Yesterday we looked at the qualitative nature of 2 as a dimension.
When we consider the importance of 1 in dimensional terms, which - literally - defines the linear logical nature of conventional mathematical understanding, we perhaps might realise that 2 likewise carries a similar importance (of a distinctive nature).
In fact 2 in this qualitative dimensional sense - as the dynamic interaction of opposite (unit) polarities that are positive (+) and negative (-) with respect to each other - has been widely used in variety of contexts.
For example it has been heavily employed in the mystical expression of the Eastern mystical religions e.g. Hinduism and Buddhism. The most famous extract from the best known Buddhist heart sutra is:
"Form is not other than Void;
Void is not Other than Form."
We could equally say in a dynamic interactive mathematical context,
"The quantitative aspect (of understanding) is not other than the qualitative;
The qualitative aspect is not other than the quantitative."
We can also add add that this important statement can have no meaning in the 1-dimensional context of present Mathematics (where symbols are defined solely in quantitative terms).
Indeed Taoism gives explicit expression to the complementarity of opposites in the notion that that the original indivisible Tao splits into two opposite aspects yin and yang with respect to all phenomenal processes.
A particularly striking earlier expression in Western thought is given by Heraclitus
in the paradoxical statement
"The way up is the way down;
the way down is the way up."
So once again we have a statement of the bi-directional nature of opposites when considered in a dynamic interactive context.
Indeed in a qualitative mathematical manner we would translate that statement
(where pure interdependence exists) as:
+ 1 = -1; - 1 = + 1
I have illustrated the crucial importance of this many times before in the context of the relationship of the primes to the natural numbers (and natural numbers to the primes).
Now in this mathematical context, the two polarities are quantitative and qualitative (which are + and - in relation to each other).
Imagine one walking in a South to North direction approaching a crossroads. We can give an unambiguous meaning (for example) to a left turn in this context.
Now imagine further that having continued "up" the road in a northerly direction one decides to turn back heading once more towards the crossroad in a southerly direction.
Again when one reaches the crossroads one can unambiguously identify a left turn from this direction. So within isolated (1-dimensional) poles of reference, an unambiguous meaning can be given to a left turn in each case.
However if we now look at this from a 2-dimensional perspective (where left and right have a merely arbitrary identity) deep paradox arises.
Though in 1-dimensional terms, as independent, both turns at the crossroads are designated left, clearly in simultaneous relationship to each other as interdependent, the turns are left and right (and right and left).
So when a unit pole (which can, depending on context, be either left or right) is designated as + 1, the opposite pole is - 1. However when we switch the polar frame of reference (in this case identified with movement "up" or "down" the road), the designation thereby changes.
Thus the very ability to recognise that turns at a crossroads must be left and right in relationship to each other, implicitly requires 2-dimensional understanding of the nature of interdependence.
If now instead of left and right as complementary opposites, we use quantitative and qualitative, let us see how this applies to conventional mathematical interpretation!
Because in formal terms such Mathematics solely recognises the quantitative aspect, this approach can be likened to someone who can only identify a left turn!
So when a mathematician studies the individual nature of prime number behaviour this can be likened to the "up" direction in our crossroads example.
Therefore, unambiguous quantitative results can arise from such interpretation.
Then once again when one now switches to the opposite reference frame in the general interpretation of prime number behaviour, again unambiguous quantitative results emerge.
However when one now simultaneously tries to understand the interdependence as between individual and general prime number behaviour (with respect to the natural numbers) deep paradox arises. For just as with our crossroads example, these two aspects now are understood as quantitative and qualitative (and qualitative and qualitative) with respect to each other.
This poses an immediate dilemma for the conventional mathematical approach which allows no formal recognition for the qualitative aspect of interpretation.
In other words, Conventional Mathematics lacks any genuine holistic method of interpretation. Whereas analytic type understanding is indeed appropriate in terms of quantitative notions of independent relationships (and then strictly only in a relative context) genuine holistic understanding is required for the authentic appreciation of the interdependence of such relationships.
So the present problem with Conventional Mathematics is the most fundamental imaginable in that is has no adequate means (within its own set of interpretations) of dealing with the key notion of interdependence (except in a grossly reduced manner that distorts its very nature).
And the very reason for this that the notion of interdependence relates directly to the qualitative aspect of understanding!
And as the non-trivial zeros relate directly to this key notion of interdependence with respect to reconciling the primes with the natural numbers (and the natural numbers with the primes) the obvious implication is that their true nature cannot be approached from within the present (1-dimensional) framework of Mathematics.
As I say, Heraclitus was one of the earliest Greek thinkers to clearly express the paradoxical nature of interdependence (which in holistic mathematical terms represents the qualitative nature of 2 as dimensional number).
Other well-known Western thinkers, who also explicitly employed such thinking, include Nicholas of Cusa with his coincidence of opposites. (Though not ranked among the very greatest of thinkers his contribution is perhaps especially interesting in that he tried to appreciate such notions in a mathematical context).
Other noteworthies include Hegel (with his dynamic interaction of thesis and antithesis) and Carl Jung (with his understanding of conscious and unconscious as dynamic interacting poles of experience).
The notion has likewise found its way into Quantum Physics (e.g. the complementary nature of light as waves and particles). However the deeper qualitative implications of such understanding have yet to be properly addressed in this context (which would require nothing less than a greatly enlarged new paradigm for science).
So the interpretation of 2 as qualitative dimension is hugely important in terms of providing the most accessible appreciation of the dynamic interactive nature of both quantitative and qualitative aspects in Mathematics.
However it represents just one of a potentially unlimited set of such interpretations (as every number in dimensional terms carries a holistic significance applicable to all mathematical relationships).
Sunday, September 30, 2012
Saturday, September 29, 2012
Incredible Nature of the Zeta Zeros (4)
We have seen that the ordinal nature of number cannot be appropriately expressed within a merely quantitative (cardinal) approach.
And as cardinal, relating to independent and ordinal, relating to interdependent aspects respectively, dynamically interact in experience, the cardinal approach cannot likewise be appropriately expressed without implicit recognition of the qualitatively distinct ordinal aspect.
So we need therefore to redefine the number system in dynamic fashion, with two aspects (Type 1 and Type 2) that are quantitative and qualitative with respect to each other. These two aspects in turn thereby can give appropriate expression to the cardinal notion of number (as relatively independent of other numbers) and the corresponding ordinal notion (based on the relative interdependence of such numbers) respectively.
