Thursday, March 22, 2012

Number Inconsistency (4)

We have arrived at the point where even the cardinal notion of number - in the context of a (potentially) infinite series - can have a purely relative meaning.

And what is fascinating about this situation is that it cannot be properly explained in the absence of the complementary qualitative (ordinal) notion of number meaning.

As we have seen, within Conventional Mathematics, the qualitative (ordinal) notion of number is reduced in quantitative terms.

This is likewise associated as we have seen with the treatment of infinite series - in effect - as an extension of linear finite notions.

Now if we have a series of positive terms, such linear extension with respect to the finite, seemingly leads to an unambiguous result (in infinite terms).

So from this perspective the sum of an an infinite series series will appear to either converge or diverge in an unambiguous manner.

So for example for in the case of the well-known geometric series 1 + 1/2 + 1/4 + 1/8 + ....., this seemingly converges to the value of 2. So as the sum consistently approximates ever closer to 2 (over a finite range), then by the logic of linear extension, if we were to take a sufficient (i.e. infinite) number of terms the answer would be 2.

Thus from this perspective (where all the terms are of the same sign) an unambiguous answer (2) results for the sum of the infinite series.

Now in the case of the harmonic series

1 + 1/2 + 1/3 + 1/4 + 1/5 +....,

though initially it may not appear obvious, it is easy enough to demonstrate that this series will diverge.

Therefore as we keep increasing the number of terms, the sum shows no sign of approaching a limiting value. Therefore once again by the reductionist process of linear extension (of what is true for the finite) we conclude unambiguously that the sum of terms of the harmonic series diverges to infinity.

However, the deeper reason why these seemingly unambiguous results arise, is of a qualitative nature.

The very definition of 1-dimensional in qualitative terms (which is complementary with its reciprocal as 1st root of 1) is that both coincide as + 1. So there is an identity of qualitative dimension (and quantitative reciprocal) with respect to the Type 2 number system as 1^1. Thus, when all terms of a series are defined unambiguously (with respect merely to the positive sign) both finite and infinite interpretation - which properly are of a quantitative and qualitative nature respectively - can seemingly be successfully reduced in terms of each other.

So significantly within Conventional Mathematics, when we use the sign for addition (+), it is given a merely quantitative interpretation!

However if we now look at the alternating version of the harmonic series we get,

1 - 1/2 + 1/3 - 1/4 + 1/5 -......

What is significant now is that we are using both positive (+) and negative (-) signs.

Now once again these are given a merely quantitative interpretation in Conventional Mathematics (defined as it is in 1-dimensional terms).

However from an appropriate 2-dimensional perspective, whereas the 1st dimension again provides the same quantitative interpretation, the 2nd dimension now provides the corresponding qualitative (ordinal) interpretation.

So, + in this context means positing (of finite meaning); - however implies the negation of such finite meaning in what is qualitatively infinite. Now in psychological terms, the finite aspect will be directly associated with (conscious)reason (using linear logic); the infinite aspect will however be associated with (unconscious) intuition which then indirectly can be expressed in a rational fashion (as circular logic).

Much as conventional mathematicians may wish to avoid this issue, the actual behaviour of the infinite alternating series now incorporates both 1-dimensional (quantitative) and 2-dimensional (qualitative) aspects, with the resulting sum of terms having a merely relative value, that crucially depends on the qualitative i.e. ordinal ranking of terms.

Now if we order the terms in a systematic way so that we add up sequentially, i.e. 1st, followed by 2nd, 3rd, 4th term and so on, the sum of the series will indeed appear to converge to a definite value i.e. the natural log of 2 (.693417..).

However as it is an infinite series, an unlimited number of other ordered arrangements are possible.

For example we could proceed by taking the first positive term and then subtracting the the first two negative terms, then adding the next positive term before again subtracting the next two negative terms and so on.

So here we would have

1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - 1/10 - 1/12 +....

= (1 - 1/2) - 1/4 + (1/3 - 1/6) - 1/8 + (1/5 - 1/10) - 1/12 +....

= 1/2 - 1/4 + 1/6 - 1/8 + 1/10 - 1/12 +...

(1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 +...)/2 = (log 2)/2

So the sum of the series again converges when uniquely ordered in this manner to another finite value (that is exactly half the first).

And there are an unlimited number of other possible arrangements with in some cases the series converging to a distinct finite value and in other cases diverging to infinity!

Now it is important to observe that when we sum up this alternating harmonic series over a finite range, an unambiguous result emerges (approximating to log 2). Here the precise ordering of terms has no impact on the eventual result (which is the same in all cases).

However this clearly is not the case for the series now continued over a (potentially) infinite range.

What this implies therefore is that the very process of linear type extension (i.e. where the infinite is treated as a quantitative extension of finite notions) is inapplicable in this case.

And if its behaviour cannot be explained through Conventional Mathematics (in 1-dimensional terms), then this clearly implies that a deeper more appropriate level of interpretation is required.

Furthermore, we have showed that its behaviour can be properly explained from a 2-dimensional perspective (combining both quantitative and qualitative interpretations of mathematical symbols).

So what I am demonstrating here is that even in the apparent context of merely quantitative type meaning (i.e. with respect to the summing of terms of an infinite series) that such behaviour cannot be properly explained in the absence of corresponding qualitative type interpretation of mathematical symbols.

So when I say that Conventional Mathematics is not fit for purpose, I mean precisely what I say.

Nov alone does it totally fail to deal with the the amazingly rich (but unrecognised) world of the qualitative meaning of mathematical symbols; it cannot explain properly the nature of its own recognised domain of the quantitative.

And the underlying reason for this is that ultimately both the quantitative and qualitative aspects of mathematical understanding - with both equally important - are inseparable.

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