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).

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.

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).

Fibonacci Type Relationships as Fractals

The mystery of the Fibonacci ratio (phi = 1.618...) is often expressed geometrically in terms of a Nautilus Shell which is a good example of a self-repeating spiral pattern (found frequently in nature).

Therefore from this perspective the very structure of the number phi is inherently dynamic and of a fractal nature (that endlessly repeats a simple pattern).

Now the Fibonacci ratio itself can be obtained as the positive solution to the simple polynomial equation,

x^2 - x - 1 = 0.

We can use a fascinating way to approximate this solution - and indeed any polynomial equation with an algebraic solution - through an easy iterative procedure.

In general terms for the equation x^2 + bx + c = 0, we start with the two numbers 0, 1 and add 1 * (- b) + 0 * (- a). So for the Fibonacci equation, this gives (1 * 1) + (0 * 1) = 1.

So we now have in the sequence 0, 1, 1 .

Continuing on in the same fashion the next term = (1 * 1) + (1 * 1) = 2.

So we now have 0, 1, 1, 2,

In this manner the well known Fibonacci sequence can be derived

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,...

Now, phi can be approximated as the ratio of a term and its preceding term (which approximation continually improves with higher terms).

So if we take the ratio of the last two terms we get 1587/977 = 1.61803444782..

This is already an extremely good approximation to the true value of phi = 1.6180339887.. .

However phi can equally be approximated in terms of a simple formula relating to the terms in the sequence that involve powers of 2.

So phi = 1/t1 + 1/t2 - 1/t4 - 1/t8 - 1/t16 - ......

Thus calculating up to t16 we get

phi = 1 + 1 - 1/3 - 1/21 - 1/987 = 1.61803444782..

So the value of this series up to t16 gives the same result for phi as t17/t16!


Just as the Golden Ratio (phi) can be represented in dynamic terms as a number fractal, in principle every (real) algebraic irrational number can be expressed in like manner as a fractal. This follows from the fact such an algebraic irrational must correspond to some polynomial equation with (real) integer coefficients. And as all such equations give rise - like the Fibonacci - to unique number sequences with recursive features, we can use these numbers to approximate (to any required degree of accuracy) the irrational numbers involved.


For example for the equation x^2 - 2x - 1 = 0, we derive the following sequence

0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ...

The solution to this equation 2.414213562373... is approximated again by ratio of successive terms.

Thus using the last two terms we obtain 985/408 = 2.41421568... (which already is a pretty good approximation).

In exactly the same manner as with the Fibonacci, we can use this sequence of numbers to drive a simple expression to derive the square root of 2.

So square root of 2 = 1/t1 + 1/t2 - 1/t4 - 1/t8 - ...

= 1 + 1/2 - 1/12 - 1/408 -....

= 1.41421568... (i.e. 2.41421568... - 1).

Partition Numbers

I recently came across the interesting fact that a new finite method for calculating partitions has been discovered by Ken Ono with some collaborators at Emory University.

This is a fascinating development with respect to an area of number theory that seems deceptively simple yet proves to be fiendishly difficult.

Partitions simply relate to the number of ways that a particular number can be broken down. It may help to initially think of this in concrete terms.

So for example the various partitions of 4 could be likened to the manner in which we could break up arrangements of - say - four pebbles.

At one extreme we could take the four together (i.e 4).
Then we could divide the four into a group of three with one left over (i.e. 3 + 1).
We could also divide into two groups of two pebbles (i.e. 2 + 2).

Then we could split one group of 2 into two single pebbles while maintaining the other group intact (i.e. 2 + 1 + 1).

Finally we could break the 4 into 4 single pebbles (i.e. 1 + 1 + 1 + 1).

So the total number of partitions of 4 is thereby demonstrated to be 5.


However whereas it is relatively easy to work out the partitions in this manner for the lower numbers, it quickly becomes increasingly more difficult so that for example the number of partitions of 100 is 190569292!

In arriving at the partitions in this manner we are considering various combinations without rearrangement. So for example 3 and 1 and 1 and 3 in this interpretation represent the same partition.

