Even as a child I had difficulties with the interpretation of the square root of 1.
Once again in conventional (Type 1) mathematical terms the answer would be given as either + 1 or - 1. Of course in practice to avoid ambiguity with respect to two possible answers the positive answer would be chosen as the principle value and in calculations involving square roots only this would be used.
So for example if you pick up a calculator, enter 1 and then the square root function the answer is 1 (and not - 1).
However there is something deeply unsatisfactory regarding such interpretation.
Mathematics prides itself on its unambiguous nature and yet here in one of the simplest operations possible, two answers that are diametrically opposite - literally poles apart - are deemed to be correct.
For example in Mathematics a proposition is deemed to be either true or false. Thus the Riemann Hypothesis for example is deemed to be either true or false. Of course no one has been able to prove either the positive or negative truth value. So the Hypothesis is still open waiting to be proved or disproved in Type 1 terms. (The great irony here is that in fact - as I have repeatedly stated - that the Riemann Hypothesis cannot be be adequately understood in Type 1 terms!)
So if we assign + 1 to the positive truth value of a proposition and - 1 to the negative truth value, then to maintain that that the proposition can be either + 1 or - 1 is to state that the proposition is both true and false!
And such ambiguity (or paradox) strikes at the very heart of mathematical type truth.
Yet, in the simple case of the square root of 1, such ambiguity is directly accepted in quantitative terms (from the conventional Type 1 mathematical perspective).
The root of this problem is very simple and relates directly to the fact that when for example we square a number, a qualitative as well as quantitative transformation in the units involved takes place. However in Type 1 terms, the qualitative transformation is simply ignored (and reduced to quantitative interpretation in 1-dimensional terms).
So when we square 1 (i.e. 1^2) the answer is indeed 1 but in square i.e. 2-dimensional units. However once again the qualitative aspect is ignored so that 1^2 is interpreted as 1^1.
Now in inverse terms if we obtain the square root of 1^2, the answer is indeed 1 i.e.1^1.
Therefore when interpreted in Type 2 mathematical terms 1^1 and 1^2 represent two distinct numbers (in qualitative terms).
So + 1 properly represents the square root of 1^2 and - 1 the square root of 1^1.
Therefore we can see quite simply that the ambiguity arising from multi varied solutions relates directly to the lack of a distinct qualitative dimension (in Type 1 terms).
We have not finished yet. So far we have shown how the square root can be interpreted without ambiguity in quantitative terms. However there is a vital qualitative corollary.
Put simply there is an inverse relationship as between quantitative and qualitative interpretations.
So if D represents the dimensional value in quantitative terms, then 1/D represents the corresponding value in a qualitative manner.
So 1^(1/2) = - 1 (in quantitative terms).
Therefore 1^2 = - 1 (in a qualitative manner).
Now - 1 (in quantitative terms) represents a point on the circle of unit radius.
Likewise therefore - 1 (in qualitative terms) represents a point on the circle of unit radius.
In other words - 1 here represents a qualitative means of interpretation (Type 2) that is circular in nature.
What this circular interpretation in turn entails is a dynamic both/and logic (rather than a static linear either/or logic).
So in dynamic terms - 1 here implies the negation of + 1 (i.e. is both + 1 and - 1 simultaneously) So 2-dimensional circular interpretation can be expressed as the dynamic complementarity of (paradoxical) opposites.
What this implies in turn is that a comprehensive (Type 3) interpretation of the square root of 1 requires not only quantitative values applying to dimensional exponents of 1 and 1/2, but equally qualitative interpretations applying to 1 and 2 dimensions respectively.
In other words properly understood in Type 3 terms, we can only understand the square root (in quantitative terms) through equally understanding 2 (as a dimension) in a corresponding qualitative manner.
Therefore to appropriately understand the quantitative value (- 1) on the circle of unit radius, we must equally provide the corresponding qualitative interpretation of - 1 (in a circular manner).
And again the relationship between both is of a direct inverse nature in dimensional terms. So the square root of 1 (i.e. through raising 1 to the dimension 1/2) is - 1. This is inversely related to the the dimension 2 (i.e. through raising 1 to the dimension 2) in qualitative terms which equally is - 1.
