Looking again at the Horizon TV programme on Fermat’s Last Theorem proved a very rewarding experience. Unlike the first time I was able to appreciate much more of the fine detail (e.g. with respect to elliptical curves and modular functions). Also it got me thinking again on a number of levels regarding my own mathematical journey.
Like Wiles as a child of about 10, I too had heard of Fermat’s Last Theorem. The problem seemed so beguilingly simple that in my naïveté I thought I would be able to solve it. However after many hours of futile endeavour I abandoned this quest in failure. Nevertheless as Mathematics remained my favourite pursuit I hoped to major in the subject at College. However after a troublesome first year when I became greatly disillusioned with the mathematical treatment of the infinite, I dropped out of the class.
Many years later I became interested again in Fermat’s Last Theorem from a very different context. I had been paying a great deal of attention to my new pursuit of Holistic Mathematics (where every mathematical symbol can be given a well-defined qualitative as well as quantitative meaning) and was looking for a problem to demonstrate its potential value. So I hit upon “The Pythagorean Dilemma” relating to the irrational nature of the square root of 2. Then to my considerable satisfaction, I felt that I was able to provide a coherent qualitative solution to this problem.
The Pythagoreans would not have been satisfied with a merely quantitative explanations that the square root of 2 is irrational. They really wanted a deeper philosophical explanation as to why in qualitative terms such a number can arise!
The implication is that there are really two aspects to proof (i) (quantitative) analytic and (ii) (qualitative) holistic.
Comprehensive mathematical interpretation then requires both types of proof.
The problem with respect to the square root of 2 arises in the context of the famous Pythagorean triangle so 1^2 + 1^2 does not result in another rational number that can be squared. Now as Fermat’s Last Theorem is closely connected with this problem, I decided to look at again (strictly from this new holistic perspective) and came up what I would see as a partial qualitative explanation for its truth.
Central to holistic mathematical appreciation is the notion that mathematical dimensions have a coherent qualitative interpretation whose structure is inversely related to the quantitative root form of those dimensions. And to see this structure we obtain the successive roots of 1!
For example 2-dimensional interpretation is linked to the two roots of 1 which are + 1 and – 1 respectively. Thus qualitative 2-dimensional interpretation is based on the complementarity of opposite poles in understanding (such as external and internal).
Now the significance of all dimensions higher than 2 is that the corresponding root structures will contain both real and imaginary parts.
This means in effect that mathematical interpretation at these dimensions entails both real (conscious) and imaginary (unconscious) aspects with the imaginary expressing the unconscious aspect in an indirect rational manner!
Real understanding in qualitative terms relates to strictly rational type understanding (corresponding to rational numbers in quantitative terms).
Thus in Conventional Mathematics, when irrational numbers are used they are given a strictly reduced interpretation.
However imaginary understanding corresponds to the rational means of conveying true holistic meaning (of a circular kind).
So the qualitative explanation for Fermat’s Last Theorem relates to the basic fact that 3-dimensional interpretation and higher can never be conducted in a merely real rational manner. Likewise in quantitative terms, when we attempt to add two quantities - which represents the simplest form of linear transformation - raised to such dimensional powers, the result likewise cannot be expressed in merely real rational terms.
We can even give a simple geometrical rationale for this.
Clearly from a linear (1-dimensional) perspective, when we add two rational numbers the result will also be rational. The very essence of the linear interpretation is that qualitative considerations are ignored. Therefore no qualitative transformation in the numbers can take place.
2-dimensional interpretation is a half-way house as between linear and circular which can be easily illustrated. Here the two roots of 1 lie on both a straight line diameter and also on its circular circumference. Therefore in quantitative terms when we add two numbers (raised to the power of 2) the answer can be either linear yielding another rational number (raised to the power of 2) or circular (i.e. an irrational number raised to the power of 2).
However for all dimensions greater than 2 the corresponding roots (of 1) cannot lie on a straight line. So both real and imaginary aspects are necessarily involved. The corresponding corollary in quantitative terms is that when we add two rational numbers (raised to such dimensions) then an inevitable qualitative transformation in the nature of the number is involved. So the resulting number (raised to the power of n > 2) must be irrational.
I think it is even possible that Fermat’s “truly marvellous demonstration” of this fact might have related to this simple insight (i.e. that all root structures greater than 2 necessarily entail imaginary as well as real components). Now a demonstration does not amount to a proof and perhaps Fermat subsequently realised how difficult it was to build on such an insight to establish a proof!
