In the last contribution, I showed the relationship as between the extended Euler Identity and the Type 2 Number System.
Thus once again,
where k = 1, e^(2*k*i*pi) = e^(2*i*pi) = 1^1
where k = 2, e^(2*k*i*pi) = e^(4*i*pi) = 1^2
where k = 3, e^(2*k*i*pi) = e^(6*i*pi) = 1^3
where k = 4, e^(2*k*i*pi) = e^(8*i*pi) = 1^4,
and so on.
As there is no recognition in Type 1 Conventional Mathematics of the qualitative dimensional aspect of interpretation, a reduced and - ultimately - faulty understanding is given of the Euler relationships.
So just as in Type 1 terms, 1^1 = 1^2 = 1^3 = 1^4 =......,
likewise in reduced quantitative terms,
e^(2*i*pi) = e^(4*i*pi) = e^(6*i*pi) = e^(8*i*pi) =.......
However this misleading interpretation can be shown to lead to a problem which is very revealing in its consequences.
Because e^(2*i*pi) = 1^1, then when we raise both sides to the power of i, we get
e^(2*i*pi)^i = 1^i
Therefore e^(- 2*pi) = 1^i
So 1/{e^(2*pi)} = 1^i
Therefore 1^i = .0018674427....
However according to Type 1 interpretation,
e^(2*i*pi) for example = e^(4*i*pi)
So therefore in Type 1 terms,
e^(4*i*pi)^i = 1^i
Thus e^(- 4*pi) = 1^i
And 1/{{e^(4*pi)} = 1^i
Thus 1^i = .00000348734...
And because in Type 1 terms,
e^(2*i*pi) = e^(4*i*pi) = e^(6*i*pi) = e^(8*i*pi) =....... ad infinitum,
this implies that we can have an infinite number of valid quantitative results for 1^i!
Now in Type 1 terms this myriad of embarrassing riches is handled in a merely pragmatic unconvincing fashion. Just as with the many possible (circular type) roots of a number the positive real numbered root is considered as the principle root (though strictly it does not represent the correct root), likewise in this situation the quantitative result pertaining to e^(2*i*pi) is taken as the principle value with the other possible values (of an infinite set) effectively ignored.
However by employing Type 2 interpretation we can easily resolve this problem
So from a Type 2 interpretation
e^(2*i*pi) ≠ e^(4*i*pi)≠ e^(6*i*pi) ≠ e^(8*i*pi)... and so on,
Rather e^(2*i*pi) = 1^1
e^(4*i*pi) = 1^2
e^(6*i*pi) = 1^3
e^(8*i*pi) = 1^4, and so on
Therefore for example whereas
e^(2*i*pi) = 1^i,
e^(4*i*pi) = 1^2i, and so on
Therefore when seen from this perspective 1^i does indeed have one unique answer.
The second answer that we calculated above i.e. .00000348734... does not correspond in fact with the value of 1^i but rather 1^2i!
What is remarkable here is that we have now used Type 2 interpretation - not alone to show a qualitative distinction - when 1 is raised to a real dimensional number, but now in reverse fashion to show that a quantitative distinction is likewise involved when 1 is raised to an imaginary dimensional number.
This also strongly hints at the true nature of the imaginary number i.e. as of a qualitative holistic nature (expressed indirectly in a real quantitative manner)!
So when we raise 1 to a real rational number (as dimension), the result will fall on the circle (of unit radius).
However when we raise 1 to an imaginary rational number (as dimension), the result will fall on the straight line!
Though I had for many years recognised that there was a qualitatively distinct approach to Mathematics (which I refer to as Type 2), For some time I considered that these two separate aspects could be conducted in relative independence of each other.
In other words I did not directly consider that Type 2 interpretation would have a direct relevance with respect to derivation of quantitative results!
However appreciation of the extended use of the Euler Identity has changed all this for its real message is that both quantitative and qualitative type interpretation are inextricably linked!
So ultimately we cannot have consistent type interpretation of quantitative results without corresponding consistency in qualitative terms.
Therefore in my own evolution appreciation of the true nature of the Euler Identity (from both a quantitative and qualitative perspective) was to prove a key landmark in eventually unravelling the true nature of the Riemann Hypothesis which is essentially the same message i.e. that both quantitative and qualitative type interpretation are inseparable!
However with respect to the Riemann Hypothesis, this poses insuperable problems as within Conventional Mathematics there is - as yet - no formal recognition of its equally important qualitative aspect!
