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)1 ,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!
Wednesday, February 4, 2015
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.
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 21 + 31 = 51.
Now the multiplication of the two number (as dimensional numbers) likewise appears simple.
So 12 * 13 = 15.
However what appears as multiplication from the Type 1 perspective represents in fact addition from the Type 2. i.e. 12 * 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. 21 * 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 21 * 31 = 61 * 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
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 21 + 31 = 51.
Now the multiplication of the two number (as dimensional numbers) likewise appears simple.
So 12 * 13 = 15.
However what appears as multiplication from the Type 1 perspective represents in fact addition from the Type 2. i.e. 12 * 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. 21 * 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 21 * 31 = 61 * 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.
Thursday, January 29, 2015
Intricacies of Addition and Multiplication (4)
We have
seen in the last entry how both the quantitative and qualitative nature of the
natural numbers is directly related to the operations of addition and
multiplication respectively (which are complementary with each other).
Thus in
Type 1 terms (where the base is defined in quantitative terms)
by
addition,
1 + 1 = 2
(i.e. 11 + 11 = 21).
Then in
Type 2 terms (where the dimension is defined in qualitative terms)
by
multiplication,
1 * 1 = 2
(i.e. 11 * 11 = 12).
Thus we
have switched from the quantitative notion of “2” as base number to the
qualitative notion of “2” (or twoness)
as dimensional number in this fashion.
Thus with
respect to the base (representing specific objects), the quantitative notion of
2 corresponds to cardinal interpretation
So 2 in
this context arises from the recognition of homogeneous independent objects
(without qualitative distinction)
Then with
respect to the dimension (representing general frameworks for objects) the
qualitative notion of 2 corresponds directly with ordinal interpretation.
So 2 (as
twoness) in this context arises from the recognition of qualitatively
distinct 1st and 2nd
dimensional frameworks (which requires seeing both as qualitatively
interdependent with each other). However
as we will indirectly demonstrate later 2 = 1st + 2nd
lacks any quantitative distinction.
In this way
we can see how addition and multiplication are directly related to both the
cardinal and ordinal interpretation of the natural numbers respectively.
However
Just like the left and right turns at a crossroads are reversed when we
approach it from the opposite direction, likewise when we switch the frame of
reference (with respect to both quantitative and qualitative) a complementary
reverse interpretation results.
So what is
addition from a Type 1 perspective, is multiplication from a Type 2 (and vice
versa). And this equally applies to both quantitative and qualitative
interpretations of base and dimensional values.
So we can equally
start with the base number defined as qualitative and the dimensional number as
quantitative respectively.
Now
addition with respect to the Type 2 aspect implies the quantitative aspect of
this dimensional number.
Thus 1 + 1
= 2 (i.e. 11 + 1 =
12).
Here number
representing dimension carries the standard cardinal meaning where 2 = two
dimensions.
Then in
complementary fashion, multiplication with respect to The Type 1 aspect implies
the qualitative aspect the application of this base number.
So 1 * 1 =
2 i.e. (11 + 11
= 21).
To distinguish
the switch in the meaning (quantitative and qualitative) that numbers now
possess, I have likewise reversed the notation, so that base numbers are now
represented with subscripts and dimensions as standard size (just as formally, base numbers were represented by normal size and dimensions with superscripts
respectively).
Though the
meaning associated with the mathematical representation of addition and
multiplication is difficult to intuitively grasp (due to the standard
identification of number with merely quantitative values) it can be expressed
quite simply in psychological terms.
In other
words, number perceptions and concepts continually interact in a dynamic
manner, whereby both rational (analytic) and intuitive (holistic) aspects are
involved.
Through this dynamic interactive process, we are thereby enabled to distinguish the
natural numbers in both cardinal and ordinal terms ,where they can represent
both (specific) objects and (general) dimensions respectively.
So for
example, we are thereby enabled to appreciate 3 as a cardinal number (applying
to specific objects); we are also enabled to appreciate 3 in cardinal terms as
applying more generally to dimensions i.e. 3 dimensions.
Equally we
are enabled to appreciate 3 in ordinal terms with respect to specific objects
(as 1st, 2nd and 3rd) and likewise with
respect to more generalised dimensions (again as 1st, 2nd
and 3rd).
The crucial
point to recognise that this crucial capacity - whereby we are enabled to keep
switching from cardinal to ordinal (and ordinal to cardinal meaning) - is directly related to the operations of
addition and multiplication (that likewise behave in a dynamic interactive
manner).
However as
long as we attempt to interpret number in a merely quantitative manner, statements regarding the true dynamic nature of addition and multiplication
can carry no resonance.
Wednesday, January 28, 2015
Intricacies of Addition and Multiplication (3)
There are obvious parallels as between the dynamic approach to number and quantum mechanics.
Indeed, I am confident that it will be ultimately understood that this dynamic approach to number serves as the starting basis for quantum mechanical understanding and in that sense is more fundamental. So properly understood, quantum mechanics is rooted in the true dynamic nature of number.
As is well-known all sub-atomic particles manifest themselves in a complementary fashion as both particles and waves.
Likewise, as we have seen, all numbers manifest themselves in a complementary manner through both quantitative (analytic) and qualitative (holistic) aspects.
It might be initially helpful to identify the particle with the quantitative aspect, and the wave with the qualitative aspect respectively.
However just as it is recognised in quantum mechanics that the particle has also a wave, and the wave a particle aspect, likewise we have seen, that when reference frames are switched with respect to number, that the quantitative aspect has a qualitative and the qualitative also a quantitative aspect respectively.
Indeed the parallels go further.
Again as is well-known the uncertainty principle apples with respect to the behaviour of sub-atomic particles. So, for instance, one cannot hope to precisely predict the position and momentum of a particle simultaneously. Rather a trade-off is involved whereby greater precision with respect to one aspect entails increasing imprecision with respect to the other.
