Wednesday, September 21, 2011

More on Euler Identity

I have used the Euler Identity in a holistic mathematical fashion (Type 2 Mathematics) to demonstrate three stages of specialised contemplative development (preceding the full radial unfolding of stages).

This can however be given a more precise expression.

So we now define e^(k*2*i*pi) = 1^x (where k = 1, 1 or 0 and x = 1, 0 and both 1 and 0 respectively.

So when k = 1,

1) e^(k*2*i*pi)= e^(2*i* pi) = 1^1

In holistic terms this corresponds to the rational linear (i.e. 1-dimensional) appreciation of (mystical) union.

Now with the 1st stage (of specialised contemplative development) a slight imbalance remains whereby appreciation remains unduly transcendent in nature.
In other words in emphasising the transcendent nature of spiritual reality as inherently empty and thereby beyond all notions of form, one to a degree still represses the corresponding immanent nature of such reality as prior and thereby inherent in all form.
This in effect leads to a subtle lingering rational attachment (of a necessarily linear nature) to the notion of unity (as phenomenal form) thereby preventing full realisation of complementary intuitive recognition (that is literally non-dimensional as empty of all form)

2) So when k = –  1,

e^(k*2*i*pi)= e^(- 2*i*pi) = 1^0

This is so as e^(2*i*pi) = 1/{e^(2*i*pi)} = (1^1)/(1^1) = 1^(1 1) = 1^0.

With the 2nd stage of such specialised development, this remaining lingering attachment to the notion of union (as a phenomenal point) is gradually negated.
Therefore as remaining involuntary rational attachment is eroded, the complementary pure intuitive realisation of the nature of union can unfold. Strictly this does not mean that the rational aspect of understanding now ceases with union but rather that it can interact with intuition in a very refined - and thereby transparent - fashion (due to the erosion of an excess element of involuntary attachment).

So whereas in the 1st stage we emphasised the refined rational linear appreciation of form (with respect to pure spiritual awareness), here in complementary fashion we are emphasising the corresponding intuitive (non-dimensional i.e. 0-dimensional) aspect of such awareness.
Put another way the emphasis here is on the immanent - as opposed to the transcendent - aspect of spiritual awareness.

3) when k = 0

e^(k*2*i*pi)= e^0 = (1^1)*(1^0)

This is so as e^0 = e^(2*i*pi)* e^(- 2*i*pi) i.e. e^(2*i*pi - 2*i*pi).

and as we have seen,

e^(2*pi*i)* e^( 2*pi*i) = (1^1)*(1^0)

Now in Type 1 Mathematics, we would simply add the powers 1 and 0. However in Type 2 Mathematics these remain of a qualitatively distinct nature.

So the significance of the third formulation is that it represents the most specialised stage of contemplative awareness, where form and emptiness are successfully united in experience. Put another way it represents the full integration of both the transcendent and immanent aspects of spiritual understanding (which in turn implies the most refined interaction possible as between both the rational and intuitive modes of understanding).

So here we have pure emptiness serving as the potential for the entire world of (actual) created phenomenal form.

In mathematical terms this represents the most complete appreciation of both the linear rational (1-dimensional) nature of dimension that serves as the basis for quantitative appreciation of Type 1 Mathematics and the purely circular intuitive (0-dimensional) appreciation that serves as the basis for corresponding qualitative appreciation of Type 2 Mathematics.

This in turn with the radial unfolding of stages allows for the growing interpenetration of both Type 1 and Type 2 (in what constitutes Type 3 Mathematics).

We can perhaps see here the truly remarkable nature of e.

In conventional Type 1 terms any number raised to the power of 0 = 1. This represents therefore a merely reduced linear interpretation of 0 (as dimension). and of course when e is raised to 0 (in this reduced quantitative sense) its value is likewise 0.

However what is unique about e is that when it raised to 0 in Type 2 terms (reflecting the circular nature of dimension) its value is likewise 0.

Now we can appreciate why this is so with reference to the fact that by its very nature e fully reconciles corresponding notions of both differentiation (linear) and integration (circular).

So in Type 1 terms both the differentiation and integration of e^x are identical.

Likewise in Type 2 terms at the the most developed level of contemplative awareness differentiated (discrete) and integrated (continuous) elements are seamless so that phenomena of form no longer appear to arise in experience.

