I was at pains then to point out that appreciation of what is involved, requires incorporation of a distinctive holistic notion (literally of the qualitative meaning of "whole") which unfortunately is completely edited out from conventional mathematical understanding.
In other words due to its reduced quantitative nature, "the whole" in any numerical context is viewed as merely the sum of its constituent parts.
It might be possible here to provide some insight into the true holistic meaning of s = – 1, with reference to - what initially might appear - a very simple example.
As we have seen in the conventional (Type 1) approach to Mathematics number is defined in linear terms i.e. with respect to the default dimensional number (as power or index) = 1.
So 3 for example in this context can be more fully expressed as 31.
Now something fascinating happens when we now negate this dimension of 1 i.e. by raising 3 to – 1.
So 3– 1 = 1/3.
Therefore enshrined in this simple relationship is the central mystery regarding the relationship between whole and part.
To make this more accessible, let us consider the example of a cake that is divided into 3 (equal) slices.
We will look initially at interpretation of the relationship between the (part) slices and (whole) cake from the conventional mathematical perspective.
As we have seen, this conventional approach is based on the notion of quantitative reductionism.
Therefore in this context the (whole) cake is equated merely as the sum of its 3 (part) slices.
Therefore in this context the whole cake (as the quantitative sum of its component slices) = 3.
Each part in turn (as a quantitative fraction of the cake) = 1/3.
However the true relationship here as between whole and part is of a much subtler nature (entailing both quantitative and qualitative notions).
The key here is the recognition that the whole has a distinctive qualitative - as well as quantitative - meaning.
The quantitative notion is related directly to the recognition of independent identity with respect to its component parts. So we little difficulty in recognising in this context that each of the slices assumes an individual quantitative identity, so that for example we can then serve each of the slices to 3 different people.
However each of the parts also necessarily enjoys a common qualitative identity, which is based on their interdependent relationship with respect to the whole cake.
For example it is this distinctive qualitative identity that enables one to identify a 1st, 2nd and 3rd slice, which depends purely on relative context. In other words, each of the 3 slices can be identified as 1st, 2nd or 3rd depending on the arbitrary method of choosing the slices.
Now, when one recognises this distinction as between the quantitative identity of the slices (as relatively independent of each other in an individual manner) and the qualitative identity of the slices (through their common shared interdependence with the whole cake), the relationship in turn as between parts and whole (and whole and parts) is crucially transformed.
So basically now in this dynamic interactive mode of understanding, the parts and the whole are understood as in complementary fashion as quantitative and qualitative with respect to each other.
Remarkably what this entails is that number itself - using the language of quantum physics - is now understood to have complementary particle and wave aspects.
Thus we can identify the cake in quantitative terms as identical with its 3 component slices.
So therefore the cake = 3 (in this particle context).
However equally we can identify the whole cake as expressive of the collective interdependence of its component slices.
So therefore the cake = 1 (in this wavelength context).
Therefore when we understand this simple example in appropriate fashion, we realise that a continual transition takes place as between the use of number with respect to both quantitative and qualitative interpretation.
Or to put it in somewhat equivalent terms, one keeps switching as between an analytic appreciation of number (in quantitative terms) and a corresponding holistic appreciation of number (in a qualitative manner).
Therefore in the dynamics of understanding, if we start with recognition of
3 quantitative slices (i.e. 3– 1, a crucial qualitative recognition is required before one can now switch to the corresponding recognition of each slice representing a fraction of the (whole) cake. And this entails implicit understanding of the qualitative meaning of s = – 1.
So in psycho spiritual terms the negation of 1 as dimension implies an unconscious intuitive recognition of the notion of the whole (in a qualitative manner). And implicit recognition of this qualitative aspect of understanding is vital in enabling one to switch from quantitative recognition of the combined 3 slices to corresponding quantitative recognition of each slice as existing as part of the combined whole.