This is equally true of number behaviour, whereby an uncertainty principle equally applies to the dynamic behaviour of number. So here for example a trade-off is also involved with respect to simultaneous knowledge of both quantitative (analytic) and qualitative (holistic) aspects.
So greater precision with respect to the quantitative aspect thereby implies greater imprecision with respect to its corresponding qualitative aspect.
This is greatly exemplified by the very nature of Conventional Mathematics. So increasing focus on the merely quantitative aspect of number behaviour has become so extreme that the qualitative aspect is not even recognised in formal interpretation.
So the misleading view that numbers have an absolute quantitative identity (without reference to their qualitative nature) has long become entrenched in accepted understanding.
In truth however numbers enjoy a merely relative identity (based on the dynamic interaction of twin complementary aspects of behaviour).
However the very appreciation of this point will require - as I continually repeat - a radical new paradigm of what Mathematics really represents. Again this "conversion" will I believe signal the greatest revolution yet in our intellectual history!
Before returning directly back to the nature of addition and multiplication, I wish to address a key feature of conventional mathematical understanding that is not properly appreciated.
This relates to its 1-dimensional nature (based on the qualitative holistic meaning 1).
So though dimensions (i.e. powers or exponents) other than 1, are of course recognised in a quantitative (analytic) manner, these are all interpreted within the standard 1-dimensional context (in qualitative terms).
As this is so important, I will comment further here on what it entails.
The 1st dimension is unique is that that it is the only dimension where qualitative and quantitative meaning are reduced in an absolute manner.
So, 1 in the Type 1 aspect of the number system, is represented as 11.
However, 1 in the Type 2 aspect of the number system, is likewise represented as 11 .
So with respect to 1, no distinction can be made as between its quantitative (Type 1) and qualitative (Type 2) interpretations. This is why a linear (i.e. 1-dimensional) rational approach to interpretation entails the reduction of qualitative to quantitative meaning in an absolute manner.
However what is not at all clearly recognised is that every number (other than 1) can equally serve as a valid means of interpretation of mathematical symbols.
The key distinction then in all these other approaches is that a dynamic relative means of appreciation ensues (entailing the interaction of both analytic and holistic aspects).
So the rigid preoccupation with 1-dimensional interpretation in Mathematics has completely blinded us to the existence of unlimited further terrains of potential meaning (where numbers ≠ 1,serve as the holistic means of interpretation).
We can easily illustrate once more the 1-dimensional approach with respect to number interpretation.
For example if we multiply two numbers, say 2 * 3, the resulting answer is given as 6.
However, if represent this expression in geometrical terms, we can quickly appreciate that a qualitative as well as quantitative conversion takes place. So in quantitative terms the result is indeed 6. However this now relates to 2-dimensional (square) rather than 1-dimensional (linear) units.
However in conventional mathematical terms, this qualitative transformation in the nature of the units is simply ignored with the result expressed in a linear (1-dimensional) manner.
Thus from this perspective 2 * 3 = 6 (i.e. 61) . So if you want to appreciate the key problem with respect to reconciling multiplication with addition, it is right here!
Therefore whereas addition (in linear terms) leads solely to a quantitative transformation in the nature of the units, multiplication - by contrast - likewise entails a qualitative transformation.
Indeed from the conventional mathematical perspective, multiplication serves as but a short-hand way of representing addition.
So 2 * 3 (from this perspective) = 2 + 2 + 2. Therefore 2 * 3 expresses the fact that we are adding 2 three times!
However this misses the point completely - as is inevitable from the conventional mathematical perspective - that multiplication essentially, relative to addition, entails a qualitative, rather than quantitative transformation.
The deeper philosophical basis of the 1-dimensional approach is the use of single polar frames of reference in the interpretation of mathematical relationships.
I frequently illustrate this with respect to the interpretation of left and right turns at a crossroads.
I one is heading N, a left turn can be unambiguously identified at the crossroads. So here a single polar frame of reference (i.e. the N direction) is used.
Now having passed through the crossroads one switched direction and then travels S, once again a left turn can be unambiguously identified (using this single pole of reference).
So both turns at the crossroads are now unambiguously identified as left (using separate poles of reference). However we know that the turns must be left and right with respect to each other!
This recognition that identification of turns is merely relative, requires the ability to appreciate N and S as complementary opposite poles (which requires simultaneously linking N and S directions).
The key insight enabling such simultaneous recognition is of a holistic intuitive nature and leads to (circular) paradoxical understanding in terms of unambiguous rational appreciation of an analytic nature.
So absolute unambiguous understanding is based on the use just one polar direction as reference frame. And this is the fundamental nature of linear (1-dimensional) interpretation.
However relative paradoxical understanding always entails simultaneous use of more than one polar reference frame (which in the simplest case entails two). And this likewise is the fundamental nature of circular (higher dimensional) interpretation.
So rather just one polar reference frame for number (i.e. analytic) we have now extended appreciation to include two such directions i.e. analytic and holistic, in dynamic relationship with each other.
However just as with the interpretation of a crossroads we keep alternating as between absolute understanding (where each turn is unambiguously defined) and relative understanding (where both turns have a merely arbitrary relative interpretation depending on context) likewise we must do the same with dynamic mathematical understanding as we continually alternate as between analytic (linear) and holistic (circular) type appreciation.
Having dealt with these important background issues let us return to the central issue of addition and multiplication.
The remarkable fact that now presents itself, is that once we define both Type 1 and Type 2 aspects of the number system, we can simply distinguish the nature of addition from multiplication.
Thus when we interpret the Type 1 aspect from the standard quantitative perspective, the natural numbers can be derived (as base values) through the continual addition of the starting unit.
