The number e which is 2.718281828... approx. is certainly one of the most important constants in Mathematics.
One issue that has long fascinated me relates to its use with respect to prime numbers as for example the general distribution of the primes which is given in its simplest form as approximating n/log n. So we would expect to find roughly n/log n primes in the first n (natural numbers)!
Once again I am mainly concerned here with the qualitative holistic approach to mathematical symbols and in this regard e is especially interesting.
In psychological terms - as we have seen in the last contribution - development is characterised by both (conscious) differentiation and (unconscious) integration. These two aspects likewise correspond with the linear (1) and circular (0) use of logic respectively.
Successful development entails both differentiation and integration. In the mystical contemplative literature as the spiritual aspirant approaches union (discrete) phenomena of form are so fleeting and short-lived that they no longer even appear to arise in experience but rather seem to have finally merged with the continual present moment. So here both conscious and unconscious are so closely related that it is no longer possible to distinguish differentiated rational form from integral i.e. holistic intuition.
Put another way it is no longer possible to separate the quantitative from the qualitative aspect of experience.
Remarkably such experience corresponds to the holistic mathematical interpretation of e (where extremely refined discrete phenomena (of a differentiated nature) can no longer be distinguished from a continual spiritual intuitive awareness (of a corresponding holistic integrated nature).
Now we can approach the analytic interpretation of e in a similar fashion.
Imagine I invest €1 for a year at 100% rate of interest. So my investment will be worth €2 at the end of the year.
Now say I am allowed to compound interest at shorter time intervals. So instead of waiting a year I can invest for six months getting a 50% return (for the 1/2 year involved) and then reinvest. Well! I will make more money this way for at the end of the year the investment will be €(1 + 1/2)^2 = €2.25. In other words I will get 75 cent additional interest on the €1.50 invested for the 2nd six months bringing the total for the year to €2.25.
If the time periods were reduced to 3 months (with 25% return over that period the investment would be worth (1 + 1/4)^4 = $2.44.
So the general formula here is (1 + 1/n)^n.
Now if we keep shortening the time periods (1/n) so that ultimately n is infinite as conventionally understood we then obtain the value of e. So our investment would be then worth at the end of the year €2.78 (to the nearest cent).
What in effect happens here is that the discrete time intervals (over which interest is calculated) eventually become so short that we can no longer distinguish them so that interest now appears to accumulate on a continual basis.
So differentiation (with respect to discrete time intervals) can no longer be distinguished from integration (of infinitesimal intervals on a continual basis).
Now there is an especially remarkable feature about e (from a Type 1 mathematical perspective) that is worth commenting on:
if y = e^x, then dy/dx = e^x.
So with respect to this simple function integration is indistinguishable from differentiation.
As we have seen in corresponding holistic (Type 2) mathematical terms integration (approaching contemplative union) cannot be distinguished from differentiation.
Or as we have seen - put another way - the quantitative aspect cannot be distinguished from the qualitative aspect.
In Type 1 terms unfortunately the qualitative aspect is inevitably reduced to quantitative interpretation. So e is merely understood in rational terms as a quantity thus obscuring its true nature. In other words e is properly a transcendental number that entails both linear (finite) and circular (infinite) aspects. So e can be approximated in value in reduced quantitative terms. However the true value remains elusive due to its infinite qualitative aspect whereby the decimal sequence continues indefinitely with no discernible pattern.
As is well known e has a vital role to play with respect to understanding the general distribution of the primes.
This immediately suggests - or at least should suggest - that inherent in the very nature of primes is the key feature that they contain quantitative and qualitative aspects that ultimately are indivisible. We will return shortly to this vital point!
Saturday, August 20, 2011
Friday, August 19, 2011
Differentiation and Integration
Differentiation and Integration are of vital importance in both physical and psychological terms.
Once again these essentially relate to two different systems of logic that are linear (1) and circular (0) with respect to each other.
In psychological terms - which is complemented in physical terms - differentiation entails conscious type understanding of reality based on the separation of opposite polarities (such as internal and external). This corresponds in turn with the linear use of either/or logic (which defines conventional mathematical understanding).
However properly speaking integration - which is again replicated in physical processes - entails unconscious understanding based on the complementarity and ultimate identity - of these same polarities. Though directly of an intuitive infinite nature, indirectly this corresponds with the circular paradoxical use of both/and logic, which in formal terms is completely ignored in conventional mathematical interpretation.
Differentiation relates directly to the quantitative means by which we can analytically interpret reality; integration by contrast relates to the corresponding qualitative means by which we can holistically interpret that same reality.
Obviously the attempt to use just one means of understanding i.e. the linear logical system, for both quantitative and qualitative type interpretation in Mathematics, leads to considerable reductionism and essentially this is the present position with Mathematics. Basically it leads to a confusion of infinite with finite type notions.
Now, differentiation and integration are likewise of considerable importance in Mathematics and perhaps surprisingly there are very close (unrecognised) links with corresponding psychological usage.
When we look at the (simplest) holistic notion of integration in psychological terms, it entails the harmonisation and interdependence in experience of opposite polarities (such as external and internal). Now the complementarity of such opposite poles (+ and -) corresponds directly with 2-dimensional understanding. So 2 here a dimension is pointing to the fact that in this qualitative context these two poles form a complementary pairing.
Then when we differentiate in experience we move from 2-dimensional (where poles are interdependent) to 1-dimensional appreciation (where they are separated). So we now move from the holistic dimension (which is qualitative) to the ability to differentiate objects as separate (in dualistic terms). So the very meaning of 2 switches from the holistic qualitative interpretation, where 2 represents a complementary pairing, to the analytical quantitative interpretation where both poles are now separated (thus enabling dualistic understanding).
