I have referred before to Matthew Watkin's book "The Mystery of the Prime Numbers".
One of the features that I especially like about this book is that he successfully converts the standard natural log notion (which has such an important bearing on the general distribution of the primes) into the much more expressive form of an equiangular spiral.
Now in geometric terms equiangular spirals are very suggestive as they combine both linear and circular notions in a systematic ordered manner.
Indeed at the two extremes of such spirals we get - what he refers to as degenerative spirals - of both the straight line and the circle.
Matthew also extends this notion of spirals into the treatment of the famous deviations in the Riemann Prime Counting Function which are intimately related in turn to the non-trivial zeros of the Zeta Function!
I look forward very much to an extended treatment of these "spiral zeros" in the second volume.
It is also worth noting that considerable attention has been given to the Prime Spiral (Ulam's Spiral) where when natural numbers are entered on a grid in spiral fashion that the prime numbers then tend to fall along diagonal lines through the spiral! However my intention here is to focus on the qualitative significance of the relationship of prime numbers to spiral wave forms!
Just as linear and circular notions have a well defined quantitative meaning in conventional (Type 1) mathematical terms, equally they have a well defined qualitative meaning in holistic (Type 2) mathematical terms. However this latter type of interpretation is totally ignored by the mathematics profession.
Thus I would strongly contend that it is the absence of this vital qualitative dimension that is preventing recognition of the true nature of prime numbers.
In other words prime numbers combine both specific (independent) aspects in their individual nature with holistic (interdependent) aspects in their overall distribution.
Actual experience of mathematical reality equally combines specific and holistic elements through the interaction of (conscious) reason and (unconscious) intuition. However once again in conventional terms the qualitative (intuitive) aspect is reduced in a quantitative (rational) manner.
Proper understanding of the Riemann Hypothesis thereby requires both linear (either/or) logic based on the clear separation of polar opposites such as external/ internal and circular (both/and) logic based on the corresponding complementarity of such opposites.
Indeed ultimately the Riemann Hypothesis relates to the vital condition necessary for the consistent relationship of both types of logic!
Wednesday, December 21, 2011
Sunday, December 18, 2011
Riemann's Zero
As is well known the Riemann Hypothesis amounts to the statement all the non-trivial zeros of the Riemann Zeta Function lie on the real line whose value = 1/2.
Now in conventional terms, mathematicians have been trying to understand this problem from a merely quantitative perspective.
My persistent point however is that Mathematics equally contains an important (largely unrecognised) qualitative aspect.
Furthermore as the Riemann Hypothesis properly relates to the ultimate reconciliation of both the quantitative and qualitative aspects of mathematical understanding, its significance cannot be appreciated in a merely (reduced) quantitative manner.
Indeed ultimately the Riemann Hypothesis points to the relationship as between dual and nondual meaning.
Imagine a circle that is drawn with unit radius. Now in conventional terms the length of the line diameter of this circle = 2 units. So the midpoint of this line at the centre of the circle from linear perspective divides the line into two equal parts. This point lies exactly halfway (1/2) on the total line.
However if we look at this midpoint from a circular perspective we would represent it as 0. In other words if the radius to the right is + 1, then the radius in the opposite direction is - 1.
Now this behaviour has a direct qualitative significance. Linear understanding is inherently dualistic (which represents the holistic meaning of 2). So the midpoint of the circle at 1 unit represents thereby half of the total line.
However to understand the measurement of the line from the centre of the circle in complementary positive and negative directions we require - literally - nondual insight (so that dual notions are rendered paradoxical). So the midpoint from this perspective is 0.
Properly understanding the Riemann Hypothesis requires both dual and nondual understanding. Indeed it points to the ultimate state where both dual and nondual notions (or alternatively quantitative and qualitative notions) of number are identical. And this state is utterly mysterious and thereby cannot be grasped in a phenomenal manner.
Thus the mysterious order governing the nature of the prime numbers is already inherent in all number behaviour representing a non-phenomenal reality (where qualitative cannot be distinguished from quantitative notions).
The attempt to prove the Riemann Hypothesis in a merely (reduced) quantitative manner is thereby utterly futile.
The significance of the Riemann Hypothesis is ultimately of a truly breath taking order in that - properly understood - phenomenal reality as we know it would not be even possible if it did not hold.
In other words underlying all of visible reality is reality is a secret code ensuring a perfect harmony of the prime numbers with respect to their specific (quantitative) and holistic (qualitative) interaction.
It is this harmony that enables all subsequent phenomenal events (representing varying interactions of this original prime number number code) to unfold.
This once again suggests that ultimately the underlying nature of reality - insofar as it can be phenomenally investigated - is purely mathematical!