So starting with 11, 21 = 11 + 11, 31 = 11 + 11 + 11, and so on.
Thus addition is here directly associated with the quantitative transformation of number.
However equally when we interpret the Type 2 aspect from the (unrecognised) qualitative perspective, the natural numbers can be derived (as dimensional values) through the continual multiplication of the starting unit.
So again starting with 11, 12 = 11 * 11, 13 = 11 * 11 * 11, and so on.
Thus multiplication is here directly associated with the qualitative transformation of number.
Therefore when correctly understood, addition and multiplication represent two fundamental operations that are quantitative (analytic) and qualitative (holistic) with respect to each other.
However this key distinction cannot be appreciated from a conventional mathematical perspective (where the quantitative aspect is solely recognised).
Indeed, I am confident that it will be ultimately understood that this dynamic approach to number serves as the starting basis for quantum mechanical understanding and in that sense is more fundamental. So properly understood, quantum mechanics is rooted in the true dynamic nature of number.
As is well-known all sub-atomic particles manifest themselves in a complementary fashion as both particles and waves.
Likewise, as we have seen, all numbers manifest themselves in a complementary manner through both quantitative (analytic) and qualitative (holistic) aspects.
It might be initially helpful to identify the particle with the quantitative aspect, and the wave with the qualitative aspect respectively.
However just as it is recognised in quantum mechanics that the particle has also a wave, and the wave a particle aspect, likewise we have seen, that when reference frames are switched with respect to number, that the quantitative aspect has a qualitative and the qualitative also a quantitative aspect respectively.
Indeed the parallels go further.
Again as is well-known the uncertainty principle apples with respect to the behaviour of sub-atomic particles. So, for instance, one cannot hope to precisely predict the position and momentum of a particle simultaneously. Rather a trade-off is involved whereby greater precision with respect to one aspect entails increasing imprecision with respect to the other.
This is equally true of number behaviour, whereby an uncertainty principle equally applies to the dynamic behaviour of number. So here for example a trade-off is also involved with respect to simultaneous knowledge of both quantitative (analytic) and qualitative (holistic) aspects.
So greater precision with respect to the quantitative aspect thereby implies greater imprecision with respect to its corresponding qualitative aspect.
This is greatly exemplified by the very nature of Conventional Mathematics. So increasing focus on the merely quantitative aspect of number behaviour has become so extreme that the qualitative aspect is not even recognised in formal interpretation.
So the misleading view that numbers have an absolute quantitative identity (without reference to their qualitative nature) has long become entrenched in accepted understanding.
In truth however numbers enjoy a merely relative identity (based on the dynamic interaction of twin complementary aspects of behaviour).
However the very appreciation of this point will require - as I continually repeat - a radical new paradigm of what Mathematics really represents. Again this "conversion" will I believe signal the greatest revolution yet in our intellectual history!
Before returning directly back to the nature of addition and multiplication, I wish to address a key feature of conventional mathematical understanding that is not properly appreciated.
This relates to its 1-dimensional nature (based on the qualitative holistic meaning 1).
So though dimensions (i.e. powers or exponents) other than 1, are of course recognised in a quantitative (analytic) manner, these are all interpreted within the standard 1-dimensional context (in qualitative terms).
As this is so important, I will comment further here on what it entails.
The 1st dimension is unique is that that it is the only dimension where qualitative and quantitative meaning are reduced in an absolute manner.
So, 1 in the Type 1 aspect of the number system, is represented as 11.
However, 1 in the Type 2 aspect of the number system, is likewise represented as 11 .
So with respect to 1, no distinction can be made as between its quantitative (Type 1) and qualitative (Type 2) interpretations. This is why a linear (i.e. 1-dimensional) rational approach to interpretation entails the reduction of qualitative to quantitative meaning in an absolute manner.
However what is not at all clearly recognised is that every number (other than 1) can equally serve as a valid means of interpretation of mathematical symbols.
The key distinction then in all these other approaches is that a dynamic relative means of appreciation ensues (entailing the interaction of both analytic and holistic aspects).
So the rigid preoccupation with 1-dimensional interpretation in Mathematics has completely blinded us to the existence of unlimited further terrains of potential meaning (where numbers ≠ 1,serve as the holistic means of interpretation).
We can easily illustrate once more the 1-dimensional approach with respect to number interpretation.
For example if we multiply two numbers, say 2 * 3, the resulting answer is given as 6.
However, if represent this expression in geometrical terms, we can quickly appreciate that a qualitative as well as quantitative conversion takes place. So in quantitative terms the result is indeed 6. However this now relates to 2-dimensional (square) rather than 1-dimensional (linear) units.
However in conventional mathematical terms, this qualitative transformation in the nature of the units is simply ignored with the result expressed in a linear (1-dimensional) manner.
Thus from this perspective 2 * 3 = 6 (i.e. 61) . So if you want to appreciate the key problem with respect to reconciling multiplication with addition, it is right here!
Therefore whereas addition (in linear terms) leads solely to a quantitative transformation in the nature of the units, multiplication - by contrast - likewise entails a qualitative transformation.
Indeed from the conventional mathematical perspective, multiplication serves as but a short-hand way of representing addition.
So 2 * 3 (from this perspective) = 2 + 2 + 2. Therefore 2 * 3 expresses the fact that we are adding 2 three times!
However this misses the point completely - as is inevitable from the conventional mathematical perspective - that multiplication essentially, relative to addition, entails a qualitative, rather than quantitative transformation.
The deeper philosophical basis of the 1-dimensional approach is the use of single polar frames of reference in the interpretation of mathematical relationships.
I frequently illustrate this with respect to the interpretation of left and right turns at a crossroads.
I one is heading N, a left turn can be unambiguously identified at the crossroads. So here a single polar frame of reference (i.e. the N direction) is used.
