The Euler Identity is generally recognised as the most beautiful theorem in Mathematics.
For example the well known "populariser" of Mathematics, Keith Devlin has stated:
""like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence".
So when it comes to the Euler Identity, mathematicians somehow realise that they are dealing with a relationship that greatly transcends their rational ability to comprehend its symbols.
Euler's Identity (or Equation) is generally given in the form,
e^iπ + 1 = 0.
e here refers to Euler's number which is the base of the natural logarithms system.
i is the imaginary unit that represents the square root of – 1.
π (or pi) represents the ratio of the circumference of a circle to its line diameter.
So along with 1 and 0, which are also included in the simple Euler equation, these represents the five most important constants that are used in Mathematics.
Now, it will suit our purposes here to rearrange the Euler Identity slightly.
So e^iπ = – 1.
Therefore when we square both sides we obtain,
e^2iπ = 1.
I refer to this as the fundamental form of the Euler Identity.
Now, one might initially wonder what this strange equation might have to do with the development process, especially with the most obscure stages approximating spiritual union!
However they key to making the appropriate connection is the realisation that when appropriately understood, in a dynamic interactive manner, all mathematical symbols have holistic - as well as analytic - interpretations.
And when one begins to appreciate the holistic significance of the mathematical symbols used in the Euler Identity, though still remaining deeply mysterious, it can be seen to intimately relate to the paradoxical nature of spiritual union.
In fact when one accepts - as I now readily do - the deep integral value of the holistic interpretation of mathematical symbols, the Euler Identity can indeed be looked on as the most appropriate scientific explanation possible of the ultimate nature of spiritual transformation.
Now, e and π represent the two best known transcendental numbers.
The transcendental numbers are in fact extremely elusive, for even though the vast majority of real numbers are of a transcendental nature, very few are known.
In "The Number Paradigms, I had concluded that the most advanced psychological structures (before pure spiritual union) are of a transcendental imaginary nature. And the dimensional number (power, exponent) of e in the fundamental Euler Identity i.e. 2iπ, is of an imaginary transcendental nature!
In analytic mathematical terms, when one raises a rational number to another rational number (as fraction) one gets an irrational (algebraic) number.
In corresponding holistic mathematical terms, I was able to us to show how development would proceed from rational to the more intuitively inspired levels of "higher" consciousness.
Then in analytic mathematical terms, when one raises a rational or (algebraic irrational number) to another irrational number, one now obtains a transcendental number.
Likewise one is able to experience the two-way relationship of irrational perceptions (associated with the refined intuitively inspired understanding of the subtle level) with corresponding irrational concepts, this leads to a corresponding transformation to the next "higher" level (with a holistic transcendental structure).
The question then arises as to what sort of transformation is possible, when one then, having further deepened in contemplative awareness, can experience in two-way fashion, the relationship of transcendental perceptions with corresponding transcendental concepts.
And remarkably, just as in analytic mathematical terms there is one special case (with respect to such relationships) that results in 1, likewise in holistic terms there is the corresponding special case, which in terms of psychological development, results in "oneness" (i.e. spiritual unity).
Once again e is a transcendental number, with a very special property in analytic mathematical terms.
So when e is raised to a power x i.e. ex, when one differentiates the expression y = ex, and then in reverse manner integrates the expression, the result remains the same
.
Now, the fundamental Euler expression is of the form ex, where x = 2iπ.
Therefore, this remains unchanged where differentiation and integration takes place. And of course when we differentiate the corresponding result on the RHS we get 0.
Equally with respect to development, in holistic psychological terms, we have the two fundamental processes of differentiation and integration respectively.
And in psychological terms, at the transcendental level of contemplative intuitive awareness, one can approach the special situation, where experience now becomes so spiritually rarefied and transparent, that one can no longer distinguish (differentiated) phenomena of form from the continued (integrated) experience of emptiness.
And this concurs when - quite literally - experience of dimensions turns full circle.
As I have repeatedly emphasised, standard analytic interpretation is of a linear (1-dimensional) nature where the dimensional aspect, which - relatively - is of a qualitative nature - is reduced in a merely quantitative manner.
This is especially true with respect to Mathematics, where for example the concept of "number" is reduced in a merely quantitative manner as applying directly to all number perceptions!
So the whole point of contemplative type awareness is to gradually move from this restricted linear interpretation of concepts to a fully circular holistic appreciation (which is completely paradoxical in dualistic terms).
With respect to the unit circle, its circumference is given as 2π. If we were to consider now the "imaginary" equivalent - which corresponds to the pure holistic appreciation of the circle - the circumference would be 2iπ.
So the dimensional value (of the fundamental Euler Identity) relates to the purely holistic appreciation of number (which is completely circular - and paradoxical - in dualistic terms).
In other words it represents the holistic intuitive extreme of awareness (as purely circular) in contrast to the corresponding analytic rational extreme (as purely linear)..
Therefore in holistic terms, the fundamental Euler Identity defines in a truly remarkable and deeply mysterious manner the phenomenon of spiritual union, which however can only be approached - rather than fully attained - in an experiential dynamic interactive manner.
And this union, as the holistic expression of "1", is now indistinguishable from the corresponding experience of nothingness (or emptiness) as the holistic expression of "0".
So once again, one may appreciate how we have come full circle.
In conventional mathematical terms (at the analytic level of interpretation, 1 is clearly separated from 0 (in a quantitative manner).
However now in corresponding holistic terms, 1 is fully interdependent with 0 (in a qualitative manner).
So the (fundamental) Euler Identity - when properly appreciated in a dynamic interactive manner - represents the mysterious marriage, where both the analytic and holistic interpretation of its symbols ultimately dissolves in an experience that is truly ineffable.
