Euler's Identity is generally expressed by the simple equation
e^(i*pi) + 1 = 0.
There is a beautiful illustrated account of Euler's Identity (and its proof) to be found on YouTube.
Perhaps more than any other relationship in Mathematics this requires a Type 2 mathematical interpretation.
Indeed indirectly the limitations of the conventional Type 1 approach in revealing the true nature of the Euler Identity is highlighted in the well known comment of Benjamin Pierce (quoted at the beginning of the clip):
"Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it and we don't know what it means. But we have proved it and therefore we must know that it is the truth."
The fact that it is absolutely paradoxical from a Type 1 perspective based on linear reason, then this clearly points to the need for the alternative (Type 2) circular logical explanation.
Also the admission that "we don't know what it means" clearly indicates that a qualitative holistic interpretation (that cannot be provided by Type 1 Mathematics) is required.
The clip end with another interesting quote from Keith Devlin:
"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the very beauty of the human form that is far more than skin deep, Euler's equation reaches down into the very depths of existence."
Once again this beautiful comment clearly hints at a meaning to Euler's Identity that greatly transcends conventional Type 1 interpretation. So there is a deep unconscious significance to the Identity, which Type 1 Mathematics is powerless to explore.
Now the first step to appreciating the true significance of Euler's Identity is to rewrite it with a very simple transformation.
So if e^(i*pi) + 1 = 0, then e^(i*pi) = - 1.
Then squaring both sides we obtain the more fundamental relationship,
e^(2*i*pi) = 1 (which I refer as the fundamental Euler Identity).
Now I have already in previous contributions pointed to the qualitative significance of these key mathematical symbols.
So e is qualitatively unique in the manner that it seamlessly combines both linear (discrete) and circular (continuous) aspects in its own identity.
Then i - from a qualitative perspective - represents an indirect rational means of conveying holistic type meaning in a linear quantitative context.
Pi which in quantitative terms represents the relationship as between the circular circumference and its line diameter again in qualitative terms points to a perfectly harmonious balance as between linear and circular type understanding.
Now as we have seen in conventional Type 1 terms, numbers when used to represent dimensions are treated in a merely reduced quantitative manner.
The key qualitative significance of the fundamental Euler Identity is that the dimensional expression i.e. 2*i*pi needs be equally interpreted from both a quantitative (Type 1) and (qualitative) Type 2 perspective.
Now I must admit that I have spent several years trying to precisely articulate the nature of the following conundrum:
as both e^0 = 1 and e^(2*i*pi) = 1, does this not imply that 2*i*pi = 0?
Well, clearly this is unsatisfactory from a conventional quantitative perspective. For it would follow from this acceptance that when we square both sides that - 4*(pi)^2 = 0 which is untenable as - 4*(pi)^2 represents a real magnitude!
However, from a qualitative (Type 2) perspective, 2*i*pi is indeed = 0!
One can perhaps begin to appreciate this fact by considering the circumference of a circle = 2*pi*r. Now in the conventional quantitative sense, where the circle is of radius = 1, then this gives an answer = 2*pi. However if we were to somehow conceive of the circle having an imaginary (rather than real) radius - which befits qualitative consideration - then the circumference would indeed be 2*pi*i (i.e. 2*i*pi).
So 2*i*pi in this context clearly points to a pure circular notion of dimension.
Just as the (reduced) linear quantitative notion of dimension is appropriate to Type 1, the circular qualitative notion is by contrast properly appropriate to Type 2 mathematical understanding.
Thus the key paradox with respect to the fundamental Euler identity relates to the fact that though expressed in a quantitative manner its true significance is of a qualitative (Type 2) rather than quantitative (Type 1) nature.
So by distinguishing the quantitative and qualitative type considerations, we are able to avoid the inappropriate quantitative conclusion that 0 = 2*i*pi.
Put another way e^0 = 1 is the quantitative (Type 1) expression of which e^(2^i^pi) = 1 is the corresponding qualitative (Type 2) equivalent. So, when we attempt to represent the pure qualitative (circular) expression for a dimension i.e. 2*i*pi in reduced (linear) quantitative terms it appears - quite literally - as 0!