In other words at the deepest level, phenomena represent the interaction of a fundamental mathematical code that governs the subsequent behaviour of all natural events. However though we can only come to knowledge of this code through phenomena, its nature - and indeed in reverse fashion what we know as nature - is ultimately ineffable.
So at some stage, Physics will have to abandon the quest for the ultimate particles before arriving at a purely mathematical appreciation of reality that underlies all manifestations of such particles.
Indeed properly understood it has already arrived at this point. As I would see it, string theory represents an elaborate fiction that the ultimate physical particles are strings. Strictly speaking however these have no manifest physical reality but really operate as the vehicle of ever purer mathematical notions.
However my key point all along is that the very scope of Mathematics needs to be radically extended to explicitly include both its qualitative as well as quantitative aspect. In this light all particles ultimately emerge as the dynamic interaction of a prime number mathematical code that is designed to preserve the perfect harmony of both its quantitative and qualitative aspects.
Though this point is largely lost because of the reductionist nature of current scientific understanding, the truly great wonder of reality is how both the finite and infinite - though utterly distinct - yet successfully coexist with each other at all levels of understanding.
Thus before we can even for example engage in conventional mathematical activity, we must already presume this meaningful correspondence of finite and infinite. Indeed as I have frequently pointed out it underlines the very notion of mathematical proof!
Looked at another way, the Riemann Hypothesis is the necessary condition for such correspondence to exist. So it represents in fact a massive act of faith in the subsequent meaning of the whole mathematical enterprise.
As the Riemann Hypothesis thereby already underlies conventional mathematical proof (as the starting condition for its meaningful interpretation), The Riemann Hypothesis cannot therefore be proven (in conventional terms).
However far from this representing a defeat, true realisation of this fact has the capacity to open up appreciation of a greatly enlarged scope for Mathematics where every number, symbol, relationship has a unique qualitative - as well as quantitative - significance. And with this will come an enormous enrichment of the true nature of both Mathematics and Science.
Now in conventional terms, mathematicians have been trying to understand this problem from a merely quantitative perspective.
My persistent point however is that Mathematics equally contains an important (largely unrecognised) qualitative aspect.
Furthermore as the Riemann Hypothesis properly relates to the ultimate reconciliation of both the quantitative and qualitative aspects of mathematical understanding, its significance cannot be appreciated in a merely (reduced) quantitative manner.
Indeed ultimately the Riemann Hypothesis points to the relationship as between dual and nondual meaning.
Imagine a circle that is drawn with unit radius. Now in conventional terms the length of the line diameter of this circle = 2 units. So the midpoint of this line at the centre of the circle from linear perspective divides the line into two equal parts. This point lies exactly halfway (1/2) on the total line.
However if we look at this midpoint from a circular perspective we would represent it as 0. In other words if the radius to the right is + 1, then the radius in the opposite direction is - 1.
Now this behaviour has a direct qualitative significance. Linear understanding is inherently dualistic (which represents the holistic meaning of 2). So the midpoint of the circle at 1 unit represents thereby half of the total line.
However to understand the measurement of the line from the centre of the circle in complementary positive and negative directions we require - literally - nondual insight (so that dual notions are rendered paradoxical). So the midpoint from this perspective is 0.
Properly understanding the Riemann Hypothesis requires both dual and nondual understanding. Indeed it points to the ultimate state where both dual and nondual notions (or alternatively quantitative and qualitative notions) of number are identical. And this state is utterly mysterious and thereby cannot be grasped in a phenomenal manner.
Thus the mysterious order governing the nature of the prime numbers is already inherent in all number behaviour representing a non-phenomenal reality (where qualitative cannot be distinguished from quantitative notions).
The attempt to prove the Riemann Hypothesis in a merely (reduced) quantitative manner is thereby utterly futile.
The significance of the Riemann Hypothesis is ultimately of a truly breath taking order in that - properly understood - phenomenal reality as we know it would not be even possible if it did not hold.
In other words underlying all of visible reality is reality is a secret code ensuring a perfect harmony of the prime numbers with respect to their specific (quantitative) and holistic (qualitative) interaction.
It is this harmony that enables all subsequent phenomenal events (representing varying interactions of this original prime number number code) to unfold.
This once again suggests that ultimately the underlying nature of reality - insofar as it can be phenomenally investigated - is purely mathematical!
In other words at the deepest level, phenomena represent the interaction of a fundamental mathematical code that governs the subsequent behaviour of all natural events. However though we can only come to knowledge of this code through phenomena, its nature - and indeed in reverse fashion what we know as nature - is ultimately ineffable.
