Wednesday, February 4, 2015

Intricacies of Addition and Multiplication (7)

As stated on several occasions every natural number enjoys both a quantitative (analytic) and qualitative (holistic) meaning that dynamically interact in experience.

So again for example the quantitative notion of 3 (as representing independent units) in cardinal terms is dynamically inseparable from the corresponding qualitative notion of 3  (as representing the interdependence of uniquely distinct units) in an ordinal manner.

And this applies by extension to all the primes and natural numbers.

However we can likewise give both quantitative and qualitative meanings to all rational fractions (which likewise interact in dynamic fashion).

Again using the reference point of 3 to illustrate let us look closely at the quantitative meaning associated with the fractions 1/3, 2/3 and 3/3.

Now 1/3 represents the reciprocal of 3.

In a more complete Type 1 manner 1/3 represents the reciprocal 31.

Now 3 represents an integer, which is - literally - a whole number.

However 1/3 as reciprocal represents its corresponding part which can be written as 3– 1.

So implicitly the change from whole to part entails the negation of the (default) 1st dimension which is the basis of all rational linear understanding.

This negation then causes to a degree an unconscious fusion of positive and negative polarities in experience which is the direct basis of all intuitive type understanding.

So the very means by which we are enabled to switch from whole to part (with respect to number) and indeed with respect to phenomenal recognition in any context implicitly requires a switch from conscious (analytic) to unconscious (holistic) type understanding.

However once this transition from whole to part has been achieved, in explicit terms the result will be interpreted in a merely (reduced) quantitative manner.

So 1/3 = (1/3)1.

Thus is we imagine a small cake divided into 3 slices, 1/3, 2/3 and 3/3 i.e. (1/3)1, (2/3)1 and (3/3),represent the various options in terms of expressing the constituent parts with respect to the whole cake (which is equally recognised as composed of 3 parts).

So the whole cake (i.e. 1 unit) = 3/3. However once again this properly entails the simultaneous recognition of the whole cake (as 1 unit) and 3 part units which entails both rational and intuitive type recognition.

Of course we can equally have 2/3 (as 2 part slices with respect to the whole) and 1/3 (as 1 part slice in relation to the whole).

Now even though in abstract terms we may use integers and fractions without respect to concrete phenomena, the basic rationale remains the same in that implicitly the change from whole to (reciprocal) part recognition implicitly entails both intuitive (unconscious) as well as (rational) conscious recognition.

However the corresponding qualitative interpretation of these same fractions is perhaps even more surprising.

In Type 2 terms, these will now be represented as 11/3, 12/3 and 13/3 .

These can then be readily identified as representing the 3 roots of unity which are
– .5 +.866i, .5   .866i and 1 respectively.

What is remarkable about these roots (though clearly not properly recognised) is that they express in a reduced quantitative manner, ordinal relations (that are directly of a qualitative nature).

So 11/3 relates in this context  of the small cake relates to the 1st of  the 3 slices.

12/3 relates to the 2nd of the 3 slices, while 13/3 relates to the 3rd of the 3 slices.

Thus once again, what is remarkable is that we can then give these ordinal notions (of a qualitative nature) an indirect quantitative interpretation.

In other words we have here a ready means of converting as between the Type 2 and Type 1 aspects of the number system (from qualitative to quantitative format).

In more general terms, in order to express the ordinal rankings with respect to any number n in an indirect quantitative manner, we simply obtain the corresponding n roots of 1.

Once again through this process, these qualitative rankings, which are unique for each cardinal value of n, can be given a unique quantitative value!  

Tuesday, February 3, 2015

Intricacies of Addition and Multiplication (6)

We now can begin to address the all important role of primes with respect to the natural number system.
However, once again as we shall see, from the more comprehensive perspective, where both the quantitative and qualitative aspects of number behaviour are explicitly recognised, conventional interpretation is found to be seriously lacking.

In conventional mathematical terms, the primes (and natural numbers) are treated in a Type 1 quantitative manner (that directly conforms to the cardinal interpretation of number).

From this perspective, the primes are viewed as the fundamental "building blocks" of the natural number system.  So every natural number (≠ 1) can be expressed through the unique combination
of  primes (as constituent factors).

In this way the primes appear as the most independent of numbers (from which the composite natural numbers in cardinal terms are derived).

However, as we have seen the primes (and natural numbers) can equally be treated in a Type 2 qualitative manner (that directly conforms to the ordinal interpretation of number).
From this perspective the primes are viewed as the numbers that provide the basis for establishing a unique relationship with every natural number (≠ 1).
In this way the primes appear as the most interdependent of numbers (from which composite natural number interrelationships in ordinal terms are derived).

Therefore from a dynamic interactive perspective, the key feature of the primes is the manner in which both quantitative (cardinal) and qualitative (ordinal) extremes of  behaviour with respect to the number system are embodied in their very nature.

