I have already pointed to the fact that - properly understood - every number expression represents a dynamic interaction as between a a base quantity and a dimensional number (that is relatively of a qualitative nature).
So what we might refer to in conventional (Type 1) terms as the number quantity 2, more accurately is expressed as 2^1 (where 2 is quantitative and 1 - relatively - of a qualitative holistic nature).
However because the very nature of linear (1-dimensional) understanding is to reduce qualitative to quantitative type meaning, from a Type 1 perspective, interpretation of numbers is invariably reduced in a mere quantitative manner.
However when correctly appreciated in holistic Type 2 terms, number expressions have an intimate bearing on the interpretation of all the main stages on the spectrum of psychological development.
As we know number is the best means we have for quantitative ordering in experience. In fact such ordering is synonymous with number appreciation. And seen from this Type 1 perspective we have various number types such as prime, natural, rational, irrational etc.
What is not recognised is that the same number types have a profound relevance for the qualitative interpretation of development.
In fact the holistic structure of each of the main levels of psychological development can be precisely matched to expressions entailing the main number types. Furthermore we can subdivide each "number" level into three coherent stages showing how the transformation into the next level on the spectrum takes place.
So, three main stages are involved with respect to unfolding of each level.
Firstly we have the unfolding of specific perceptions (characteristic of the level in question). In the second stage this gives way to unfolding of the more generalised conceptual understanding (again characterising the level). Now the relationship of such perceptions and concepts is as quantitative and qualitative with respect to each other!
Then in the third stage increasing dynamic interaction takes place as between perceptions and concepts causing a transformation to a new level (whose structure in turn corresponds with a new distinctive number type).
Development commences from a completely undifferentiated state where form is indistinguishable from emptiness. This relates in qualitative terms to - what I refer to as - the original numbers. So unity (1) is initially indistinguishable from nothingness (0). Then when development commences the first initial differentiation takes place leading to the birth of duality (2) in experience.
Though we still have great confusion, in a qualified sense we can identify three stages (1) where duality takes place with respect to incipient perception (2) when duality now takes place with respect to the incipient formation of concepts and finally where some level of interaction of both takes place. This can be identified with the archaic level.
All going well, the dynamic interaction of understanding that is quantitative and qualitative with respect to each other, leads to a transformation to the next level.
This level is then identified in terms of the holistic interpretation of the prime numbers that entails an intimate relationship of both conscious and unconscious aspects of experience. Indeed we can accurately use the word primitive to refer to such experience where holistic collective notions (pertaining to the unconscious) are continually confused with distinct specific notions (pertaining to the conscious). We can identify this with the magical level.
Again in a qualified manner we have the unfolding of three stages firstly with respect to specific perceptions of a primitive kind, then more general concepts (of a primitive nature) and finally the growing dynamic interaction of both perceptions and concepts.
When we raise a prime number to a prime number dimension in quantitative terms, a transformation takes place in that we derive a new (composite) natural number.
Likewise in holistic psychological terms when we relate specific perceptions of a primitive quantitative nature to their general concepts (that relatively are qualitative in nature) a transformation to a new level of understanding takes place which correlates with the holistic nature of natural numbers.
One of the key characteristics of natural numbers is that numbers are whole (and not yet divided into fractional components).
Likewise at this level (which relates to mythic development) wholes - especially with respect to concepts - cannot be properly broken into parts. Put another way, abstract ability is not yet sufficiently developed so that objects are still given a (whole) personality. Only later when the conscious aspect is further differentiated from the unconscious does true abstract ability of an impersonal nature properly unfold.
One fascinating feature of this level is that one learns to - temporarily - negate both perceptions and concepts (while holding them in memory). This is made possible through the greater level of phenomenal constancy characteristic of this level.
This literally means therefore that such dynamic negation of concepts in experience relates to negative rather than positive dimensions (in number terms).
Once again we can identify three stages 1) where natural perceptions (and their corresponding temporary negation) takes place 2) natural concepts (and their temporary negation occurs and finally (3) the growing interaction of both perceptions and concepts in both a positive and negative sense.
So during this level the natural numbers give way to the integers (where numbers can be both positive and negative) in holistic terms. And the scientific structure of the level is based on such understanding.
When we look at the simple number expression 4^ (- 1) we have an integer that is raised to the negative of 1 (as dimensional power). And this number expression in Type 1 terms leads to the generation of a fraction i.e. 1/4.
Likewise in development, when whole perceptions are related to concepts (that can be temporarily negated in experience) we generate analytic understanding of a fractional kind.
This leads to an important transformation in development whereby the mythic gives rise to the rational stages (that are so relevant in conventional scientific understanding).
What is remarkable here is that appropriate holistic mathematical interpretation implies that even the simplest rational task implies the ability to temporarily negate the linear (1-dimensional) mode of understanding.
Let us illustrate with respect to the task of cutting a cake into four slices.
Now, the cake obviously represents a whole and each slice another unique whole object. So the ability to recognise each slice also as a part requires the temporary negation of its whole status. This then enables the link of the slice to the whole status of the cake whereby it is now recognised as a part of that whole. So, initially each slice is posited as a whole perception and then negated (with respect to this status) enabling it to be thereby understood as a part of the greater whole of the cake.
So here we establish the link between whole and part perceptions which implicitly requires the ability to temporarily negate their respective identities while still holding them in memory. And the same ability also implicitly develops with respect to concepts. For example one can initially recognises a specific number such as 2 as a perception. However the recognition that this number belongs to the universal number class i.e concept of number requires the temporary ability to negate its specific status in recognition of its general identity. Then the reverse procedure of once again recognising 2 as a specific perception in turn requires the temporary negation of its number concept. Therefore - though still operating at a merely implicit level the continual dynamic positing and negating of both perceptions (as quantities) and concepts (in qualitative terms) takes place. Once again the (linear) rational level - which dominates conventional understanding of Mathematics and Science - has 3 stages.
The first corresponds to the rational appreciation of perceptions in the ability to break object perceptions into smaller parts and rearranging them again in composite wholes. This is generally referred to in Piagetian terms as conop.
The second stage corresponds to the ability to break general universal concepts into sub-categories and then synthesise them with respect to the original concepts. In Piagetian terms this is formop.
Finally the third stage entails the growing interaction of rational perceptions and concepts. When suitably refined - though this is not generally the case - this leads to a dynamic intuitive based form of rational understanding i.e. vision-logic.
Another remarkable transformation occurs in Type 1 quantitative terms when a rational number is raised to a fractional dimension (as power) in that an irrational number results.
For example in the well known case when 2 is raised to 1/2 we obtain the square root of 2 = 1.4142... which is an irrational number.