Returning to the programme on Wiles, one striking paradox that hit me was how much the very process of his discovery runs counter to the established mathematical notion of rational proof.
So for example Wiles through his early discovery of Fermat’s Last Theorem was inspired by a powerful childhood dream i.e. one day to find the proof to this great puzzle. So his initial motivation relates more to the holistic unconscious (than rational thought). Also his subsequent voyage of discovery in many ways paralleled that of a spiritual contemplative seeking union with God.
Indeed with his thin frame and quiet demeanour he very much fitted the part of the religious ascetic. He then pursued a very uncertain journey in considerable isolation for seven years gaining total immersion in his problem. For much of the time he wandered in darkness, hanging on in faith and hope of an eventual resolution.
Then finally after long and painful endeavour he received a special Euraka moment of illumination when he finally resolved his problem. So great was his emotion at this final revelation that he could not even attempt to describe the feeling but only to say that he would never experience a moment like it again!
Wiles proof is rightly hailed as a truly remarkable mathematical achievement. However the point that I am making is that the actual experience of discovering such a proof entails much more than what is formally recognised.
So Mathematics - especially with truly creative discoveries - entails both intuitive (unconscious) as well as rational (conscious) processes. In particular Wiles’ decisive final insight was of a (holistic) intuitive nature. However this important holistic aspect is screened out totally from formal mathematical interpretation!
Though properly speaking there are two aspects to Mathematics that are quantitative and qualitative, only one is recognised. Thus the interpretations of Conventional Mathematics are of a highly reduced nature (and are only strictly valid within the 1-dimensional mode of qualitative interpretation adopted).
This intimately applies to the nature of mathematical proof.
It is only within a linear (1-dimensional) interpretation that mathematical proof can be given an absolute meaning. So from this perspective Fermat’s Last Theorem is either true or not true. And the popular belief is that Wiles has now finally resolved the matter for once and for all by proving that it is indeed true!
However the actual process by which the validity of his proof was decided shows that such absolute interpretation is not strictly valid.
Indeed when Wiles first presented his “proof” in 1993, it was widely accepted by the mathematical community. It was only later that a referee found a flaw in reasoning at an important juncture. Even when this was pointed out to Wiles, it took him some time for him to recognise the true importance of the difficulty. So more than a year of further investigation was required (with the help of a talented student) before Wiles was finally able to amend his proof.
So in truth “mathematical proof” in such cases – indeed in all cases - represents but a special form of social consensus among the mathematical community and is strictly of a probable nature. In other words as time goes by with no other questions being raised regarding its validity it can be accepted with an ever greater degree of confidence as true. But this truth will still always remain of a merely probable nature.
There is an even bigger challenge with the accepted notion of proof that I am here raising. This relates to the fact that with the passage of time, our very understanding of the nature of “proof” is likely to become considerably more refined.
So when properly understood, each dimensional number in qualitative terms corresponds with a unique mode of interpretation with respect to mathematical symbols.
So there is not just one valid mode - as presently believed - of acceptable mathematical interpretation (but rather potentially an unlimited set).
This understanding can then be applied directly to the status of the proof of Fermat’s Last Theorem. Just as it is not possible to add two rational numbers (raised to the dimensional power of 3 or higher) together to obtain another rational number (raised to the same power), equally it is not possible to maintain a strictly linear interpretation with respect to Fermat’s Last Theorem when understanding takes place in 3 (or higher) dimensional terms.
In other words - in terms of the qualitative understanding of such dimensions - interpretation is subject to the Uncertainty Principle. What this means in effect is that Fermat’s Last Theorem would now be given both a conventional (quantitative) and holistic (qualitative) interpretation. Thus inevitably there would be a trade-off necessary with respect to both types of appreciation. Therefore the more definite the merely quantitative, the more fuzzy would remain the corresponding holistic dimension. Equally the more definite the holistic, the more fuzzy would remain the quantitative aspect.
Now it might be maintained that my own appreciation with respect to the Wiles’ quantitative proof of Fermat’s Last Theorem is still fuzzy; however while accepting this observation, it only helps to confirm the general point I am making with respect to the true nature of mathematical proof in the wider context of interpretation (where both quantitative and qualitative aspects are formally incorporated).
