Wednesday, July 20, 2011

Holistic Mathematical Dimensions (1)

As we have seen dimensional numbers as used in conventional Type 1 Mathematics are given a reduced quantitative meaning (that conceals their true nature).

Therefore to unravel the true qualitative meaning of dimensions requires the use of a holistic Type 2 mathematical approach.

Furthermore this holistic approach has direct implications for appreciation of the true nature of space and time in both physical and psychological terms (which from this qualitative perspective are inherently complementary).

The starting basis for this new appreciation of dimensions is the realisation that all phenomenal reality is conditioned by 3 sets of polarities.

The first of these relates to the fact that in experience phenomena necessarily entail the interaction of polarities that are - relatively - external and internal with respect to each other. Thus to experience any phenomenon (e.g. mathematical symbol objectively as external requires a corresponding subjective mental construct that is thereby - relatively - of an internal nature.

Now if we denote conscious phenomena in holistic mathematical terms as "real" then all phenomena are conditioned by opposite polarities that are - relatively - positive and negative (with respect to each other).

The second set of polarities relates to the further fact that in experience all phenomena switch as between opposite aspects that are whole and part with respect to each other.

Now in conscious experience a basic reductionism takes place whereby wholes and parts are reduced in quantitative terms (with the whole in any context thereby representing the sum of its constituent parts).
This indeed leads to the experience of holons i.e. whole/parts where the qualitative distinction of whole and part is not maintained.

To properly distinguish whole and part in qualitative terms thereby requires unconscious (as well as conscious) recognition.
Ultimately qualitative recognition entails a spiritual element to understanding. Then with pure intuitive type recognition from one perspective each individual part can uniquely mediate Spirit (as an archetypal individual symbol). Likewise from the other perspective the collective whole can likewise mediate spirit (as an archetypal collective symbol).

In holistic mathematical terms the relationship as between conscious and unconscious is real to imaginary.

So now when we combine the two sets of polarities with respect to external/internal and individual/collective aspects we have - relatively - both real and imaginary distinctions in a positive and negative manner.

The third set of polarities relates directly to the distinction as between form and emptiness in recognition of the fact that - properly understood - all phenomenal experience entails both material and spiritual aspects.

This third set actually entails a special case of the two former sets which can be explained as follows.

The geometrical way of representing these polarity sets is with respect to a circle of unit radius (in the complex plane) .

Now to represent the first set we draw a straight horizontal line through the centre to meet the circumference on both sides. So if the right hand radius is + 1, then the left hand radius is - 1.

To represent the second set we now draw a straight line vertically through the centre so as to again meet the circumference of the circle on both sides.

So if the upper (vertical) radius is + i, then - relatively - the bottom radius is - i.

Finally to represent the third set we draw a straight line diagonally (from left to right) through the centre (at an equal distance from both horizontal and vertical lines).

Now if the upper diagonal line (or radius) is 1/k (1 + i) where k = square root of 2, then the lower part (as radius) is 1/k (- 1 - i).

Now if we choose the other direction for the diagonal line from bottom right to upper left, then the corresponding representations will be 1/k(1 - i) and 1/k (- 1 + i) respectively.

So the holistic mathematical representation of the 3 polarity sets corresponds in structural terms to the 2nd, 4th and 8th roots of 1 respectively.

The key to qualitative dimensional appreciation is the recognition that these quantitative roots bear an inverse relationship with their corresponding dimensions in qualitative terms.

So in this sense all experience is fundamentally structured according to the holistic qualitative meaning of mathematical dimensions.

And as such experience relates to both physical and psychological aspects of understanding, this thereby entails both physical and psychological reality (which are complementary) are fundamentally structured in accordance with Type 2 mathematical appreciation of dimensional numbers.

Monday, July 18, 2011

Importance of Dimensions

As we have seen, numbers representing dimensions (or powers) are of a qualitatively different nature than corresponding number quantities.

These dimensions then give rise to a circular number system (of a qualitative nature) that is inversely related to their corresponding roots (in quantitative terms).

The importance of this number system is that it provides the appropriate basis for a true appreciation of the nature of space and time.