However, in Western culture there is very little evidence of the sustained growth of further more advanced stages of understanding (beyond the rational level).
One important reason for this is quite revealing. The source of the dynamic transformation that takes place in the emergence of each new psychological level results from the interaction of perceptions and concepts, that are implicitly experienced as quantitative and qualitative with respect to each other. This enables therefore a fruitful interaction of conscious and unconscious to emerge.
In formal terms the interpretations of Mathematics and Science directly reduce the qualitative in quantitative terms. As this leads to emphasis solely on the conscious aspect of experience, a considerable diminution in true holistic type qualitative understanding can result. For example Holistic (Type 2) Mathematics, which I am outlining in these contributions, is totally unrecognised at present by the mathematical profession!
In experiential terms therefore this can reduce the unconscious intuitive aspect of understanding (within which true qualitative appreciation is embodied) to such a significant degree that sustained progress beyond the rational level is not possible.
In other words the specialisation of rational understanding, which is so characteristic of Western culture can greatly reduce the role of the unconscious in experience with the consequence that dynamic transformation beyond the rational level is significantly impeded.
Wednesday, October 5, 2011
Sunday, October 2, 2011
Multiplication and Addition
The key problem in reconciling addition with multiplication is that they represent mathematical processes that are quantitative and qualitative with respect to each other. And as Conventional (type 1) Mathematics is based on a merely reduced quantitative approach this creates enormous difficulties in properly appreciating the nature of the problem.
As we have seen properly we have two number systems that are quantitative and qualitative with respect to each other.
1) In the conventional (Type 1) system, the natural numbers 1, 2, 3, 4, 5,.... for example respect quantities are defined with respect to a (default) dimensional value of 1.
So written in full, this system is represented as:
1^1, 2^1, 3^1, 4^1, 5^1,........
2) In the unrecognised (Type 2) system, the same natural numbers 1, 2, 3, 4, 5,.... represent qualitative dimension that are defined with respect to a (default) base quantity of 1.
So written in full, this alternative system is represented as:
1^1, 1^2, 1^3, 1^4, 1^5,........
Now with respect to the first system when we add two numbers, say, 2 + 3,
this is fully represented as
(2^1) + (3^1) = 5^1
However when we add the same two numbers, 2 + 3, with respect to the second system, this is fully represented as
(1^2) * (1^3) = 1^5.
So whereas addition of these two numbers is involved with respect to the first (quantitative) system, multiplication of the same two numbers (now representing dimensions) is entailed with respect to the second (qualitative) system.
This clearly entails that whereas pure addition (i.e. with respect to numbers that are all defined with respect to 1 as default dimension) is of a direct quantitative nature. Pure multiplication (i.e. with respect to numbers that are all defined with respect to 1 as default base) is by contrast of a direct qualitative nature.
Therefore we cannot ultimately hope to reconcile addition and multiplication without equal recognition of both Type 1 (quantitative) and Type 2 (qualitative) numerical systems.
In practice therefore where non-unitary values are given to both (quantitative) base and (qualitative) dimensional numbers, Type 3 Mathematics (representing the coherent interaction of both Type 1 and Type 2 systems) must be used for comprehensive understanding.
A further problem relates to the reconciliation of multiplication and exponentiation.
If multiplication is now treated in a (Type 1) quantitative manner, then
2 * 3 for example is represented as (2^1) * (3^1) = 6^1.
However with respect to the second (Type 2) dimensional system 2 * 3 is represented as
(1^2)^3
So multiplication with respect to the two numbers in the Type 1 (quantitative) system represents exponentiation with respect to the same two numbers in the second.
And just as (2^1) * (3^1) = (3^1) * (2^1) (with respect to the first)
(1^2)^3 = (1^3)^2 (with respect to the second).
As we have seen properly we have two number systems that are quantitative and qualitative with respect to each other.
1) In the conventional (Type 1) system, the natural numbers 1, 2, 3, 4, 5,.... for example respect quantities are defined with respect to a (default) dimensional value of 1.
So written in full, this system is represented as:
1^1, 2^1, 3^1, 4^1, 5^1,........
2) In the unrecognised (Type 2) system, the same natural numbers 1, 2, 3, 4, 5,.... represent qualitative dimension that are defined with respect to a (default) base quantity of 1.
So written in full, this alternative system is represented as:
1^1, 1^2, 1^3, 1^4, 1^5,........
Now with respect to the first system when we add two numbers, say, 2 + 3,
this is fully represented as
(2^1) + (3^1) = 5^1
However when we add the same two numbers, 2 + 3, with respect to the second system, this is fully represented as
(1^2) * (1^3) = 1^5.
So whereas addition of these two numbers is involved with respect to the first (quantitative) system, multiplication of the same two numbers (now representing dimensions) is entailed with respect to the second (qualitative) system.
This clearly entails that whereas pure addition (i.e. with respect to numbers that are all defined with respect to 1 as default dimension) is of a direct quantitative nature. Pure multiplication (i.e. with respect to numbers that are all defined with respect to 1 as default base) is by contrast of a direct qualitative nature.
Therefore we cannot ultimately hope to reconcile addition and multiplication without equal recognition of both Type 1 (quantitative) and Type 2 (qualitative) numerical systems.
In practice therefore where non-unitary values are given to both (quantitative) base and (qualitative) dimensional numbers, Type 3 Mathematics (representing the coherent interaction of both Type 1 and Type 2 systems) must be used for comprehensive understanding.
A further problem relates to the reconciliation of multiplication and exponentiation.
If multiplication is now treated in a (Type 1) quantitative manner, then
2 * 3 for example is represented as (2^1) * (3^1) = 6^1.
However with respect to the second (Type 2) dimensional system 2 * 3 is represented as
(1^2)^3
So multiplication with respect to the two numbers in the Type 1 (quantitative) system represents exponentiation with respect to the same two numbers in the second.
And just as (2^1) * (3^1) = (3^1) * (2^1) (with respect to the first)
(1^2)^3 = (1^3)^2 (with respect to the second).
Saturday, October 1, 2011
Nature of Number as Dimension
I have been thinking again in a deeper manner regarding the nature of number representing a dimension (or power).
Let us start with the convenient (default) case of 1.
Now clearly 1 can represent a unit quantity. So implicit therefore in the recognition of any specific object is the number 1 (as an actual finite quantity).
However when used to represent a dimension the number 1 takes on a distinctive holistic meaning (in a potential infinite manner).
So for example if we attempt to represent the number system on a straight line this automatically presumes a linear (1-dimensional) background that is - potentially - infinite.
Therefore though we can use the same symbol 1 to represent a base unit quantity or alternatively the linear dimension (within which such a number is expressed) clearly the meaning is very different in each case.