For example the well known "populariser" of Mathematics, Keith Devlin has stated:
""like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence".
So when it comes to the Euler Identity, mathematicians somehow realise that they are dealing with a relationship that greatly transcends their rational ability to comprehend its symbols.
Euler's Identity (or Equation) is generally given in the form,
e^iπ + 1 = 0.
e here refers to Euler's number which is the base of the natural logarithms system.
i is the imaginary unit that represents the square root of – 1.
π (or pi) represents the ratio of the circumference of a circle to its line diameter.
So along with 1 and 0, which are also included in the simple Euler equation, these represents the five most important constants that are used in Mathematics.
Now, it will suit our purposes here to rearrange the Euler Identity slightly.
So e^iπ = – 1.
Therefore when we square both sides we obtain,
e^2iπ = 1.
I refer to this as the fundamental form of the Euler Identity.
Now, one might initially wonder what this strange equation might have to do with the development process, especially with the most obscure stages approximating spiritual union!
However they key to making the appropriate connection is the realisation that when appropriately understood, in a dynamic interactive manner, all mathematical symbols have holistic - as well as analytic - interpretations.
And when one begins to appreciate the holistic significance of the mathematical symbols used in the Euler Identity, though still remaining deeply mysterious, it can be seen to intimately relate to the paradoxical nature of spiritual union.
In fact when one accepts - as I now readily do - the deep integral value of the holistic interpretation of mathematical symbols, the Euler Identity can indeed be looked on as the most appropriate scientific explanation possible of the ultimate nature of spiritual transformation.
Now, e and π represent the two best known transcendental numbers.
The transcendental numbers are in fact extremely elusive, for even though the vast majority of real numbers are of a transcendental nature, very few are known.
In "The Number Paradigms, I had concluded that the most advanced psychological structures (before pure spiritual union) are of a transcendental imaginary nature. And the dimensional number (power, exponent) of e in the fundamental Euler Identity i.e. 2iπ, is of an imaginary transcendental nature!
In analytic mathematical terms, when one raises a rational number to another rational number (as fraction) one gets an irrational (algebraic) number.
In corresponding holistic mathematical terms, I was able to us to show how development would proceed from rational to the more intuitively inspired levels of "higher" consciousness.
Then in analytic mathematical terms, when one raises a rational or (algebraic irrational number) to another irrational number, one now obtains a transcendental number.
Likewise one is able to experience the two-way relationship of irrational perceptions (associated with the refined intuitively inspired understanding of the subtle level) with corresponding irrational concepts, this leads to a corresponding transformation to the next "higher" level (with a holistic transcendental structure).
The question then arises as to what sort of transformation is possible, when one then, having further deepened in contemplative awareness, can experience in two-way fashion, the relationship of transcendental perceptions with corresponding transcendental concepts.
And remarkably, just as in analytic mathematical terms there is one special case (with respect to such relationships) that results in 1, likewise in holistic terms there is the corresponding special case, which in terms of psychological development, results in "oneness" (i.e. spiritual unity).
Once again e is a transcendental number, with a very special property in analytic mathematical terms.
So when e is raised to a power x i.e. ex, when one differentiates the expression y = ex, and then in reverse manner integrates the expression, the result remains the same
Now, the fundamental Euler expression is of the form ex, where x = 2iπ.
Therefore, this remains unchanged where differentiation and integration takes place. And of course when we differentiate the corresponding result on the RHS we get 0.
Equally with respect to development, in holistic psychological terms, we have the two fundamental processes of differentiation and integration respectively.
And in psychological terms, at the transcendental level of contemplative intuitive awareness, one can approach the special situation, where experience now becomes so spiritually rarefied and transparent, that one can no longer distinguish (differentiated) phenomena of form from the continued (integrated) experience of emptiness.
And this concurs when - quite literally - experience of dimensions turns full circle.
As I have repeatedly emphasised, standard analytic interpretation is of a linear (1-dimensional) nature where the dimensional aspect, which - relatively - is of a qualitative nature - is reduced in a merely quantitative manner.
This is especially true with respect to Mathematics, where for example the concept of "number" is reduced in a merely quantitative manner as applying directly to all number perceptions!
So the whole point of contemplative type awareness is to gradually move from this restricted linear interpretation of concepts to a fully circular holistic appreciation (which is completely paradoxical in dualistic terms).
With respect to the unit circle, its circumference is given as 2π. If we were to consider now the "imaginary" equivalent - which corresponds to the pure holistic appreciation of the circle - the circumference would be 2iπ.
So the dimensional value (of the fundamental Euler Identity) relates to the purely holistic appreciation of number (which is completely circular - and paradoxical - in dualistic terms).
In other words it represents the holistic intuitive extreme of awareness (as purely circular) in contrast to the corresponding analytic rational extreme (as purely linear)..
Therefore in holistic terms, the fundamental Euler Identity defines in a truly remarkable and deeply mysterious manner the phenomenon of spiritual union, which however can only be approached - rather than fully attained - in an experiential dynamic interactive manner.
And this union, as the holistic expression of "1", is now indistinguishable from the corresponding experience of nothingness (or emptiness) as the holistic expression of "0".
So once again, one may appreciate how we have come full circle.
In conventional mathematical terms (at the analytic level of interpretation, 1 is clearly separated from 0 (in a quantitative manner).
However now in corresponding holistic terms, 1 is fully interdependent with 0 (in a qualitative manner).
So the (fundamental) Euler Identity - when properly appreciated in a dynamic interactive manner - represents the mysterious marriage, where both the analytic and holistic interpretation of its symbols ultimately dissolves in an experience that is truly ineffable.
Comments
Post a Comment