So e uniquely combines in its inherent nature both quantitative (discrete) and qualitative (continuous) aspects.

Now I have long identified the inherent nature of a prime number in that it too combines both extreme quantitative and qualitative aspects in its nature. Thus from a discrete independent nature prime numbers seemingly display no pattern. However from a continuous holistic perspective they display (en bloc) a truly remarkable regularity.

Not surprisingly therefore e has a vital role with respect to understanding the nature of primes.

Quite simply if we want to find the average space between primes (in any region of the number system) we simply find the dimensional power to which e must be raised to attain that number.

So for example to obtain 1,000,000, one must raise e to 13.8155... (i.e. the natural log of 1,000,000).

That means in the region of 1,000,000 we would expect the average space between primes to be just less than 14!

Friday, September 2, 2011

Parallel Riemann Hypothesis!

We concluded the last contribution with the remarkable finding that

i^i = e^(- pi/2), which is a real number!

Now if we take natural logs of each side

then i(log i) = - pi/2,

therefore 1/i(log i) = - 2/pi.

So, - i/log i = - 2/pi

Thus i/log i = 2/pi.

As we know the prime number theorem relating to the general frequency of the primes among the natural numbers is most simply expressed as n/log n (with the proportionate frequency increasing as n becomes larger).

So by allowing n to become progressively larger we have the linear quantitative attempt to reach the infinite (in an actual manner).

Now properly understood i represents the corresponding holistic notion of the infinite where one attempts to appropriate it (in a potential manner).

We can see this in the common psychological appreciation of the imaginary as something that emanates from the holistic unconscious to be embodied in an actual (conscious) manner.

Therefore understood in this light i/log i represents the qualitative correspondent to the prime number theorem!

Now in quantitative terms, we attempt to understand prime numbers from a Type 1 perspective as base quantities i.e.

2^1, 3^1, 5^1, 7^1,.......

However the prime numbers have in Type 2 terms a corresponding qualitative interpretation as dimensions i.e.

1^2, 1^3, 1^5, 1^7,......

In an inverse quantitative manner we can obtain the circular structure of these dimensions through obtaining the reciprocal roots.

So therefore we can attempt to find the 2 roots, 3 roots, 5 roots, 7 roots of unity and so on for each of the prime numbers.

Now in this approach we consider all roots which will have both a real and imaginary component.

With respect to both parts we take the values in an absolute manner (ignoring negative signs). Then we sum up both parts (both real and imaginary taken separately) and then obtain the average.

We can demonstrate simply here for p = 3.

There are 3 corresponding roots of 1 involved i.e. 1, - .5 +.866i and - .5 - .866i

Now ignoring negative signs the sum of the real part here = 1 + .5 + .5 = 2.

Therefore the mean average = 2/3 = .6666.. .

Then taking the magnitude of the imaginary part (ignoring the i) the sum = .866 + .866 = 1.732

Therefore the mean average = 1.7321/3 = .57735...

Now the remarkable finding here as the value of p increases is that the mean value of the absolute quantitative value for both the real and imaginary parts converges on 2/pi = .636619772...
We can readily find all these values through use of the Euler Identity,

e^(2*i^pi) = cos (2*pi) + i sin (2*pi).
So the 3 roots of 1 - where the dimensional numbers are 1/3, 2/3 and 3/3 respectively are calculated in this manner as

cos {(2/3)*pi} + i {sin(2/3)*pi},
cos {(4/3)*pi} + i {sin(42/3)*pi}, and
cos {(6/3)*pi} + i {sin(6/3)*pi}.

As p becomes ever larger the mean value (for both parts) approximates ever closer to i/log i.

So we seem here in fact to have a circular number equivalent to the prime number theorem (that is couched in a linear quantitative manner).

However it does not end here!

We can see from our example above that the mean value for the real part = .6666.. and the imaginary part = .57735... respectively.

Therefore the mean value for the real part exceeds 2/pi and the corresponding value for the imaginary part is less than 2/pi respectively.

In fact looking at the absolute differences the value is .6666... - .636619772..

= .03005 (approx)... for the real part

and .636619772... - .57735... = .05927(approx)

Now the ratio of this difference real/imaginary = .03005/.05927 = .507 (approx)

This already seems very close to .5 (sound familiar!)