So at some stage, Physics will have to abandon the quest for the ultimate particles before arriving at a purely mathematical appreciation of reality that underlies all manifestations of such particles.
Indeed properly understood it has already arrived at this point. As I would see it, string theory represents an elaborate fiction that the ultimate physical particles are strings. Strictly speaking however these have no manifest physical reality but really operate as the vehicle of ever purer mathematical notions.
However my key point all along is that the very scope of Mathematics needs to be radically extended to explicitly include both its qualitative as well as quantitative aspect. In this light all particles ultimately emerge as the dynamic interaction of a prime number mathematical code that is designed to preserve the perfect harmony of both its quantitative and qualitative aspects.
Though this point is largely lost because of the reductionist nature of current scientific understanding, the truly great wonder of reality is how both the finite and infinite - though utterly distinct - yet successfully coexist with each other at all levels of understanding.
Thus before we can even for example engage in conventional mathematical activity, we must already presume this meaningful correspondence of finite and infinite. Indeed as I have frequently pointed out it underlines the very notion of mathematical proof!
Looked at another way, the Riemann Hypothesis is the necessary condition for such correspondence to exist. So it represents in fact a massive act of faith in the subsequent meaning of the whole mathematical enterprise.
As the Riemann Hypothesis thereby already underlies conventional mathematical proof (as the starting condition for its meaningful interpretation), The Riemann Hypothesis cannot therefore be proven (in conventional terms).
However far from this representing a defeat, true realisation of this fact has the capacity to open up appreciation of a greatly enlarged scope for Mathematics where every number, symbol, relationship has a unique qualitative - as well as quantitative - significance. And with this will come an enormous enrichment of the true nature of both Mathematics and Science.
Thursday, December 15, 2011
The Harmonic Series Again!
As I have repeatedly stated Conventional (Type 1) Mathematics is formally based on a linear i.e. 1-dimensional rational approach (in qualitative terms).
However the considerable problem that exists is that actual understanding of all mathematical processes entails an interaction of both rational and intuitive type understanding.
Thus the conventional approach simply reduces the intuitive aspect in rational terms.
Alternatively it inevitably reduces - in any context - qualitative to quantitative type interpretation.
In qualitative terms, rational understanding always implies the positing of phenomena in a conscious manner.
Intuitive understanding - by contrast - implies the corresponding dynamic negation of such understanding in an unconscious holistic manner.
Thus a full account of the nature of linear understanding (that entails both rational and intuitive type understanding) requires recognition of the 1st dimension in both a positive and negative manner.
I have pointed already to the important fact that the very notion of a fraction implicitly requires transition from the positive (rational) to negative (intuitive) recognition with respect to the 1st dimension.
Thus in quantitative terms 4 is more fully represented as 4^1.
In qualitative terms this represents rational (conscious) understanding with respect to 1st dimension (positive).
However the related fraction 1/4 is initially represented as 4^(- 1)and in qualitative terms this relates to intuitive (unconscious) understanding through negation of the 1st dimension (negative).
So the very process of switching from whole to part recognition in experience implicitly requires switching from rational to intuitive type recognition.
However because Conventional Mathematics is formally defined in qualitative terms with respect to the positive 1st dimension, the significance of this qualitative change in understanding is overlooked with the result interpreted in a merely (reduced) quantitative manner i.e. 1/4 as (1/4)^1.
So though a qualitative holistic shift of consciousness is necessary to enable the transformation from whole to part recognition, both whole and part are then interpreted in a merely quantitative manner (i.e. with respect to the positive 1st dimension).
Many indications of what I am saying here are provided by simple mathematical results.
Once again from where conventional rational understanding is of a linear nature, holistic intuitive recognition is qualitatively of a circular nature.
Prime numbers from a rational perspective are the most linear of all numbers and - literally 1-dimensional in nature (with no composite factors).
So 7^1 is truly linear in this regard with no factors.
4 by contrast (as composite) is inherently 2-dimensional in nature i.e. 2^2.
However when we raise a prime number to - 1, a remarkable transformation takes place whereby it now exhibits highly circular characteristics.
So 7^(- 1) = 1/7 = .142857...
Here the 6 digits 142857 have some notable circular properties. For example they recur indefinitely in the same manner. Also when we multiply by 2, 3, 4, 5 and 6 the same digits appear (that maintain the same cyclical order).
So 1/7 is perhaps the best known of the cyclic primes (that exhibit such circular properties).
Then the sum of the natural numbers 1 + 2 + 3 + 4 + 5 +... represents the archetypal linear series of numbers i.e. as numbers naturally marked off on a straight line.
So all these numbers are implicitly 1-dimensional i.e. raised to the power of 1.
