Yesterday we looked briefly at the qualitative nature of time (and space) from an (algebraic) irrational perspective.
Now an (algebraic) irrational number arises as the solution to a polynomial equation with rational coefficients. The famed square root of 2 - which is the best known example of an irrational number - arises from the simple polynomial expression x^2 = 2!
What this implies with respect to the nature of time (and space) is that a hybrid dynamic mix of the two logical systems (linear and circular) are involved, whereby relative notions are continually reduced in somewhat absolute terms and - in reverse - absolute notions quickly transformed in a relative manner.
Once again, we see this clearly in nature at the sub-atomic level where energy is continually reduced in terms of mass and mass once more transformed into energy.
So in holistic mathematical terms, such interactions properly take place in an environment characterised by irrational notions of time (and space).
In corresponding psychological terms, understanding of these dimensions typically unfolds through authentic contemplative development, where nondual notions of reality are continually reduced in a dualistic manner and then likewise such dualistic notions continually transformed again in a nondual manner.
And this leads to a a more refined dynamic type of understanding whereby reason and intuition continually interact in experience.
We have already looked at the differentiated nature of experience that corresponds with the odd integers and the corresponding integrated nature of the even dimensions. So in a very accurate sense, irrational understanding arises when both odd and even dimensions are combined. So once the first two dimensions unfold in experience, then irrational type understanding (in this strict mathematical sense) will then arise through the process of attempting to relate both dimensions.
As with rational, all irrational numbers can be given both a positive and negative identity.
So this then raises the question as to what is implied by the nature of time (and space) in negative irrational dimensions.
Now, perhaps initially this can be more easily approached from the psychological perspective. We have already seen how with the odd dimensions (as positive) i.e. where one attempts to understand in a direct rational manner, that the main problem relates to unrecognised projections (of an unconscious intuitive kind).
By contrast with the even dimensions (as positive) i.e. where one attempts to understand in a directly intuitive manner, the main problem arising is that of (unrecognised) rational attachments.
Therefore negation with respect to the irrational number dimensions, implies the dual attempt to erode unwanted attachments of both an unconscious and conscious nature.
When successful therefore, this leads to both a purer rational and intuitive appreciation of the dynamic nature of reality involved.
However an important limitation attaches to (algebraic) irrational understanding in that the (imaginary) unconscious nature of personality initially remains comparatively undeveloped. This then sets limitations to the extent to which dynamic negation can be successful in eroding all unnecessary confusion.
This is where we come to another remarkable holistic mathematical finding.
When Hilbert in his famous address named his 23 important - and yet unsolved -mathematical problems one of these related to the status of a number such as 2^(square root of 2). It was believed to be of a transcendental nature but this had not yet been proved. Indeed Hilbert mistakenly believed that this problem would take longer to solve than the Riemann Hypothesis!
In fact it was proved within Hilbert's lifetime. However it demonstrates once more how the the very nature of a number is transformed in quantitative terms through relating a base expression to a dimensional number (as power).
So we saw yesterday with respect to a^b, when both a and b are rational (with b a fraction, an (algebraic) irrational number arises.
We can take this one step further by showing how when b is now irrational (and a either rational or irrational) that a transcendental number arises.
In conventional mathematical terms, a transcendental number is expressed as an irrational number that cannot arise as a solution to a polynomial equation with rational coefficients. The most famous examples of such numbers are π and e.
However as always we can give such a number a qualitative as well as quantitative meaning.
As we have seen the earlier stages with respect to authentic contemplative development (in what is sometimes is referred to as the subtle realm) imply the irrational interpretation of dimensions (from the holistic qualitative perspective).
Typically one's perceptions of reality are much more fluid where both dual (rational) and nondual (intuitive) aspects increasingly interpenetrate. Later in development more deep rooted concepts likewise unfold with again dual and nondual aspects interpenetrating.
However, as one now begins to increasingly match both perceptions and concepts of this nature a further important transformation in development is required, whereby experience now becomes transcendental (in a precise qualitative mathematical manner)
Now it would be valuable to probe more closely here what such transcendental experience entails.
Putting in bluntly, at the earlier irrational (subtle) stages, a certain mismatch of conscious and unconscious is in evidence. From one perspective, one is still too ready to reduce what is properly unconscious (and nondual) in rational terms. Likewise from the other perspective one is equally too ready to elevate what is properly conscious and nondual in an intuitive manner.
