We have seen that number has in fact two distinct meanings:
1) The Type 1 (cardinal) interpretation where each number is composed of independent units.
2) The Type 2 (ordinal) interpretation where each number is composed of interdependent i.e. related units.
Now this has a crucial bearing on the nature of the primes.
From the Type 1 (cardinal) perspective, the primes are considered as the unique building blocks from which the natural numbers are formed.
However from the Type 2 (ordinal) perspective, each prime is uniquely defined by its natural number members.
So in the former (Type 1) case, 5 as a prime, constitutes one of the essential building blocks from which the natural numbers are derived.
However in the latter (Type 2) case, 5 is already defined by its 1st, 2nd, 3rd, 4th and 5th natural number members.
So already included in this notion of a prime is the composite natural number 4!
Now because Conventional Mathematics is defined exclusively in Type 1 terms, with ordinal notions - as I have carefully explained - in effect reduced to cardinal, this issue of the necessary two-way interdependence of the primes and natural numbers is completely overlooked!
Because cardinal identity is solely considered in a quantitative manner, an utterly misleading picture emerges, whereby the relationship as between primes and natural numbers is considered to be solely one-way (with the natural numbers unambiguously determined by the primes)
So the first step in moving to a truly coherent dynamic interactive nature of the number system, is to recognise the equal importance of both the Type 1 and Type 2 aspects.
The major issue that then arises is that of mutual conversion of each aspect in terms of the other.
So from one perspective, how do we convert the Type 2 (qualitative) aspect in a consistent Type 1 (quantitative) manner?
Equally from the complementary perspective, how do we convert the Type 1 (quantitative) aspect in a consistent Type 2 (qualitative) manner?
And this is where the Zeta 2 zeros are so important.
Now in general terms for any prime number t, the Zeta 2 zeros are given as the solutions to the finite equation,
1 + s1 + s2 + s3 + ... + st – 1 = 0
These zeros express the truly relative i.e. circular nature of ordinal positions as the t roots of 1, (excluding the default case of 1 where ordinal becomes inseparable from cardinal identity in an absolute manner).
Therefore in the case of our example of the prime 5, the solutions to
1 + s1 + s2 + s3 + s4 = 0,
express in a Type 1 quantitative manner, the notions of 1st, 2nd, 3rd and 4th in the context of 5 members.
Again these four solutions are given as .309 + .951 i, – .809 + .588 i, – .809 + .588 i and .309 – 951 i (correct to 3 decimal places).
Then we combine these 4 values with the default value of 1 (representing 5th in the context of 5) the total sum = 0, expressing the fact, that as these values are expressing qualitative notions of relative interdependence, their collective sum has no quantitative value.
So (.309 + .951 i) + ( – .809 + .588 i) + ( – .809 + .588 i) + (.309 – 951 i) + 1 = 0
And remember again these quantitative values represent the conversion of the qualitative ordinal notions of,
1st + 2nd + 3rd + 4th + 5th = 0
We also have the complementary problem of converting our standard Type 1 notion of number consistently in a Type 2 manner!
This in fact represents the same set of values that we have already obtained. However it now requires that these values (the five roots) be understood in a true holistic fashion. This requires moving from a 1-dimensional to 5-dimensional appreciation, which requires a specialisation of intuitive ability that is yet not yet remotely recognised (certainly within Mathematics)!
However again it is perhaps possible to express what is required with reference to the simplest case.
In other words how do we convert the standard (Type 1) linear quantitative notion of 2 in a coherent Type 2 qualitative manner?
So we start with the 2 roots of 1 i.e. + 1 and – 1. However the task is now to understand these in a genuine holistic qualitative manner. This in turn requires authentic 2-dimensional appreciation which entails the ability to see number reality as representing the interaction of opposite poles, that are positive and negative in relation to each other.
And as I have explained many times before, this relates to the manner in which number "objects" (as external) continually interact in experience with mental constructs (as - relatively - internal).
Thus we no longer view mathematical reality (in 2-dimensional) terms as an abstract objective world (independent of the enquirer) but rather as a dynamic interactive process entailing both external and internal poles that are + 1 and – 1 with respect to each other.
So in raising i.e. transforming through intuitive insight the qualitative nature of these two roots (in a true 2-dimensional fashion) we obtain 11 and 12 representing 1st and 2nd (of 2 dimensions).
When we can additionally combine the two-way interactive nature of whole and part, we now have 4 polarities (external/internal and whole/part) that can be viewed like the 4 directions of a compass.
So in general terms all "higher" dimensional appreciation relates to a distinctive manner in which one configures experience with respect to these 4 co-ordinates. However obtaining specialisation with respect to appreciation of such interaction will require considerable evolution in our (unconscious) intuitive abilities that have not yet been remotely tapped!
However the truly important thing to appreciate at this stage is the fundamental two-way role of the Zeta 2 zeros.
Put simply, they enable the seamless consistent conversion as between both aspects of the primes.
Thus again from one perspective, we are able to convert the Type 2 (ordinal) natural number members of each prime in a consistent Type 1 quantitative manner.
Then equally from the other perspective, we are able to convert the Type 1 (cardinal) notion of each prime in a consistent Type 2 qualitative manner, where its true dimensional nature (as related ordinal members) can be readily appreciated.
However none of this can have any resonance, while we insist on interpreting mathematical reality in the present greatly reduced manner (that solely recognises the Type 1 quantitative aspect).
Clearly a massive revolution is now required with respect to mathematical perspective, for at present through our collective blindness, we are completely misinterpreting the true nature of number, and thereby just about everything else in Mathematics and Science.
Monday, August 31, 2015
Sunday, August 30, 2015
Zeta Zeros and the Changing Nature of Number (3)
Yesterday we looked at the Type 1 notion of number with respect to our example of 5 chairs.
Again in this context 5 has a (reduced) quantitative meaning as 5 = 1 + 1 + 1 + 1 + 1.
However in the dynamics of understanding, 5 keeps switching from its "part" notion of 5 individual items to its "whole" notion of 1 collective group of items (and vice versa). And these are strictly quantitative as to qualitative (and qualitative as to quantitative) with respect to each other.
In this way we are able to recognise the chairs both as whole units in their own right and yet parts with respect to the single group!
Once again in conventional interpretation, this dynamic two -way interactive relationship as between whole and parts (in quantitative and qualitative terms) is reduced in an absolute quantitative manner.
So in Type 1 terms, when we say,
5 = 1 + 1 + 1 + 1 + 1,
each of the individual units is homogeneous in nature and thereby lacking any qualitative distinction!
However there is an alternative Type 2 complementary manner of defining this relationship as,
5 = 1st + 2nd + 3rd + 4th + 5th.
In this case, whereas each of the individual units now possesses a unique qualitative distinction in ordinal terms, the collective sum of the units lacks any quantitative distinction!
Thus 5 - as indeed all numbers and mathematical symbols - has a Type 1 analytic meaning (without qualitative distinction) and a Type 2 holistic meaning (without quantitative distinction).
Indirectly this Type 2 meaning can be converted in a Type 1 quantitative manner.
So in Type 2 terms the 5 fractions 1/5, 2/5, 3/5, 4/5 and 5/5 are expressed as,
Depending on the choices made with respect to position, any of the 4 results can be chosen for each of any 4 members of the group (with the 5th = 1), with the others interchanging in circular manner as required so that the overall sum of the 5 = 0.
The relative nature of what is involved can be most easily understood in the case of a number group of just 2 members.
Now what is 1st or 2nd in this group is purely arbitrary before the initial choice is made! In this sense it is similar to the quantum world whereby a particle can exist as a superposition of states before its actual existence is determined through making an arbitrary decision as to location.
