Wednesday, January 25, 2012

The Square Root of 1

Even as a child I had difficulties with the interpretation of the square root of 1.

Once again in conventional (Type 1) mathematical terms the answer would be given as either + 1 or - 1. Of course in practice to avoid ambiguity with respect to two possible answers the positive answer would be chosen as the principle value and in calculations involving square roots only this would be used.

So for example if you pick up a calculator, enter 1 and then the square root function the answer is 1 (and not - 1).

However there is something deeply unsatisfactory regarding such interpretation.

Mathematics prides itself on its unambiguous nature and yet here in one of the simplest operations possible, two answers that are diametrically opposite - literally poles apart - are deemed to be correct.

For example in Mathematics a proposition is deemed to be either true or false. Thus the Riemann Hypothesis for example is deemed to be either true or false. Of course no one has been able to prove either the positive or negative truth value. So the Hypothesis is still open waiting to be proved or disproved in Type 1 terms. (The great irony here is that in fact - as I have repeatedly stated - that the Riemann Hypothesis cannot be be adequately understood in Type 1 terms!)

So if we assign + 1 to the positive truth value of a proposition and - 1 to the negative truth value, then to maintain that that the proposition can be either + 1 or - 1 is to state that the proposition is both true and false!
And such ambiguity (or paradox) strikes at the very heart of mathematical type truth.

Yet, in the simple case of the square root of 1, such ambiguity is directly accepted in quantitative terms (from the conventional Type 1 mathematical perspective).

The root of this problem is very simple and relates directly to the fact that when for example we square a number, a qualitative as well as quantitative transformation in the units involved takes place. However in Type 1 terms, the qualitative transformation is simply ignored (and reduced to quantitative interpretation in 1-dimensional terms).

So when we square 1 (i.e. 1^2) the answer is indeed 1 but in square i.e. 2-dimensional units. However once again the qualitative aspect is ignored so that 1^2 is interpreted as 1^1.

Now in inverse terms if we obtain the square root of 1^2, the answer is indeed 1 i.e.1^1.

Therefore when interpreted in Type 2 mathematical terms 1^1 and 1^2 represent two distinct numbers (in qualitative terms).

So + 1 properly represents the square root of 1^2 and - 1 the square root of 1^1.

Therefore we can see quite simply that the ambiguity arising from multi varied solutions relates directly to the lack of a distinct qualitative dimension (in Type 1 terms).

We have not finished yet. So far we have shown how the square root can be interpreted without ambiguity in quantitative terms. However there is a vital qualitative corollary.

Put simply there is an inverse relationship as between quantitative and qualitative interpretations.

So if D represents the dimensional value in quantitative terms, then 1/D represents the corresponding value in a qualitative manner.

So 1^(1/2) = - 1 (in quantitative terms).

Therefore 1^2 = - 1 (in a qualitative manner).

Now - 1 (in quantitative terms) represents a point on the circle of unit radius.

Likewise therefore - 1 (in qualitative terms) represents a point on the circle of unit radius.

In other words - 1 here represents a qualitative means of interpretation (Type 2) that is circular in nature.

What this circular interpretation in turn entails is a dynamic both/and logic (rather than a static linear either/or logic).

So in dynamic terms - 1 here implies the negation of + 1 (i.e. is both + 1 and - 1 simultaneously) So 2-dimensional circular interpretation can be expressed as the dynamic complementarity of (paradoxical) opposites.

What this implies in turn is that a comprehensive (Type 3) interpretation of the square root of 1 requires not only quantitative values applying to dimensional exponents of 1 and 1/2, but equally qualitative interpretations applying to 1 and 2 dimensions respectively.

In other words properly understood in Type 3 terms, we can only understand the square root (in quantitative terms) through equally understanding 2 (as a dimension) in a corresponding qualitative manner.

Therefore to appropriately understand the quantitative value (- 1) on the circle of unit radius, we must equally provide the corresponding qualitative interpretation of - 1 (in a circular manner).

And again the relationship between both is of a direct inverse nature in dimensional terms. So the square root of 1 (i.e. through raising 1 to the dimension 1/2) is - 1. This is inversely related to the the dimension 2 (i.e. through raising 1 to the dimension 2) in qualitative terms which equally is - 1.

Finally we can perhaps now see the root of the confusion that arises from the conventional (Type 1) interpretation.

Maintaining that the square root of 1 can be either + 1 or - 1 arises from the simple failure to recognise the qualitative role of dimension. Because only the default 1st dimension is recognised (in qualitative terms) in Type 1 terms, then when using this approach, paradoxical meaning (pertaining to the 2nd dimension) is misleadingly portrayed in a reduced linear manner.

And this is the root of the whole problem of multi varied solutions in Type 1 Mathematics which cannot be satisfactorily resolved till the qualitative (Type 2) aspect is explicitly recognised.

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