Wednesday, November 26, 2014

Do Numbers Evolve? (4)

I have referred repeatedly to the dynamic interaction as between quantitative and qualitative aspects with respect to number.

Ultimately this interaction relates to the interplay of both the finite (actual) and infinite (potential) notions, which in psychological terms relate to both conscious and unconscious aspects of understanding respectively.

So mathematical objects such as numbers possess an actual existence from a finite (conscious) perspective directly mediated in rational terms; however equally they possess a potential existence from an infinite (unconscious) perspective that is directly mediated in an intuitive manner. And both of these ceaselessly interact dynamically in experience leading to continual transformation with respect to such objects.

So properly, i.e. in a dynamic interactive manner, number thereby necessarily evolves. And this relates not just to the nature of (internal) psychological understanding, but also to the external objects (both of which - by definition - are now necessarily relative to each other).

However as an alternative to the sole use of  quantitative and qualitative terms, I would suggest the corresponding pairing of analytic and holistic (which perhaps appears a little more scientific).

However it is important to point out that I am using analytic in the broader sense in which the terms is commonly used in science, which equates directly with a reduced quantitative interpretation of relationships!
Now analytic has also a well-defined narrower meaning within Mathematics in relation to the treatment of infinite series and limits. However suffice it to say that within Mathematics, more restricted use of the terms "analytic" (and "analysis") are also analytic in the broader sense of the term (in that they are defined solely within a reduced quantitative context).

Therefore to return to my basic position, properly understood all number has both analytic and holistic aspects (in dynamic relationship with each other).

From one important perspective, this is true internally for each number. So, as we have seen the number "2" for example entails both the analytic aspect of "2" as a specific number quantity in cardinal terms, and the holistic aspect of "2" (i.e. twoness) as collectively applying to all possible instances of "2").
So properly understood these two notions are actual and potential with respect to each other.

And because in the dynamics of experience (like approaching a crossroads from opposite directions) polar reference frames continually switch) there is also an important sense, where "2" now refers to a specific number quality (i.e. in the ordinal notion of 2nd) while "2" now attains a collective meaning in the cardinal notion of dimension that now actually applies to all numbers.

Thus in the dynamics of the experience of each number, there is a ceaseless two-way interplay of both analytic and holistic type understanding, through which we are enabled to switch seamlessly as between cardinal and ordinal type appreciation (with respect to both objects and dimensions).

Then from the other important perspective, similar dynamics apply to the number system as a whole.
This then enables us to consistently combine both the cardinal and ordinal identities of all numbers (not is relative isolation) but in full relationship with other numbers.

Now the precondition for such consistency is that a seamless means exists for switching as between both the Type 1 and Type 2 aspects of the number system.

Thus from one perspective we need to be able to seamlessly convert the Type 2 aspect in a Type 1 manner.
Then equally from the alternative perspective we need to be able to seamlessly convert the Type 1 aspect in a Type 2 manner.

Though its significance seems to me to be completely missed by the mathematical community, I will start with the first of these conversions (which in fact is relatively easy to appreciate).

Now we will illustrate here again for convenience with respect to the number "2".

So the standard analytic definition of "2" (as a specific number quantity) is given through the Type 1 aspect as 21. So once again the Type 1 aspect is always defined with respect to the default dimensional value of 1.

The corresponding holistic definition of "2" (as the collective number quality of twoness) is given through the Type 2 aspect as 12. "2" now refers directly to a number dimension (rather than a base quantity).

Thus to convert this Type 2 aspect in Type 2 terms, we need in effect to obtain the square root.

So in general terms x= 1 with in this case x= 1. So x = + 1 and – 1.

We have now moved to a circular definition of number (with both + 1 and – 1 lying on the unit circle in the complex plane).

However these two results are given but an analytic quantitative interpretation in conventional mathematical terms.
However the corresponding holistic meaning is highly revealing, requiring in effect a uniquely distinctive manner of mathematical interpretation.

+ in this context entails the psychological notion of positing (i.e. making conscious).
– however entails the corresponding notion of negation (i.e. of what is unconscious) thereby representing unconscious understanding.

When understanding is especially refined, as with the fusion of matter and anti-matter particles in physics, unconscious negation (of what is consciously posited) will approach full attainment resulting in a pure intuitive understanding (representing a psycho spiritual energy state).

So strictly speaking the holistic appreciation of each number represents a pure energy state (with complementary physical and psychological meanings).

Thus in effect we have two extremes with respect to the understanding of number (and remember in dynamic terms number as object has no strict meaning independent of such understanding)!

Thus we can appreciate number in the standard analytic fashion as an absolutely existing quantity form (that never changes). Here it is viewed as nothing in qualitative terms

However from the opposite extreme we can appreciate number in the unrecognised holistic fashion as approaching a pure energy state (where it is nothing in quantitative terms).

However properly understood, number experience entails an interaction somewhere between both extremes, where both quantitative aspects (as form) and qualitative aspects (as energy) ceaselessly interact leading to a continual transformation thereby in the nature of each number.

So once again we have the analytic quantitative extreme (recognised through the Type 1 aspect)

Here 2 = 1 + 1 (Strictly 21 =  1+ 11).

So here the two units are defined in a homogeneous quantitative manner (i.e. without any distinctive quality)

Then in the Type 2 system 2 = 1st + 2nd (so both units are now defined as without any quantitative distinction!)

Then when we convert the Type 2 to the Type 1, we can indirectly represent this important reality consistently in a quantitative manner.

So 1st and 2nd are now represented as + 1 and .– 1 respectively.

And + 1 .– 1 = 0!

So the task of converting consistently from Type 2 to Type 1 implies that we can represent the ordinal members of each group uniquely by a set of circular numbers (lying as roots on the unit circle) that always add up to zero.

And this is where the prime numbers can be seen to have an equally valid Type 2 (as well as Type 1) identity.

From the Type 1 perspective, the unique importance of the primes comes from viewing them as the "building blocks" of the natural number system.

So all natural numbers (other than 1) can be uniquely expressed as the product of prime factors.  

However, the primes have an equally important role in Type 2 terms, where however their directional link to the natural numbers is completely reversed.

So from the Type 2 perspective, each prime can be uniquely expressed in an ordinal natural number fashion by its various roots (again except 1).

So for example if we take 5 as a prime, it can be uniquely expressed in terms of its 5 roots (excluding 1 which is common to all roots).

Now these 5 roots provide an indirect (Type 1) means of uniquely expressing in quantitative terms   the various natural number members of 5 (i.e. 1st , 2nd , 3rd, 4th and 5th respectively) in an ordinal manner.

However there is an obvious paradox with respect to the Type 1 and Type 2 approaches.

In the first case, each natural number (except 1) is uniquely defined by its prime members in cardinal terms.

In the 2nd case, each prime is uniquely defined by its natural number members (except 1) in ordinal terms (indirectly expressed in a quantitative manner through its prime roots).

This leads directly to the holistic qualitative recognition of the two-way interdependence of primes and natural numbers in both cardinal and ordinal terms.

In other words a holistic synchronicity entailing the two-way interaction of primes and natural numbers (which is directly qualitative in nature) underlies the deepest workings of the number system.

However though obvious when viewed from the appropriate perspective, the realisation of  this simple fact will permanently elude a mathematical profession that reduces interpretation of number in a merely quantitative fashion.

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