It must be stated at this stage that there are two recognised ways in conventional mathematical terms of deriving the natural numbers.

The first is based on addition as the simple one where every natural number can be expressed through the repeated addition of 1. So 2 for example from this perspective = 1 + 1.

The second is based on multiplication where now every natural number ((≠ 1) can be uniquely expressed as the product of primes. So once again for example 6 = 2 * 3.

However, this approach, as we have seen is reserved solely for the Type 1 quantitative treatment of number (in cardinal terms).
So equally, as we have seen, this must be extended to the Type 2 qualitative treatment of number (in an ordinal manner).

So 2 from this additive perspective 2 (now reflecting the qualitative notion of  2 as "twoness") = 1st + 2nd.

 And 6 from this multiplicative perspective (reflecting the qualitative notion of 6 as "sixness" = 2 (as "twoness") * 3 (as "threeeness").

Now when we recognise the corresponding Type 2 ordinal aspect of the primes, an obvious problem with respect to conventional interpretation immediately arises.

For example from a quantitative (Type 1) perspective, 7 represents a prime (in cardinal terms) and therefore represents an (independent) "building block" of the natural number system.

However from a qualitative (Type 2) perspective, 7 (as "sevenness") already implies 1st, 2nd, 3rd, 4th, 5th, 6th and 7th members). But already two of these members (i.e. 4th and 6th) represent composite number notions.

What this really entails is that the very notion of primes as sole "building blocks" is meaningless from a dynamic perspective.  Rather we must now look at both the primes and natural numbers as being mutually interdependent with each other in a two-way interactive manner.

In any case, the primes are not truly independent - even in a restricted cardinal manner - as they are already dependent for their composition on the Peano based additive system.

However when one equally recognises the Type 2 qualitative aspect of prime behaviour, then it is obvious that they embody both independent and interdependent aspects of number behaviour, that can only be properly appreciated in a dynamic complementary manner.

So from the Type 1 quantitative perspective, the primes appear as the independent building blocks of the natural number system (in cardinal terms).

However equally from the Type 2 qualitative perspective, the natural numbers appear as the interdependent "building blocks" of the primes (in ordinal terms).

So remarkable the ultimate relationship of the primes to the natural numbers (and natural numbers to the primes) is one approaching total synchronicity!

There is another important point which must be made at this juncture.

Once again the conventional treatment of the (Type 1) natural number system (as quantitatively viewed in a cardinal manner) is in terms of a line reflecting the linear (1-dimensional) mode of interpretation adopted.

However when we properly allow for the (Type 2)  treatment of the natural number system (as qualitatively viewed in an ordinal manner) the appropriate means of representation is in in terms of a circle.

In this way both the quantitative and qualitative aspects  of the number system are seen in dynamic terms as linear and circular with respect to each other.

However, understanding must be very refined, for when reference frames switch the linear can attain a qualitative and the circular a  quantitative meaning respectively.

Monday, February 2, 2015

Intricacies of Addition and Multiplication (5)

So far, we have dealt with numbers in their simplest form i.e. as individual natural numbers, which can be defined with two aspects that are Type 1 (quantitative) and Type 2 (qualitative) with respect to each other.

Thus, when the base number in Type 1 terms - which can vary with respect to the natural numbers - is quantitative, the default dimensional number of 1 is defined in Type 2 terms as qualitative.

Then in reverse terms when the dimensional number from a Type 2 perspective - which now can equally vary with respect to the natural numbers - is qualitative, the default base number of 1 is defined in Type 1 terms as quantitative.

However through the dynamics of interaction reference frames keep switching.
Therefore from this complementary perspective when the base number in Type 2 terms - which can likewise vary with respect to the natural numbers - is qualitative, the default dimensional number is defined in Type 1 terms as quantitative.
Then in reverse terms when the dimensional number in Type 1 terms - which varies with respect to the natural numbers - is quantitative, the default base number is defined in Type 2 terms as qualitative.

Therefore, every natural number, through the dynamics of experiential interaction, acquires both quantitative (analytic) and qualitative (holistic) meanings, with respect to numbers representing (specific) base objects and (general) dimensional categories respectively.

So once again the number 3 for example can be given a quantitative (analytic) meaning with respect to either (specific) objects or (general) dimensions e.g. 3 cars or 3 mathematical dimensions. This indeed coincides with the cardinal interpretation of the (collective) number as comprised of homogeneous independent units (without qualitative distinction) where 3 = 1 + 1 + 1.

However the number 3 can equally be given a qualitative (holistic) meaning with respect to either (specific) objects or (general) dimensions as "threeness" (i.e. quality of 3) with respect to the distinct individual identification of the cars and dimensions in question.
This likewise coincides with the ordinal interpretation of the (individual) numbers, which comprises of unique identity of each unit (without quantitative distinction) where 3 (as threeness) = 1st + 2nd + 3rd.

