Tuesday, July 2, 2013

The Central Issue

I keep returning to this central point.

All experience – including of course mathematical – is conditioned by key fundamental polarities which dynamically interact with each other in a relative manner.

So for example what is understood as objective (and external) has no meaning in the absence of what is - relatively - subjective (and internal). This entails for example that the objective notion of number has no strict meaning in the absence of the corresponding mental construct of number (which – relatively – is subjective in nature).

Likewise what is understood as quantitative (as parts) has no meaning in the absence of what is - relatively - qualitative (as whole).

The important implication here is that the quantitative notion of number (as independent) has no strict meaning in the absence of an overall context for relating numbers (as interdependent with each other) which - relatively - is of a qualitative whole nature.

So properly understood, all numbers - and indeed by extension all mathematical notions - represent dynamic interaction patterns with a merely relative meaning (depending on context).

Now the reason why this is not readily appreciated in our present culture is due to a gross form of reductionism (which effectively remains totally unquestioned).

So remarkably what is by far the most fundamental issue for the proper appreciation of its symbols is not even recognised as a legitimate issue for concern within Conventional Mathematics.

The philosopher Herbert Marcuse wrote an influential book in the 1960’s entitled “One Dimensional Man” and quite literally Conventional Mathematics is defined by its linear i.e. one-dimensional paradigm.

What this entails in effect is a gross form of reductionism where dynamic interaction as between the key polarities is frozen in an absolute manner.
So the dynamic interaction as between external and internal polarities is reduced with numbers for example misleadingly given an absolute independent existence in objective terms. Of course if pressed on the matter a professional mathematician may well concede that numbers cannot be understood in the absence of mental constructs. However it will somehow be then assumed that these mental constructs absolutely correspond with their objective identities.

So just as with an absolute number we ignore the sign (+ or –), with absolute type understanding we likewise ignore the sign with external and internal directions (which are relatively + and –) assumed to directly correspond with each other.

This leads for example in Conventional Mathematics to an utterly mistaken notion of numbers as abstract unchanging identities in objective terms (independent of our relationship with them).

In truth our understanding of numbers has no strict meaning in the absence of the corresponding means through which they are interpreted.

As we have seen the interpretation of number as independent in absolute terms simply corresponds to interpretation that is 1-dimensional in nature, where just one pole (such as objective) is used as an absolute reference frame.

Thus the huge failing within Mathematics is the recognition that potentially associated with every number (as dimension) is a unique interpretation of mathematical symbols (with a relative partial validity).

So - rather than one reduced absolute interpretation - properly understood, an unlimited number of possible alternative interpretations exist for all mathematical symbols (with each possessing a partial relative validity). Again such reduced interpretation corresponds with 1-dimensional understanding (in qualitative terms). However associated with each other number (as dimension) is a unique dynamic configuration of mathematical symbols (with a merely relative meaning depending on context). 

Thus - what we recognise as - Mathematics represents just one possible interpretation (i.e. 1-dimensional). Unfortunately, because of its reductionist absolute nature, it therefore blinds us to the reality of an unlimited set of alternative interpretations (that apply within a dynamic interactive context of appreciation). 

The confusion within Conventional Mathematics is possibly even more profound with respect to the second key polarity set i.e. quantitative and qualitative.

We are so accustomed now to viewing numbers in merely quantitative terms that we have lost any real notion of what the - equally important - qualitative aspect entails. 

Once again in conventional mathematical terms numbers are viewed in quantitative terms as absolutely independent. However if numbers are absolutely independent this begs the huge question of how they can then be placed in a relational context whereby they can be successfully related with other numbers (as interdependent).

So again we can only proceed in this manner i.e. to relate numbers to other numbers through a process of gross reductionism.

If we look at the natural numbers for example, the quantitative aspect of number can then be easily identified with their cardinal definition.

So the number 3 for example is effectively treated as a whole indivisible unit (in quantitative terms). So if we attempt to break this into part units the whole will be treated as the sum of individual homogenous units (without qualitative distinction) as i.e. 1 + 1 + 1. 

However if we are to relate the 3 members of a group this implies making qualitative distinctions in ordinal terms whereby we can unambiguously identify a 1st, 2nd and 3rd member.  

However such qualitative distinctions are not implied by the cardinal definition of number. In other words the treatment of ordinal number notions in Conventional Mathematics is based on this massive reductionism whereby the qualitative aspect is misleadingly assumed to directly correspond to the quantitative.

So when mathematicians for example attempt to probe the relationship as between the primes and the natural numbers they reduce this to a merely cardinal notion of number (as quantitative).

However properly understood the qualitative aspect (as ordinal) is of a distinctive nature. For example momentary reflection on the meaning of 2 in ordinal terms (i.e. 2nd) leads quickly to the view that it is of a merely relative nature. So 2nd  in the context of 2 is distinct from 2nd in the context of 3 and so on.

And if the ordinal notion of number is of a relative nature, then the cardinal notion must likewise be relative.

So, cardinal notions of number (as quantitative) are independent in a merely relative sense; likewise ordinal notions of number (as qualitative) are interdependent in a merely relative sense. Put another way both the quantitative and qualitative aspects of number necessarily interact with each other in dynamic terms.

Thus we must define two distinct aspects to the number system (which I define Type 1 and Type 2 respectively).

Type 1 represents the quantitative aspect (with now, a merely relative definition).