We have already looked briefly at the Type 1 aspect geared to cardinal interpretation in quantitative terms.
We have also addressed the crucial limitation with such interpretation as to why it cannot provide an adequate basis (without resolving to gross reductionism) regarding an ordinal ranking of numbers.
We have also looked at an important problem with respect to raising a number to another number (representing a dimensional power).
So once again for example, when we attempt to obtain the value of a simple number expression such as 2^2, strictly both a qualitative as well as quantitative change is involved.
Therefore when we express 2^2 = 4 (i.e. 4^1) we are simply reducing its result in a quantitative manner.
Thus to isolate the merely qualitative nature of number transformation, we thereby need to define a new Type 2 aspect (that is the direct inverse of the Type 1 cardinal variety).
With Type 1, when we refer to the natural numbers 1, 2, 3, 4,.... this implies that all these numbers are implicitly defined with respect to the dimensional power of 1.
So the Type 1 system (allowing for the relationship of two numbers which are quantitative and qualitative with respect to each other) is defined again as,
1^1, 2^1, 3^1, 4^1,.......
So in this case - properly understood in dynamic terms - if the base number is defined in quantitative terms, the correspoding dimensional number is of a qualitative nature.
However in Conventional Mathematics where this Type 1 aspect is treated in a static absolute fashion, both base and dimensional numbers are misleadingly defined in a merely quantitative manner.
Therefore the result of any expression relating a base number with its dimensional power is interpreted in a merely reduced quantitative manner.
So once again 2^2 in the conventional approach = 4 (i.e. 4^1) with the result interpreted in quantitative terms.
Therefore when we view the conventional approach (with its absolute number interpretation) from the more comprehensive dynamic perspective (entailing the dynamic interaction of two complementary number aspects), this conventional treatment reveals itself as the limiting special case, with 1 is the default dimension (representing the qualitative nature of number).
And the very basis of this conventional approach is that it is qualitatively defined - literally - in a linear (1-dimensional) fashion.
So Conventional Mathematics is accurately defined, from a holistic qualitative perspective, by this approach. And once again its very essence is that the qualitative aspect of number is thereby directly reduced in quantitative terms!
Now the Type 2 aspect (of the more comprehensive mathematical treatment) represents the direct complementary inverse of the Type 1 aspect.
So with respect to the natural numbers we again have,
1, 2, 3, 4,.....
However these now represent dimensional numbers, which vary with respect to a fixed number quantity as base i.e. 1.
Thus the natural number system is defined in the Type 2 approach as,
1^1, 1^2, 1^3, 1^4,....
Now, one can immediately note that from the (cardinal) quantitative perspective, such a number system seems trivial as the quantitative value of each number expression = 1.
However the very point of this second aspect is to indicate the ordinal (qualitative) rather than the cardinal (quantitative) nature of the number system.
So, to appreciate its ordinal nature, we need to convert from a linear (which befits the Type 1 aspect) to a corresponding circular approach to number, which serves as the true home of ordinal understanding.
Remember the qualitative aspect relates to a quality of interdependence as between numbers (thereby enabling a consistent set of ordinal rankings)!
Now in popular language the expression "a circle of friends" is commonly used. We also speak of political circles, sporting circles - even mathematical circles - as representative of communities of like-minded individuals that share a common interest.
Thus the very notion of a circle in popular language implies an interdependence of individuals thereby defining in an appropriate context an identifiable community with a shared interest .
Not surprisingly it is similar also in the context of number, where ordinal notions are approached in a circular fashion (thereby enabling their interdependent nature to be appreciated).
Quite simply to define this circular system with respect to any dimensional number power of 1 (i.e. n) we obtain the corresponding n roots of 1.
Therefore to define the qualitative nature of 2 (as a dimensional number) we thereby obtain the 2 roots of 1 (in quantitative terms).
Now the key to understanding this system is that the qualitative nature of the number (as dimension) and its corresponding roots (in quantitative terms) are dynamically related.
So for example in this case the qualitative nature of 2 (as a number) is intimately related to the two roots of 1 (i.e. + 1 and - 1) in quantitative terms. And in geometrical terms these are represented as two equidistant points on the unit circle in the complex plane!
However, the crucial point to appreciate is that whereas quantitative interpretation is in accordance with a linear (either/or) logic , the corresponding qualitative interpretation is in accordance with a circular (both/and) logic.
Thus whereas in linear terms + 1 and - 1 are understood as separate number quantities, from a circular perspective + 1 and - 1 are understood as complementary opposite poles of understanding (in qualitative terms).
We will develop this further in future blog entries.
However for the moment let us return to the problem that we faced with in the cardinal approach (according to Type 1 understanding).
So in Type 1 terms
2 = 1 + 1 (understood in a quantitative manner).
This left us with no means of making a qualitative distinction.
However now in ordinal terms the same two terms
= (+) 1 - 1.
Now interestingly when we add these two numbers the result is zero.
This directly implies that the ordinal (interdependent) nature of the relationship between terms strictly has no quantitative meaning!
And if take the n roots of 1 (where n is any natural number other than 1) again the sum = 0.
The one exception of course is where the dimension = 1
So when take the 1 root of 1 we obtain 1 (which does indeed have a cardinal meaning).
Once again this represents another way of stating that Conventional Mathematics is defined in qualitative terms by a linear (1-dimensional) logical approach, where qualitative notions are reduced to quantitative.
However once we appreciate that each number, representing a dimension, possesses a unique circular expression (through obtaining the corresponding roots of 1), this directly implies that potentially an unlimited set of qualitative interpretations can be given to mathematical symbols.
Thus the special case where the dimensional number = 1, accurately defines the approach of Conventional Mathematics (where qualitative is reduced to quantitative meaning).
Sadly - and so misleadingly - this one limited interpretation is widely assumed in our culture as synonymous with valid Mathematics.
However in truth a potentially unlimited set of other interpretations – all with a valid applicability in relative terms - exist.
And in all of these cases, quantitative is related to qualitative understanding in a dynamic interactive manner.
When the truth of this gradually begins to dawn, which I believe is inevitable at some stage, then we will begin the greatest revolution yet in mathematical history (with potentially vast ramifications for all of the sciences).
And as cardinal, relating to independent and ordinal, relating to interdependent aspects respectively, dynamically interact in experience, the cardinal approach cannot likewise be appropriately expressed without implicit recognition of the qualitatively distinct ordinal aspect.