Now, remarkably this is replicated in mathematical terms.
If we start with the simple expression y = x^2, 2 here as number properly represents a (qualitative) dimension. However when we differentiate y with respect to x, dy/dx = 2x. So the dimension has now been reduced to 1 (as befits differentiation) with the 2 now representing a base quantity!
In the same manner as sub-atomic phenomena in physics - where all particles likewise have a wave aspect and all waves a particle aspect - likewise properly understood in mathematics all numbers as qualitative terms representing dimensions likewise have quantitative aspects; and all numbers as quantities likewise have quantitative effects.
So it is true that numbers (as dimensions) also have a legitimate quantitative aspect - a fact that is widely exploited in conventional interpretation!
However this obscures the important truth that in relation to each other, numbers as base values (raised to a particular dimension) and numbers as dimensions (to which a particular number is raised) are properly quantitative as to qualitative (and alternatively qualitative as to quantitative) in relation to each other.
This is of vital significance for example in understanding the true nature of prime numbers!
Once again these essentially relate to two different systems of logic that are linear (1) and circular (0) with respect to each other.
In psychological terms - which is complemented in physical terms - differentiation entails conscious type understanding of reality based on the separation of opposite polarities (such as internal and external). This corresponds in turn with the linear use of either/or logic (which defines conventional mathematical understanding).
However properly speaking integration - which is again replicated in physical processes - entails unconscious understanding based on the complementarity and ultimate identity - of these same polarities. Though directly of an intuitive infinite nature, indirectly this corresponds with the circular paradoxical use of both/and logic, which in formal terms is completely ignored in conventional mathematical interpretation.
Differentiation relates directly to the quantitative means by which we can analytically interpret reality; integration by contrast relates to the corresponding qualitative means by which we can holistically interpret that same reality.
Obviously the attempt to use just one means of understanding i.e. the linear logical system, for both quantitative and qualitative type interpretation in Mathematics, leads to considerable reductionism and essentially this is the present position with Mathematics. Basically it leads to a confusion of infinite with finite type notions.
Now, differentiation and integration are likewise of considerable importance in Mathematics and perhaps surprisingly there are very close (unrecognised) links with corresponding psychological usage.
When we look at the (simplest) holistic notion of integration in psychological terms, it entails the harmonisation and interdependence in experience of opposite polarities (such as external and internal). Now the complementarity of such opposite poles (+ and -) corresponds directly with 2-dimensional understanding. So 2 here a dimension is pointing to the fact that in this qualitative context these two poles form a complementary pairing.
Then when we differentiate in experience we move from 2-dimensional (where poles are interdependent) to 1-dimensional appreciation (where they are separated). So we now move from the holistic dimension (which is qualitative) to the ability to differentiate objects as separate (in dualistic terms). So the very meaning of 2 switches from the holistic qualitative interpretation, where 2 represents a complementary pairing, to the analytical quantitative interpretation where both poles are now separated (thus enabling dualistic understanding).
Now, remarkably this is replicated in mathematical terms.
If we start with the simple expression y = x^2, 2 here as number properly represents a (qualitative) dimension. However when we differentiate y with respect to x, dy/dx = 2x. So the dimension has now been reduced to 1 (as befits differentiation) with the 2 now representing a base quantity!
In the same manner as sub-atomic phenomena in physics - where all particles likewise have a wave aspect and all waves a particle aspect - likewise properly understood in mathematics all numbers as qualitative terms representing dimensions likewise have quantitative aspects; and all numbers as quantities likewise have quantitative effects.
So it is true that numbers (as dimensions) also have a legitimate quantitative aspect - a fact that is widely exploited in conventional interpretation!
However this obscures the important truth that in relation to each other, numbers as base values (raised to a particular dimension) and numbers as dimensions (to which a particular number is raised) are properly quantitative as to qualitative (and alternatively qualitative as to quantitative) in relation to each other.
This is of vital significance for example in understanding the true nature of prime numbers!
Tuesday, August 16, 2011
Holistic Values
In the Riemann Zeta Function, values of the function are provided for s (i.e. the power or dimensional values to which the function is defined) even though such values are often meaningless from a conventional perspective.
For example when the value of s = 0 the Riemann Zeta Function = 1 + 1 + 1 + ... which of course diverges to infinity (as conventionally understood).
However it is possible to give the function a value in the following manner.
The Zeta Function is defined as
1 + 1/(2^s) + 1/(3^s) + 1/(4^s) +......
Now if we consider just the even values terms and subtract double of each of these terms from the original series we obtain the well known Eta Function which is defined in terms of alternating terms
1 - 1/(2^s) + 1/(3^s) - 1/(4^s) +......
Now through dividing each of the even valued terms by 2^s we can derive the original terms in the Zeta Function.
This therefore enables us to establish a simple relationship as between the two Functions so that the Zeta = Eta Function divided by {1 -1/{2^(s - 1)}}
When s = 0 the Eta function results in the alternating sequence of terms
1 - 1 + 1 - 1 + 1 - ...
The sum of this sequence does not properly converge in conventional terms.
When we add up an even number of terms the value = 0; however when we add an odd number the value = 1. Thus by taking the average of these two results we can derive a single answer = 1/2.
And then from this Eta value the corresponding Zeta value can be easily calculated = - 1/2.
However the qualitative problem of explaining why the Zeta Function now has a simple finite value, when in conventional terms it diverges to infinity, needs to be explained.