So properly understood in this context, the nature of space and time is of a direct mathematical nature (relating however to Type 2 rather than Type 1 appreciation).

Furthermore such appreciation relates to both the physical and psychological aspects of reality (both of which are complementary in nature).

Indeed we can use Type 2 appreciation to explain conventional interpretation of the nature of space and time.

From a qualitative Type 2 perspective, Type 1 Mathematics is of a linear (1-dimensional) form.

Therefore when this is applied to space and time, it leads to one of these being treated as qualitative (with the remaining 3 seen as quantitative).

This therefore is consistent with the conventional perspective whereby objects are treated as 3-dimensional in quantitative spatial terms, with the remaining dimension as - relatively - qualitative as time (though of course time also has an indirect quantitative aspect which can be measured).

However once we realise that every number (apart from 1) equally has a qualitative dimensional inetrpretation, this opens up the way for an entirely new appreciation of space and time (of which conventional interpretation represents but one limited case)!

And in all of these other cases a dynamic complementary type relationship exists as between both the physical and psychological aspects of space and time.

Thursday, July 14, 2011

The Number System Revisited

Earlier we outlined two number systems that are quantitative (linear) and qualitative (circular) with respect to each other.

The first natural number system which defines Type 1 Mathematics is of a quantitative nature (with a default 1-dimensional interpretation).

1^1, 2^1, 3^1, 4^1,.......

The second natural number system which defines Type 2 Mathematics is of a qualitative nature (with the default number 1 raised to successive dimensional numbers).

1^1, 1^2, 1^3, 1^4,.......

This leads to a circular number system that structurally is obtained through taking successive roots of 1 (and then defining results in an appropriate qualitative manner).

Right away this suggests that the two systems are in fact interdependent for the quantitative interpretation of roots is dynamically inseparable from the qualitative interpretation of notion of their corresponding dimensions. So for example the second (square) root of 1 cannot be properly conceived in the absence of corresponding 2-dimensional interpretation!

However we can now widen both Type 1 and Type 2 Mathematics to include imaginary - as well as real - numbers.

So in Type 1 terms the quantitative number system now includes both real and imaginary natural number terms:

1^1, 2^1, 3^1, 4^1,....... and

i^1, 2i^1, 3i^1, 4i^1,.......

Likewise in Type 2 terms the qualitative number system now includes both real and imaginary natural number terms (as dimensions):

1^1, 1^2, 1^3, 1^4,....... and

1^i, 1^2i, 1^3i, 1^4i,.......

So both real an imaginary numbers can be given both a quantitative and qualitative interpretation!

What is remarkable however is how - in dynamic interactive terms - interpretations alternate as between both their quantitative and qualitative aspects.

So dynamically speaking, real and imaginary are quantitative and qualitative with respect to each other.

We can see this readily from the fact in Type 2 Mathematics when 1 is raised to a real integer dimensional power it results in a qualitative number interpretation; however when in Type 1 Mathematics, 1 is raised to an imaginary integer dimensional power it results in a corresponding quantitative interpretation.

This once again strongly suggests that Type 1 and Type 2 Mathematics cannot be properly understood in isolation and in fact are interdependent.

So the full integration of both Type 1 and Type 2 Mathematics leads to the most comprehensive approach (which is Type 3 Mathematics).

Wednesday, July 13, 2011

Imaginary - Hidden Qualitative Notion

When appropriately understood a number representing a dimension or power is of qualitative nature relative to the number to which it is raised (which is thereby quantitative).

So if we take for example 1^2, in this context 1 is quantitative, whereby 2 is - relatively - of a qualitative nature.

And again in Type 2 Mathematics (i.e. where 1 is raised to a dimensional power) associated with each dimensional number is a unique qualitative result (that is similar in structure to the corresponding quantitative root of the number).

Thus in the important case where the dimension is 2 in Type 2 Mathematics,
1^2 = - 1 and when it is 4, 1^4 = i.

So both - 1 and i thereby have important qualitative interpretations.

Once again, - 1 relates to the negation of rational (unitary) form which is the basis in experience for generating empty (0) intuition. And in a more comprehensive mathematical understanding both reason and intuition must be formally incorporated!