So we need therefore to redefine the number system in dynamic fashion, with two aspects (Type 1 and Type 2) that are quantitative and qualitative with respect to each other. These two aspects in turn thereby can give appropriate expression to the cardinal notion of number (as relatively independent of other numbers) and the corresponding ordinal notion (based on the relative interdependence of such numbers) respectively.
We have already looked briefly at the Type 1 aspect geared to cardinal interpretation in quantitative terms.
We have also addressed the crucial limitation with such interpretation as to why it cannot provide an adequate basis (without resolving to gross reductionism) regarding an ordinal ranking of numbers.
We have also looked at an important problem with respect to raising a number to another number (representing a dimensional power).
So once again for example, when we attempt to obtain the value of a simple number expression such as 2^2, strictly both a qualitative as well as quantitative change is involved.
Therefore when we express 2^2 = 4 (i.e. 4^1) we are simply reducing its result in a quantitative manner.
Thus to isolate the merely qualitative nature of number transformation, we thereby need to define a new Type 2 aspect (that is the direct inverse of the Type 1 cardinal variety).
With Type 1, when we refer to the natural numbers 1, 2, 3, 4,.... this implies that all these numbers are implicitly defined with respect to the dimensional power of 1.
So the Type 1 system (allowing for the relationship of two numbers which are quantitative and qualitative with respect to each other) is defined again as,
1^1, 2^1, 3^1, 4^1,.......
So in this case - properly understood in dynamic terms - if the base number is defined in quantitative terms, the correspoding dimensional number is of a qualitative nature.
However in Conventional Mathematics where this Type 1 aspect is treated in a static absolute fashion, both base and dimensional numbers are misleadingly defined in a merely quantitative manner.
Therefore the result of any expression relating a base number with its dimensional power is interpreted in a merely reduced quantitative manner.
So once again 2^2 in the conventional approach = 4 (i.e. 4^1) with the result interpreted in quantitative terms.
Therefore when we view the conventional approach (with its absolute number interpretation) from the more comprehensive dynamic perspective (entailing the dynamic interaction of two complementary number aspects), this conventional treatment reveals itself as the limiting special case, with 1 is the default dimension (representing the qualitative nature of number).
And the very basis of this conventional approach is that it is qualitatively defined - literally - in a linear (1-dimensional) fashion.
So Conventional Mathematics is accurately defined, from a holistic qualitative perspective, by this approach. And once again its very essence is that the qualitative aspect of number is thereby directly reduced in quantitative terms!
Now the Type 2 aspect (of the more comprehensive mathematical treatment) represents the direct complementary inverse of the Type 1 aspect.
So with respect to the natural numbers we again have,
1, 2, 3, 4,.....
However these now represent dimensional numbers, which vary with respect to a fixed number quantity as base i.e. 1.
Thus the natural number system is defined in the Type 2 approach as,
1^1, 1^2, 1^3, 1^4,....
Now, one can immediately note that from the (cardinal) quantitative perspective, such a number system seems trivial as the quantitative value of each number expression = 1.
However the very point of this second aspect is to indicate the ordinal (qualitative) rather than the cardinal (quantitative) nature of the number system.
So, to appreciate its ordinal nature, we need to convert from a linear (which befits the Type 1 aspect) to a corresponding circular approach to number, which serves as the true home of ordinal understanding.
Remember the qualitative aspect relates to a quality of interdependence as between numbers (thereby enabling a consistent set of ordinal rankings)!
Now in popular language the expression "a circle of friends" is commonly used. We also speak of political circles, sporting circles - even mathematical circles - as representative of communities of like-minded individuals that share a common interest.
Thus the very notion of a circle in popular language implies an interdependence of individuals thereby defining in an appropriate context an identifiable community with a shared interest .
Not surprisingly it is similar also in the context of number, where ordinal notions are approached in a circular fashion (thereby enabling their interdependent nature to be appreciated).
Quite simply to define this circular system with respect to any dimensional number power of 1 (i.e. n) we obtain the corresponding n roots of 1.
Therefore to define the qualitative nature of 2 (as a dimensional number) we thereby obtain the 2 roots of 1 (in quantitative terms).
Now the key to understanding this system is that the qualitative nature of the number (as dimension) and its corresponding roots (in quantitative terms) are dynamically related.
So for example in this case the qualitative nature of 2 (as a number) is intimately related to the two roots of 1 (i.e. + 1 and - 1) in quantitative terms. And in geometrical terms these are represented as two equidistant points on the unit circle in the complex plane!
However, the crucial point to appreciate is that whereas quantitative interpretation is in accordance with a linear (either/or) logic , the corresponding qualitative interpretation is in accordance with a circular (both/and) logic.
Thus whereas in linear terms + 1 and - 1 are understood as separate number quantities, from a circular perspective + 1 and - 1 are understood as complementary opposite poles of understanding (in qualitative terms).
We will develop this further in future blog entries.
However for the moment let us return to the problem that we faced with in the cardinal approach (according to Type 1 understanding).
So in Type 1 terms
2 = 1 + 1 (understood in a quantitative manner).
This left us with no means of making a qualitative distinction.
However now in ordinal terms the same two terms
= (+) 1 - 1.
Now interestingly when we add these two numbers the result is zero.
This directly implies that the ordinal (interdependent) nature of the relationship between terms strictly has no quantitative meaning!
And if take the n roots of 1 (where n is any natural number other than 1) again the sum = 0.
The one exception of course is where the dimension = 1
So when take the 1 root of 1 we obtain 1 (which does indeed have a cardinal meaning).
Once again this represents another way of stating that Conventional Mathematics is defined in qualitative terms by a linear (1-dimensional) logical approach, where qualitative notions are reduced to quantitative.
However once we appreciate that each number, representing a dimension, possesses a unique circular expression (through obtaining the corresponding roots of 1), this directly implies that potentially an unlimited set of qualitative interpretations can be given to mathematical symbols.
Thus the special case where the dimensional number = 1, accurately defines the approach of Conventional Mathematics (where qualitative is reduced to quantitative meaning).
Sadly - and so misleadingly - this one limited interpretation is widely assumed in our culture as synonymous with valid Mathematics.
However in truth a potentially unlimited set of other interpretations – all with a valid applicability in relative terms - exist.
And in all of these cases, quantitative is related to qualitative understanding in a dynamic interactive manner.