Happily, Fermat’s Last Theorem is at least capable of proof (in the 1-dimensional sense of current mathematical interpretation). However I believe there are other outstanding problems (such as Riemann’s Hypothesis) where even this type of proof (or disproof) will not be possible.
Like Wiles as a child of about 10, I too had heard of Fermat’s Last Theorem. The problem seemed so beguilingly simple that in my naïveté I thought I would be able to solve it. However after many hours of futile endeavour I abandoned this quest in failure. Nevertheless as Mathematics remained my favourite pursuit I hoped to major in the subject at College. However after a troublesome first year when I became greatly disillusioned with the mathematical treatment of the infinite, I dropped out of the class.
Many years later I became interested again in Fermat’s Last Theorem from a very different context. I had been paying a great deal of attention to my new pursuit of Holistic Mathematics (where every mathematical symbol can be given a well-defined qualitative as well as quantitative meaning) and was looking for a problem to demonstrate its potential value. So I hit upon “The Pythagorean Dilemma” relating to the irrational nature of the square root of 2. Then to my considerable satisfaction, I felt that I was able to provide a coherent qualitative solution to this problem.
The Pythagoreans would not have been satisfied with a merely quantitative explanations that the square root of 2 is irrational. They really wanted a deeper philosophical explanation as to why in qualitative terms such a number can arise!
The implication is that there are really two aspects to proof (i) (quantitative) analytic and (ii) (qualitative) holistic.
Comprehensive mathematical interpretation then requires both types of proof.
The problem with respect to the square root of 2 arises in the context of the famous Pythagorean triangle so 1^2 + 1^2 does not result in another rational number that can be squared. Now as Fermat’s Last Theorem is closely connected with this problem, I decided to look at again (strictly from this new holistic perspective) and came up what I would see as a partial qualitative explanation for its truth.
Central to holistic mathematical appreciation is the notion that mathematical dimensions have a coherent qualitative interpretation whose structure is inversely related to the quantitative root form of those dimensions. And to see this structure we obtain the successive roots of 1!
For example 2-dimensional interpretation is linked to the two roots of 1 which are + 1 and – 1 respectively. Thus qualitative 2-dimensional interpretation is based on the complementarity of opposite poles in understanding (such as external and internal).
Now the significance of all dimensions higher than 2 is that the corresponding root structures will contain both real and imaginary parts.
This means in effect that mathematical interpretation at these dimensions entails both real (conscious) and imaginary (unconscious) aspects with the imaginary expressing the unconscious aspect in an indirect rational manner!
Real understanding in qualitative terms relates to strictly rational type understanding (corresponding to rational numbers in quantitative terms).
Thus in Conventional Mathematics, when irrational numbers are used they are given a strictly reduced interpretation.
However imaginary understanding corresponds to the rational means of conveying true holistic meaning (of a circular kind).
So the qualitative explanation for Fermat’s Last Theorem relates to the basic fact that 3-dimensional interpretation and higher can never be conducted in a merely real rational manner. Likewise in quantitative terms, when we attempt to add two quantities - which represents the simplest form of linear transformation - raised to such dimensional powers, the result likewise cannot be expressed in merely real rational terms.
We can even give a simple geometrical rationale for this.
Clearly from a linear (1-dimensional) perspective, when we add two rational numbers the result will also be rational. The very essence of the linear interpretation is that qualitative considerations are ignored. Therefore no qualitative transformation in the numbers can take place.
2-dimensional interpretation is a half-way house as between linear and circular which can be easily illustrated. Here the two roots of 1 lie on both a straight line diameter and also on its circular circumference. Therefore in quantitative terms when we add two numbers (raised to the power of 2) the answer can be either linear yielding another rational number (raised to the power of 2) or circular (i.e. an irrational number raised to the power of 2).
However for all dimensions greater than 2 the corresponding roots (of 1) cannot lie on a straight line. So both real and imaginary aspects are necessarily involved. The corresponding corollary in quantitative terms is that when we add two rational numbers (raised to such dimensions) then an inevitable qualitative transformation in the nature of the number is involved. So the resulting number (raised to the power of n > 2) must be irrational.
I think it is even possible that Fermat’s “truly marvellous demonstration” of this fact might have related to this simple insight (i.e. that all root structures greater than 2 necessarily entail imaginary as well as real components). Now a demonstration does not amount to a proof and perhaps Fermat subsequently realised how difficult it was to build on such an insight to establish a proof!