Now having passed through the crossroads one switched direction and then travels S, once again a left turn can be unambiguously identified (using this single pole of reference).
So both turns at the crossroads are now unambiguously identified as left (using separate poles of reference). However we know that the turns must be left and right with respect to each other!
This recognition that identification of turns is merely relative, requires the ability to appreciate N and S as complementary opposite poles (which requires simultaneously linking N and S directions).
The key insight enabling such simultaneous recognition is of a holistic intuitive nature and leads to (circular) paradoxical understanding in terms of unambiguous rational appreciation of an analytic nature.
So absolute unambiguous understanding is based on the use just one polar direction as reference frame. And this is the fundamental nature of linear (1-dimensional) interpretation.
However relative paradoxical understanding always entails simultaneous use of more than one polar reference frame (which in the simplest case entails two). And this likewise is the fundamental nature of circular (higher dimensional) interpretation.
So rather just one polar reference frame for number (i.e. analytic) we have now extended appreciation to include two such directions i.e. analytic and holistic, in dynamic relationship with each other.
However just as with the interpretation of a crossroads we keep alternating as between absolute understanding (where each turn is unambiguously defined) and relative understanding (where both turns have a merely arbitrary relative interpretation depending on context) likewise we must do the same with dynamic mathematical understanding as we continually alternate as between analytic (linear) and holistic (circular) type appreciation.
Having dealt with these important background issues let us return to the central issue of addition and multiplication.
The remarkable fact that now presents itself, is that once we define both Type 1 and Type 2 aspects of the number system, we can simply distinguish the nature of addition from multiplication.
Thus when we interpret the Type 1 aspect from the standard quantitative perspective, the natural numbers can be derived (as base values) through the continual addition of the starting unit.
So starting with 11, 21 = 11 + 11, 31 = 11 + 11 + 11, and so on.
Thus addition is here directly associated with the quantitative transformation of number.
However equally when we interpret the Type 2 aspect from the (unrecognised) qualitative perspective, the natural numbers can be derived (as dimensional values) through the continual multiplication of the starting unit.
So again starting with 11, 12 = 11 * 11, 13 = 11 * 11 * 11, and so on.
Thus multiplication is here directly associated with the qualitative transformation of number.
Therefore when correctly understood, addition and multiplication represent two fundamental operations that are quantitative (analytic) and qualitative (holistic) with respect to each other.
However this key distinction cannot be appreciated from a conventional mathematical perspective (where the quantitative aspect is solely recognised).
Tuesday, January 27, 2015
Intricacies of Addition and Multiplication (2)
We saw yesterday that all numbers - indeed all mathematical notions - possess both quantitative (analytic) and qualitative (holistic) aspects, which dynamically interact with each other in complementary fashion.
Therefore from a comprehensive perspective, every number is thereby defined in terms of both a base and dimensional value (where with respect to the expression ab, a is the base and b the dimensional value respectively.
From this perspective the natural numbers can be defined with respect to both Type 1 and Type 2 aspects respectively.
With the Type 1 aspect, where the base aspect varies (as quantity), the (default) dimensional number is 1; with the Type 2 aspect, by contrast, where the dimensional aspect (as quality) varies, the default base number is 1.
The cardinal notion of number is then directly associated from this perspective with the Type 1 aspect; the ordinal notion is then directly associated with the Type 2.
Crucially therefore both the cardinal notion (implying independence) with respect to number quantities and the ordinal notion (implying, by contrast, interdependence with respect to the qualitative relationship between numbers) represent two distinct types of number appreciation, which are dynamically relative in nature with respect to each other.
However it is equally possible to switch reference frames. Then with respect to the natural numbers of the Type 1 aspect, the base value now varies and is defined in qualitative (holistic) terms with respect to a (default) dimensional value of 1; then, in relative terms, the dimensional aspect varies, and is now defined in quantitative (analytic) terms with respect to a default base value of 1.
In this way, both base and dimensional values take on quantitative and qualitative meanings that dynamically interact with each other in a complementary fashion.
And if you think that this seems somewhat far-fetched, let me remind you that this is what continually takes place with respect to our experience of number!
So from one perspective we are able to recognise objects in natural number fashion with respect to their cardinal (quantitative) characteristics. So if I point to a group of 3 people on the street, such recognition entails quantitative recognition of number. If however I now specifically identify - say in terms of a criterion such as height - 1st, 2nd and 3rd members of that group - this now entails corresponding ordinal recognition of a qualitative nature.
And likewise it is similar with respect to numbers, now serving as dimensions. So the recognition of 3 dimensions (now serving as more general categories within which specif number objects can be defined) implies cardinal recognition of a quantitative kind. However the distinction as between the 1st, 2nd and 3rd dimensions (according to some criterion) implies ordinal recognition of a qualitative kind.
So with respect to actual experience we effortlessly switch as between both quantitative (analytic) and qualitative (holistic) recognition with respect to natural numbers serving as both objects and dimensions respectively.
However when it comes to conventional mathematical understanding, we are offered but a reduced - and ultimately highly distorted - interpretation of the nature of number.
So rather than two aspects - quantitative and qualitative - being explicitly recognised that are of equal importance, the qualitative is effectively reduced to the quantitative aspect in a very limited manner.
And then rather that the more accurate relative appreciation of number, as the consequent dynamic interaction as between opposite polarities, the inaccurate notion of number as absolutely existing is an abstract manner has now unfortunately become firmly embedded in consciousness. Indeed it will take the greatest revolution yet witnessed in our intellectual history to enable the required conversion in viewpoint to a truly dynamic perspective.
For what I am really addressing here is the urgent need now now for a completely new kind of mathematical appreciation that can in no way can be incorporated as some kind of extension of existing understanding.
