As stated so often when properly understood as the very nature of experience, Mathematics has both quantitative and qualitative aspects in dynamic interaction with each other. So from this perspective one does not understand symbols in static terms as absolute forms, but rather in dynamic interactive terms as symbols of transformation!
I will now attempt to illustrate one extremely important example of this new understanding (with intimate parallels to the nature of psychological development).
As befits the dynamic approach, in a number expression such as a^b, if we designate the base number a in quantitative terms - the dimensional number b is - relatively of a qualitative nature.
And it is this interaction as between quantitative and qualitative aspects that can then be used to explain how the nature of number itself evolves to "higher" forms.
So for example if we start with the simplest of prime numbers 2 and then raise this to 2 (i.e. 2^2), the result is a natural number integer (which is not prime).
So we can see how the very nature of the number has thereby changed.
Now to obtain the appropriate corresponding situation in psychological terms, we must remember that all experience necessarily entails the dynamic interaction of perceptions and concepts which are - relatively - quantitative and qualitative with respect to each other.
Therefore if we designate the perceptions as quantitative (which is the standard approach in Conventional Science) then corresponding concepts - are relatively - of a qualitative nature. (Of course in formal scientific and mathematical interpretation, concepts are misleadingly also treated as quantitative leading to a merely reduced interpretation of subsequent dynamics).
So in other words an infant at the primitive stage of development initially will develop primitive perceptions and later primitive concepts (both of a transient nature) . It is then in the successful fusion of such perceptions and concepts that development reaches the next natural stage (i.e. where natural objects with a greater degree of constancy emerge in experience).
So in this sense we see how psychological development - in line with the nature of number - evolves from a prime to a natural stage. So what we are really showing here is how number possesses both a qualitative and quantitative relationship to order (with the qualitative aspect of number directly relevant to ordering the various stages of development).
Now switching back to the quantitative nature of natural numbers, the next development is to recognise a negative as well as positive identity in the generation of all the integers.
Then when we raise - as for example the number 2 to - 1 i.e. 2^(- 1) a remarkable number transformation takes place whereby we generate a new type of rational number (i.e. a fraction).
Now again there are direct correspondents on the psychological side. The negative integers here refer to the increasing ability of the child to hold objects in memory even when temporarily absent (giving them a greater absolute constancy).
This in turn enables the child to experience concepts in negative terms i.e. where they can be held in memory as a basis for organising experience when dealing with corresponding perceptions.
And this is the very basis of rational ability whereby both object perceptions and concepts can be repeatedly subdivided in analytical terms and rearranged into new aggregate wholes.
And once again Conventional Science (and Mathematics) is defined by the specialisation to the nth degree of such ability.
However now we come to the interesting part!
If we take again the simple number 2 and now raise it to its reciprocal fraction 1/2, we generate a new type of number that is (algebraically) irrational in nature. Indeed this is the famed square root of 2 that caused so much difficulty for the Pythagoreans many years ago!.
The psychological correspondent implies that if we now try to dynamically relate rational perceptions with rational concepts, which is the very nature of scientific and mathematical experience, we should generate a new (qualitative) form of irrational understanding in holistic terms!
The obvious question then arises as to why this does not typically happen and the answer is - as we have seen - due to the misleading manner in which both perceptions and concepts are interpreted in formal terms with respect merely to their quantitative aspect. Therefore qualitative understanding, in the form of supporting intuition, remains of a merely implicit nature that is quickly reduced in rational terms.
So here we are giving a demonstration using the simplest of numbers to highlight an extremely important limitation of standard scientific and mathematical practice.
Thus even from the quantitative perspective, we cannot properly understood the nature of an irrational number without likewise also explicitly recognising a qualitative aspect! Now the Pythagoreans recognised this and their consternation arose from the inability to properly explain this qualitative aspect. However such appreciation has subsequently become lost through a greatly reduced - merely quantitative - interpretation of mathematical symbols in Western culture.
Therefore once again, because the (qualitative) dimensional nature of number relates holistically to the nature of both the physical and psychological dimensions, we must thereby recognise that time (and space) can necessarily be given an (algebraic) irrational meaning.
Now the square root of 2 has two irrational roots i.e. that are positive and negative with respect to each other.
I have attempted before to explain the nature of the corresponding experience of time and space with respect to the appreciation of a flower such as a rose.
Now formerly one would have largely experienced this object as largely separate and discrete in experience. Initially this would be of a linear (1-dimensional) nature where the rose is viewed as a separate object in space and time. Then later with 2-dimensional understanding, greater subtlety would pertain with an appreciation of both mental and object perception of the rose as - relatively - external and internal with respect to each other.
These two directions would equally apply with this new irrational appreciation. However in relation to both the external and internal directions, a mixture of rational and intuitive appreciation would now be involved. Thus from the rational perspective, one would still appreciate the rose as a finite discrete object; however from the holistic intuitive perspective, one would recognise its continuity with all creation (as an archetype) whereby it radiates an infinite quality.
So quite simply the irrational nature of time and space arises when both discrete (finite) and continuous (infinite) aspects are so intertwined.