Returning to the programme on Wiles, one striking paradox that hit me was how much the very process of his discovery runs counter to the established mathematical notion of rational proof.
So for example Wiles through his early discovery of Fermat’s Last Theorem was inspired by a powerful childhood dream i.e. one day to find the proof to this great puzzle. So his initial motivation relates more to the holistic unconscious (than rational thought). Also his subsequent voyage of discovery in many ways paralleled that of a spiritual contemplative seeking union with God.
Indeed with his thin frame and quiet demeanour he very much fitted the part of the religious ascetic. He then pursued a very uncertain journey in considerable isolation for seven years gaining total immersion in his problem. For much of the time he wandered in darkness, hanging on in faith and hope of an eventual resolution.
Then finally after long and painful endeavour he received a special Euraka moment of illumination when he finally resolved his problem. So great was his emotion at this final revelation that he could not even attempt to describe the feeling but only to say that he would never experience a moment like it again!
Wiles proof is rightly hailed as a truly remarkable mathematical achievement. However the point that I am making is that the actual experience of discovering such a proof entails much more than what is formally recognised.
So Mathematics - especially with truly creative discoveries - entails both intuitive (unconscious) as well as rational (conscious) processes. In particular Wiles’ decisive final insight was of a (holistic) intuitive nature. However this important holistic aspect is screened out totally from formal mathematical interpretation!
Though properly speaking there are two aspects to Mathematics that are quantitative and qualitative, only one is recognised. Thus the interpretations of Conventional Mathematics are of a highly reduced nature (and are only strictly valid within the 1-dimensional mode of qualitative interpretation adopted).
This intimately applies to the nature of mathematical proof.
It is only within a linear (1-dimensional) interpretation that mathematical proof can be given an absolute meaning. So from this perspective Fermat’s Last Theorem is either true or not true. And the popular belief is that Wiles has now finally resolved the matter for once and for all by proving that it is indeed true!
However the actual process by which the validity of his proof was decided shows that such absolute interpretation is not strictly valid.
Indeed when Wiles first presented his “proof” in 1993, it was widely accepted by the mathematical community. It was only later that a referee found a flaw in reasoning at an important juncture. Even when this was pointed out to Wiles, it took him some time for him to recognise the true importance of the difficulty. So more than a year of further investigation was required (with the help of a talented student) before Wiles was finally able to amend his proof.
So in truth “mathematical proof” in such cases – indeed in all cases - represents but a special form of social consensus among the mathematical community and is strictly of a probable nature. In other words as time goes by with no other questions being raised regarding its validity it can be accepted with an ever greater degree of confidence as true. But this truth will still always remain of a merely probable nature.
There is an even bigger challenge with the accepted notion of proof that I am here raising. This relates to the fact that with the passage of time, our very understanding of the nature of “proof” is likely to become considerably more refined.
So when properly understood, each dimensional number in qualitative terms corresponds with a unique mode of interpretation with respect to mathematical symbols.
So there is not just one valid mode - as presently believed - of acceptable mathematical interpretation (but rather potentially an unlimited set).
This understanding can then be applied directly to the status of the proof of Fermat’s Last Theorem. Just as it is not possible to add two rational numbers (raised to the dimensional power of 3 or higher) together to obtain another rational number (raised to the same power), equally it is not possible to maintain a strictly linear interpretation with respect to Fermat’s Last Theorem when understanding takes place in 3 (or higher) dimensional terms.
In other words - in terms of the qualitative understanding of such dimensions - interpretation is subject to the Uncertainty Principle. What this means in effect is that Fermat’s Last Theorem would now be given both a conventional (quantitative) and holistic (qualitative) interpretation. Thus inevitably there would be a trade-off necessary with respect to both types of appreciation. Therefore the more definite the merely quantitative, the more fuzzy would remain the corresponding holistic dimension. Equally the more definite the holistic, the more fuzzy would remain the quantitative aspect.
Now it might be maintained that my own appreciation with respect to the Wiles’ quantitative proof of Fermat’s Last Theorem is still fuzzy; however while accepting this observation, it only helps to confirm the general point I am making with respect to the true nature of mathematical proof in the wider context of interpretation (where both quantitative and qualitative aspects are formally incorporated).
Happily, Fermat’s Last Theorem is at least capable of proof (in the 1-dimensional sense of current mathematical interpretation). However I believe there are other outstanding problems (such as Riemann’s Hypothesis) where even this type of proof (or disproof) will not be possible.
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