We now come to the discuss the crucially important role of the primes with respect to the natural number system.
In the conventional mathematical approach - that is solely geared to absolute type appreciation of the quantitative (Type 1) aspect of the number system, the primes are customarily viewed as the essential building blocks of the natural numbers.
Thus - apart from 1 - every natural number is either prime or composed as a unique product of primes.
So for example 6 = 2 * 3 (and cannot be derived from any other combination of prime factors).
Therefore from this quantitative perspective the direction of causation with respect to the primes and natural numbers is strictly one-way, with the natural numbers necessarily derived from the primes.
However, even very early on, I could see a big problem with this way of thinking, as the natural numbers are already implicit in the very use of the primes!
For example, if I refer to 7 as a prime (in a cardinal sense) then this implies that I can meaningfully distinguish its 1st, 2nd, 3rd, 4th, 5th, 6th and 7th terms.
Thus the very definition of a prime in cardinal terms, implicitly implies an ordered set of natural numbers (in ordinal terms).
So we insist on viewing 7 as an independent prime building block of the (cardinal) natural numbers; however it is already implicitly defined in terms of a set of (ordinal) natural numbers.
This therefore suggests that once we incorporate ordinal - as well as cardinal - meaning, that the relationship as between primes and natural numbers is necessarily bi-directional in a circular paradoxical manner.
Seen from another angle, implicit in the very listing of the primes (in cardinal terms) as 2, 3, 5, 7, ...is a corresponding set of natural numbers (in ordinal terms) as 1st, 2nd, 3rd, 4th,...
We could equally say from the opposite perspective, that implicit in the very listing of the ordinal natural numbers 1st, 2nd, 3rd, 4th,..., are the corresponding prime numbers 2, 3, 5, 7,...., (which are necessary to drive the natural numbers).
Therefore from this more comprehensive number perspective (incorporating both cardinal and ordinal notions) the relationship between the primes and natural numbers is necessarily one of two-way interdependence.
In fact this is just another way of stating once again that the number system necessarily entails both quantitative and qualitative aspects in dynamic relationship with each other. Thus, if we designate the cardinal numbers in a quantitative manner, then the ordinal numbers are thereby - relatively - of a qualitative nature.
When on reflects for a moment on this matter, it should appear somewhat obvious.
So again for example when refers to "3" in a cardinal manner, it thereby is interpreted in quantitative terms as independent of other numbers. However when one then refers to 3rd (in an ordinal fashion) it necessarily entails a qualitative relationship (with respect to a corresponding group of numbers).
However, when one looks at the conventional treatment of ordinal numbers, this qualitative aspect is rarely if ever referred to in explanation. Rather the more neutral term of "rankings" is used, thereby subtly reducing the qualitative aspect in quantitative terms.
However when one properly grasps the true qualitative nature of ordinal numbers with respect to cardinal, then one realises that - rather than rigid absolute entities defined in a merely quantitative manner - the number system by contrast necessarily represents a two-way dynamic interactive process entailing both cardinal and ordinal aspects (that are - relatively - quantitative and qualitative with respect to each other).
This then leads to a new enhanced appreciation of the true relationship of the primes and natural numbers, which is now seen to ultimately operate in a two-way synchronous manner.
So with respect to the origins of the number system, the primes and natural numbers ultimately "cause" each other. In other words, they mutually arise in a dynamic synchronous manner.
And the importance of this relationship is that through the interaction of the primes and natural numbers, both the quantitative and qualitative aspects of the number system are mutually transmitted (in a two-way interactive fashion).
One can hardly over state the significance of this realisation, for it opens up the way for a vastly enhanced appreciation of the true significance of the number system.
So far we have sought to understand the number system in a merely quantitative manner (in a greatly distorted reduced manner).
However we are now on the verge of appreciating that the true role of number equally embraces the quantitative as well as quantitative realms. And we are in a position to show how these are truly related (which will greatly enhance appreciation of both aspects).
So from this new dynamic perspective. the number system is now seen as inseparable from the course of all phenomenal evolution (in quantitative and qualitative terms). Indeed, quite simply, all phenomenal form (in space and time) is ultimately encoded in a dynamic number pattern as its most fundamental "genes". In this sense, manifest evolution simply represents the decoded nature of number!
Another related insight regarding the nature of primes, was equally to lead to this new dynamic appreciation of their nature.
As I have stated before, every mathematical symbol with a standard analytic interpretation can equally be given a corresponding (unrecognised) holistic interpretation.
Some years ago, when I was seriously developing my holistic mathematical understanding, I spent a considerable time establishing all the holistic equivalents to the recognised number types e.g. prime, natural, positive and negative, rational, irrational (algebraic and transcendental), imaginary, complex etc.
It struck me forcibly that the holistic notion of "prime" bore a very close relationship with the nature of what we identify as "primitive" in psychological terms.
Thus a primitive instinct represents a direct confusion of conscious with unconscious type meaning i.e. where the holistic desire for overall meaning is confused with immediate localised phenomena in experience.
From a mathematical perspective, this could be expressed as the confusion of holistic and analytic (or likewise quantitative and qualitative).
Therefore unraveling such confusion in coming to realise the true nature of the primes, would be inseparable from the psychological process, whereby both conscious and unconscious become properly differentiated from each other (before being properly integrated).
Thus in mathematical terms, the first stage of conscious differentiation relates to Type 1 (analytic) mathematical interpretation.
The second stage of unconscious differentiation relates to Type 2 (holistic) mathematical interpretation.
The third stage (by which conscious and unconscious can be maturely integrated relates to Type 3 (radial) mathematical interpretation.
And this is the stage that we are now at, in attempting to unravel the mystery of the primes and zeta zeros.
Friday, June 26, 2015
Thursday, June 25, 2015
Zeta Zeros Made Simple (4)
So again we have seen how the natural numbers can be initially interpreted in two relatively separate ways
1) the standard conventional analytic manner, where each number is viewed in an individual independent manner as quantitative. This can be referred to as the Type 1 aspect of the (cardinal) number system.
2) the largely unrecognised holistic manner, where the relationship as between all numbers is now viewed by contrast in a collective interdependent manner as qualitative. This can be referred to - in relative terms - as the Type 2 aspect of the number system.
In dynamic interactive terms, both the Type 1 and Type 2 aspects are ultimately fully complementary with each other.
However a major issue relates to the compatibility of both the Type 1 and Type 2 aspects which speak - as it were - in different languages.
This then leads on to a more refined appreciation of the number system incorporating the Type 3 aspect, where both Type 1 and Type 2 aspects are simultaneously reconciled with each other.
The Type 3 aspect simply refers to the Zeta 1 (i.e. Riemann) zeros.
Now the key fundamental role of these zeros is that they provide a common means of conversion as between the Type 1 and Type 2 aspects (that are not directly compatible with each other).
Thus from one valid perspective, through the Zeta 1 zeros we can convert the Type 1 aspect of the number system (relating to the standard analytic interpretation of number quantities) in a Type 2 holistic manner.
