- In the case where n = 1, no qualitative transformation is involved. A quantitative transformation is solely involved so that c is always another natural number.

This is another posting extending on the qualitative dimension of the
Pythagorean approach. (Earlier I posted "The Pythagorean Dilemma" which
offered a holistic interpretation of irrational numbers. This is closely
related and offers a holistic interpretation of "Fermat's Last Theorem".
I plan a third to demonstrate the truly multi-level nature of mathematical
interpretation (using the Pythagorean theorem as an illustration).

Until its solution by Andrew Wiles in 1995, Fermat's Last Theorem had become the most famous unsolved problem in Mathematics.

The roots of the problem lie deep in the famous Pythagorean Theorem (where the square on the hypotenuse of a right angled triangle is equal to the sum of squares of the two other sides).

The Pythagoreans had discovered that there are innumerable natural number solutions for the three sides of the right angled triangle.

For example in the simplest case where opposite and adjacent sides are 3 and 4 units respectively, the hypotenuse is equal to 5 units.

Expressed in algebraic terms, if a, b represent the opposite and adjacent sides and c the hypotenuse then the equation a (raised to the power of n) + b (raised to the power of n) = c (raised to the power of n), has innumerable natural number solutions (e.g. a = 3, b = 4, c =5) for n = 2.

Fermat then considered this simple general equation for all integer powers of n greater than 2, and postulated that there were no simultaneous natural number solutions for a, b and c . Thus if we raise our three variables a, b and c to the power of 3, 4, 5, 6, 7, continuing on to infinity, no natural number solutions for a, b, and c will exist in any power. Though he claimed to have found a "marvellous proof" for this hypothesis unfortunately he did not provide it in his notebooks.

Though appearing relatively simple, a solution eluded all attempts for some 350 years. Then finally, to the satisfaction of the mathematical community, Andrew Wiles succeeded in proving Fermat's Last theorem in 1995.

This indeed was a remarkable achievement which evoked unusual popular interest. A Horizon documentary on the topic won a documentary of the year award at the annual (British) BAFTA awards. A book by Simon Singh (based on this documentary) was on the "best sellers" list for several weeks (in Ireland and the UK).

I do not intend here to attempt to go into the general nature of Wiles' proof for Fermat's Last Theorem (which itself is quite abstruse). However it does raise interesting questions.

Wiles' proof demonstrates the relative nature of mathematical proof. We cannot actually say that the theorem has been absolutely proven. Some doubt therefore as to its validity - however small - must still remain. Indeed when Wiles initially "proved" the theorem in 1993 a subsequent fault in the analysis was found. For some time Wiles himself did not see the significance of his error (when it was pointed out to him). His proof is long and intricate and can be understood in its entirety by very few mathematicians. There must be still a possibility that some important - though subtle - flaw in analysis still remains. Thus Wiles' proof is of a probable nature though the longer his proof sustains the intense scrutiny of fellow mathematicians, the probability that it is valid will of course increase. However strictly speaking its truth-value - as with all mathematical "proofs" is relative (rather than absolute).

Fermat believed that he had found an ingenious proof for the Theorem (which presumably was very short). Certainly he must have had some important insight leading him to formulate his famous hypothesis. Thus though Wiles' achievement is magnificent, his proof is extremely long and difficult (and certainly not of the kind Fermat had in mind).

This brings me to the purpose of this posting.

Fermat's Last Theorem has its origins in the Pythagorean Triangle. It is vital to appreciate that the Pythagoreans had a richer view of proof than that accepted in modern mathematics. They were interested in both quantitative (analytic) and qualitative (holistic) aspects of a problem.

In this context it is important to appreciate why the discovery of irrational numbers was so devastating for their approach.

In the simplest case of the right angled triangle (where opposite and adjacent sides are both = 1, the hypotenuse is = the square root of 2 (which is an irrational number).

Now the quantitative (analytical) proof of why the square root of 2 is irrational in itself would not have satisfied the Pythagoreans. (An ingenious proof was indeed provided by Euclid in his "Elements"). Their approach equally required a deeper qualitative (holistic) explanation.