In qualitative terms, i is of a more refined though extremely important nature.

Basically it represents a linear rational attempt to (indirectly) incorporate intuition in interpretation.

So quite literally if we are to attempt to express the 2nd dimension in (reduced) 1-dimensional terms then we obtain the square root of - 1 (which in fact is the definition of an imaginary number).

Therefore i in qualitative terms represents the incorporation of circular 2-dimensional type logic indirectly in a linear rational matter.

So an alternative way to express the fact that comprehensive interpretation requires both rational and intuitive modes of understanding is to say that comprehensive interpretation necessarily requires both real and imaginary rational components!

Now remarkably when 1 is raised to the power of i (i.e. 1^i) a whole range of quantitative type results ensues.

Now again in Type 1 Mathematics - in a complementary manner to the handling of roots - only the 1st result is generally considered and taken as the principle value.

So 1^i = {e^(2*pi*i)}^i = e^(- 2*pi) = .00186744...

However because in Type 1 Mathematics e^(2*pi*i) = 1, therefore {e^(2*pi*i)}^n = 1, where n = 1,2,3,4,....

So for example if n = 2, then 1^i = {e^(2*pi*i)}^i = {e^(4*pi*i)}^i

Therefore in the first case 1^i = .00186744...

However in the second case 1^i = .00000348734....

And a potentially unlimited set of answers can be generated in this manner by using alternative natural number values for n!

However once again from a Type 2 perspective {e^(2*pi*i)} ≠ {e^(4*pi*i)}.

Put another way 1^i ≠ 1^2i.

What is fascinating about this is that failure to recognise the qualitatively distinct nature of a number as dimension, ultimately leads to confusion not only in qualitative but equally in quantitative terms.

Once again both 1^i and 1^2i are distinct expressions with unique results in quantitative terms. However because of the lack of a coherent qualitative basis, it is led to misleadingly concluding that they are equivalent expressions!

Now it is also worth pointing out that i in qualitative terms represents an indirect linear rational means of representing what is inherently of a qualitative nature in quantitative terms.

Therefore in this context when i is used as a dimension (i.e. as power of 1) it takes on a reverse quantitative meaning which leads therefore to the generation of a quantitative result.

So 1 (raised to a real integral power) generates a result that is inherently of a qualitative nature!

However 1 (raised to an imaginary integral power) generates a result that is inherently of a quantitative nature!

And as the number and the dimension to which it is raised are always quantitative and qualitative (and qualitative and quantitative) with respect to each other, this means that when appropriately understood in Type 2 mathematical terms, in the expression,

1^ni (where n = 1, 2, 3,.....), 1 is now inherently of a qualitative nature with its corresponding imaginary power - relatively - quantitative.

So ultimately in a fully comprehensive mathematical understanding (Type 3), both Type 1 and Type 2 interpretation must be integrated to maintain consistency in both quantitative and qualitative terms.

Tuesday, July 12, 2011

Reformulating Basic Notions

We have seen how the notion of a root is problematic from a Type 1 mathematical perspective.

The reason for this is that such Mathematics attempts to view relationships from a merely (reduced) quantitative perspective.

However whenever powers (or roots) or multiplication (and division) are involved both qualitative and quantitative transformations in the nature of the variables take place.

We illustrated this point at length with respect to the simple expression 1^2 showing that here 2 (as the dimensional number) relates in qualitative terms to a new holistic manner of qualitatively interpreting relationships in logical terms.
This logical system - which has a direct relevance to interpreting quantum mechanical relationships - is commonly referred to as the complementarity of opposites and is based therefore on the dynamic interaction of opposite polarities (each of which has an arbitrary definition depending on context).

When one attempts to view relationships from a Type 1 perspective, recognising merely the quantitative aspect of numbers, it is easy to show that this leads quickly to logical confusion whereby for example we maintain that + 1 as a result is the same as either + 1 or - 1.

In practice in Type 1 calculations this problem is avoided by using only principle roots. So even though the square root of 1 from a Type 1 perspective can be + 1 or - 1, if we confine ourselves to the principle root, then it is unambiguously + 1. And of course this is the value that will be provided for the square root of 1 on any calculator!

