What is accepted as "proof" in conventional mathematical terms represents but a limited notion (where once again the qualitative notion of number is reduced to the quantitative).
As we have seen, properly understood - in dynamic interactive terms - if the base number is quantitative, then the dimensional number is - relatively - of a qualitative nature.
So for example in the expression 21, if 2 is quantitative, then the dimensional number 1 is - relatively - qualitative in nature.
The psychological counterpart refers to the relationship as between a particular perception of number and the general concept of number (to which it is related).
So for example if the particular perception of the number "2" is deemed as quantitative, then the concept of number to which it is related is - relatively - qualitative in nature.
Now what does this mean precisely?
Well, "2" represents an actual finite number in quantitative terms. However the general concept of number has a potential infinite meaning in qualitative terms i.e. as applying to all possible numbers.
The crucial point to grasp in this context is that finite and infinite are quite distinct notions that again are quantitative and qualitative with respect to each other.
Strictly, whereas the finite is a (conscious) rational notion that is analytic, the infinite - by contrast - is directly an (unconscious) intuitive notion that is holistic in nature.
So, in the actual recognition of number, both rational and intuitive capacities are necessarily involved.
However the very essence of standard (1-dimensional) interpretation is that the qualitative is effectively reduced in mere quantitative rational terms.
So in this context, the intuitive notion of the infinite is thereby reduced in rational terms, as actually - rather than potentially - applying to all specific numbers.
Now there is a big problem here in maintaining that the concept of number applies to "all" specific numbers (as strictly "all" cannot be defined in an actual finite manner).
In other words because the number system is inexhaustible in finite terms, the very recognition of a set of numbers that can be actually identified, implies an indeterminate set that cannot be identified in this manner.
Thus an inevitable indeterminacy applies to the definition of "all" numbers in a finite manner.
However this issue - like so many other crucial issues - is simply glossed over in conventional interpretation. Therefore in effect, even though strictly nonsense, the infinite is thereby misleadingly treated as a linear extension of the finite.
In this way finite and infinite notions can be related to each other in a rational manner!
Now this fundamental problem intimately applies to very nature of proof in Conventional Mathematics.
Again the purpose of a general proof - say - of the Pythagorean Theorem - is that it should apply in "all" cases.
So in correct terms we can validly maintain that an accepted general proof potentially applies to all cases within its class (in an infinite manner).
So for example the assertion that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides, potentially applies to all such triangles (in an infinite manner).
However this strictly does not establish the applicability of the theorem in any actual case. Thus, to assert that the theorem applies in actual cases is simply to confuse the infinite with finite notions (or alternatively to reduce the qualitative in merely quantitative terms).
Now this is by no means intended to suggest there is no value therefore to the accepted notion of mathematical proof! Rather it is to raise the even deeper issue (which underlines all mathematical relationships) as to how we can establish a consistent relationship as between quantitative and qualitative notions! And the relationship as between finite and infinite in the context of mathematical proof represents just one important example of this key problem.
Once again the Riemann Hypothesis - when correctly interpreted - points to this very issue as the requirement, in the context of number, for the ultimate identification of its quantitative (cardinal) and qualitative (ordinal) aspects.
Properly dealing with the notion of mathematical proof will require moving to a balanced dynamic interactive approach with respect to all mathematical relationships (where both quantitative and qualitative aspects are equally recognised).
In my own terminology this will require therefore both Type 1 and Type 2 - relatively separate - aspects to mathematical interpretation which then are comprehensive integrated in Type 3 terms.
Just as in Quantum Mechanics, this will lead to a new "Uncertainty Principle" with respect to mathematical proof.
In other words in such dynamic terms, proof will be understood in a merely relative manner, with both Type 1 (quantitative) and Type 2 (qualitative) aspects of interpretation applying in a relative approximate manner.
Current (1-dimensional) proof represents an extreme in terms of the mere Type 1 (quantitative) aspect leading to a mistaken absolutist view of its nature.
Even momentary reflection on the nature of proof will lead one to see it as representing a special form of social consensus.
On occasion, a mistaken view with respect to such consensus can exist. For example it was initially accepted in 1993 that Andrew Wiles had proved "Fermat's Last Theorem" only for an important error to be subsequently found. Now. happily this has since been corrected with an unchallenged consensus as to the validity of the proof existing since 1995. However the possibility - however small - that additional problems may subsequently arise cannot be ruled out completely.
Indeed acceptance that Fermat's Last Theorem" has been proved, for most people represents an act of faith (rather than reason). So we trust that the small number of mathematicians competent enough to check the details have done so correctly.
However I am making here the much more fundamental point that - by definition - all conventional mathematical proof represents a basic form of reductionism (whereby qualitative notions are reduced in merely quantitative terms).
Thus correcting this reductionism requires from the onset acceptance of the merely approximate relative nature of all proof.
I have already mentioned the Pythagorean Hypothesis! This led to an important episode in mathematical history which illustrates my point very well.
As is well-known, the Pythagoreans quickly discovered that their theorem led to the existence of irrational numbers. For example in the simplest case where both the adjacent and opposite sides = 1, the hypotenuse = the square root of 2 (which is irrational).
Now apparently the Pythagoreans were able to demonstrate that the square root of 2 was indeed irrational. So they had an accepted Type 1 proof at their disposal.
However they were not happy to leave it at that ! They wanted to understand the deeper reason as to why irrational number quantities, such as the square root of 2, can arise in a world that they believed was scientifically governed in qualitative terms by the rational paradigm.
So - using my terminology - they were looking for the Type 2 aspect of mathematical proof for irrational numbers (which they were unable to produce).
Though Mathematics has certainly become much more specialised in quantitative terms since the time of the Pythagoreans, in some ways it is the poorer, for appreciation of the vital qualitative aspect has by now completely been lost.
I attempted some years ago to provide precisely this Type 2 aspect of proof for the irrationality of the square root of 2 (which is summarised in the blog entry "The Pythagorean Dilemma").
So like the identification of particle and wave aspects of matter, we can see that all proof represents a certain compromise as between Type 1 (quantitative) and Type 2 (qualitative) aspects.
Too much focus on one aspect creates increasing fuzziness therefore in terms of identification of the other aspect.
So with conventional interpretation, focus merely on the Type 1 aspect has become so extreme that it has blotted out recognition entirely of its complementary Type 2 aspect.
Thus in future in a more comprehensive mathematical understanding, all proof will require equal attention to both Type 1 and Type 2 aspects, where both the analytic and holistic appreciation of mathematical relationships can develop side by side.
In this new enriched world, in some cases the initial unfolding of Type 1 recognition will lead to a corresponding search for complementary Type 2 understanding; in other cases it will be the reverse with holistic Type 2 understanding preceding proper recognition in Type 1 (analytic) terms.
Present mathematical understanding is therefore totally lop-sided (with holistic recognition completely overlooked).
Of course not all problems can have a Type 1 proof (even in the accepted 1-dimensional sense) and the Riemann Hypothesis falls clearly into this category.
Once again, it points to the ultimate identity as between the quantitative and qualitative aspects of number. Acceptance of this fundamental identity is more a matter of faith than reason.
This thereby entails that acceptance of the entire mathematical edifice requires likewise such an initial act of faith.