In scientific terms the Pythagoreans adopted what is frequently termed "the rational paradigm". This paradigm is philosophically consistent with the generation of rational number quantities. Quite simply however they were not able to explain how a (qualitative) rational approach could generate irrational quantities.

The holistic (qualitative) proof of irrational number quantities requires the construction of a qualitative irrational paradigm. So just as rational quantities can be seen as consistent with a rational (qualitative) paradigm, now in like manner irrational quantities can be seen to be consistent with an irrational (qualitative) paradigm.

In an earlier posting "The Pythagorean Dilemma" I attempted to provide this holistic (qualitative) explanation by explaining the mathematical structure of this irrational paradigm.

If we apply the Pythagorean method to Fermat's Last Theorem (which has its roots in the Pythagorean Triangle), then we need equally to provide a holistic (qualitative) explanation. Indeed I believe that the provision of such an explanation would provide the essential insight (presently missing) to formulate a simpler analytical proof that is intuitively satisfying and closer to what Fermat had in mind.

I have long maintained that Fermat's Last Theorem possesses immense qualitative significance. (Indeed it proved inspirational in terms of my development of Holistic Mathematics!)

The current mathematical paradigm is a product of the rational linear level and is literally one-dimensional. Thus in this approach the representation of the number system is as a straight line. In fact all numbers are implicitly defined (in quantitative terms) with respect to the first dimension and are represented on the same straight line.

Thus if we refer to the number "4" implicit in this is the first dimension. In other words 4 is strictly spealing 4 raised to the power of 1).

If initially a number is defined with respect to a different power, its value will ultimately be obtained in reduced one-dimensional terms. Thus if we start with the number 4 (raised to the power of 2) then its reduced (one-dimensional) value will be expressed as 16 (which is implicitly raised to the power of 1).

The essential point here is that there are really two aspects to number.

We have the (horizontal) aspect that is quantitative. Equally we have the (vertical) aspect as dimension, power or exponent that is – relatively - qualitative. However in (conventional) mathematics a merely (reduced) quantitative (i.e. one-dimensional) interpretation of number is given. Thus the true qualitative understanding of the vertical aspect (i.e. power or dimension) is thereby lost and simply reduced to the quantitative.

Thus though numbers have both quantitative and qualitative significance conventional mathematics however treats all numbers as quantities.

Now when numbers are expressed in one-dimensional format the qualitative aspect does not arise. So if we add any two natural numbers (that are already defined in one-dimensional terms), the answer by definition is another natural number (that is one-dimensional).

When we add two numbers that are defined in one-dimensional terms a horizontal quantitative transformation takes place. However a vertical qualitative transformation is not involved (In other words the number type does not change). Thus if we add the two natural numbers e.g. 5 and 6, (that are implicitly raised to the power of 1), a quantitative transformation takes place whereby we get a new numerical result i.e. 11. However the qualitative aspect of number remains unchanged. In other words by definition the result is still a natural number.

However when we add two natural numbers that are raised to higher powers a qualitative (vertical) as well as quantitative (horizontal) transformation in number behaviour is involved whereby the very nature of the number type may change. In other words the result may well be a different number type (raised to the same dimension).

Basically this is what lies behind the explanation of Fermat’s Last Theorem.

Conventional Mathematics by definition does not provide a qualitative interpretation of number dimensions. It reduces such numbers to quantitative format.

The first task therefore in explaining Fermat’s Last theorem is to provide a holistic model of number where the qualitative aspect is made explicit. We can then demonstrate exact complementary behaviour in terms of quantitative number behaviour.

In Holistic Mathematics I basically define the Spectrum of Consciousness in terms of different levels of understanding that are precisely defined in terms of directions. Now these directions are identical with the mathematical notion of dimensions.

In this holistic formulation of the Spectrum, the important "middle" level of L0 is the "real" rational linear level. This is the paradigm on which (conventional) mathematical understanding is based. It is "real" in that it is geared to conscious (rather than unconscious) recognition of phenomena. It is rational in that it is based on the either/or logic based on the separation of polar opposites. It is linear in that it is based on one-directional (i.e. one-dimensional) understanding.