In the former case 1 represents a specific finite notion that is inherently quantitative in nature; in the latter case it represents a holistic - potentially - infinite notion that is of an inherently qualitative nature.
And as all numbers representing quantitative values must implicitly be expressed with respect to a corresponding number dimension (with the default value = 1), then every number expression - when properly appreciated - necessarily entails a relationship between two aspects which are quantitative and qualitative with respect to each other.
As we have seen, the default dimensional state of a number is 1. And as it is the very nature of linear (1-dimensional) understanding to reduce the qualitative aspect to the quantitative, this means in effect that the qualitative notion of number is effectively always ignored in Conventional (Type 1) Mathematics.
This also causes an important difficulty when dealing with dimensional values (other than 1) which are inevitably treated in a reduced linear manner.
For example 2-dimensional reality would relate to a potentially infinite plane (within which a 2-dimensional object can be placed). However because the qualitative nature of logical understanding remains 1-dimensional, in Type 1 Maths this entails considering the plane as (linearly) extended in two directions that are horizontal and vertical with respect to each other.
However once we depart from 1-dimensional qualitative interpretation, the true nature of dimension is revealed to be of a circular nature.
In fact - when again appropriately understood - this is actually demonstrated in Type 1 Mathematics through the notion of roots.
If we obtain the two roots of unity, they will lie as equidistant points on the circle of unit radius (in the complex plane). Now we can of course in quantitative terms recognise these as + 1 and - 1 respectively. However if we are to give an appropriate 2-dimensional interpretation in qualitative terms (as is appropriate) then we require a logical means of combining + 1 and - 1 as being both true. Now this is done through the paradoxical (both/and) logic of the complementary opposites where each pole like the left and right turns on a road has a merely relative validity.
So we can see here an important inverse relationship as between the 2 quantitative roots of the number 1 and the corresponding 2-dimensional qualitative interpretation (with which they are consistent).
Strictly speaking we do not have 2 roots of 1 i.e 1^1.
- 1 is indeed the (unique) square root of 1. + 1 is however the (unique) square root of 1^2. And 1^1 and 1^2 relate to distinct qualitative numbers (representing dimensions).
Now in a comprehensive appreciation, an even more subtle dynamic interactive understanding is required. Thus when we start with the base - say of 1^2 - here 1 is quantitative and 2 as dimension - relatively - qualitative. However equally the base number 1 can be given a quantitative meaning with the dimensional number 2, thereby in relative terms quantitative. For example all natural logs (representing numbers as powers) clearly have a quantitative meaning as do for example the values of s in the Riemann Zeta Function!
So in relative terms, where the base number has a qualitative meaning this implies that the conceptual nature of the number is highlighted in understanding.
Thus 1 for example can be seen in quantitative terms as a number perception (in relation to the qualitative concept of number). However in reverse terms it can be seen as the number concept (to which 1 relates). So in experiential terms both aspects necessarily interact with all numbers thereby possessing both quantitative (specific) and qualitative (holistic) aspects.
However this poses severe limitations on a mathematical approach that solely recognises the quantitative aspect. And this reduced interpretation is what we misleadingly refer to as Mathematics.
More properly it refers to Type 1 Mathematics. So enormous scope remains for the proper development of Type 2 Mathematics (focussing on the neglected qualitative aspect) and Type 3 Mathematics (where both aspects - quantitative and qualitative - are coherently related).
Let us start with the convenient (default) case of 1.
Now clearly 1 can represent a unit quantity. So implicit therefore in the recognition of any specific object is the number 1 (as an actual finite quantity).
However when used to represent a dimension the number 1 takes on a distinctive holistic meaning (in a potential infinite manner).
So for example if we attempt to represent the number system on a straight line this automatically presumes a linear (1-dimensional) background that is - potentially - infinite.
Therefore though we can use the same symbol 1 to represent a base unit quantity or alternatively the linear dimension (within which such a number is expressed) clearly the meaning is very different in each case.
In the former case 1 represents a specific finite notion that is inherently quantitative in nature; in the latter case it represents a holistic - potentially - infinite notion that is of an inherently qualitative nature.
And as all numbers representing quantitative values must implicitly be expressed with respect to a corresponding number dimension (with the default value = 1), then every number expression - when properly appreciated - necessarily entails a relationship between two aspects which are quantitative and qualitative with respect to each other.
As we have seen, the default dimensional state of a number is 1. And as it is the very nature of linear (1-dimensional) understanding to reduce the qualitative aspect to the quantitative, this means in effect that the qualitative notion of number is effectively always ignored in Conventional (Type 1) Mathematics.
This also causes an important difficulty when dealing with dimensional values (other than 1) which are inevitably treated in a reduced linear manner.
For example 2-dimensional reality would relate to a potentially infinite plane (within which a 2-dimensional object can be placed). However because the qualitative nature of logical understanding remains 1-dimensional, in Type 1 Maths this entails considering the plane as (linearly) extended in two directions that are horizontal and vertical with respect to each other.
However once we depart from 1-dimensional qualitative interpretation, the true nature of dimension is revealed to be of a circular nature.
In fact - when again appropriately understood - this is actually demonstrated in Type 1 Mathematics through the notion of roots.
If we obtain the two roots of unity, they will lie as equidistant points on the circle of unit radius (in the complex plane). Now we can of course in quantitative terms recognise these as + 1 and - 1 respectively. However if we are to give an appropriate 2-dimensional interpretation in qualitative terms (as is appropriate) then we require a logical means of combining + 1 and - 1 as being both true. Now this is done through the paradoxical (both/and) logic of the complementary opposites where each pole like the left and right turns on a road has a merely relative validity.
So we can see here an important inverse relationship as between the 2 quantitative roots of the number 1 and the corresponding 2-dimensional qualitative interpretation (with which they are consistent).
Strictly speaking we do not have 2 roots of 1 i.e 1^1.
- 1 is indeed the (unique) square root of 1. + 1 is however the (unique) square root of 1^2. And 1^1 and 1^2 relate to distinct qualitative numbers (representing dimensions).
Now in a comprehensive appreciation, an even more subtle dynamic interactive understanding is required. Thus when we start with the base - say of 1^2 - here 1 is quantitative and 2 as dimension - relatively - qualitative. However equally the base number 1 can be given a quantitative meaning with the dimensional number 2, thereby in relative terms quantitative. For example all natural logs (representing numbers as powers) clearly have a quantitative meaning as do for example the values of s in the Riemann Zeta Function!
So in relative terms, where the base number has a qualitative meaning this implies that the conceptual nature of the number is highlighted in understanding.
