In yesterday's blog entry, I discusses - using the number 3 to illustrate - how its four distinctive meanings could be represented in mathematical terms.
This implies a dynamic interactive form of understanding where both a base and dimensional number are used in conjunction with each other. When the number i.e. 3 in this case is used to denote the base aspect, the corresponding dimensional aspect is given as 1; however when 3 is then used to denote the dimensional aspect, the corresponding base aspect is now 1.
Then in each case either the base or dimensional number is explicit in conscious understanding, with the other aspect - which is dynamically complementary - playing a merely implicit role in an unconscious manner.
So understanding keeps switching as between both conscious and unconscious recognition with respect to base and dimensional number respectively.
Here I wish to trace out more precisely the psychological dynamics that are involved with respect to such number recognition.
First we have the rational analytic perception of "3", which equates directly with 1), in yesterday's blog entry. This corresponds to the specific recognition in an explicit conscious quantitative manner of the number object "3" (as having a distinct individual identity).
Again I have denoted this as 31. This entails however the implicit unconscious recognition of 1 representing a dimension. In other words without this implicit recognition of the 1st dimension as potentially applying to all possible natural numbers, we would not be able to identify "3" in an explicit manner.
Next we have the intuitive holistic perception of "3", which equates directly with 2), in yesterday's entry. This corresponds to the explicit general recognition in an unconscious qualitative manner of the number object "3" (as applying to all actual classes of 3).
I have identified this in inverse fashion as 31. This again entails the implicit conscious recognition of 1 as representing the 1st dimension now identified in rational fashion as applying to all finite natural numbers. So once again without this implicit recognition of the 1st dimension, we could not collectively identify 3 with different classes (of 3 objects).
Then we have the rational analytic concept of "3" which equates directly with 3) in yesterday's entry.
This corresponds to the explicit general recognition in a conscious quantitative manner of the number "3" as representing dimension (i.e. as comprising 3 linear dimensions).
I have denoted this as 13. This again entails the explicit conscious recognition of "3" representing 3 (linear) dimensions. However once again without the implicit recognition of 1, where 1 now has a qualitative holistic meaning giving each unit an individual uniqueness this recognition of 3 dimensions would not be possible (as this requires uniquely identifying each dimension as length width and height respectively
Finally we have the intuitive holistic concept of "3"which equates directly with 4) in yesterday's entry. This corresponds to the explicit general recognition in an unconscious qualitative manner of "3" representing dimension of potentially applying in an infinite fashion to each of its 3 directions.
I have denoted this as 13. This again entails the explicit unconscious recognition of 3 representing 3 (circular) dimensions. However without the implicit recognition of 1, the arbitrary relative position of each dimension (as 1st, 2nd and 3rd respectively) would not be possible. In other words the 1st dimension must be fixed in a relatively - independent fashion before the other two dimensions can be related to it.
Therefore to conclude each number continually alternates in dynamic interactive fashion as between its analytic and holistic expression with respect to both base and dimensional aspects; this corresponds directly in complementary fashion with the likewise interaction of rational and intuitive expression with respect to both perceptions and concepts respectively.
Friday, April 24, 2015
Thursday, April 23, 2015
Reflections on Number (3)
We will show here how each of the four meanings of a number can be represented in mathematical terms.
What is crucial here - in what properly represents a dynamic interactive form of understanding - is to express every number with respect to a dimension.
So in the general case ab, a represents the base and b the dimensional number respectively.
Thus now using once again the number 3 to illustrate we will go through the 4 distinct meanings
1) This again is the standard quantitative interpretation of 3 as representing a cardinal number.
This can be written as 31. So here the emphasis is explicitly on 3 as the base quantity.
Because of complementarity this means that the dimensional number 1 is merely implicit enabling 3 to be uniquely identified (from all possible members on number line).
So 3 as base is quantitative (in explicit terms); 1 as dimension is qualitative (in implicit terms).
2) This corresponds to our second definition in yesterday's entry where 3 as base number now explicitly takes on a holistic qualitative meaning as the notion of "threeness" which enables the collective identification of any group containing 3 members.
This can be written as 31. So here the emphasis is explicitly on 1 as the dimensional quantity (i.e. applying to all members on the number line).