However when we raise these numbers same numbers to - 1
i.e. 1^ (- 1), 2^(- 1), 3^(- 1), 4^(- 1), 5^(- 1) +... we generate the harmonic series 1 + 1/2 + 1/3 + 1/4 + 1/5 + ...
Now remember my basic starting about regarding the nature of the primes is that they combine extreme characteristics with respect to linear and circular aspects!
So the individual prime numbers are the most independent and linear of all numbers (with no constituent factors).
However the general behaviour of the primes (in the frequency of their overall distribution) involves the other extreme of a holistic circular tendency.
Now the very manner in which both linear and circular aspect are involved in experience goes back to the way in which whole and part interact.
Therefore to switch from the whole say 4 (4^1) to part 1/4 the dimensional number switches to - 1. So 1/4 = 4^(- 1).
Thus the decisive switch from whole to part requires that conscious (linear) understanding that is defined with respect to the positive 1st dimension be dynamically negated in an unconscious (circular) intuitive manner.
So similar dynamics are involved with respect to the harmonic series (by comparison with the natural number series).
And remarkably the harmonic series gives the simplest answer to the general nature of prime number distribution. In other words the measure of the average spread as between successive prime numbers (which becomes progressively larger as the natural numbers increase) is given by the sum of the harmonic series!
Indeed the change in the average gap between these primes as n i.e n^(+ 1) increases by 1 is given as 1/n i.e. n^(- 1).
So here in the general behaviour of the prime numbers we have an intimate relationship as between the positing and negating respectively (with respect to the 1st dimension) of the number n.
Thus moving from the specific independent linear notion of an individual prime number to the holistic interdependent circular notion of prime number distribution directly involves the positing (in conscious rational manner) and the corresponding negation (in an unconscious intuitive manner) of linear (1-dimensional) understanding.
So the proper understanding in conventional (Type 1) mathematical terms of the quantitative nature of prime number behaviour is inseparable from corresponding holistic (Type 2) qualitative mathematical interpretation (of such behaviour).
And such understanding requires that we can combine both linear and circular type notions equally in both a quantitative and qualitative manner.
This circular nature of the harmonic series can be demonstrated in yet another striking manner.
Now the harmonic series represents one important example of the Zeta function
i.e. 1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + ... where s = 1.
As is well known the for all (positive) even integer values of s the resulting sum of the series can be given in terms of a expression involving pi.
And as pi serves as the direct relation of circular and linear quantitative notions in the relationship of the (circular) circumference to its (line) diameter, likewise pi serves as the archetypal relation of corresponding circular and linear notions (understood in a qualitative manner).
For example when s = 2, the sum of the series = {(pi)^2}/6.
The harmonic series can be expressed in terms of the combination of the sum of values corresponding with even integer values for the Zeta Function.
So 1 + 1/2 + 1/3 + 1/4 + ... = 2{ζ(2)/2 + ζ(4)/4 + ζ(6)/6 + ζ(8)/8 + ...}.
And this infinite series can therefore be expressed as the continual sum of terms that consist of powers of pi (that are multiplied by a rational fraction)!
It is again notable that the harmonic series is intimately associated with musical sound.
So just as the natural numbers can again be seen as the supreme expression of linear quantitative understanding, the harmonic series can be seen in a sense as a supreme expression of qualitative type appreciation.
And ultimately this is related to the fact that in switching from the positive 1st dimension to its negative one likewise switches from (conscious) rational to (unconscious) intuition, which is the very means by which we switch from quantitative to qualitative type appreciation.
Wednesday, December 14, 2011
A Strikingly Simple Prime Number Relationship
As Euler showed the harmonic series 1 + 1/2 + 1/3 + 1/4 +..., is approximated as log n + λ (for a finite value of n).
And as λ (the Euler-Masceroni constant) is a constant = .5772 approx this means that for large n, log n is approximated by the harmonic series.
This means therefore that perhaps the simplest expression for the frequency of prime distribution is given as the sum of the harmonic series for n terms divided by n (which becomes increasingly accurate for larger n).
Looked at another way the sum of the harmonic series (for large n) approximates the average spread or gap as between prime numbers in the region of n.
Therefore as the sum of the first million terms for example of the harmonic series = 14.384 (approx). Therefore the average gap as between primes in the region of 1,000,000 is roughly 14. Though this approximation is not yet very accurate, the approximation would greatly improve (in relative terms) as the value of n increases.
Now it is well known that the average spread as between primes continually increases as the value of n increases.
What I find particularly striking in this regard is that the increase in the average spread (or gap) as between primes as we increase n by 1 is given by 1/n.
So for example as we increase n 1,000,000 to 1,000,001 the increase in the average gap as between primes is 1/1,000,000.