From the equally valid opposite perspective, again through the Zeta 1 zeros, we can convert the Type 2 aspect of the number system (relating to its collective holistic qualitative nature) in a Type 1 analytic manner.
Put another way, from this perspective (where we start from the natural numbers) the truly fundamental role of the Zeta 1 zeros is to ensure the consistent interaction of this cardinal system in both quantitative and qualitative terms.
Now remarkably, this point cannot be even recognised from the conventional mathematical standpoint (as qualitative interpretation is already reduced in quantitative terms).
Thus the very consistency of the natural number system - to which the Riemann zeros relate - is thereby necessarily already assumed to exist in a conventional mathematical manner.
Therefore it is utterly futile to attempt to prove (or disprove) the Riemann Hypothesis, which deals with a central condition for this consistency to hold, from the conventional mathematical perspective.
However, though we have started in this exploration with the natural numbers, to derive the zeta zeros, we can switch the frames of reference in a dynamic interactive manner to start with the zeta zeros (now interpreted in an analytic manner) to then derive the natural numbers in their holistic state.
So the process is necessarily bi-directional with no prior causation with respect to the number system.
Therefore the remarkable feature to appreciate regarding the number system is its two-way synchronicity, through which both the natural numbers and zeta zeros both mutually arise as its very origin. Thus the marvellous consistency which forms the basis not only for the harmonious relationship of quantitative and qualitative within the number system itself, but subsequently throughout all phenomenal creation, is already embedded in phenomenal form at its very inception.
However the subsequent full realisation of its nature brings us to the very limit of pure spiritual meaning. So what is already implicit in the number system at the very start of evolution can only fully unfold, through evolution itself fulfilling its eternal destiny.
Thus the number system lies at the very threshold of incomprehensible mystery!
I have long maintained that Mathematics as currently understood is still of a very limed nature, relating to just one strand in an overall comprehensive system.
The comprehensive vision of Mathematics therefore would include the three following major areas.
1) Analytic (which I have identified here with the Type 1 quantitative aspect of the number system). Conventional Mathematics is still entirely defined in Type 1 terms.
2) Holistic (which I have identified as the Type 2 qualitative aspect of the number system).
Here every symbol with an accepted analytic interpretation in Type 1 terms can equally be given an important holistic interpretation in a Type 2 manner. This open up an entirely new appreciation of a scientific qualitative nature that has not yet been remotely tapped in our culture.
3) Radial (which I associate with the Type 3 aspect).
This will allow for the fullest possible appreciation of the true nature of Mathematics that can be both immensely creative and highly productive combining in harmonious manner both the the Type 1 and Type 2 aspects.
Indeed, strictly all Mathematics is Type 3 entailing dynamically interacting relationships.
However, we are still very much stuck in an entirely reduced vision of its nature, where its symbols are viewed in a misleading reduced manner.
Of course there is a considerable limited value in this reduced interpretation. However to identify Mathematics entirely with this reduced perspective represents mental myopia of the most extreme kind.
In the next entry, we will deal with the crucial role of the primes to the natural numbers in this system.
1) the standard conventional analytic manner, where each number is viewed in an individual independent manner as quantitative. This can be referred to as the Type 1 aspect of the (cardinal) number system.
2) the largely unrecognised holistic manner, where the relationship as between all numbers is now viewed by contrast in a collective interdependent manner as qualitative. This can be referred to - in relative terms - as the Type 2 aspect of the number system.
In dynamic interactive terms, both the Type 1 and Type 2 aspects are ultimately fully complementary with each other.
However a major issue relates to the compatibility of both the Type 1 and Type 2 aspects which speak - as it were - in different languages.
This then leads on to a more refined appreciation of the number system incorporating the Type 3 aspect, where both Type 1 and Type 2 aspects are simultaneously reconciled with each other.
The Type 3 aspect simply refers to the Zeta 1 (i.e. Riemann) zeros.
Now the key fundamental role of these zeros is that they provide a common means of conversion as between the Type 1 and Type 2 aspects (that are not directly compatible with each other).
Thus from one valid perspective, through the Zeta 1 zeros we can convert the Type 1 aspect of the number system (relating to the standard analytic interpretation of number quantities) in a Type 2 holistic manner.
From the equally valid opposite perspective, again through the Zeta 1 zeros, we can convert the Type 2 aspect of the number system (relating to its collective holistic qualitative nature) in a Type 1 analytic manner.
Put another way, from this perspective (where we start from the natural numbers) the truly fundamental role of the Zeta 1 zeros is to ensure the consistent interaction of this cardinal system in both quantitative and qualitative terms.
Now remarkably, this point cannot be even recognised from the conventional mathematical standpoint (as qualitative interpretation is already reduced in quantitative terms).
Thus the very consistency of the natural number system - to which the Riemann zeros relate - is thereby necessarily already assumed to exist in a conventional mathematical manner.
Therefore it is utterly futile to attempt to prove (or disprove) the Riemann Hypothesis, which deals with a central condition for this consistency to hold, from the conventional mathematical perspective.
However, though we have started in this exploration with the natural numbers, to derive the zeta zeros, we can switch the frames of reference in a dynamic interactive manner to start with the zeta zeros (now interpreted in an analytic manner) to then derive the natural numbers in their holistic state.
So the process is necessarily bi-directional with no prior causation with respect to the number system.
Therefore the remarkable feature to appreciate regarding the number system is its two-way synchronicity, through which both the natural numbers and zeta zeros both mutually arise as its very origin. Thus the marvellous consistency which forms the basis not only for the harmonious relationship of quantitative and qualitative within the number system itself, but subsequently throughout all phenomenal creation, is already embedded in phenomenal form at its very inception.
However the subsequent full realisation of its nature brings us to the very limit of pure spiritual meaning. So what is already implicit in the number system at the very start of evolution can only fully unfold, through evolution itself fulfilling its eternal destiny.
Thus the number system lies at the very threshold of incomprehensible mystery!
I have long maintained that Mathematics as currently understood is still of a very limed nature, relating to just one strand in an overall comprehensive system.
The comprehensive vision of Mathematics therefore would include the three following major areas.
1) Analytic (which I have identified here with the Type 1 quantitative aspect of the number system). Conventional Mathematics is still entirely defined in Type 1 terms.
2) Holistic (which I have identified as the Type 2 qualitative aspect of the number system).
Here every symbol with an accepted analytic interpretation in Type 1 terms can equally be given an important holistic interpretation in a Type 2 manner. This open up an entirely new appreciation of a scientific qualitative nature that has not yet been remotely tapped in our culture.
3) Radial (which I associate with the Type 3 aspect).
This will allow for the fullest possible appreciation of the true nature of Mathematics that can be both immensely creative and highly productive combining in harmonious manner both the the Type 1 and Type 2 aspects.
Indeed, strictly all Mathematics is Type 3 entailing dynamically interacting relationships.
However, we are still very much stuck in an entirely reduced vision of its nature, where its symbols are viewed in a misleading reduced manner.