Strictly speaking we cannot have quantitative appreciation (of a cardinal nature) without corresponding qualitative appreciation (of and ordinal nature) and vice versa.

However it is vital once again to appreciate that both of these aspects correspond to two distinct interpretations of numbers, that are quantitative and qualitative with respect to each other.
Thus the nature of number is inherently dynamic and interactive comprising twin opposite poles that behave in a complementary manner.

We now move on to the more intricate situation of properly explaining the nature of addition and multiplication (from this new dynamic interactive perspective).

The addition of two number quantities (as base numbers) in standard Type 1 terms seems quite simple.

So for example  2+ 31  = 51.

Now the multiplication of the two number (as dimensional numbers) likewise appears simple.

So 1* 13  = 15.

However  what appears as multiplication from the Type 1 perspective represents in fact addition from the Type 2. i.e.  1* 13  =  12 + 3.

This simply indicates how addition and multiplication are themselves directly linked to the Type 1 and Type 2 aspects of number respectively. So what represents addition from one perspective, represents multiplication from the other (and vice versa).

Now we will consider the more difficult case where two natural numbers (with base ≠ 1) are multiplied.

Indeed this consideration is of the utmost importance as all natural numbers (≠ 1) can be uniquely expressed as the product of prime factors!

Let us therefore in this context consider the case of 2 * 3 i.e. 2* 31.

In conventional mathematical terms a merely reduced quantitative result is given.

So from this perspective, .

2 * 3  = 6 (i.e. 61)

However this unsatisfactory from a simple geometrical interpretation will indicate that the result should be represented in square (2-dimensional) rather than linear (1-dimensional) units.

So therefore, properly understood the multiplication of these two numbers requires both Type 1 and Type 2 aspects.

Therefore 2* 31 = 6* 12

Thus in quantitative Type 1 terms, the result is 6 (i.e. 61). However in qualitative Type 2 terms the result is 2 i.e. 12 .

The combination of these two aspects reflects the fact that the quantitative result (6) is now expressed in 2-dimensional terms.

However though relative to the base number result (i.e. 6), the dimensional number 2 is qualitative, within its own frame of reference (as applying to dimensional rather than base numbers) it is quantitative in nature.

So the base result from a quantitative perspective is 6 (i.e. 6 unit objects); and  the dimensional result from a quantitative perspective is 2 (i.e. 2-dimensional).

We now equally need to reflect the corresponding qualitative meaning of the symbols.

Again in conventional mathematical terms we strictly must represent 2 * 3 as representing separate
independent objects. For example imagine the 6 objects are circular rings as represented below.

Then we let 3 represent each row (containing 3 objects) and 2 each column (containing 2 objects) as below.

 O O O
 O O O

Now by treating each ring as independent we can con count 3 in each row so that the total result = 3 + 3 = 6.

However to represent 3 + 3 as 2 * 3 , a decisive qualitative transformation is required whereby rather than seeing the objects in each row as independent (in cardinal terms) that they be rather seen in qualitative ordinal terms as composed of 1st, 2nd and 3rd members.

Without such recognition we could not meaningfully identify the two separate rows (which requires a common recognition with respect to the 3 members involved).

So the identification of each row requires the qualitative recognition of the notion of 3 (as "threeness") thereby establishing the common shared identity of the 3 members involved.

Then the key insight by which the operator 2 can be used to multiply the number in each row (i.e. 3) requires the common recognition of the identity of members in both rows.
So each member in the 1st row thereby is seen to share a common identity with each member in the 2nd row and it is this qualitative recognition of interdependence that thereby enables us to use the operator 2 in a multiplicative sense.

So overall the multiplication of 2 * 3 contains quantitative aspects respect to both the base and dimensional nature of the units involved.

So we thereby can identify 6 independent units (in quantitative terms) that are expressed in a 2-dimensional fashion (as square units).

However equally we have the qualitative recognition of 3 (through the ordinal recognition of 1st, 2nd and 3rd members in each row and the qualitative recognition of 2 (through the ordinal recognition of a 1st and 2nd row).

Ultimately therefore the resulting recognition of 6 (through multiplication) entails the quantitative recognition of each member as independent and the qualitative recognition of the six members sharing a common identity through the one to one correspondence of the 3 members in each of the 2 rows.

So in base terms not alone do we establish the quantitative recognition of 6 as cardinal, we now equally establish in qualitative terms ordinal recognition of 1st, 2nd, 3rd, 4th, 5th and 6th members.  

Likewise in dimensional terms, not alone do we establish the quantitative recognition of 2 (i.e. as 2-dimensional units), we equally establish the qualitative recognition of 2 enabling us in this context to distinguish 1st and 2nd dimensions in an ordinal fashion.