualitative terms is inseparable from e^0 in quantitative terms!
Therefore in qualitative terms - when finally experience becomes of a pure formless nature as transcendence - it thereby equally becomes immanent as inherent in all form (represented qualitatively as 1).
In the various mystical traditions extensive attention has been given to the nature of this key transformation.
Perhaps the most famous expression is given in the Buddhist sutra:
"Form is nor other than Emptiness
Emptiness is not other than Form."
Well in a precise mathematical manner (where symbols are appropriately understand in the qualitative manner) the Euler Identity describes the same transformation.
However even this is not the end of the mathematical story of number.