Of course there is a considerable limited value in this reduced interpretation. However to identify Mathematics entirely with this reduced perspective represents mental myopia of the most extreme kind.
In the next entry, we will deal with the crucial role of the primes to the natural numbers in this system.
Wednesday, June 24, 2015
Zeta Zeros Made Simple (3)
Once again, we have shown how each number can be given both quantitative and qualitative interpretations, which interact in the dynamics of experience.
The quantitative aspect is where each number is treated as a part i.e. (independent of other numbers).
The qualitative aspect, by contrast, is where each number is given a whole dimensional meaning (as interdependent with all other numbers).
So once more, "3" for example, can be treated in an independent quantitative manner ; however equally it can be used to represent what is common to different classes (which can be increased without limit).
In addition, each number involved is used in the first quantitative sense as for example with 3 + 2; however strictly in multiplication the second qualitative meaning is likewise necessarily involved.
Therefore when we multiply 3 * 2, the use of "3" here signifies a number that is common with respect to the 2 classes involved (as for example 2 rows with 3 items in each row).
And of course we can go on to associate any other natural number in the same manner with "3".
It might help further to refer in this context to the first quantitative use of "3" as the Type 1 interpretation, and then - in relative manner - to the second qualitative use as the Type 2 interpretation.
However this then creates the big problem of how to ensure consistency as between the two distinctive meanings of number (in this case illustrated by "3").
And this problem remains completely unrecognised from the conventional mathematical interpretation (where the qualitative interpretation of number is reduced to the quantitative).
Put another way - though again unrecognised in conventional terms - every number, by its very nature, necessarily speaks in two distinctive languages (that are not directly compatible in terms of each other).
Now n terms of the number system as a whole, this issue can be expressed in the following manner.
From one perspective, each of the natural numbers is independent of every other number (in an absolute quantitative manner).
However all the natural numbers as a collective set, are likewise interdependent with each other (in a relative qualitative manner).
Therefore the fundamental key issue with respect to the number system is to show how both of these meanings (quantitative and qualitative) can be reconciled with each other.
This is where the Riemann zeta function is of such importance.
In its simplest form it is written as
Therefore we can see that each natural number is included here in sequential order to be raised to the negative of a dimensional power (written as s).
Then in finding the solutions of the equation for s = 0, i.e.
ζ (s) = 1 – s + 2 – s + 3 – s + 4 – s +….. = 0,
we can derive a crucially important new set of numbers (as solutions for s) which are known as the non-trivial zeta zeros (i.e. the Riemann zeros) which I refer to as the Zeta 1 zeros.
Now the true significance of these zeros is that they provide a set of numbers where both the quantitative and qualitative aspects of the cardinal system of natural numbers are reconciled.
What this means in effect is that at each of these zeros, both the quantitative and qualitative interpretations of number are now identical. However, it is important to appreciate that this can only be understood in a dynamic relative manner where such identity can only be approximated in a phenomenal manner.
The zeros occur as pairs in the form of .5 + it. and .5 – it respectively.
However t represents a transcendental number (which can only be approximated in a rational manner). Likewise the set of zeros is unlimited; however we can always know but a limited finite number of such zeros.
What this entails is that we must imply an entirely distinctive paradoxical type of appreciation when interpreting the zeros.
In conventional mathematical terms, a linear either/or logic is used in appreciation of the quantitative nature of number. Thus a number is either prime or composite; numbers are either random or distributed in an orderly manner; multiplication can be distinguished from addition; finally we can identify either the quantitative or qualitative aspect of number in a separate manner.
However, remarkably the zeta zeros, properly speaking, operate according to a circular both/and logic that is paradoxical from a conventional perspective.
Thus the very nature of the zeros is that with these numbers the distinction between what is prime and composite is eroded; likewise the distinction as between what is random and ordered does not operate; crucially at each zero, addition is reconciled with multiplication and finally and most importantly the distinction as between quantitative and qualitative aspects is likewise now removed.
Though all this seems counter intuitive, it should be easier to appreciate with reference to everyday appreciation of the nature of a crossroads. Thus if one approaches a crossroads heading North, an unambiguous either/or distinction can be made as between left and right turns; likewise if one approaches the crossroads from the opposite direction heading South, again an unambiguous distinction can be made as between left and right turns.
However, when one simultaneously views the turns from both North and South directions, what is left and what is right (separately) is rendered deeply paradoxical. For what is left from one direction is right from the other and vice versa. So depending on context each turn can be left or right!
Fundamentally, the zeros represent the same logic. Though I have not yet dealt with the ordinal nature of number yet in this series of entries, number can indeed be approached from both cardinal and ordinal directions. Though the identification of quantitative and qualitative aspects is ambiguous when taken from just one direction (e.g. cardinal) when both ordinal and cardinal directions are combined, the polar opposites we customarily associate with number interpretation (esp. quantitative and qualitative) are rendered paradoxical.
However, it is simply impossible to grasp this from the conventional mathematical perspective.
In the case of our crossroads, we clearly cannot identify left and right turns, if we only recognise one of these directions! The realisation that the turns can be both left and right (when approached from two opposite directions) would be meaningless in this context.
Likewise, the recognition that a number can simultaneously be both quantitative and qualitative (in an approximate relative manner) is likewise impossible without the clear recognition of separate quantitative (Type 1) and qualitative (Type 2) interpretations in the first place.
So the simultaneous recognition of number as both quantitative and qualitative relates to the most refined form of mathematical understanding (which I refer to as Type 3).
So the zeta zeros from this new perspective, can therefore be expressed succinctly as representing the Type 3 aspect of the natural number system (in cardinal terms).
The quantitative aspect is where each number is treated as a part i.e. (independent of other numbers).
The qualitative aspect, by contrast, is where each number is given a whole dimensional meaning (as interdependent with all other numbers).
So once more, "3" for example, can be treated in an independent quantitative manner ; however equally it can be used to represent what is common to different classes (which can be increased without limit).
In addition, each number involved is used in the first quantitative sense as for example with 3 + 2; however strictly in multiplication the second qualitative meaning is likewise necessarily involved.
Therefore when we multiply 3 * 2, the use of "3" here signifies a number that is common with respect to the 2 classes involved (as for example 2 rows with 3 items in each row).
And of course we can go on to associate any other natural number in the same manner with "3".
It might help further to refer in this context to the first quantitative use of "3" as the Type 1 interpretation, and then - in relative manner - to the second qualitative use as the Type 2 interpretation.
However this then creates the big problem of how to ensure consistency as between the two distinctive meanings of number (in this case illustrated by "3").
And this problem remains completely unrecognised from the conventional mathematical interpretation (where the qualitative interpretation of number is reduced to the quantitative).
Put another way - though again unrecognised in conventional terms - every number, by its very nature, necessarily speaks in two distinctive languages (that are not directly compatible in terms of each other).
Now n terms of the number system as a whole, this issue can be expressed in the following manner.
From one perspective, each of the natural numbers is independent of every other number (in an absolute quantitative manner).
However all the natural numbers as a collective set, are likewise interdependent with each other (in a relative qualitative manner).