So in potential terms, if one particular item is chosen as the 1st (i.e. whereby it is posited as the 1st) this thereby negates the 2nd item (as 1st).
However equally if the other item is now posited as 1st, then the remaining item is thereby negated with respect to this position.
We could say therefore that the two positions are represented by the two roots of 1 i.e. + 1 and – 1 .
And the sum of these roots = 0, which expresses the merely relative notions of these positions!
Now I am already using + 1 and – 1 in a holistic manner that relates directly to the (intuitive) unconscious aspect of relative interdependence . This is in striking contrast to the corresponding analytic manner that relates directly to the (rational) conscious aspect of absolute independence.
Though of course we can never in experience totally separate both poles as the notion of interdependence can only meaningfully start from what is already seen as independent (and vice versa).
Though the unconscious dynamics are harder to appreciate, with respect to our example of a number group of 5, we would now determine the relative ordinal positions of the 5 members through obtaining the 5 roots of 1.
In experiential terms, this enables one to give a relatively independent meaning to each ordinal position, while recognising that in collective interdependent terms they cancel each other out!
In this sense therefore ordinal meaning lies at the other extreme from cardinal.
Whereas cardinal meaning is understood in an analytic quantitative manner with numbers independent of each other, ordinal meaning is by contrast understood in a holistic qualitative manner with all numbers strictly interdependent with each other!
Again in this context 5 has a (reduced) quantitative meaning as 5 = 1 + 1 + 1 + 1 + 1.
However in the dynamics of understanding, 5 keeps switching from its "part" notion of 5 individual items to its "whole" notion of 1 collective group of items (and vice versa). And these are strictly quantitative as to qualitative (and qualitative as to quantitative) with respect to each other.
In this way we are able to recognise the chairs both as whole units in their own right and yet parts with respect to the single group!
Once again in conventional interpretation, this dynamic two -way interactive relationship as between whole and parts (in quantitative and qualitative terms) is reduced in an absolute quantitative manner.
So in Type 1 terms, when we say,
5 = 1 + 1 + 1 + 1 + 1,
each of the individual units is homogeneous in nature and thereby lacking any qualitative distinction!
However there is an alternative Type 2 complementary manner of defining this relationship as,
5 = 1st + 2nd + 3rd + 4th + 5th.
In this case, whereas each of the individual units now possesses a unique qualitative distinction in ordinal terms, the collective sum of the units lacks any quantitative distinction!
Thus 5 - as indeed all numbers and mathematical symbols - has a Type 1 analytic meaning (without qualitative distinction) and a Type 2 holistic meaning (without quantitative distinction).
Indirectly this Type 2 meaning can be converted in a Type 1 quantitative manner.
So in Type 2 terms the 5 fractions 1/5, 2/5, 3/5, 4/5 and 5/5 are expressed as,
11/5,
Now the reason we divide by 5 is because we are attempting to express a 5-dimensional notion (i.e. the related notion of 5) in a 1-dimensional (linear) manner through which the conventional independent notion of number is interpreted!
Now the reason we divide by 5 is because we are attempting to express a 5-dimensional notion (i.e. the related notion of 5) in a 1-dimensional (linear) manner through which the conventional independent notion of number is interpreted!
Now apart from the last 15/5 = 1, all the others have a merely relative meaning.
For example, if I identify again a group of 5 chairs and identify 4 of these chairs as the 1st, 2nd, 3rd and 4th respectively, then - by definition - the one remaining chair is unambiguously the 5th member in this case.
Therefore whenever we identify a member of a group as the nth (of a group of n) ordinal meaning is reduced in a cardinal manner.
So now 5 = 1 + 1 + 1 + 1 + 1 = 1st + 2nd + 3rd + 4th + 5th!
However if we leave the initial choice open, all ordinal positions - depending on context - can be associated with each of the 5 members.
As I explained in an earlier blog entry (the 1st of this series), when we isolate this last case as the one trivial solution, the other 4 non-trivial roots will be expressed by the equation;
1 + s1 + s2 + s3 + s4 = 0
These 4 solutions, .309 + .951 i, – .809 + .588 i, – .809 + .588 i and .309 – 951 i (correct to 3 decimal places), thereby express (indirectly in quantitative manner) the 1st, 2nd, 3rd and 4th relative ordinal positions (in the context of 5)
What is amazing here is that number is now serving a - holistic - rather than analytic role or alternatively a relative rather than absolute meaning.
Depending on the choices made with respect to position, any of the 4 results can be chosen for each of any 4 members of the group (with the 5th = 1), with the others interchanging in circular manner as required so that the overall sum of the 5 = 0.
The relative nature of what is involved can be most easily understood in the case of a number group of just 2 members.
Now what is 1st or 2nd in this group is purely arbitrary before the initial choice is made! In this sense it is similar to the quantum world whereby a particle can exist as a superposition of states before its actual existence is determined through making an arbitrary decision as to location.
So in potential terms, if one particular item is chosen as the 1st (i.e. whereby it is posited as the 1st) this thereby negates the 2nd item (as 1st).
However equally if the other item is now posited as 1st, then the remaining item is thereby negated with respect to this position.
We could say therefore that the two positions are represented by the two roots of 1 i.e. + 1 and – 1 .
And the sum of these roots = 0, which expresses the merely relative notions of these positions!
Now I am already using + 1 and – 1 in a holistic manner that relates directly to the (intuitive) unconscious aspect of relative interdependence . This is in striking contrast to the corresponding analytic manner that relates directly to the (rational) conscious aspect of absolute independence.
Though of course we can never in experience totally separate both poles as the notion of interdependence can only meaningfully start from what is already seen as independent (and vice versa).
Though the unconscious dynamics are harder to appreciate, with respect to our example of a number group of 5, we would now determine the relative ordinal positions of the 5 members through obtaining the 5 roots of 1.
In experiential terms, this enables one to give a relatively independent meaning to each ordinal position, while recognising that in collective interdependent terms they cancel each other out!
In this sense therefore ordinal meaning lies at the other extreme from cardinal.
Whereas cardinal meaning is understood in an analytic quantitative manner with numbers independent of each other, ordinal meaning is by contrast understood in a holistic qualitative manner with all numbers strictly interdependent with each other!
Saturday, August 29, 2015
Zeta Zeros and the Changing Nature of Number (2)
We have seen how number keeps switching between two distinctive nations that are quantitative (independent) and qualitative (interdependent) with respect to each other.
This bears remarkable comparison to the wave/particle complementarity of quantum mechanics.
So for example if we define a number in Type 1 terms e.g. 5 (51) as representing the particle aspect, then in Type 2 terms 5 ( 15) represents the corresponding wave aspect. So here 5 switches as between both its particle and wave aspects in Type 1 and Type 2 terms.
Remember again that when the Type 1 aspect is associated with the cardinal aspect, then - relatively the Type 2 aspect is thereby associated with the ordinal aspect!
Because in the dynamics of experience, we continually switch in two-way fashion as between cardinal and ordinal notions with respect to natural numbers, this likewise implies therefore that we keep switching likewise in two-way fashion as between particle and wave aspects.
And these aspects themselves can switch depending on the point of reference. So the Type 1 equally can be associated with wave and the Type 2 with particle aspects respectively.
In fact, what is not all realised - and which will cause utter consternation when eventually grasped - is that the quantum mechanical behaviour that is apparent at the sub-atomic regions of matter, is an inherent aspect of the true dynamic nature of the number system.
We will now illustrate this wave/particle like behaviour of number with respect to the recognition of 5 objects - say 5 chairs. Now in quantitative terms, the recognition of 5 implies the recognition of 4 which implies the recognition of 3 which in turn implies the recognition of 2 which finally implies the recognition of 1. So 5 ultimately represents 1 + 1 + 1 + 1 + 1.
