Wednesday, August 24, 2011

Critical Region (3)

It has to be clearly recognised that the infinite is a qualitatively distinct concept from that of the finite. Ultimately it points to the fact that with respect to reality we have both actual and potential aspects. The actual is always made manifest in finite terms whereas the infinite more correctly pertains to the potential (from which the actual emerges). Again in direct scientific terms, the finite is appropriated in a (conscious) rational manner whereas the infinite is appropriated in an (unconscious) intuitive manner.

This in turn poses great difficulties for the conventional (Type 1) mathematical approach which in formal terms is confined to solely (linear) rational modes of interpretation.

Therefore the notion of the infinite that applies in Type 1 Mathematics is but a reduced quantitative notion that in many respects is entirely inadequate.

Somehow the mistaken view is perpetuated - as for example with series - that if we keep increasing the number of terms that we eventually accumulate an "infinite" no. (of such terms). However this is strictly speaking nonsense! Once again the infinite is a qualitatively distinct notion from the finite and cannot be "reached" therefore in finite manner. So if we keep increasing the terms of - for example - the natural number series without limit as might be said, we always obtain a finite number of terms. At no stage does the actual number of terms become infinite!

There are direct indications - even in Type 1 Mathematics - that the infinite is indeed qualitatively different from the finite. For example if we add two numbers in finite terms (say 1 + 2) a quantitative transformation is involved. Likewise if we multiply two non unitary numbers (say 2 * 3) again a quantitative transformation takes place.

However if we add two infinite numbers (or indeed a finite to a infinite number) no quantitative transformation is involved. Again if we multiply two infinite numbers or (an infinite by a finite) again no transformation is involved. Now, Cantor did indeed explore the notion of different types of infinities (within the confines of Type 1 Mathematics) but essentially his work - though presented in a reduced quantitative manner - draws attention to the fact that there is inevitably a qualitative aspect also involved. So ultimately the conclusion for example that the set of transcendental numbers is "bigger" than than that of the rationals is a reduced quantitative way of expressing the fact that the transcendentals are qualitatively distinct from the rationals (combining both discrete and continuous notions) whereas the rational are based on merely discrete notions!

All of this is of vital importance with respect to values for the Riemann Zeta Function which once again is defined (in Type 1 terms) as the infinite series

1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + ...... where s is any complex number a + bi.

Now it must be stressed again that this use of the infinite - which defines Type 1 Mathematics - represents but a reduced quantitative notion (i.e. where the potential notion of the infinite is reduced in actual finite terms).

In this context this approach does seem to work in certain cases. As is well known Euler had earlier shown that the Function indeed converges for all real values of s > 1. and this result can be exended to all complex numbers where again the real part a > 1.

Riemann however managed to extend the domain of definition for the Function to all values of s (except 1).

However this creates the immediate problem that the results for a wide range of these values make no sense from the conventional linear perspective.

My main concern here is not to show precisely how Riemann managed to achieve this "magical" transformation. What I am more concerned to deal with are the key philosophical implications that are involved which ultimately requires that an entirely distinctive type of Mathematics (Type 2) is used.

Ultimately this dramatically changes the very nature of the Riemann Hypothesis from a hypothesis in Type 1 Mathematics to a new fundamental hypothesis that serves as the cornerstone of the reconciliation of Type 1 and Type 2 Mathematics (which is the basis for the more comprehensive Type 3 Mathematics).

In other words all mathematical symbols possess both quantitative and qualitative aspects corresponding to two distinct types of interpretation. So the fundamental requirement is that consistency can be maintained with respect to both requirements. And the Riemann Hypothesis provides the very condition necessary for such consistency.

This of course implies that prime numbers inherently combine such quantitative and qualitative aspects in a very special manner!

Now we are looking here at the critical region of the Riemann Zeta Hypothesis for all values of s between 0 and 1.

Now the Function is undefined for s = 1 where the harmonic series results which in conventional terms sums to infinity.

The question then arises as to why the domain of definition cannot be stretched so as to include s = 1 (when in fact it can include every other value).

Now the qualitative mathematical reason (Type 2) is very illuminating in this regard.

Type 1 Mathematics is qualitatively based on the linear rational approach which is - literally - 1-dimensional in nature. Thus when for example we square the number 2 a qualitative as well as quantitative transformation takes place. So strictly the answer is now 4 square (i.e. 2-dimensional) units. However in Type 1 terms we simply ignore this qualitative dimensional change and express the result in reduced - merely - quantitative terms. So in Type 1 calculations all dimensional charges in the units involved are reduced to 1. Thus 2 * 2 = 4 (i.e. 4 ^ 1).