Therefore the fundamental key issue with respect to the number system is to show how both of these meanings (quantitative and qualitative) can be reconciled with each other.
This is where the Riemann zeta function is of such importance.
In its simplest form it is written as
ζ (s) = 1 –
s + 2 – s + 3 – s + 4 – s +…..
Therefore we can see that each natural number is included here in sequential order to be raised to the negative of a dimensional power (written as s).
Then in finding the solutions of the equation for s = 0, i.e.
ζ (s) = 1 – s + 2 – s + 3 – s + 4 – s +….. = 0,
we can derive a crucially important new set of numbers (as solutions for s) which are known as the non-trivial zeta zeros (i.e. the Riemann zeros) which I refer to as the Zeta 1 zeros.
Now the true significance of these zeros is that they provide a set of numbers where both the quantitative and qualitative aspects of the cardinal system of natural numbers are reconciled.
What this means in effect is that at each of these zeros, both the quantitative and qualitative interpretations of number are now identical. However, it is important to appreciate that this can only be understood in a dynamic relative manner where such identity can only be approximated in a phenomenal manner.
The zeros occur as pairs in the form of .5 + it. and .5 – it respectively.
However t represents a transcendental number (which can only be approximated in a rational manner). Likewise the set of zeros is unlimited; however we can always know but a limited finite number of such zeros.
What this entails is that we must imply an entirely distinctive paradoxical type of appreciation when interpreting the zeros.
In conventional mathematical terms, a linear either/or logic is used in appreciation of the quantitative nature of number. Thus a number is either prime or composite; numbers are either random or distributed in an orderly manner; multiplication can be distinguished from addition; finally we can identify either the quantitative or qualitative aspect of number in a separate manner.
However, remarkably the zeta zeros, properly speaking, operate according to a circular both/and logic that is paradoxical from a conventional perspective.
Thus the very nature of the zeros is that with these numbers the distinction between what is prime and composite is eroded; likewise the distinction as between what is random and ordered does not operate; crucially at each zero, addition is reconciled with multiplication and finally and most importantly the distinction as between quantitative and qualitative aspects is likewise now removed.
Though all this seems counter intuitive, it should be easier to appreciate with reference to everyday appreciation of the nature of a crossroads. Thus if one approaches a crossroads heading North, an unambiguous either/or distinction can be made as between left and right turns; likewise if one approaches the crossroads from the opposite direction heading South, again an unambiguous distinction can be made as between left and right turns.
However, when one simultaneously views the turns from both North and South directions, what is left and what is right (separately) is rendered deeply paradoxical. For what is left from one direction is right from the other and vice versa. So depending on context each turn can be left or right!
Fundamentally, the zeros represent the same logic. Though I have not yet dealt with the ordinal nature of number yet in this series of entries, number can indeed be approached from both cardinal and ordinal directions. Though the identification of quantitative and qualitative aspects is ambiguous when taken from just one direction (e.g. cardinal) when both ordinal and cardinal directions are combined, the polar opposites we customarily associate with number interpretation (esp. quantitative and qualitative) are rendered paradoxical.
However, it is simply impossible to grasp this from the conventional mathematical perspective.
In the case of our crossroads, we clearly cannot identify left and right turns, if we only recognise one of these directions! The realisation that the turns can be both left and right (when approached from two opposite directions) would be meaningless in this context.
Likewise, the recognition that a number can simultaneously be both quantitative and qualitative (in an approximate relative manner) is likewise impossible without the clear recognition of separate quantitative (Type 1) and qualitative (Type 2) interpretations in the first place.
So the simultaneous recognition of number as both quantitative and qualitative relates to the most refined form of mathematical understanding (which I refer to as Type 3).
So the zeta zeros from this new perspective, can therefore be expressed succinctly as representing the Type 3 aspect of the natural number system (in cardinal terms).
Tuesday, June 23, 2015
Zeta Zeros Made Simple (2)
Yesterday, we saw how both quantitative and qualitative aspects necessarily attach in a dynamic interactive manner to all natural numbers.
Therefore, through the quantitative aspect, we recognise the distinct independence of each number (from all other numbers). By contrast through the qualitative aspect we recognise the corresponding interdependence of each of these (with all other numbers).
With addition, we are explicitly concerned with the former quantitative notion of number as representing independent units; with multiplication, by contrast, we are now explicitly concerned with the latter qualitative notion of number interdependence with respect to its common identity with various classes (the number of which can be extended without limit).
In addition, no (qualitative) dimensional change in the nature of units takes place; however with multiplication a dimensional - as well as quantitative - change necessarily takes place in units (though this fact is conveniently edited out of the conventional mathematical interpretation of multiplication).
Using the close analogy with quantum physics, number exhibits both particle and wave like characteristics. So which aspect reveals itself depends on the particular context of investigation.
Thus if we associate the operation of addition with its particle like attributes (as independent numbers) the operation of multiplication by contrast will then reveal its corresponding wave like attributes (as interdependent with other numbers).
Clearly therefore, from a dynamic interactive perspective, the nature of number is necessarily relative (with respect to both its independent and interdependent aspects).
So, if we now look at the natural number system as a whole, we have two related perspectives through which it can be viewed.
1) we can view it in the customary analytic manner, where both quantitative and qualitative poles are clearly separated in an absolute type manner (with the qualitative aspect thereby reduced to the quantitative).
Therefore from this perspective, the natural numbers 1, 2, 3, 4,.... are viewed as independent number entities in a quantitative manner.
2) we can view it - in the largely unrecognised - qualitative manner, where the natural numbers 1, 2, 3, 4 in interdependent terms (in the recognition of what is common to different number classes).
So again if we have two rows of items with 3 items in each row, the number "3" now represents what is common to both rows. Therefore, in this sense we are making reference directly to its qualitative - rather than quantitative - meaning.
Of course initially in identifying each row we must recognise "3" in the customary independent manner (as quantitative). However, when we then recognise its commonality with respect to both rows, we switch to the qualitative aspect.
This in fact explains the deeper reason why multiplication necessarily entails both quantitative and qualitative aspects of number transformation.
Thus 2 * 3 = 6 in quantitative terms). However 2 * 3 equally entails a qualitative transformation in a 2-dimensional fashion (though this then is edited out of conventional mathematical interpretation).
Therefore the deeper reason as to why multiplication also entails a qualitative transformation in the nature of units is due to the necessary switch that is involved with respect to the recognition of number (from its independent to its interdependent nature).
There is another helpful way of viewing this qualitative aspect of number.
Once again, to illustrate, we can use "3" to refer to the quantitative aspect of number.
However when we use "3" in the qualitative sense, it can be referred to as "threeness" (i.e. the quality of 3).
Therefore, in operations, a continual switching takes place as between number as representing a quantity, and its corresponding qualitative nature as "numberness".
In psychological terms, quantitative and qualitative aspects respectively are related to the manner in which both conscious and unconscious interact.
In direct terms we can identify the quantitative aspect with (conscious) rational recognition; however the qualitative aspect relates directly to (unconscious) intuitive recognition.