Indeed there is an even simpler - and equally puzzling - expression that needs clarification.
In Type 1 terms both (- 1)^2 and (+ 1)^2 = 1.
Therefore we might be tempted to erroneously conclude that - 1 = + 1!
In fact this is a problem of the first magnitude that is not dealt with at all in conventional Type 1 terms!
It can only be resolved by introducing Type 2 notions.
So properly expressed from a Type 2 perspective (- 1)^2 = 1^1 whereas (+ 1)^2 = 1^2.
And in qualitative (Type 2) mathematical terms, 1^1 and 1^2 are distinct expressions.
However if mathematicians used the Type 1 approach consistently then they should be concluding by their own logic that - 1 = + 1!
Now this is issue is fudged in practice by confining calculations to the "principle" root, though strictly this is not the correct root.
So for example the true square root of 1 is - 1 (and not + 1).
Remarkably as we shall see this conclusion is supported by the extended use of the fundamental Euler Identity.
Friday, August 26, 2011
Tuesday, August 23, 2011
The Imaginary Concept
Once again when I maintain that every mathematical notion has a qualitative as well as (recognised) quantitative interpretation this relates to a dynamic holistic form of understanding that is directly based on intuitive type recognition. However the mathematical basis of such understanding comes from the indirect rational means through which such understanding is then appropriately conveyed. And as we have seen such understanding is of a circular logical nature.
Now of course all this might appear very strange to anyone accustomed to look at Mathematics from a merely Type 1 perspective. And it is has to be said that specialisation in this respect has been so strong that it has all but blotted out recognition of the alternative Type 2 perspective.
However the simple fact remains that it is not possible - for example - to properly understand the nature of prime numbers in the absence of Type 2 understanding.
Clearly imaginary numbers are of great importance in conventional Type 1 terms.
However once again - because of the lack of any adequate Type 2 understanding - a merely reduced form of interpretation operates (leaving us completely blind to their true philosophical significance).
In fact imaginary numbers entail the incorporation of the alternative circular logical approach that is expressed in a reduced linear rational manner. In this way imaginary numbers seemingly can be successfully incorporated within the standard Type 1 approach (that is characterised by the linear logical approach).
In conventional (Type 1) terms, the imaginary number i is defined as the square root of - 1. Thus when we square i (to get a 2-dimensional expression of 1^2 which is - 1)) me move back into the world of real numbers.
So what is imaginary here i.e. i actually results from attempting to express - 1 (which is of a real nature and pertaining to a 2-dimensional interpretation) in a reduced 1-dimensional manner.
Thus in this way we can see directly how the imaginary amounts to a linear (1-dimensional) way of expressing a circular (2-dimensional) notion.
In this way the 2nd dimension - and by extension all higher dimensional expressions - can be successfully incorporated through the use of complex numbers in the standard linear Type 1 framework that characterises the conventional approach to Mathematics.
However the imaginary notion - as with all mathematical concepts - can equally be given a true holistic interpretation in a qualitative (Type 2) manner.
As we seen we have already defined the 2nd dimension in qualitative holistic terms as - 1 (literally relating in this context to the dynamic negation of the unitary notion of form). In other words the very means through which we move from (linear) conscious understanding (that is characterised by the separation of polar opposites in experience) to (circular) unconscious appreciation (where opposite polarities are understood as complementary) is through the negation of the positive pole (identified as conscious).
So properly understood in a true qualitative sense the imaginary concept in Mathematics points directly to the vital role of unconscious (as well as conscious) appreciation in Mathematics.
Informally this is to a degree recognised. For example one could attempt for example to rationally explain the Pythagorean theorem to a student. However without supporting intuition, that student would lack - literally -the imagination to see what is implied by the rational connections.
Likewise in any mathematical work of a creative nature, intuition is a vital ingredient in developing the key insights necessary to sustain the whole endeavour.
However when in comes to formal interpretation in conventional terms, the role of intuition in mathematical understanding is completely ignored. Thus the standard - merely - rational interpretations that are offered, in a very important sense misrepresent the true nature of mathematical experience.