Now once we accept the true qualitative nature of a dimension, then numerical calculations (involving transformational changes) can be given an alternative result based on qualitative rather than quantitative considerations.

In the context of the Riemann Zeta Function therefore we give the Function - when it diverges from a Type 1 perspective - a finite Type 2 interpretation (where the result converges).

Remember Type 1 and Type 2 are finite and infinite with respect to each other! Therefore a finite result in quantitative terms is infinite from a qualitative perspective. Likewise what appears infinite in quantitative terms will now appear finite from a qualitative perspective. Put another way, if the Zeta Function diverges in standard Type 1 terms it necessarily converges from the alternative Type 2 perspective.

However the one obvious exception here is where s = 1. Because this series is already defined in terms of the default qualitative dimension used in Type 1 Mathematics, it therefore has no alternative meaning from a Type 2 perspective. However, for all other dimensional values of s, an alternative does necessarily exist.

Thursday, August 18, 2011

The Critical Region (2)

We can perhaps illustrate more the nature of qualitative - as opposed to quantitative - type interpretation of numerical values with respect to the Riemann Zeta function for the case where s = 1.

This results again in the well known harmonic series,

1 + 1/2 + 1/3 + 1/4 + ..... which in standard interpretation diverges to infinity.

Now by subtracting the even terms (*2) from the original series we come up with the corresponding Eta series,

1 - 1/2 + 1/3 - 1/4 + ..... which does indeed converge in conventional terms giving the well known result i.e. Ln 2.

However on closer expression what we have subtracted to derive the Eta result seemingly results in the original harmonic series!

So when we multiply the even terms i.e. 1/2 + 1/4 + 1/6 + 1/8 +... etc. by 2 we thereby obtain 1 + 1/2 + 1/3 + 1/4 +... which is the original harmonic series.

Therefore the Eta series which sums to Ln 2 - on this basis - has been derived by subtracting the harmonic series from itself.

So on this logic Ln 2 = 0.

Well what does this actually mean?

Once again, we have obtained this seemingly nonsensical result through mixing 1-dimensional and 2-dimensional notions of terms. When all the terms are of the same sign, linear 1-dimensional notions apply. However when alternating positive and negative terms are involved 2-dimensional notions (in the balancing of successive positive and negative terms) apply.

So on closer inspection the quantitative result of ln 2, corresponding to the Eta series, requires that we follow one set way of ordering terms. For example if we reordered the terms consistently adding two successive even terms before subtracting

i.e. (1/2 + 1/4) - 1/3 + (1/6 + 1/8) - 1/5 +..... we would get a different result!

So when we initially subtracted the even terms (* 2) from the original harmonic series, we were using a linear (1-dimensional) logic. However in interpreting the resulting Eta Series we are using a strict 2-dimensional logic where successive terms are taken as complementary pairings.

Now the very essence of non-dimensional understanding is that it entails the direct confusion of linear and circular notions. In psychological terms we would say that such understanding remains both undifferentiated and non integrated representing mere potential for the subsequent unfolding in experience of both aspects i.e. (linear) differentiation and (circular) integration. So with successful development the first main requirement is the specialised development of linear (1-dimensional) understanding which dominates conventional scientific and mathematical understanding. Then where authentic spiritual contemplative development unfolds higher dimesnions of understanding can thereby take place.

Geometrically we could envisage the 0th dimension as the point at the centre of a circle (which equally is the centre of its line diameter).

So when we say that ln 2 = 0, we are referring to its qualitative dimensional nature (rather than its quantitative value). In other words 0 in dimensional terms relates to the direct embedding of linear (single term) interpretation that is unambiguously of one sign, with complementary (two term) interpretation where values are taken as a pair where they alternate with successive positive and negative values.

Wednesday, August 17, 2011

The Critical Region (1)

As we have seen the Euler Zeta Function

1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) +..... is defined for all values of s > 1.

However in standard linear terms we cannot give numerical meaning to the function for other values of s.

For example when s = 0, we generate the series 1 + 1 + 1 + 1 +.... which - again in conventional terms - diverges to infinity.

However Riemann showed that in his treatment of the function where s can take on any complex value that it is possible to extend the domain of definition of the Function for all values of x (except 1).

The critical region involves values of s from 0 to 1. It has long been known that all the non-trivial zeros must lie in this region (with the Riemann Hypothesis suggesting that they all lie on the line (for real part of s = .5)

Now this is where qualitative - as opposed to mere quantitative - interpretation of numerical becomes extremely important.

Once again it is not possible to give a finite meaning to the sum of a series such as 1 + 1 + 1 + 1 +.... which clearly gets larger and larger and in conventional terminology diverges to infinity.