It may help to provide further context for this dynamic interpretation of number to now look at the issue more closely again in terms of the psychological dynamics that underlie number experience.
All phenomenal experience - including of course mathematical - is conditioned by fundamental sets of opposite polarities.
The first set relates to external and internal poles. So what we identify as objectively existing strictly has no meaning independent of the internal mental constructs used to interpret such experience.
Thus in truth, experience represents a continual dynamic interaction with respect to objects as external and corresponding mental constructs, which relatively of an internal nature.
Thus all objective understanding necessarily reflects a certain (arbitrary) interpretation with respect to such experience.
So for example the belief that numbers represent absolute objects enjoying an abstract existence in a universal mathematical Heaven, ultimately reflects a distorted interpretation (i.e. that numbers can have an objective existence independent of interpretation).
Therefore once we recognise the necessary interaction of (external) objects with (internal) interpretation, mathematical reality must then be understood in a dynamic relative fashion.
The second set of key polarities relates to the interaction of whole and part, which equally manifests itself in terms of quantitative and qualitative aspects that are general and specific with respect to each other.
It is this second set that is directly relevant to the dynamic interpretation of number here outlined.
In the experiential interaction of polarities, we always necessarily experience these poles - to a degree that can greatly vary - as both independent from us and yet interdependent with us in experience.
For example to recognise and object as external, we thereby need to experience it as - relatively - independent from us. However it cannot be experienced as completely independent as this would exclude any mental interaction (thereby eliminating the possibility of experience).
Therefore, with respect to our conscious recognition, we experience the object as independent; however with respect to our unconscious recognition (of which we may not be explicitly aware) both external and internal aspects are recognised as interdependent with us.
Then with respect to mathematical understanding, the independent aspect (of conscious recognition) is directly identified in analytic fashion with rational understanding (strictly linear rational understanding).
However the corresponding interdependent aspect (of unconscious recognition) is directly identified in holistic manner with intuitive appreciation (which indirectly can be given a paradoxical circular rational expression) .
Therefore the recognition that number has dual aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other, equally implies the incorporation of (conscious) reason with unconscious (intuition).
Therefore Mathematics can no longer be identified as merely a conscious rational pursuit. Rather the much greater requirement now exists to successfully reconcile both (conscious) reason with (unconscious) intuition with respect to all mathematical understanding .
Though intuition may indeed be informally recognised as important for creative mathematical discovery, in formal terms it is reduced to reason. Thus the failure to recognise both quantitative (analytic) and qualitative (holistic) aspects of number as clearly distinct, directly parallels the corresponding failure to explicitly recognise (conscious) reason and (unconscious) intuition as distinct with respect to understanding.
The relationship between number as representing base and dimensional objects respectively, corresponds directly in psychological terms with the corresponding relationship as between perceptions and concepts.
In the dynamics of understanding, when a number perception takes place in quantitative, terms, a complementary conceptual recognition of that number takes place in a qualitative manner.
So when the natural number perception is of a cardinal nature, the corresponding conceptual recognition is - relatively - ordinal.
However. equally when the number perception takes place in a qualitative manner, the complementary conceptual recognition is qualitative.
Therefore when the natural number perception is now ordinal in nature, the corresponding conceptual recognition is - relatively - cardinal.
In this manner, through the dynamics of experience we keep switching as between both cardinal and ordinal recognition of natural numbers in base and dimensional number terms.
In other words we can recognise natural numbers as applying to both specific objects and general dimensions in both a cardinal and ordinal manner!
However when a distorted interpretation is imposed on this number process, whereby only the quantitative aspects is recognised (thereby explicitly conforming to linear rational interpretation), it sets severe limits with respect to the nature of interaction than can take place. Remember smooth interaction requires recognition of complementary quantitative and qualitative aspects!
So in conventional mathematical terms, the interaction between opposite polarities (quantitative and qualitative) becomes so rigid that the qualitative aspect is no longer even recognised.
The mistaken belief in a merely quantitative interpretation (corresponding to linear rational interpretation) then prevails.
This has become so ingrained - representing the deep unrecognised shadow of Conventional Mathematics - that I would thereby expect enormous resistance with respect to the appropriate dynamic interpretation of number (that properly concurs with experience) .
Therefore from a comprehensive perspective, every number is thereby defined in terms of both a base and dimensional value (where with respect to the expression ab, a is the base and b the dimensional value respectively.
From this perspective the natural numbers can be defined with respect to both Type 1 and Type 2 aspects respectively.
With the Type 1 aspect, where the base aspect varies (as quantity), the (default) dimensional number is 1; with the Type 2 aspect, by contrast, where the dimensional aspect (as quality) varies, the default base number is 1.
The cardinal notion of number is then directly associated from this perspective with the Type 1 aspect; the ordinal notion is then directly associated with the Type 2.
Crucially therefore both the cardinal notion (implying independence) with respect to number quantities and the ordinal notion (implying, by contrast, interdependence with respect to the qualitative relationship between numbers) represent two distinct types of number appreciation, which are dynamically relative in nature with respect to each other.
However it is equally possible to switch reference frames. Then with respect to the natural numbers of the Type 1 aspect, the base value now varies and is defined in qualitative (holistic) terms with respect to a (default) dimensional value of 1; then, in relative terms, the dimensional aspect varies, and is now defined in quantitative (analytic) terms with respect to a default base value of 1.
In this way, both base and dimensional values take on quantitative and qualitative meanings that dynamically interact with each other in a complementary fashion.
And if you think that this seems somewhat far-fetched, let me remind you that this is what continually takes place with respect to our experience of number!
So from one perspective we are able to recognise objects in natural number fashion with respect to their cardinal (quantitative) characteristics. So if I point to a group of 3 people on the street, such recognition entails quantitative recognition of number. If however I now specifically identify - say in terms of a criterion such as height - 1st, 2nd and 3rd members of that group - this now entails corresponding ordinal recognition of a qualitative nature.