Wednesday, August 31, 2011
Tuesday, August 30, 2011
A New Number System (1)
Let me first clarify the distinction I make as between linear and circular (with respect to number systems) on the one hand and quantitative and qualitative on the other.
Now in the conventional Type 1 mathematical approach - though the overall qualitative approach is decidedly linear - both linear and circular notions can be dealt with from a quantitative perspective. The real number system for example is viewed in a linear fashion (as points laid out in a line). Indeed the imaginary number system is viewed in a similar fashion (lying on a line vertical to those of the real). However when it comes to the roots of unity, these lie in the circle of unit radius (in the complex plane). Now the quantitative nature of these roots can be dealt with (without however their true significance being realised).
In the Type 2 mathematical approach both linear and circular notions of logical interpretation can also be brought to bear on these same number systems. Also one clear implication of this approach is that circular notions (in quantitative terms) cannot be properly interpreted in the absence of circular type interpretation (from the qualitative perspective).
Thus the full use of both linear and circular notions requires that they be given both Type 1 (quantitative) and Type 2 (qualitative) interpretations.
The key significance of the Euler Identity is that - though seemingly arising in the context of the Type 1 approach to Mathematics - it actually gives rise to the need for the Type 2 approach.
In a very true sense therefore it arises at the very intersection of Type 1 and Type 2 approaches (where both quantitative and qualitative mathematical notions are fully interdependent).
I have already defined the natural number system with respect to Type 1 and Type 2 interpretation.
The Type 1 approach is qualitatively linear in nature and is literally defined in 1-dimensional terms,
1^1, 2^1, 3^1, 4^1,.......
So here the base number quantity keeps changing while the default dimensional number quality remains fixed as 1.
The Type 2 approach is by contrast quantitatively linear in nature, where the dimensional number quality keeps changing,
1^1, 1^2, 1^3, 1^4,.......
The circular nature of this alternative number system can indirectly be shown
in quantitative terms through obtaining the corresponding root (i.e. the reciprocal of the dimension in question).
Therefore in the circular number system, there is an inverse relationship as between a qualitative dimensional interpretation and corresponding quantitative root.
So once again for example, to properly explain the square root of 1 i.e. 1^(1/2), we need the corresponding qualitative dimensional interpretation of 2 i.e. 1^2!
Now what is fascinating about the fundamental Euler Identity is that it leads directly to this Type 2 Number system
So e^(2*i*pi) = 1 i.e. 1^1
Now all other numbers for example in the Type 2 natural number system can be obtained from the expression
e^(2*k*i*pi) where k = 1, 2, 3, 4,....
So in fact e^(2*i*pi) is just the special case where k = 1.
Thus where k = 1, e^(2*k*i*pi) = e^(2*i*pi) = 1^1
where k = 2, e^(2*k*i*pi) = e^(4*i*pi) = 1^2
where k = 3, e^(2*k*i*pi) = e^(6*i*pi) = 1^3
where k = 4, e^(2*k*i*pi) = e^(8*i*pi) = 1^4,
and so on.
Now we have already dealt with the ambiguity in terms of conventional interpretation of roots where for example but + 1 and - 1 are both given as the square root of 1, i.e. 1^(1/2).
Indeed I have already argued using Type 2 interpretation that properly - 1 is (unambiguously) the square root of 1 i.e.1^(1/2).
Now this is born out directly through extension of the fundamental Euler Identity.
So when k = 1/2, then
e^(2*k*i*pi) = e^(i*pi) = - 1.
So one valid way of interpreting the Euler Identity (as it is generally presented) is that the square root of 1 i.e. 1^(1/2) = - 1.
However this conflicts directly with conventional reduced (Type 1) interpretation whereby the principle square root of 1 = + 1.
So for example if you take out any calculator, input 1 and then raise this to .5, you will be given the result of 1, which the Euler Identity tells us is erroneous!
One of teh great advantages of the extended Euler Identity is that it provides a ready means for calculating all roots of 1.
So e^(2*k*i*pi) = cos(2*k*pi) + i sin(2*k*pi)
Therefore for example to obtain the cube root of 1 i.e. 1^(1/3) we let k = 1/3
So e^(2*k*i*pi) = e^(2*i*pi)/3 = cos(2*pi)/3 + i sin(2*pi)/3
which represented in degrees (rather than radians) = cos 120 + i sin 120
= {- 1 + 3^(1/2)i}/2 = 0.5 + .8660i (correct to 4 decimal places)
Note once again that just one unambiguous answer corresponds with the cube root of 1 i.e 1^(1/3)!