However in the Riemann Zeta Function the sum of this series (and a vast range of other divergent series) are indeed given a definite finite value. So this raises the very obvious question as to what such a result can mean. And the fascinating answer is that it points in all cases to an additional holistic qualitative interpretation in accordance with Type 2 Mathematics.

The ultimate implication is that we cannot properly understand the very meaning of the Riemann Hypothesis in the absence of Type 2 mathematical understanding. And once we do establish the true meaning of the Hypothesis it becomes readily apparent that it can neither be proved nor disproved in standard mathematical terms (i.e. in accordance with Type 1 interpretation).

It may be helpful at this stage to raise an area in relation to Fibonacci type number sequences that initially provided for me many of the insights regarding qualitative interpretation of numerical values that are so useful with respect to the Riemann Zeta function.

Yesterday, in my related blog on "The Spectrum of Mathematics" I briefly mentioned these. So I will myself here:

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 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).

The equation x^2 - 1 = 0 gives the correspondent to the pure 2-dimensional case where the values for x = + 1 and - 1.

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,....

Now what is interesting here is that we cannot approximate the (positive) 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 (occuring as each second term in sequence). In this we get either 1/1 or 0/0. The first would give us the conventional rational quantitative interpretation using linear (1-dimensional) logic. However the second actually corresponds to the qualitative holistic interpretation according to circular (2-dimensional) logic. This can be expressed as the complementarity of opposite poles so that 0, which numerically is given here could equally be represented as 1 - 1 where both aspects must be taken as a pairing.

So to sum up:

Thee initial equation x^2 - 1 = 0 i.e. x^2 = 1, is of a pure 2-dimensional nature. Therefore we can only obtain meaningful quantitative solutions by taking the ratio - not of successive terms as in linear terms - but rather the ratio of every second term (as befits 2-dimensional interpretation).

Two results now arise. The first, 1/1 gives us the (positive) quantitative result of the equation (i.e. the square root of 1).
The second 0/0 gives the qualitative basis of this result based on circular 2-dimensional understanding (that entails the complementarity of opposite poles).

When we attempt to obtain the root in linear terms through the ratio of opposite terms we get either 1/0 or 0/1. What both of these indicate is a relationship between two different interpretations, 1/0 (as between 1-dimensional and 2-dimensional and in reverse fashion 0/1 (as between 2-dimensional and 1-dimensional).

Both of these result from the attempt to split up what is inherently of a 2-dimensional nature (in qualitative terms) in a manner amenable to 1-dimensional linear understanding (which is not appropriate in this context).

So we cannot interpret the behaviour of such a sequence without reference to its qualitative dimensional characteristics. Because of the merely reduced quantitative interpretation of symbols employed in Type 1 Mathematics, these qualitative aspects are never properly investigated.

So when s = 0 the Riemann Zeta Function results in the sum of terms,

1 + 1 + 1 + 1 +...... which diverges in linear (1-dimensional) quantitative terms.

However it is possible to provide a finite value for this series. In my piece on "Holistic Values" on "The Spectrum of Mathematics" blog I explain how this is done:

The Zeta Function is defined as

1 + 1/(2^s) + 1/(3^s) + 1/(4^s) +......

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 multiplying 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 Function = 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 - ...

Now 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 come up with a single answer = 1/2.

And then from this Eta value the corresponding Zeta value can be easily calculated = -1/2.

There is in fact another way of doing this: If we attempt to obtain the value of 1/(1 - x) we generate the infinite series:

1 + x + x^2 + x^3 + ....

Clearly this latter series only converges for values of x between - 1 and + 1.

But the former expression can be defined for all values of x (except where x = 1).

Now when x = - 1, 1/(1 -x) = 1/2;

If we attempt to express the equivalent series in terms of x = - 1, we obtain

1 - 1 + 1 - 1 +.... which gives us the result that we have already calculated through another means.

However we have already used this Eta value to calculate the corresponding value for the Zeta series where s = 0

i.e. 1 + 1 + 1 + 1 +..... = - 1/2

Now this series can equally be generated by letting x = 1 in our series

1 + x + x^2 + x^3 + ....

So 1 + x + x^2 + x^3 + .... = - 1/2;

However 1/(1 - x) = 1 + x + x^2 + x^3 + .... = - 1/2 (when x = 1).

However 1/(1 - x) = 1/0 (when x = 1.

What this establishes therefore is that the famed result for the Riemann Zeta Function where s = 0 involves the relationship as between linear and circular type interpretation.

In other words the result, - 1/2 is actually the attempt to express the qualitative nature of circular (2-dimensional) understanding in a linear (1-dimensional) manner. Once again 2-dimensional interpretation involves two poles as an inherent pairing that are positive and negative with respect to each other. So if we take the negative pole and attempt to express it as a fraction of the pairing we get - 1/2.