Wednesday, October 27, 2010

More on Finite and Infinite

I have already stated that the conventional interpretation of the infinite leads to a reduced notion whereby it is viewed as an extension of the finite.

For example we still routinely refer to series as infinite e.g. the natural number series. Here the mistaken impression is given that with the terms getting progressively larger that "in the limit" they become infinite.

However strictly this is just nonsense. As terms get larger they always remain finite and at no stage pass over to a so-called infinite state.
It would be valid to say that we cannot set a limit to the size of possible terms in the series (while still accepting that they always remain finite). But this is quite different from saying that they become infinite!

A similar problem results in the treatment of "infinitesmals" which again is a meaningless notion.
As a quantity becomes progressively smaller, it still remains finite and at no stage becomes infinitesmal.
Though it may be quite appropriate to approximate such quantities as 0 when obtaining the numerical value of a calculation, an important qualitative distinction remains in that such calculations are always of a merely relative nature.

Put yet another way, the distinction as between infinite and finite concepts relates to the corresponding distinction as between potential and actual notions.

For example the number concept is of an infinite nature potentially relating to any number (not actually specified). However a specific number perception e.g. 2 is then actualised in a finite manner. Thus the dynamic interaction that necessarily exists in experience as between the number concept and related perceptions inevitably combines both finite and infinite notions.

Thus to avoid confusing the infinite with the finite we need to recognise that - again in dynamic terms - the finite determination of any number always implies that other finite numbers thereby remain indeterminate.

Thus what is loosely - and inaccurately - referred to as an infinite series relates to a situation where the progressive determination of certain terms implies that other terms remain actually indeterminate!

Therefore when correctly understood in dynamic experiential terms, the uncertainty principle applies to all mathematical interpretation.
Once again this relates to the fact that (discrete) rational and (continuous) intuitive aspects are always entailed with respect to understanding. Thus we can only make greater clarity with respect to one aspect through accepting corresponding fuzziness with respect to the other.

And as I have been demonstrating the apparent absolute nature of Conventional Mathematics stems from the - mistaken - view that mathematical interpretation is merely of a rational nature!

Once again when we properly accept the true interactive nature of mathematical possibility of "proof" in an absolute sense disappears.

The "proof" of a general hypothesis strictly relates to its merely potential nature as applying to all (non-specified) cases which is of an infinite continuous nature. However the application of the "proof" to any specific case entails actualisation in a finite discrete manner. So to maintain that the general "proof" logically applies to the specific case entails a basic confusion (whereby the infinite is reduced to finite notions).
This is not to suggest that mathematical "proof" is thereby of no value. Rather it is to suggest that it is subject to the uncertainty principle and thereby of a merely probable nature. Indeed momentary reflection on the issue will reveal that the very acceptance of mathematical "proof" entails a certain form of social consensus that can later transpire to have been mistaken. For example Andrew Wiles first proof of "Fermat's Last Theorem" was found to be in error after it had already passed through a rigorous process of verification. So the present status of the revised proof is strictly of a probable nature. In other words as time goes by with no further errors being found we can accept its probable truth with an ever greater degree of confidence!

So in dynamic experiential terms (which represents the true nature of mathematical understanding) all "proof" is subject to the uncertainty principle.

However even within the reduced assumptions of Conventional Mathematics there are certain problems that in principle can be shown to have no proof (or disproof).
And chief among these problems is the Riemann Hypothesis.

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