When the truth of this gradually begins to dawn, which I believe is inevitable at some stage, then we will begin the greatest revolution yet in mathematical history (with potentially vast ramifications for all of the sciences).
Friday, September 28, 2012
Incredible Nature of the Zeta Zeros (3)
I am aware in these blogs that I am often repeating points that I have made in other entries. However as I am attempting to open up what I believe is entirely new ground with the most fundamental consequences possible for the true nature of Mathematics, I understand that initially considerable resistance may exist to accepting the implications of what is being conveyed. So in continually approaching the same problem from a number of different angles it may therefore better enable interested readers to grasp the central argument.
Yesterday I pointed to the crucially important fact that two notions of number exist (cardinal and ordinal) that are quantitative and qualitative with respect to each other.
Secondly I stated that we must define the number system in a dynamic interactive manner to preserve the distinctive nature of both aspects.
And as the conventional mathematical approach attempts to define number in a static absolute fashion this leads to gross reductionism with respect to ordinal notions; indeed ultimately - because cardinal and ordinal are interdependent - it also leads to considerable distortion even with respect to the cardinal notion of number!
And as it is so important, let me once again return to the central point.
In actual experience when we use a number - say the number "4", an inevitable interaction with respect to two distinctive type of meaning are involved.
So from the cardinal aspect we understand 4 in a collective whole manner. This is indeed why we refer to the natural numbers as wholes i.e. integers.
However a second distinctive ordinal aspect is likewise involved. From this perspective 4 is seen - not directly in its collective indivisible state - but rather in terms of its individual members.
So 4 now is defined ordinally in terms of its 1st, 2nd, 3rd and 4th members.
Now we saw yesterday that there is a fundamental problem with the cardinal approach to this issue.
From this perspective 4 = 1 + 1 + 1 + 1.
However this attempt to express the collective whole number 4 (in cardinal terms) as the sum of individual units (that are also cardinal) robs the ordinal notion of number of any coherent meaning.
For if all the units of 4 are homogenous (in quantitative terms), then we have no means of consistently ranking these numbers as 1st, 2nd 3rd and 4th which applies some unique means of applying qualitative distinction.
For example in the Premier league the top 4 clubs qualify to play (though last year was an exception) in the European Champions League.
So when we say that 4 clubs can qualify, we are using 4 in its collective cardinal sense (that is quantitative). However when we look at the 1st, 2nd, 3rd and 4th members we have now switched to a unique ordinal sense of interpretation (that is qualitative in nature).
Thus the very means of distinguishing these members requires identifying unique qualitative features. for quite simply if all members are the same - as the cardinal approach would suggest - then no means exist for attempting any unique ranking!
So this problem with respect to distinguishing cardinal and ordinal relates to the fundamental issue of properly defining whole and part notions.
In conventional Mathematics a reduced approach is adopted, whereby the whole is viewed as merely the sum of its quantitative parts. In this way ordinal notions are directly confused with cardinal in the mistaken view that number can be interpreted in a merely quantitative fashion! So a notion that strictly applies to the independent nature of number is mistakenly applied to the interpretation of the relationship between numbers.
When one looks more closely at the true ordinal nature of number one quickly realises that it has a merely relative identity that crucially depends on context.
For example if I attempt to identify the 2nd of two members, then it may indeed appear to have an unambiguous identity. However if I now identify the 2nd of 3, the context has now changed so in fact a different relationship of the ordinal member with other members is involved.
And of course ultimately a potentially unlimited set of such definitions of 2 as ordinal exist as the 2nd member can be defined in terms of 2, 3, 4, ....n members (where n has no finite limit).
We are now accustomed to use numbers (to refer to ordinal rankings) in a wide variety of contexts. Thus in sport for example we have the PGA rankings in golf. We have similar rankings in tennis, football, boxing - indeed whatever sport you might consider. Then we have economic rankings of shares, GDP per capita and so on.
For example we might say that Tiger Woods is ranked no. 2 in men's golf at present.
However this use of 2 here refers to an ordinal rather than cardinal meaning, the interpretation of which changes (depending on context). If there were only two golfers ranked in the world it might appear that we can give 2 (in this ordinal context) an unambiguous interpretation. However with world golf now so competitive with a great many professionals seeking a ranking, 2 in this context thereby conveys a very different meaning.
Therefore the key point being made here is that the meaning of 2 - when used in an ordinal sense - is merely of a relative nature depending on context. It literally depends therefore on its relationship with other numbers.
And because cardinal and ordinal interpretations are inevitably linked in a dynamic interactive manner, this implies that the cardinal notion of number is likewise of a merely relative nature. Thus though we focus directly on each number (as relatively independent) in a cardinal setting, the very ability to do so, implies that we can implicitly extract it from an interdependent setting (where it exists in relation to other numbers).
However the considerable task still exists as to how to give this ordinal notion of number a precise mathematical identity (which we will look at closely in the next blog entry).
Yesterday I pointed to the crucially important fact that two notions of number exist (cardinal and ordinal) that are quantitative and qualitative with respect to each other.
Secondly I stated that we must define the number system in a dynamic interactive manner to preserve the distinctive nature of both aspects.
And as the conventional mathematical approach attempts to define number in a static absolute fashion this leads to gross reductionism with respect to ordinal notions; indeed ultimately - because cardinal and ordinal are interdependent - it also leads to considerable distortion even with respect to the cardinal notion of number!
And as it is so important, let me once again return to the central point.
In actual experience when we use a number - say the number "4", an inevitable interaction with respect to two distinctive type of meaning are involved.
So from the cardinal aspect we understand 4 in a collective whole manner. This is indeed why we refer to the natural numbers as wholes i.e. integers.
However a second distinctive ordinal aspect is likewise involved. From this perspective 4 is seen - not directly in its collective indivisible state - but rather in terms of its individual members.
So 4 now is defined ordinally in terms of its 1st, 2nd, 3rd and 4th members.
Now we saw yesterday that there is a fundamental problem with the cardinal approach to this issue.
From this perspective 4 = 1 + 1 + 1 + 1.
However this attempt to express the collective whole number 4 (in cardinal terms) as the sum of individual units (that are also cardinal) robs the ordinal notion of number of any coherent meaning.
For if all the units of 4 are homogenous (in quantitative terms), then we have no means of consistently ranking these numbers as 1st, 2nd 3rd and 4th which applies some unique means of applying qualitative distinction.
For example in the Premier league the top 4 clubs qualify to play (though last year was an exception) in the European Champions League.