(More accurately as we increase n from 999,999.5 to 1,000,000.5 the average gap between primes increases by 1/1,000,000).
This result can easily be demonstrated through differentiation of log n + λ (with respect to n) which results in 1/n.
Now if we multiply the simple expression for the general frequency of primes i.e. n/log n by 1/n we obtain 1/log n (which represents the probability that n is prime).
Thus, we can say that the product of the general frequency of prime distribution and the change in the average gap as between primes (for large n) approximates well the probability that n is prime.
And as λ (the Euler-Masceroni constant) is a constant = .5772 approx this means that for large n, log n is approximated by the harmonic series.
This means therefore that perhaps the simplest expression for the frequency of prime distribution is given as the sum of the harmonic series for n terms divided by n (which becomes increasingly accurate for larger n).
Looked at another way the sum of the harmonic series (for large n) approximates the average spread or gap as between prime numbers in the region of n.
Therefore as the sum of the first million terms for example of the harmonic series = 14.384 (approx). Therefore the average gap as between primes in the region of 1,000,000 is roughly 14. Though this approximation is not yet very accurate, the approximation would greatly improve (in relative terms) as the value of n increases.
Now it is well known that the average spread as between primes continually increases as the value of n increases.
What I find particularly striking in this regard is that the increase in the average spread (or gap) as between primes as we increase n by 1 is given by 1/n.
So for example as we increase n 1,000,000 to 1,000,001 the increase in the average gap as between primes is 1/1,000,000.
(More accurately as we increase n from 999,999.5 to 1,000,000.5 the average gap between primes increases by 1/1,000,000).
This result can easily be demonstrated through differentiation of log n + λ (with respect to n) which results in 1/n.
Now if we multiply the simple expression for the general frequency of primes i.e. n/log n by 1/n we obtain 1/log n (which represents the probability that n is prime).
Thus, we can say that the product of the general frequency of prime distribution and the change in the average gap as between primes (for large n) approximates well the probability that n is prime.
Friday, November 18, 2011
Interesting Square Connection
If we just focus on the absolute value of the denominator of the Riemann's Zeta Function for s = - 1, - 3, - 5, - 7 and - 9, which are 12, 120, 252, 240 and 132 respectively we can find an interesting square connection.
So each of these numbers can be expressed as the product of two numbers which differ in ascending order by consecutive powers of 2.
So 12 = 4 * 3 with the difference (1) = 2^0.
120 = 12 * 10 with the difference (2) = 2^1.
252 = 18 * 14 with the difference (4) = 2^2.
240 = 20 * 12 with the difference (8) = 2^3.
132 = 22 * 6 with the difference (16) = 2^4.
After this the pattern begins to break down!
The absolute value corresponding to the denominator for s = - 11 = 32760. This can indeed be expressed as the product of two numbers that differ by a square of 2 but not (but not 2^5).
So 32760 = 182 * 180 with the difference (2) = 2^1.
However with the next number corresponding to s = - 13, no such relationship exists as between the product of two numbers i.e. involving the difference of a square of 2.
So each of these numbers can be expressed as the product of two numbers which differ in ascending order by consecutive powers of 2.
So 12 = 4 * 3 with the difference (1) = 2^0.
120 = 12 * 10 with the difference (2) = 2^1.
252 = 18 * 14 with the difference (4) = 2^2.
240 = 20 * 12 with the difference (8) = 2^3.
132 = 22 * 6 with the difference (16) = 2^4.
After this the pattern begins to break down!
The absolute value corresponding to the denominator for s = - 11 = 32760. This can indeed be expressed as the product of two numbers that differ by a square of 2 but not (but not 2^5).
So 32760 = 182 * 180 with the difference (2) = 2^1.
However with the next number corresponding to s = - 13, no such relationship exists as between the product of two numbers i.e. involving the difference of a square of 2.
Sunday, November 6, 2011
Finite and Infinite
I have stated many times that conventional mathematical appreciation is based on a merely reduced notion of the infinite (where effectively it is treated as an extension of the finite).
This for example defines the nature of conventional proof where what is true for the general case (potentially applying to the infinite) is thereby assumed to apply to all actual (finite) cases.
And I have referred to this basic reductionism as interpretation that is linear (i.e. 1-dimensional) in qualitative terms.
This linear approach is also very much to the fore in the treatment of series where in most cases once again a seemingly unambiguous relationship as between finite and infinite emerges.
For example from a finite perspective we can see that a geometrical series such as 1 + 1/2 + 1/4 + ... converges towards some finite limit (getting ever closer to 2 without actually reaching this limiting value).