However though implicitly this qualitative recognition (entailing unconscious intuitive insight) is involved, explicitly in conventional mathematical terms, a merely reduced interpretation is given.
So the great mystery of the true relationship as between whole and part is lost here with merely a quantitative reductionist interpretation remaining.
Of course it is equally true in reverse. When we are now aware in quantitative terms of each slice (as a fractional part of the cake) it implicitly requires a corresponding qualitative recognition of the whole, so as to be able to switch from quantitative recognition of each individual part to corresponding quantitative recognition of the combined 3 slices.
So (1/3)– 1 = 3 1.
In fact all this opens up an entirely new holistic appreciation in physical (and psychological) terms of the true nature of space and time.
So quite remarkably, every number (and sign) with a recognised quantitative meaning in conventional mathematical terms, can equally be given a corresponding qualitative interpretation which intimately relates to the dynamic manner in which space and time are experienced (again in both physical and psychological terms).
Thus in this context, I have explained the holistic psychological significance of the negative 1st dimension as intimately involved in the - apparently - simple act of concrete number recognition.
Indeed one could sum up true appreciation of the holistic mathematical significance of the negative 1st dimension, as the keen realisation that the whole (in any context) has a distinctive qualitative meaning that cannot be reduced in terms of its quantitative parts.
As I stated in another context (which in fact is related), this is deeply relevant in terms of the spiritual life. Here with the onset of authentic contemplative progress, customary dualistic distinctions (through which quantitatively reduced notions are expressed) begin to break down in dramatic fashion. So through a greatly refined intuitive recognition, one begins to truly discover a new qualitative appreciation of "the whole".
Again this is deeply relevant in terms of appreciating the Riemann zeta function for s = – 1, where we are confronted with a series (i.e. the sum of the natural numbers).
Now due to conventional training (in unquestioned acceptance of the reduced quantitative approach) one immediately tries to interpret this series in quantitative terms and this clearly appears (from this perspective) to diverge to infinity.
However the clue as to its finite interpretation = – 1/12, lies in holistic rather than strict analytic appreciation.
Here is yet another clue as to what dimensional interpretation according to the negative 1st dimension entails.
Now in conventional mathematical terms, there is strong belief in the abstract validity of mathematical "objects" such as numbers (and especially the primes).
And in holistic mathematical terms, this equates with the conscious (positive) direction of experience.
However strictly speaking mathematical "objects" can have no meaning independent of our mental interaction with them. And relative to these "objects", the subjective mental constructs we use in their understanding equate with the negative (unconscious) direction.
So the development in such "negative consciousness" leads to a growing appreciation of the holistic dimension underlying conventional analytic type understanding.
In particular one recognises that what is known objectively as "mathematical truth" necessarily reflects a particular limited interpretation of reality (in which such truth appears unambiguous).
And as we have seen Mathematics is characterised to an extraordinary degree by interpretation that corresponds qualitatively to the positive 1st dimension, in what perhaps more simply can be expressed as linear rational understanding. For example this is well represented through the very notion of a number line, on which all real numbers are supposed to lie!
However when one begins to understand with respect to the negative 1st dimension, one realises that this standard version of mathematical truth represents just one limited interpretation (i.e. according to the positive 1st dimension). In particular, one now recognises that what chiefly characterises such understanding is whole/part reductionism in quantitative terms i.e. where the whole in any context is viewed as the quantitative total of its various parts.
Thus when one starts experiencing in terms of the negative 1st dimension, an important inversion takes place whereby now truth becomes defined in true holistic (i.e. whole) terms as the relationship of interdependent parts to an overall qualitative whole.
And this is directly relevant to the Riemann zeta function ζ(s ) where s = – 1.
So the correct way of interpreting the sum of the resulting series i.e. 1 + 2 + 3 + 4 + ... result is not in terms of its quantitative sum, but rather in terms of the relationship in some unique context of constituent parts (of an interdependent nature) to an overall collective whole in qualitative terms.