So when we say that 4 clubs can qualify, we are using 4 in its collective cardinal sense (that is quantitative). However when we look at the 1st, 2nd, 3rd and 4th members we have now switched to a unique ordinal sense of interpretation (that is qualitative in nature).
Thus the very means of distinguishing these members requires identifying unique qualitative features. for quite simply if all members are the same - as the cardinal approach would suggest - then no means exist for attempting any unique ranking!
So this problem with respect to distinguishing cardinal and ordinal relates to the fundamental issue of properly defining whole and part notions.
In conventional Mathematics a reduced approach is adopted, whereby the whole is viewed as merely the sum of its quantitative parts. In this way ordinal notions are directly confused with cardinal in the mistaken view that number can be interpreted in a merely quantitative fashion! So a notion that strictly applies to the independent nature of number is mistakenly applied to the interpretation of the relationship between numbers.
When one looks more closely at the true ordinal nature of number one quickly realises that it has a merely relative identity that crucially depends on context.
For example if I attempt to identify the 2nd of two members, then it may indeed appear to have an unambiguous identity. However if I now identify the 2nd of 3, the context has now changed so in fact a different relationship of the ordinal member with other members is involved.
And of course ultimately a potentially unlimited set of such definitions of 2 as ordinal exist as the 2nd member can be defined in terms of 2, 3, 4, ....n members (where n has no finite limit).
We are now accustomed to use numbers (to refer to ordinal rankings) in a wide variety of contexts. Thus in sport for example we have the PGA rankings in golf. We have similar rankings in tennis, football, boxing - indeed whatever sport you might consider. Then we have economic rankings of shares, GDP per capita and so on.
For example we might say that Tiger Woods is ranked no. 2 in men's golf at present.
However this use of 2 here refers to an ordinal rather than cardinal meaning, the interpretation of which changes (depending on context). If there were only two golfers ranked in the world it might appear that we can give 2 (in this ordinal context) an unambiguous interpretation. However with world golf now so competitive with a great many professionals seeking a ranking, 2 in this context thereby conveys a very different meaning.
Therefore the key point being made here is that the meaning of 2 - when used in an ordinal sense - is merely of a relative nature depending on context. It literally depends therefore on its relationship with other numbers.
And because cardinal and ordinal interpretations are inevitably linked in a dynamic interactive manner, this implies that the cardinal notion of number is likewise of a merely relative nature. Thus though we focus directly on each number (as relatively independent) in a cardinal setting, the very ability to do so, implies that we can implicitly extract it from an interdependent setting (where it exists in relation to other numbers).
However the considerable task still exists as to how to give this ordinal notion of number a precise mathematical identity (which we will look at closely in the next blog entry).
Thursday, September 27, 2012
Incredible Nature of the Zeta Zeros (2)
The basic problem with the conventional approach to Mathematics is quite simple to state.
All experience - including mathematical - is of a dynamic interactive nature entailing the relative independence of distinct phenomena with their overall interdependence in holistic terms.
However as Conventional Mathematics is formally defined in terms of isolated reference frames, e.g. where objective and subjective are clearly separated, it treats mathematical objects abstractly in an independent manner.
So for example numbers (such as prime and natural) are treated in this absolute fashion as possessing an objective independent identity.
However if such numbers did indeed possess such an absolute nature, then strictly it would be impossible to recognise numbers in relation to other numbers as interdependent!
This directly implies therefore implies that such interdependent relationships can only be treated in a reduced manner. This masks therefore problems which on closer examination can be shown to be of the most fundamental nature.
Now when appropriately understood - again in dynamic interactive terms - the relative independent nature of number can be (initially) identified with its cardinal aspect in quantitative terms; the corresponding interdependent aspect, whereby numbers can be related with other numbers, can then be identified with its ordinal aspect in a qualitative manner.
Therefore from this perspective, the quantitative and qualitative aspects of number relate to cardinal and ordinal interpretation respectively.
And because the conventional approach to number - based on absolute notions of independence - is formally defined in a merely quantitative manner, this directly implies that it is not possible to deal with the corresponding ordinal aspect except in a reduced manner that distorts its very meaning.
And we will illustrate the extremely important relevance of this finding shortly!
If we start with the natural number system from the conventional (absolute) mathematical perspective these will be defined directly in a cardinal (quantitative) manner as,
1, 2, 3, 4,.....
However properly understood - in relative terms - all numbers contain two aspects which are quantitative and qualitative with respect to each other.
From this new perspective the cardinal number system is defined in terms of a default (qualitative) dimension of 1,
i.e. 1^1, 2^1, 3^1, 4^1,....
In this context I refer to the first number (that varies) as the base and the fixed invariant number as the dimension and these two numbers are quantitative and qualitative with respect to each other.
The significance of this can be easily illustrated. From the conventional perspective for example when a number is squared we concern ourselves solely in the quantitative transformation thereby involved.
So 2^2 = 4 (i.e. 4^1).
Now if we think of this in geometrical terms, a qualitative change is likewise involved whereby we move from linear (1-dimensional) to square (2-dimensional) units. So a square of 4 square units (with each side 2 units) is qualitatively distinct from a straight line that is 4 units! However from a reduced quantitative perspective this important qualitative distinction is ignored (with literally the square result expressed in 1-dimensional terms).
So when we say that 2*2 = 4, what this implies is that the reduced quantitative value of this number expression = 4. In other words we have ignored the corresponding qualitative change in the number that has thereby occurred.
Now this procedure is valid insofar as the cardinal (quantitative) aspect of number is involved (and strictly only then in a relative sense); however ultimately it leads to total confusion when we explore the corresponding ordinal (qualitative) aspect!
And this leads directly to the most fundamental issue possible with respect to number, for momentary reflection on the matter will immediately make it obvious that we cannot use cardinal notions without ordinal, or ordinal without cardinal. Put simply, we cannot attain a coherent interpretation of number without both quantitative and qualitative aspects equally incorporated in a dynamic relative manner.
So 1, 2. 3. 4 etc. in cardinal terms imply the corresponding notions of 1st, 2nd, 3rd, 4th etc. (from an ordinal perspective).
And as cardinal and ordinal notions are quantitative and qualitative with respect to each other, we therefore cannot properly formulate an interpretation of number (that is consistent) in a merely absolute quantitative manner!
Indeed not alone can we not formulate an ordinal system of number (that is coherent) in this manner, strictly we cannot even formulate a cardinal system that is consistent!