Therefore when we say that in the limit the value of the series = 2 (where the no. of terms is infinite) this again seems to comply with linear type interpretation (i.e. where the infinite is treated as an extension of finite notions).
However there are other cases where what appears true in finite terms does not readily comply with infinite notions.
For example
1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where the number of terms in linear terms is assumed infinite).
This would imply therefore that when x = - 2 that
1/3 = 1 - 2 + 4 - 8 + .....
However when we view the R.H.S in finite terms, we can see that the terms get progressively larger with the value of the series diverging. Now it is true that the terms (and consequent sum of terms) alternate between positive and negative values. However from a linear perspective we cannot say that its sum will converge to a definite finite value.
However the - apparent - equivalent L.H.S expression suggests that this is precisely what happens.
So once again from a linear perspective we obtain a result that is intuitively not in keeping with its rational mode of interpretation.
This strongly suggests that a different form of rational interpretation is required to explain the nature of the result.
However because it is qualitatively defined in terms of 1-dimensional interpretation, Conventional (Type 1) Mathematics is not appropriate for this task.
Now when we return to our example we can see what is the problem
If we consider 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1), where the series is defined in terms of a finite number (n) of terms,
then 1 - {x ^(n - 1)}/(1 - x) = 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1)
So the conclusion that
1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where n is infinite), is based on the the assumption that n - 1 = n terms (when n is infinite).
Thus the logic that applies for the infinite case i.e. n - 1 = n is directly confused with standard finite logic i.e. n - 1 ≠ n.
This then leads to the non-intuitive result (in linear rational terms) that for example
1 - 2 + 4 - 8 + .... = 1/3
Though mathematicians are of course aware of this anomaly, they attempt to explain it in terms of two results that comply with differing domains of definition.
However this avoids the deeper qualitative question of what such non-intuitive results actually entail!
Facing up to this issue requires accepting the radical conclusion that just as numbers such as 1, 2, 3, 4, ... etc. have a well-defined meaning in quantitative terms, equally they have an - as yet - unrecognised meaning in qualitative terms whereby they refer to unique modes of rational interpretation of symbols.
Once again Conventional (Type 1) Mathematics attempts to confine interpretation to 1-dimensional logic in qualitative terms. However potentially an unlimited set of other qualitative logical interpretations can be given.
And the series that I have used to illustrate this point itself points to the need for a different means of rational interpretation (so that results can then intuitively concur with the correct rational mode adopted).
This for example defines the nature of conventional proof where what is true for the general case (potentially applying to the infinite) is thereby assumed to apply to all actual (finite) cases.
And I have referred to this basic reductionism as interpretation that is linear (i.e. 1-dimensional) in qualitative terms.
This linear approach is also very much to the fore in the treatment of series where in most cases once again a seemingly unambiguous relationship as between finite and infinite emerges.
For example from a finite perspective we can see that a geometrical series such as 1 + 1/2 + 1/4 + ... converges towards some finite limit (getting ever closer to 2 without actually reaching this limiting value).
Therefore when we say that in the limit the value of the series = 2 (where the no. of terms is infinite) this again seems to comply with linear type interpretation (i.e. where the infinite is treated as an extension of finite notions).
However there are other cases where what appears true in finite terms does not readily comply with infinite notions.
For example
1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where the number of terms in linear terms is assumed infinite).
This would imply therefore that when x = - 2 that
1/3 = 1 - 2 + 4 - 8 + .....
However when we view the R.H.S in finite terms, we can see that the terms get progressively larger with the value of the series diverging. Now it is true that the terms (and consequent sum of terms) alternate between positive and negative values. However from a linear perspective we cannot say that its sum will converge to a definite finite value.
However the - apparent - equivalent L.H.S expression suggests that this is precisely what happens.
So once again from a linear perspective we obtain a result that is intuitively not in keeping with its rational mode of interpretation.
This strongly suggests that a different form of rational interpretation is required to explain the nature of the result.
However because it is qualitatively defined in terms of 1-dimensional interpretation, Conventional (Type 1) Mathematics is not appropriate for this task.
Now when we return to our example we can see what is the problem
If we consider 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1), where the series is defined in terms of a finite number (n) of terms,
then 1 - {x ^(n - 1)}/(1 - x) = 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1)
So the conclusion that
1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where n is infinite), is based on the the assumption that n - 1 = n terms (when n is infinite).
Thus the logic that applies for the infinite case i.e. n - 1 = n is directly confused with standard finite logic i.e. n - 1 ≠ n.
This then leads to the non-intuitive result (in linear rational terms) that for example
1 - 2 + 4 - 8 + .... = 1/3
Though mathematicians are of course aware of this anomaly, they attempt to explain it in terms of two results that comply with differing domains of definition.