However if we are to properly solve the logical inconsistency raised by Type 1 Mathematics, then we need to use Type 2 understanding.

So we have already explained at length why the use of 2 (as dimension) leads to a uniquely distinct manner of logically interpreting relationships (as opposed to the default dimension of 1 as used in Type 1 Mathematics).

With Type 2 Mathematics, a unique result is associated with each number (when used as a dimensional power).

Therefore from this perspective whereas 1^1 = + 1, 1^2 by contrast = - 1 (from a qualitative perspective).

What this entails is that, whereas the former relates to the conscious positing of unitary form as in linear rational understanding, the latter relates to the corresponding dynamic negation of such form (which is the very basis through which unconscious type intuition is generated).

Then in corresponding inverse quantitative terms 1^1 = + 1 and likewise 1^(1/2) = - 1.

Thus from a Type 2 perspective, it is strictly inaccurate to refer to the square root of 1 as 1. This is but a reduced interpretation (where the corresponding inverse dimension is likewise reduced to 1).

Remarkably there is direct support for this latter Type 2 approach given by use of the modified Euler identity.

As is well known e^(2*pi*i) = 1.

This expression therefore provides the means for obtaining the value of any root of 1.

So therefore to obtain the square root of 1 we raise the RHS to the power of 1/2 which results in the well known result

e^(pi*i) = - 1.

So using the (modified) Euler Identity we can obtain an unambiguous answer in quantitative terms that is associated with any fractional power of 1 (as dimension).

Likewise in corresponding inverse terms we equally have an unambiguous qualitative interpretation associated with each dimension as number.

Thus for example e^(4*pi*i) = 1^2.

Therefore once again the qualitative interpretation associated here with the use of 2 (as dimension) corresponds in direct inverse terms with the quantitative result associated with 1/2 (as dimension).

So in qualitative terms + 1 (linear reason) is directly associated with the 1st dimension.

By contrast - 1 (circular reason i.e. as the indirect expression of intuition) is directly associated with the 2nd dimension.

However 2-dimensional interpretation then necessarily requires combining both the 1st and 2nd dimensions (in this context).

So in a combined 2-dimensional logical approach we employ both linear (either/or) logic and circular (both/and) logic.

Friday, July 8, 2011

Addendum on Squares

I first raised the qualitative issue with respect to mathematical interpretation through indicating that the square of 1 (2-dimensional) is clearly qualitatively different from the line segment of 1 (1-dimensional) though quantitatively the result is similar.

Further contributions then showed how the 2-dimensional qualitative structure is based on the complementarity of opposite (unitary) polarities that are positive (+) and negative (-) with respect to each other.

Now initially one might have difficulty in relating this to the quantitative geometrical notion of the square (of 1 unit).

However this complementary 2-dimensional nature can in fact be illustrated with reference to the geometrical square.

If one starts in one position - say - the top right hand corner and then goes around the square till one arrives back in the same position, this will require that the parallel lines of the square must be traversed in opposite directions (both horizontal and vertical).

So if the forward direction of one of these lines is + 1, then the reverse direction of the parallel line will be - 1.

Thus to draw the square geometrically in linear terms requires that both horizontal and vertical lines be drawn from two opposite directions!

Resolving Inconsistency

Once again we have shown how in Type 1 Mathematics numbers when used as dimensions - though inherently of a qualitative nature relative to the (quantitative) number to which they are raised - are given but a reduced quantitative meaning.

Therefore from this perspective 1^2 is indistinguishable from 1, i.e. 1^1 (in quantitative terms).

However this assumption subsequently gives way to basic logical inconsistency when we attempt to find the 2nd (square) root of each expression.

Thus in the former case the square root of 1^2 (which is obtained through raising to the power of 1/2 = 1, 1^1. However the square root in the latter case = 1^(1/2) which cane be given two equally valid answers in Type 1 terms, i.e. + 1 and - 1.

So in the first case we obtain just one answer which is unambiguous. however in the second case we obtain two possible answers that are directly paradoxical in quantitative terms!