Not surprisingly, these problems lie at the very root of the problem with respect to proper recognition of the relationship between the primes and the natural numbers (and the natural numbers and the primes).
The conventional mathematical approach to looking at this relationship is unbalanced and ultimately untenable.
Approaching the issue from the (absolute) cardinal perspective, it does indeed appear that the relationship is one-way with the primes serving as the building blocks of the natural numbers (excluding 1).
So form this perspective every natural number (again other than 1) can be expressed as the unique product of prime number factors.
So for example 30 = 2*3*5 represents a unique combination of prime number factors and therefore cannot be expressed through any other combination!
However there is a key problem with this approach which is largely overlooked.
When we use a prime factor such as 5 in a cardinal sense it is taken as a single whole. However in any meaningful sense this collective whole set represents the sum of individual number objects.
So 5 = 1 + 1 + 1 + 1 + 1.
However this attempt to define 5 in cardinal terms (i.e. as the sum of individual members that are also cardinal) leads to a crucial problem.
Now clearly in common language we would readily accept that a collection of 5 necessarily includes a 1st, 2nd, 3rd, 4th and 5th member!
However when one reflects on the matter, this represents an ordinal type distinction that cannot be meaningfully derived from 5 = 1 + 1 + 1 + 1 + 1.
In other words the basis of this cardinal definition is to attempt to render number (as without qualitative distinction).
However the very notion of ordinal ranking directly implies that we can somehow distinguish each member (which thereby implies such qualitative distinction).
So, if we insist that all units are homogenous in a merely quantitative manner, then we have no means of attempting any ordinal ranking.
In other words the capacity to make ordinal distinctions comes from the holistic relationship with respect to individual members that are in some sense perceived as possessing a unique quality.
Thus the (absolute) cardinal approach where number is abstractly understood in absolute quantitative terms (as independent) cannot therefore explain the overall holistic relationship as between numbers. Therefore, it has no means within its own definitions of explaining the corresponding ordinal notion of number (without gross reductionism being involved).
And as we cannot even begin to properly deal with cardinal without equally implying ordinal notions, then it cannot provide a coherent interpretation of the cardinal aspect (again without reductionism)!
As I say these represent the most fundamental issues possible with respect to the number system.
Again putting it bluntly I have come to see clearly over the years that the present accepted approach to Mathematics is quite simply not fit for purpose.
So in the next blog entry we will come back to exploring more closely the ordinal nature of number and how it can be given a coherent qualitative interpretation.
All experience - including mathematical - is of a dynamic interactive nature entailing the relative independence of distinct phenomena with their overall interdependence in holistic terms.
However as Conventional Mathematics is formally defined in terms of isolated reference frames, e.g. where objective and subjective are clearly separated, it treats mathematical objects abstractly in an independent manner.
So for example numbers (such as prime and natural) are treated in this absolute fashion as possessing an objective independent identity.
However if such numbers did indeed possess such an absolute nature, then strictly it would be impossible to recognise numbers in relation to other numbers as interdependent!
This directly implies therefore implies that such interdependent relationships can only be treated in a reduced manner. This masks therefore problems which on closer examination can be shown to be of the most fundamental nature.
Now when appropriately understood - again in dynamic interactive terms - the relative independent nature of number can be (initially) identified with its cardinal aspect in quantitative terms; the corresponding interdependent aspect, whereby numbers can be related with other numbers, can then be identified with its ordinal aspect in a qualitative manner.
Therefore from this perspective, the quantitative and qualitative aspects of number relate to cardinal and ordinal interpretation respectively.
And because the conventional approach to number - based on absolute notions of independence - is formally defined in a merely quantitative manner, this directly implies that it is not possible to deal with the corresponding ordinal aspect except in a reduced manner that distorts its very meaning.
And we will illustrate the extremely important relevance of this finding shortly!
If we start with the natural number system from the conventional (absolute) mathematical perspective these will be defined directly in a cardinal (quantitative) manner as,
1, 2, 3, 4,.....
However properly understood - in relative terms - all numbers contain two aspects which are quantitative and qualitative with respect to each other.
From this new perspective the cardinal number system is defined in terms of a default (qualitative) dimension of 1,
i.e. 1^1, 2^1, 3^1, 4^1,....
In this context I refer to the first number (that varies) as the base and the fixed invariant number as the dimension and these two numbers are quantitative and qualitative with respect to each other.
The significance of this can be easily illustrated. From the conventional perspective for example when a number is squared we concern ourselves solely in the quantitative transformation thereby involved.
So 2^2 = 4 (i.e. 4^1).
Now if we think of this in geometrical terms, a qualitative change is likewise involved whereby we move from linear (1-dimensional) to square (2-dimensional) units. So a square of 4 square units (with each side 2 units) is qualitatively distinct from a straight line that is 4 units! However from a reduced quantitative perspective this important qualitative distinction is ignored (with literally the square result expressed in 1-dimensional terms).
So when we say that 2*2 = 4, what this implies is that the reduced quantitative value of this number expression = 4. In other words we have ignored the corresponding qualitative change in the number that has thereby occurred.
Now this procedure is valid insofar as the cardinal (quantitative) aspect of number is involved (and strictly only then in a relative sense); however ultimately it leads to total confusion when we explore the corresponding ordinal (qualitative) aspect!
And this leads directly to the most fundamental issue possible with respect to number, for momentary reflection on the matter will immediately make it obvious that we cannot use cardinal notions without ordinal, or ordinal without cardinal. Put simply, we cannot attain a coherent interpretation of number without both quantitative and qualitative aspects equally incorporated in a dynamic relative manner.
So 1, 2. 3. 4 etc. in cardinal terms imply the corresponding notions of 1st, 2nd, 3rd, 4th etc. (from an ordinal perspective).
And as cardinal and ordinal notions are quantitative and qualitative with respect to each other, we therefore cannot properly formulate an interpretation of number (that is consistent) in a merely absolute quantitative manner!
Indeed not alone can we not formulate an ordinal system of number (that is coherent) in this manner, strictly we cannot even formulate a cardinal system that is consistent!
Not surprisingly, these problems lie at the very root of the problem with respect to proper recognition of the relationship between the primes and the natural numbers (and the natural numbers and the primes).
The conventional mathematical approach to looking at this relationship is unbalanced and ultimately untenable.