Finally we can perhaps now see the root of the confusion that arises from the conventional (Type 1) interpretation.
Maintaining that the square root of 1 can be either + 1 or - 1 arises from the simple failure to recognise the qualitative role of dimension. Because only the default 1st dimension is recognised (in qualitative terms) in Type 1 terms, then when using this approach, paradoxical meaning (pertaining to the 2nd dimension) is misleadingly portrayed in a reduced linear manner.
And this is the root of the whole problem of multi varied solutions in Type 1 Mathematics which cannot be satisfactorily resolved till the qualitative (Type 2) aspect is explicitly recognised.
Wednesday, January 25, 2012
Monday, January 23, 2012
Generalising New Approach
As we have seen number can have both quantitative and qualitative characteristics. Therefore in a comprehensive (Type 3) approach a number is longer a number in static terms but rather has twin aspects that are quantitative and qualitative respectively.
So as we saw in the last post 2^2 = 4, 2.
In other words the 1st number here refers to its quantitative characteristic (which defines the conventional approach).
The 2nd number defines its qualitative characteristic (which in this case is 2-dimensional).
Now the quantitative and qualitative characteristics are linear and circular with respect to each other. Thus 4 (as quantity) is a linear number (lying on the real number line).
2 (as quality) is a circular number with its corresponding square root (lying on the circle of unit radius).
In Type 1 terms only the quantitative aspect is recognised.
So 2^2 = 4
More fully this could be written as 2^2 = 4, 1.
So the Type 1 approach is characterised by a situation where the default qualitative (or dimensional) number = 1.
In Type 2 terms only the qualitative aspect is recognised.
So 2^2 = 2.
More fully this would be written as 1, 2
In other words the Type 2 approach is characterised by the situation where the default quantitative number is always 1!
Now one might ask what these qualitative or (dimensional) numbers actually signify!
Well, each number actually defines a unique logical approach with which to interpret mathematical relationships.
Once again default qualitative case of 1 defines the conventional linear rational approach to Mathematics (which is one-dimensional or more precisely one-directional).
For example it is assumed in conventional terms that a proposition is either true or false. However this approach simply defines 1-dimensional logic.
For all other (higher) dimensions a merely relative truth value results. For example in the simplest case of 2-dimensional logic we have the complementarity of opposite polar directions.
For example if truth is defined by the two-way interaction of external and internal poles, then objective truth (which implies only one of these poles) has no strict meaning. Rather truth now represents a dynamic (switching) relationship as between both poles.
All higher dimensions involve a circular relationship as between poles. For example in 4-dimensional terms we have complementary opposites in both real (horizontal) and imaginary (vertical) directions.
So we here have experience switching between positive and negative poles with respect to real (conscious) and imaginary (indirectly conscious i.e. as representation of the unconscious) respectively. So we can perhaps appreciate here that the 4 holistic dimensions here are the qualitative counterpart of the four roots of unity (in quantitative terms).
So as we saw in the last post 2^2 = 4, 2.
In other words the 1st number here refers to its quantitative characteristic (which defines the conventional approach).
The 2nd number defines its qualitative characteristic (which in this case is 2-dimensional).
Now the quantitative and qualitative characteristics are linear and circular with respect to each other. Thus 4 (as quantity) is a linear number (lying on the real number line).
2 (as quality) is a circular number with its corresponding square root (lying on the circle of unit radius).
In Type 1 terms only the quantitative aspect is recognised.
So 2^2 = 4
More fully this could be written as 2^2 = 4, 1.
So the Type 1 approach is characterised by a situation where the default qualitative (or dimensional) number = 1.
In Type 2 terms only the qualitative aspect is recognised.
So 2^2 = 2.
More fully this would be written as 1, 2
In other words the Type 2 approach is characterised by the situation where the default quantitative number is always 1!
Now one might ask what these qualitative or (dimensional) numbers actually signify!
Well, each number actually defines a unique logical approach with which to interpret mathematical relationships.
Once again default qualitative case of 1 defines the conventional linear rational approach to Mathematics (which is one-dimensional or more precisely one-directional).
For example it is assumed in conventional terms that a proposition is either true or false. However this approach simply defines 1-dimensional logic.