Then the emphasis on 3 is now implicit where 3 has a unique qualitative meaning that is potentially infinite.
Notice how in the case of 1) 3 represents a specific number quantity; however by contrast in the case of 2), 3 now represents a holistic number quality (applicable to all possible groups of 3 members).
3) We now switch to ordinal notions
3 now takes on the meaning of a distinct group of 3 members that is explicitly defined in terms of its 1st, 2nd and 3rd members. This thereby represents a qualitative meaning of 3 that is actually finite.
This can be represented as 13. So the emphasis here is explicitly on 3 as the dimensional number in qualitative terms which implies that implicitly the base number of 1 is understood in a quantitative manner. What this implies is that before we can rank members of a group ordinally (i.e. in qualitative terms) we must implicitly recognise each as a separate unit (in a quantitative manner).
4) We finally have the notion of ordinal identity that can be applies collectively to any number of groups (with 3 members).
This is written as 13. Here each group of 3 is identified explicitly as separate unit (which then is implicitly recognised as containing members that are arranged in an ordinal fashion). This in fact represents a quantitative meaning of 1 that is potentially infinite.
Therefore what happens in the dynamics of experience is that the number 3 here keeps switching as between its cardinal and ordinal meanings in both an actual finite and potentially infinite manner.
Alternatively it keeps switching as between quantitative and qualitative meanings in both an analytic and holistic fashion collectively.
What is crucial here - in what properly represents a dynamic interactive form of understanding - is to express every number with respect to a dimension.
So in the general case ab, a represents the base and b the dimensional number respectively.
Thus now using once again the number 3 to illustrate we will go through the 4 distinct meanings
1) This again is the standard quantitative interpretation of 3 as representing a cardinal number.
This can be written as 31. So here the emphasis is explicitly on 3 as the base quantity.
Because of complementarity this means that the dimensional number 1 is merely implicit enabling 3 to be uniquely identified (from all possible members on number line).
So 3 as base is quantitative (in explicit terms); 1 as dimension is qualitative (in implicit terms).
2) This corresponds to our second definition in yesterday's entry where 3 as base number now explicitly takes on a holistic qualitative meaning as the notion of "threeness" which enables the collective identification of any group containing 3 members.
This can be written as 31. So here the emphasis is explicitly on 1 as the dimensional quantity (i.e. applying to all members on the number line).
Then the emphasis on 3 is now implicit where 3 has a unique qualitative meaning that is potentially infinite.
Notice how in the case of 1) 3 represents a specific number quantity; however by contrast in the case of 2), 3 now represents a holistic number quality (applicable to all possible groups of 3 members).
3) We now switch to ordinal notions
3 now takes on the meaning of a distinct group of 3 members that is explicitly defined in terms of its 1st, 2nd and 3rd members. This thereby represents a qualitative meaning of 3 that is actually finite.
This can be represented as 13. So the emphasis here is explicitly on 3 as the dimensional number in qualitative terms which implies that implicitly the base number of 1 is understood in a quantitative manner. What this implies is that before we can rank members of a group ordinally (i.e. in qualitative terms) we must implicitly recognise each as a separate unit (in a quantitative manner).
4) We finally have the notion of ordinal identity that can be applies collectively to any number of groups (with 3 members).
This is written as 13. Here each group of 3 is identified explicitly as separate unit (which then is implicitly recognised as containing members that are arranged in an ordinal fashion). This in fact represents a quantitative meaning of 1 that is potentially infinite.
Therefore what happens in the dynamics of experience is that the number 3 here keeps switching as between its cardinal and ordinal meanings in both an actual finite and potentially infinite manner.
Alternatively it keeps switching as between quantitative and qualitative meanings in both an analytic and holistic fashion collectively.
Wednesday, April 22, 2015
Reflections on Number (2)
Once again I am going to illustrate 4 distinct meanings of number - illustrating with respect to the number 3 - before then showing that are in all inextricably linked in experience.
1) We start with the standard cardinal notion of 3 which represents the accepted quantitative notion of number e.g. 3 cups on a table. Number here is given an analytic independent identity (without qualitative distinction).
2) Here we have a very distinctive notion of 3 as now collectively applying to all groups (containing 3 members). So 3 can apply to 1, 2, 3, 4,.... groups without limit.