In general terms a key problem generally involved with domain stretching is that finite and infinite notions are mixed indiscriminately. This is properly associated therefore with a qualitative change in the nature of interpretation involved (which however due to the reduced nature of Type 1 Mathematics is overlooked).
Once again in conventional terms numerical understanding is based on 1-dimensional linear interpretation.
Now I already have defined 2-dimensional logical interpretation as involving the complementarity of opposites which is defined in holistic terms as + 1 - 1 (taken as a complementary pair).
In corresponding quantitative fashion when we take the terms in our sequence as complementary pairs we obtain the sum of 0.
However when we take the sum of terms in a single fashion (thereby using an odd number) we obtain the sum of 1.
Therefore in qualitative terms we would explain the resulting average of 1/2 in qualitative terms as resulting from the balanced mix of both 1-dimensional (linear) and 2-dimensional (circular) interpretation.
Now remember once again in conventional terms quantitative calculations that seem to make intuitive sense are always defined by a merely linear qualitative interpretation.
So the key factor in now explaining why we can come up with this non-intuitive value for the Riemann Zeta Function where s = 0 is precisely because it actually involves in qualitative terms both 1-dimensional and 2-dimensional interpretation.
And as the Riemann Transformation formula establishes important links as between values of the Function with conventional and non-conventional numerical values, we cannot possibly hope to understand the proper nature of the Function in the absence of corresponding qualitative interpretation.
Indeed ultimately this is what the Riemann Hypothesis is all about i.e. establishing a key condition for consistency with respect to both quantitative and qualitative type mathematical interpretation.
In other words it establishes the key condition for consistency as between both Type 1 and Type 2 Mathematics which can be seen therefore as the fundamental axiom necessary for Type 3 Mathematics (which is the most comprehensive of all Types where both quantitative and qualitative aspects dynamically interact).
A fascinating further example of this qualitative issue can be given with reference to Fibonacci type sequences.
For example the Fibonacci Sequence can be obtained with reference to the simple quadratic equation x^2 - x - 1 = 0.
What we do here is to start with 0 and 1 and then combine the second term (* 1) with the first term (*1) to get 1. Now these two values are obtained as the negative of the coefficients of the last 2 terms in the quadratic expression. So the last 2 terms in the sequence are now 1 and 1. So again combining the second of these (*1) with the first (*1) we now obtain the next term in the sequence i.e. 2 So the final 2 terms are now 1 and 2 and we continue on in the same manner to obtain further terms.
Now a fascinating aspect of such sequences is that we can then approximate the positive value for x in the original equation (i.e. phi) through the ratio of the last 2 terms in the sequence (taking the larger over the smaller).
Now the equation x^2 - 1 = 0 gives the correspondent to the pure 2-dimensional case where the values for x = + 1 and - 1.
Noe this corresponds to the general quadratic equation x^2 + bx + c = 0 where b = 0 and c = - 1.
So in starting with 0 and 1 we keep adding zero times the second term to 1 times the first to get 0 as the next term. And it continues in this fashion so that we get 0, 1, 0, 1, 0, 1,....
What is interesting here is that we cannot approximate the value of x directly through getting the ratio of successive terms which will give us either 0/1 or alternatively 1/0.
However we can obtain the value directly through concentrating on the ratios of terms (occurring as each second term in sequence). In this we get either 1/1 or 0/0.
Now the first would give us the conventional rational quantitative interpretation. However the second actually corresponds to the qualitative holistic relationship.
So we cannot interpret the behaviour of such a sequence without reference to its qualitative dimensional characteristics. Once again because of the merely reduced quantitative interpretation of symbols employed in Type 1 Mathematics, these qualitative aspects are never properly investigated.
For example when the value of s = 0 the Riemann Zeta Function = 1 + 1 + 1 + ... which of course diverges to infinity (as conventionally understood).
However it is possible to give the function a value in the following manner.
The Zeta Function is defined as
1 + 1/(2^s) + 1/(3^s) + 1/(4^s) +......
Now if we consider just the even values terms and subtract double of each of these terms from the original series we obtain the well known Eta Function which is defined in terms of alternating terms
1 - 1/(2^s) + 1/(3^s) - 1/(4^s) +......
Now through dividing each of the even valued terms by 2^s we can derive the original terms in the Zeta Function.
This therefore enables us to establish a simple relationship as between the two Functions so that the Zeta = Eta Function divided by {1 -1/{2^(s - 1)}}
When s = 0 the Eta function results in the alternating sequence of terms
1 - 1 + 1 - 1 + 1 - ...
The sum of this sequence does not properly converge in conventional terms.
When we add up an even number of terms the value = 0; however when we add an odd number the value = 1. Thus by taking the average of these two results we can derive a single answer = 1/2.
And then from this Eta value the corresponding Zeta value can be easily calculated = - 1/2.
However the qualitative problem of explaining why the Zeta Function now has a simple finite value, when in conventional terms it diverges to infinity, needs to be explained.
In general terms a key problem generally involved with domain stretching is that finite and infinite notions are mixed indiscriminately. This is properly associated therefore with a qualitative change in the nature of interpretation involved (which however due to the reduced nature of Type 1 Mathematics is overlooked).
Once again in conventional terms numerical understanding is based on 1-dimensional linear interpretation.
Now I already have defined 2-dimensional logical interpretation as involving the complementarity of opposites which is defined in holistic terms as + 1 - 1 (taken as a complementary pair).
In corresponding quantitative fashion when we take the terms in our sequence as complementary pairs we obtain the sum of 0.