Therefore the customary reduction of qualitative to quantitative interpretation in Conventional Mathematics corresponds directly with the corresponding reduction of intuitive to rational type meaning.
This is certainly true in formal terms, where Mathematics is presented as a body of rational type truths. However indirectly of course, the intuitive aspect must be always involved, as mathematical operations in quantitative terms are strictly speaking meaningless in the absence of the qualitative notion of a common relationship.
The difficulty in dynamic interactive terms is that the qualitative can always be shown to have a corresponding quantitative aspect (just as the quantitative can be shown to have a qualitative aspect).
Again this is directly analogous with quantum physics where particles exhibit wave like characteristics and waves particle like characteristics, respectively.
Thus when one merely concentrates on the quantitative like aspect (associated with qualitative type understanding) then it becomes easy to simply reduce the qualitative in a quantitative manner.
In psychological terms, one switches to the conceptual notion of number through dynamic negation of number perceptions. Now when the initial number perceptions are identified in a quantitative manner, the initial conceptual recognition will be of a qualitative intuitive nature (where it is seen as potentially applying in an infinite manner to all numbers).
However in practice the understanding of the number concept is then quickly reduced in a quantitative rational manner (where it is now understood as actually applying in a finite manner to all numbers).
In reverse fashion, the recognition of distinct number perceptions, implies the dynamic negation of number concepts. Again the initial realisation is implicitly of an intuitive nature. However this is quickly reduced in a rational manner.
Therefore, though in the dynamics of understanding both finite (rational) and infinite (intuitive) notions ceaselessly interact in both quantitative and qualitative terms, standard mathematical interpretation reduces all this in a finite rational fashion relating merely to the quantitative aspect.
In this approach the infinite is then misleadingly viewed as merely a linear extension of the finite notion.
However, when one properly appreciates the unique distinctiveness of both the quantitative and qualitative aspects of number, then the key issue relates as to consistent use with respect to the combination of both aspects.
And this is where the zeta zeros play a crucial role!
Therefore, through the quantitative aspect, we recognise the distinct independence of each number (from all other numbers). By contrast through the qualitative aspect we recognise the corresponding interdependence of each of these (with all other numbers).
With addition, we are explicitly concerned with the former quantitative notion of number as representing independent units; with multiplication, by contrast, we are now explicitly concerned with the latter qualitative notion of number interdependence with respect to its common identity with various classes (the number of which can be extended without limit).
In addition, no (qualitative) dimensional change in the nature of units takes place; however with multiplication a dimensional - as well as quantitative - change necessarily takes place in units (though this fact is conveniently edited out of the conventional mathematical interpretation of multiplication).
Using the close analogy with quantum physics, number exhibits both particle and wave like characteristics. So which aspect reveals itself depends on the particular context of investigation.
Thus if we associate the operation of addition with its particle like attributes (as independent numbers) the operation of multiplication by contrast will then reveal its corresponding wave like attributes (as interdependent with other numbers).
Clearly therefore, from a dynamic interactive perspective, the nature of number is necessarily relative (with respect to both its independent and interdependent aspects).
So, if we now look at the natural number system as a whole, we have two related perspectives through which it can be viewed.
1) we can view it in the customary analytic manner, where both quantitative and qualitative poles are clearly separated in an absolute type manner (with the qualitative aspect thereby reduced to the quantitative).
Therefore from this perspective, the natural numbers 1, 2, 3, 4,.... are viewed as independent number entities in a quantitative manner.
2) we can view it - in the largely unrecognised - qualitative manner, where the natural numbers 1, 2, 3, 4 in interdependent terms (in the recognition of what is common to different number classes).
So again if we have two rows of items with 3 items in each row, the number "3" now represents what is common to both rows. Therefore, in this sense we are making reference directly to its qualitative - rather than quantitative - meaning.
Of course initially in identifying each row we must recognise "3" in the customary independent manner (as quantitative). However, when we then recognise its commonality with respect to both rows, we switch to the qualitative aspect.
This in fact explains the deeper reason why multiplication necessarily entails both quantitative and qualitative aspects of number transformation.
Thus 2 * 3 = 6 in quantitative terms). However 2 * 3 equally entails a qualitative transformation in a 2-dimensional fashion (though this then is edited out of conventional mathematical interpretation).
Therefore the deeper reason as to why multiplication also entails a qualitative transformation in the nature of units is due to the necessary switch that is involved with respect to the recognition of number (from its independent to its interdependent nature).
There is another helpful way of viewing this qualitative aspect of number.
Once again, to illustrate, we can use "3" to refer to the quantitative aspect of number.
However when we use "3" in the qualitative sense, it can be referred to as "threeness" (i.e. the quality of 3).
Therefore, in operations, a continual switching takes place as between number as representing a quantity, and its corresponding qualitative nature as "numberness".
In psychological terms, quantitative and qualitative aspects respectively are related to the manner in which both conscious and unconscious interact.
In direct terms we can identify the quantitative aspect with (conscious) rational recognition; however the qualitative aspect relates directly to (unconscious) intuitive recognition.
Therefore the customary reduction of qualitative to quantitative interpretation in Conventional Mathematics corresponds directly with the corresponding reduction of intuitive to rational type meaning.
This is certainly true in formal terms, where Mathematics is presented as a body of rational type truths. However indirectly of course, the intuitive aspect must be always involved, as mathematical operations in quantitative terms are strictly speaking meaningless in the absence of the qualitative notion of a common relationship.
The difficulty in dynamic interactive terms is that the qualitative can always be shown to have a corresponding quantitative aspect (just as the quantitative can be shown to have a qualitative aspect).
Again this is directly analogous with quantum physics where particles exhibit wave like characteristics and waves particle like characteristics, respectively.
Thus when one merely concentrates on the quantitative like aspect (associated with qualitative type understanding) then it becomes easy to simply reduce the qualitative in a quantitative manner.
In psychological terms, one switches to the conceptual notion of number through dynamic negation of number perceptions. Now when the initial number perceptions are identified in a quantitative manner, the initial conceptual recognition will be of a qualitative intuitive nature (where it is seen as potentially applying in an infinite manner to all numbers).
However in practice the understanding of the number concept is then quickly reduced in a quantitative rational manner (where it is now understood as actually applying in a finite manner to all numbers).
In reverse fashion, the recognition of distinct number perceptions, implies the dynamic negation of number concepts. Again the initial realisation is implicitly of an intuitive nature. However this is quickly reduced in a rational manner.
Therefore, though in the dynamics of understanding both finite (rational) and infinite (intuitive) notions ceaselessly interact in both quantitative and qualitative terms, standard mathematical interpretation reduces all this in a finite rational fashion relating merely to the quantitative aspect.
In this approach the infinite is then misleadingly viewed as merely a linear extension of the finite notion.
However, when one properly appreciates the unique distinctiveness of both the quantitative and qualitative aspects of number, then the key issue relates as to consistent use with respect to the combination of both aspects.
And this is where the zeta zeros play a crucial role!
