## Thursday, May 5, 2011

### 2-Dimensional Proof

I have already pointed to the remarkable fact that the square root of 1 yields - according to conventional interpretation - two equally valid answers i.e. either + 1 and - 1 that are diametrically opposite to each other.

The (unrecognised) qualitative corollary of this is that mathematical proof at the (inverse) 2-dimensional level of understanding yields a totally paradoxical appreciation whereby a proposition that is both positive (+) and thereby true is equally negative (-) and thereby false.

Put another way a merely relative - rather than absolute - truth value applies at this level of understanding.

To see more clearly what is involved here, we must remember that in dynamic experiential terms, all mathematical interpretation involves both external (objective) and internal (subjective) poles. Whereas in linear (1-dimensional) terms these are clearly separated in corresponding circular (2-dimensional) terms these are complementary and ultimately identical.

So in conventional mathematical terms a mathematical proof may be viewed in two distinct ways:

(i) as what is considered true in an unambiguous objective sense (that is not altered through psychological interaction for its confirmation). In this sense for example one would maintain that the Pythagorean Theorem is objectively true (irrespective of the nature of one's psychological appreciation);

(ii) as what is considered true in an unambiguous internal mental sense (that is not altered through interaction with the objective referents of such mental constructs). So from this alternative perspective the truth of the proof is identified directly with the (internal) mental interpretation involved.

Now it might be readily admitted by mathematicians that external and internal aspects are necessarily positive (+) and negative (-) with respect to each other.

However in conventional terms this distinction is ignored resulting - literally - in a qualitatively absolute interpretation.

In other words the crucial assumption is made that a direct correspondence necessarily exists as between the (internal) mental constructs used in interpretation and the external reality (to which they relate).

However once we accept that these opposite poles in experience (external and internal) necessarily interact then a different form of interpretation results.

In other words from this new dynamic perspective external and internal do not enjoy an (absolutely) independent existence but only obtain meaning through relationship with each other.

Therefore if we attempt to identify this interaction (of external and internal) with either pole as independent then this excludes the equal truth of its opposite pole.

So in dynamic interactive terms if we maintain the proof of a mathematical proposition in a merely (external) objective sense, this thereby excludes the equal validity of its (internal) mental aspect; in corresponding fashion if we then maintain the the proof in a merely (internal) mental fashion this thereby excludes the equal validity of its (external) objective aspect.

Therefore from a balanced interactive perspective, a proof of any proposition must necessarily be both true (+) and not true (-).

In fact this merely exemplifies in the context of experience the simple example of road directions that I have so frequently given.

When we use an independent frame of reference (e.g. where movement is either up or down a road) a turn off the road will have an unambiguous meaning (as either left or right). However when we then attempt to combine both frames simultaneously (as interdependent), then a turn has a merely paradoxical meaning (as both left and right).

So in the context of mathematical interpretation, conventional (1-dimensional) appreciation is based on independent reference frames (where external and internal aspects are considered as separate).

However where - in qualitative terms - a circular 2-dimensional approach is employed interdependent reference frames are required (where external and internal are now considered as complementary).

And just as a turn off a road has a merely paradoxical interpretation in this context, likewise the truth of any mathematical proposition likewise has a merely paradoxical interpretation.

Therefore from a 2-dimensional qualitative perspective all mathematical interpretation (which of course includes all mathematical proof) is of a paradoxical nature.

What this implies is that meaning at this level is understood in a dynamic interactive manner that entails continual transformation (in both the external and internal aspects) of experience.

Put another, way truth is now of a merely relative nature or to use an analogy from quantum physics, all mathematical truth is now subject to the uncertainty principle.

So the true meaning of a dimension (which is inversely related to its corresponding root number structure) implies direction.

With linear (1-dimensional) interpretation only one direction is admitted leading to unambiguous interpretation. So crucially for example from this perspective a proposition is either true or false. In other words in has just one truth value (which is interpreted in an absolute fashion). Also in physical terms, this is identified with the conventional assumption that movement in both time and space takes place in a forward (positive) direction.

With circular - which in its simples manifestation implies 2-dimensional - interpretation, two directions are always involves (that are polar opposites of each other). So from this perspective for example a mathematical proposition is always both true and false (when communicated in reduced linear terms). Likewise in physical terms movements in space and time now necessarily have both positive (forward) and negative (backward) directions.

Of course an unlimited number of such "higher" dimensional interpretations potentially exist.

For example at the extremely important 4-dimensional level, all mathematical propositions would be given four directions.

So from this perspective a proposition would have a positive and negative truth value in both real and imaginary terms. The imaginary in this qualitative context arises from the true relationship as between wholes and parts (which in dynamic interactive terms are "real" and "imaginary" with respect to each other). In psychological terms this relates to the distinction as between perceptions and concepts. Now when one reflects on it one cannot deal with any perception (such as specific number) without implying the corresponding concept of number. However crucially in qualitative terms the relationship of "perception" to "concept" is as of "real" to "imaginary".

And at the 4-dimensional level of interpretation, understanding is now qualitatively so refined that this distinction can be made.

Though potentially as I have stated an unlimited number of qualitative dimensional inetrpretations exist, the most important would be confined to the earlier numbers (esp. 1 - 8).

In general terms it is much easier here to explain the even numbered dimensions (especially those that are powers of 2 such as 2, 4 and 8).

In turn this all has a fascinating connection with the Riemann Zeta Function.

So we will now turn to consideration of the odd integer values for s (both positive and negative) with respect to this function.