However when we take the sum of terms in a single fashion (thereby using an odd number) we obtain the sum of 1.
Therefore in qualitative terms we would explain the resulting average of 1/2 in qualitative terms as resulting from the balanced mix of both 1-dimensional (linear) and 2-dimensional (circular) interpretation.
Now remember once again in conventional terms quantitative calculations that seem to make intuitive sense are always defined by a merely linear qualitative interpretation.
So the key factor in now explaining why we can come up with this non-intuitive value for the Riemann Zeta Function where s = 0 is precisely because it actually involves in qualitative terms both 1-dimensional and 2-dimensional interpretation.
And as the Riemann Transformation formula establishes important links as between values of the Function with conventional and non-conventional numerical values, we cannot possibly hope to understand the proper nature of the Function in the absence of corresponding qualitative interpretation.
Indeed ultimately this is what the Riemann Hypothesis is all about i.e. establishing a key condition for consistency with respect to both quantitative and qualitative type mathematical interpretation.
In other words it establishes the key condition for consistency as between both Type 1 and Type 2 Mathematics which can be seen therefore as the fundamental axiom necessary for Type 3 Mathematics (which is the most comprehensive of all Types where both quantitative and qualitative aspects dynamically interact).
A fascinating further example of this qualitative issue can be given with reference to Fibonacci type sequences.
For example the Fibonacci Sequence can be obtained with reference to the simple quadratic equation x^2 - x - 1 = 0.
What we do here is to start with 0 and 1 and then combine the second term (* 1) with the first term (*1) to get 1. Now these two values are obtained as the negative of the coefficients of the last 2 terms in the quadratic expression. So the last 2 terms in the sequence are now 1 and 1. So again combining the second of these (*1) with the first (*1) we now obtain the next term in the sequence i.e. 2 So the final 2 terms are now 1 and 2 and we continue on in the same manner to obtain further terms.
Now a fascinating aspect of such sequences is that we can then approximate the positive value for x in the original equation (i.e. phi) through the ratio of the last 2 terms in the sequence (taking the larger over the smaller).
Now the equation x^2 - 1 = 0 gives the correspondent to the pure 2-dimensional case where the values for x = + 1 and - 1.
Noe this corresponds to the general quadratic equation x^2 + bx + c = 0 where b = 0 and c = - 1.
So in starting with 0 and 1 we keep adding zero times the second term to 1 times the first to get 0 as the next term. And it continues in this fashion so that we get 0, 1, 0, 1, 0, 1,....
What is interesting here is that we cannot approximate the value of x directly through getting the ratio of successive terms which will give us either 0/1 or alternatively 1/0.
However we can obtain the value directly through concentrating on the ratios of terms (occurring as each second term in sequence). In this we get either 1/1 or 0/0.
Now the first would give us the conventional rational quantitative interpretation. However the second actually corresponds to the qualitative holistic relationship.
So we cannot interpret the behaviour of such a sequence without reference to its qualitative dimensional characteristics. Once again because of the merely reduced quantitative interpretation of symbols employed in Type 1 Mathematics, these qualitative aspects are never properly investigated.
Thursday, August 11, 2011
Nature of Prime Numbers
I have already dealt with the important issue of the square root of 1 commenting on how but a reduced explanation is given through the conventional quantitative approach of Type 1 Mathematics. Strictly speaking this leads to logical inconsistency that amazingly is just brushed aside (as it cannot be resolved from a Type 1 perspective).
So using the Type 2 qualitative holistic treatment of mathematical symbols it is possible to demonstrate that corresponding inversely with the quantitative notion of the 2nd root of unity is a unique qualitative interpretation (relating to a differing logical system).
So remarkably corresponding then with each number as dimension is a unique logical system of interpretation. Therefore there are an infinite number of possible interpretations with the default approach of Type 1 Mathematics corresponding with the use of 1 (as dimension). And as we have seen this in turn corresponds with the standard linear rational approach that seeks to make unambiguous either/or distinctions.
However an even more direct example of the confusion lurking in the Type 1 approach arises in the context of the Riemann Zeta Function that gives rise to the most important unsolved problem in Type 1 Mathematics i.e. the Riemann Hypothesis.
Again from a Type 1 perspective when one squares a number a single valued result with no ambiguity results.
So for example the square of 1 is 1, the square of 2 is 4 and so on.
Therefore when one adds the squares of the natural number series i.e. 1 + 4 + 9 + 16 + ... the result should clearly diverge (from a Type 1 perspective) to infinity.
However remarkably according to the Riemann Zeta Function (in what represents the first of the so called trivial zeros) 1 + 4 + 9 + 16 +.... = 0
No satisfactory explanation can be offered within Type 1 mathematical appreciation as to to the legitimacy of such a result. One can of course attempt to explain how it arises as the result of extending through analytic continuation a function to all regions of the complex plane.
However this in itself does not deal with the direct problem of how two results (that are contradictory from a Type 1 perspective) can arise. It is like asking someone to believe that now 1 + 1 = 3 (rather than the accepted result of 2) though perhaps even more ridiculous.
The startling resolution of this problem is however provided through Type 2 appreciation (where one now understands through the logical structure of the 2nd rather than the 1st dimension).
Therefore to properly comprehend the Riemann Zeta Function (applying to both positive and negative values of the dimensional power s) one requires a combination of both Type 1 and Type 2 Mathematics.
Indeed the important Riemann Functional Equation can then be correctly seen as showing the relationship as between Type 1 (quantitative) and Type 2 (qualitative) numerical values.