However such recognition in fact is very subtle, in that through experience we constantly switch as between whole and part notions (and vice versa).
So implicitly in recognising the 5 chairs as independent part items, we must also recognise the overall collection of these chairs as a whole group (= 1).
Therefore the switch from the 5 individual chairs to the collective recognition of the 1 set of chairs (as the whole group) entails the corresponding switch from part to whole aspects respectively. And then in like manner, to switch back from the recognition of the 1 set of chairs to the 5 individual chairs, requires the corresponding reverse switch from whole to part aspects.
Now, this switch from part to whole (and whole to part) recognition of number, strictly entails the dynamic interaction of both the quantitative and qualitative (and qualitative and quantitative), in the two-way interaction of the Type 1 (particle) and Type 2 (wave) aspects of number respectively.
However in conventional mathematical explanation a reduced interpretation is given solely in terms of the Type 1 quantitative aspect.
So in effect the notion of the unitary whole (with respect to the group of five chairs) is reduced to the part notion of the 5 chairs in a merely quantitative manner.
Now if we look at a group of five chairs, what we see in quantitative terms are the 5 individual chairs.
However the very ability to see this group of 5, constituting in this context a unique whole (as a set) directly entails qualitative - rather than quantitative recognition. Thus in the dynamics of understanding, an intuitive recognition of the interdependence of whole and part is required.
This then enables the switch from the part recognition (of the 5 individual chairs) to the subtler whole notion of these chairs representing a group.
Now without this implicit recognition of whole/part interdependence, which is directly intuitive in nature, there would no way of making this important switch in recognition, with the important connection as between the part individual chairs (as 5) and the whole collective group (as 1) impossible to make
Now we can equally see this in reverse. We could start with the five chairs as 1 collective unit through imagining them perhaps wrapped up together in a transparent bag. So this bag (of chairs) now represents 1 in a quantitative whole manner. Now to recognise an individual chair as a part unit (as 1/5) of the whole, we have to make the opposite transition from whole to parts notions which implicitly involves the switch from quantitative to qualitative. This again is provided through the intuitive recognition of the interdependence of part and whole, enabling the decisive switch in recognition to be made.
However in explicit terms we now recognise the individual 5 chairs again in a merely quantitative manner.
Thus the frames of reference with respect to whole and part (and part and whole) recognition keep changing in the dynamics of experience. This implicitly requires the intuitive recognition of the interdependence of whole and part (and part and whole) for these switches to be made in a two-way fashion. However explicitly this is quickly reduced with both whole and part notions interpreted merely in a rational quantitative manner (as independent).
So in effect the whole is reduced to its parts in a quantitative manner.
And this gross reductionism is the most fundamental problem imaginable which pervades the entire field of Mathematics and all its related sciences!
Dealing with this problem will entail the most radical intellectual revolution in thought yet in our history.
So rather than number - all all its related mathematical notions - being understood in a merely reduced absolute manner amenable to the conscious use of (linear) reason, we will have to move to a new approach, inherently dynamic in nature. This will entail the balanced use of both (conscious) reason and (unconscious) intuition in a manner where both the quantitative and qualitative aspects of all mathematical notions can be explicitly recognised.
And then the knock-on effects of this new appreciation of number (and extended mathematical relationships) for all the sciences will be truly enormous!
Thus to follow on from our example, though we can refer to 1/5, 2/5, 3/5 4.5 and 5/5 in an absolute type quantitative manner i.e.as (1/5)1, (2/5)1, (3/5)1, (4/5)1 and (5/5)1 , as we have seen this completely ignores whole/part interaction entailing the qualitative aspect.
Thus strictly the interpretation of all fractions - as all numbers - is of a dynamic relative nature.
Thus the absolute nature of these fractions should be viewed as a limit to which the truly relative interpretation approximates. Put another way a necessary uncertainty principle applies to all numbers.
Thus an irrational number such as √2 has a relative value that cannot be represented in an absolute discrete manner.
However, all absolute numbers (of a discrete nature) are in turn representations of number interactions of a strictly relative nature!
This bears remarkable comparison to the wave/particle complementarity of quantum mechanics.
So for example if we define a number in Type 1 terms e.g. 5 (51) as representing the particle aspect, then in Type 2 terms 5 ( 15) represents the corresponding wave aspect. So here 5 switches as between both its particle and wave aspects in Type 1 and Type 2 terms.
Remember again that when the Type 1 aspect is associated with the cardinal aspect, then - relatively the Type 2 aspect is thereby associated with the ordinal aspect!
Because in the dynamics of experience, we continually switch in two-way fashion as between cardinal and ordinal notions with respect to natural numbers, this likewise implies therefore that we keep switching likewise in two-way fashion as between particle and wave aspects.
And these aspects themselves can switch depending on the point of reference. So the Type 1 equally can be associated with wave and the Type 2 with particle aspects respectively.
In fact, what is not all realised - and which will cause utter consternation when eventually grasped - is that the quantum mechanical behaviour that is apparent at the sub-atomic regions of matter, is an inherent aspect of the true dynamic nature of the number system.
We will now illustrate this wave/particle like behaviour of number with respect to the recognition of 5 objects - say 5 chairs. Now in quantitative terms, the recognition of 5 implies the recognition of 4 which implies the recognition of 3 which in turn implies the recognition of 2 which finally implies the recognition of 1. So 5 ultimately represents 1 + 1 + 1 + 1 + 1.
However such recognition in fact is very subtle, in that through experience we constantly switch as between whole and part notions (and vice versa).
So implicitly in recognising the 5 chairs as independent part items, we must also recognise the overall collection of these chairs as a whole group (= 1).
Therefore the switch from the 5 individual chairs to the collective recognition of the 1 set of chairs (as the whole group) entails the corresponding switch from part to whole aspects respectively. And then in like manner, to switch back from the recognition of the 1 set of chairs to the 5 individual chairs, requires the corresponding reverse switch from whole to part aspects.
Now, this switch from part to whole (and whole to part) recognition of number, strictly entails the dynamic interaction of both the quantitative and qualitative (and qualitative and quantitative), in the two-way interaction of the Type 1 (particle) and Type 2 (wave) aspects of number respectively.
However in conventional mathematical explanation a reduced interpretation is given solely in terms of the Type 1 quantitative aspect.
So in effect the notion of the unitary whole (with respect to the group of five chairs) is reduced to the part notion of the 5 chairs in a merely quantitative manner.
Now if we look at a group of five chairs, what we see in quantitative terms are the 5 individual chairs.
However the very ability to see this group of 5, constituting in this context a unique whole (as a set) directly entails qualitative - rather than quantitative recognition. Thus in the dynamics of understanding, an intuitive recognition of the interdependence of whole and part is required.
This then enables the switch from the part recognition (of the 5 individual chairs) to the subtler whole notion of these chairs representing a group.
Now without this implicit recognition of whole/part interdependence, which is directly intuitive in nature, there would no way of making this important switch in recognition, with the important connection as between the part individual chairs (as 5) and the whole collective group (as 1) impossible to make
Now we can equally see this in reverse. We could start with the five chairs as 1 collective unit through imagining them perhaps wrapped up together in a transparent bag. So this bag (of chairs) now represents 1 in a quantitative whole manner. Now to recognise an individual chair as a part unit (as 1/5) of the whole, we have to make the opposite transition from whole to parts notions which implicitly involves the switch from quantitative to qualitative. This again is provided through the intuitive recognition of the interdependence of part and whole, enabling the decisive switch in recognition to be made.