Thus 1 for example can be seen in quantitative terms as a number perception (in relation to the qualitative concept of number). However in reverse terms it can be seen as the number concept (to which 1 relates). So in experiential terms both aspects necessarily interact with all numbers thereby possessing both quantitative (specific) and qualitative (holistic) aspects.
However this poses severe limitations on a mathematical approach that solely recognises the quantitative aspect. And this reduced interpretation is what we misleadingly refer to as Mathematics.
More properly it refers to Type 1 Mathematics. So enormous scope remains for the proper development of Type 2 Mathematics (focussing on the neglected qualitative aspect) and Type 3 Mathematics (where both aspects - quantitative and qualitative - are coherently related).
Wednesday, September 21, 2011
More on Euler Identity
I have used the Euler Identity in a holistic mathematical fashion (Type 2 Mathematics) to demonstrate three stages of specialised contemplative development (preceding the full radial unfolding of stages).
This can however be given a more precise expression.
So we now define e^(k*2*i*pi) = 1^x (where k = 1, – 1 or 0 and x = 1, 0 and both 1 and 0 respectively.
So when k = 1,
1) e^(k*2*i*pi)= e^(2*i* pi) = 1^1
In holistic terms this corresponds to the rational linear (i.e. 1-dimensional) appreciation of (mystical) union.
Now with the 1st stage (of specialised contemplative development) a slight imbalance remains whereby appreciation remains unduly transcendent in nature.
In other words in emphasising the transcendent nature of spiritual reality as inherently empty and thereby beyond all notions of form, one to a degree still represses the corresponding immanent nature of such reality as prior and thereby inherent in all form.
This in effect leads to a subtle lingering rational attachment (of a necessarily linear nature) to the notion of unity (as phenomenal form) thereby preventing full realisation of complementary intuitive recognition (that is literally non-dimensional as empty of all form)
2) So when k = – 1,
e^(k*2*i*pi)= e^(- 2*i*pi) = 1^0
This is so as e^(– 2*i*pi) = 1/{e^(2*i*pi)} = (1^1)/(1^1) = 1^(1 – 1) = 1^0.
With the 2nd stage of such specialised development, this remaining lingering attachment to the notion of union (as a phenomenal point) is gradually negated.
Therefore as remaining involuntary rational attachment is eroded, the complementary pure intuitive realisation of the nature of union can unfold. Strictly this does not mean that the rational aspect of understanding now ceases with union but rather that it can interact with intuition in a very refined - and thereby transparent - fashion (due to the erosion of an excess element of involuntary attachment).
So whereas in the 1st stage we emphasised the refined rational linear appreciation of form (with respect to pure spiritual awareness), here in complementary fashion we are emphasising the corresponding intuitive (non-dimensional i.e. 0-dimensional) aspect of such awareness.
Put another way the emphasis here is on the immanent - as opposed to the transcendent - aspect of spiritual awareness.
3) when k = 0
e^(k*2*i*pi)= e^0 = (1^1)*(1^0)
This is so as e^0 = e^(2*i*pi)* e^(- 2*i*pi) i.e. e^(2*i*pi - 2*i*pi).
and as we have seen,
e^(2*pi*i)* e^(– 2*pi*i) = (1^1)*(1^0)
Now in Type 1 Mathematics, we would simply add the powers 1 and 0. However in Type 2 Mathematics these remain of a qualitatively distinct nature.
So the significance of the third formulation is that it represents the most specialised stage of contemplative awareness, where form and emptiness are successfully united in experience. Put another way it represents the full integration of both the transcendent and immanent aspects of spiritual understanding (which in turn implies the most refined interaction possible as between both the rational and intuitive modes of understanding).
So here we have pure emptiness serving as the potential for the entire world of (actual) created phenomenal form.
In mathematical terms this represents the most complete appreciation of both the linear rational (1-dimensional) nature of dimension that serves as the basis for quantitative appreciation of Type 1 Mathematics and the purely circular intuitive (0-dimensional) appreciation that serves as the basis for corresponding qualitative appreciation of Type 2 Mathematics.
This in turn with the radial unfolding of stages allows for the growing interpenetration of both Type 1 and Type 2 (in what constitutes Type 3 Mathematics).
We can perhaps see here the truly remarkable nature of e.
In conventional Type 1 terms any number raised to the power of 0 = 1. This represents therefore a merely reduced linear interpretation of 0 (as dimension). and of course when e is raised to 0 (in this reduced quantitative sense) its value is likewise 0.
However what is unique about e is that when it raised to 0 in Type 2 terms (reflecting the circular nature of dimension) its value is likewise 0.
Now we can appreciate why this is so with reference to the fact that by its very nature e fully reconciles corresponding notions of both differentiation (linear) and integration (circular).
So in Type 1 terms both the differentiation and integration of e^x are identical.
Likewise in Type 2 terms at the the most developed level of contemplative awareness differentiated (discrete) and integrated (continuous) elements are seamless so that phenomena of form no longer appear to arise in experience.
So e uniquely combines in its inherent nature both quantitative (discrete) and qualitative (continuous) aspects.
Now I have long identified the inherent nature of a prime number in that it too combines both extreme quantitative and qualitative aspects in its nature. Thus from a discrete independent nature prime numbers seemingly display no pattern. However from a continuous holistic perspective they display (en bloc) a truly remarkable regularity.
Not surprisingly therefore e has a vital role with respect to understanding the nature of primes.
Quite simply if we want to find the average space between primes (in any region of the number system) we simply find the dimensional power to which e must be raised to attain that number.
So for example to obtain 1,000,000, one must raise e to 13.8155... (i.e. the natural log of 1,000,000).
That means in the region of 1,000,000 we would expect the average space between primes to be just less than 14!
This can however be given a more precise expression.
So we now define e^(k*2*i*pi) = 1^x (where k = 1, – 1 or 0 and x = 1, 0 and both 1 and 0 respectively.
So when k = 1,
1) e^(k*2*i*pi)= e^(2*i* pi) = 1^1
In holistic terms this corresponds to the rational linear (i.e. 1-dimensional) appreciation of (mystical) union.
Now with the 1st stage (of specialised contemplative development) a slight imbalance remains whereby appreciation remains unduly transcendent in nature.
In other words in emphasising the transcendent nature of spiritual reality as inherently empty and thereby beyond all notions of form, one to a degree still represses the corresponding immanent nature of such reality as prior and thereby inherent in all form.
This in effect leads to a subtle lingering rational attachment (of a necessarily linear nature) to the notion of unity (as phenomenal form) thereby preventing full realisation of complementary intuitive recognition (that is literally non-dimensional as empty of all form)
2) So when k = – 1,
e^(k*2*i*pi)= e^(- 2*i*pi) = 1^0
This is so as e^(– 2*i*pi) = 1/{e^(2*i*pi)} = (1^1)/(1^1) = 1^(1 – 1) = 1^0.