The importance of this finding in turn for the Riemann Hypothesis is that it establishes the key condition for ensuring the consistency of both Type 1 and Type 2 mathematical interpretation.
This then serves as the key axiom for the pursuit of the more comprehensive Type 3 Mathematics (where both quantitative and qualitative aspects continually interpenetrate in understanding).
Of course this finding thereby removes the possibility of proving the Riemann Hypothesis within the accepted Type 1 mathematical interpretation.
In other words the truth to which the Riemann Hypothesis relates already precedes the axioms of Type 1 Mathematics. This provides the initial condition - literally of faith - underlying both Type 1 and Type 2 Mathematics i.e. that the truths derived from both types of understanding can be trusted as meaningful in their distinctive domains.
So clearly this initial axiom underlying belief in the logical consistency of Type 1 Mathematics cannot itself be proven from Type 1 interpretation!
For anyone wishing to see there are other ample hints showing that prime numbers cannot be understood in merely quantitative terms.
Resulting from Riemann's work is the finding that associated with each of the non-trivial zeros of the Zeta Function is a characteristic wave pattern. The accumulation of these wave patterns can in turn enable a more exact distribution of the no. of primes (within a given natural number magnitude).
These wave patterns are therefore essential in appreciating the true harmony of the primes. Indeed the Zeta Function in itself is ultimately rooted in the harmonic series (which Pythagoras demonstrated has close connections with the harmony we experience in musical sounds).
However though there is a marked quantitative basis to music, clearly it entails also (in the overall relationship of different notes to each other) a true qualitative appreciation.
Likewise though obviously there is a marked quantitative basis to individual prime numbers, there is likewise a distinctive qualitative basis in the overall holistic relationship which the primes bear to each other.
The big limitation in Type 1 Mathematics is that both the individual and collective nature of primes can only be investigated in a merely reduced quantitative manner.
However in dynamic terms the correct relationship as between the individual and collective aspects is as quantitative to qualitative (and qualitative to quantitative) respectively.
And this is the central truth embodied in the Riemann Hypothesis!
There are other interesting issues worthy of investigation e.g. as to why a Type 1 approach in the context of the Riemann Zeta Function can generate results that are applicable to Type 2! So I will return to this in a future contribution.
This reveals an even deeper truth regarding the nature of Mathematics in that it essentially entails a dynamic living interactive process (pertaining to both the physical and psychological realms).
Thus the truths embodied in prime numbers already reflect the fundamental manner in which wholes and parts are related to each other in nature.
Prime numbers are therefore equally of both a quantitative (analytic) and qualitative (holistic) nature. So we have prime numbers as base quantities (that can be raised to dimensions or powers that are - relatively - qualitative in nature).
The big limitation of Type 1 Mathematics is that it has no means of dealing with primes as representing dimensional numbers except in a reduced quantitative manner!
Ultimately this likewise has huge implications for physics in that the starting point for the emergence of material phenomena derives from the dynamic relationship of the prime numbers (with respect to both their quantitative and qualitative aspects).
So using the Type 2 qualitative holistic treatment of mathematical symbols it is possible to demonstrate that corresponding inversely with the quantitative notion of the 2nd root of unity is a unique qualitative interpretation (relating to a differing logical system).
So remarkably corresponding then with each number as dimension is a unique logical system of interpretation. Therefore there are an infinite number of possible interpretations with the default approach of Type 1 Mathematics corresponding with the use of 1 (as dimension). And as we have seen this in turn corresponds with the standard linear rational approach that seeks to make unambiguous either/or distinctions.
However an even more direct example of the confusion lurking in the Type 1 approach arises in the context of the Riemann Zeta Function that gives rise to the most important unsolved problem in Type 1 Mathematics i.e. the Riemann Hypothesis.
Again from a Type 1 perspective when one squares a number a single valued result with no ambiguity results.
So for example the square of 1 is 1, the square of 2 is 4 and so on.
Therefore when one adds the squares of the natural number series i.e. 1 + 4 + 9 + 16 + ... the result should clearly diverge (from a Type 1 perspective) to infinity.
However remarkably according to the Riemann Zeta Function (in what represents the first of the so called trivial zeros) 1 + 4 + 9 + 16 +.... = 0
No satisfactory explanation can be offered within Type 1 mathematical appreciation as to to the legitimacy of such a result. One can of course attempt to explain how it arises as the result of extending through analytic continuation a function to all regions of the complex plane.
However this in itself does not deal with the direct problem of how two results (that are contradictory from a Type 1 perspective) can arise. It is like asking someone to believe that now 1 + 1 = 3 (rather than the accepted result of 2) though perhaps even more ridiculous.
The startling resolution of this problem is however provided through Type 2 appreciation (where one now understands through the logical structure of the 2nd rather than the 1st dimension).
Therefore to properly comprehend the Riemann Zeta Function (applying to both positive and negative values of the dimensional power s) one requires a combination of both Type 1 and Type 2 Mathematics.
Indeed the important Riemann Functional Equation can then be correctly seen as showing the relationship as between Type 1 (quantitative) and Type 2 (qualitative) numerical values.
The importance of this finding in turn for the Riemann Hypothesis is that it establishes the key condition for ensuring the consistency of both Type 1 and Type 2 mathematical interpretation.
This then serves as the key axiom for the pursuit of the more comprehensive Type 3 Mathematics (where both quantitative and qualitative aspects continually interpenetrate in understanding).
Of course this finding thereby removes the possibility of proving the Riemann Hypothesis within the accepted Type 1 mathematical interpretation.