This problem points directly to the missing (qualitative) dimensional aspect of interpretation which is developed through Type 2 Mathematics.

So from a Type 2 perspective 1^2 ≠ 1^1 (in qualitative terms). In this context each number as dimension refers to a unique logical manner of holistically interpreting mathematical relationships. And the structure of this logical system is inversely related to the corresponding root number of 1.

In the case of 1, the qualitative problem does not even arise as clearly the 1st root of 1 - if we are to conceive of such a thing is identical to 1 (raised to the power of ).

However for all other dimensional numbers ≠ 1, a unique qualitative logical system (for that number) will arise.

So where the dimensional number = 2 (as in our example) the structure of the logical system involved will be inversely related to the two roots of unity which are + 1 and - 1 respectively.

However whereas in quantitative terms, these roots are understood in linear (either/or) terms, in corresponding qualitative fashion they are given a circular (both/and) interpretation.

So the logical system of interpretation associated with the number 2 (as dimension) is based on the dynamic complementarity of opposites, i.e that involves opposite polarities of (unitary) form that are positive and negative with respect to each other.

Now when we try and explain the nature of such circular complementary interpretation in reduced linear terms, the opposite polarities split in either/or unambiguous terms. Therefore if one root is designated as + 1, then other root is - 1.

To see more clearly what is involved here I will illustrate with respect to the frequently used example of turns on a road.

As we know whether a turn on a road is designated as left or right, depends on the direction from which it is approached. So if a turn is designated as left, walking up the road, then it will appear right when approached from the opposite direction (walking down the road).

So if we designate a situation where a road can be approached from both and up and down directions, clearly any turn will thereby be both left and right. And this is precisely what the 2-dimensional logical system implies! However if we now unambiguously fix movement with just one direction, then each turn can be given an unambiguous answer. However in this situation there are two independent directions that can be taken i.e. either up or down the road.

So if we designate left as positive and a designated turn is indeed left when approached from walking up the road, then it corresponds with the positive root; however this inevitably implies that when taking the opposite direction down the road that we will encounter the negative of left (i.e. a right turn).

Thus in this reduced linear (1-dimensional) situation where we can take a direction up or down as independent, either a positive or negative answer can apply with respect to a specific turn.

However in the former circular (2-dimensional) situation where both up and down are simultaneously seen as interdependent, any turn - by definition - is both positive and negative.

Therefore it is only by introducing the neglected qualitative aspect of Mathematics (Type 2), can we resolve a fundamental logical inconsistency with respect to Type 1 Mathematics in showing clearly the relationship between a quantitative result (the two roots of 1) and its corresponding qualitative interpretation (based on 2 as dimensional number).

Thursday, July 7, 2011

A New Number System

For communication purposes, attention will be initially confined here to the natural number system. The purpose of the exercise is to demonstrate that - appropriately understood - there are in fact two distinct natural number systems relating to the quantitative and qualitative aspects of number respectively.

In Type 1 Mathematics the natural number system would be expressed as

1, 2, 3, 4, 5, ..... and in conventional terms diverges to infinity.

However implicit within each of these natural numbers is a default dimensional number of 1.
Properly understood - relative to each of the natural numbers (as quantitatively understood) - the default dimensional number is of a qualitative nature.

Therefore, because the very nature of Type 1 Mathematics is ultimately defined in 1-dimensional (linear) terms whereby the qualitative aspect of understanding is reduced to the quantitative, the dimensional aspect is conveniently ignored.

Thus expressed in a more comprehensive manner the natural number system for Type 1 Mathematics would be written

1^1, 2^1, 3^1, 4^1, 5^1, .....

However there is an alternative natural number system - with a direct qualitative significance - that equally can be constructed. In this context we have a fixed default base quantity = 1, with each successive number involving a change in the dimensional number involved.

Therefore expressed in a more comprehensive manner the number system for Type 2 Mathematics would be written:

1^1, 1^2, 1^3, 1^4, 1^5, .....

Now when we try to view this latter number system from a Type 1 perspective it seems of little use as the quantitative value in each case remains unchanged as 1.