However this avoids the deeper qualitative question of what such non-intuitive results actually entail!
Facing up to this issue requires accepting the radical conclusion that just as numbers such as 1, 2, 3, 4, ... etc. have a well-defined meaning in quantitative terms, equally they have an - as yet - unrecognised meaning in qualitative terms whereby they refer to unique modes of rational interpretation of symbols.
Once again Conventional (Type 1) Mathematics attempts to confine interpretation to 1-dimensional logic in qualitative terms. However potentially an unlimited set of other qualitative logical interpretations can be given.
And the series that I have used to illustrate this point itself points to the need for a different means of rational interpretation (so that results can then intuitively concur with the correct rational mode adopted).
Wednesday, November 2, 2011
The Strange Case of η( - 1)
The Eta Series for s = - 1 is,
η( - 1) = 1/1^(- 1) - 1/2^(- 1) + 1/3^(- 1) - 1/4^(- 1) + ....
Therefore
η( - 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....
Withe reference to the Riemann Zeta Function the value for this Eta series = 1/4.
The question then arises as what meaning can we give this result!
It is perhaps better in illustrating to start with η(0) = 1 - 1 + 1 - 1 + 1 - 1 + ...
The value of this alternating series = 1/2
It is easy enough in this case to see how this value might arise!
If we take an even number of terms, the sum of the series = 0.
However if we take an odd number of terms the sum = 1.
Therefore it seems reasonable - where the number of terms is unspecified - to average the two values as 1/2. However in illustrating this we need to consider a finite number of terms.
In more general terms the answer here is n/2 (where the nth term when it is odd = 1).
However what is interesting is that when we now consider the series (in Type 1 mathematical terms) as infinite
i.e. 1/(1 + x) = 1 + x + x^2 + x^3 + x^4 + ...,
by setting x = - 1 we obtain 1/2 unambiguously as the correct answer.
i.e. 1/2 = 1 - 1 + 1 - 1 + 1 - ..
So the value here from the infinite perspective as 1/2 can be simply considered as n/2 (where the quantitative value of n is set at a default value of 1).
What I am getting at here is very significant indeed in terms of properly interpreting the nature of the Riemann Hypothesis!
Conventional (Type 1) Mathematics is inherently defined with respect to a default dimensional (qualitative) value of 1. In other words the nature of Type 1 Mathematics is qualitatively linear (1-dimensional) where all quantitative values are ultimately reduced in 1-dimensional terms.
So for example 2^2 = 4^1 (in Type 1 terms).
However Holistic (Type 2) Mathematics - in inverse fashion - is inherently defined with respect to a default base quantitative value of 1. So in concentrating on the nature of qualitative transformation (as with the switch from finite to infinite series) Type 2 interpretation is not directly concerned with the quantitative nature of number but rather as its representation of a holistic transformation (where dimensional powers other than + 1 are entailed)!
Now this will perhaps become clearer when we look at both the finite and infinite interpretation of terms corresponding to η(- 1).
η(- 1) = 1 - 2 + 3 - 4 + .....
We consider this series initially as finite and attempt to sum its value in linear (Type 1) terms.
For example we will initially derive η(0) with 10 terms.
i.e. y = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9
Then when we differentiate y with respect to x we obtain 9 remaining terms on the RHS
i.e. 1 - 2x + 3(x^2) - 4(x^3) + 5(x^4) - 6(x^5) + 7(x^6) - 8(x^7) + 9(x^8)
Setting x = - 1 we obtain
1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9
Then in summing this finite series by grouping terms in pairs
the sum of the first 8 terms is - 1 - 1 - 1 - 1
So, if the number of terms is even the sum of the series = - (n - 2)/2
Thus as we originally started with n = 10 the sum of the first 8 terms = - (8/2) = - 4.
However if we sum the first n - 1 terms (which is now odd) we obtain - (n - 2)/2 + (n - 1).
Thus the sum of the first 9 terms = - 4 + 9 = 5.
As we did before for η(0), since there is a 50:50 chance of obtaining the positive value for the sum (associated with an odd number of terms). So the average = {(n - 1) - (n - 2)/2}/2 = (2n - 2 - n + 2)/4 = n/4
So again with originally n = 10, the sum of the first 9 terms = 5. As there is a 50:50 chance of getting this value, the expected value is therefore 2.5. And this is the value corresponding to n/4 (where n = 10).
What is fascinating is that when we then consider the series in infinite terms through differentiating both sides of original expression we get
1/(1 + x)^2 = 1 + 2^x + 3(x^2) + 4(x^3) + .....
By setting x = - 1,
1/4 = 1 - 2 + 3 - 4 + 5 - .....