In other words the truth to which the Riemann Hypothesis relates already precedes the axioms of Type 1 Mathematics. This provides the initial condition - literally of faith - underlying both Type 1 and Type 2 Mathematics i.e. that the truths derived from both types of understanding can be trusted as meaningful in their distinctive domains.
So clearly this initial axiom underlying belief in the logical consistency of Type 1 Mathematics cannot itself be proven from Type 1 interpretation!
For anyone wishing to see there are other ample hints showing that prime numbers cannot be understood in merely quantitative terms.
Resulting from Riemann's work is the finding that associated with each of the non-trivial zeros of the Zeta Function is a characteristic wave pattern. The accumulation of these wave patterns can in turn enable a more exact distribution of the no. of primes (within a given natural number magnitude).
These wave patterns are therefore essential in appreciating the true harmony of the primes. Indeed the Zeta Function in itself is ultimately rooted in the harmonic series (which Pythagoras demonstrated has close connections with the harmony we experience in musical sounds).
However though there is a marked quantitative basis to music, clearly it entails also (in the overall relationship of different notes to each other) a true qualitative appreciation.
Likewise though obviously there is a marked quantitative basis to individual prime numbers, there is likewise a distinctive qualitative basis in the overall holistic relationship which the primes bear to each other.
The big limitation in Type 1 Mathematics is that both the individual and collective nature of primes can only be investigated in a merely reduced quantitative manner.
However in dynamic terms the correct relationship as between the individual and collective aspects is as quantitative to qualitative (and qualitative to quantitative) respectively.
And this is the central truth embodied in the Riemann Hypothesis!
There are other interesting issues worthy of investigation e.g. as to why a Type 1 approach in the context of the Riemann Zeta Function can generate results that are applicable to Type 2! So I will return to this in a future contribution.
This reveals an even deeper truth regarding the nature of Mathematics in that it essentially entails a dynamic living interactive process (pertaining to both the physical and psychological realms).
Thus the truths embodied in prime numbers already reflect the fundamental manner in which wholes and parts are related to each other in nature.
Prime numbers are therefore equally of both a quantitative (analytic) and qualitative (holistic) nature. So we have prime numbers as base quantities (that can be raised to dimensions or powers that are - relatively - qualitative in nature).
The big limitation of Type 1 Mathematics is that it has no means of dealing with primes as representing dimensional numbers except in a reduced quantitative manner!
Ultimately this likewise has huge implications for physics in that the starting point for the emergence of material phenomena derives from the dynamic relationship of the prime numbers (with respect to both their quantitative and qualitative aspects).
Wednesday, July 20, 2011
Holistic Mathematical Dimensions (1)
As we have seen dimensional numbers as used in conventional Type 1 Mathematics are given a reduced quantitative meaning (that conceals their true nature).
Therefore to unravel the true qualitative meaning of dimensions requires the use of a holistic Type 2 mathematical approach.
Furthermore this holistic approach has direct implications for appreciation of the true nature of space and time in both physical and psychological terms (which from this qualitative perspective are inherently complementary).
The starting basis for this new appreciation of dimensions is the realisation that all phenomenal reality is conditioned by 3 sets of polarities.
The first of these relates to the fact that in experience phenomena necessarily entail the interaction of polarities that are - relatively - external and internal with respect to each other. Thus to experience any phenomenon (e.g. mathematical symbol objectively as external requires a corresponding subjective mental construct that is thereby - relatively - of an internal nature.
Now if we denote conscious phenomena in holistic mathematical terms as "real" then all phenomena are conditioned by opposite polarities that are - relatively - positive and negative (with respect to each other).
The second set of polarities relates to the further fact that in experience all phenomena switch as between opposite aspects that are whole and part with respect to each other.
Now in conscious experience a basic reductionism takes place whereby wholes and parts are reduced in quantitative terms (with the whole in any context thereby representing the sum of its constituent parts).
This indeed leads to the experience of holons i.e. whole/parts where the qualitative distinction of whole and part is not maintained.
To properly distinguish whole and part in qualitative terms thereby requires unconscious (as well as conscious) recognition.
Ultimately qualitative recognition entails a spiritual element to understanding. Then with pure intuitive type recognition from one perspective each individual part can uniquely mediate Spirit (as an archetypal individual symbol). Likewise from the other perspective the collective whole can likewise mediate spirit (as an archetypal collective symbol).
In holistic mathematical terms the relationship as between conscious and unconscious is real to imaginary.
So now when we combine the two sets of polarities with respect to external/internal and individual/collective aspects we have - relatively - both real and imaginary distinctions in a positive and negative manner.
The third set of polarities relates directly to the distinction as between form and emptiness in recognition of the fact that - properly understood - all phenomenal experience entails both material and spiritual aspects.
This third set actually entails a special case of the two former sets which can be explained as follows.
The geometrical way of representing these polarity sets is with respect to a circle of unit radius (in the complex plane) .
Now to represent the first set we draw a straight horizontal line through the centre to meet the circumference on both sides. So if the right hand radius is + 1, then the left hand radius is - 1.
To represent the second set we now draw a straight line vertically through the centre so as to again meet the circumference of the circle on both sides.
So if the upper (vertical) radius is + i, then - relatively - the bottom radius is - i.
Finally to represent the third set we draw a straight line diagonally (from left to right) through the centre (at an equal distance from both horizontal and vertical lines).
Now if the upper diagonal line (or radius) is 1/k (1 + i) where k = square root of 2, then the lower part (as radius) is 1/k (- 1 - i).
Now if we choose the other direction for the diagonal line from bottom right to upper left, then the corresponding representations will be 1/k(1 - i) and 1/k (- 1 + i) respectively.