However in explicit terms we now recognise the individual 5 chairs again in a merely quantitative manner.
Thus the frames of reference with respect to whole and part (and part and whole) recognition keep changing in the dynamics of experience. This implicitly requires the intuitive recognition of the interdependence of whole and part (and part and whole) for these switches to be made in a two-way fashion. However explicitly this is quickly reduced with both whole and part notions interpreted merely in a rational quantitative manner (as independent).
So in effect the whole is reduced to its parts in a quantitative manner.
And this gross reductionism is the most fundamental problem imaginable which pervades the entire field of Mathematics and all its related sciences!
Dealing with this problem will entail the most radical intellectual revolution in thought yet in our history.
So rather than number - all all its related mathematical notions - being understood in a merely reduced absolute manner amenable to the conscious use of (linear) reason, we will have to move to a new approach, inherently dynamic in nature. This will entail the balanced use of both (conscious) reason and (unconscious) intuition in a manner where both the quantitative and qualitative aspects of all mathematical notions can be explicitly recognised.
And then the knock-on effects of this new appreciation of number (and extended mathematical relationships) for all the sciences will be truly enormous!
Thus to follow on from our example, though we can refer to 1/5, 2/5, 3/5 4.5 and 5/5 in an absolute type quantitative manner i.e.as (1/5)1, (2/5)1, (3/5)1, (4/5)1 and (5/5)1 , as we have seen this completely ignores whole/part interaction entailing the qualitative aspect.
Thus strictly the interpretation of all fractions - as all numbers - is of a dynamic relative nature.
Thus the absolute nature of these fractions should be viewed as a limit to which the truly relative interpretation approximates. Put another way a necessary uncertainty principle applies to all numbers.
Thus an irrational number such as √2 has a relative value that cannot be represented in an absolute discrete manner.
However, all absolute numbers (of a discrete nature) are in turn representations of number interactions of a strictly relative nature!
Thursday, August 27, 2015
Zeta Zeros and the Changing Nature of Number (1)
In conventional terms, we think of number in an absolute manner as possessing an independent quantitative identity that remains unambiguous in every context.
This is especially the case in respect to cardinal numbers, where each number e.g. "2", has a definite fixed meaning in this sense..
This is also the case with respect to the ordinal notion of number, where each ordinal number is interpreted with respect to the last unit of a number group.
So 1st is - by definition - the last unit of a group of 1 member. 2nd is then defined as the last unit of a group of 2, 3rd as the last unit of a group of 3 and so on.
In this way, the ordinal notion of number is in effect reduced in a cardinal manner.
However there is a crucial difference as between the notion of number (defined as a specific number with respect to a given dimension) and the corresponding notion (where number refers directly to the dimension which now generally applies to all specific numbers).
In the first case number has an independent meaning. So when we define numbers in a 1-dimensional manner (as lying on the number line), each number e.g. 3 is given an absolute independent identity.
However when we probe into the dimensional notion of number, we are required to accept a corresponding interdependent notion of number (where each dimension is related to the others in an organised manner).
Thus to move from the notion of a 1-dimensional representation of an object (i.e. the line) to a 2-dimensional representation (e.g. a square) the 2nd dimension must be clearly related to the 1st.
So if the 1st dimension represents length, then the second (drawn at right angles will represent width).
So they clearly are not independent of each other but related in a definite manner as interdependent.
And then if we proceed further to 3-dimensional representation the 3rd dimension (the height) must again be related in definite manner (as interdependent with the other 2 dimensions).
Therefore though the base notion of number (as within a given dimension) is independent in a quantitative manner, the corresponding dimensional notion is interdependent - as the relationship of each of its dimensions - in a strictly qualitative manner.
Thus rather than being absolute, the true notion of number is strictly of a relative nature, with aspects that are quantitative (independent) and qualitative (interdependent) with respect to each other.
For in quantitative terms a number is defined as the sum of its unit parts.
So 3 (for example) = 1 + 1 + 1.
So here the units are all homogeneous (literally without qualitative distinction).
However the very notion of ordinal implies that 1st, 2nd and 3rd can be thereby distinguished in a qualitative manner.
However by defining 1st, 2nd and 3rd as the last units of a group of 1, 2 and 3 members respectively
1st + 2nd + 3rd thereby can be expressed as 1 + 1 + 1 (which reduces these ordinal notions to cardinal definition).
What this means in effect is that no choice is left with respect to the dimension chosen. For example with respect to 3 dimensions, when we identify 3rd with the last dimension of 3, then this means that the other two dimensions must have already been chosen. So for example if the 1st dimension is identified with length and the second dimension with width, then the last dimension thereby relates to the depth. If however we had identified 2nd - not with the last unit of 2 but - as the 2nd of 3 dimensions, then if the 1st dimension had been chosen as the length, the 2nd dimension would not have been fixed in meaning, but could have been chosen as either the width or height in this case.
Therefore the requirement of identifying the nth dimension in any context as the last dimension of n, in effect reduces ordinal notions in cardinal terms (where each dimension in effect is seen as clearly separate from the other dimensions).
Thus, when we remove this restriction of equating each ordinal number with the last member of the corresponding cardinal group, then a new relative notion of each ordinal number emerges.
For example instead of defining 1st as the first - which is also the last - of a group of 1, we could define it as the 1st of a group of 2, 3, 4, .....n members.
Therefore in this new context the meaning of 1st is now of a merely arbitrary nature, with an unlimited number of relative interpretations.
Indeed we can easily recognise this relative nature of ordinal numbers (without perhaps equal recognition of its mathematical significance!
So in a 1 horse race, coming in 1st would not be much significance. However if it comes in 1st in a race with -say - 40 horses, then - relatively - this is a much greater achievement. Therefore as the number of the cardinal group increases, the relative importance of any earlier ordinal member likewise increases. And this process is ultimately without limit.
Now, remarkably there is a simple mathematical way of giving expression to all these relative interpretations of ordinal numbers, which in effect amounts to the dimensional notion of fractions.
We can for example easily imagine a circular cake that is divided into 5 equal pieces.
The fractions 1/5, 2/5, 3/5, 4/5 and 5/5 then represent the fraction of the whole cake represented by 1, 2, 3, 4 and 5 slices respectively. And in the final case where we have 5 part slices with respect to the total of 5 parts this represents 1 unit (now representing the whole cake).
However even here the true situation is extremely subtle as we move from part to whole notions which strictly entails the relationship of quantitative and qualitative aspects.
However we represent this in (reduced) Type 1 terms, where all fractions are expressed with respect to the dimensional power of 1 i.e. (1/5)1, (2/5)1, (3/5)1, (4/5)1 and (5/5)1.
However we can equally give a Type 2 meaning to these fractions, where now in an inverse manner, they represents dimensional powers with respect to a default base number of 1.
So in Type 2 terms we have 11/5, 12/5, 13/5, 14/5 and 15/5. These in fact represent the 5 roots of 1 that give rise to a circular - rather than linear - number system.
They now acquire a fascinating qualitative type meaning where,
11/5 represents the 1st (in the context of 5), 12/5, the 2nd (in the context of 5), 13/5, the 3rd (in the context of 5), 14/5, the 4th (in the context of 5) and 15/5 the 5th (in the context of 5),
Now of course the last here i.e. the 5th (in the context of 5) is always 1. This corresponds to the fact that the default root of 1 = 0. We can for our purposes refer to this as the trivial root.
So to obtain the t roots of 1, we can set 1 = st , i.e. 1 – st = 0.
Now the default absolute root is represented as 1 – s = 0. Therefore dividing 1 – st = 0 by 1 – s = 0, we thereby obtain the remaining t – 1 roots.