So once again the sum for the infinite series is the same as derived for the finite, with the important difference that n is here given a default value of 1.
This strongly suggests that in transforming from finite to infinite expressions the very representation of number itself switches from a specific (quantitative) to a holistic (qualitative) meaning. In other words the expected value of the nth term (when n is odd) is n/4. So when referring to the general ratio (rather than its specific quantitative value) we get 1/4!
η( - 1) = 1/1^(- 1) - 1/2^(- 1) + 1/3^(- 1) - 1/4^(- 1) + ....
Therefore
η( - 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....
Withe reference to the Riemann Zeta Function the value for this Eta series = 1/4.
The question then arises as what meaning can we give this result!
It is perhaps better in illustrating to start with η(0) = 1 - 1 + 1 - 1 + 1 - 1 + ...
The value of this alternating series = 1/2
It is easy enough in this case to see how this value might arise!
If we take an even number of terms, the sum of the series = 0.
However if we take an odd number of terms the sum = 1.
Therefore it seems reasonable - where the number of terms is unspecified - to average the two values as 1/2. However in illustrating this we need to consider a finite number of terms.
In more general terms the answer here is n/2 (where the nth term when it is odd = 1).
However what is interesting is that when we now consider the series (in Type 1 mathematical terms) as infinite
i.e. 1/(1 + x) = 1 + x + x^2 + x^3 + x^4 + ...,
by setting x = - 1 we obtain 1/2 unambiguously as the correct answer.
i.e. 1/2 = 1 - 1 + 1 - 1 + 1 - ..
So the value here from the infinite perspective as 1/2 can be simply considered as n/2 (where the quantitative value of n is set at a default value of 1).
What I am getting at here is very significant indeed in terms of properly interpreting the nature of the Riemann Hypothesis!
Conventional (Type 1) Mathematics is inherently defined with respect to a default dimensional (qualitative) value of 1. In other words the nature of Type 1 Mathematics is qualitatively linear (1-dimensional) where all quantitative values are ultimately reduced in 1-dimensional terms.
So for example 2^2 = 4^1 (in Type 1 terms).
However Holistic (Type 2) Mathematics - in inverse fashion - is inherently defined with respect to a default base quantitative value of 1. So in concentrating on the nature of qualitative transformation (as with the switch from finite to infinite series) Type 2 interpretation is not directly concerned with the quantitative nature of number but rather as its representation of a holistic transformation (where dimensional powers other than + 1 are entailed)!
Now this will perhaps become clearer when we look at both the finite and infinite interpretation of terms corresponding to η(- 1).
η(- 1) = 1 - 2 + 3 - 4 + .....
We consider this series initially as finite and attempt to sum its value in linear (Type 1) terms.
For example we will initially derive η(0) with 10 terms.
i.e. y = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9
Then when we differentiate y with respect to x we obtain 9 remaining terms on the RHS
i.e. 1 - 2x + 3(x^2) - 4(x^3) + 5(x^4) - 6(x^5) + 7(x^6) - 8(x^7) + 9(x^8)
Setting x = - 1 we obtain
1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9
Then in summing this finite series by grouping terms in pairs
the sum of the first 8 terms is - 1 - 1 - 1 - 1
So, if the number of terms is even the sum of the series = - (n - 2)/2
Thus as we originally started with n = 10 the sum of the first 8 terms = - (8/2) = - 4.
However if we sum the first n - 1 terms (which is now odd) we obtain - (n - 2)/2 + (n - 1).
Thus the sum of the first 9 terms = - 4 + 9 = 5.
As we did before for η(0), since there is a 50:50 chance of obtaining the positive value for the sum (associated with an odd number of terms). So the average = {(n - 1) - (n - 2)/2}/2 = (2n - 2 - n + 2)/4 = n/4
So again with originally n = 10, the sum of the first 9 terms = 5. As there is a 50:50 chance of getting this value, the expected value is therefore 2.5. And this is the value corresponding to n/4 (where n = 10).
What is fascinating is that when we then consider the series in infinite terms through differentiating both sides of original expression we get
1/(1 + x)^2 = 1 + 2^x + 3(x^2) + 4(x^3) + .....
By setting x = - 1,
1/4 = 1 - 2 + 3 - 4 + 5 - .....
So once again the sum for the infinite series is the same as derived for the finite, with the important difference that n is here given a default value of 1.
This strongly suggests that in transforming from finite to infinite expressions the very representation of number itself switches from a specific (quantitative) to a holistic (qualitative) meaning. In other words the expected value of the nth term (when n is odd) is n/4. So when referring to the general ratio (rather than its specific quantitative value) we get 1/4!
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