Now in the conventional Type 1 mathematical approach - though the overall qualitative approach is decidedly linear - both linear and circular notions can be dealt with from a quantitative perspective. The real number system for example is viewed in a linear fashion (as points laid out in a line). Indeed the imaginary number system is viewed in a similar fashion (lying on a line vertical to those of the real). However when it comes to the roots of unity, these lie in the circle of unit radius (in the complex plane). Now the quantitative nature of these roots can be dealt with (without however their true significance being realised).
In the Type 2 mathematical approach both linear and circular notions of logical interpretation can also be brought to bear on these same number systems. Also one clear implication of this approach is that circular notions (in quantitative terms) cannot be properly interpreted in the absence of circular type interpretation (from the qualitative perspective).
Thus the full use of both linear and circular notions requires that they be given both Type 1 (quantitative) and Type 2 (qualitative) interpretations.
The key significance of the Euler Identity is that - though seemingly arising in the context of the Type 1 approach to Mathematics - it actually gives rise to the need for the Type 2 approach.
In a very true sense therefore it arises at the very intersection of Type 1 and Type 2 approaches (where both quantitative and qualitative mathematical notions are fully interdependent).
I have already defined the natural number system with respect to Type 1 and Type 2 interpretation.
The Type 1 approach is qualitatively linear in nature and is literally defined in 1-dimensional terms,
1^1, 2^1, 3^1, 4^1,.......
So here the base number quantity keeps changing while the default dimensional number quality remains fixed as 1.
The Type 2 approach is by contrast quantitatively linear in nature, where the dimensional number quality keeps changing,
1^1, 1^2, 1^3, 1^4,.......
The circular nature of this alternative number system can indirectly be shown
in quantitative terms through obtaining the corresponding root (i.e. the reciprocal of the dimension in question).
Therefore in the circular number system, there is an inverse relationship as between a qualitative dimensional interpretation and corresponding quantitative root.
So once again for example, to properly explain the square root of 1 i.e. 1^(1/2), we need the corresponding qualitative dimensional interpretation of 2 i.e. 1^2!
Now what is fascinating about the fundamental Euler Identity is that it leads directly to this Type 2 Number system
So e^(2*i*pi) = 1 i.e. 1^1
Now all other numbers for example in the Type 2 natural number system can be obtained from the expression
e^(2*k*i*pi) where k = 1, 2, 3, 4,....
So in fact e^(2*i*pi) is just the special case where k = 1.
Thus where k = 1, e^(2*k*i*pi) = e^(2*i*pi) = 1^1
where k = 2, e^(2*k*i*pi) = e^(4*i*pi) = 1^2
where k = 3, e^(2*k*i*pi) = e^(6*i*pi) = 1^3
where k = 4, e^(2*k*i*pi) = e^(8*i*pi) = 1^4,
and so on.
Now we have already dealt with the ambiguity in terms of conventional interpretation of roots where for example but + 1 and - 1 are both given as the square root of 1, i.e. 1^(1/2).
Indeed I have already argued using Type 2 interpretation that properly - 1 is (unambiguously) the square root of 1 i.e.1^(1/2).
Now this is born out directly through extension of the fundamental Euler Identity.
So when k = 1/2, then
e^(2*k*i*pi) = e^(i*pi) = - 1.
So one valid way of interpreting the Euler Identity (as it is generally presented) is that the square root of 1 i.e. 1^(1/2) = - 1.
However this conflicts directly with conventional reduced (Type 1) interpretation whereby the principle square root of 1 = + 1.
So for example if you take out any calculator, input 1 and then raise this to .5, you will be given the result of 1, which the Euler Identity tells us is erroneous!
One of teh great advantages of the extended Euler Identity is that it provides a ready means for calculating all roots of 1.
So e^(2*k*i*pi) = cos(2*k*pi) + i sin(2*k*pi)
Therefore for example to obtain the cube root of 1 i.e. 1^(1/3) we let k = 1/3
So e^(2*k*i*pi) = e^(2*i*pi)/3 = cos(2*pi)/3 + i sin(2*pi)/3
which represented in degrees (rather than radians) = cos 120 + i sin 120
= {- 1 + 3^(1/2)i}/2 = 0.5 + .8660i (correct to 4 decimal places)
Note once again that just one unambiguous answer corresponds with the cube root of 1 i.e 1^(1/3)!