With the 2nd stage of such specialised development, this remaining lingering attachment to the notion of union (as a phenomenal point) is gradually negated.
Therefore as remaining involuntary rational attachment is eroded, the complementary pure intuitive realisation of the nature of union can unfold. Strictly this does not mean that the rational aspect of understanding now ceases with union but rather that it can interact with intuition in a very refined - and thereby transparent - fashion (due to the erosion of an excess element of involuntary attachment).
So whereas in the 1st stage we emphasised the refined rational linear appreciation of form (with respect to pure spiritual awareness), here in complementary fashion we are emphasising the corresponding intuitive (non-dimensional i.e. 0-dimensional) aspect of such awareness.
Put another way the emphasis here is on the immanent - as opposed to the transcendent - aspect of spiritual awareness.
3) when k = 0
e^(k*2*i*pi)= e^0 = (1^1)*(1^0)
This is so as e^0 = e^(2*i*pi)* e^(- 2*i*pi) i.e. e^(2*i*pi - 2*i*pi).
and as we have seen,
e^(2*pi*i)* e^(– 2*pi*i) = (1^1)*(1^0)
Now in Type 1 Mathematics, we would simply add the powers 1 and 0. However in Type 2 Mathematics these remain of a qualitatively distinct nature.
So the significance of the third formulation is that it represents the most specialised stage of contemplative awareness, where form and emptiness are successfully united in experience. Put another way it represents the full integration of both the transcendent and immanent aspects of spiritual understanding (which in turn implies the most refined interaction possible as between both the rational and intuitive modes of understanding).
So here we have pure emptiness serving as the potential for the entire world of (actual) created phenomenal form.
In mathematical terms this represents the most complete appreciation of both the linear rational (1-dimensional) nature of dimension that serves as the basis for quantitative appreciation of Type 1 Mathematics and the purely circular intuitive (0-dimensional) appreciation that serves as the basis for corresponding qualitative appreciation of Type 2 Mathematics.
This in turn with the radial unfolding of stages allows for the growing interpenetration of both Type 1 and Type 2 (in what constitutes Type 3 Mathematics).
We can perhaps see here the truly remarkable nature of e.
In conventional Type 1 terms any number raised to the power of 0 = 1. This represents therefore a merely reduced linear interpretation of 0 (as dimension). and of course when e is raised to 0 (in this reduced quantitative sense) its value is likewise 0.
However what is unique about e is that when it raised to 0 in Type 2 terms (reflecting the circular nature of dimension) its value is likewise 0.
Now we can appreciate why this is so with reference to the fact that by its very nature e fully reconciles corresponding notions of both differentiation (linear) and integration (circular).
So in Type 1 terms both the differentiation and integration of e^x are identical.
Likewise in Type 2 terms at the the most developed level of contemplative awareness differentiated (discrete) and integrated (continuous) elements are seamless so that phenomena of form no longer appear to arise in experience.
So e uniquely combines in its inherent nature both quantitative (discrete) and qualitative (continuous) aspects.
Now I have long identified the inherent nature of a prime number in that it too combines both extreme quantitative and qualitative aspects in its nature. Thus from a discrete independent nature prime numbers seemingly display no pattern. However from a continuous holistic perspective they display (en bloc) a truly remarkable regularity.
Not surprisingly therefore e has a vital role with respect to understanding the nature of primes.
Quite simply if we want to find the average space between primes (in any region of the number system) we simply find the dimensional power to which e must be raised to attain that number.
So for example to obtain 1,000,000, one must raise e to 13.8155... (i.e. the natural log of 1,000,000).
That means in the region of 1,000,000 we would expect the average space between primes to be just less than 14!
Friday, September 2, 2011
Parallel Riemann Hypothesis!
We concluded the last contribution with the remarkable finding that
i^i = e^(- pi/2), which is a real number!
Now if we take natural logs of each side
then i(log i) = - pi/2,
therefore 1/i(log i) = - 2/pi.
So, - i/log i = - 2/pi
Thus i/log i = 2/pi.
As we know the prime number theorem relating to the general frequency of the primes among the natural numbers is most simply expressed as n/log n (with the proportionate frequency increasing as n becomes larger).
So by allowing n to become progressively larger we have the linear quantitative attempt to reach the infinite (in an actual manner).
Now properly understood i represents the corresponding holistic notion of the infinite where one attempts to appropriate it (in a potential manner).
We can see this in the common psychological appreciation of the imaginary as something that emanates from the holistic unconscious to be embodied in an actual (conscious) manner.
Therefore understood in this light i/log i represents the qualitative correspondent to the prime number theorem!
In quantitative terms, we attempt to understand prime numbers from a Type 1 perspective as base quantities i.e.
2^1, 3^1, 5^1, 7^1,.......
However the prime numbers have in Type 2 terms a corresponding qualitative interpretation as dimensions i.e.
1^2, 1^3, 1^5, 1^7,......
In an inverse quantitative manner we can obtain the circular structure of these dimensions through obtaining the reciprocal roots.
So therefore we can attempt to find the 2 roots, 3 roots, 5 roots, 7 roots of unity and so on for each of the prime numbers.
In this approach we consider all roots which will have both a real and imaginary component.
With respect to both parts we take the values in an absolute manner (ignoring negative signs). Then we sum up both parts (both real and imaginary taken separately) and then obtain the average.
We can demonstrate simply here for p = 3.
There are 3 corresponding roots of 1 involved i.e. 1, - .5 +.866i and - .5 - .866i
Ignoring negative signs the sum of the real part here = 1 + .5 + .5 = 2.
Therefore the mean average = 2/3 = .6666.. .
Then taking the magnitude of the imaginary part (ignoring the i) the sum = .866 + .866 = 1.732
Therefore the mean average = 1.7321/3 = .57735...
i^i = e^(- pi/2), which is a real number!
Now if we take natural logs of each side
then i(log i) = - pi/2,
therefore 1/i(log i) = - 2/pi.
So, - i/log i = - 2/pi
Thus i/log i = 2/pi.
As we know the prime number theorem relating to the general frequency of the primes among the natural numbers is most simply expressed as n/log n (with the proportionate frequency increasing as n becomes larger).
So by allowing n to become progressively larger we have the linear quantitative attempt to reach the infinite (in an actual manner).
Now properly understood i represents the corresponding holistic notion of the infinite where one attempts to appropriate it (in a potential manner).
We can see this in the common psychological appreciation of the imaginary as something that emanates from the holistic unconscious to be embodied in an actual (conscious) manner.