Monday, June 22, 2015
Zeta Zeros Made Simple (1)
My intention in these blog entries is to keep approaching the zeta zeros from a variety of perspectives, so as to impress upon the reader their fundamental importance in a manner that ultimately becomes intuitively obvious with respect to experience.
Once again however, I emphasise two sets of zeros i.e. Zeta 1 and Zeta 2, relating to both the cardinal and ordinal aspects of the number system respectively.
The first of these - frequently referred to as the Riemann zeros - relating to the cardinal aspect, is intimately tied up with the distinction as between addition and multiplication.
Because of the reductionist (1-dimensional) rational nature of Conventional Mathematics, no proper means exists for the distinction of the quantitative and qualitative aspects of number, with the qualitative aspect in every context necessarily reduced in a merely quantitative manner.
So for example, the natural numbers 1, 2, 3,..., can be used in two distinctive senses that are quantitative and qualitative with respect to each other.
The first sense is the accepted quantitative meaning.
So, again if I take the number "3" to illustrate, in quantitative terms this is understood unambiguously in an absolute manner as independent.
This then could be represented as 1 + 1 + 1, i.e. as composed in quantitative terms of homogeneous units (that literally lack any qualitative distinction).
When we we are engaged in addition, this first notion of number is directly involved.
Thus 3 + 2 for example = (1 + 1 + 1) + (1 + 1) = 1 + 1+ 1 + 1 + 1 i,e 5!
Now what is often forgotten however with respect to conventional addition is that strictly speaking, no number can have an absolute independent identity. Rather implicitly - even with respect to addition - the recognition must exist that each number necessarily has a qualitative relationship to the common notion of number (as dimension).
In other words all the natural numbers lie on a common line (which is thereby 1-dimensional).
Therefore it is through the common membership of each natural number with this common line that we are thereby enabled to relate numbers with each other (as in addition).
However this common relationship with the number line, whereby we are enabled to meaningfully give order to numbers, is strictly of a qualitative, rather than quantitative nature.
Therefore to preserve the relative distinction as between quantitative and qualitative aspects, each number is now defined with respect to this default 1st dimension.
Therefore we now define the number "3" with respect to its quantitative meaning as 31..
Because the explicit focus is on the quantitative aspect of number, it is highlighted in black, whereas the dimensional aspect, which is relatively qualitative and merely implicit, is shown in a light grey colour.
Therefore, in a more refined manner, we see that though the quantitative and qualitative aspects of number are now both recognised, that addition entails that explicit emphasis be placed on the quantitative aspect.
However the significance of this is that - strictly - even addition must now be understood in a dynamic relative manner. This is very much analogous to the designation of a turn at a crossroads as either left and right (which necessarily depends on the direction from which the crossroads is approached).
In other words, we can still make unambiguous distinctions as to a turn i.e. as either left or right, but now this clearly enjoys but an arbitrary relative meaning (depending on context).
Now what is not all clearly recognised from the conventional mathematical perspective is that when we switch to multiplication, the sense in which number is used, likewise changes.
We have already explained the addition of 3 and 2 i.e. 3 + 2.
We now look at the corresponding multiplication of 3 and 2, i.e. 3 * 2.
To illustrate this operation we can imagine two rows (with 3 items in each row).
If we treated this operation as just an extended form of addition then clearly we could represent 3 * 2 as 3 + 3 = 6.
However, this entails a hidden form of reductionism that completely conceals the true distinction of multiplication from addition.
So we have two rows (with 3 times in each row).
Therefore what is crucial with respect to multiplication is the recognition that these 3 items are common to both rows. So therefore instead of adding all individual items (as independent of each other), through this recognition of mutual commonality (with respect to the items in each row) we can thereby multiply 3 by the number of rows involved (which in this case is 2).
Therefore crucially the operation of multiplication entails the opposite complementary notion of number, as what is common (i.e. interdependent) as between different classes.
So in this case we have two classes (i.e rows) with 3 the number that is common in both cases.
However we are now using the number "3" is an altogether different sense than before - not in an independent quantitative manner - but rather in the complementary qualitative interdependent sense (where its relationship with other similar classes is explicitly recognised).
The clear recognition of this point is of the greatest utmost importance, for when one grasps its significance one then clearly realises that the entire conventional edifice of Mathematics is simply not fit for purpose.
And this is ably demonstrated by the fact that Conventional Mathematics - by its very nature - lacks the means for clearly recognisiing the complementary quantitative and qualitative aspects of number, which alternate when one switches from the operation of addition to multiplication.
Again n this simple example that I have used, we have associated the number "3" with 2 (in the recognition of two common classes of 3 members).
However the number "3" - which I am using here to illustrate - can be associated with any other natural number (on the dimensional line).
So what really has happened here is that the relative emphasis on base number (3) and dimensional number (1) has now switched in complementary fashion.
Therefore in the case of multiplication, we now place explicit emphasis on the dimensional aspect of 1 (as representing the common relationship between numbers) while the emphasis on the quantitative nature of 3 is now merely implicit, though of course we must be able to implicitly recognise 3 in quantitative terms, before we can explicitly realise its common identity with other classes of 3 (in a qualitative manner).
Therefore we represent this latter qualitative meaning of 3 as 31.
Therefore we have now demonstrated how the operations of addition and multiplication entail two different notions of number (that in dynamic interactive terms are complementary in experience).
With addition the meaning of a number such as "3" is explicitly quantitative in nature, while the 1-dimensional line, that is relatively of a qualitative nature - establishing its relationship with other numbers - remains merely implicit.
Once again this notion of 3 is represented as 31.
With multiplication, by contrast the meaning of "3" has now switched in complementary fashion, so that it is explicitly qualitative in nature (relating directly to its dimensional nature), while the quantitative meaning of "3" is now merely implicit.
This notion of 3 is represented as 31 (with the dimensional number of 1 now highlighted in black and the number "3" represented in contrasting light grey fashion).
In the next entry, I will establish why these distinctions are crucial for appreciation of the true role of the Riemann (i.e. Zeta 1) zeros.
Once again however, I emphasise two sets of zeros i.e. Zeta 1 and Zeta 2, relating to both the cardinal and ordinal aspects of the number system respectively.
The first of these - frequently referred to as the Riemann zeros - relating to the cardinal aspect, is intimately tied up with the distinction as between addition and multiplication.
Because of the reductionist (1-dimensional) rational nature of Conventional Mathematics, no proper means exists for the distinction of the quantitative and qualitative aspects of number, with the qualitative aspect in every context necessarily reduced in a merely quantitative manner.
So for example, the natural numbers 1, 2, 3,..., can be used in two distinctive senses that are quantitative and qualitative with respect to each other.
The first sense is the accepted quantitative meaning.
So, again if I take the number "3" to illustrate, in quantitative terms this is understood unambiguously in an absolute manner as independent.
This then could be represented as 1 + 1 + 1, i.e. as composed in quantitative terms of homogeneous units (that literally lack any qualitative distinction).
When we we are engaged in addition, this first notion of number is directly involved.