So the holistic mathematical representation of the 3 polarity sets corresponds in structural terms to the 2nd, 4th and 8th roots of 1 respectively.
The key to qualitative dimensional appreciation is the recognition that these quantitative roots bear an inverse relationship with their corresponding dimensions in qualitative terms.
So in this sense all experience is fundamentally structured according to the holistic qualitative meaning of mathematical dimensions.
And as such experience relates to both physical and psychological aspects of understanding, this thereby entails both physical and psychological reality (which are complementary) are fundamentally structured in accordance with Type 2 mathematical appreciation of dimensional numbers.
Therefore to unravel the true qualitative meaning of dimensions requires the use of a holistic Type 2 mathematical approach.
Furthermore this holistic approach has direct implications for appreciation of the true nature of space and time in both physical and psychological terms (which from this qualitative perspective are inherently complementary).
The starting basis for this new appreciation of dimensions is the realisation that all phenomenal reality is conditioned by 3 sets of polarities.
The first of these relates to the fact that in experience phenomena necessarily entail the interaction of polarities that are - relatively - external and internal with respect to each other. Thus to experience any phenomenon (e.g. mathematical symbol objectively as external requires a corresponding subjective mental construct that is thereby - relatively - of an internal nature.
Now if we denote conscious phenomena in holistic mathematical terms as "real" then all phenomena are conditioned by opposite polarities that are - relatively - positive and negative (with respect to each other).
The second set of polarities relates to the further fact that in experience all phenomena switch as between opposite aspects that are whole and part with respect to each other.
Now in conscious experience a basic reductionism takes place whereby wholes and parts are reduced in quantitative terms (with the whole in any context thereby representing the sum of its constituent parts).
This indeed leads to the experience of holons i.e. whole/parts where the qualitative distinction of whole and part is not maintained.
To properly distinguish whole and part in qualitative terms thereby requires unconscious (as well as conscious) recognition.
Ultimately qualitative recognition entails a spiritual element to understanding. Then with pure intuitive type recognition from one perspective each individual part can uniquely mediate Spirit (as an archetypal individual symbol). Likewise from the other perspective the collective whole can likewise mediate spirit (as an archetypal collective symbol).
In holistic mathematical terms the relationship as between conscious and unconscious is real to imaginary.
So now when we combine the two sets of polarities with respect to external/internal and individual/collective aspects we have - relatively - both real and imaginary distinctions in a positive and negative manner.
The third set of polarities relates directly to the distinction as between form and emptiness in recognition of the fact that - properly understood - all phenomenal experience entails both material and spiritual aspects.
This third set actually entails a special case of the two former sets which can be explained as follows.
The geometrical way of representing these polarity sets is with respect to a circle of unit radius (in the complex plane) .
Now to represent the first set we draw a straight horizontal line through the centre to meet the circumference on both sides. So if the right hand radius is + 1, then the left hand radius is - 1.
To represent the second set we now draw a straight line vertically through the centre so as to again meet the circumference of the circle on both sides.
So if the upper (vertical) radius is + i, then - relatively - the bottom radius is - i.
Finally to represent the third set we draw a straight line diagonally (from left to right) through the centre (at an equal distance from both horizontal and vertical lines).
Now if the upper diagonal line (or radius) is 1/k (1 + i) where k = square root of 2, then the lower part (as radius) is 1/k (- 1 - i).
Now if we choose the other direction for the diagonal line from bottom right to upper left, then the corresponding representations will be 1/k(1 - i) and 1/k (- 1 + i) respectively.
So the holistic mathematical representation of the 3 polarity sets corresponds in structural terms to the 2nd, 4th and 8th roots of 1 respectively.
The key to qualitative dimensional appreciation is the recognition that these quantitative roots bear an inverse relationship with their corresponding dimensions in qualitative terms.
So in this sense all experience is fundamentally structured according to the holistic qualitative meaning of mathematical dimensions.
And as such experience relates to both physical and psychological aspects of understanding, this thereby entails both physical and psychological reality (which are complementary) are fundamentally structured in accordance with Type 2 mathematical appreciation of dimensional numbers.
Monday, July 18, 2011
Importance of Dimensions
As we have seen, numbers representing dimensions (or powers) are of a qualitatively different nature than corresponding number quantities.
These dimensions then give rise to a circular number system (of a qualitative nature) that is inversely related to their corresponding roots (in quantitative terms).
The importance of this number system is that it provides the appropriate basis for a true appreciation of the nature of space and time.
So properly understood in this context, the nature of space and time is of a direct mathematical nature (relating however to Type 2 rather than Type 1 appreciation).
Furthermore such appreciation relates to both the physical and psychological aspects of reality (both of which are complementary in nature).
Indeed we can use Type 2 appreciation to explain conventional interpretation of the nature of space and time.
From a qualitative Type 2 perspective, Type 1 Mathematics is of a linear (1-dimensional) form.
Therefore when this is applied to space and time, it leads to one of these being treated as qualitative (with the remaining 3 seen as quantitative).
This therefore is consistent with the conventional perspective whereby objects are treated as 3-dimensional in quantitative spatial terms, with the remaining dimension as - relatively - qualitative as time (though of course time also has an indirect quantitative aspect which can be measured).
However once we realise that every number (apart from 1) equally has a qualitative dimensional inetrpretation, this opens up the way for an entirely new appreciation of space and time (of which conventional interpretation represents but one limited case)!
And in all of these other cases a dynamic complementary type relationship exists as between both the physical and psychological aspects of space and time.