And likewise it is similar with respect to numbers, now serving as dimensions. So the recognition of 3 dimensions (now serving as more general categories within which specif number objects can be defined) implies cardinal recognition of a quantitative kind. However the distinction as between the 1st, 2nd and 3rd dimensions (according to some criterion) implies ordinal recognition of a qualitative kind.
So with respect to actual experience we effortlessly switch as between both quantitative (analytic) and qualitative (holistic) recognition with respect to natural numbers serving as both objects and dimensions respectively.
However when it comes to conventional mathematical understanding, we are offered but a reduced - and ultimately highly distorted - interpretation of the nature of number.
So rather than two aspects - quantitative and qualitative - being explicitly recognised that are of equal importance, the qualitative is effectively reduced to the quantitative aspect in a very limited manner.
And then rather that the more accurate relative appreciation of number, as the consequent dynamic interaction as between opposite polarities, the inaccurate notion of number as absolutely existing is an abstract manner has now unfortunately become firmly embedded in consciousness. Indeed it will take the greatest revolution yet witnessed in our intellectual history to enable the required conversion in viewpoint to a truly dynamic perspective.
For what I am really addressing here is the urgent need now now for a completely new kind of mathematical appreciation that can in no way can be incorporated as some kind of extension of existing understanding.
It may help to provide further context for this dynamic interpretation of number to now look at the issue more closely again in terms of the psychological dynamics that underlie number experience.
All phenomenal experience - including of course mathematical - is conditioned by fundamental sets of opposite polarities.
The first set relates to external and internal poles. So what we identify as objectively existing strictly has no meaning independent of the internal mental constructs used to interpret such experience.
Thus in truth, experience represents a continual dynamic interaction with respect to objects as external and corresponding mental constructs, which relatively of an internal nature.
Thus all objective understanding necessarily reflects a certain (arbitrary) interpretation with respect to such experience.
So for example the belief that numbers represent absolute objects enjoying an abstract existence in a universal mathematical Heaven, ultimately reflects a distorted interpretation (i.e. that numbers can have an objective existence independent of interpretation).
Therefore once we recognise the necessary interaction of (external) objects with (internal) interpretation, mathematical reality must then be understood in a dynamic relative fashion.
The second set of key polarities relates to the interaction of whole and part, which equally manifests itself in terms of quantitative and qualitative aspects that are general and specific with respect to each other.
It is this second set that is directly relevant to the dynamic interpretation of number here outlined.
In the experiential interaction of polarities, we always necessarily experience these poles - to a degree that can greatly vary - as both independent from us and yet interdependent with us in experience.
For example to recognise and object as external, we thereby need to experience it as - relatively - independent from us. However it cannot be experienced as completely independent as this would exclude any mental interaction (thereby eliminating the possibility of experience).
Therefore, with respect to our conscious recognition, we experience the object as independent; however with respect to our unconscious recognition (of which we may not be explicitly aware) both external and internal aspects are recognised as interdependent with us.
Then with respect to mathematical understanding, the independent aspect (of conscious recognition) is directly identified in analytic fashion with rational understanding (strictly linear rational understanding).
However the corresponding interdependent aspect (of unconscious recognition) is directly identified in holistic manner with intuitive appreciation (which indirectly can be given a paradoxical circular rational expression) .
Therefore the recognition that number has dual aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other, equally implies the incorporation of (conscious) reason with unconscious (intuition).
Therefore Mathematics can no longer be identified as merely a conscious rational pursuit. Rather the much greater requirement now exists to successfully reconcile both (conscious) reason with (unconscious) intuition with respect to all mathematical understanding .
Though intuition may indeed be informally recognised as important for creative mathematical discovery, in formal terms it is reduced to reason. Thus the failure to recognise both quantitative (analytic) and qualitative (holistic) aspects of number as clearly distinct, directly parallels the corresponding failure to explicitly recognise (conscious) reason and (unconscious) intuition as distinct with respect to understanding.
The relationship between number as representing base and dimensional objects respectively, corresponds directly in psychological terms with the corresponding relationship as between perceptions and concepts.
In the dynamics of understanding, when a number perception takes place in quantitative, terms, a complementary conceptual recognition of that number takes place in a qualitative manner.
So when the natural number perception is of a cardinal nature, the corresponding conceptual recognition is - relatively - ordinal.
However. equally when the number perception takes place in a qualitative manner, the complementary conceptual recognition is qualitative.
Therefore when the natural number perception is now ordinal in nature, the corresponding conceptual recognition is - relatively - cardinal.
In this manner, through the dynamics of experience we keep switching as between both cardinal and ordinal recognition of natural numbers in base and dimensional number terms.
In other words we can recognise natural numbers as applying to both specific objects and general dimensions in both a cardinal and ordinal manner!
However when a distorted interpretation is imposed on this number process, whereby only the quantitative aspects is recognised (thereby explicitly conforming to linear rational interpretation), it sets severe limits with respect to the nature of interaction than can take place. Remember smooth interaction requires recognition of complementary quantitative and qualitative aspects!
So in conventional mathematical terms, the interaction between opposite polarities (quantitative and qualitative) becomes so rigid that the qualitative aspect is no longer even recognised.
The mistaken belief in a merely quantitative interpretation (corresponding to linear rational interpretation) then prevails.
This has become so ingrained - representing the deep unrecognised shadow of Conventional Mathematics - that I would thereby expect enormous resistance with respect to the appropriate dynamic interpretation of number (that properly concurs with experience) .
Monday, January 26, 2015
Intricacies of Addition and Multiplication (1)
The apparently simple operation of multiplication reveals in a glaring fashion the stark inadequacies of conventional mathematical interpretation. In fact the true nature of multiplication cannot be properly explained from this perspective!