Now the problem with concrete macro objects is that parts immediately assume a recognisable independence in their own right, thus encouraging our typical understanding of the relationship as between whole and parts in a reduced quantitative manner.
So when we divided in our example the whole cake into 3 constituent slices, each individual slice can be seen to possess a recognisable quantitative independence. Therefore we are enabled to overlook the fact that a corresponding qualitative interdependence must necessarily apply to the relationship between the three slices.
However at the quantum physical level, it is very different in that sub-atomic particles lose any distinct independent identity and therefore can only be properly understood in relation to an overall qualitative whole context.
Therefore, despite the great advances made at the analytic level in terms of the understanding of quantum mechanical relationships, a truly major problem remains a failure to properly introduce holistic qualitative notions into the interpretation of particle interactions.
Thus quantum mechanics continues to appear so non-intuitive, precisely because we insist on an interpretation that has been heavily conditioned by the conventional mathematical paradigm (that is of a limited 1-dimensional nature).
However indirectly, holistic ideas can shown to be at work.
For example the understanding of the behaviour of strings (in the case of bosonic strings) is strongly linked with the Riemann zeta function for s = – 1.
Now here a string can vibrate in 24 different vertical directions based on a 2-dimensional horizontal surface. And this is associated with the finding that this earlier version of strings was found to be only consistent in 26 dimensions.
So these dimensions here provide an overall holistic context in which the physical behaviour of the strings can operate in a consistent manner.
Thus, if one then attempted to break up this whole in terms of a single string, this would represent 1/24 of the total (which is just half of 1/12).
The negative sign here could then signify the inverted understanding where the part is now being derived from the overall whole (in qualitative dimensional terms) whereas conventionally in linear (1-dimensional terms) the whole is derived from the constituent parts (in a quantitative manner).
So 12 and 24 are "magic" numbers that arise in a variety of physical circumstances such as quantum physics and the monster group. Thus special properties of holistic symmetry (which are of a qualitative nature) are associated with these numbers.
And the Riemann zeta function (for s = – 1) is directly related to these holistic qualitative notions of symmetry.
Some time ago I became aware of a fascinating alternative example with direct relevance to psychology.
I had been studying Jung and become especially interested in his treatment of personality types.
Though Jung was not a mathematician in the conventional sense, I found that many of his ideas could be readily formulated in a holistic mathematical manner.
Now Jung defined 8 personality types which in the well known Myers-Briggs type Indicator was then enlarged to 16 (that was heavily based on his own work).
However, having studied it deeply for some time, I gradually became convinced that there were 8 more "missing" personality types.
So I then devised a new system of 24 Personality Types based directly on holistic mathematical notions with each offering a valid - though necessarily limited - perspective on the understanding of reality.
Basically in this approach I start with 4 fundamental dimensions (in accordance with the holistic complex meaning of the 4 roots of 1).
Then each personality type is viewed as a unique permutation with respect to the four original dimensions. And as 4P4 = 24, this entails that 24 relatively distinct Personality Types can be defined.
In then struck me that each of these types represents a way of configuring the experience of space and time.
This then led to a new appreciation of dimension (in this context) as a unique configuration with respect to the original 4 dimensions.
I then realised that this would necessarily have a complementary interpretation in physical terms. So the 24 vibrations of the bosonic string, would thereby likewise represent unique permutations with respect to the 4 original dimensions. Clearly at the string level, space and time would remain entangled with each other in an interdependent fashion. Only later would they become sufficiently separated to be viewed as 4 dimensions in an independent manner.
So here we have two domains physical and psychological connected through the same holistic notion of dimensions.
Interestingly in the Jungian approach and the later Myers- Briggs approach, an even balance is maintained as between the positive (conscious) types (designated as S types) and the negative (unconscious) types (designated as N types). So S types identify directly with actual reality in a conscious manner; N types relate to potential (holistic) reality in a more unconscious manner.