Now enormous confusion exists in Mathematics with both 1) and 2) generally confused with each other.
This is a crucially important point as the proper understanding of multiplication depends on this distinction.
Now again using 3 in the first sense might notice 3 cups and later - say - 3 letters in the hallway and perhaps then 3 cars in the driveway.
However strictly the recognition of 3 in each case would necessarily remain independent of each other.
So therefore the crucial factor in being able to establish a connection as between each group is the recognition that 3 now plays - as in 2) - a collective role (i.e. as what is common to each group).
As I say these two meanings with respect to number are intimately tied up with the process of multiplication.
Imagine two rows of coins laid out in rectangular fashion with 3 coins in each row.
Now from a multiplicative perspective we would represent this as 3 * 2.
So what is involved here is the initial recognition of 3 coins (in each row) in an independent manner.
Now if we only recognised the notion of 3 as independent as in 1) then we could only represent the total number of coins in an additive manner as 3 + 3 (where again both are interpreted in an independent manner).
However multiplication requires that we likewise recognise 3 as interdependent in a collective sense. This thereby enables us to see the common relationship as between each row.
Therefore in the relationship 3 * 2, 2 now indicates the common rather than independent notion of 2.
In this sense multiplication necessarily entails both 1) and 2) with respect to the interpretation of number with the first number relating to 1) and all subsequent multipliers to 2).
However we also have two further meanings of 3.
3) in this case 3 represents an individual group where the 3 members are all related to each other in an ordinal manner. This is the corresponding qualitative notion of number where the members of the group are identified as 1st, 2nd and 3rd respectively.
So with 1) 3 = 1 + 1 + 1; however with 3) 3 = 1st,+ 2nd + 3rd
4) In this case 3 no takes on a collective meaning where again it refers to the common recognition of all groups of 3 (where each is defined in a qualitative ordinal manner).
Thus in its simplest terms 4) represents at an ordinal level what 2) was earlier seen to represent at a cardinal level.
Now in dynamic experiential terms, it is impossible to separate these meanings, for they all imply each other in a complementary manner.
Thus the recognition of 3 explicitly in a cardinal manner implies the corresponding implicit recognition of 3 in an ordinal fashion.
Equally in reverse the explicit recognition of 3 in ordinal terms, implies the corresponding implicit recognition of 3 in a cardinal manner.
Likewise in a similar fashion, explicit recognition in individual terms implies implicit recognition in collective terms and explicit recognition in collective terms implies implicit recognition in an individual manner.
1) We start with the standard cardinal notion of 3 which represents the accepted quantitative notion of number e.g. 3 cups on a table. Number here is given an analytic independent identity (without qualitative distinction).
2) Here we have a very distinctive notion of 3 as now collectively applying to all groups (containing 3 members). So 3 can apply to 1, 2, 3, 4,.... groups without limit.
Now enormous confusion exists in Mathematics with both 1) and 2) generally confused with each other.
This is a crucially important point as the proper understanding of multiplication depends on this distinction.
Now again using 3 in the first sense might notice 3 cups and later - say - 3 letters in the hallway and perhaps then 3 cars in the driveway.
However strictly the recognition of 3 in each case would necessarily remain independent of each other.
So therefore the crucial factor in being able to establish a connection as between each group is the recognition that 3 now plays - as in 2) - a collective role (i.e. as what is common to each group).
As I say these two meanings with respect to number are intimately tied up with the process of multiplication.
Imagine two rows of coins laid out in rectangular fashion with 3 coins in each row.
Now from a multiplicative perspective we would represent this as 3 * 2.
So what is involved here is the initial recognition of 3 coins (in each row) in an independent manner.
Now if we only recognised the notion of 3 as independent as in 1) then we could only represent the total number of coins in an additive manner as 3 + 3 (where again both are interpreted in an independent manner).
However multiplication requires that we likewise recognise 3 as interdependent in a collective sense. This thereby enables us to see the common relationship as between each row.
Therefore in the relationship 3 * 2, 2 now indicates the common rather than independent notion of 2.
In this sense multiplication necessarily entails both 1) and 2) with respect to the interpretation of number with the first number relating to 1) and all subsequent multipliers to 2).
However we also have two further meanings of 3.