For all other (higher) dimensions a merely relative truth value results. For example in the simplest case of 2-dimensional logic we have the complementarity of opposite polar directions.
For example if truth is defined by the two-way interaction of external and internal poles, then objective truth (which implies only one of these poles) has no strict meaning. Rather truth now represents a dynamic (switching) relationship as between both poles.
All higher dimensions involve a circular relationship as between poles. For example in 4-dimensional terms we have complementary opposites in both real (horizontal) and imaginary (vertical) directions.
So we here have experience switching between positive and negative poles with respect to real (conscious) and imaginary (indirectly conscious i.e. as representation of the unconscious) respectively. So we can perhaps appreciate here that the 4 holistic dimensions here are the qualitative counterpart of the four roots of unity (in quantitative terms).
Saturday, January 21, 2012
Three Mathematical Interpretations
I will now attempt to demonstrate how the the result of the simple expression,
2^2 would be interpreted in Type 1 (Conventional), Type 2 (Holistic) and Type 3 (Comprehensive) mathematical terms.
Now in Type 1 terms 2^2 = 4 (i.e. 4^1).
So what we obtain here is a numerical result in a reduced quantitative manner.
Now if we were to represent this in geometrical terms, 2^2 would represent a (2-dimensional) square of side 2 units. Therefore its area would = 4 square units.
In other words, though a qualitative change in the nature of the units involved, results from squaring (i.e. in moving from a 1-dimensional to a 2-dimensional format, this is simply ignored in Type 1 terms. Thus from a reduced quantitative perspective the result of 2^2 is indeed 4.
So correctly we could write the result 4, 2 with 4 representing the quantitative and 2 the qualitative dimensional aspect respectively the answer is given as 4, 1 (where the 1 which is merely a default value can be ignored).
In Type 2 terms 2^2 = 2 (i.e. 1^2).
Here - in reverse fashion - we look at the qualitative change that has taken place (while ignoring the quantitative value).
Now though we used a square to illustrate 2 dimensional reality in the previous part, this itself is but a reduced notion based on linear extension of what corresponds to 1-dimensional interpretation.
Now as we have no direct interest in the quantitative transformation brought about from 2^2 but rather in the qualitative transformation involved we can replace the quantity by 1 to obtain 1^2. So whereas Type 1 Mathematics is 1-dimensional in nature, Type 2 Mathematics is by contrast 1-quantitative so that the dimensional number involved is expressed with respect to the default base quantity 1!
Now the key to mathematically expressing the significance of the dimensional number D is to express 1 with respect to its inverse i.e. 1/D.
Thus when we replace in this case D with 1/D we obtain 1^(1/2) = - 1. Now this represents a quantitative value on the circle of unit radius (in the complex plane).
Now this same number - 1 can be given a coherent qualitative meaning that corresponds to 2 as a dimension (i.e. 2-dimensional interpretation).
In holistic mathematical terms + 1 implies the positing of conscious form; conventional mathematical reason implies such positing!
However - 1 holistically implies the negation of unitary form and relates directly to the unconscious intuitive aspect of understanding.
So 2-dimensional understanding relates directly to the intuitive aspect of understanding by which qualitative holistic connections are made in understanding. However this is inevitably reduced in Type 1 Mathematics to linear (1-dimensional) rational format.
In Type 3 terms 2^2 = 4, 2
In other words Type 3 Mathematics combines both Type 1 and Type 2 understanding.
Therefore form one perspective we recognise the quantitative transformation implied by the expression 2^2 i.e. 4 (which is expressed with respect to the default dimensional number of 1). However equally we recognise the qualitative transformation implied by 2^2 i.e. 2 as dimensional number (which is expressed with respect to the default quantitative number of 1).
So understanding here becomes of an increasingly interactive nature whereby one continually moves from quantitative to qualitative (and qualitative to quantitative understanding). In psychological terms this would imply the balanced mix of reason and (higher-level) intuition. Remember again that in Type 1 Mathematics intuition - though present - is inevitably reduced to reason!
The implications here are truly enormous!
Type 1 Mathematics is entirely defined in terms of just one logical system of interpretation (which in precise holistic mathematical terms is 1-dimensional).