In fact as the value of p increases the ratio of this difference does indeed tend ever more closely to .5!

So once again what we are stating is this!

As p becomes larger, the absolute mean value of both real and imaginary prime roots of 1 converges ever closer to 2/pi (i.e. i/log i).

Insofar as a difference remains the ratio of absolute deviation of real/imaginary value converges ever closer to .5.

Just as Riemann came up with improvements to prediction of the general frequency of the primes, I experimented with my own improvements.

Now let's say that we wish to calculate the deviation of the absolute mean value of the real part from 2/pi for a larger value of p (say 127).

What we do here is to multiply the deviation (for p = 3) by (p/p1)^2 where p = 3 and p1 = 127.

So this gives us .03005 * (3/127)^2 = .000016768 (approx)

Considering that we are using such an early prime number = 3, this compares extremely well with the true deviation = .000016232 (approx).

In fact this and any other calculation can be significantly improved by then dividing the result by (1 + d) where again in this case d = .03005.

So in in this case we can then approximate the true deviation as .000016277..

Thus our answer is already correct to 3 significant figures!

Predictions can be greatly improved through using the deviations of later prime numbers.

For example if we use the deviation associated with p = 61, we can calculate the corresponding deviation associated with p = 127 correct to 6 significant figures!

Variations of this approach can be used likewise to predict corresponding deviations associated with the imaginary part!

Now in principle just as the non-trivial zeros of the Riemann Zeta function can be used to correct the deviations from the actual with respect to the general distribution of the primes, a corresponding method should exist enabling - ultimately - an exact mean of absolute prime root values for both real and imaginary parts.

So just as the Riemann Hypothesis is used to accurately calculate the average number of primes (in a linear context), this latter approach is used to calculate the average value of these primes in a circular context.

And in each case .5 plays a key role. In fact the circular version provides a key indication of what the .5 actually represents.

As we have seen in the circular context .5 represents ratio of (real) cos to (imaginary) sin values which indicates in turn a quantitative (analytic) to qualitative (holistic) connection.

And this is really what The Riemann Hypothesis is all about i.e. in establishing the condition necessary for full reconciliation of both quantitative and qualitative aspects of interpretation!

A New Number System (3)

There are really two components to this new number system (where numbers are interpreted with respect to their pure dimensional (as opposed to their base quantitative) characteristics.

Once again the linear system - for base quantities - is defined in terms of a fixed dimensional number i.e. 1.

So the natural numbers 1, 2, 3, 4,... in this system are more fully represented as

1^1, 2^1, 3^1, 4^1,.....

However the corresponding circular system - for dimensional qualitative values - is defined in terms of a fixed base quantity 1.

So the natural numbers 1, 2, 3, 4,... in this system are more fully represented as

1^1, 1^2, 1^3, 1^4,.....

The circular nature of this latter system comes through raising 1 to the reciprocal of each dimension thus obtaining a quantitative value that lies on the circle of unit radius.

So when we raise 1 for example to the reciprocal of 4 i.e. 1/4 we obtain in quantitative terms i, which lies on the circle of unit radius!

Because there is a direct relationship as between each dimension (as quality) and its reciprocal (in quantitative terms), this means that the 4 as dimension is associated with the qualitative (i.e. holistic) interpretation of i.

Thus rather that just one valid interpretation of mathematical symbols, which in conventional terms is associated with the default value of 1, potentially an infinite set of possible interpretations exists for all all mathematical symbols, relationships etc.

So whereas Type 1 Mathematics is associated merely with the (reduced) quantitative aspects of mathematical symbols, Type 2 is associated with appropriate qualitative interpretation of these same symbols.

Thus i for example has not merely a quantitative, but also an important qualitative meaning. However this qualitative dimension is completely ignored in Type 1 conventional terms.

This makes no sense for ultimately the quantitative results that are derived for example in complex analysis are somewhat meaningless in the absence of appropriate qualitative interpretation!

Type 3 Mathematics - which is easily the most refined and demanding in nature, then involves consistently relating both quantitative (Type 1) and qualitative (Type 2) interpretation.

However just as the base quantitative system has an imaginary counterpart, likewise the dimensional qualitative counterpart has an imaginary counterpart.