The starting point in rectifying this problem is the clear recognition that - far from being static absolute identities - all numbers are inherently dynamic in nature entailing the interaction of twin complementary aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other.
For example the number 2 is a cardinal number with a recognised independent quantitative meaning.
This implies that it is defined in a manner that renders it as without any meaningful qualitative (i.e. relational) context.
If numbers were indeed independent in an absolute manner, then by definition, no means would exist for establishing their interdependence (i.e. relationship) with other numbers!
So what we have in conventional mathematical terms is but a reduced interpretation of number. Thus the critical issue of establishing a relationship with other numbers - which is inherently of a qualitative nature - is effectively ignored!
So what is completely overlooked in formal conventional terms, is that every number - which I am illustrating here in the specific case of 2 - has strictly but a relative meaning with both quantitative and qualitative aspects that dynamically interact with each other in a complementary (opposite) manner.
So once again the number "2" has a quantitative (analytic) meaning. So for example, if I look out my front window and see two cars in the driveway, I am using "2" in its accepted quantitative sense.
However "2" has an equally important meaning that is of a qualitative (holistic) nature which can be referred to as "twoness" (i.e. the quality of "2").
Now whereas the former aspect relates to the independent nature of number recognition in a quantitative manner, the latter aspect - by contrast - relates to the corresponding interdependent nature of number which thereby provides for number its crucial relationship context.
In this way through the interaction of both quantitative and qualitative aspects, we are enabled to recognise number relationships in relative fashion, entailing both independence (from other numbers as distinct entities) and interdependence (through common relationship with these other numbers) respectively.
Now when we define the component "building blocks" of "2" in a quantitative manner, we do so in a manner that strictly lacks any qualitative distinction. To facilitate further exposition, I will refer to this as the Type 1 aspect of number!
So from the Type 1 perspective, 2 = 1 + 1, Here each unit is defined in a strictly homogeneous manner (where neither can be meaningfully distinguished from each other).
This begs the very fundamental question is how one is enabled to form number recognition of "2", given that its unit components are defined in an independent manner! So once again the all important qualitative aspect,whereby meaningful relationship as between separate units can be established,is entirely missing from conventional mathematical interpretation.
When we look at the latter aspect of number (i.e. Type 2), we arrive at a complementary appreciation of number that is qualitative (and thereby strictly lacking any quantitative distinction).
Then from the Type 2 perspective 2 (now reflecting its qualitative nature as "twoness") enables us to clearly distinguish the unique nature of both its units.
So here, 2 = 1st + 2nd, where each unit is now uniquely defined in an ordinal manner.
.
It is vitally important to realise in this context that both the cardinal and ordinal nature of number are directly identified with the two aspects of number (Type 1 and Type 2) respectively, which are crucially distinct from each other. This once again highlights the significantly reduced accepted interpretation, where both cardinal and ordinal aspects are absolutely treated in a merely (Type 1)) quantitative manner.
We are now ready to make the all important leap, whereby the true distinctive nature of addition and multiplication can be clearly revealed.
Because we are now recognising two aspects to number (that dynamically interact in experiential understanding), all numbers are defined in a twin manner containing both a base and a dimensional aspect.
Furthermore to move from Type 1 to Type 2 recognition we invert both aspects in a complementary manner.
So in Type 1 terms the number "2" is more comprehensively defined as 21. So the base here is 2 and the dimensional aspect 1.
The conventional treatment of natural numbers effectively views numbers solely in Type 1 terms (where they are given an absolute interpretation).
So 1, 2, 3, 4, ... can be more fully defined as 11, 21, 31, 41...
However because the reduced quantitative value in each case remains unchanged (when the dimensional number is 1), the implied default dimension (i.e. 1) is omitted altogether.
Furthermore whenever number expressions entail dimensional values (powers or exponents) other than 1, the ultimate value is given in a reduced quantitative manner (defined in terms of 1 as dimension).
So for example, in conventional mathematical terms 22 = 4 (i.e.41). So though in geometrical terms, we can easily see that this would represent 4 square (i.e. 2-dimensional) units - rather than 4 linear (1-dimensional) units, this qualitative change in the nature of the units is simply ignored, with the resulting value i.e. 4 given in a merely reduced quantitative manner (i.e. in 1-dimensional terms).
In Type 2 terms however the number "2" is defined in a complementary manner as 12 (where base and dimensional aspects are switched).
Thus the dimensional number here now varies (with respect to a fixed base number of 1) directly indicating the true qualitative nature of the number "2". Now once again this relates to the qualitative recognition of 2 as a number group whereby both 1st and 2nd members can be uniquely distinguished in an ordinal fashion.
However from a conventional perspective, the Type 2 aspect of number seems pointless, as the reduced quantitative value of each number = 1 (i.e. 11).
Finally in this entry, I wish to highlight the dynamic nature of interpretation that is now required (when we attempt to reconcile both Type 1 and Type 2 aspects with each other.
Expressing it simply, the numbers representing base and dimensional values respectively are always opposite to each other.
Therefore if - as we have seen with the Type 1 aspect - the base number is defined in a quantitative (analytic) manner, the corresponding (default) dimensional number is now - relatively - of a qualitative (holistic) nature.
However if by contrast - as we have seen with the Type 2 aspect - the dimensional number is defined in a qualitative (holistic) manner, the corresponding (default) base number (i.e. 1) is then of a quantitative (analytic) nature.
However, just as in the manner that the directions of left and right turns at a crossroads are reversed when we approach it from an opposite direction, likewise, when the polar frame of reference switches with respect to number the base number can now take on a qualitative (holistic) meaning, while the dimensional number - now relatively - is of a quantitative (analytic) nature.