So we can now perhaps appreciate how the imaginary concept - as used in conventional terms - actually represents an indirect way of incorporating holistic mathematical notions within the standard analytic framework.
However when appreciated from a full qualitative perspective, the imaginary notion points directly to the need for inclusion of an entirely distinctive form of understanding (which I refer to as Type 2).
Let me clarify here this very important point.
Conventional (Type 1) Mathematics does indeed include both real and imaginary notions. However it can only deal with both in a merely reduced quantitative manner.
Thus in qualitative terms, Type 1 Mathematics confines itself to the merely real (conscious rational) aspect of mathematical understanding.
Thus Type 2 Mathematics - as I define it - is directly designed to deal with the imaginary aspect of mathematical understanding in qualitative terms.
So again in this qualitative context we can define Type 1 as related to the real aspect and Type 2 to the imaginary aspect of mathematical understanding respectively.
Clearly in the most comprehensive context both the real and imaginary aspects of mathematical understanding will be combined in both a quantitative and qualitative manner.
And I refer to this most comprehensive form as Type 3 Mathematics!
However before leaving, I wish to point to a key difference as between the quantitative and qualitative notion of the imaginary!
Because in linear terms opposite polarities are clearly separated this implies that positive (+) is of course clearly separated from (-).
However in circular terms (for the 2-dimensional case) opposite polarities exhibit perfect complementarity. So in this context the positive (+) is interdependent and indeed ultimately identical with the negative (-) pole.
Thus whereas the imaginary in quantitative terms is defined as the square root of - 1, in the corresponding qualitative interpretation - 1 has a dynamic meaning that implies negation of what is positive. So the negative in this context necessarily requires the positive pole to dynamically operate!
Now of course all this might appear very strange to anyone accustomed to look at Mathematics from a merely Type 1 perspective. And it is has to be said that specialisation in this respect has been so strong that it has all but blotted out recognition of the alternative Type 2 perspective.
However the simple fact remains that it is not possible - for example - to properly understand the nature of prime numbers in the absence of Type 2 understanding.
Clearly imaginary numbers are of great importance in conventional Type 1 terms.
However once again - because of the lack of any adequate Type 2 understanding - a merely reduced form of interpretation operates (leaving us completely blind to their true philosophical significance).
In fact imaginary numbers entail the incorporation of the alternative circular logical approach that is expressed in a reduced linear rational manner. In this way imaginary numbers seemingly can be successfully incorporated within the standard Type 1 approach (that is characterised by the linear logical approach).
In conventional (Type 1) terms, the imaginary number i is defined as the square root of - 1. Thus when we square i (to get a 2-dimensional expression of 1^2 which is - 1)) me move back into the world of real numbers.
So what is imaginary here i.e. i actually results from attempting to express - 1 (which is of a real nature and pertaining to a 2-dimensional interpretation) in a reduced 1-dimensional manner.
Thus in this way we can see directly how the imaginary amounts to a linear (1-dimensional) way of expressing a circular (2-dimensional) notion.
In this way the 2nd dimension - and by extension all higher dimensional expressions - can be successfully incorporated through the use of complex numbers in the standard linear Type 1 framework that characterises the conventional approach to Mathematics.
However the imaginary notion - as with all mathematical concepts - can equally be given a true holistic interpretation in a qualitative (Type 2) manner.
As we seen we have already defined the 2nd dimension in qualitative holistic terms as - 1 (literally relating in this context to the dynamic negation of the unitary notion of form). In other words the very means through which we move from (linear) conscious understanding (that is characterised by the separation of polar opposites in experience) to (circular) unconscious appreciation (where opposite polarities are understood as complementary) is through the negation of the positive pole (identified as conscious).
So properly understood in a true qualitative sense the imaginary concept in Mathematics points directly to the vital role of unconscious (as well as conscious) appreciation in Mathematics.
Informally this is to a degree recognised. For example one could attempt for example to rationally explain the Pythagorean theorem to a student. However without supporting intuition, that student would lack - literally -the imagination to see what is implied by the rational connections.
Likewise in any mathematical work of a creative nature, intuition is a vital ingredient in developing the key insights necessary to sustain the whole endeavour.
