I have commented before on - what I refer to as - the Pythagorean Dilemma.
In other words the significance of the discovery that the square root of 2 is an (algebraic) irrational number, was as much of a qualitative as a quantitative nature.
As I have stated, the Pythagoreans recognised an important qualitative significance to number. Prior to their discovery of the irrational nature of 2, they had assumed that all number quantities were of a rational nature. Happily this complemented well the scientific paradigm they used to interpret this reality which qualitatively was also of a rational nature.
So the true significance of the irrational nature of 2, is that the Pythagoreans lacked the qualitative holistic means to explain how it could arise, thus shattering the harmonious balance they sougth to preserve with respect to mathematical activity.
The rational paradigm which still dominates present scientific and mathematical thinking is basically suited to interpretation of meaning that is of a finite discrete nature.
However an irrational number by its very nature combines both finite (discrete) and infinite (continuous) aspects. Thus its quantitative value can be approximated as a rational number to any required degree of accuracy. However its qualitative nature leads to a continuous unending decimal sequence (with no fixed pattern).
Therefore though the very nature of an irrational number - literally - transcends the mere rational perspective, Conventional Mathematics can only attempt to deal with such a number in a reduced quantitative manner.
Now once again the (linear) rational paradigm is 1-dimensional in nature (where all number quantities are ultimately expressed in 1-dimensional terms).
Over the past 14 blog entries however I have been at pains to point out that a complementary (circular) rational paradigm exists where every dimensional number is defined in items of the same base quantity of 1. And as we have seen these dimensions then bear an important inverse relationship with their corresponding roots (in quantitative terms).
So in these contributions, I have shown how all rational numbers (positive and negative) possess a unique qualitative significance that intimately applies to the nature of time and space (in both physical and psychological terms).
However just as an irrational number properly combines both finite (discrete) and infinite (continuous) aspects in its very nature, the same equally applies to an irrational number when given its appropriate qualitative interpretation.
So the upshot of this is that from both the quantitative and qualitative perspectives, irrational numbers are of a hybrid nature truly requiring both Type 1 (analytic) and Type 2 (holistic) mathematical interpretation.
And when this is done we can then give meaning to the irrational nature of time and space (in physical and psychological terms).