Now when we allow for rearrangement, the calculation of the number of partitions is surprisingly simple.

Thus taking once again the number 4 we can include here as additional partitions 1 + 3, 1 + 1 + 2 and 1 + 2 + 1 giving eight partitions in all.

In fact the general formula for sum of partitions (with rearrangement) is 2^(n - 1).

So the answer of 8 represents the case where n = 4.

Now this result of the number of partitions (with rearrangement allowed) can be expressed as the sum 1 + 2^0 + 2^1 + 2^2 + ... + 2^(n - 2).

Interestingly - dating from Euler - the number of unrestricted partitions (without rearrangement) can be expressed as a generating function entailing the partition numbers.


One fruitful exercise would be the exploration of the relationship as between unrestricted partitions (without rearrangement) and restricted partitions (with rearrangement).

Clearly the number of unrestricted is considerably less than restricted for large n.

It struck me that Mersenne primes can be seen to represent a unique relationship with 2^(n - 1).
Thus all Mersenne primes therefore are related to (appropriate) restricted partition numbers (through the subtraction of 1).

So using the restricted formula for partition numbers i.e. 2^(n - 1) where once again rearrangement is allowed, the first Mersenne prime when n = 3 is 2^2 - 1 = 3.

The second Mersenne prime for n = 4 is 2^3 - 1 = 7. The third Mersenne prime for n = 6 is 2^5 - 1 = 7 and the fourth for n = 8 is 2^7 - 1 = 127.


It also struck me that perhaps a more general relationship involving the relationship of the prime to natural numbers also pertains to the relationship as between the unrestricted (without arrangement) and restricted partitions (with arrangement).

For example the 100th restricted partition number = 2^99 and the 100th unrestricted partition number (190569292) lies between 2^29 and 2^30. Now as the number of primes contained in the first 99 natural numbers = 25, it is tempting to believe that perhaps there is some link here with the general distribution of the prime numbers.


However for much higher values of n this apparent relationship breaks down.

In other words when we express an unrestricted partition number n, as a power of 2, the power of this number ultimately bears very little relationship with the frequency of primes up to n - 1!

However it is still tempting to surmise that - even if less apparent - that an important relationship relating to the distribution of primes (among the natural numbers) underlies the relationship of unrestricted to restricted partition numbers.

In this context, Ono and his team demonstrated a pronounced recurrence pattern to partition numbers whereby - ultimately - all terms could in principle be shown to recur at regular intervals (as multiples of the original term) in the sequence of partition numbers.

Ramanujan had already demonstrated this recurrence pattern for 5, 7 and 11. However though much less obvious this can be extended to the other partition numbers!

For example the first 30 terms of the (unrestricted) partition number sequence are

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718 and 4565.

Now if we look at 5 in this sequence we can see that a multiple of this number recurs with every 5th succeeding number. So 30 is clearly a multiple as are 135, 490, 1575 and 4565.

If we now look at 7 we can see that a multiple of this occurs with every 7th succeeding number. So 77, 490 and 2436 are all multiples of 7!

Now finally if we look at 11, we can see that a multiple of this number occurs with every 11th succeeding number. So 297 and 3718 (in this sequence) are multiples of 11!

Not surprisingly these recurrence patterns lead to the notion of the partition numbers as very interesting examples of fractals. So partition numbers in their inherent structure give rise to fractals.

From a related piece of work that I am investigating at present, I have come to the conclusion that all algebraic irrational numbers are inherently dynamic in their very structure exhibiting unique fractal patterns.

The deeper significance of this finding is that all such numbers entail the relationship as between discrete and continuous notions (that in qualitative terms are linear and circular with respect to each other).

So ultimately the very nature of partition numbers entails this same relationship!

Cardinal and Ordinal Numbers (2)

As we have seen every number can be given both a cardinal and ordinal meaning which are quantitative and qualitative with respect to each other.
And where the ordinal number is the last in a group of numbers, the ordinal can be represented as the reciprocal of the cardinal.

So if we write 1^4 this can be given a quantitative meaning - where 4 is a cardinal number representing the dimension (or power) in question whereby 1^1^1^1 = 1 (in reduced quantitative terms).