Monday, August 29, 2011
Avoiding Contradiction
From the Type 1 mathematical perspective
1) e^(2*i*pi) = 1
2) e^(- 2*1*pi) = 1
3) e^(0) = 1.
One might thereby be attempted to conclude therefore that the dimensional expressions in each case must also be equal
i.e. that 2*i*pi = - 2*i*pi = 0!
This clearly is not permissible from a Type 1 (quantitative) perspective. However it requires inclusion of the Type 2 (qualitative) perspective to properly show why this is the case.
As we have seen in Type 1 terms 1^1 = 1^2 = 1^3 .... = 1^n. So from this merely quantitative perspective each of these terms = 1.
We have already shown that properly distinguishing - 1 (as the root of 1) from + 1 requires a Type 2 explanation. So in qualitative terms 1^1 is recognised as distinct from 1^2. So strictly speaking therefore - 1 is the square root of 1^1, whereas + 1 is the square root of 1^2!
Appropriately distinguishing 2*i*pi, - 2*i*pi and 0 from each other (as dimensional expressions) requires a similar qualitative interpretation!
1) e^(2*i*pi) = 1 (i.e. 1^1)
2) e^(- 2*1*pi) = 1 (i.e. 1^0)
This is easily seen by the fact that 2) is the inverse of 1) so that
e^(- 2*1*pi) = 1/e^(2*i*pi) = 1^1/(1^1) = 1^(1 - 1) = 1^0.
3) e^(0) = 1) * 2) = e^(2*i*pi) * e^(- 2*1*pi) = e^(2*i*pi - 2*i*pi) = e^0.
So to conclude e^0 = 1^1 * 1^0.
Now of course from a merely Type 1 perspective 1^1 * 1^0 = 1^1. However the very point in this context is that we are using a Type 2 (qualitative) interpretation.
Thus in a qualitative context, the three different dimensional expressions 2*i*pi, - 2*i*pi and 0 have a subtly distinct interpretation.
So 2*i*pi = 1^1 corresponds to the linear rational component of non-dimensional understanding (insofar as we can give the the non-dimensional notion a finite actual meaning)!
- 2*i*pi = 1^0 corresponds to the circular intuitive component of non-dimensional understanding (where we are relate to a formless non-phenomenal experience).
0 = 2*i*pi - 2*i*pi combines both rational and intuitive comprehension. So in dynamic experiential terms, our notion of 0 as in the expression e^0, necessarily entails both quantitative and qualitative aspects that are interdependent. However in terms of the the two number systems (quantitative and qualitative) these are necessarily split up.
To conclude the notion of 0 as a dimension literally relates to the concept of a point (which is non-dimensional). This in turn implies the identity of both linear and circular interpretations (in both quantitative and qualitative terms).
Clearly as Type 1 Mathematics is devoid of a circular qualitative dimension, it lacks the means to adequately interpret the Euler Identity. It can indeed provide the quantitative demonstration of its validity, but then lacks the means to convey its deeper significance (which is of a qualitative nature).
The real mystery of the Euler Identity is that it beautifully combines both the quantitative and qualitative meanings of its symbols (in a manner where they become indivisible). However this requires that both Type 1 and Type 2 mathematical interpretations be coherently combined!
1) e^(2*i*pi) = 1
2) e^(- 2*1*pi) = 1
3) e^(0) = 1.
One might thereby be attempted to conclude therefore that the dimensional expressions in each case must also be equal
i.e. that 2*i*pi = - 2*i*pi = 0!
This clearly is not permissible from a Type 1 (quantitative) perspective. However it requires inclusion of the Type 2 (qualitative) perspective to properly show why this is the case.
As we have seen in Type 1 terms 1^1 = 1^2 = 1^3 .... = 1^n. So from this merely quantitative perspective each of these terms = 1.
We have already shown that properly distinguishing - 1 (as the root of 1) from + 1 requires a Type 2 explanation. So in qualitative terms 1^1 is recognised as distinct from 1^2. So strictly speaking therefore - 1 is the square root of 1^1, whereas + 1 is the square root of 1^2!
Appropriately distinguishing 2*i*pi, - 2*i*pi and 0 from each other (as dimensional expressions) requires a similar qualitative interpretation!