Now if we extend this to the 24 types then 12 would lean towards conscious and another 12 to unconscious orientation. And in holistic mathematical terms these would be + and – with respect to each other.
Thus if we start from standard appreciation of the world (especially in mathematical terms) as S based, true holistic appreciation would be N based.
So for the 12 types that identify with the conscious aspect (+ 12) there would be another 12 that are identify more with the corresponding unconscious aspect (– 12).
–1/12 would then arise in this context as the attempt to express one of the 12 personality types (geared to holistic type appreciation) as an analytic fraction of all 12 (which in truth are highly interrelated as one integral whole).
So the key point here is that the numerical result – 1/12 properly relates to a holistic type relationship of a qualitative nature (though indirectly employing the analytic notion of a fraction).
This is likewise the case with the bosonic string. When the string is postulated to vibrate in 24 directions, this relates directly to a holistic qualitative capacity (associated with the string).
Therefore the fractional relationship of the vibration of one string in terms of the total (where all are highly interrelated) represents directly a qualitative - rather than quantitative - relationship.
We have dealt here with the important case of ζ(s ) where s = – 1.
And what we have found is that a unique interpretation, entailing holistic qualitative and (indirect) analytic quantitative notions are associated with this relationship.
So a unique interpretation attaches to s = – 1 in the context of the Riemann zeta function.
Equally a unique holistic relationship attaches to s for every odd integer value. And in general, the resulting fractional results that emerge are associated with high levels of symmetry with respect to holistic qualitative type relationships (i.e. where the whole in any context can only be meaningfully understood in terms of the interdependence of its constituent parts).
Though in some respect the values of ζ(s) for the negative odd integers appear somewhat arbitrary, this is not the case.
In fact I believe that a remarkable feature is associated with the denominators of all these results that directly relates to the very nature of the primes
Once again in conventional terms, we can demonstrate that a number is prime, by attempting to break it into constituent factors. Then if no factors can be found other that the number itself and 1, then that number is prime.
However the Riemann zeta function for the negative odd integers turns this process on its head, whereby we can now - in principle - show that a number is prime if it is a factor of a certain number (which is given by the denominator of the corresponding values for the Riemann zeta function).
Now through the Riemann zeta function, the value of ζ(1 – s) is related to the corresponding value of ζ(s).
So for example the value of ζ(– 1) is related to ζ(2).
Then in general terms, I postulate this simple rule.
In the case of the odd integers if the absolute value of the denominator of ζ(1 – s) is divisible by s + 1, then s + 1 is prime.
So in the case of s = 2, ζ(1 – s) = ζ(– 1) with the absolute value of its denominator = 12.
And as 12 is divisible by 3, therefore 3 is a prime!
Then in the case of s = 4, ζ(1 – s) = ζ(– 3) with the absolute value of its denominator = 120.
And 120 is divisible by s + 1 (= 5). So 5 is prime
And continuing, in the case of s = 6, ζ(1 – s) = ζ(– 5) with the absolute value of its denominator = 252.
And as 252 is divisible by s + 1 (= 7), then 7 is a prime.
Finally to illustrate in the case of s = 8, ζ(1 – s) = ζ(– 7) with the absolute value of its denominator = 240.
And as 240 is not divisible by s + 1 (= 9), then 9 is not prime!
I was able to check these values for values of s up to 200, with no exceptions arising. However as the numerators get truly enormous for higher odd integer values of s, it is not possible to keep checking at higher values.
So as a practical way of finding primes, this approach would be of little value.
However in principle, it remains very important as it seems that the unique nature of denominator values of the Riemann zeta function (for odd integers) is determined by this inverse qualitative nature of prime identity.
In other words, as I have pointed out before from the quantitative analytic perspective, we view the primes as the quantitative building blocks of the natural numbers.
However from the corresponding holistic perspective, this relationship is reversed with the qualitative nature of the primes (through their unique relationship as factors) depending on the natural numbers!