Approaching the issue from the (absolute) cardinal perspective, it does indeed appear that the relationship is one-way with the primes serving as the building blocks of the natural numbers (excluding 1).
So form this perspective every natural number (again other than 1) can be expressed as the unique product of prime number factors.
So for example 30 = 2*3*5 represents a unique combination of prime number factors and therefore cannot be expressed through any other combination!
However there is a key problem with this approach which is largely overlooked.
When we use a prime factor such as 5 in a cardinal sense it is taken as a single whole. However in any meaningful sense this collective whole set represents the sum of individual number objects.
So 5 = 1 + 1 + 1 + 1 + 1.
However this attempt to define 5 in cardinal terms (i.e. as the sum of individual members that are also cardinal) leads to a crucial problem.
Now clearly in common language we would readily accept that a collection of 5 necessarily includes a 1st, 2nd, 3rd, 4th and 5th member!
However when one reflects on the matter, this represents an ordinal type distinction that cannot be meaningfully derived from 5 = 1 + 1 + 1 + 1 + 1.
In other words the basis of this cardinal definition is to attempt to render number (as without qualitative distinction).
However the very notion of ordinal ranking directly implies that we can somehow distinguish each member (which thereby implies such qualitative distinction).
So, if we insist that all units are homogenous in a merely quantitative manner, then we have no means of attempting any ordinal ranking.
In other words the capacity to make ordinal distinctions comes from the holistic relationship with respect to individual members that are in some sense perceived as possessing a unique quality.
Thus the (absolute) cardinal approach where number is abstractly understood in absolute quantitative terms (as independent) cannot therefore explain the overall holistic relationship as between numbers. Therefore, it has no means within its own definitions of explaining the corresponding ordinal notion of number (without gross reductionism being involved).
And as we cannot even begin to properly deal with cardinal without equally implying ordinal notions, then it cannot provide a coherent interpretation of the cardinal aspect (again without reductionism)!
As I say these represent the most fundamental issues possible with respect to the number system.
Again putting it bluntly I have come to see clearly over the years that the present accepted approach to Mathematics is quite simply not fit for purpose.
So in the next blog entry we will come back to exploring more closely the ordinal nature of number and how it can be given a coherent qualitative interpretation.
Wednesday, September 26, 2012
Incredible Nature of the Zeta Zeros (1)
From the onset let us abandon any notion that the significance of the non-trivial zeros with respect to Riemann's Zeta Function can be understood in a linear rational manner.
And as their true nature greatly transcends such understanding this directly poses a dilemma for conventional mathematical interpretation (which is formally based on linear reason).
Therefore, as Hilbert rightly suggested, these zeros, when appropriately appreciated, relate to what is truly the greatest secret underlying the nature of the phenomenal - not alone the mathematical - universe, their very nature serves as an entry into the direct experience of ineffable mystery.
Though in fairness, there is now a growing appreciation of the possible physical significance of these zeros, there is little or no realisation yet of their enormous potential psychological implications.
Indeed I have little doubt that in future times the non-trivial zeros will serve an extremely important holistic scientific role in preparing spiritual aspirants to attain a pure state of meditation consistent with maintaining the highest degree of involvement in phenomenal concerns. In this way the zeros will be seen as an invaluable aid in the quest for the fullest expression of life!
Putting it bluntly the present (standard) interpretation of what constitutes Mathematics is of an extremely limited nature.
So in effect we have taken an important special case of linear (1-dimensional) understanding, where qualitative is reduced to quantitative interpretation and misleadingly elevated this as solely synonymous with valid Mathematics.
Nothing however could be further from the truth. In fact a potentially unlimited set of possible alternative interpretations of mathematical relationships exists (each one of which possesses a partial relative validity). And in all of these cases both quantitative and qualitative type aspects are related in a dynamically interactive manner.
So properly understand all mathematical activity is of a dynamic relative nature, where quantitative and qualitative aspects of understanding (which are complementary) interact.
Quite simply therefore appropriate interpretation of such activity must also formally recognise that like the blades of a scissors, Mathematics possesses two equally important aspects that are quantitative and qualitative (and qualitative and quantitative) with respect to each other.
Now, when seen from this wider dynamic perspective, there is indeed an especially important limiting case where the focus is placed primarily on quantitative meaning.
But rather like the role of Newtonian Physics, this special case should be seen as but a useful approximation with respect to mathematical reality, which inherently is of a dynamic nature and necessarily subject to uncertainty.
Indeed because of the extreme quantitative bias that defines the formal presentation of mathematical ideas, we have increasingly lost any clear notion in our society of what the qualitative aspect of Mathematics might even entail!
Though from one perspective, I can greatly admire the incredible abstract ability and indeed sheer brilliance of so many professional mathematicians, I would also see that in the main they remain blind to obvious deficiencies in the procedures that they adopt without question.
So the first requirement in approaching the nature of the non-trivial zeros is acceptance of the key fact that the number system itself must be interpreted in a dynamic interactive manner (with quantitative and qualitative aspects that are complementary).
Therefore we will first start by considering both of these aspects as relatively independent of each other.
So we have - what I term - the Type 1 aspect that is geared to the quantitative interpretation of mathematical symbols (in the standard analytical type manner using linear reason).
Then we equally have the unrecognised Type 2 aspect where the same mathematical symbols are now interpreted in a new holistic manner (using a more paradoxical circular type reasoning).
All mathematical understanding entails both reason and intuition. In brief the quantitative aspect relates to the (linear) rational and the qualitative directly to the holistic intuitive aspect respectively.
However this intuitive aspect can then be translated indirectly in a circular rational manner (using a paradoxical form of logic).
So the Type 1 aspect entails the relative specialisation in (linear) reason though intuition must also necessarily be involved.
The Type 2 aspect entails the relative specialisation of intuition, indirectly expressed in a paradoxical rational manner, though (linear) reason must also be employed.
Thus both aspects are complementary, with the fullest Type 3 expression entailing the most refined interaction of both intuitive and rational understanding.
Now the key relationship of how the primes are related to the natural numbers (and natural numbers to the primes) can be coherently constructed in isolation through both the Type 1 and Type 2 aspects. However as we shall see, they lead to interpretations - while consistent within their own frames of reference - that are in fact paradoxical in terms of each other!
So it is through the remarkable resolution of this paradox (i.e. of the two way-relationships of the primes and natural numbers) that the non-trivial zeros attain their key importance.