1) e^(2*i*pi) = 1 (i.e. 1^1)
2) e^(- 2*1*pi) = 1 (i.e. 1^0)
This is easily seen by the fact that 2) is the inverse of 1) so that
e^(- 2*1*pi) = 1/e^(2*i*pi) = 1^1/(1^1) = 1^(1 - 1) = 1^0.
3) e^(0) = 1) * 2) = e^(2*i*pi) * e^(- 2*1*pi) = e^(2*i*pi - 2*i*pi) = e^0.
So to conclude e^0 = 1^1 * 1^0.
Now of course from a merely Type 1 perspective 1^1 * 1^0 = 1^1. However the very point in this context is that we are using a Type 2 (qualitative) interpretation.
Thus in a qualitative context, the three different dimensional expressions 2*i*pi, - 2*i*pi and 0 have a subtly distinct interpretation.
So 2*i*pi = 1^1 corresponds to the linear rational component of non-dimensional understanding (insofar as we can give the the non-dimensional notion a finite actual meaning)!
- 2*i*pi = 1^0 corresponds to the circular intuitive component of non-dimensional understanding (where we are relate to a formless non-phenomenal experience).
0 = 2*i*pi - 2*i*pi combines both rational and intuitive comprehension. So in dynamic experiential terms, our notion of 0 as in the expression e^0, necessarily entails both quantitative and qualitative aspects that are interdependent. However in terms of the the two number systems (quantitative and qualitative) these are necessarily split up.
To conclude the notion of 0 as a dimension literally relates to the concept of a point (which is non-dimensional). This in turn implies the identity of both linear and circular interpretations (in both quantitative and qualitative terms).
Clearly as Type 1 Mathematics is devoid of a circular qualitative dimension, it lacks the means to adequately interpret the Euler Identity. It can indeed provide the quantitative demonstration of its validity, but then lacks the means to convey its deeper significance (which is of a qualitative nature).
The real mystery of the Euler Identity is that it beautifully combines both the quantitative and qualitative meanings of its symbols (in a manner where they become indivisible). However this requires that both Type 1 and Type 2 mathematical interpretations be coherently combined!
When 0 is not Nothing!
The significance of the dimensional expression 2*i*pi is highly elusive. In fact there are direct connections to the mystical notion of "bindu" or point centredness or even more recently the physical notion of a singularity.
The way I look at it is as the centre point of the "imaginary" circle.
We can approach this initially with reference to the "real" circle.
Now the centre of this circle is equally the centre point of its line diameter. So in this sense it is at this point that both linear and circular notions are reconciled as identical. Put another way we could say that the quantitative and qualitative notions of interpretation (characterising Type 1 and Type 2 Mathematics respectively) are reconciled.
However the problem with the real circle is that it cannot be conceived in the absence of linear notions (relating to the size its radius). So though with the central line diameter we can depict the middle point as 0 with the right hand side in positive units and the corresponding left hand side in negative units (with both equal) in actual fact we represent both positively geometrically as lines (with equal extension). Now if we could imagine the circle in more dynamic terms where positive and negative aspects cancel each other out the circle would shrink to a point (with no linear extension). Likewise of course the line diameter would shrink to the same point. So at this point the line and the circle would be identical. Now inherent in the qualitative notion of the imaginary is this complementarity of positive and negative polarities. So the imaginary unit circle which is represented by 2*i*pi is therefore identical with this point (where in a sense the circle and line have collapsed to a point where they are indistinguishable from each other).
Now quantitative interpretation is of a linear rational nature based on finite notions of form; qualitative interpretation is directly of a circular intuitive nature based on infinite notions of emptiness.
So in fact we can have 3 different expressions for this non-dimensional point.
First we can have the rational interpretation where it is approached from the perspective of actual form i.e. as being represented by a quantity - rather like the infinitesimal notion - that is in the process of becoming nothing i.e. 0.
Secondly we can have the intuitive interpretation where it approached from the perspective of emptiness (through negation of the quantitative aspect). So from this perspective when we have surrendered any rational notion of nothingness we are intuitively left with the experiential awareness of its true qualitative nature.
Thirdly we can have the combined appreciation of both quantitative (rational) and qualitative (intuitive) notions.
As I say this area is in fact dealt with at length in the mystical contemplative literature where the three notions I have mentioned represent three stages in the specialised growth of pure spiritual understanding.