This category of irrational dimensions likewise has deep implications for the true nature of sub-atomic particles, where again total independence (from other particles) does not strictly pertain, but rather a hybrid existence combining both discrete and continuous aspects.
I will now attempt to illustrate one extremely important example of this new understanding (with intimate parallels to the nature of psychological development).
As befits the dynamic approach, in a number expression such as a^b, if we designate the base number a in quantitative terms - the dimensional number b is - relatively of a qualitative nature.
And it is this interaction as between quantitative and qualitative aspects that can then be used to explain how the nature of number itself evolves to "higher" forms.
So for example if we start with the simplest of prime numbers 2 and then raise this to 2 (i.e. 2^2), the result is a natural number integer (which is not prime).
So we can see how the very nature of the number has thereby changed.
Now to obtain the appropriate corresponding situation in psychological terms, we must remember that all experience necessarily entails the dynamic interaction of perceptions and concepts which are - relatively - quantitative and qualitative with respect to each other.
Therefore if we designate the perceptions as quantitative (which is the standard approach in Conventional Science) then corresponding concepts - are relatively - of a qualitative nature. (Of course in formal scientific and mathematical interpretation, concepts are misleadingly also treated as quantitative leading to a merely reduced interpretation of subsequent dynamics).
So in other words an infant at the primitive stage of development initially will develop primitive perceptions and later primitive concepts (both of a transient nature) . It is then in the successful fusion of such perceptions and concepts that development reaches the next natural stage (i.e. where natural objects with a greater degree of constancy emerge in experience).
So in this sense we see how psychological development - in line with the nature of number - evolves from a prime to a natural stage. So what we are really showing here is how number possesses both a qualitative and quantitative relationship to order (with the qualitative aspect of number directly relevant to ordering the various stages of development).
Now switching back to the quantitative nature of natural numbers, the next development is to recognise a negative as well as positive identity in the generation of all the integers.
Then when we raise - as for example the number 2 to - 1 i.e. 2^(- 1) a remarkable number transformation takes place whereby we generate a new type of rational number (i.e. a fraction).
Now again there are direct correspondents on the psychological side. The negative integers here refer to the increasing ability of the child to hold objects in memory even when temporarily absent (giving them a greater absolute constancy).
This in turn enables the child to experience concepts in negative terms i.e. where they can be held in memory as a basis for organising experience when dealing with corresponding perceptions.
And this is the very basis of rational ability whereby both object perceptions and concepts can be repeatedly subdivided in analytical terms and rearranged into new aggregate wholes.
And once again Conventional Science (and Mathematics) is defined by the specialisation to the nth degree of such ability.
However now we come to the interesting part!
If we take again the simple number 2 and now raise it to its reciprocal fraction 1/2, we generate a new type of number that is (algebraically) irrational in nature. Indeed this is the famed square root of 2 that caused so much difficulty for the Pythagoreans many years ago!.
The psychological correspondent implies that if we now try to dynamically relate rational perceptions with rational concepts, which is the very nature of scientific and mathematical experience, we should generate a new (qualitative) form of irrational understanding in holistic terms!
The obvious question then arises as to why this does not typically happen and the answer is - as we have seen - due to the misleading manner in which both perceptions and concepts are interpreted in formal terms with respect merely to their quantitative aspect. Therefore qualitative understanding, in the form of supporting intuition, remains of a merely implicit nature that is quickly reduced in rational terms.
So here we are giving a demonstration using the simplest of numbers to highlight an extremely important limitation of standard scientific and mathematical practice.
Thus even from the quantitative perspective, we cannot properly understood the nature of an irrational number without likewise also explicitly recognising a qualitative aspect! Now the Pythagoreans recognised this and their consternation arose from the inability to properly explain this qualitative aspect. However such appreciation has subsequently become lost through a greatly reduced - merely quantitative - interpretation of mathematical symbols in Western culture.
Therefore once again, because the (qualitative) dimensional nature of number relates holistically to the nature of both the physical and psychological dimensions, we must thereby recognise that time (and space) can necessarily be given an (algebraic) irrational meaning.
Now the square root of 2 has two irrational roots i.e. that are positive and negative with respect to each other.
I have attempted before to explain the nature of the corresponding experience of time and space with respect to the appreciation of a flower such as a rose.
Now formerly one would have largely experienced this object as largely separate and discrete in experience. Initially this would be of a linear (1-dimensional) nature where the rose is viewed as a separate object in space and time. Then later with 2-dimensional understanding, greater subtlety would pertain with an appreciation of both mental and object perception of the rose as - relatively - external and internal with respect to each other.
These two directions would equally apply with this new irrational appreciation. However in relation to both the external and internal directions, a mixture of rational and intuitive appreciation would now be involved. Thus from the rational perspective, one would still appreciate the rose as a finite discrete object; however from the holistic intuitive perspective, one would recognise its continuity with all creation (as an archetype) whereby it radiates an infinite quality.
So quite simply the irrational nature of time and space arises when both discrete (finite) and continuous (infinite) aspects are so intertwined.
This category of irrational dimensions likewise has deep implications for the true nature of sub-atomic particles, where again total independence (from other particles) does not strictly pertain, but rather a hybrid existence combining both discrete and continuous aspects.
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