They in fact lead to a new distinctive Type 3 interpretation of number that is inherently of the most dynamically interactive possible, where both quantitative and qualitative aspects are reconciled.
From this new perspective, the very significance of the non-trivial zeros relates to the fundamental requirement of reconciling two modes of interpretation (that are uniquely distinct in isolated terms). Clearly when seen in this light, the true nature of the non-trivial zeros cannot be approached in a conventional mathematical manner. Not alone is this defined solely in Type 1 terms (with mere focus on quantitative interpretation) but in an absolute - rather that relative - independent fashion.
I have stated the direct consequence of this numerous times before without a hint of hyperbole. Not alone can the Riemann Hypothesis neither be proved (nor disproved) in conventional mathematical terms; its true significance cannot even be approached from this perspective.
In fact the non-trivial zeros serve as the fundamental requirement for the most advanced from of mathematical understanding (which I refer to as Type 3).
Both the Type 1 and Type 2 aspects can then be clearly seen as merely relatively independent expressions of what is interactively combined in Type 3.
Sometime in the future, Mathematics will be directly understood in Type 3 terms.
We are a long way off from that day. However the very fact that I am discussing such a development in this blog, indicates that the huge transformation process thereby required with respect to Mathematics has already started.
And as their true nature greatly transcends such understanding this directly poses a dilemma for conventional mathematical interpretation (which is formally based on linear reason).
Therefore, as Hilbert rightly suggested, these zeros, when appropriately appreciated, relate to what is truly the greatest secret underlying the nature of the phenomenal - not alone the mathematical - universe, their very nature serves as an entry into the direct experience of ineffable mystery.
Though in fairness, there is now a growing appreciation of the possible physical significance of these zeros, there is little or no realisation yet of their enormous potential psychological implications.
Indeed I have little doubt that in future times the non-trivial zeros will serve an extremely important holistic scientific role in preparing spiritual aspirants to attain a pure state of meditation consistent with maintaining the highest degree of involvement in phenomenal concerns. In this way the zeros will be seen as an invaluable aid in the quest for the fullest expression of life!
Putting it bluntly the present (standard) interpretation of what constitutes Mathematics is of an extremely limited nature.
So in effect we have taken an important special case of linear (1-dimensional) understanding, where qualitative is reduced to quantitative interpretation and misleadingly elevated this as solely synonymous with valid Mathematics.
Nothing however could be further from the truth. In fact a potentially unlimited set of possible alternative interpretations of mathematical relationships exists (each one of which possesses a partial relative validity). And in all of these cases both quantitative and qualitative type aspects are related in a dynamically interactive manner.
So properly understand all mathematical activity is of a dynamic relative nature, where quantitative and qualitative aspects of understanding (which are complementary) interact.
Quite simply therefore appropriate interpretation of such activity must also formally recognise that like the blades of a scissors, Mathematics possesses two equally important aspects that are quantitative and qualitative (and qualitative and quantitative) with respect to each other.
Now, when seen from this wider dynamic perspective, there is indeed an especially important limiting case where the focus is placed primarily on quantitative meaning.
But rather like the role of Newtonian Physics, this special case should be seen as but a useful approximation with respect to mathematical reality, which inherently is of a dynamic nature and necessarily subject to uncertainty.
Indeed because of the extreme quantitative bias that defines the formal presentation of mathematical ideas, we have increasingly lost any clear notion in our society of what the qualitative aspect of Mathematics might even entail!
Though from one perspective, I can greatly admire the incredible abstract ability and indeed sheer brilliance of so many professional mathematicians, I would also see that in the main they remain blind to obvious deficiencies in the procedures that they adopt without question.
So the first requirement in approaching the nature of the non-trivial zeros is acceptance of the key fact that the number system itself must be interpreted in a dynamic interactive manner (with quantitative and qualitative aspects that are complementary).
Therefore we will first start by considering both of these aspects as relatively independent of each other.
So we have - what I term - the Type 1 aspect that is geared to the quantitative interpretation of mathematical symbols (in the standard analytical type manner using linear reason).
Then we equally have the unrecognised Type 2 aspect where the same mathematical symbols are now interpreted in a new holistic manner (using a more paradoxical circular type reasoning).
All mathematical understanding entails both reason and intuition. In brief the quantitative aspect relates to the (linear) rational and the qualitative directly to the holistic intuitive aspect respectively.
However this intuitive aspect can then be translated indirectly in a circular rational manner (using a paradoxical form of logic).
So the Type 1 aspect entails the relative specialisation in (linear) reason though intuition must also necessarily be involved.
The Type 2 aspect entails the relative specialisation of intuition, indirectly expressed in a paradoxical rational manner, though (linear) reason must also be employed.
Thus both aspects are complementary, with the fullest Type 3 expression entailing the most refined interaction of both intuitive and rational understanding.
Now the key relationship of how the primes are related to the natural numbers (and natural numbers to the primes) can be coherently constructed in isolation through both the Type 1 and Type 2 aspects. However as we shall see, they lead to interpretations - while consistent within their own frames of reference - that are in fact paradoxical in terms of each other!
So it is through the remarkable resolution of this paradox (i.e. of the two way-relationships of the primes and natural numbers) that the non-trivial zeros attain their key importance.
They in fact lead to a new distinctive Type 3 interpretation of number that is inherently of the most dynamically interactive possible, where both quantitative and qualitative aspects are reconciled.
From this new perspective, the very significance of the non-trivial zeros relates to the fundamental requirement of reconciling two modes of interpretation (that are uniquely distinct in isolated terms). Clearly when seen in this light, the true nature of the non-trivial zeros cannot be approached in a conventional mathematical manner. Not alone is this defined solely in Type 1 terms (with mere focus on quantitative interpretation) but in an absolute - rather that relative - independent fashion.
I have stated the direct consequence of this numerous times before without a hint of hyperbole. Not alone can the Riemann Hypothesis neither be proved (nor disproved) in conventional mathematical terms; its true significance cannot even be approached from this perspective.
In fact the non-trivial zeros serve as the fundamental requirement for the most advanced from of mathematical understanding (which I refer to as Type 3).
Both the Type 1 and Type 2 aspects can then be clearly seen as merely relatively independent expressions of what is interactively combined in Type 3.
Sometime in the future, Mathematics will be directly understood in Type 3 terms.
We are a long way off from that day. However the very fact that I am discussing such a development in this blog, indicates that the huge transformation process thereby required with respect to Mathematics has already started.
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