Monday, February 13, 2012

Imaginary Numbers and Qualitative Understanding

Once again I would like to restate my central purpose. Though, conventionally Mathematics currently recognises (in formal terms) only the quantitative aspect of interpretation (Type 1), in truth a complementary - and equally important - qualitative aspect (Type 2) also exists.

And a fully comprehensive mathematical understanding entails the dynamic interaction of both aspects of interpretation (Type 3).

Unfortunately mathematicians who do to some degree recognise a qualitative aspect often do so in a somewhat trivial manner (where it does not directly have relevance for what they really understand as Mathematics).

For example in addressing the qualitative side, reference may be made to numbers having a "personality" and "lucky" or "superstitious" numbers. Also we may hear of cultures where numbers possess feminine connotations. And of course most practicing mathematicians will recognise that there is indeed a qualitative side to mathematics in the sense for example of the "beauty" or "elegance" of a proof. Also clearly aesthetic appreciation with respect to great paintings, architecture, painting, sculpture etc. frequently has a mathematical dimension.

However what is greatly missing is any direct realisation that - properly understood - this qualitative side of understanding is intimately tied up in experience with every mathematical symbol and relationship but then screened out altogether from formal interpretation.

So in the context of our present discussion, how many mathematicians would accept that the qualitative aspect of mathematics is related to appreciation of the true nature of the Riemann Hypothesis?

So the proper translation of mathematical experience requires the realisation that every symbol, relationship, hypothesis etc with meaning in the accepted quantitative manner equally has meaning in - as yet unrecognised - qualitative fashion.

So, ultimately comprehensive mathematical understanding, in any context, requires a dual interpretation according to two distinctive aspects (i.e. quantitative and qualitative).

In the last few blogs I have already attempted to convey what this qualitative aspect of interpretation implies.

So we started with the number 1 (representing its holistic dimensional meaning) and showed how this has an a extremely important relevance as an overall qualitative manner of interpreting mathematical symbols. And in a very precise manner Conventional (Type 1) Mathematics is defined by a linear (1-dimensional) rational approach.

We also saw that the + sign equally has an important holistic significance in referring to the (conscious) positing of understanding which again typifies Type 1 interpretation. So here, symbols are literally posited (without respect to a complementary pole) in an absolute manner. In other words, though mathematical experience necessarily entails the interaction of both external (positive) and internal (negative) poles in relation to each other, Type 1 Mathematics breaks this two-way relationship between poles - literally - viewing mathematical entities as absolute "objects". So 1-dimensional in this qualitative context entails uni-polar understanding (where a pole such as objective is separated from its opposite subjective pole).

We then went on to indicate the holistic significance of the number "2" (again in holistic dimensional terms).

And just as 1 implies uni-polar, 2 now implies bi-polar. In other words all interpretation from a 2-dimensional perspective implies the recognition - again in any mathematical context - of the necessary interaction of two complementary poles as dynamically relative to each other.

So for example external (objective) and internal (subjective) represent important examples of such interaction; also whole and part represents another vitally important pairing.

And just as + (in a holistic sense) is tied up with 1-dimensional interpretation, - is tied up with 2-dimensional appreciation.

So properly understood, the ability to recognise two opposite poles as interdependent, requires (unconscious) intuition. And the very means by which this is liberated in experience is through the dynamic negation of the rational conscious (thereby making what was conscious now unconscious!).

Thus with the acceptance of both 1 and 2 as qualitative holistic numbers and the corresponding acceptance of + and - equally in a holistic sense as the positing and negation respectively of conscious reason, we have already made sufficient progress to ultimately appreciate the true nature of the Riemann Hypothesis.

I then used the instructive example of a crossroads to show the intersection of both 1-dimensional and 2-dimensional interpretation. Thus, using an isolated (unipolar) frame of reference as I move up a road, I can identify a left turn at a crossroads in unambiguous terms. Then when moving down the (from past the crossroads) I can again identify a left turn in unambiguous terms. So using this labelling system consistent with 1-dimensional interpretation, both turns at the crossroads are identified as left.

However when I now switch to 2-dimensional (bi-polar) understanding, I intuitively recognise that every pole identification has a complementary partner (that is opposite). Therefore in relation to each other the two turns are left and right and right and left (in relation to each other). Thus, left or right here have a merely relative meaning which keeps switching (depending on the direction from which they are approached).