The resulting equation for the non-trivial roots is given as:
1 + s1 + s2 + x3 + .... + st – 1 = 0.
This is what I refer to as the Zeta 2 function, which complements the well-known Zeta 1 (i.e. Riemann) zeta function.
Whereas the latter is related to the hidden holistic expression of the cardinal primes, the latter is related to the corresponding hidden holistic expression of the ordinal nature of each prime.
So if t is prime, then 1 + s1 + s2 + s3 + .... + st – 1 = 0 expresses the non trivial zeros for that prime.
For example if t = 5, then 1 + s1 + s2 + s3 + s4 = 0 expresses indirectly in quantitative terms, 1st, 2nd, 3rd and 4th (in the context of 5 members).
So whereas the Zeta 2 function starts from the premise that the cardinal natural numbers express unique combinations of the primes, the Zeta 1 function starts from the complementary premise that each prime consist of a unique combination of ordinal natural numbers!
This is especially the case in respect to cardinal numbers, where each number e.g. "2", has a definite fixed meaning in this sense..
This is also the case with respect to the ordinal notion of number, where each ordinal number is interpreted with respect to the last unit of a number group.
So 1st is - by definition - the last unit of a group of 1 member. 2nd is then defined as the last unit of a group of 2, 3rd as the last unit of a group of 3 and so on.
In this way, the ordinal notion of number is in effect reduced in a cardinal manner.
However there is a crucial difference as between the notion of number (defined as a specific number with respect to a given dimension) and the corresponding notion (where number refers directly to the dimension which now generally applies to all specific numbers).
In the first case number has an independent meaning. So when we define numbers in a 1-dimensional manner (as lying on the number line), each number e.g. 3 is given an absolute independent identity.
However when we probe into the dimensional notion of number, we are required to accept a corresponding interdependent notion of number (where each dimension is related to the others in an organised manner).
Thus to move from the notion of a 1-dimensional representation of an object (i.e. the line) to a 2-dimensional representation (e.g. a square) the 2nd dimension must be clearly related to the 1st.
So if the 1st dimension represents length, then the second (drawn at right angles will represent width).
So they clearly are not independent of each other but related in a definite manner as interdependent.
And then if we proceed further to 3-dimensional representation the 3rd dimension (the height) must again be related in definite manner (as interdependent with the other 2 dimensions).
Therefore though the base notion of number (as within a given dimension) is independent in a quantitative manner, the corresponding dimensional notion is interdependent - as the relationship of each of its dimensions - in a strictly qualitative manner.
Thus rather than being absolute, the true notion of number is strictly of a relative nature, with aspects that are quantitative (independent) and qualitative (interdependent) with respect to each other.
For in quantitative terms a number is defined as the sum of its unit parts.
So 3 (for example) = 1 + 1 + 1.
So here the units are all homogeneous (literally without qualitative distinction).
However the very notion of ordinal implies that 1st, 2nd and 3rd can be thereby distinguished in a qualitative manner.
However by defining 1st, 2nd and 3rd as the last units of a group of 1, 2 and 3 members respectively
1st + 2nd + 3rd thereby can be expressed as 1 + 1 + 1 (which reduces these ordinal notions to cardinal definition).
What this means in effect is that no choice is left with respect to the dimension chosen. For example with respect to 3 dimensions, when we identify 3rd with the last dimension of 3, then this means that the other two dimensions must have already been chosen. So for example if the 1st dimension is identified with length and the second dimension with width, then the last dimension thereby relates to the depth. If however we had identified 2nd - not with the last unit of 2 but - as the 2nd of 3 dimensions, then if the 1st dimension had been chosen as the length, the 2nd dimension would not have been fixed in meaning, but could have been chosen as either the width or height in this case.
Therefore the requirement of identifying the nth dimension in any context as the last dimension of n, in effect reduces ordinal notions in cardinal terms (where each dimension in effect is seen as clearly separate from the other dimensions).
Thus, when we remove this restriction of equating each ordinal number with the last member of the corresponding cardinal group, then a new relative notion of each ordinal number emerges.
For example instead of defining 1st as the first - which is also the last - of a group of 1, we could define it as the 1st of a group of 2, 3, 4, .....n members.
Therefore in this new context the meaning of 1st is now of a merely arbitrary nature, with an unlimited number of relative interpretations.
Indeed we can easily recognise this relative nature of ordinal numbers (without perhaps equal recognition of its mathematical significance!
So in a 1 horse race, coming in 1st would not be much significance. However if it comes in 1st in a race with -say - 40 horses, then - relatively - this is a much greater achievement. Therefore as the number of the cardinal group increases, the relative importance of any earlier ordinal member likewise increases. And this process is ultimately without limit.
Now, remarkably there is a simple mathematical way of giving expression to all these relative interpretations of ordinal numbers, which in effect amounts to the dimensional notion of fractions.
We can for example easily imagine a circular cake that is divided into 5 equal pieces.
The fractions 1/5, 2/5, 3/5, 4/5 and 5/5 then represent the fraction of the whole cake represented by 1, 2, 3, 4 and 5 slices respectively. And in the final case where we have 5 part slices with respect to the total of 5 parts this represents 1 unit (now representing the whole cake).
However even here the true situation is extremely subtle as we move from part to whole notions which strictly entails the relationship of quantitative and qualitative aspects.
However we represent this in (reduced) Type 1 terms, where all fractions are expressed with respect to the dimensional power of 1 i.e. (1/5)1, (2/5)1, (3/5)1, (4/5)1 and (5/5)1.
However we can equally give a Type 2 meaning to these fractions, where now in an inverse manner, they represents dimensional powers with respect to a default base number of 1.
So in Type 2 terms we have 11/5, 12/5, 13/5, 14/5 and 15/5. These in fact represent the 5 roots of 1 that give rise to a circular - rather than linear - number system.
They now acquire a fascinating qualitative type meaning where,
11/5 represents the 1st (in the context of 5), 12/5, the 2nd (in the context of 5), 13/5, the 3rd (in the context of 5), 14/5, the 4th (in the context of 5) and 15/5 the 5th (in the context of 5),
Now of course the last here i.e. the 5th (in the context of 5) is always 1. This corresponds to the fact that the default root of 1 = 0. We can for our purposes refer to this as the trivial root.
So to obtain the t roots of 1, we can set 1 = st , i.e. 1 – st = 0.
Now the default absolute root is represented as 1 – s = 0. Therefore dividing 1 – st = 0 by 1 – s = 0, we thereby obtain the remaining t – 1 roots.
The resulting equation for the non-trivial roots is given as:
1 + s1 + s2 + x3 + .... + st – 1 = 0.
This is what I refer to as the Zeta 2 function, which complements the well-known Zeta 1 (i.e. Riemann) zeta function.
Whereas the latter is related to the hidden holistic expression of the cardinal primes, the latter is related to the corresponding hidden holistic expression of the ordinal nature of each prime.
So if t is prime, then 1 + s1 + s2 + s3 + .... + st – 1 = 0 expresses the non trivial zeros for that prime.
For example if t = 5, then 1 + s1 + s2 + s3 + s4 = 0 expresses indirectly in quantitative terms, 1st, 2nd, 3rd and 4th (in the context of 5 members).
So whereas the Zeta 2 function starts from the premise that the cardinal natural numbers express unique combinations of the primes, the Zeta 1 function starts from the complementary premise that each prime consist of a unique combination of ordinal natural numbers!
Thursday, July 23, 2015
A Brief Addendum
As is well known log n or the more accurate log n – 1 provides a good approximation of the average spread or gap as between primes in the region of n.
However what is not properly known is that log n - 1 equally provides a good approximation of the average amount of natural factors of a number (also in the region of n).