However 4 equally here has an ordinal meaning as the 4th dimension (where 4 dimensions overall are considered). So here the 4th dimension represents 1/4 (of all four dimensions)
Thus to express 4 with respect to the ordinal number 4 (as the the 4th dimension) we obtain the value of 1^(1/4) = i.

So i here has a qualitative interpretation as imaginary i.e. the indirect expression of holistic unconscious meaning in conscious terms.

Of course 1/4 here can be given a quantitative meaning so that 1^(1/4) = i represents a number on the circle of unit radius.

Now interestingly if we now consider the 4th dimension as one of 5 dimensions it no longer can be represented by 1/4!


Whenever a number ≠ 1, a complementary relationship exists between the dimensional power and its reciprocal (that are quantitative and qualitative with respect to each other).

Once again in the default case = 1, both this dimensional number and its reciprocal are identical (in Type 1 terms) so that qualitative is reduced to quantitative meaning.

Cardinal and Ordinal Numbers

I have frequently referred to the fact that all numbers have both quantitative and qualitative aspects which are dynamically interdependent in experience.

However Conventional (Type 1) Mathematics, in giving formal expression to the merely quantitative aspect, thereby distorts appreciation of the true nature of number.

One manifestation of the relationship of quantitative to qualitative aspects pertains to the relationship as between the cardinal and ordinal use of number.

The very nature of the conventional linear (1-dimensional) approach is that it reduces qualitative to quantitative type interpretation.

This is exemplified by the manner in which we look at natural numbers.

In cardinal terms we can speak of the natural numbers as 1, 2, 3, 4,... as marked along the standard number scale.

However equally in ordinal terms we can speak of the 1st, 2nd, 3rd, 4th .... numbers as marked along the same scale.

So here there is a direct correspondence as between each successive number (as used in both a cardinal and ordinal sense).
Thus 1 corresponds with the 1st number, 2 with with the 2nd number, 3 with the 3rd number, 4 with the 4th number etc.

Indeed, without implicit recognition of the ordinal use of number, it would be strictly impossible to define the cardinal numbers.

However there is a subtle difference with respect to both types. The cardinal number 4 for example suggests a multiple of part units (that are literally 1).

So in concrete terms 4 could be used to denominate a collection of any four units (e.g. 4 stones).

However 4 in ordinal terms i.e. as the 4th unit inherently conveys the opposite relationship of part to whole. So the 4th unit specifically refers to just one unit which is seen as part of - what in cardinal terms - is a whole that comprises 4 units. So implicitly to give a ranking of 4 (as the 4th) necessarily implies recognition of 3 other units that are 1st, 2nd and 3rd. In this sense it represents the inverse (i.e. reciprocal) of the cardinal notion (i.e. 1/4).

Therefore when seen in this light, the relationship as between the cardinal and ordinal notion of number in experience represents the dynamic interaction as between whole and part (and part and whole).

Now, when written more fully 4 is represented as 4^1.
In other words the default quantitative interpretation of number (where powers or dimensions are considered redundant) is with respect to 1. And in qualitative terms this corresponds directly with linear (1-dimensional) understanding.

However the reciprocal 1/4 can be represented as 4^(- 1). So from a qualitative perspective the very manner through which we switch from the cardinal to the ordinal notion of number implies the negation of the 1st dimension.

As we have seen before, this entails switching from rational to intuitive type understanding.

At a deeper level this implies that the proper understanding of number requires an approach that combines both rational (specific) and intuitive (holistic) type interpretation (relating to quantitative and qualitative aspects respectively).

In Type 1 quantitative terms, the absolute value of a number arises when the sign is ignored. So in absolute terms - 1 is thereby indistinguishable from + 1.

Likewise in Type 2 qualitative terms, absolute interpretation arises when we ignore the sign (i.e. direction) of understanding. So linear understanding ( + 1) is thereby indistinguishable from its negation i.e. - 1.

Put another way with absolute type interpretation (as befits Type 1 Mathematics), quantitative cannot be clearly distinguished from qualitative meaning!