However as a unique system of interpretation is associated with every number (as dimension), this implies that Mathematics can be given an unlimited number of logical interpretations (with each interpretation implying a unique manner of configuring reason and intuition).
So in the expression 2^2 this implies 2-dimensional interpretation. Now as conscious rational (1-dimensional) implies positing (+), corresponding unconscious intuitive understanding comes from the negation (of what has been posited). So correctly 2-dimensional interpretation implies - in paradoxical rational terms - the complementarity of opposites (where what is true is merely relative, as positive and negative at the same time.
2-dimensional mathematical interpretation thereby implies a more refined appreciation of mathematical symbols where for example a number has now two directions in understanding (which are positive and negative with respect to each other).
In other words the number 2 can be looked on as an external object (which is posited in experience) or as an internal construct (which arises through negation of the external aspect). In other words with 2-dimensional understanding one keeps switching as between external object and internal perception in a dynamic interactive manner.
So a number - by definition - at this level of understanding does not statically exist (as an independent object) but rather has a merely relative existence through the dynamic interaction of both positive and negative polarities of experience.
2^2 would be interpreted in Type 1 (Conventional), Type 2 (Holistic) and Type 3 (Comprehensive) mathematical terms.
Now in Type 1 terms 2^2 = 4 (i.e. 4^1).
So what we obtain here is a numerical result in a reduced quantitative manner.
Now if we were to represent this in geometrical terms, 2^2 would represent a (2-dimensional) square of side 2 units. Therefore its area would = 4 square units.
In other words, though a qualitative change in the nature of the units involved, results from squaring (i.e. in moving from a 1-dimensional to a 2-dimensional format, this is simply ignored in Type 1 terms. Thus from a reduced quantitative perspective the result of 2^2 is indeed 4.
So correctly we could write the result 4, 2 with 4 representing the quantitative and 2 the qualitative dimensional aspect respectively the answer is given as 4, 1 (where the 1 which is merely a default value can be ignored).
In Type 2 terms 2^2 = 2 (i.e. 1^2).
Here - in reverse fashion - we look at the qualitative change that has taken place (while ignoring the quantitative value).
Now though we used a square to illustrate 2 dimensional reality in the previous part, this itself is but a reduced notion based on linear extension of what corresponds to 1-dimensional interpretation.
Now as we have no direct interest in the quantitative transformation brought about from 2^2 but rather in the qualitative transformation involved we can replace the quantity by 1 to obtain 1^2. So whereas Type 1 Mathematics is 1-dimensional in nature, Type 2 Mathematics is by contrast 1-quantitative so that the dimensional number involved is expressed with respect to the default base quantity 1!
Now the key to mathematically expressing the significance of the dimensional number D is to express 1 with respect to its inverse i.e. 1/D.
Thus when we replace in this case D with 1/D we obtain 1^(1/2) = - 1. Now this represents a quantitative value on the circle of unit radius (in the complex plane).
Now this same number - 1 can be given a coherent qualitative meaning that corresponds to 2 as a dimension (i.e. 2-dimensional interpretation).
In holistic mathematical terms + 1 implies the positing of conscious form; conventional mathematical reason implies such positing!
However - 1 holistically implies the negation of unitary form and relates directly to the unconscious intuitive aspect of understanding.
So 2-dimensional understanding relates directly to the intuitive aspect of understanding by which qualitative holistic connections are made in understanding. However this is inevitably reduced in Type 1 Mathematics to linear (1-dimensional) rational format.
In Type 3 terms 2^2 = 4, 2
In other words Type 3 Mathematics combines both Type 1 and Type 2 understanding.
Therefore form one perspective we recognise the quantitative transformation implied by the expression 2^2 i.e. 4 (which is expressed with respect to the default dimensional number of 1). However equally we recognise the qualitative transformation implied by 2^2 i.e. 2 as dimensional number (which is expressed with respect to the default quantitative number of 1).
So understanding here becomes of an increasingly interactive nature whereby one continually moves from quantitative to qualitative (and qualitative to quantitative understanding). In psychological terms this would imply the balanced mix of reason and (higher-level) intuition. Remember again that in Type 1 Mathematics intuition - though present - is inevitably reduced to reason!