However when in comes to formal interpretation in conventional terms, the role of intuition in mathematical understanding is completely ignored. Thus the standard - merely - rational interpretations that are offered, in a very important sense misrepresent the true nature of mathematical experience.
So we can now perhaps appreciate how the imaginary concept - as used in conventional terms - actually represents an indirect way of incorporating holistic mathematical notions within the standard analytic framework.
However when appreciated from a full qualitative perspective, the imaginary notion points directly to the need for inclusion of an entirely distinctive form of understanding (which I refer to as Type 2).
Let me clarify here this very important point.
Conventional (Type 1) Mathematics does indeed include both real and imaginary notions. However it can only deal with both in a merely reduced quantitative manner.
Thus in qualitative terms, Type 1 Mathematics confines itself to the merely real (conscious rational) aspect of mathematical understanding.
Thus Type 2 Mathematics - as I define it - is directly designed to deal with the imaginary aspect of mathematical understanding in qualitative terms.
So again in this qualitative context we can define Type 1 as related to the real aspect and Type 2 to the imaginary aspect of mathematical understanding respectively.
Clearly in the most comprehensive context both the real and imaginary aspects of mathematical understanding will be combined in both a quantitative and qualitative manner.
And I refer to this most comprehensive form as Type 3 Mathematics!
However before leaving, I wish to point to a key difference as between the quantitative and qualitative notion of the imaginary!
Because in linear terms opposite polarities are clearly separated this implies that positive (+) is of course clearly separated from (-).
However in circular terms (for the 2-dimensional case) opposite polarities exhibit perfect complementarity. So in this context the positive (+) is interdependent and indeed ultimately identical with the negative (-) pole.
Thus whereas the imaginary in quantitative terms is defined as the square root of - 1, in the corresponding qualitative interpretation - 1 has a dynamic meaning that implies negation of what is positive. So the negative in this context necessarily requires the positive pole to dynamically operate!
Monday, August 22, 2011
Natural Logs
If the natural log of a number n is x, this implies that e^x = n.
So the important point to grasp here therefore is that the log always points to a number that represents a dimension (i.e. power or exponent).
And as I have previously related numbers that represent quantities (raised to a particular power) and numbers that directly represent dimensions (to which base quantities are raised) are properly quantitative and qualitative with respect to each other.
Once again this can be illustrated through the simple example of a number expression such as 1^(1/3). Now 1 and 1/3 are here rational numbers that in isolation can be considered as linear quantities (that are represented by discrete intervals on the straight line). 1 represents the base quantity and 1/3 the dimensional power or exponent.
However though each number can be considered as a linear quantity in isolation, in relation to each other they are as quantitative to qualitative (and qualitative to quantitative).
This in this interdependent fashion if 1 represents the quantitative aspect of number, then 1/3 represents the corresponding qualitative aspect.
And the truth of this observation is then - indirectly - demonstrated by the result of the expression which lies on the circle (of unit radius).
Though this of course is well recognised in conventional (Type 1) Mathematics, because of a merely quantitative bias, its significance is greatly overlooked.
In other words the true (root) reason why this shift from linear to circular interpretation takes place is because the interaction is of a base number quantity with a dimensional number that - relatively - is of a qualitative nature.
Once again when we shift the frame from reference from up to down on a street the turns at a crossroads may both appear similar. So one one of these turns for example moving up the road may appear as left and then coming down the road in the opposite direction the other turn also appear as left.
However in relation the each other they are clearly left and right (and right and left).
So the position in Type 1 Mathematics of recognising a merely quantitative basis for both linear and circular type number quantities is exactly analogous to one who labels in our example the two turns at a crossroads as both left!. Again this arises through attempting to treat numbers (representing both quantities and dimensions) in an isolated fashion.
So once again in a combined expression the number representing the base and dimensional aspects are properly quantitative and qualitative (and qualitative and quantitative) with respect to each other.
Now we have already seen (in the last contribution) that inherent in the very definition of a prime number are both quantitative (analytic) and qualitative (holistic) aspects of equal importance and which in the dynamic generation of prime number behaviour (both with respect to the individual primes and their general distribution) simultaneously arise.