So again the natural numbers in the first system would be

i^1, 2i^1, 3i^1, 4i^1,....,

whereas in the second system the corresponding imaginary version is

1^i, 1^2i, 1^3i, 1^4i,....

Now because in Type 1 Mathematics the second system is not formally recognised this entails with respect to the real part that

1^1 = 1^2 = 1^3 = 1^4 =.....= 1^n.

As we have seen this leads to the misleading conclusion that for example + 1 and - 1 are both the square root of 1.
(Through use of the Type 2 system we can see that - 1 is the square root of 1^1 and + 1 the square root of 1^2 (which are distinct in Type 2 terms).
In other words we cannot properly divorce here proper quantitative from proper qualitative interpretation!)

Also because in Type 1 Mathematics the second system is not recognised this entails with respect to the imaginary part that

1^i = 1^2i= 1^3i = 1^4i =.....= 1^n.

This comes from the corresponding assumption that

e^(2*i*pi) = e^(4*i*pi) = e^(6*i*pi) = e^(8*i*pi) =....= e^(2n*i*pi).

This then leads to the misleading conclusion that 1^i for example can have an unlimited number of possible quantitative solutions.

However because properly speaking in a Type 2 approach

1^i, 1^2i, 1^3i, 1^4i, =.....= 1^n are all distinct,

this means that 1^i has indeed just one unique quantitative value!

So once again - this type directly with respect to a quantitative result - we cannot properly divorce here quantitative from qualitative interpretation!

As we have seen 1^i = e^(- 2pi) = .00186744....

However as i = 1^(1/4), this means that i^i = 1^(i/4) = e^{(- pi)/2} = .2078795763...

It must be stressed that in accordance with Type 1 Mathematics than - as with 1^i - an infinite set of possible values exists for i^i.

So in Type 1 terms, 1^i = e^(2*i*pi)^i = e^(4*i*pi)^i = e^(6*i*pi)^i = e^(8*i*pi)^i =....

By this logic, for example 1^i = e^(2*i*pi)^i = e^(8*i*pi)^i

= e^(- 2*pi) = e^(- 8*pi)

So i^i = e^(- pi/2) = e^(- 2pi)

However this would therefore suggest that i^i = 1^i (which makes little sense).

Therefore Type 2 interpretation needs to be included to avoid such confusion!

Thursday, September 1, 2011

Alain Connes

This time when reading Karl Sabbagh's "Dr. Riemann's Zeros" I came across an interesting quote from Alain Connes on P.205. In commenting on the relationship between Geometry and Algebra he states:

'It really is fantastic step' he said, to understand that the square of a number - which is just a geometrical square - and the cube, which is just a geometrical cube - can be added together, even though you would say, "But one has dimension the length squared and the other the length cubed" and you would never add things which have different dimensions. So algebra is an amazing achievement, and once you have formulated things in an algebraic terms then they take on a life of their own.'

Algebra of course is incredibly important. However the price that has been paid in terms of such rational abstraction is that a basic form of reductionism is involved. In other words we can see clearly in geometrical terms that 2-dimensional is qualitatively distinct from 3-dimensional reality. However in algebraic terms this qualitative distinction is quickly lost with variables interpreted with respect to their mere quantitative meaning.

Indeed it is rather ironic that Connes in speaking about the Riemann Hypothesis in an a later quote on p.208 states:

"It is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication' Connes said. It's a gaping hole in our understanding, because until we really understand it we cannot say that we understand the line. Even the line itself is extraordinarily mysterious.'

May I suggest once again that the very reason why the link between addition and multiplication seems so intractable is precisely because the qualitative nature of variable transformation - necessarily involved in all multiplication - is formally ignored in present Type 1 Mathematics.

So properly understood there are both Type 1 (quantitative) and Type 2 (qualitative) aspects to all mathematical interpretation.

The Riemann Hypothesis in fact is a key statement regarding the relationship as between these two aspects.

Likewise with the line, Type 1 Mathematics, due to the same lack of a qualitative dimension, one cannot properly distinguish finite (discrete) from infinite (continuous) notions. So once again though the infinite notion is qualitatively distinct from the finite, in Type 1 interpretation it is necessarily reduced quantitatively in a finite manner!