With linear understanding only the positive direction of understanding is recognised. This results directly from the separation of polar opposites in experience. Thus the objective world is viewed as independent of subjective reality. Time is then viewed as moving unambiguously forward for objective phenomena. Logical linkages and causal connections as between phenomena are then likewise taken in one direction.

The one-dimensional nature of L0 results directly therefore from the clear separation of polar opposites in experience.

Thus in horizontal terms objective is separated from subjective. The subjective aspect is then either ignored or understood in a reduced objective manner.

In vertical terms quantitative is separated from qualitative. The qualitative aspect is then again ignored or understood in a reduced quantitative manner.

Finally in diagonal terms actual is separated from potential. The potential (infinite) aspect of experience is once more either ignored or understood in reduced actual (finite) fashion.

As conventional mathematics is built on the one-dimensional rational paradigm, it can only deal with the qualitative aspect of number (i.e. number is dimension) by reducing it in quantitative terms.

HL1 ("Higher Level 1") which is what I call the circular level (i.e. subtle realm) is precisely defined in holistic mathematical terms as two-directional (i.e. two-dimensional). Reality is now understood in terms of the dynamic interaction of horizontal polarities (i.e. objective and subjective). In mathematical terms all relationships have both a positive (objective) and negative (subjective) direction. (These are understood in relative terms).

Thus the defining characteristic of HL1 is that qualitative understanding is two-dimensional (with both positive and negative polarities).

Now this is perfectly replicated in terms of (reduced) quantitative number behaviour (in complementary fashion). In other words when we attempt to extract the square root of any number it always exhibits two directions (i.e. has either a positive or negative value).

Thus when we add two natural numbers (defined in two-dimensional terms) a qualitative as well as quantitative transformation takes place. The resulting number (again defined in two-dimensional terms) can be either a rational number or irrational.

This requires careful explanation.

HL1 can be seen as a half-way house as between reason and intuition. On the one hand it is characterised by an outpouring of spiritual illumination (intuition). When we try to express the nature of such intuition in (indirect) rational fashion we must do so in terms of the complementarity of polar opposites (both positive and negative). Now in terms of former rational understanding where such opposites are clearly separated, this understanding is irrational. Thus for example in rational logic a proposition is either true or false (i.e. has an absolute truth-value). However in terms of irrational logic a proposition is both true and false (i.e. has a purely relative truth-value).

Though HL1 is characterised by high level intuition, there is still a marked tendency to try and understand in reduced rational terms (thereby becoming attached to the rational translations). Indeed the very dynamics of development throughout the entire level are driven by this confusion as between two types of understanding (i.e. rational and intuitive). When translated in rational terms, intuition appears as irrational. Thus the understanding of HL1 is both rational and irrational in "real" terms.

Now again there is perfect complementarity here in terms of the qualitative dynamics of number behaviour. When we add two numbers (raised to the power of 2), the result can either be a rational number (raised to the power of 2) or an irrational number (raised to the power of 2).

Thus if we add 3 and 4 (both raised to the power of 2) then the answer is 5 (raised to the power of 2). Here the resulting quantity base is a natural number (i.e. rational).

However if we add 1 and 1 (both raised to the power of 2) then the answer will be (1.4142…) raised to the power of 2 (which is an irrational number). This indeed is the famous square root of 2, which caused the Pythagoreans so much difficulty.

Now the nature of this dilemma by which we can generate both rational and irrational numbers (when adding natural numbers in two dimensions) can be easily demonstrated in geometric terms.

The square root of any number can be represented as either two points (on a straight line diameter) or alternatively two points (on the circumscibed circle). Thus when we add two numbers (raised to the power of 2) and then extract the square root we always get either a linear (rational) or circular (irrational) solution.

Now we have disposed of the one-dimensional and two-dimensional cases.