And notice the complementarity as between both estimates! In one case we are referring to primes and in the other case to natural numbers; then again in the first case we are referring to the Type 1 notion of number (i.e. defined in a 1-dimensional manner); then in the second case we are referring to the corresponding dimensional notion of number (relating to its factor components).
This complementarity once again points to the truly dynamic interactive nature of the number system.
Thus the behaviour of the primes (with respect to the average gap as between primes in the region of n) is dynamically inseparable from the corresponding behaviour of natural number factors (this time with respect to their combined ratio in relation to n).
Put another way the behaviour in both cases is ultimately determined in a holistic synchronous manner.
Whereas great attention has been placed on one aspect of this behaviour (i.e. with respect to prime number frequency), precious little attention has been devoted to the complementary form of behaviour with respect to the frequency of the natural factors of numbers.
So in principle, just as it is is possible through use of the Riemann (Zeta) 1 zeros to eventually determine the exact frequency of the primes to any number, equally it should be possible to determine the exact cumulative frequency of natural number factors of the composites. through corresponding adjustments using the Zeta 2 zeros.
Another interesting fact!
n/log n – 1 approximates the frequency of primes to n.
The connection then as between the primes and natural numbers is of an additive nature,
n/log n - 1 equally can be used to measure the ratio as between n and the corresponding average number of factors of n.
However the connection here is of a multiplicative (rather than additive) nature.
Thus the two uses of the same formula illustrate the complementary relation as between addition and multiplication (which represent the Type 1 and Type 2 aspects of number with respect to each other).
Also it should be be observed that the average frequency of prime factors of a number is given approximately as log log n.
Therefore the ratio of natural number to prime factors is given as log n/ log log n.
If we now let log n = n1, then this ratio can be given as n1/log n1.
Or if we wish to use the more accurate expression already employed, this would be given as n1/log n1 – 1 .
So now we can see clearly how a derivation of the formula for calculation of frequency of primes (with respect to the natural numbers) can be used for calculation of the ratio of natural number to prime factors.
And once again whereas the connection as between primes and natural numbers in the first (Type 1) is of an additive nature, in the second (Type 2) case it is of a complementary multiplicative nature.
Just one more observation!
log log n (or alternatively log log n – 1) in Type 2 terms can be used to approximate the average number of prime factors of n.
Therefore log log n equally can be given a Type 1 meaning.
Thus whereas log n measures the average gap between primes at n, log log n measures the average gap as between primes whose gap is log n..
So for example the average gap as between primes at n =1,000,000 = 13.815...
Therefore log log n measures the average gap of primes to 14 (approx).
However what is not properly known is that log n - 1 equally provides a good approximation of the average amount of natural factors of a number (also in the region of n).
And notice the complementarity as between both estimates! In one case we are referring to primes and in the other case to natural numbers; then again in the first case we are referring to the Type 1 notion of number (i.e. defined in a 1-dimensional manner); then in the second case we are referring to the corresponding dimensional notion of number (relating to its factor components).
This complementarity once again points to the truly dynamic interactive nature of the number system.
Thus the behaviour of the primes (with respect to the average gap as between primes in the region of n) is dynamically inseparable from the corresponding behaviour of natural number factors (this time with respect to their combined ratio in relation to n).
Put another way the behaviour in both cases is ultimately determined in a holistic synchronous manner.
Whereas great attention has been placed on one aspect of this behaviour (i.e. with respect to prime number frequency), precious little attention has been devoted to the complementary form of behaviour with respect to the frequency of the natural factors of numbers.
So in principle, just as it is is possible through use of the Riemann (Zeta) 1 zeros to eventually determine the exact frequency of the primes to any number, equally it should be possible to determine the exact cumulative frequency of natural number factors of the composites. through corresponding adjustments using the Zeta 2 zeros.
Another interesting fact!
n/log n – 1 approximates the frequency of primes to n.
The connection then as between the primes and natural numbers is of an additive nature,
n/log n - 1 equally can be used to measure the ratio as between n and the corresponding average number of factors of n.
However the connection here is of a multiplicative (rather than additive) nature.
Thus the two uses of the same formula illustrate the complementary relation as between addition and multiplication (which represent the Type 1 and Type 2 aspects of number with respect to each other).
Also it should be be observed that the average frequency of prime factors of a number is given approximately as log log n.
Therefore the ratio of natural number to prime factors is given as log n/ log log n.
If we now let log n = n1, then this ratio can be given as n1/log n1.
Or if we wish to use the more accurate expression already employed, this would be given as n1/log n1 – 1 .
So now we can see clearly how a derivation of the formula for calculation of frequency of primes (with respect to the natural numbers) can be used for calculation of the ratio of natural number to prime factors.
And once again whereas the connection as between primes and natural numbers in the first (Type 1) is of an additive nature, in the second (Type 2) case it is of a complementary multiplicative nature.
Just one more observation!
log log n (or alternatively log log n – 1) in Type 2 terms can be used to approximate the average number of prime factors of n.
Therefore log log n equally can be given a Type 1 meaning.
Thus whereas log n measures the average gap between primes at n, log log n measures the average gap as between primes whose gap is log n..
So for example the average gap as between primes at n =1,000,000 = 13.815...
Therefore log log n measures the average gap of primes to 14 (approx).
Wednesday, July 22, 2015
Zeta Zeros Made Simple (12)
We have seen that in cardinal (Type 1) terms, each natural number can be uniquely expressed as a product of the primes.
Then in complementary ordinal (Type 2) terms, each prime can itself be uniquely expressed as an ordered sequence of natural numbers.
Thus again from the cardinal perspective, 3 is viewed as prime building block with respect to the natural number system.
However, when viewed from the corresponding ordinal perspective, 3 is now viewed as uniquely composed of its natural number ordinal members i.e. 1st, 2nd and 3rd that indirectly can be represented in a quantitative fashion. (Now again the final member which in general terms is the pth of p members is not unique as it is always equal to 1).
However all other expressions for 1st, 2nd, 3rd, 4th,......(p – 1)th members are indeed unique!.
Thus, when we consider both the (Type 1) cardinal and (Type 2) ordinal nature of number, then the relationship between the primes and natural numbers is clearly seen as circular (in a two-way fashion).
Thus again from the cardinal perspective (based on the multiplicative approach), the primes are seen as the unique building blocks of the natural numbers; however when seen from the corresponding ordinal perspective (based on the additive approach) the natural numbers are seen as the unique building blocks of each prime!
Thus the very nature of the number system, though which both the primes and natural numbers are mutually generated, is of a dynamic nature based on the bi-directional interaction of complementary opposites (that necessarily are of a relative nature).
And as this system approaches perfect synchronicity, the primes and natural numbers are increasingly seen as perfect mirrors of each other in a formless ineffable manner.
Thus if we initially conceive - as is the standard practice - of the primes and natural numbers in an analytic fashion, then the two sets of zeros (Zeta 1 and Zeta 2) can only be properly understood in a corresponding holistic fashion (where the interaction of both quantitative and qualitative aspects are mutually harmonised).
So the very essence of analytic interpretation, once again, is that polar opposites such as quantitative and qualitative are clearly separated in an absolute type manner.
However the corresponding essence of holistic interpretation is that that these same opposites are now fully harmonised in a relative type manner.
So once again in the standard cardinal interpretation, approaching from the additive perspective, the natural numbers 1, 2 and 3, for example, are clearly identified in analytic fashion as absolute quantities (that are independent of each other).
However properly speaking in corresponding ordinal interpretation 1st, 2nd and 3rd are identified in holistic fashion as relative notions (and thereby interdependent with each other).
