Wednesday, November 19, 2014

Do Numbers Evolve? (1)

On first impression, this might seem to most people as a somewhat ridiculous question.

Indeed the conventional view - which gives great comfort to so many practitioners in our fast changing world - is that number represents the only thing we can rely on to remain absolutely the same. So from this perspective, the prime numbers for example were the same yesterday, today and will forever remain so here and indeed anywhere else in the Universe (where intelligent beings exist to discover them)!

However on closer examination, strictly this conventional view can be convincingly shown to represent but an illusion (which admittedly however in a reduced quantitative sense has proved of enormous benefit).

In physical terms to accurately classify any object we must be able to identify it with a universal class to which it belongs.
To give  a trivial example, to speak unambiguously with respect to a fruit such as a strawberry, we need to be able to define accurately a universal class to which all strawberries belong.

However we will eventually discover that at the margins difficult problems of identification will exist with a certain degree of arbitrariness as to whether a particular example correctly falls into the relevant class. So the boundaries of our definition are necessarily vague and approximate.

Now we might initially think that this problem does not exist in the mathematical world of "abstract" objects that thereby free us from such physical restraints.

However, paradoxically on closer reflection a much greater degree of mystery attaches to the universal class constituting "number" than any physical class (such as strawberries).

So to unambiguously recognise a particular number we should be able to define the universal class of "number". However this is a far more difficult task that one might imagine.

So for example the development of Mathematics has seen a steady increase in the somewhat exotic objects that are universally recognised as numbers.

Initially, number was identified solely with the natural (counting) numbers which are solely positive.. Then gradually, after much resistance their negative counterparts also cam to be included.
A further advance then led to the inclusion of rational fractions (such as 1/2) in the number system. Then the Pythagoreans in investigating the square root of 2 were, to their horror, to discover a new type of irrational number. Such irrational numbers have now been further refined to include both algebraic (such as √2) and transcendental numbers (such as π).

Another major development with respect to the solutions of polynomial equations led to the recognition of imaginary numbers (based on i as the square root of  – 1).

And in more recent times, further developments led to the inclusion of transfinite numbers and a whole new strange class of numbers (based on the primes) referred to as p-adic numbers.

So over the millennia we have seen a remarkable evolution in the objects that are now recognised as legitimately belonging to the number system. Thus it seems to me reasonable to assume that further extension is likely to take place in the future with as yet unknown number objects becoming included.

Thus there is clearly very fuzzy boundaries existing as to what might be considered as number. To put it more bluntly we are unable to properly define what is number and yet attempt to claim an absolute unambiguous identity for every specific number object encountered.

Now though there is no proper (epistemological) justification for such certainty.
So what really characterises - as I hope to presently  demonstrate at length - the apparent absolute nature of mathematical objects such as numbers, is a largely unquestioned mass consensus, which at bottom is geared to the preservation of a considerable illusion regarding their true nature and indeed the true nature of Mathematics generally.

Let me illustrate this now with respect to one of the simplest, most important and best known numbers i.e. "2".

Now again according to the general mathematical consensus, 2 has an absolute rigid identity that can be successfully abstracted from our changing everyday physical world.

This view, expressed in its extreme fashion for example by G. H. Hardy, looks on numbers as eternally existing in some kind of mathematical Heaven (ungoverned by the laws of space and time).

However on closer reflection this view can be shown to be quite untenable.

The starting point here for more authentic understanding is the recognition that Mathematics is intimately bound up with experience. So therefore we start by examining how the recognition of number experientially unfolds.

Now all experience - including of course mathematical - is governed by twin sets of fundamental polarities that dynamically interact.

The first of these relates to external (objective) and internal (mental subjective) polarities.

Therefore the experience of the number "2" entails both an external pole (i.e. as object) and a corresponding internal pole (as mental perception).

So the experience of the number "2" entails a dynamic interaction of both object and perception (which cannot be meaningfully abstracted in absolute manner from each other).

Put another way, strictly speaking a mathematical object such as "2" has no meaning independent of the corresponding mental perception of "2" with both poles in tandem properly constituting an interactive dialogue of number meaning.

In other words all number understanding has a merely relative validity.

So Conventional mathematics is in fact directly based on a reduced interpretation of such experience. Here the two poles are viewed with respect to their absolute separation (though implicitly in experiential terms this is not possible). Thus the number object (in this case"2") is misleadingly given an abstract absolute objective identity.
Interpretation is then misleadingly viewed as simply mirroring in mental terms (again in an absolute manner) this absolute identity.

In this way in conventional mathematical terms, interaction as between opposite poles (external and internal) is thereby completely edited out of the picture (in explicit terms).

This then is misleadingly associated with the considerable illusion that numbers thereby enjoy an absolute rigid identity (unrelated to time).

However because such reductionism is so entrenched in our mathematical thought processes (conditioned now though several millennia) it is extremely difficult to get mathematicians to address this issue.

Even on the rare occasions when I have seen mathematicians seriously question the  basis of such procedures (in philosophical reflection on their discipline), they still seemed in a sense to operate with split personalities, readily accepting all such reductionism (without question) when operating as mathematicians.
And I accept that there is enormous pressure on professional mathematicians (in maintaining the respect of their peers) to operate precisely in this manner.

This is why I have long considered that paradoxically the blunt message that Mathematics (as presently understood) is not in fact fit for purpose can only be properly preached by someone standing outside the profession altogether (while still remaining deeply interested in Mathematics).

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