3) in this case 3 represents an individual group where the 3 members are all related to each other in an ordinal manner. This is the corresponding qualitative notion of number where the members of the group are identified as 1st, 2nd and 3rd respectively.
So with 1) 3 = 1 + 1 + 1; however with 3) 3 = 1st,+ 2nd + 3rd
4) In this case 3 no takes on a collective meaning where again it refers to the common recognition of all groups of 3 (where each is defined in a qualitative ordinal manner).
Thus in its simplest terms 4) represents at an ordinal level what 2) was earlier seen to represent at a cardinal level.
Now in dynamic experiential terms, it is impossible to separate these meanings, for they all imply each other in a complementary manner.
Thus the recognition of 3 explicitly in a cardinal manner implies the corresponding implicit recognition of 3 in an ordinal fashion.
Equally in reverse the explicit recognition of 3 in ordinal terms, implies the corresponding implicit recognition of 3 in a cardinal manner.
Likewise in a similar fashion, explicit recognition in individual terms implies implicit recognition in collective terms and explicit recognition in collective terms implies implicit recognition in an individual manner.
Monday, April 13, 2015
Reflections on Number (1)
Once again we return to the crucially important notion of number to discover that it has laden with a great deal of hidden subtlety which needs to be carefully deciphered before coherent mathematical interpretation is possible.
Let us illustrate for example with respect to the number "2".
1) Now 2 has an accepted quantitative meaning in specific analytic terms.
So for example I identify 2 cars in my driveway, I am using number in this conventional sense.
Thus here 2 = 1 + 1 where the individual units are literally understood as homogeneous, without qualitative distinction (i.e. no unique relationship to each other).
Put another way, this represents the interpretation of the number as an independent entity (integer) in an impersonal individual manner.
2) However 2 equally has a quantitative meaning in general holistic terms. This equates with the dimensional (rather than the base notion of 2). So in the dimensional expression ab, a is the base and b the dimensional number respectively!
So if I for example identify classes of objects with respect to the stipulation that each contains 2 members (e.g. 2 cars, 2 chairs, 2 names etc, then I am using 2 in this collective holistic sense where it can apply to any number of object groups (of 2).
Thus the crucial distinction here is that 2 now serves a collective - rather than individual - role in identifying a number property (i.e. 2) that is common to all classes defined in an impersonal collective manner.
In the two examples so far we have defined the number 2 in a quantitative manner (with respect to both its specific (analytic) and collective (holistic) properties.
However we can now equally define 2 in qualitative terms with respect to both aspects.
3) So 2 now is a number with a qualitative meaning in specific analytic terms. We could refer to this quality of 2 as "twoness" which thereby gives the number a unique personal identity.
Now whereas the quantitative counterpart notion of 2 is defined in a cardinal manner so that 2 = 1 + 1, this corresponding qualitative notion is defined by contrast in an ordinal manner.
Thus it is understood here that 2 = 1st + 2nd members (that are qualitatively distinct).
Therefore whereas the quantitative notion of 2 (as an independent integer) entails no unique relationship between units) the qualitative notion by contrast implies a relationship of interdependence as between units (where each is uniquely distinct).
4) Finally 2 equally has a qualitative meaning in general holistic terms.
So we are now referring to the number 2 once again in a dimensional sense, but where it now is identified in ordinal terms as a number identifier with respect to a collection of groups.
In other words according to agreed criteria we could identify a number of different groups with respect to unique 1st and 2nd members respectively. So in this sense all the groups share the same qualitative identity of "twoness".
Put more simply, numbers representing both base and dimensional aspects respectively, repeatedly switch as between cardinal and ordinal meaning (in a quantitative and qualitative manner).
So again 2 as the base aspect has a cardinal interpretation (in quantitative terms) with a specific application to an independent individual entity.
However 2 also representing a dimension (power or exponent) has a cardinal interpretation (in quantitative terms) with a holistic application (as applying in common to all instances of 2).
Then 2, again as base aspect, has an ordinal interpretation (in qualitative terms) with application to the two distinct members of an individual group (as 1st and 2nd respectively).
Finally 2 now representing a dimension has an ordinal interpretation in qualitative terms with a holistic aspect (as applying to all distinct instances of two unique members.