Thus we can give these a relatively independent quantitative identity (as represented by the 3 roots of 1). However the qualitative interdependence of these roots is then expressed through their collective sum = 0.
In other words, though individually each ordinal position can be given a relatively independent quantitative identity, collectively all of these positions have but a qualitative identity = 0 (in quantitative terms).
Thus we can see that very interpretation of roots is here of a holistic nature.
And then the Zeta 2 zeros simply represent the same roots with the one common (i.e. trivial root of 1) temporarily excluded.
Then again in the standard cardinal interpretation, this time approaching from the multiplicative perspective, the primes 2, 3 and 5 for example are again clearly identified in analytic fashion as absolute quantities (that are independent of each other).
However properly speaking the corresponding "ordinal" interpretation (whereby the primes are uniquely combined to form the composite natural numbers) are again identified in holistic fashion as relative notions (and thereby interdependent with each other).
Thus from one perspective, we have the individual primes, as independent of each other; on the other hand we have the collective relationship of the primes that uniquely generates each composite natural number throughout the entire system.
Thus from the analytic perspective, the primes again appear as absolutely independent of the composite natural numbers in a quantitative manner. However from the holistic perspective, the combined relationship of primes is now viewed as fully interdependent with the natural numbers in a qualitative manner.
So now in complementary fashion each Riemann (Zeta 1) zero wonderfully expresses a location (on an imaginary axis) where both the primes and natural numbers are relatively identical with each other in a qualitative fashion. Here now the collective sum of zeros relatively yields an independent quantitative identity that can then be used to completely unravel the distortion in the general (continuous) estimate of prime number frequency.
Therefore the very role of the zeta zeros (Zeta 1 and Zeta 2) is to dynamically enable in a fully coherent and consistent manner, the continual switching in the number system as between both quantitative and qualitative aspects, In other words, the zeta zeros enable this two way switching as between Type 1 and Type 2 aspects, from the notion of number as absolutely independent (of other numbers) to the corresponding notion of number as consistently related to all other numbers in the system (and vice versa).
And as we have seen this switching as between quantitative and qualitative aspects (and qualitative and quantitative) is mediated through the corresponding two-way relationship as between the primes and the natural numbers (and the natural numbers and primes).
In conclusion, I cannot stress strongly enough how reduced - and thereby distorted - is the conventional mathematical approach to the number system.
At its very heart lies the totally unwarranted attempt to exclude all qualitative type considerations from quantitative type relationships. So there is no explicit notion of a holistic aspect to Mathematics that is distinct from the analytic! Therefore there is likewise no realisation that ultimately both analytic and holistic aspects must be combined in an integrated fashion to provide the appropriate framework for either aspect (in isolation).
In truth, the quantitative obsession in Conventional Mathematics is utter madness and yet this this is the "rock" on which such Mathematics is built.
Some time in the future, it will be clearly realised that as quantitative and qualitative notions are fully complementary in mathematical terms, that any attempt to coherently understand the quantitative, must thereby likewise include the qualitative!
However this conversion urgently needs to start now, which can thereby eventually lead to an unimaginable transformation in the entire scientific and intellectual landscape.
Then in complementary ordinal (Type 2) terms, each prime can itself be uniquely expressed as an ordered sequence of natural numbers.
Thus again from the cardinal perspective, 3 is viewed as prime building block with respect to the natural number system.
However, when viewed from the corresponding ordinal perspective, 3 is now viewed as uniquely composed of its natural number ordinal members i.e. 1st, 2nd and 3rd that indirectly can be represented in a quantitative fashion. (Now again the final member which in general terms is the pth of p members is not unique as it is always equal to 1).
However all other expressions for 1st, 2nd, 3rd, 4th,......(p – 1)th members are indeed unique!.
Thus, when we consider both the (Type 1) cardinal and (Type 2) ordinal nature of number, then the relationship between the primes and natural numbers is clearly seen as circular (in a two-way fashion).
Thus again from the cardinal perspective (based on the multiplicative approach), the primes are seen as the unique building blocks of the natural numbers; however when seen from the corresponding ordinal perspective (based on the additive approach) the natural numbers are seen as the unique building blocks of each prime!
Thus the very nature of the number system, though which both the primes and natural numbers are mutually generated, is of a dynamic nature based on the bi-directional interaction of complementary opposites (that necessarily are of a relative nature).
And as this system approaches perfect synchronicity, the primes and natural numbers are increasingly seen as perfect mirrors of each other in a formless ineffable manner.
Thus if we initially conceive - as is the standard practice - of the primes and natural numbers in an analytic fashion, then the two sets of zeros (Zeta 1 and Zeta 2) can only be properly understood in a corresponding holistic fashion (where the interaction of both quantitative and qualitative aspects are mutually harmonised).
So the very essence of analytic interpretation, once again, is that polar opposites such as quantitative and qualitative are clearly separated in an absolute type manner.
However the corresponding essence of holistic interpretation is that that these same opposites are now fully harmonised in a relative type manner.
So once again in the standard cardinal interpretation, approaching from the additive perspective, the natural numbers 1, 2 and 3, for example, are clearly identified in analytic fashion as absolute quantities (that are independent of each other).
However properly speaking in corresponding ordinal interpretation 1st, 2nd and 3rd are identified in holistic fashion as relative notions (and thereby interdependent with each other).
Thus we can give these a relatively independent quantitative identity (as represented by the 3 roots of 1). However the qualitative interdependence of these roots is then expressed through their collective sum = 0.
In other words, though individually each ordinal position can be given a relatively independent quantitative identity, collectively all of these positions have but a qualitative identity = 0 (in quantitative terms).
Thus we can see that very interpretation of roots is here of a holistic nature.
And then the Zeta 2 zeros simply represent the same roots with the one common (i.e. trivial root of 1) temporarily excluded.
Then again in the standard cardinal interpretation, this time approaching from the multiplicative perspective, the primes 2, 3 and 5 for example are again clearly identified in analytic fashion as absolute quantities (that are independent of each other).
However properly speaking the corresponding "ordinal" interpretation (whereby the primes are uniquely combined to form the composite natural numbers) are again identified in holistic fashion as relative notions (and thereby interdependent with each other).
Thus from one perspective, we have the individual primes, as independent of each other; on the other hand we have the collective relationship of the primes that uniquely generates each composite natural number throughout the entire system.
Thus from the analytic perspective, the primes again appear as absolutely independent of the composite natural numbers in a quantitative manner. However from the holistic perspective, the combined relationship of primes is now viewed as fully interdependent with the natural numbers in a qualitative manner.
So now in complementary fashion each Riemann (Zeta 1) zero wonderfully expresses a location (on an imaginary axis) where both the primes and natural numbers are relatively identical with each other in a qualitative fashion. Here now the collective sum of zeros relatively yields an independent quantitative identity that can then be used to completely unravel the distortion in the general (continuous) estimate of prime number frequency.
Therefore the very role of the zeta zeros (Zeta 1 and Zeta 2) is to dynamically enable in a fully coherent and consistent manner, the continual switching in the number system as between both quantitative and qualitative aspects, In other words, the zeta zeros enable this two way switching as between Type 1 and Type 2 aspects, from the notion of number as absolutely independent (of other numbers) to the corresponding notion of number as consistently related to all other numbers in the system (and vice versa).
And as we have seen this switching as between quantitative and qualitative aspects (and qualitative and quantitative) is mediated through the corresponding two-way relationship as between the primes and the natural numbers (and the natural numbers and primes).
In conclusion, I cannot stress strongly enough how reduced - and thereby distorted - is the conventional mathematical approach to the number system.
