In my previous entries, I have stressed that every number can be given both an analytic and holistic interpretation respectively and that in the dynamics of experience, both aspects are inevitably intertwined, with one made explicit in conscious manner, with the other remaining - relatively - implicit in unconscious fashion. .
And each number can likewise be given a base identity or a dimensional identity respectively.
So once more illustrating with respect to the dimensional aspect, the number 3 has an analytic interpretation with respect to 3 (referring to 3 dimensions in a quantitative manner). However 3 equally has a qualitative interpretation as the "threeness" or the quality of 3, which thereby enables the common identification of all members relating to a class of 3 dimensions (such as the length width and height measurements of different rooms).
However one might wish to probe further as to the precise difference as between the quantitative and qualitative interpretations.
So again, if I for example refer to the 3 dimensions with respect to the room of a house (length, width and height) this represents the accepted quantitative view.
However in conventional terms the distinct identity of a number (such as 3) used with respect to objects is not properly distinguished from what is used for dimensions.
But there are crucial differences. 3 as used for objects has a finite specific meaning i.e. as 3 unit objects). i.e. 3 = 1 + 1 + 1
However 3 as used for dimensions has by contrast a collective general meaning. Here each unit (i.e. separate dimension) applies potentially to every possible natural number in an infinite manner). So one more, length, width and height measurements could apply to 1, 2, 3, 4,.......rooms.
There is also another key difference:
When we use 3 in the restricted finite sense (where each unit applies to just one actual object) the units are treated as independent and homogeneous.
So when 3 = 1 + 1 + 1, the relationship between units is not considered.
However as far as dimensional "units" are concerned, this is not really the case. Here the units are not in fact independent but are related to each other in an ordered fashion as length, width and height respectively.
Thus treating the units as independent gives them a reduced meaning. Now it is true that from a quantitative perspective, that if we have 3 dimensions for a room, as length width and height respectively, the total volume will be the same (irrespective of the order in which they are taken).
So in fact when we multiply numbers a dimensional aspect is always involved. However in reduced quantitative terms this is ignored so that 2 * 3 * 5 for example = 30 (with no reference to the dimensional change involved).
In other words when we multiply 2 * 3 * 5 in this way, it is as if we accept that these measurements thereby belong to to the same dimension. So for example if we recognise the length as the only dimension, then 2 * 3 * 5 thereby represents the 3 numbers multiplied with respect to the same dimension. So the answer is thereby given in 1-dimensional terms.
So the very key to recognising higher dimensions (> 1) is that such dimensions by their very nature are not absolutely independent of each other, but must exist with respect to each other in an orderly manner.
So what we are faced with all the time is a constant dialectic as between notions of independence and interdependence respectively.
With independence, we view the units as quantitative in a cardinal manner.
So again 3 = 1 + 1 + 1.
However with interdependence, we view the units as qualitative in an ordinal fashion.
So here 3 = 1st + 2nd + 3rd.
And in experiential terms with respect to understanding, these two notions are necessarily of a relative nature in a dynamic complementary manner.
Thus we can only explicitly recognise cardinal units as independent (in an explicit quantitative manner), if we already implicitly recognise a corresponding ordinal relationship between units (in a qualitative manner).
Likewise we can only explicitly recognise ordinal units as interdependent (in a qualitative manner) if we already implicitly recognise a corresponding cardinal relationship between units (in a quantitative manner).
So the key issue then relates to how we can successfully convert as between qualitative and quantitative notions respectively.
Tuesday, April 28, 2015
Friday, April 24, 2015
Reflections on Number (4)
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.
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.
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.
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