And the simplest expression expressing the general distribution of the primes is given as n/log n (where we are using the natural log base of e).
Now n properly relates to a number representing a base quantity whereas log n properly relates to a dimensional number (that relatively is of a qualitative nature).
So once again - when appropriately interpreted - this simple expression (for the general distribution of primes) demonstrates the inherent connection (in the very nature of primes) as between its quantitative and qualitative aspects.
So the important point to grasp here therefore is that the log always points to a number that represents a dimension (i.e. power or exponent).
And as I have previously related numbers that represent quantities (raised to a particular power) and numbers that directly represent dimensions (to which base quantities are raised) are properly quantitative and qualitative with respect to each other.
Once again this can be illustrated through the simple example of a number expression such as 1^(1/3). Now 1 and 1/3 are here rational numbers that in isolation can be considered as linear quantities (that are represented by discrete intervals on the straight line). 1 represents the base quantity and 1/3 the dimensional power or exponent.
However though each number can be considered as a linear quantity in isolation, in relation to each other they are as quantitative to qualitative (and qualitative to quantitative).
This in this interdependent fashion if 1 represents the quantitative aspect of number, then 1/3 represents the corresponding qualitative aspect.
And the truth of this observation is then - indirectly - demonstrated by the result of the expression which lies on the circle (of unit radius).
Though this of course is well recognised in conventional (Type 1) Mathematics, because of a merely quantitative bias, its significance is greatly overlooked.
In other words the true (root) reason why this shift from linear to circular interpretation takes place is because the interaction is of a base number quantity with a dimensional number that - relatively - is of a qualitative nature.
Once again when we shift the frame from reference from up to down on a street the turns at a crossroads may both appear similar. So one one of these turns for example moving up the road may appear as left and then coming down the road in the opposite direction the other turn also appear as left.
However in relation the each other they are clearly left and right (and right and left).
So the position in Type 1 Mathematics of recognising a merely quantitative basis for both linear and circular type number quantities is exactly analogous to one who labels in our example the two turns at a crossroads as both left!. Again this arises through attempting to treat numbers (representing both quantities and dimensions) in an isolated fashion.
So once again in a combined expression the number representing the base and dimensional aspects are properly quantitative and qualitative (and qualitative and quantitative) with respect to each other.
Now we have already seen (in the last contribution) that inherent in the very definition of a prime number are both quantitative (analytic) and qualitative (holistic) aspects of equal importance and which in the dynamic generation of prime number behaviour (both with respect to the individual primes and their general distribution) simultaneously arise.
And the simplest expression expressing the general distribution of the primes is given as n/log n (where we are using the natural log base of e).
Now n properly relates to a number representing a base quantity whereas log n properly relates to a dimensional number (that relatively is of a qualitative nature).
So once again - when appropriately interpreted - this simple expression (for the general distribution of primes) demonstrates the inherent connection (in the very nature of primes) as between its quantitative and qualitative aspects.
Sunday, August 21, 2011
The True Nature of Prime Numbers
It is essential to recognise from the onset that prime numbers have both quantitative and qualitative aspects that are of equal importance.
Now the quantitative aspect is readily recognised in Type 1 Mathematics, where prime numbers are viewed as the building blocks of the number system.
So from this perspective, we have a one way link whereby the natural numbers are derived from the primes.
However the primes have an equally important qualitative aspect in their overall holistic relationship to the natural numbers. So from this perspective we have the reverse direction whereby the primes with respect to their general distribution intimately depend on the natural numbers.
Now it might be argued that Type 1 Mathematics is intimately concerned likewise with this latter aspect to which for example the Prime Number Theorem and Riemann Hypothesis directly relate!
However as befits Type 1 Mathematics, both the individual and collective aspect of primes is approached from a merely (reduced) quantitative perspective.
Whereas in isolation both the individual prime numbers and their general distribution do indeed have a valid quantitative aspect, in dynamic relation to each other the relationship between them is necessarily quantitative to qualitative (and qualitative to quantitative) respectively.
I have illustrated this important point countless times before. However as it is so vital in grasping the essential nature of the primes, I will do so again.