Therefore understood in this light i/log i represents the qualitative correspondent to the prime number theorem!
In quantitative terms, we attempt to understand prime numbers from a Type 1 perspective as base quantities i.e.
2^1, 3^1, 5^1, 7^1,.......
However the prime numbers have in Type 2 terms a corresponding qualitative interpretation as dimensions i.e.
1^2, 1^3, 1^5, 1^7,......
In an inverse quantitative manner we can obtain the circular structure of these dimensions through obtaining the reciprocal roots.
So therefore we can attempt to find the 2 roots, 3 roots, 5 roots, 7 roots of unity and so on for each of the prime numbers.
In this approach we consider all roots which will have both a real and imaginary component.
With respect to both parts we take the values in an absolute manner (ignoring negative signs). Then we sum up both parts (both real and imaginary taken separately) and then obtain the average.
We can demonstrate simply here for p = 3.
There are 3 corresponding roots of 1 involved i.e. 1, - .5 +.866i and - .5 - .866i
Ignoring negative signs the sum of the real part here = 1 + .5 + .5 = 2.
Therefore the mean average = 2/3 = .6666.. .
Then taking the magnitude of the imaginary part (ignoring the i) the sum = .866 + .866 = 1.732
Therefore the mean average = 1.7321/3 = .57735...
The remarkable finding here as the value of p increases is that the mean value of the absolute quantitative value for both the real and imaginary parts converges on 2/pi = .636619772...
We can readily find all these values through use of the Euler Identity,
e^(2*i^pi) = cos (2*pi) + i sin (2*pi).
So the 3 roots of 1 - where the dimensional numbers are 1/3, 2/3 and 3/3 respectively are calculated in this manner as
cos {(2/3)*pi} + i {sin(2/3)*pi},
cos {(4/3)*pi} + i {sin(42/3)*pi}, and
cos {(6/3)*pi} + i {sin(6/3)*pi}.
As p becomes ever larger the mean value (for both parts) approximates ever closer to i/log i.
So we seem here in fact to have a circular number equivalent to the prime number theorem (that is couched in a linear quantitative manner).
However it does not end here!
We can see from our example above that the mean value for the real part = .6666.. and the imaginary part = .57735... respectively.
Therefore the mean value for the real part exceeds 2/pi and the corresponding value for the imaginary part is less than 2/pi respectively.
In fact looking at the absolute differences the value is .6666... - .636619772..
= .03005 (approx)... for the real part
and .636619772... - .57735... = .05927(approx)
Now the ratio of this difference real/imaginary = .03005/.05927 = .507 (approx)
This already seems very close to .5 (sound familiar!)
In fact as the value of p increases the ratio of this difference does indeed tend ever more closely to .5!
So once again what we are stating is this!
As p becomes larger, the absolute mean value of both real and imaginary prime roots of 1 converges ever closer to 2/pi (i.e. i/log i).
Insofar as a difference remains the ratio of absolute deviation of real/imaginary value converges ever closer to .5.
Just as Riemann came up with improvements to prediction of the general frequency of the primes, I experimented with my own improvements.
Now let's say that we wish to calculate the deviation of the absolute mean value of the real part from 2/pi for a larger value of p (say 127).
What we do here is to multiply the deviation (for p = 3) by (p/p1)^2 where p = 3 and p1 = 127.
So this gives us .03005 * (3/127)^2 = .000016768 (approx)
Considering that we are using such an early prime number = 3, this compares extremely well with the true deviation = .000016232 (approx).
In fact this and any other calculation can be significantly improved by then dividing the result by (1 + d) where again in this case d = .03005.
So in in this case we can then approximate the true deviation as .000016277..
Thus our answer is already correct to 3 significant figures!
Predictions can be greatly improved through using the deviations of later prime numbers.
For example if we use the deviation associated with p = 61, we can calculate the corresponding deviation associated with p = 127 correct to 6 significant figures!
Variations of this approach can be used likewise to predict corresponding deviations associated with the imaginary part!
Now in principle just as the non-trivial zeros of the Riemann Zeta function can be used to correct the deviations from the actual with respect to the general distribution of the primes, a corresponding method should exist enabling - ultimately - an exact mean of absolute prime root values for both real and imaginary parts.
So just as the Riemann Hypothesis is used to accurately calculate the average number of primes (in a linear context), this latter approach is used to calculate the average value of these primes in a circular context.
And in each case .5 plays a key role. In fact the circular version provides a key indication of what the .5 actually represents.
As we have seen in the circular context .5 represents ratio of (real) cos to (imaginary) sin values which indicates in turn a quantitative (analytic) to qualitative (holistic) connection.
And this is really what the Riemann Hypothesis is all about i.e. in establishing the condition necessary for full reconciliation of both quantitative and qualitative aspects of interpretation!
We can readily find all these values through use of the Euler Identity,
e^(2*i^pi) = cos (2*pi) + i sin (2*pi).
So the 3 roots of 1 - where the dimensional numbers are 1/3, 2/3 and 3/3 respectively are calculated in this manner as
cos {(2/3)*pi} + i {sin(2/3)*pi},
cos {(4/3)*pi} + i {sin(42/3)*pi}, and
cos {(6/3)*pi} + i {sin(6/3)*pi}.
As p becomes ever larger the mean value (for both parts) approximates ever closer to i/log i.
So we seem here in fact to have a circular number equivalent to the prime number theorem (that is couched in a linear quantitative manner).
However it does not end here!
We can see from our example above that the mean value for the real part = .6666.. and the imaginary part = .57735... respectively.
Therefore the mean value for the real part exceeds 2/pi and the corresponding value for the imaginary part is less than 2/pi respectively.
In fact looking at the absolute differences the value is .6666... - .636619772..
= .03005 (approx)... for the real part
and .636619772... - .57735... = .05927(approx)
Now the ratio of this difference real/imaginary = .03005/.05927 = .507 (approx)
This already seems very close to .5 (sound familiar!)
In fact as the value of p increases the ratio of this difference does indeed tend ever more closely to .5!
So once again what we are stating is this!
As p becomes larger, the absolute mean value of both real and imaginary prime roots of 1 converges ever closer to 2/pi (i.e. i/log i).
Insofar as a difference remains the ratio of absolute deviation of real/imaginary value converges ever closer to .5.
Just as Riemann came up with improvements to prediction of the general frequency of the primes, I experimented with my own improvements.
Now let's say that we wish to calculate the deviation of the absolute mean value of the real part from 2/pi for a larger value of p (say 127).
What we do here is to multiply the deviation (for p = 3) by (p/p1)^2 where p = 3 and p1 = 127.
So this gives us .03005 * (3/127)^2 = .000016768 (approx)
Considering that we are using such an early prime number = 3, this compares extremely well with the true deviation = .000016232 (approx).