We will elaborate further in the next entry!
The starting point in rectifying this problem is the clear recognition that - far from being static absolute identities - all numbers are inherently dynamic in nature entailing the interaction of twin complementary aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other.
For example the number 2 is a cardinal number with a recognised independent quantitative meaning.
This implies that it is defined in a manner that renders it as without any meaningful qualitative (i.e. relational) context.
If numbers were indeed independent in an absolute manner, then by definition, no means would exist for establishing their interdependence (i.e. relationship) with other numbers!
So what we have in conventional mathematical terms is but a reduced interpretation of number. Thus the critical issue of establishing a relationship with other numbers - which is inherently of a qualitative nature - is effectively ignored!
So what is completely overlooked in formal conventional terms, is that every number - which I am illustrating here in the specific case of 2 - has strictly but a relative meaning with both quantitative and qualitative aspects that dynamically interact with each other in a complementary (opposite) manner.
So once again the number "2" has a quantitative (analytic) meaning. So for example, if I look out my front window and see two cars in the driveway, I am using "2" in its accepted quantitative sense.
However "2" has an equally important meaning that is of a qualitative (holistic) nature which can be referred to as "twoness" (i.e. the quality of "2").
Now whereas the former aspect relates to the independent nature of number recognition in a quantitative manner, the latter aspect - by contrast - relates to the corresponding interdependent nature of number which thereby provides for number its crucial relationship context.
In this way through the interaction of both quantitative and qualitative aspects, we are enabled to recognise number relationships in relative fashion, entailing both independence (from other numbers as distinct entities) and interdependence (through common relationship with these other numbers) respectively.
Now when we define the component "building blocks" of "2" in a quantitative manner, we do so in a manner that strictly lacks any qualitative distinction. To facilitate further exposition, I will refer to this as the Type 1 aspect of number!
So from the Type 1 perspective, 2 = 1 + 1, Here each unit is defined in a strictly homogeneous manner (where neither can be meaningfully distinguished from each other).
This begs the very fundamental question is how one is enabled to form number recognition of "2", given that its unit components are defined in an independent manner! So once again the all important qualitative aspect,whereby meaningful relationship as between separate units can be established,is entirely missing from conventional mathematical interpretation.
When we look at the latter aspect of number (i.e. Type 2), we arrive at a complementary appreciation of number that is qualitative (and thereby strictly lacking any quantitative distinction).
Then from the Type 2 perspective 2 (now reflecting its qualitative nature as "twoness") enables us to clearly distinguish the unique nature of both its units.
So here, 2 = 1st + 2nd, where each unit is now uniquely defined in an ordinal manner.
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It is vitally important to realise in this context that both the cardinal and ordinal nature of number are directly identified with the two aspects of number (Type 1 and Type 2) respectively, which are crucially distinct from each other. This once again highlights the significantly reduced accepted interpretation, where both cardinal and ordinal aspects are absolutely treated in a merely (Type 1)) quantitative manner.
We are now ready to make the all important leap, whereby the true distinctive nature of addition and multiplication can be clearly revealed.
Because we are now recognising two aspects to number (that dynamically interact in experiential understanding), all numbers are defined in a twin manner containing both a base and a dimensional aspect.
Furthermore to move from Type 1 to Type 2 recognition we invert both aspects in a complementary manner.
So in Type 1 terms the number "2" is more comprehensively defined as 21. So the base here is 2 and the dimensional aspect 1.
The conventional treatment of natural numbers effectively views numbers solely in Type 1 terms (where they are given an absolute interpretation).
So 1, 2, 3, 4, ... can be more fully defined as 11, 21, 31, 41...
However because the reduced quantitative value in each case remains unchanged (when the dimensional number is 1), the implied default dimension (i.e. 1) is omitted altogether.
Furthermore whenever number expressions entail dimensional values (powers or exponents) other than 1, the ultimate value is given in a reduced quantitative manner (defined in terms of 1 as dimension).
So for example, in conventional mathematical terms 22 = 4 (i.e.41). So though in geometrical terms, we can easily see that this would represent 4 square (i.e. 2-dimensional) units - rather than 4 linear (1-dimensional) units, this qualitative change in the nature of the units is simply ignored, with the resulting value i.e. 4 given in a merely reduced quantitative manner (i.e. in 1-dimensional terms).
In Type 2 terms however the number "2" is defined in a complementary manner as 12 (where base and dimensional aspects are switched).
Thus the dimensional number here now varies (with respect to a fixed base number of 1) directly indicating the true qualitative nature of the number "2". Now once again this relates to the qualitative recognition of 2 as a number group whereby both 1st and 2nd members can be uniquely distinguished in an ordinal fashion.
However from a conventional perspective, the Type 2 aspect of number seems pointless, as the reduced quantitative value of each number = 1 (i.e. 11).
Finally in this entry, I wish to highlight the dynamic nature of interpretation that is now required (when we attempt to reconcile both Type 1 and Type 2 aspects with each other.
Expressing it simply, the numbers representing base and dimensional values respectively are always opposite to each other.
Therefore if - as we have seen with the Type 1 aspect - the base number is defined in a quantitative (analytic) manner, the corresponding (default) dimensional number is now - relatively - of a qualitative (holistic) nature.
However if by contrast - as we have seen with the Type 2 aspect - the dimensional number is defined in a qualitative (holistic) manner, the corresponding (default) base number (i.e. 1) is then of a quantitative (analytic) nature.
However, just as in the manner that the directions of left and right turns at a crossroads are reversed when we approach it from an opposite direction, likewise, when the polar frame of reference switches with respect to number the base number can now take on a qualitative (holistic) meaning, while the dimensional number - now relatively - is of a quantitative (analytic) nature.
We will elaborate further in the next entry!
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