The first is often referred to a the arrival at a pure state of concentrated transcendent awareness i.e. as spirit as the centre of one's being. Here spirit is understood as beyond all phenomenal form.
However because initially there is lack of corresponding immanent awareness i.e. where spirit is seen as equally inherent in all form, this leads to a lingering phenomenal attachment to the very notion of this point. Put another way though maintaining the absolute (separate) nature of ineffable spirit one thereby still understands the notion in an unduly phenomenal manner.
So the second stage requires the negation of this lingering - merely - phenomenal understanding a spiritual point. This - when successful - leads in turn to corresponding free intuitive awareness of its qualitative nature.
However because both rational and intuitive are themselves dynamically complementary in experience, ultimately both refined rational and intuitive interpretation must be combined in the realisation of creation as both form and emptiness. This thereby entails both the immanent aspect (where spirit is inherent in matter) and the transcendent aspect (where spirit is beyond all form).
And by this third stage both the quantitative and qualitative aspects of understanding are coherently related.
We will see how these notions are related to the mathematical treatment of 2*i*pi in the next contribution.
The way I look at it is as the centre point of the "imaginary" circle.
We can approach this initially with reference to the "real" circle.
Now the centre of this circle is equally the centre point of its line diameter. So in this sense it is at this point that both linear and circular notions are reconciled as identical. Put another way we could say that the quantitative and qualitative notions of interpretation (characterising Type 1 and Type 2 Mathematics respectively) are reconciled.
However the problem with the real circle is that it cannot be conceived in the absence of linear notions (relating to the size its radius). So though with the central line diameter we can depict the middle point as 0 with the right hand side in positive units and the corresponding left hand side in negative units (with both equal) in actual fact we represent both positively geometrically as lines (with equal extension). Now if we could imagine the circle in more dynamic terms where positive and negative aspects cancel each other out the circle would shrink to a point (with no linear extension). Likewise of course the line diameter would shrink to the same point. So at this point the line and the circle would be identical. Now inherent in the qualitative notion of the imaginary is this complementarity of positive and negative polarities. So the imaginary unit circle which is represented by 2*i*pi is therefore identical with this point (where in a sense the circle and line have collapsed to a point where they are indistinguishable from each other).
Now quantitative interpretation is of a linear rational nature based on finite notions of form; qualitative interpretation is directly of a circular intuitive nature based on infinite notions of emptiness.
So in fact we can have 3 different expressions for this non-dimensional point.
First we can have the rational interpretation where it is approached from the perspective of actual form i.e. as being represented by a quantity - rather like the infinitesimal notion - that is in the process of becoming nothing i.e. 0.
Secondly we can have the intuitive interpretation where it approached from the perspective of emptiness (through negation of the quantitative aspect). So from this perspective when we have surrendered any rational notion of nothingness we are intuitively left with the experiential awareness of its true qualitative nature.
Thirdly we can have the combined appreciation of both quantitative (rational) and qualitative (intuitive) notions.
As I say this area is in fact dealt with at length in the mystical contemplative literature where the three notions I have mentioned represent three stages in the specialised growth of pure spiritual understanding.
The first is often referred to a the arrival at a pure state of concentrated transcendent awareness i.e. as spirit as the centre of one's being. Here spirit is understood as beyond all phenomenal form.
However because initially there is lack of corresponding immanent awareness i.e. where spirit is seen as equally inherent in all form, this leads to a lingering phenomenal attachment to the very notion of this point. Put another way though maintaining the absolute (separate) nature of ineffable spirit one thereby still understands the notion in an unduly phenomenal manner.
So the second stage requires the negation of this lingering - merely - phenomenal understanding a spiritual point. This - when successful - leads in turn to corresponding free intuitive awareness of its qualitative nature.
However because both rational and intuitive are themselves dynamically complementary in experience, ultimately both refined rational and intuitive interpretation must be combined in the realisation of creation as both form and emptiness. This thereby entails both the immanent aspect (where spirit is inherent in matter) and the transcendent aspect (where spirit is beyond all form).
And by this third stage both the quantitative and qualitative aspects of understanding are coherently related.
We will see how these notions are related to the mathematical treatment of 2*i*pi in the next contribution.
Sunday, August 28, 2011
Spiritual Illumination
Again with reference to Marcus du Sautoy's "The Music of the Primes", I found an interesting quote on P.297 attributed to Andre Weil.
