Thursday, March 15, 2012

Number Inconsistency (1)

I had already accepted that Mathematics (in its current form) is not fit for purpose some 40 years ago while at University.

At that time - and for many years later - I would have couched my arguments in a somewhat philosophical manner, regarding the nature of finite and infinite and how Conventional Mathematics necessarily entailed a reduction - in any context - of infinite with finite - or alternatively from the other perspective - finite with infinite notions.

The first form is evident in the manner in which series are viewed, where in effect the infinite notion is treated as a linear extension of what is finite. This for example is the case with convergent and divergent series. So in the first when a finite series converges on a limiting value, the finite notion is then extended in linear terms to the infinite. Alternatively when it diverges away from a limiting value, again this notion, derived from finite experience, is extended in linear terms to the infinite.

However though seems to accord with common sense intuition, a fundamental inconsistency is thereby involved.

Likewise we can have the opposite case, where what is true for the infinite (in the reduced sense that it is used in Mathematics) is then considered true for the finite case. So, whenever a theorem has been proven for the general case (i.e. as applying to "all") is then assumed to apply in a finite context to specific examples. So for example the Pythagorean Theorem (that in a right angled triangle the square on the hypotenuse is equal to the sum of squares on the other two sides) has been proven as true in "all" cases (as infinite). It is then assumed to apply in any individual case (which is finite). However again a subtle - but fundamental - inconsistency is involved.

Now practicing mathematicians might be inclined to dismiss all this as mere philosophical supposition (thereby remaining fully committed to the status quo).

However recently, I have come to realise that the same basic argument can be expressed in a manner that mathematicians will find much harder to refute, relating to the distinction as between the cardinal and ordinal use of number. And it is the very clarification of these issues that played a large role in my own development in leading me on to discover the true relationship of prime to natural (and natural to prime) numbers.

This basically involves interpreting number consistently both with respect to cardinal and ordinal usage. And when this is done, the wonderful mystery of how the prime numbers simply are a reflection of the natural (and the natural simply in turn a reflection of the primes) is then revealed in all its glorious splendour!

We can indeed express the basic argument very simply!

With respect to Conventional Mathematics, number cannot be consistently interpreted in both a cardinal and ordinal fashion! Indeed this is the root of the uncertainty principle I have mentioned on several occasions recently, and lies at the centre of all mathematical procedures.

Now the fundamental philosophical reason behind this is that cardinal and ordinal refer - in relative terms - to the quantitative and qualitative aspects of number respectively (which in turn reflects the distinction as between finite and infinite notions).

The corollary of this is that a mathematical approach that does not in fact recognise a clear qualitative distinction (with respect to finite and infinite notions) likewise cannot deal satisfactorily with the distinction as between the cardinal and ordinal use of number. So these two notions of number - though fundamentally distinct - are continually confused with each other in conventional mathematical interpretation.

When we use the cardinal notion of number in the conventional mathematical fashion, it implies an absolute whole existence.

So in this sense the number 5 does not contain unique parts. So if we were to crack open the the number 5 as it were - say - with reference to a collection of 5 objects - we could represent each object without qualitative distinction as 1 (in a strictly quantitative manner).

Thus 5 = 1 + 1 + 1 + 1 + 1. So here we have the very hallmark of the linear approach whereby whole numbers are treated as the collection of unitary parts (without any qualitative distinction).

Indeed it is this very approach that is the basis of the string theory approach in physics where the fundamental "objects" of nature are viewed in terms of 1-dimensional strings (without qualitative distinction).

It is this cardinal mindset therefore that enables conventional mathematicians to view prime numbers as the "basic building blocks" of the number system. So the primes are treated somewhat as individual atoms without further substructure. Then the natural numbers arise through a unique product of such constituent prime factors.

However the fundamental problem with viewing numbers in a merely cardinal sense is that we thereby treat them in an absolute independent manner.

The very capacity to combine numbers with each other (as in the derivation of natural from prime number factors) is that it presupposes a relational capacity (which is of a qualitative nature).

When we allow for this qualitative relational aspect of numbers, then we necessarily must interpret them in an ordinal - rather than strict - cardinal sense.

Thus such a relational aspect thereby implies the ordinal use of number! And on investigation this proves to be quite problematic (with no fixed identity).

For example when we use 2 in an ordinal sense to imply second, its meaning depends on the context (which can vary endlessly). So with 2 items the second item may seem unambiguous (as the second of 2). However if 2 is used as a ranking with respect to 3 items it now changes (as the second of 3). And of course, as we can rank 2 in an unlimited manner, the relational meaning of 2 is likewise unlimited.