These dimensions then give rise to a circular number system (of a qualitative nature) that is inversely related to their corresponding roots (in quantitative terms).
The importance of this number system is that it provides the appropriate basis for a true appreciation of the nature of space and time.
So properly understood in this context, the nature of space and time is of a direct mathematical nature (relating however to Type 2 rather than Type 1 appreciation).
Furthermore such appreciation relates to both the physical and psychological aspects of reality (both of which are complementary in nature).
Indeed we can use Type 2 appreciation to explain conventional interpretation of the nature of space and time.
From a qualitative Type 2 perspective, Type 1 Mathematics is of a linear (1-dimensional) form.
Therefore when this is applied to space and time, it leads to one of these being treated as qualitative (with the remaining 3 seen as quantitative).
This therefore is consistent with the conventional perspective whereby objects are treated as 3-dimensional in quantitative spatial terms, with the remaining dimension as - relatively - qualitative as time (though of course time also has an indirect quantitative aspect which can be measured).
However once we realise that every number (apart from 1) equally has a qualitative dimensional inetrpretation, this opens up the way for an entirely new appreciation of space and time (of which conventional interpretation represents but one limited case)!
And in all of these other cases a dynamic complementary type relationship exists as between both the physical and psychological aspects of space and time.
Thursday, July 14, 2011
The Number System Revisited
Earlier we outlined two number systems that are quantitative (linear) and qualitative (circular) with respect to each other.
The first natural number system which defines Type 1 Mathematics is of a quantitative nature (with a default 1-dimensional interpretation).
1^1, 2^1, 3^1, 4^1,.......
The second natural number system which defines Type 2 Mathematics is of a qualitative nature (with the default number 1 raised to successive dimensional numbers).
1^1, 1^2, 1^3, 1^4,.......
This leads to a circular number system that structurally is obtained through taking successive roots of 1 (and then defining results in an appropriate qualitative manner).
Right away this suggests that the two systems are in fact interdependent for the quantitative interpretation of roots is dynamically inseparable from the qualitative interpretation of notion of their corresponding dimensions. So for example the second (square) root of 1 cannot be properly conceived in the absence of corresponding 2-dimensional interpretation!
However we can now widen both Type 1 and Type 2 Mathematics to include imaginary - as well as real - numbers.
So in Type 1 terms the quantitative number system now includes both real and imaginary natural number terms:
1^1, 2^1, 3^1, 4^1,....... and
i^1, 2i^1, 3i^1, 4i^1,.......
Likewise in Type 2 terms the qualitative number system now includes both real and imaginary natural number terms (as dimensions):
1^1, 1^2, 1^3, 1^4,....... and
1^i, 1^2i, 1^3i, 1^4i,.......
So both real an imaginary numbers can be given both a quantitative and qualitative interpretation!
What is remarkable however is how - in dynamic interactive terms - interpretations alternate as between both their quantitative and qualitative aspects.
So dynamically speaking, real and imaginary are quantitative and qualitative with respect to each other.
We can see this readily from the fact in Type 2 Mathematics when 1 is raised to a real integer dimensional power it results in a qualitative number interpretation; however when in Type 1 Mathematics, 1 is raised to an imaginary integer dimensional power it results in a corresponding quantitative interpretation.
This once again strongly suggests that Type 1 and Type 2 Mathematics cannot be properly understood in isolation and in fact are interdependent.
So the full integration of both Type 1 and Type 2 Mathematics leads to the most comprehensive approach (which is Type 3 Mathematics).
The first natural number system which defines Type 1 Mathematics is of a quantitative nature (with a default 1-dimensional interpretation).
1^1, 2^1, 3^1, 4^1,.......
The second natural number system which defines Type 2 Mathematics is of a qualitative nature (with the default number 1 raised to successive dimensional numbers).
1^1, 1^2, 1^3, 1^4,.......
This leads to a circular number system that structurally is obtained through taking successive roots of 1 (and then defining results in an appropriate qualitative manner).
Right away this suggests that the two systems are in fact interdependent for the quantitative interpretation of roots is dynamically inseparable from the qualitative interpretation of notion of their corresponding dimensions. So for example the second (square) root of 1 cannot be properly conceived in the absence of corresponding 2-dimensional interpretation!
However we can now widen both Type 1 and Type 2 Mathematics to include imaginary - as well as real - numbers.
So in Type 1 terms the quantitative number system now includes both real and imaginary natural number terms:
1^1, 2^1, 3^1, 4^1,....... and
i^1, 2i^1, 3i^1, 4i^1,.......
Likewise in Type 2 terms the qualitative number system now includes both real and imaginary natural number terms (as dimensions):
1^1, 1^2, 1^3, 1^4,....... and
1^i, 1^2i, 1^3i, 1^4i,.......
So both real an imaginary numbers can be given both a quantitative and qualitative interpretation!
What is remarkable however is how - in dynamic interactive terms - interpretations alternate as between both their quantitative and qualitative aspects.
So dynamically speaking, real and imaginary are quantitative and qualitative with respect to each other.
We can see this readily from the fact in Type 2 Mathematics when 1 is raised to a real integer dimensional power it results in a qualitative number interpretation; however when in Type 1 Mathematics, 1 is raised to an imaginary integer dimensional power it results in a corresponding quantitative interpretation.
This once again strongly suggests that Type 1 and Type 2 Mathematics cannot be properly understood in isolation and in fact are interdependent.
So the full integration of both Type 1 and Type 2 Mathematics leads to the most comprehensive approach (which is Type 3 Mathematics).
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