The Pythagorean School is very interesting in that it pursued a much more comprehensive notion of Mathematics than what now conventionally exists.
Basically for the Pythagoreans, mathematical symbols possessed both an (intuitive) qualitative as well as (rational) quantitative significance.
Put another way their appreciation combined holistic as well as analytic aspects. Through this holistic aspect, certain mathematical symbols were seen to possess universal archetypal properties. Thus the proper nature of mathematical activity greatly transcended mere rational knowledge opening up the way to authentic spiritual contemplative awareness.
For the Pythagoreans, numbers (i.e. the integers) were especially important mathematical symbols, through which the secrets of reality, as they believed, were encoded.
They also believed that all numbers were rational (i.e. could be written as ratios of the integers).
So there was an assumed correspondence as between rational numbers (as quantities) and - what might be called - the rational paradigm (in qualitative terms).
However the application of the famed Pythagorean triangle was to shatter this belief. In the simplest case of the right angled triangle, where both adjacent and opposite sides = 1, the hypotenuse = the square root of 2.
Now the Pythagoreans knew that this did not represent a rational number but rather an irrational quantity (i.e. that could not be expressed as the ratio of two integers).
The reason therefore why this discovery was so significant was that it undermined the vital correspondence they believed to exist as between both the quantitative and qualitative aspects of Mathematics. In order words, the Pythagoreans in their enquiry of nature adopted then - as now - the rational paradigm. So they believed that all scientific investigation would conform in qualitative terms to the rational approach; however the right angled triangle clearly demonstrated the existence of irrational quantities. So the Pythagorean Dilemma - as I refer to it - relates to the fact that they lacked the qualitative means of explaining why irrational number quantities could arise.
Subsequently in Western Mathematics the Pythagorean Dilemma has simply been ignored rather than resolved through a basic form of reductionism whereby its activity is now identified solely with its quantitative aspect. From this perspective there is no need to explain the deeper why with respect to the nature of irrational numbers. Rather they are treated in reduced fashion where their quantitative value can be approximated in finite terms to any required degree of accuracy.
However, though it must be readily admitted that Western Mathematics has indeed made enormous advances with respect to its specialised quantitative development that equally, from an overall comprehensive perspective, it has become hugely unbalanced (with its important holistic aspect no longer formally recognised).
So an important task in redressing this imbalance is to show now how the Pythagorean Dilemma can be properly resolved.
The key to answering this problem is to demonstrate that just as we can have irrational as well as rational quantities, likewise in qualitative scientific terms we can define an irrational as well as rational paradigm with which to interpret phenomenal reality.
The very essence of an irrational number (such as the square root of 2) is that it combines both (discrete) finite and (continuous) infinite aspects in its very identity. Thus for example we can approximate the square root of 2 (thereby giving it a merely reduced discrete rational identity) to any given level of accuracy. So correct to 4 decimal places its value is 1.4142 (which is rational).
However in truth an irrational number possesses an irreducible qualitative aspect in that its decimal sequence continues indefinitely (and can never be represented as a fraction).
So the very nature of an irrational number therefore is that it combines both finite and infinite aspects (which cannot be reduced in terms of each other).
The corollary of this in qualitative terms is that irrational understanding likewise combines both finite and infinite aspects. And as we have seen this entails the combination of both rational and intuitive modes of interpretation.
And as discussed in the previous blog, whereas the rational is conveyed through the standard linear logical mode, the intuitive aspect is indirectly conveyed through the circular mode (based on the complementarity of opposite polarities).
Now to put this in perspective we need to consider what actually happens in experience when authentic contemplative development unfolds!
Initially in the spiritual life this entails considerable detachment (wich in holistic mathematical terms entails the dynamic negation of linear rational forms). This negation of conscious phenomena then causes a decisive switch in experience whereby a new type of meaning incubates in the unconscious. Then when the time is ripe it bursts forth as authentic intuitive awareness in a brilliant spiritual illumination.
Quite literally experience now becomes 2-dimensional and transformed in an irrational qualitative manner. Once again the 1st dimension relates to the conscious (posited) direction as linear understanding and the 2nd to the unconscious (negated) direction (as intuitive awareness); Whereas previously a phenomenon e.g. a flower was given a discrete local identity in experience, now it enjoys a two-fold identity. At one level one can again identify it (in more refined fashion) as a local phenomenon; however it now equally possesses an archetypal holistic identity (as mediator of a divine spiritual light). And this holistic aspect in turn relates to the dynamic complementarity of opposites in the fusion of both external and internal aspects of understanding.
In other words, all phenomena are understood to possess both finite and infinite aspects (that cannot be reduced in terms of each other).
The implication therefore for science is that the irrational (2-dimensional) paradigm itself must necessarily combine understanding of phenomena according to both linear logic (where opposites are clearly separated) and circular logic (where they are considered as complementary).
In this way the Pythagorean Dilemma is thereby solved.
In quantitative terms, obtaining the square root of a number entails raising that number to a dimension of 1/2 (i.e. power of 1/2).
And the holistic interpretation of what is involved here equally requires raising understanding in inverse qualitative terms to the dimension 2 (or power of 2).
So 1/D (relating to the dimension or power of a number) in quantitative terms entails the corresponding interpretation of D (in qualitative terms) with the nature of both structurally similar.
The key limitation of the Pythagorean worldview is that the relationship as between both quantitative and qualitative aspects of understanding is viewed in merely 1-dimensional terms.
So just as from a 2-dimensional perspective we see reality in qualitative holistic terms as representing both positive and negative polarities (that are dynamically interdependent) in corresponding quantitative analytic terms we see the square root of a number, such as 1, as representing both positive and negative polarities i.e. + 1 and - 1 (that are statically independent).
And again just as 2-dimensional understanding qualitatively combines both local discrete understanding of a finite and holistic continuous awareness of an infinite nature (where mathematical symbols operate as archetypes) respectively, likewise with an irrational square root number quantity such as 2 (obtained by raising to the inverse dimension of 1/2) again both discrete finite and continuous infinite aspects are combined in its nature.
Of course when - as in Conventional Mathematics - the holistic aspect is disregarded, mathematical symbols greatly lose their qualitative numinous properties and become treated as merely reduced quantities.
Again one may be tempted to question what relevance any of this can have for understanding the Riemann Hypothesis!
Well you see it is all a matter of perspective and from the holistic mathematical perspective it is indeed of paramount significance.
Once again the Riemann Hypothesis relates to the fact that all the non-trivial zeros of the zeta function lie on the real line = 1/2.
Now 1/2 in this context relates to a dimensional number. And of course a square root relates equally to the same dimensional number.
So the whole point about this exercise is that actual understanding of the Riemann Hypothesis requires corresponding qualitative interpretation relating to the 2nd dimension. This in turn entails that it cannot be properly interpreted using 1-dimensional linear logic but must also incorporate circular understanding (corresponding to the 2nd dimension).
In fact as we shall see later the very nature of a prime number (when properly interpreted) requires the incorporation of both linear and circular aspects.
For example from the linear aspect it is customary to view prime numbers as independent building blocks from which all the natural numbers are obtained. However from the equally important circular perspective prime numbers are fully interdependent with the natural numbers (with their general distribution intimately depending on these same numbers).
Indeed ultimately the Riemann Hypothesis represents the fundamental condition required for the consistent use of both logical systems (which is inherent in the very nature of prime numbers).
And this simple truth cannot be appreciated from within the standard 1-dimensional interpretation of Conventional Mathematics!