At its very heart lies the totally unwarranted attempt to exclude all qualitative type considerations from quantitative type relationships. So there is no explicit notion of a holistic aspect to Mathematics that is distinct from the analytic! Therefore there is likewise no realisation that ultimately both analytic and holistic aspects must be combined in an integrated fashion to provide the appropriate framework for either aspect (in isolation).
In truth, the quantitative obsession in Conventional Mathematics is utter madness and yet this this is the "rock" on which such Mathematics is built.
Some time in the future, it will be clearly realised that as quantitative and qualitative notions are fully complementary in mathematical terms, that any attempt to coherently understand the quantitative, must thereby likewise include the qualitative!
However this conversion urgently needs to start now, which can thereby eventually lead to an unimaginable transformation in the entire scientific and intellectual landscape.
Tuesday, July 21, 2015
Zeta Zeros Made Simple (11)
We have seen that in holistic terms, the various roots of 1 provide the appropriate means of expressing the unique ordinal nature of each number in a group (indirectly in a quantitative manner).
So once again, for example, in a prime group of 5 members, the 5 roots of 1, i.e. 11/5, 12/5, 13/5, 14/5, 15/5 holistically express the notions of 1st, 2nd, 3rd, 4th and 5th in the context of 5 members.
And of course the final root 15/5 = 1, expresses the default case of the 5th (in the context of 5), which directly relates to the last unit of 5, just as 14/4 represents the last unit of 4, 13/3 the last unit of 3, 12/2 the last unit of 2 and 11/1 the last unit of 1 respectively.
And these holistic expressions relate directly to a circular - rather than linear - notion of number, that geometrically is represented as the equidistant points on the unit circle (in the complex plane).
In conventional Type 1 terms, the primes are unique in that they provide the basic building blocks from which all other composite natural numbers are derived (in a quantitative manner) .
However, here in reverse complementary manner, we can see the alternative nature of primes, in that each prime is composed of a unique set of natural number roots (which cannot be replicated for any other prime).
Now strictly this statement needs qualification, in that one root i.e. 1n/n is always non-unique as 1.
In fact, in this context, we could refer to this default root as the trivial root of 1.
So the t roots of 1 are expressed as 1 = st, i.e. 1 - st = 0.
Therefore to find the expression for the non-trivial roots, we divide the expreby the trivial root i.e. 1 - s = 0.
This then yields the expression 1 + s1 + s2 + s 3 + ..... + st – 1 = 0.
I refer to this as the Zeta 2 Function, which complements the Riemann (Zeta 1) Function,
i.e. 1 – s + 2 – s + 3 – s + 4 – s + ...... = 0
Whereas the Zeta 1 Function is infinite, the Zeta 2 function is finite; then whereas the natural numbers appear as base numbers with respect to the Zeta 1, they appear as dimensional numbers with respect to the Zeta 2; finally whereas the unknown s appears as a (negative) dimensional number in the Zeta 1, it appears as a (positive) base number in the Zeta 2!
If we initially, with respect to the Zeta 2, confine the value of t to the primes, then the solutions for s represent all the unique ordinal natural number solutions for s.
For example when t = 3, the Zeta 2 Function is
1 + s1 + s2 = 0.
Therefore the 2 unique solutions (i.e. non-trivial zeros) in this case are
– .5 + .866i and – .5 – .866i respectively.
And these solutions represent the indirect quantitative expression of the qualitative notions of 1st and 2nd (in the context of 3 members). Because quantitative and qualitative notions are thereby related, these properly require holistic mathematical understanding.
And these solutions which uniquely express ordinal notions (in the context of different sized groups), I refer to as the Zeta 2 zeros.
So each non-trivial natural numbered root (within a prime group), can be given a unique quantitative expression (in a relative manner).
However - by definition - the sum of all roots (including trivial) = 0.
What this entails, in holistic mathematical terms, is that the collective sum of roots has no quantitative significance.
So notice the complementarity!
In cardinal Type 1 terms 1 + 1 + 1 = 3. Here, though the collective sum of components has a quantitative significance, each of the individual component parts (as homogeneous) has no qualitative significance.
Then in Type 2 terms 1st + 2nd + 3rd, indirectly represented quantitatively as – .5 + .866i – .5 – .866i + 1= 0.
Here though the individual components have an (indirect) quantitative significance, the collective sum has no quantitative significance. In other words it has a merely qualitative interdependent significance (where all arbitrary relative connections as between quantitative variables cancel out).
So once again, for example, in a prime group of 5 members, the 5 roots of 1, i.e. 11/5, 12/5, 13/5, 14/5, 15/5 holistically express the notions of 1st, 2nd, 3rd, 4th and 5th in the context of 5 members.
And of course the final root 15/5 = 1, expresses the default case of the 5th (in the context of 5), which directly relates to the last unit of 5, just as 14/4 represents the last unit of 4, 13/3 the last unit of 3, 12/2 the last unit of 2 and 11/1 the last unit of 1 respectively.
And these holistic expressions relate directly to a circular - rather than linear - notion of number, that geometrically is represented as the equidistant points on the unit circle (in the complex plane).
In conventional Type 1 terms, the primes are unique in that they provide the basic building blocks from which all other composite natural numbers are derived (in a quantitative manner) .
However, here in reverse complementary manner, we can see the alternative nature of primes, in that each prime is composed of a unique set of natural number roots (which cannot be replicated for any other prime).
Now strictly this statement needs qualification, in that one root i.e. 1n/n is always non-unique as 1.
In fact, in this context, we could refer to this default root as the trivial root of 1.
So the t roots of 1 are expressed as 1 = st, i.e. 1 - st = 0.
Therefore to find the expression for the non-trivial roots, we divide the expreby the trivial root i.e. 1 - s = 0.
This then yields the expression 1 + s1 + s2 + s 3 + ..... + st – 1 = 0.
I refer to this as the Zeta 2 Function, which complements the Riemann (Zeta 1) Function,
i.e. 1 – s + 2 – s + 3 – s + 4 – s + ...... = 0
Whereas the Zeta 1 Function is infinite, the Zeta 2 function is finite; then whereas the natural numbers appear as base numbers with respect to the Zeta 1, they appear as dimensional numbers with respect to the Zeta 2; finally whereas the unknown s appears as a (negative) dimensional number in the Zeta 1, it appears as a (positive) base number in the Zeta 2!
If we initially, with respect to the Zeta 2, confine the value of t to the primes, then the solutions for s represent all the unique ordinal natural number solutions for s.
For example when t = 3, the Zeta 2 Function is
1 + s1 + s2 = 0.
Therefore the 2 unique solutions (i.e. non-trivial zeros) in this case are
– .5 + .866i and – .5 – .866i respectively.
And these solutions represent the indirect quantitative expression of the qualitative notions of 1st and 2nd (in the context of 3 members). Because quantitative and qualitative notions are thereby related, these properly require holistic mathematical understanding.
And these solutions which uniquely express ordinal notions (in the context of different sized groups), I refer to as the Zeta 2 zeros.
So each non-trivial natural numbered root (within a prime group), can be given a unique quantitative expression (in a relative manner).
However - by definition - the sum of all roots (including trivial) = 0.
What this entails, in holistic mathematical terms, is that the collective sum of roots has no quantitative significance.
So notice the complementarity!
In cardinal Type 1 terms 1 + 1 + 1 = 3. Here, though the collective sum of components has a quantitative significance, each of the individual component parts (as homogeneous) has no qualitative significance.
Then in Type 2 terms 1st + 2nd + 3rd, indirectly represented quantitatively as – .5 + .866i – .5 – .866i + 1= 0.
Here though the individual components have an (indirect) quantitative significance, the collective sum has no quantitative significance. In other words it has a merely qualitative interdependent significance (where all arbitrary relative connections as between quantitative variables cancel out).
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