In fact this and any other calculation can be significantly improved by then dividing the result by (1 + d) where again in this case d = .03005.
So in in this case we can then approximate the true deviation as .000016277..
Thus our answer is already correct to 3 significant figures!
Predictions can be greatly improved through using the deviations of later prime numbers.
For example if we use the deviation associated with p = 61, we can calculate the corresponding deviation associated with p = 127 correct to 6 significant figures!
Variations of this approach can be used likewise to predict corresponding deviations associated with the imaginary part!
Now in principle just as the non-trivial zeros of the Riemann Zeta function can be used to correct the deviations from the actual with respect to the general distribution of the primes, a corresponding method should exist enabling - ultimately - an exact mean of absolute prime root values for both real and imaginary parts.
So just as the Riemann Hypothesis is used to accurately calculate the average number of primes (in a linear context), this latter approach is used to calculate the average value of these primes in a circular context.
And in each case .5 plays a key role. In fact the circular version provides a key indication of what the .5 actually represents.
As we have seen in the circular context .5 represents ratio of (real) cos to (imaginary) sin values which indicates in turn a quantitative (analytic) to qualitative (holistic) connection.
And this is really what the Riemann Hypothesis is all about i.e. in establishing the condition necessary for full reconciliation of both quantitative and qualitative aspects of interpretation!
A New Number System (3)
There are really two components to this new number system (where numbers are interpreted with respect to their pure dimensional (as opposed to their base quantitative) characteristics.
Once again the linear system - for base quantities - is defined in terms of a fixed dimensional number i.e. 1.
So the natural numbers 1, 2, 3, 4,... in this system are more fully represented as
1^1, 2^1, 3^1, 4^1,.....
However the corresponding circular system - for dimensional qualitative values - is defined in terms of a fixed base quantity 1.
So the natural numbers 1, 2, 3, 4,... in this system are more fully represented as
1^1, 1^2, 1^3, 1^4,.....
The circular nature of this latter system comes through raising 1 to the reciprocal of each dimension thus obtaining a quantitative value that lies on the circle of unit radius.
So when we raise 1 for example to the reciprocal of 4 i.e. 1/4 we obtain in quantitative terms i, which lies on the circle of unit radius!
Because there is a direct relationship as between each dimension (as quality) and its reciprocal (in quantitative terms), this means that the 4 as dimension is associated with the qualitative (i.e. holistic) interpretation of i.
Thus rather that just one valid interpretation of mathematical symbols, which in conventional terms is associated with the default value of 1, potentially an infinite set of possible interpretations exists for all all mathematical symbols, relationships etc.
So whereas Type 1 Mathematics is associated merely with the (reduced) quantitative aspects of mathematical symbols, Type 2 is associated with appropriate qualitative interpretation of these same symbols.
Thus i for example has not merely a quantitative, but also an important qualitative meaning. However this qualitative dimension is completely ignored in Type 1 conventional terms.
This makes no sense for ultimately the quantitative results that are derived for example in complex analysis are somewhat meaningless in the absence of appropriate qualitative interpretation!
Type 3 Mathematics - which is easily the most refined and demanding in nature, then involves consistently relating both quantitative (Type 1) and qualitative (Type 2) interpretation.
However just as the base quantitative system has an imaginary counterpart, likewise the dimensional qualitative counterpart has an imaginary counterpart.
So again the natural numbers in the first system would be
i^1, 2i^1, 3i^1, 4i^1,....,
whereas in the second system the corresponding imaginary version is
1^i, 1^2i, 1^3i, 1^4i,....
Now because in Type 1 Mathematics the second system is not formally recognised this entails with respect to the real part that
1^1 = 1^2 = 1^3 = 1^4 =.....= 1^n.
As we have seen this leads to the misleading conclusion that for example + 1 and - 1 are both the square root of 1.
(Through use of the Type 2 system we can see that - 1 is the square root of 1^1 and + 1 the square root of 1^2 (which are distinct in Type 2 terms).
In other words we cannot properly divorce here proper quantitative from proper qualitative interpretation!)
Also because in Type 1 Mathematics the second system is not recognised this entails with respect to the imaginary part that
1^i = 1^2i= 1^3i = 1^4i =.....= 1^n.
This comes from the corresponding assumption that
e^(2*i*pi) = e^(4*i*pi) = e^(6*i*pi) = e^(8*i*pi) =....= e^(2n*i*pi).
This then leads to the misleading conclusion that 1^i for example can have an unlimited number of possible quantitative solutions.
However because properly speaking in a Type 2 approach
1^i, 1^2i, 1^3i, 1^4i, =.....= 1^n are all distinct,
this means that 1^i has indeed just one unique quantitative value!
So once again - this type directly with respect to a quantitative result - we cannot properly divorce here quantitative from qualitative interpretation!
As we have seen 1^i = e^(- 2pi) = .00186744....
However as i = 1^(1/4), this means that i^i = 1^(i/4) = e^{(- pi)/2} = .2078795763...
It must be stressed that in accordance with Type 1 Mathematics than - as with 1^i - an infinite set of possible values exists for i^i.
So in Type 1 terms, 1^i = e^(2*i*pi)^i = e^(4*i*pi)^i = e^(6*i*pi)^i = e^(8*i*pi)^i =....
By this logic, for example 1^i = e^(2*i*pi)^i = e^(8*i*pi)^i
= e^(- 2*pi) = e^(- 8*pi)
So i^i = e^(- pi/2) = e^(- 2pi)
However this would therefore suggest that i^i = 1^i (which makes little sense).
Therefore Type 2 interpretation needs to be included to avoid such confusion!
Once again the linear system - for base quantities - is defined in terms of a fixed dimensional number i.e. 1.
So the natural numbers 1, 2, 3, 4,... in this system are more fully represented as
1^1, 2^1, 3^1, 4^1,.....
However the corresponding circular system - for dimensional qualitative values - is defined in terms of a fixed base quantity 1.
So the natural numbers 1, 2, 3, 4,... in this system are more fully represented as
1^1, 1^2, 1^3, 1^4,.....
The circular nature of this latter system comes through raising 1 to the reciprocal of each dimension thus obtaining a quantitative value that lies on the circle of unit radius.
So when we raise 1 for example to the reciprocal of 4 i.e. 1/4 we obtain in quantitative terms i, which lies on the circle of unit radius!
Because there is a direct relationship as between each dimension (as quality) and its reciprocal (in quantitative terms), this means that the 4 as dimension is associated with the qualitative (i.e. holistic) interpretation of i.