Conventional Mathematics however is riddled throughout with a gross form of reductionism, whereby the qualitative aspect of appreciation is continually interpreted in a quantitative manner (indicating a corresponding failure to properly distinguish finite and infinite notions).
Likewise the holistic aspect of appreciation (where number carries a collective sense) likewise is reduced in a merely analytic type manner (with a merely individual interpretation).
Thus instead of the number 2, as in my example, being given at least 4 distinctive meanings (that dynamically interact in experience), in conventional mathematical terms it is given but a grossly reduced interpretation (i.e.where the qualitative aspect is reduced to the quantitative and the holistic aspect to the analytic).
We will develop these insights further in the next entry.
Let us illustrate for example with respect to the number "2".
1) Now 2 has an accepted quantitative meaning in specific analytic terms.
So for example I identify 2 cars in my driveway, I am using number in this conventional sense.
Thus here 2 = 1 + 1 where the individual units are literally understood as homogeneous, without qualitative distinction (i.e. no unique relationship to each other).
Put another way, this represents the interpretation of the number as an independent entity (integer) in an impersonal individual manner.
2) However 2 equally has a quantitative meaning in general holistic terms. This equates with the dimensional (rather than the base notion of 2). So in the dimensional expression ab, a is the base and b the dimensional number respectively!
So if I for example identify classes of objects with respect to the stipulation that each contains 2 members (e.g. 2 cars, 2 chairs, 2 names etc, then I am using 2 in this collective holistic sense where it can apply to any number of object groups (of 2).
Thus the crucial distinction here is that 2 now serves a collective - rather than individual - role in identifying a number property (i.e. 2) that is common to all classes defined in an impersonal collective manner.
In the two examples so far we have defined the number 2 in a quantitative manner (with respect to both its specific (analytic) and collective (holistic) properties.
However we can now equally define 2 in qualitative terms with respect to both aspects.
3) So 2 now is a number with a qualitative meaning in specific analytic terms. We could refer to this quality of 2 as "twoness" which thereby gives the number a unique personal identity.
Now whereas the quantitative counterpart notion of 2 is defined in a cardinal manner so that 2 = 1 + 1, this corresponding qualitative notion is defined by contrast in an ordinal manner.
Thus it is understood here that 2 = 1st + 2nd members (that are qualitatively distinct).
Therefore whereas the quantitative notion of 2 (as an independent integer) entails no unique relationship between units) the qualitative notion by contrast implies a relationship of interdependence as between units (where each is uniquely distinct).
4) Finally 2 equally has a qualitative meaning in general holistic terms.
So we are now referring to the number 2 once again in a dimensional sense, but where it now is identified in ordinal terms as a number identifier with respect to a collection of groups.
In other words according to agreed criteria we could identify a number of different groups with respect to unique 1st and 2nd members respectively. So in this sense all the groups share the same qualitative identity of "twoness".
Put more simply, numbers representing both base and dimensional aspects respectively, repeatedly switch as between cardinal and ordinal meaning (in a quantitative and qualitative manner).
So again 2 as the base aspect has a cardinal interpretation (in quantitative terms) with a specific application to an independent individual entity.
However 2 also representing a dimension (power or exponent) has a cardinal interpretation (in quantitative terms) with a holistic application (as applying in common to all instances of 2).
Then 2, again as base aspect, has an ordinal interpretation (in qualitative terms) with application to the two distinct members of an individual group (as 1st and 2nd respectively).
Finally 2 now representing a dimension has an ordinal interpretation in qualitative terms with a holistic aspect (as applying to all distinct instances of two unique members.
Conventional Mathematics however is riddled throughout with a gross form of reductionism, whereby the qualitative aspect of appreciation is continually interpreted in a quantitative manner (indicating a corresponding failure to properly distinguish finite and infinite notions).
Likewise the holistic aspect of appreciation (where number carries a collective sense) likewise is reduced in a merely analytic type manner (with a merely individual interpretation).
Thus instead of the number 2, as in my example, being given at least 4 distinctive meanings (that dynamically interact in experience), in conventional mathematical terms it is given but a grossly reduced interpretation (i.e.where the qualitative aspect is reduced to the quantitative and the holistic aspect to the analytic).
We will develop these insights further in the next entry.
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)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!
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!
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.
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