Thus to see the significance of this from a Type 2 perspective we switch to a circular - rather than linear - appreciation.
And the secret to this circular dimensional appreciation (of a qualitative nature) is the recognition that such dimensions have a close inverse relationship with their corresponding roots in quantitative terms.

We dealt for example with the dimension 2 in the last blog.

Thus to appreciate the qualitative nature of 2-dimensional interpretation we obtain the two roots of unity (which as we have seen are + 1 and - 1 respectively).

Now whereas in quantitative terms these two roots are interpreted with respect to linear either/or logic so that the square root can be either + 1 or - 1 respectively, in corresponding qualitative dimensional terms terms this same relationship is interpreted with respect to circular both/and logic which is often expressed as the complementarity of opposites. This is what I refer to in my writings as Integral 1 understanding which is the minimum required for true integral appreciation.

So from a 2-dimensional logical perspective, all interpretation is seen to combine positive and negative poles of understanding (which are considered complementary).

What this means in effect for Mathematics is that - when appreciated in a more refined cognitive manner - any mathematical object such as a number that is given an objective existence (as external to the enquiring must equally be given a corresponding mental interpretation (through the perception of that number).

So for example one cannot form knowledge of the number 2 (as an external mathematical object) without the mental perception corresponding to the number 2, which is - relatively - subjective and internal in direction.

So in the inevitable interaction of all mathematical experience from this context two poles are inevitably always involved that are positive and negative with respect to each other. And such is the essence of 2-dimensional interpretation.

So what I have been at pains to demonstrate here is how the number 2 - when appreciated in its true qualitative dimensional sense, has an intimate holistic relevance for the overall manner in which we logically interpret mathematical relationships.

So there is not just one such manner corresponding to the standard either/or logical system of 1-dimensional interpretation but potentially an infinite number!

Expressed in an alternative manner what this entails is that rather than being based on merely (conscious) rational means of interpretation that Mathematics properly combines both (conscious) rational and (unconscious) intuitive means of recognition.

And clearly from a more comprehensive perspective both intuition and reason should be combined in formal interpretation!

So in qualitative terms the holistic significance of any number (as representing a dimension) is that it can be directly associated with a unique means of configuring the relationship between conscious and unconscious modes of understanding. And each of such configurations thereby defines a unique logical manner of interpreting mathematical relationships.

Fro example the important number 4 (as dimension) is inversely associated with the 4 corresponding roots of 1 which are + 1, - 1, + i and - i respectively.

Therefore 4-dimensional interpretation entails a more refined type of understanding that is based on the complementarity of both real and imaginary opposites in experience (which I have developed at length in my writings and refer to as Integral 2 understanding).

However, once again what is truly remarkable is that in principle a unique logical interpretation (reflecting a special mix of refined rational and intuitive type appreciation) can be associated with every number as dimension.

So just as Type 1 Mathematics can be shown to have a remarkable resonance with reality (as quantitatively understood), Type 2 Mathematics will eventually be shown to have similar remarkable resonance with reality (as qualitatively understood).

Wednesday, July 6, 2011

Basic Problem

As a child I was fascinated with the processes of addition (and subtraction), multiplication (and division) and powers (and roots) in a way that even then hinted at present interests.

Looking back I sensed clearly even then that the latter two processes involved both a qualitative and quantitative transformation of number. However Mathematics in effect ignored the qualitative aspect of this transformation altogether.

This problem is so easy to state that it still amazes me how it is still - seemingly - completely overlooked by the mathematical profession.

Once again if we take the simple case of squaring a number, both quantitative and qualitative aspects are involved.

Thus in the simplest case when we square the number 1, in quantitative terms the result remains unchanged = 1.

However clearly a qualitative transformation has taken place in the nature of units involved. So for example if are to represent the original number 1 i.e. 1^1 we would represent it geometrically by a line segment of 1 unit. However we would then represent 1^2 geometrically as a square (with each side = 1 unit).

So through the process of squaring the number, a qualitative transformation has thereby taken place. In other words we move from a 1-dimensional to a 2-dimensional interpretation!

However, from a qualitative perspective Conventional (i.e. Type 1) Mathematics is defined in linear (1-dimensional) terms whereby the qualitative attributes of numbers (and indeed all symbols) are reduced to mere quantitative interpretation.