Thus 3 + 2 for example = (1 + 1 + 1) + (1 + 1) = 1 + 1+ 1 + 1 + 1 i,e 5!
Now what is often forgotten however with respect to conventional addition is that strictly speaking, no number can have an absolute independent identity. Rather implicitly - even with respect to addition - the recognition must exist that each number necessarily has a qualitative relationship to the common notion of number (as dimension).
In other words all the natural numbers lie on a common line (which is thereby 1-dimensional).
Therefore it is through the common membership of each natural number with this common line that we are thereby enabled to relate numbers with each other (as in addition).
However this common relationship with the number line, whereby we are enabled to meaningfully give order to numbers, is strictly of a qualitative, rather than quantitative nature.
Therefore to preserve the relative distinction as between quantitative and qualitative aspects, each number is now defined with respect to this default 1st dimension.
Therefore we now define the number "3" with respect to its quantitative meaning as 31..
Because the explicit focus is on the quantitative aspect of number, it is highlighted in black, whereas the dimensional aspect, which is relatively qualitative and merely implicit, is shown in a light grey colour.
Therefore, in a more refined manner, we see that though the quantitative and qualitative aspects of number are now both recognised, that addition entails that explicit emphasis be placed on the quantitative aspect.
However the significance of this is that - strictly - even addition must now be understood in a dynamic relative manner. This is very much analogous to the designation of a turn at a crossroads as either left and right (which necessarily depends on the direction from which the crossroads is approached).
In other words, we can still make unambiguous distinctions as to a turn i.e. as either left or right, but now this clearly enjoys but an arbitrary relative meaning (depending on context).
Now what is not all clearly recognised from the conventional mathematical perspective is that when we switch to multiplication, the sense in which number is used, likewise changes.
We have already explained the addition of 3 and 2 i.e. 3 + 2.
We now look at the corresponding multiplication of 3 and 2, i.e. 3 * 2.
To illustrate this operation we can imagine two rows (with 3 items in each row).
If we treated this operation as just an extended form of addition then clearly we could represent 3 * 2 as 3 + 3 = 6.
However, this entails a hidden form of reductionism that completely conceals the true distinction of multiplication from addition.
So we have two rows (with 3 times in each row).
Therefore what is crucial with respect to multiplication is the recognition that these 3 items are common to both rows. So therefore instead of adding all individual items (as independent of each other), through this recognition of mutual commonality (with respect to the items in each row) we can thereby multiply 3 by the number of rows involved (which in this case is 2).
Therefore crucially the operation of multiplication entails the opposite complementary notion of number, as what is common (i.e. interdependent) as between different classes.
So in this case we have two classes (i.e rows) with 3 the number that is common in both cases.
However we are now using the number "3" is an altogether different sense than before - not in an independent quantitative manner - but rather in the complementary qualitative interdependent sense (where its relationship with other similar classes is explicitly recognised).
The clear recognition of this point is of the greatest utmost importance, for when one grasps its significance one then clearly realises that the entire conventional edifice of Mathematics is simply not fit for purpose.
And this is ably demonstrated by the fact that Conventional Mathematics - by its very nature - lacks the means for clearly recognisiing the complementary quantitative and qualitative aspects of number, which alternate when one switches from the operation of addition to multiplication.
Again n this simple example that I have used, we have associated the number "3" with 2 (in the recognition of two common classes of 3 members).
However the number "3" - which I am using here to illustrate - can be associated with any other natural number (on the dimensional line).
So what really has happened here is that the relative emphasis on base number (3) and dimensional number (1) has now switched in complementary fashion.
Therefore in the case of multiplication, we now place explicit emphasis on the dimensional aspect of 1 (as representing the common relationship between numbers) while the emphasis on the quantitative nature of 3 is now merely implicit, though of course we must be able to implicitly recognise 3 in quantitative terms, before we can explicitly realise its common identity with other classes of 3 (in a qualitative manner).
Therefore we represent this latter qualitative meaning of 3 as 31.
Therefore we have now demonstrated how the operations of addition and multiplication entail two different notions of number (that in dynamic interactive terms are complementary in experience).
With addition the meaning of a number such as "3" is explicitly quantitative in nature, while the 1-dimensional line, that is relatively of a qualitative nature - establishing its relationship with other numbers - remains merely implicit.
Once again this notion of 3 is represented as 31.
With multiplication, by contrast the meaning of "3" has now switched in complementary fashion, so that it is explicitly qualitative in nature (relating directly to its dimensional nature), while the quantitative meaning of "3" is now merely implicit.
This notion of 3 is represented as 31 (with the dimensional number of 1 now highlighted in black and the number "3" represented in contrasting light grey fashion).
In the next entry, I will establish why these distinctions are crucial for appreciation of the true role of the Riemann (i.e. Zeta 1) zeros.
Monday, June 1, 2015
More on Dynamic Nature of Number (5)
What we are concerned with here is the key issue of how to convert the qualitative nature of number consistently in a (reduced) quantitative manner.
Therefore when a number, such as 3 (representing the base) is defined in an independent quantitative cardinal manner, the corresponding qualitative dimensional notion of 3 is defined in an interdependent ordinal manner incorporating the relationship of 1st, 2nd and 3rd respectively.
This the former notion of 3 (as cardinal) is represented in Type 1 terms as 31.
Then the latter notion of 3 (as ordinal) is represented in Type 2 terms, as 11, 12 and 13 respectively.
Therefore to express 1st, 2nd and 3rd in the context of 3 members we obtain the cube root of each
i.e. 11/3, 12/3 and 13/3.
We have already seen how the notion of nth in the context of n equates to the cardinal notion of 1n/n = 1.
So once again 1st (in the context of 1) + 2nd (in the context of 2) + 3rd (in the context of 3) = 1 + 1 + 1 = 3. Thus in this limited case ordinal reduces successfully to direct cardinal interpretation.
However in now considering 1st, 2nd and 3rd, in relation to each other (representing the 3 members of a group), we move from a linear to a circular notion of identity.
Thus 11/3, 12/3 and 13/3 are represented in geometrical terms as 3 equidistant points on the circle of unit radius (in the complex plane).
So 11/3 represents the 1st (in the context of 3) in an indirect quantitative manner as the complex number – .5 + .866i
12/3 represents the 2nd (in the context of 3) in an indirect quantitative manner as the complex number – .5 – .866i.
And 13/3 as we have seen represents the 3rd (in the context of 3) directly in a quantitative manner as 1.
What is fascinating here is that we have now used the Type 2 aspect of the number system - where explicit emphasis is on the dimensional aspect of number - to provide an alternative interpretation of rational numbers as fractions.
In conventional terms 1/3, 2/3 and 3/3 are given a standard quantitative meaning (which however betrays an unrecognised form of qualitative reductionism).
For example if we have a small cake divided into 3 slices we would represent 1 slice as 1/3 of the cake, 2 slices as 2/3 of the cake and 3 slices as 3/3 of the cake i.e. identical to the entire cake as a unit.
