A Special Case (3)

In the last blog entry, specifically with reference to the number “2”, I showed how there are two equally important ways in which the number can be defined.

Thus in standard linear terms, where its sub-units are considered in an absolute independent manner, 2 is given an analytic quantitative identity (in cardinal terms).

However in paradoxical circular terms, where its sub-units are now considered as relatively interdependent - and thereby fully interchangeable with each other - 2 is given a holistic qualitative identity (in an ordinal manner).

Though the ordinal nature of number is of course recognised in conventional mathematical terms, it is invariably reduced in a merely quantitative manner (whereby each position is given a fixed identity).

Thus using a physical analogy from quantum physics, every number can manifest itself in both a particle and wave-like manner, and these two aspects necessarily keep switching with each other in the dynamics of understanding.

However the conventional mathematical approach grossly misrepresents the true nature of number by attempting to (formally) interpret it in a static fixed manner.

Now, I have frequently referred to the two aspect of number as Type 1 and Type 2 respectively!

Type 1 corresponds with the linear conventional interpretation of number in analytic quantitative terms.

The natural numbers from this perspective (1, 2, 3, 4, 5, …) are more fully represented as   

1¹, 2¹, 3¹, 4¹, 5¹, …

Type 2 corresponds (indirectly) with the circular paradoxical interpretation of numbers in a holistic qualitative manner

The natural numbers from this perspective are more fully represented as

1¹, 1², 1³, 1⁴, 1⁵, …

Now these have no distinctive meaning in quantitative terms. However from a holistic qualitative perspective 1² represents the interdependence of two related units.

Then indirectly this interdependence can be expressed in a holistic circular manner as + 1 and  – 1, where both units are interchangeable.

(Of course + 1 and – 1 can equally be expressed - as in conventional mathematical terms - in an absolute manner where they are clearly separated in an analytic manner).

However it is the continued failure to recognise the true holistic aspect of number, which I am addressing in these blogs.

So mathematical understanding necessarily contains both analytic and holistic aspects, which in direct terms are represented by reason and intuition respectively.

For example I have just read this quote from Poincare.

“It is by logic, we prove, by intuition we invent”

Logic - in the sense that Poincare intends - represents the analytic aspect of understanding which is of an unambiguous rational nature.

However intuition represents the corresponding holistic aspect, which inherently is of a paradoxical circular nature.

Therefore though analytic reason and holistic intuition are clearly distinct from each other, in conventional mathematical terms the holistic aspect is inevitably reduced in formal terms to the analytic.

In this sense formal mathematical interpretation is of a grossly reduced nature (and ultimately not fit for purpose).

But before this crucial point can be properly grasped, the distinctive holistic aspect of all mathematical understanding must be properly recognised.

Now if we go back to the original simple expression i.e. x = 1, we can perhaps now better appreciate what happens when both sides are raised to the power of n.

So xⁿ = 1ⁿ.

Therefore when n = 2, x² = 1².

Therefore, the number on the right side of the equation properly belong to the Type 2 aspect of the number system.

However in conventional mathematical terms, as 1² has no distinctive quantitative value, it is inevitably reduced to 1 (i.e. 1¹) where it is interpreted as the first natural number in the Type 1 system.

Thus a number that should be treated in a holistic manner - and indirectly expressed in a circular interdependent fashion - is now treated in analytic terms.

And this is what then creates the conflict with earlier analytic type interpretation.

So once again when x = 1, x – 1 = 0.

Therefore squaring both sides (x – 1)² = 0.

And this equation in Type 1 terms has two “linear” solutions i.e. + 1 and + 1 (as independent).

However equally when x = 1, x² = 1² .

And this equation in Type 2 terms has two “circular” solutions i.e. + 1 and – 1 (as interdependent).

Thus we obtain two “different” answers because both correspond respectively to different notions of dimensions (that are analytic and holistic with respect to each other).

However again, these two sets of answers cannot be reconciled satisfactorily through conventional mathematical interpretation (that solely recognises the analytic aspect of number as quantitative).