So therefore when we obtain the power of a number or - indeed - multiply numbers together, the qualitative transformation in the nature of the numbers is simply ignored with the result expressed in reduced quantitative terms.

So for example 3^3 = 27 (i.e. 27^1).

Thus in a very precise manner I have always referred to the logical nature of Type 1 Mathematics - which misleadingly is viewed as synonymous with all Mathematics - as linear i.e. 1-dimensional from a qualitative perspective. And as repeatedly stated this effectively in any context reduces qualitative to quantitative type interpretation.

Now it can be simply demonstrated that when seen from a more comprehensive perspective that Type 1 Mathematics is characterised by fundamental logical confusion.

Let us return to our simple example to demonstrate.

Thus 1^2 = 1 (i.e. 1^1) from a Type 1 perspective.

Therefore when we now raise each side to the power of 1/2 we obtain

1^1 = 1^(1/2).

However whereas in Type 1 terms, the RHS yields an unambiguous answer + 1, the LHS has two diametrically opposite possible answers (in Type 1 terms).

So from this perspective the square root of 1 can paradoxically be either + 1 or - 1.

This simple illustration shows that there is a fundamental unrecognised problem with mathematical reasoning (which occurs whenever a qualitative change in the nature of numerical units takes place).

As this problem therefore relates to all multiplication (and division) and all powers (and roots) it is therefore of the utmost significance.

In order to maintain consistency in quantitative terms, Type 1 Mathematics effectively ignores all but the first root (which is thereby referred to as the principle root). So for example if you use a calculator to find the square root of 1 it will give merely this principle root (= + 1). In this way the logical inconsistency that I have demonstrated is thereby conveniently avoided (not solved).

In the next contribution, I will show how the proper solving of this problem requires giving number (as dimension) a new qualitative interpretation.

Friday, July 1, 2011

Three Types of Mathematics (3)

I have stated that psychological development in Western culture largely plateaus at the 2nd Band (Middle) which is associated with Type 1 Mathematics.

I have also stated that Type 2 and Type 3 Mathematics are associated with the intuitively refined rational structures that characterise the other 5 Bands on the Spectrum (with Type relating to Bands 3 and 4 and Type 3 to Bands 5, 6 and 7 respectively).

The question might be validly posed as to the relevance of Type 2 and Type 3 Mathematics for a culture that has not sufficiently attained development at the appropriate Bands required for such mathematical understanding.

However a more nuanced appreciation of the dynamic nature of development would indeed allow at least for limited appreciation at present of these latter two Types of Mathematics.

Though there is indeed a certain sense in which the stages of development unfold in a linear manner, of necessity a complementary type relationship also applies e.g. as between lower and higher stages. Thus some access to higher and indeed radial stages opens up through mastery of lower and middle levels. So in a dynamic interactive sense one necessarily has access to all Bands on the Spectrum. However mature access to each level requires a process of sustained exposure to each level through a process that largely occurs in a linear sequence.
Therefore though development in rational terms may indeed for most individuals largely plateau (in a specialised sense) at the 2nd Band, varying degrees of access might still remain to the higher numbered Bands thus providing some limited basis for appreciation of Type 2 and type 3 Mathematics.

Development in any case rarely takes place in an even balanced fashion. So once again it is certainly possible for one who in most respects operates at a Band 2 level of appreciation to attain significantly higher Band understanding with respect to some modes of understanding. Frequently for example higher attainment with respect to rational structures will unfold which then would provide the ready basis for an even deeper appreciation of Type 2 and Type 3 Mathematics.

Another argument which could be made is that as we have reached a stage of very rapid evolution with respect to technology (especially IT) that this will call forth the need very soon for dramatic shifts in consciousness so as to absorb the repercussions of such developments. And this could therefore lead to rapid evolution for many of at least some of the higher Bands on the Spectrum which again would facilitate Type 2 and Type 3 mathematical understanding.
In support of this last point I would specifically mention present key issues in both Mathematics and Physics that will require higher Type understanding.

Indeed most of my work in recent years has been geared to demonstrating this very fact!