Thus here, 1 (in the context of 3) = 1/3 i.e. (1/3)1.
2 (in the context of 3) = 2/3 i.e. (2/3)1.
3 (in the context of 3) = 3/3 i.e. (3/3)1 = 1.
In this way, seemingly we can deal with fractional notions in an unambiguous quantitative manner.
However this conceals a big difficulty.
If we look at the 3 slices in an independent quantitative manner, they represent individual whole units.
And the cake itself represents an individual whole unit.
However to see the slices as a fraction of the whole cake, we need to switch from the notion of whole to part (with respect to slices) in relation to the overall cake which is now seen in complementary terms as whole.
Therefore as the notions of whole and part are quantitative as to qualitative (and qualitative as to quantitative) in relation to each other.
Therefore implicitly involved in this quantitative recognition of fractions is a corresponding qualitative recognition of an ordinal nature.
In fact the qualitative aspect of what is involved in obtaining (quantitative) fractions can be compellingly demonstrated in holistic mathematical terms.
Thus we start with the cardinal number 3 represented in Type 1 terms as 31.
However 1/3 = 3 – 1.
Thus moving from a natural number to its reciprocal (as a fraction) entails in holistic mathematical terms, the negation of the 1st dimension i.e. the negation of rational linear type understanding at a conscious level..
This then makes way for the complementary intuitive recognition at an unconscious level of the simultaneous correspondence of quantitative and qualitative notions which then enables quantitative recognition to switch from initial recognition of 3 as whole units to the corresponding recognition of these units as part.
Thus in the example of the cake, the ability to recognise the 3 part slices as equal to the 1 whole cake entails the corresponding ability to switch as between whole and part (and part and whole notions).
In reverse manner we have shown that through the Type 2 aspect of the number system that the same fractions now explicitly acquire a qualitative meaning as 1st, 2nd and 3rd respectively.
However again this qualitative understanding implicitly requires the corresponding quantitative recognition of 3 .
Thus once more, in terms of the Type 1 interpretation of fractions 1/3, 2/3 and 3/3 respectively represent the relationship between parts and whole (where 3 e.g. 3 part slices = 1 whole cake).
So though here quantitative recognition is explicit, corresponding qualitative recognition (of the two-way relationship between wholes and parts) implicitly is also required.
Then in terms of the Type 2 interpretation, 1/3, 2/3 and 3/3 now respectively represent the explicit qualitative relationship as between the 1st, 2nd and 3rd members of a group. However implicitly this requires the corresponding quantitative recognition of the group as containing 3 members.
Therefore when a number, such as 3 (representing the base) is defined in an independent quantitative cardinal manner, the corresponding qualitative dimensional notion of 3 is defined in an interdependent ordinal manner incorporating the relationship of 1st, 2nd and 3rd respectively.
This the former notion of 3 (as cardinal) is represented in Type 1 terms as 31.
Then the latter notion of 3 (as ordinal) is represented in Type 2 terms, as 11, 12 and 13 respectively.
Therefore to express 1st, 2nd and 3rd in the context of 3 members we obtain the cube root of each
i.e. 11/3, 12/3 and 13/3.
We have already seen how the notion of nth in the context of n equates to the cardinal notion of 1n/n = 1.
So once again 1st (in the context of 1) + 2nd (in the context of 2) + 3rd (in the context of 3) = 1 + 1 + 1 = 3. Thus in this limited case ordinal reduces successfully to direct cardinal interpretation.
However in now considering 1st, 2nd and 3rd, in relation to each other (representing the 3 members of a group), we move from a linear to a circular notion of identity.
Thus 11/3, 12/3 and 13/3 are represented in geometrical terms as 3 equidistant points on the circle of unit radius (in the complex plane).
So 11/3 represents the 1st (in the context of 3) in an indirect quantitative manner as the complex number – .5 + .866i
12/3 represents the 2nd (in the context of 3) in an indirect quantitative manner as the complex number – .5 – .866i.
And 13/3 as we have seen represents the 3rd (in the context of 3) directly in a quantitative manner as 1.
What is fascinating here is that we have now used the Type 2 aspect of the number system - where explicit emphasis is on the dimensional aspect of number - to provide an alternative interpretation of rational numbers as fractions.
In conventional terms 1/3, 2/3 and 3/3 are given a standard quantitative meaning (which however betrays an unrecognised form of qualitative reductionism).
For example if we have a small cake divided into 3 slices we would represent 1 slice as 1/3 of the cake, 2 slices as 2/3 of the cake and 3 slices as 3/3 of the cake i.e. identical to the entire cake as a unit.
Thus here, 1 (in the context of 3) = 1/3 i.e. (1/3)1.
2 (in the context of 3) = 2/3 i.e. (2/3)1.
3 (in the context of 3) = 3/3 i.e. (3/3)1 = 1.
In this way, seemingly we can deal with fractional notions in an unambiguous quantitative manner.
However this conceals a big difficulty.
If we look at the 3 slices in an independent quantitative manner, they represent individual whole units.
And the cake itself represents an individual whole unit.
However to see the slices as a fraction of the whole cake, we need to switch from the notion of whole to part (with respect to slices) in relation to the overall cake which is now seen in complementary terms as whole.
Therefore as the notions of whole and part are quantitative as to qualitative (and qualitative as to quantitative) in relation to each other.
Therefore implicitly involved in this quantitative recognition of fractions is a corresponding qualitative recognition of an ordinal nature.
In fact the qualitative aspect of what is involved in obtaining (quantitative) fractions can be compellingly demonstrated in holistic mathematical terms.
Thus we start with the cardinal number 3 represented in Type 1 terms as 31.
However 1/3 = 3 – 1.
Thus moving from a natural number to its reciprocal (as a fraction) entails in holistic mathematical terms, the negation of the 1st dimension i.e. the negation of rational linear type understanding at a conscious level..
This then makes way for the complementary intuitive recognition at an unconscious level of the simultaneous correspondence of quantitative and qualitative notions which then enables quantitative recognition to switch from initial recognition of 3 as whole units to the corresponding recognition of these units as part.
Thus in the example of the cake, the ability to recognise the 3 part slices as equal to the 1 whole cake entails the corresponding ability to switch as between whole and part (and part and whole notions).
In reverse manner we have shown that through the Type 2 aspect of the number system that the same fractions now explicitly acquire a qualitative meaning as 1st, 2nd and 3rd respectively.
However again this qualitative understanding implicitly requires the corresponding quantitative recognition of 3 .
Thus once more, in terms of the Type 1 interpretation of fractions 1/3, 2/3 and 3/3 respectively represent the relationship between parts and whole (where 3 e.g. 3 part slices = 1 whole cake).
So though here quantitative recognition is explicit, corresponding qualitative recognition (of the two-way relationship between wholes and parts) implicitly is also required.
Then in terms of the Type 2 interpretation, 1/3, 2/3 and 3/3 now respectively represent the explicit qualitative relationship as between the 1st, 2nd and 3rd members of a group. However implicitly this requires the corresponding quantitative recognition of the group as containing 3 members.
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