So the full understanding of what is involved requires both 1-dimensional and 2-dimensional logic. And in terms of the second type, results that are unambiguous in terms of linear logic, now seem deeply paradoxical.

And this arises through mixing rational understanding (which is of a limited dual nature) with intuitive understanding which though it can be given an indirect rational expression is inherently holistic and nondual in nature.

And this simple illustration - which however is extremely subtle in its full implications - is enough to explain why the Riemann Hypothesis can have no solution (in conventional Type 1 mathematical terms).

So instead of left and right in our illustration, we have the alternative polarity set, quantitative and qualitative. And in terms of movement up the road we have the corresponding mathematical approach of understanding with respect to particular parts (with respect to individual prime numbers) and in reverse direction to the general whole (with respect to the frequency of primes).

Thus when mathematicians study the individual nature of primes this is amenable to quantitative interpretation. Then when, in reverse, they study the general distribution of primes, again this seems amenable to quantitative interpretation.

However, when we now attempt to view both aspects as interdependent - like the left and right turns at a crossroads - we must admit a complementary qualitative - as well as quantitative - aspect to prime interpretation.

The interdependent nature of the primes is especially evident in the way in which the non-trivial zeros of the Zeta Function are encoded in the primes (and in reverse the manner in which the primes are encoded in the non-trivial zeros).

Therefore appreciation of the nature of the Riemann Hypothesis (which stands at the centre of this mathematical crossroads) clearly requires both 1-dimensional and 2-dimensional interpretation.

And as Conventional Type 1 Mathematics is formally based on mere 1-dimensional understanding, not alone can the Riemann Hypothesis not be solved within its axiomatic system, it cannot even be properly appreciated in this manner!

However for a fuller understanding of what is involved from a qualitative perspective, with the Riemann Zeta Function, we need to move on to the qualitative interpretation of 4 (as a holistic dimension).

Now as we have seen the qualitative meaning of 4 (with respect to the second aspect of our number system) is directly related to its inverse in quantitative terms i.e. 1/4.

And 1^(1/4) = i.

Therefore 4 as a dimension relates to the highly important qualitative holistic interpretation of i!

At one level, it seems truly astonishing to my mind that mathematicians make widespread use of imaginary numbers, without apparently having any interest in the philosophical meaning of what this might entail!

Now in quantitative terms i represents the square root of - 1.

Now we have already defined - 1 in holistic terms as intuitive understanding (corresponding the the dynamic interaction of two complementary poles).

So in effect the qualitative interpretation of i represents an indirect rational attempt to encode holistic intuition in a linear (1-dimensional) manner.

Now all of this has a deep resonance with notions in Jungian psychology which I will deal with in a later post.

However the really important thing to appreciate is that the use of i (and by extension other imaginary numbers) represents intuitive understanding conveyed in an indirect rational manner.

So just as we can have real and imaginary numbers in quantitative terms, equally we can have real and imaginary numbers (in a qualitative manner).

And the true significance of this is that we can now incorporate both conscious and unconscious in an acceptable scientific fashion with the "real" aspect corresponding to rational understanding (of a linear kind) and the imaginary to intuitive appreciation (of a 2-dimensional nature) that indirectly is expressed rationally in a linear manner as what is holistically embodied by rational symbols. In other words the "imaginary" serves as an indirect way of conveying the qualitative aspect of understanding.

So the use of mathematical symbols in a precise quantitative manner (Type 1), refers to the "real" aspect of mathematical interpretation; Type 2 then indirectly refers to the "imaginary" (qualitative) aspect of such interpretation; Type 3 which is both "real" and "imaginary" combines both quantitative and qualitative interpretation in a comprehensive manner.

So in holistic mathematical terms, truly comprehensive (Type 3) interpretation requires a complex rational approach i.e. with both "real" and "imaginary" aspects (in qualitative terms).


  1. Even though complex numbers are fed into the zeta function, the function nevertheless privileges real numbers over imaginary numbers by having 1 as a numerator. This function ought to be partnered with another zeta function with the imaginary number i as the numerator (let’s call this the imaginary zeta function), which will be orthogonal to the other function.

    Hockney, Mike

  2. Thomas,

    I appreciate your contribution.