Once again in one-dimensional terms when we add two natural numbers a quantitative transformation is solely involved. The quantitative magnitude thus changes but the qualitative number type (i.e. natural) is unchanged.

In two-dimensional terms when we add two natural numbers a qualitative as well as a

quantitative transformation may be involved. Just as all square roots can be defined as lying either on a straight line or circle, when we extract the square root of our result it will either be linear (rational) or circular (irrational).

Fermat's Last Theorem is concerned directly with higher dimensions (greater than 2).

Now all dimensions of order 3 or higher are qualitatively different from order 2.

I will explain this important point firstly from the psychological perspective.

One dimensional understanding is characteristic of the rational linear level (where the positive direction of experience is alone recognised)

Two-dimensional understanding is characteristic of HL1 (where both positive and negative directions are recognised). This is understanding based on the complementarity of "real" opposites.

Three-dimensional understanding would correspond with the very important transition from HL1 to HL2 (the point level or causal realm).

This is the characterised by a "new" kind of "imaginary" understanding.

"Imaginary" understanding represents essentially the projection of the unconscious indirectly in conscious manner. Phenomena thereby become the expression of an archetypal spiritual meaning.

Once again conscious understanding is identified with the positive direction of experience (where opposite polarities are separated).

Unconscious understanding is identified with the negative direction (where opposite polarities are united).

Imaginary understanding is the attempt to express this negative direction (which is inherently two-dimensional) in one-dimensional terms. It thereby complements exactly in qualitative terms the notion of an imaginary quantity in (conventional) mathematics.

All "higher" dimensions (3 or higher) are characterised by both "real"(analytic) and "imaginary" (holistic) understanding.

For example four-dimensional understanding (which culminates with the completion of the point level or causal realm) has both "real" (horizontal) aspects (positive and negative) and "imaginary" (vertical) aspects (positive and negative).

Eight-dimensional understanding – which is what I refer to as "The Theory of Everything" – culminates with the null level (nondual reality) and has both "real" (horizontal) "imaginary" (vertical) and "complex (diagonal) aspects (with positive and negative directions). The intermediate dimensional (five, six and seven) can be seen as transitional states between the causal and null levels.

Multidimensional reality (beyond eight) arises through the increasing dynamic interaction as between polarities (horizontal, vertical and diagonal) and is characteristic of what I term Radial Reality.

Basically the holistic ("imaginary") aspect of experience relates to (intuitive) infinite experience that is indirectly translated in (rational) finite terms. When the relationship of the "real" and "imaginary" is expressed in reduced "real" manner it becomes relative and indeterminate (as it contains both finite and infinite elements).

This qualitative behaviour is exactly replicated in complementary quantitative terms.

All roots (3 or higher) involve "imaginary" as well as "real" solutions.

When we add two numbers therefore (raised to the power of 3 or higher) a qualitative as well as quantitative transformation takes place (which has "imaginary" as well as "real" aspects). When we express this in reduced "real" terms our solution will display both finite (quantitative) and infinite (qualitative) aspects. In other words the number will be irrational.

The holistic explanation therefore of Fermat’s Last Theorem is so fundamental as to be tautological.

Geometrically it can be expressed very simply. Whereas the roots of a any number (3 or higher) have a circular interpretation (i.e. lie as equidistant points on a circle, they do not have a simple linear interpretation (i.e. cannot be connected by a straight line). In like manner when we combine numbers (raised to the power of 3 or higher), the result has a circular result (that is raised to the same power). It does not have however a linear result. In other words the resulting number (raised to the same power) is irrational.

I have argued before that there are really three aspects in terms of truly comprehensive mathematical "proof".

Firstly there is (quantitative) analytical proof. This is what is conventionally understood as "proof". A "proof" for Fermat’s Last Theorem is therefore now available at this level.

Secondly there is (qualitative) holistic proof. This type of proof was implicit in the Pythagorean approach. This post has attempted to provide such a proof.

Thirdly there is comprehensive (what I call "radial"). This shows the relationships as between both quantitative (analytic) and qualitative (holistic) proofs.