The implications here are truly enormous!
Type 1 Mathematics is entirely defined in terms of just one logical system of interpretation (which in precise holistic mathematical terms is 1-dimensional).
However as a unique system of interpretation is associated with every number (as dimension), this implies that Mathematics can be given an unlimited number of logical interpretations (with each interpretation implying a unique manner of configuring reason and intuition).
So in the expression 2^2 this implies 2-dimensional interpretation. Now as conscious rational (1-dimensional) implies positing (+), corresponding unconscious intuitive understanding comes from the negation (of what has been posited). So correctly 2-dimensional interpretation implies - in paradoxical rational terms - the complementarity of opposites (where what is true is merely relative, as positive and negative at the same time.
2-dimensional mathematical interpretation thereby implies a more refined appreciation of mathematical symbols where for example a number has now two directions in understanding (which are positive and negative with respect to each other).
In other words the number 2 can be looked on as an external object (which is posited in experience) or as an internal construct (which arises through negation of the external aspect). In other words with 2-dimensional understanding one keeps switching as between external object and internal perception in a dynamic interactive manner.
So a number - by definition - at this level of understanding does not statically exist (as an independent object) but rather has a merely relative existence through the dynamic interaction of both positive and negative polarities of experience.
Thursday, January 12, 2012
New Role of Mathematics
What I am attempting to portray here is a greatly enlarged vision of Mathematics where both standard (quantitative) and the - as yet - unrecognised holistic (qualitative) aspects are involved.
Indeed I outline three broad areas with respect to this enlarged vision.
Type 1 relates to the existing established mathematical approach which is specifically geared to deal in a direct manner with quantitative type relationships,
Type 2 then relates to the unrecognised holistic mathematical approach which is geared to deal with qualitative type relationships to in an indirect rational manner.
Whereas Type 1 Mathematics is based on the linear (1-dimensional) use of reason in an unambiguous either/or fashion, by contrast Type 2 Mathematics is based on circular (higher dimensional) use of reason in a paradoxical both/and manner. So the direct basis of holistic type understanding is intuitive type insight. Now in standard linear rational terms such intuition is merely reduced to reason with Mathematics viewed formally as a merely rational (1-dimensional) discipline.
However in higher dimensional understanding, intuition cannot be reduced in this manner. However quite amazingly all mathematical symbols now acquire a new holistic meaning as a means of indirectly conveying the qualitative relationships involved.
Type 3 is then the most comprehensive form of Mathematics where both Type 1 (quantitative) and the Type 2 (qualitative) aspects dynamically interact.
Now there are a couple of interesting observations that can be made on this New Mathematics.
In a certain sense one can validly claim that all reality is indeed - ultimately - of a mathematical nature i.e. when one includes both the quantitative and qualitative interpretation of its symbols (such as number).
Now this might appear to be reductionist in the sense that affective artistic type appreciation is allowed no role.
However this is not the case! Whereas Type 1 Mathematics is directly of a scientific nature (and indirectly artistic), Type 2 Mathematics is directly of an artistic nature (and indirectly scientific).
In other words the kind of intuitive experience that directly informs holistic mathematical appreciation itself directly arises from artistic type sensibility.
So one cannot successfully hope to encode holistic type appreciation of reality (in an indirect circular rational manner) without already attaining direct affective experience of such reality.
Therefore, science and art are intimately related at a dynamic level of understanding.
Indeed one could also include the religious quest in this mix.
When one understands properly the true nature of Mathematics it becomes clear that is all based on a massive act of faith in the validity of its procedures.
Now this is missed in conventional mathematical terms (as it ignores the true nature of infinite holistic type relationships) this simply entails - as in all mathematical proof - that the infinite is simply reduced to finite interpretation!
However recently, I have been at pains to emphasise that the true implication of the Riemann Hypothesis (which cannot be conventionally proved or disproved) is that for Mathematics to ultimately proceed, a massive act of faith must be placed in the ultimate identity of both the quantitative and qualitative interpretation of its symbols.