Thus rather that just one valid interpretation of mathematical symbols, which in conventional terms is associated with the default value of 1, potentially an infinite set of possible interpretations exists for all all mathematical symbols, relationships etc.
So whereas Type 1 Mathematics is associated merely with the (reduced) quantitative aspects of mathematical symbols, Type 2 is associated with appropriate qualitative interpretation of these same symbols.
Thus i for example has not merely a quantitative, but also an important qualitative meaning. However this qualitative dimension is completely ignored in Type 1 conventional terms.
This makes no sense for ultimately the quantitative results that are derived for example in complex analysis are somewhat meaningless in the absence of appropriate qualitative interpretation!
Type 3 Mathematics - which is easily the most refined and demanding in nature, then involves consistently relating both quantitative (Type 1) and qualitative (Type 2) interpretation.
However just as the base quantitative system has an imaginary counterpart, likewise the dimensional qualitative counterpart has an imaginary counterpart.
So again the natural numbers in the first system would be
i^1, 2i^1, 3i^1, 4i^1,....,
whereas in the second system the corresponding imaginary version is
1^i, 1^2i, 1^3i, 1^4i,....
Now because in Type 1 Mathematics the second system is not formally recognised this entails with respect to the real part that
1^1 = 1^2 = 1^3 = 1^4 =.....= 1^n.
As we have seen this leads to the misleading conclusion that for example + 1 and - 1 are both the square root of 1.
(Through use of the Type 2 system we can see that - 1 is the square root of 1^1 and + 1 the square root of 1^2 (which are distinct in Type 2 terms).
In other words we cannot properly divorce here proper quantitative from proper qualitative interpretation!)
Also because in Type 1 Mathematics the second system is not recognised this entails with respect to the imaginary part that
1^i = 1^2i= 1^3i = 1^4i =.....= 1^n.
This comes from the corresponding assumption that
e^(2*i*pi) = e^(4*i*pi) = e^(6*i*pi) = e^(8*i*pi) =....= e^(2n*i*pi).
This then leads to the misleading conclusion that 1^i for example can have an unlimited number of possible quantitative solutions.
However because properly speaking in a Type 2 approach
1^i, 1^2i, 1^3i, 1^4i, =.....= 1^n are all distinct,
this means that 1^i has indeed just one unique quantitative value!
So once again - this type directly with respect to a quantitative result - we cannot properly divorce here quantitative from qualitative interpretation!
As we have seen 1^i = e^(- 2pi) = .00186744....
However as i = 1^(1/4), this means that i^i = 1^(i/4) = e^{(- pi)/2} = .2078795763...
It must be stressed that in accordance with Type 1 Mathematics than - as with 1^i - an infinite set of possible values exists for i^i.
So in Type 1 terms, 1^i = e^(2*i*pi)^i = e^(4*i*pi)^i = e^(6*i*pi)^i = e^(8*i*pi)^i =....
By this logic, for example 1^i = e^(2*i*pi)^i = e^(8*i*pi)^i
= e^(- 2*pi) = e^(- 8*pi)
So i^i = e^(- pi/2) = e^(- 2pi)
However this would therefore suggest that i^i = 1^i (which makes little sense).
Therefore Type 2 interpretation needs to be included to avoid such confusion!
Thursday, September 1, 2011
Alain Connes
This time when reading Karl Sabbagh's "Dr. Riemann's Zeros" I came across an interesting quote from Alain Connes on P.205. In commenting on the relationship between Geometry and Algebra he states:
'It really is fantastic step' he said, to understand that the square of a number - which is just a geometrical square - and the cube, which is just a geometrical cube - can be added together, even though you would say, "But one has dimension the length squared and the other the length cubed" and you would never add things which have different dimensions. So algebra is an amazing achievement, and once you have formulated things in an algebraic terms then they take on a life of their own.'
Algebra of course is incredibly important. However the price that has been paid in terms of such rational abstraction is that a basic form of reductionism is involved. In other words we can see clearly in geometrical terms that 2-dimensional is qualitatively distinct from 3-dimensional reality. However in algebraic terms this qualitative distinction is quickly lost with variables interpreted with respect to their mere quantitative meaning.
Indeed it is rather ironic that Connes in speaking about the Riemann Hypothesis in an a later quote on p.208 states:
"It is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication' Connes said. It's a gaping hole in our understanding, because until we really understand it we cannot say that we understand the line. Even the line itself is extraordinarily mysterious.'
May I suggest once again that the very reason why the link between addition and multiplication seems so intractable is precisely because the qualitative nature of variable transformation - necessarily involved in all multiplication - is formally ignored in present Type 1 Mathematics.
So properly understood there are both Type 1 (quantitative) and Type 2 (qualitative) aspects to all mathematical interpretation.
The Riemann Hypothesis in fact is a key statement regarding the relationship as between these two aspects.
Likewise with the line, Type 1 Mathematics, due to the same lack of a qualitative dimension, one cannot properly distinguish finite (discrete) from infinite (continuous) notions. So once again though the infinite notion is qualitatively distinct from the finite, in Type 1 interpretation it is necessarily reduced quantitatively in a finite manner!
'It really is fantastic step' he said, to understand that the square of a number - which is just a geometrical square - and the cube, which is just a geometrical cube - can be added together, even though you would say, "But one has dimension the length squared and the other the length cubed" and you would never add things which have different dimensions. So algebra is an amazing achievement, and once you have formulated things in an algebraic terms then they take on a life of their own.'
Algebra of course is incredibly important. However the price that has been paid in terms of such rational abstraction is that a basic form of reductionism is involved. In other words we can see clearly in geometrical terms that 2-dimensional is qualitatively distinct from 3-dimensional reality. However in algebraic terms this qualitative distinction is quickly lost with variables interpreted with respect to their mere quantitative meaning.
Indeed it is rather ironic that Connes in speaking about the Riemann Hypothesis in an a later quote on p.208 states:
"It is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication' Connes said. It's a gaping hole in our understanding, because until we really understand it we cannot say that we understand the line. Even the line itself is extraordinarily mysterious.'
May I suggest once again that the very reason why the link between addition and multiplication seems so intractable is precisely because the qualitative nature of variable transformation - necessarily involved in all multiplication - is formally ignored in present Type 1 Mathematics.
So properly understood there are both Type 1 (quantitative) and Type 2 (qualitative) aspects to all mathematical interpretation.
The Riemann Hypothesis in fact is a key statement regarding the relationship as between these two aspects.
Likewise with the line, Type 1 Mathematics, due to the same lack of a qualitative dimension, one cannot properly distinguish finite (discrete) from infinite (continuous) notions. So once again though the infinite notion is qualitatively distinct from the finite, in Type 1 interpretation it is necessarily reduced quantitatively in a finite manner!
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