So we can see here that the ordinal notion of a number is of a qualitative relational nature with its precise interpretation depending on context.

Now once we accept this relational ordinal notion as in this case with respect to 2), it poses a fundamental problem for the absolute notion of number as cardinal.

Quite simply if we define cardinal numbers in an absolute independent sense, then it is strictly impossible to subsequently relate them to other numbers!

So coming back to the conventional view of primes as building blocks. If we define the primes initially in cardinal terms, then the unique relational capacity as the product of prime numbers, cannot come from their cardinal identity! In other words we cannot therefore explain the derivation of natural numbers (from prime factors) in a strictly quantitative manner!

And then there is the further problem that the very ability to meaningfully refer to the primes already implies an ordinal number ranking that is natural!

So when we refer to 7 for example as the 4th prime, the very notion of a (composite) natural number is already inherent in the prime, even though we had sought earlier to explain prime numbers as the "building blocks" of the natural numbers.

Thus when we subject it to close scrutiny, the conventional treatment of number is full of confusion where no clear distinction is made as between its respective cardinal and ordinal use.

And again, the key problem is the reductionist attempt to define number merely with respect to its quantitative aspects.

We can see this dilemma in an even better light when we look at number in a circular context with respect to the Type 2 system (where the qualitative nature of number is explicitly recognised).

On a personal level, I struggled with this issue for many years before reaching a satisfactory resolution. For example in referring to 2 as dimension, I was conscious of the fact that I kept using it in two different ways (without fully reconciling thr relationship between them).

So for example the 2nd dimension of 1, in this Type 2 context (as 1^2) implied in complementary quantitative fashion the 2nd root of 1 (as 1^1/2) = - 1.

However I also frequently used 2 in a cardinal sense which then implied the 2 roots of 1 (i.e. 1^1 and 1^2) respectively.

However I was aware of an inherent problem here for in the context - say - of the 3 roots of 1, we would also have a 2nd root (as the 2nd of 3). However this would have a different meaning than the 2nd root (in the context of 2 roots). And therefore likewise in complementary manner the notion of 2 as 2nd would have a distinctive meaning in the context of 3 dimensions!

And then in clarifying this distinction, I realised that it inevitably meant that the cardinal notion of number would likewise have a merely relative meaning in this context.

So for example when I refer to the 2 roots of unity, these roots ( + 1 and - 1) have a cardinal meaning (with the second of these i.e. 2 as ordinal in the sense of second of 2) corresponding with - 1 (in cardinal terms).

However if I now refer to the 3 roots of unity, these roots + 1, {-1 + 3^(1/2)i}/2 and {-1 - 3^(1/2)i}/2 have a cardinal meaning. But now the second of these (i.e. 2 as 2nd of 3) is associated with a different cardinal number expression (as the 2nd corresponding root).

So strictly speaking once we properly unravel the nature of cardinal and ordinal interpretation, both can change identity! In other words, correctly understood, the very nature of number, in this interactive context (of cardinal and ordinal meaning) is of a merely relative nature.

So from this context the very approach in Conventional Mathematics (in explicitly recognising solely the quantitative aspect) appears as simple untenable. In other words number has both quantitative and qualitative aspects (which dynamically interact in understanding).

So we originally - in a linear context - cracked open the prime number 5 to find it made up of homogeneous units i.e. devoid of qualitative significance.

However now from the complementary circular perspective, when we crack open 5 (as representing both the qualitative dimensional meaning of 5 and its corresponding 5 roots of 1 in quantitative terms) we find that each of these constituent ordinal units (as the 1st, 2nd, 3rd, 4th and 5th roots respectively) is uniquely distinct.

Furthermore we can see here clearly the manner in which the natural numbers (as ordinal) are contained in the primes.

So of course the upshot of all this is that both the primes and the natural numbers have both cardinal and ordinal interpretations which ultimately are purely interchangeable. And it is in this recognition that the mystery of the primes (which is inseparable from the mystery of the natural numbers) is ultimately resolved.

And on reflection the implications of this are even graver for Conventional Mathematics than I had once imagined.

I have recently come to the realisation that even with respect to the Type 1 aspect of Mathematics there are two distinct approaches.

The Conventional approach is explicitly based on absolute quantitative type interpretation which strictly is quite untenable in giving a satisfactory interpretation of the nature of number.

The alternative Type 1 approach - which is what I would advocate - while concentrating mainly on the quantitative properties of numbers, crucially interprets them in a merely relative manner (thus implicitly recognising their equal qualitative nature).

And then when later in the Type 3 approach the incorporation of Type 1 and Type 2 is required, the Type 1 aspect can then be readily adapted to this task.

No comments:

Post a Comment