I will finish this series of blog entries with a brief comparison as between the standard mathematical notions of differentiation and integration and the corresponding psychological notions as used in a developmental context.
It is illuminating in this context to start with perhaps the best known simple example of differentiation in mathematics i.e. where y = x2. So here dy/dx (i.e. the differential) = 2x.
What has happened in effect is that the number "2", which initially was designed to represent a dimensional notion now has been in effect reduced to its base number meaning.
(Once again with reference to the general expression ab, a represent the base and b the dimensional number respectively).
What is fascinating here, as I am once again demonstrating at present on my Riemann Hypothesis blog, is that both dimensional and base numbers are, in dynamic interactive terms, qualitative and quantitative with respect to each
Therefore if as in conventional terms, we view "2" as a (base) number quantity, then - relatively - "2" when then used to represent a dimensional number, is of a qualitative nature. In other words "2" as dimension, strictly relates to the quality of "twoness" which all numbers - in a 2-dimensional context - share. So for example the areas of different fields share this quality of "twoness" (in being of a 2-dimensional nature) which thereby enables us to relate their quantitative measurements.
Now in psychological terms the process of differentiation is fundamentally of a similar nature.
Thus when we differentiate, we thereby reduce the qualitative nature of experience (that expresses the relational aspect of interdependence) in a merely quantitative manner.
So for example, when differentiated, the objects of experience - especially in a scientific context - appear to assume an independent identity (as separate from everything else).
In this way, the processes of differentiation in both a holistic mathematical (where dynamic interaction is recognised) and psychological developmental sense are similar, in that they both entail the reduction of the qualitative (interdependent) aspect of experience in a quantitative (independent) manner.
Just as integration in Mathematics can be viewed as the reverse of differentiation, likewise it is the same in psychological terms.
Therefore in mathematical terms when we integrate, in reverse fashion, 2x, we obtain x2. So now we have transformed, as it were, from the meaning of "2" as representing a base quantity, to the corresponding meaning of "2" as representing a dimensional quality!
Likewise in psychological terms, when we integrate in experience, we move from the notion of separate independent phenomena to the corresponding qualitative notion of interdependence. So integration in this sense always implicitly implies the notion of qualitative interdependence.
A special case attaches in mathematical terms to the differentiation of ex, which remains unchanged.
It equally means that that reverse procedure of integrating ex likewise will leave it unchanged.
This therefore conveys a special meaning on ex in holistic mathematical terms, as it entails that it is likewise invariant with respect to differentiation and integration.
This implies a greatly advanced state of spiritual contemplation, where dualistic phenomena are now so refined that they immediately dissolve in experience (and do not even appear therefore to arise). This is then compatible with a continual nondual spiritual presence (representing the integral dimension). In this way both dual and nondual aspects remain fully compatible with each other. In other words the differentiated aspect of experience becomes completely harmonised with the integral aspect.
Then for a particular value of x = 2iπ, ex = 1. In other words, when the purely circular qualitative notion of dimension is attained (approaching 0) in experience, emptiness (as 0) becomes compatible with the unity of all form (as 1).
In this sense, in holistic mathematical terms, the Euler Identity, which is widely regarded as the most remarkable equation in Mathematics, can be associated with the great spiritual mystery, where form and emptiness coincide in one's spiritual realisation.
It is illuminating in this context to start with perhaps the best known simple example of differentiation in mathematics i.e. where y = x2. So here dy/dx (i.e. the differential) = 2x.
What has happened in effect is that the number "2", which initially was designed to represent a dimensional notion now has been in effect reduced to its base number meaning.
(Once again with reference to the general expression ab, a represent the base and b the dimensional number respectively).
What is fascinating here, as I am once again demonstrating at present on my Riemann Hypothesis blog, is that both dimensional and base numbers are, in dynamic interactive terms, qualitative and quantitative with respect to each
Therefore if as in conventional terms, we view "2" as a (base) number quantity, then - relatively - "2" when then used to represent a dimensional number, is of a qualitative nature. In other words "2" as dimension, strictly relates to the quality of "twoness" which all numbers - in a 2-dimensional context - share. So for example the areas of different fields share this quality of "twoness" (in being of a 2-dimensional nature) which thereby enables us to relate their quantitative measurements.
Now in psychological terms the process of differentiation is fundamentally of a similar nature.
Thus when we differentiate, we thereby reduce the qualitative nature of experience (that expresses the relational aspect of interdependence) in a merely quantitative manner.
So for example, when differentiated, the objects of experience - especially in a scientific context - appear to assume an independent identity (as separate from everything else).
In this way, the processes of differentiation in both a holistic mathematical (where dynamic interaction is recognised) and psychological developmental sense are similar, in that they both entail the reduction of the qualitative (interdependent) aspect of experience in a quantitative (independent) manner.
Just as integration in Mathematics can be viewed as the reverse of differentiation, likewise it is the same in psychological terms.
Therefore in mathematical terms when we integrate, in reverse fashion, 2x, we obtain x2. So now we have transformed, as it were, from the meaning of "2" as representing a base quantity, to the corresponding meaning of "2" as representing a dimensional quality!
Likewise in psychological terms, when we integrate in experience, we move from the notion of separate independent phenomena to the corresponding qualitative notion of interdependence. So integration in this sense always implicitly implies the notion of qualitative interdependence.
A special case attaches in mathematical terms to the differentiation of ex, which remains unchanged.
It equally means that that reverse procedure of integrating ex likewise will leave it unchanged.
This therefore conveys a special meaning on ex in holistic mathematical terms, as it entails that it is likewise invariant with respect to differentiation and integration.
This implies a greatly advanced state of spiritual contemplation, where dualistic phenomena are now so refined that they immediately dissolve in experience (and do not even appear therefore to arise). This is then compatible with a continual nondual spiritual presence (representing the integral dimension). In this way both dual and nondual aspects remain fully compatible with each other. In other words the differentiated aspect of experience becomes completely harmonised with the integral aspect.
Then for a particular value of x = 2iπ, ex = 1. In other words, when the purely circular qualitative notion of dimension is attained (approaching 0) in experience, emptiness (as 0) becomes compatible with the unity of all form (as 1).
In this sense, in holistic mathematical terms, the Euler Identity, which is widely regarded as the most remarkable equation in Mathematics, can be associated with the great spiritual mystery, where form and emptiness coincide in one's spiritual realisation.
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