The problem with the existing quantitative proof is that it is unrelated to its qualitative counterpart. This suggests that a much simpler quantitative "proof" for Fermat’s Last Theorem is available (with close relationship to the qualitative proof).

We really have two logical systems involved. Qualitative understanding is based on a both/and logic. In direct terms it is intuitive (though it has an indirect rational aspect).

Quantitative understanding is based on an either/or logic. In direct terms it is rational (though it has an indirect intuitive aspect).

Once again in one-dimensional terms, qualitative understanding does not arise (More correctly it is reduced directly to its quantitative aspect).

In two-dimensional terms, understanding – as we have seen – has both positive and negative aspects. In qualitative terms these comprise a unity (that is intuitively grasped).

Indirectly this can be expressed rationally as the complementarity of polar opposites.

In quantitative terms these poles split into two distinct aspects (which are the expression of the two roots of a number). We then use an either/or logic and say that the answer can be either positive or negative.

The key insight here is that every number can have both a linear and circular definition.

In the first dimension the circular definition does not arise.

However in all other cases a distinctive circular definition is involved.

Thus 1 in one-dimensional terms is simply 1 i.e. has only one (positive) direction.

However 1 in two-dimensional terms has two directions (positive and negative). This is the circular definition (as both points lie on the circumference of a circle of unit radius)

In qualitative terms these are both involved in the understanding of the number.

In reduced quantitative terms, they separate as either positive or negative poles (corresponding to the two roots of unity).

1 in the important four-dimensional case has four directions (real and imaginary with positive and negative aspects. (Again this is the circular definition as all four numbers lie as equidistant points on the circle of unit radius).

Again in qualitative terms these four directions are involved together in the four-dimensional understanding of number (i.e. a number raised to the power of 4).

In reduced quantitative terms these poles separate as either real or imaginary solutions (that can be either positive or negative).

Thus the four roots of unity are 1, -1 i and –i.

Likewise the 100 roots of units will appear as 100 equal points on the same circle of unit radius.

Thus higher dimensional numbers give rise qualitatively and quantitatively to circular definitions).

So let us now conclude by giving a simple comprehensive "proof"of Fermat’s Last Theorem (which combines both quantitative and qualitative understanding).

Again let a (raised to the power of n) + b (raised to the power of n)
= c (raised to the power of n), where a , b and n are natural numbers.

- In the case where n = 2, when we add a and b a qualitative transformation in the number c may be involved. Understanding combines elements that are both linear and circular (i.e. rational and irrational).
- In the case of n greater than 2, a qualitative transformation in the number c will always be involved (leading to a change in number type). Understanding now combines elements that are circular (with no simple "real" linear definition). Understanding is irrational in this sense.

In corresponding fashion when we obtain the two roots of c the answer can either be linear (i.e. rational) or circular (irrational).

Geometrically this is easily demonstrated as the fact that the two roots of a number can always be expressed as two points lying either on a straight line diameter or (corresponding) circular circumference.

In corresponding fashion when we obtain these "higher" roots of c the answer will always be circular (i.e. irrational).

Geometrically this is again demonstrated by the fact that though these"higher" order roots can always be expressed as points on a circular circumference they cannot be equally be all expressed as points on a straight line. Alternatively such roots cannot be expressed in solely "real" terms.

I am confident that it was such a simple insight that quickly led Fermat to postulate his famous Theorem. However it cannot be properly appreciated in merely quantitative terms. All "higher" roots (greater than 2) are qualitatively distinct in that they involve "imaginary" as well as "real" solutions. As the Pythagoreans would have realised this requires a satisfactory holistic interpretation.

Though conventional mathematics of course deals with "higher" order powers and roots of numbers, it does so within the context of a paradigm that is linear.

Thus it can only proceed by a process of reductionism, where non-linear quantitative behaviour is interpreted within the qualitative context of linear understanding.

Fermat’s Last theorem really demonstrates the need for non-linear qualitative understanding. Again in this context the interpretation of Fermat’s Last Theorem becomes so simple as to be tautological.