So when we look at it in this manner - properly understood - Mathematics in its true enlarged state, entails the balanced mix of both cognitive (scientific), affective (artistic) and volitional (spiritual/religious) meaning.
One other crucial area related to the role of Mathematics and Physics.
As we have seen the ultimate nature of reality is mathematical (in this expanded sense). This is event in the manner in which the mystical traditions speak of such reality as union (1) or emptiness (0).
Physical reality - where phenomena emerge - relates to the breaking of the original perfect mathematical symmetry (where no distinction exists as between the quantitative and qualitative aspects of its symbols).
However as soon as this perfect symmetry is broken (where quantitative considerations become to a degree separated), the physical reality of relative phenomena in space and time comes into existence.
So one way of looking at the goal of physical science is to recognise ultimately its pure mathematical nature (in an ineffable manner where quantitative and qualitative aspects are identical).
Indeed I outline three broad areas with respect to this enlarged vision.
Type 1 relates to the existing established mathematical approach which is specifically geared to deal in a direct manner with quantitative type relationships,
Type 2 then relates to the unrecognised holistic mathematical approach which is geared to deal with qualitative type relationships to in an indirect rational manner.
Whereas Type 1 Mathematics is based on the linear (1-dimensional) use of reason in an unambiguous either/or fashion, by contrast Type 2 Mathematics is based on circular (higher dimensional) use of reason in a paradoxical both/and manner. So the direct basis of holistic type understanding is intuitive type insight. Now in standard linear rational terms such intuition is merely reduced to reason with Mathematics viewed formally as a merely rational (1-dimensional) discipline.
However in higher dimensional understanding, intuition cannot be reduced in this manner. However quite amazingly all mathematical symbols now acquire a new holistic meaning as a means of indirectly conveying the qualitative relationships involved.
Type 3 is then the most comprehensive form of Mathematics where both Type 1 (quantitative) and the Type 2 (qualitative) aspects dynamically interact.
Now there are a couple of interesting observations that can be made on this New Mathematics.
In a certain sense one can validly claim that all reality is indeed - ultimately - of a mathematical nature i.e. when one includes both the quantitative and qualitative interpretation of its symbols (such as number).
Now this might appear to be reductionist in the sense that affective artistic type appreciation is allowed no role.
However this is not the case! Whereas Type 1 Mathematics is directly of a scientific nature (and indirectly artistic), Type 2 Mathematics is directly of an artistic nature (and indirectly scientific).
In other words the kind of intuitive experience that directly informs holistic mathematical appreciation itself directly arises from artistic type sensibility.
So one cannot successfully hope to encode holistic type appreciation of reality (in an indirect circular rational manner) without already attaining direct affective experience of such reality.
Therefore, science and art are intimately related at a dynamic level of understanding.
Indeed one could also include the religious quest in this mix.
When one understands properly the true nature of Mathematics it becomes clear that is all based on a massive act of faith in the validity of its procedures.
Now this is missed in conventional mathematical terms (as it ignores the true nature of infinite holistic type relationships) this simply entails - as in all mathematical proof - that the infinite is simply reduced to finite interpretation!
However recently, I have been at pains to emphasise that the true implication of the Riemann Hypothesis (which cannot be conventionally proved or disproved) is that for Mathematics to ultimately proceed, a massive act of faith must be placed in the ultimate identity of both the quantitative and qualitative interpretation of its symbols.
So when we look at it in this manner - properly understood - Mathematics in its true enlarged state, entails the balanced mix of both cognitive (scientific), affective (artistic) and volitional (spiritual/religious) meaning.
One other crucial area related to the role of Mathematics and Physics.
As we have seen the ultimate nature of reality is mathematical (in this expanded sense). This is event in the manner in which the mystical traditions speak of such reality as union (1) or emptiness (0).
Physical reality - where phenomena emerge - relates to the breaking of the original perfect mathematical symmetry (where no distinction exists as between the quantitative and qualitative aspects of its symbols).
However as soon as this perfect symmetry is broken (where quantitative considerations become to a degree separated), the physical reality of relative phenomena in space and time comes into existence.
So one way of looking at the goal of physical science is to recognise ultimately its pure mathematical nature (in an ineffable manner where quantitative and qualitative aspects are identical).
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