However because rational and intuitive aspects are complementary in nature, the proper balancing of both irrational perceptions and concepts requires that both conscious and unconscious aspects come into equal balance.
Thus when the new transcendental type knowledge unfolds (in what - again - is sometimes in Eastern terms referred to as the causal realm) it is of a new even more refined nature. So with respect to the nature of reality, one does not emphasise either dual or nondual aspects (as separate) but rather as the relationship between both dual and nondual aspects. Attaining such a position requires that attention focus more directly on the harmonising nature of the will (as a means of reconciling both conscious and unconscious).
This also provides a fascinating qualitative insight into why a transcendental number cannot be the solution of a polynomial equation.
Such a solution always entails a reduction of a higher dimensional value in 1-dimensional terms.
So once again if we have x^2 = 2, the higher dimensional value here (corresponding to 2 as dimension) = 2. Then we obtain x = the square root of 2, it thereby corresponds to the reduced 1-dimensional value.
However the very nature of transcendental is that reality essentially represents the relationship as between dual and nondual. Therefore we cannot attempt to either reduce or elevate one in terms of the other.
So the transcendental nature of time (and space) is now of an extremely subtle variety as representing the essential relationship as between actual (finite) and potential (infinite) aspects of understanding. This corresponds in experience with a highly refined and dynamic understanding where both reason and intuition interpenetrate in pretty equal measure.
In fact perhaps the best representation of the nature of such understanding is with respect to most famous transcendental number π.
Now π in quantitative terms represents the (perfect) relationship as between the circular circumference and its line diameter.
Equally in qualitative terms, π represents the (perfect) relationship as between both circular and linear type understanding. And what is common to both is the point at the centre of the circle which equally is at the centre of the line diameter.
So pure transcendental understanding, therefore can be expressed as the ineffable midpoint (or singularity) where linear or circular understanding of a separate nature no longer remains.
Just one further observation is worth making here!
I have mentioned before how from a higher dimensional perspective the very nature of mathematical proof is inherently subject to an Uncertainty Principle.
What this entails is that - properly understood - such proof represents an inevitable dynamic interaction as between two aspects which are quantitative and qualitative with respect to each other.
As we have seen elsewhere, implicitly the Pythagoreans were searching for this type of proof. From a quantitative perspective they were indeed able to show why the square root of 2 is irrational! However what really troubled them was that they were unable to provide a satisfactory qualitative rationale as why this was the case!
So now we have yet another example with respect to the nature of a transcendental number. From a quantitative perspective, it can be proved why any rational (or irrational number) other than 1 raised to an irrational dimensional power will result in a transcendental number.
And from my own holistic mathematical investigation of the nature of the stages of contemplative development, I have been able to provide a corresponding qualitative explanation as why to this behaviour occurs.
Therefore, a comprehensive understanding of this relationship entails both quantitative and qualitative aspects (with an inevitable uncertainty attaching to both). Thus current mathematical proof with respect to the merely quantitative nature of transcendental numbers, reveals a subtle confusion (for in quantitative terms we can never precisely determine the value of any transcendental number).
So for example the wide held belief that π is a constant, strictly has no meaning in - merely - quantitative terms (as we can never precisely determine its value).
When we look at Mathematics in a more comprehensive manner we realise that the quantitative is always balanced by a corresponding (holistic) qualitative aspect.
So a rational number therefore (in quantitative terms) corresponds to rational type understanding (from a holistic qualitative perspective).
Equally however a transcendental number (in quantitative terms) corresponds to transcendental type understanding (from a holistic qualitative perspective) And as we have see transcendental in this qualitative sense relates to the the highly refined understanding where both linear (rational) and circular (intuitive) type understanding are both explicitly recognised and kept in a certain balance to each other. (And as we have seen with the purest development of such understanding they are kept in perfect balance!)
Therefore one cannot properly attempt - without gross reductionism - a rational proof (in qualitative terms) of what is transcendental (from a quantitative perspective).
So once again for example we cannot give a - merely - rational meaning to the notion that π is a constant because it is not a rational number (with it's value strictly indeterminate from a merely quantitative perspective!)