"Every mathematician worthy of the name has experienced ... the state of lucid exaltation in which one thought succeeds another as if miraculously...this feeling may last for hours at a time, even for days. Once you have experienced it you are eager to repeat it but unable to do it at will, unless perhaps by dogged work..."
What Weil is desribing here is in fact - what spiritual writers refer to as - "illumination" where for a while one has a peak experience of holistic intuitive insight. It is in such moments the truly great mathematical insights are obtained and those decisive creative breakthroughs where for a brief moment one is able to "see" certain important relationships - perhaps for the first time - in an enhanced manner.
In fact quite clearly such moments relate directly to the qualitative - rather than quantitative - aspect of mathematical appreciation. Though the intuitive insights obtained may indeed be later expressed in a (reduced) rational manner that wins the acceptance of the mathematical community, the initial intuitive realisation properly remains of a qualitative nature.
Remarkably however, the qualitative aspect of Mathematics is given no formal recognition at all in conventional terms.
In other words though every mathematical symbol, relationship, hypothesis etc. has a distinctive (holistic) qualitative as well as (analytical) quantitative interpretation in Type 1 Mathematics only the the latter is recognised.
Thus in a comprehesive treatment we should have both Type 1 (quantitative) and Type 2 (qualitative) aspects that initially are developed in relative independence from each other.
Then when both of these aspects have achieved appropriate degrees of specialisation they can be fruitfully combined with each other in the most advanced form of Mathematics (i.e. Type 3).
"Every mathematician worthy of the name has experienced ... the state of lucid exaltation in which one thought succeeds another as if miraculously...this feeling may last for hours at a time, even for days. Once you have experienced it you are eager to repeat it but unable to do it at will, unless perhaps by dogged work..."
What Weil is desribing here is in fact - what spiritual writers refer to as - "illumination" where for a while one has a peak experience of holistic intuitive insight. It is in such moments the truly great mathematical insights are obtained and those decisive creative breakthroughs where for a brief moment one is able to "see" certain important relationships - perhaps for the first time - in an enhanced manner.
In fact quite clearly such moments relate directly to the qualitative - rather than quantitative - aspect of mathematical appreciation. Though the intuitive insights obtained may indeed be later expressed in a (reduced) rational manner that wins the acceptance of the mathematical community, the initial intuitive realisation properly remains of a qualitative nature.
Remarkably however, the qualitative aspect of Mathematics is given no formal recognition at all in conventional terms.
In other words though every mathematical symbol, relationship, hypothesis etc. has a distinctive (holistic) qualitative as well as (analytical) quantitative interpretation in Type 1 Mathematics only the the latter is recognised.
Thus in a comprehesive treatment we should have both Type 1 (quantitative) and Type 2 (qualitative) aspects that initially are developed in relative independence from each other.
Then when both of these aspects have achieved appropriate degrees of specialisation they can be fruitfully combined with each other in the most advanced form of Mathematics (i.e. Type 3).
Saturday, August 27, 2011
Interesting Comments
I was just re-reading Du Sautoy's "Music of the Primes" in the Chapter where he links up the Riemann Zeros with findings from quantum mechanics. In one paragraph he makes statements that link up perfectly with the qualitative approach that I have adopted.
For example he sees on P.267. "For as long as the quantum world remains unobserved it exists only in the world of imaginary numbers".
Now I have long maintained that actual observed and potential unobserved reality are properly interpreted according to two logical systems that are linear and circular with respect to each other. And in the simplest version we use both 1-dimensional (linear) and 2-dimensional (circular) interpretation!
Also I have maintained that the imaginary concept - as used in Mathematics - represents an indirect means of converting 2-dimensional to 1-dimensional format.
So not surprisingly therefore the 2-dimensional world of potential (unobserved) reality would be represented through imaginary numbers.
He the says later in the same paragraph
"When we observe an event in the quantum world, it is as though we are not seeing the event itself in its natural domain, but a shadow of the event projected into our "real" world of natural numbers"
This statement in fact perfectly equates with the physical complement of Jungian notions of the unconscious.
So when experience is directly of a holistic unconscious nature (which cannot be phenomenally appropriated in its own domain) it becomes projected into the conscious world (where it is generally confused with the specific phenomena involved). And of course in Jungian terms this is the very means by which the "shadow" (unconscious) personality expresses itself.
Now the direct implication of this for the world of quantum reality is that - properly understood - it entails the physical aspect as complementary to the notion of the unconscious in psychological terms.