If one travels up a road a turn will have an unambiguous designation (as left or right). So say for example we come to a crossroads where we have a left turn (in the Western direction) and a right turn (in the Eastern direction).
Now it is possible here to operate through recognising both turns as left in the following manner. If we mark one turn as left travelling up the road, by switching reference frames and now moving down the road, what was right from the up direction will now be left from the down direction.
So by switching reference frames and treating each situation in isolation, a merely left designation now applies to both turns.
It is precisely the same with Type 1 Mathematics. When behaviour with respect to the individual primes is adopted, a quantitative approach applies. Then when attention switches to the general nature of the primes again a - merely - quantitative approach is adopted.
Now of course where both aspects are studied in isolation this is perfectly valid.
However when we attempt to combine both aspects to show the necessary interdependence as between individual and general behaviour, then both quantitative and qualitative aspects must be formally incorporated. And this requires Type 1 and Type 2 Mathematics.
So the behaviour of primes is properly of a dynamic interactive nature, where both the individual primes and their collective behaviour (with respect to the natural numbers) are simultaneously determined.
And this clearly implies the equal importance of both the quantitative and qualitative aspects of the primes.
So the differentiation of the individual primes (in a discrete manner) cannot be abstracted from their corresponding integration with the natural numbers (that is continuous).
So the behaviour of prime numbers in correct dynamic terms is intimately related to natural life processes (which have both complementary physical and psychological aspects).
However the ultimate secrets governing behaviour are already inherent in such processes prior to all such phenomenal investigation.
Now the quantitative aspect is readily recognised in Type 1 Mathematics, where prime numbers are viewed as the building blocks of the number system.
So from this perspective, we have a one way link whereby the natural numbers are derived from the primes.
However the primes have an equally important qualitative aspect in their overall holistic relationship to the natural numbers. So from this perspective we have the reverse direction whereby the primes with respect to their general distribution intimately depend on the natural numbers.
Now it might be argued that Type 1 Mathematics is intimately concerned likewise with this latter aspect to which for example the Prime Number Theorem and Riemann Hypothesis directly relate!
However as befits Type 1 Mathematics, both the individual and collective aspect of primes is approached from a merely (reduced) quantitative perspective.
Whereas in isolation both the individual prime numbers and their general distribution do indeed have a valid quantitative aspect, in dynamic relation to each other the relationship between them is necessarily quantitative to qualitative (and qualitative to quantitative) respectively.
I have illustrated this important point countless times before. However as it is so vital in grasping the essential nature of the primes, I will do so again.
If one travels up a road a turn will have an unambiguous designation (as left or right). So say for example we come to a crossroads where we have a left turn (in the Western direction) and a right turn (in the Eastern direction).
Now it is possible here to operate through recognising both turns as left in the following manner. If we mark one turn as left travelling up the road, by switching reference frames and now moving down the road, what was right from the up direction will now be left from the down direction.
So by switching reference frames and treating each situation in isolation, a merely left designation now applies to both turns.
It is precisely the same with Type 1 Mathematics. When behaviour with respect to the individual primes is adopted, a quantitative approach applies. Then when attention switches to the general nature of the primes again a - merely - quantitative approach is adopted.
Now of course where both aspects are studied in isolation this is perfectly valid.
However when we attempt to combine both aspects to show the necessary interdependence as between individual and general behaviour, then both quantitative and qualitative aspects must be formally incorporated. And this requires Type 1 and Type 2 Mathematics.
So the behaviour of primes is properly of a dynamic interactive nature, where both the individual primes and their collective behaviour (with respect to the natural numbers) are simultaneously determined.
And this clearly implies the equal importance of both the quantitative and qualitative aspects of the primes.
So the differentiation of the individual primes (in a discrete manner) cannot be abstracted from their corresponding integration with the natural numbers (that is continuous).
So the behaviour of prime numbers in correct dynamic terms is intimately related to natural life processes (which have both complementary physical and psychological aspects).
However the ultimate secrets governing behaviour are already inherent in such processes prior to all such phenomenal investigation.
Saturday, August 20, 2011
The Number e
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!
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!
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.
Subscribe to:
Posts (Atom)