Here all numbers (as representing base quantities) are defined with respect to the default dimensional number 1 (representing a qualitative context). So to give number quantities a meaning (as relatively independent), we must initially define them with respect to a fixed dimensional (qualitative) context as 1. So number quantities are therefore typically represented with respect to a number line.

So in the expression a1, if a (representing the base number) is quantitative then 1 (representing the dimensional number) is of a qualitative nature.

Now as we know in Conventional Mathematics numerical notions are expressed with respect merely to their reduced quantitative values.

Thus if we take for example the numerical expression 32, its value is given in a merely reduced quantitative manner as 9 (i.e. 91).

In fact in conventional terms the default number 1 (representing the dimensional value) is left out altogether furthering the illusion that numbers can be successfully interpreted in a merely quantitative manner.

Pure addition is related directly to this Type 1 system

So 1 + 1 + 1 = 3 i.e. 11 +11 +11  = 31.

Thus in the Type 1 system numbers are given a cardinal whole identity in quantitative terms whereby individual part units thereby lack a qualitative (ordinal) distinction.

In the Type 2 system, we have the reverse situation. Here the base quantity is fixed as the default unit (1) in quantitative terms, while the number representing the dimension can vary.

Pure multiplication is related directly to this Type 2 system.

So 1 * 1 * 1 = 3, i.e. 11 *11 *11  = 13

Therefore with the Type 2  system, the quantitative aspect remains unchanged while the qualitative context to which it is related can change.

So for example, if I mark out 1 inch on ruler this represent 1 expressed in a linear (1-dimensional) context = 11.

Then when I now look at a cube of side 1 inch, the volume (in quantitative terms) remains unchanged as 1. However the dimensional context in which the 1 is expressed has now clearly changed to 3. So this strictly represents a qualitative rather than quantitative transformation in units.

13  = 1(1 + 1 + 1) .

So what represents pure multiplication (from the Type 1 perspective) represents pure addition (from the Type 2 perspective).   

Whereas in Type 1 terms 3 has a cardinal whole identity (in quantitative terms), in Type 2 terms 3 is now expressed in terms of its individual members (in qualitative terms).
So 3 now is defined in terms of distinctive 1st, 2nd and 3rd members in an ordinal manner (which correspond to 1st, 2nd and 3rd dimensions).

In Type 3 terms number is defined in terms of both base and dimensional values (that can differ from 1).

So for example in the expression 32, base and dimensional numbers differ from 1. 

Therefore, correctly interpreted this numerical expression entails both quantitative and qualitative aspects of numerical transformation. So if we start with 3 as quantitatively interpreted then 2 (as dimensional) number will be - relatively - of a qualitative nature.

However these meanings actually switch in experience; so 3 (as base number) has a qualitative while 2 (as dimensional number) has likewise a quantitative aspect.

So a comprehensive Type 3 understanding of number allows for the interchange as between quantitative and qualitative aspects (with respect to both base and dimensional numbers).  

Now remarkably we must implicitly do all this unconsciously in experience. In other words to recognise a specific number in quantitative terms we must implicitly relate it to other numbers (in a qualitative relational manner).
Then to recognise the more general aspect of number in abstract terms (as in algebraic understanding) we must be able again to provide a general relational context (of a qualitative nature).

Conventional Mathematics therefore represents a significant misrepresentation of the actual dynamics of experience where the quantitative/qualitative interaction entailing both rational (conscious) and intuitive (unconscious) aspects is reduced in a merely rational quantitative manner.
When one appreciates the nature of these two necessary aspects of the number system (with respect to quantitative and qualitative aspects) it utterly transforms one’s understanding of the relationship between the prime and natural numbers.

Thus from the Type 1 perspective, each cardinal natural number is uniquely expressed as the combination of prime number factors (in quantitative terms). So from this perspective the primes are seen as the cardinal building blocks of the natural numbers (in quantitative terms).
However from the Type 2 perspective this picture is completely inverted with now each prime number uniquely expressed as the combination of ordinal natural numbers (in qualitative terms).

So again from this perspective the prime number 3 for example is expressed uniquely by its 1st, 2nd and 3rd members. So from this perspective the natural numbers are seen as the ordinal building blocks of the prime numbers (in qualitative terms). 

Then when one simultaneously brings together both Type 1 and Type 2 (in Type 3 understanding) the primes and natural numbers are seen to mutually generate each other in a manner that is ultimately ineffable.
When one begins to look at number in the right manner (i.e. as dynamically interactive) then it becomes quickly apparent that not alone is the Riemann Hypothesis incapable of proof (or disproof)in conventional terms, but much more importantly it cannot be properly understood in this manner.

The crucial lesson to learn is that our understanding of number - and indeed by extension all mathematics - remains deeply flawed. We have elevated one limiting special case (i.e. where qualitative is reduced to quantitative meaning) as representing all of valid Mathematics. And by its very nature this special case has blinded us to seeing the incomparably richer territory that properly should represent Mathematics. 

At present mathematicians misinterpret the significance of the Riemann Hypothesis in believing that its eventual proof will have major consequences for Mathematics (as presently interpreted).

The truth is that the fact that the Riemann Hypothesis cannot be proven, or even properly understood, in conventional mathematical terms has altogether more fundamental consequences in requiring a radical new appreciation of the proper nature of Mathematics.

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