We can refer to this physical complement as the holistic ground or - perhaps - holistic dimensional ground of reality. However the implication is clear!
Just as psychological experience entails the continual dynamic interaction of two aspects that are conscious and unconscious with respect to each other, likewise physical reality - especially at the quantum level - entails the continual dynamic interaction of manifest (observed) reality and a hidden (unobserved) holistic potential from which the observed phenomena emerge.
Now the problem in physics is that because no recognition is given to the intuitive mode of the unconscious in formal interpretation, it has great difficulties in accurately portraying the true nature of quantum reality (from a qualitative perspective).
It is now recognised that the non-trivial zeros of the Riemann Hypothesis bear a close relationship to certain quantum chaotic energy vibrations in the physical world.
The clear implication therefore that adequate interpretation of the Riemann Hypothesis entails a qualitative (Type 2) as well as quantitative (Type 1) approach.
And indeed once again the key significance of the Hypothesis - as I have repeatedly stated - is that it expresses the condition necessary for the consistent reconciliation of both aspects.
For example he sees on P.267. "For as long as the quantum world remains unobserved it exists only in the world of imaginary numbers".
Now I have long maintained that actual observed and potential unobserved reality are properly interpreted according to two logical systems that are linear and circular with respect to each other. And in the simplest version we use both 1-dimensional (linear) and 2-dimensional (circular) interpretation!
Also I have maintained that the imaginary concept - as used in Mathematics - represents an indirect means of converting 2-dimensional to 1-dimensional format.
So not surprisingly therefore the 2-dimensional world of potential (unobserved) reality would be represented through imaginary numbers.
He the says later in the same paragraph
"When we observe an event in the quantum world, it is as though we are not seeing the event itself in its natural domain, but a shadow of the event projected into our "real" world of natural numbers"
This statement in fact perfectly equates with the physical complement of Jungian notions of the unconscious.
So when experience is directly of a holistic unconscious nature (which cannot be phenomenally appropriated in its own domain) it becomes projected into the conscious world (where it is generally confused with the specific phenomena involved). And of course in Jungian terms this is the very means by which the "shadow" (unconscious) personality expresses itself.
Now the direct implication of this for the world of quantum reality is that - properly understood - it entails the physical aspect as complementary to the notion of the unconscious in psychological terms.
We can refer to this physical complement as the holistic ground or - perhaps - holistic dimensional ground of reality. However the implication is clear!
Just as psychological experience entails the continual dynamic interaction of two aspects that are conscious and unconscious with respect to each other, likewise physical reality - especially at the quantum level - entails the continual dynamic interaction of manifest (observed) reality and a hidden (unobserved) holistic potential from which the observed phenomena emerge.
Now the problem in physics is that because no recognition is given to the intuitive mode of the unconscious in formal interpretation, it has great difficulties in accurately portraying the true nature of quantum reality (from a qualitative perspective).
It is now recognised that the non-trivial zeros of the Riemann Hypothesis bear a close relationship to certain quantum chaotic energy vibrations in the physical world.
The clear implication therefore that adequate interpretation of the Riemann Hypothesis entails a qualitative (Type 2) as well as quantitative (Type 1) approach.
And indeed once again the key significance of the Hypothesis - as I have repeatedly stated - is that it expresses the condition necessary for the consistent reconciliation of both aspects.
Friday, August 26, 2011
Euler's Identity
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.
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.
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 - 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 withing 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 - 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 withing 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.
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
Now, 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 appreciates through the logical structure of the 2nd rather than the 1st dimension).
Therefore to properly understand the Riemann Zeta Function (applying to both positive and negative values of the dimensional power s) 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 understanding.
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 interpretation).
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 thse wave patterns can in turn enable a more exact distribution of the no. of primes (within a given natural number magnitude).
These waves 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 contriubution.
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 fundmamental 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 phenommena 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
Now, 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 appreciates through the logical structure of the 2nd rather than the 1st dimension).
Therefore to properly understand the Riemann Zeta Function (applying to both positive and negative values of the dimensional power s) 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 understanding.
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 interpretation).
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 thse wave patterns can in turn enable a more exact distribution of the no. of primes (within a given natural number magnitude).
These waves 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 contriubution.
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 fundmamental 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 phenommena derives from the dynamic relationship of the prime numbers (with respect to both their quantitative and qualitative aspects).
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