From a Type 3 mathematical perspective - quite literally - every mathematical symbol, relationship, theorem etc. that can be given a Type 1 interpretation (in quantitative terms) must be interpreted in a complementary interactive manner with respect to both quantitative and qualitative aspects.
Indeed strictly speaking mathematical meaning now resides in the combined dynamic relationship as between aspects (rather than in either aspect as separate).
This therefore has great implications for the mathematical notion of proof.
From a (linear) quantitative perspective, a proposition is accepted as "proved" once the appropriate Type 1 explanation has been given.
So from a Type 1 perspective therefore, the Pythagorean Theorem has indeed been proved. In fact a wide variety of different valid proofs are available in this regard.
However, as we have seen there is also - as I have been demonstrating - with respect to the Riemann Zeta Function - an equally valid qualitative significance associated with every mathematical entity.
So in principle for every Type 1 proposition that has been proved in Type 1 terms, a coherent holistic qualitative interpretation can be given.
In broad terms that might be helpful, the Type 1 aspect essentially deals with "how" a proposition is true (or untrue) as the case may be.
The Type 2 aspect then uncovers the deeper "why" of the truth value of the proposition.
And then finally the Type 3 approach (which combines the previous two aspects as interdependent) marries both the "how" and the "why" in a manner that is truly meaningful in both a rational and holistic intuitive manner.
Now it is important to stress that the very manner of understanding from the rational and intuitive perspectives requires very distinctive capacities.
Therefore one who specialises in an unduly narrow fashion in linear type reason is highly unlikely to possess a developed holistic capacity (at a Type 2 level); likewise one who naturally operates in a holistic manner with respect to understanding again is very unlikely to possess sufficient rigour with respect to the linear rational (Type 1) aspect.
Indeed the true position is much worse than that; because of the extreme specialisation of Mathematics as a discipline, it is highly unlikely that most mathematicians would even want to recognise the (hitherto) unrecognised Type 2 aspect (even when shown to them)!
Thankfully, there will always be some notable exceptions at the margins, as it were, who possess significant mathematical talent while remaining open to a variety of other influences. So I would expect that most "specialists" would simply dismiss a blog such as this as quite irrelevant in terms of what they consider "Mathematics". However there may be yet a few generous souls who can recognise that I am indeed seriously addressing key fundamental mathematical issues, while offering a radical vision of a greatly enhanced mathematical future. And it is that small audience of open minded individuals that I principally wish to address in these blogs!
So to sum up all propositions, that can be proved in Type 1 terms, can be given a corresponding holistic rational (in Type 2 terms) which essentially implies a distinctive type of "proof".
Now, proof in Type 1 terms implies a linear sequence of logic steps (based on an initial set of axioms) so as to eventually arrive at the desired conclusion in this manner.
So the proof in effect is shown to be implied by the initial axioms, as demonstrated through independent logical steps taken to arrive at the conclusion.
However "proof" in Type 2 terms operates in a distinctive circular manner, where the interdependence of all aspects contained is holistically demonstrated through a dynamic set of two-way complementary type relationships. Ultimately arriving at such a "proof" relates to a required level of intuitive capacity (that can enable such connections). Indirectly however the connections can then be rationally demonstrated in a circular logical fashion.
Now it must be clarified right away, that we are not here speaking about the intuitive approach as understood in Conventional Mathematics. Riemann and Ramanujan for example were both highly intuitive mathematicians, but they still operated within the confines of Type 1 (quantitative) Mathematics.
I am speaking about the intuitive capacities required for true qualitative (i.e. Type 2) appreciation which is of an entirely different order altogether. Type 1 Mathematics is strictly 1-dimensional from a qualitative perspective. So I am talking about the intuition that potentially applies to all dimensions other than 1.
So the Type 1 aspect is designed to provide the quantitative aspect of mathematical proof (based on a linear type logical approach).
The Type 2 aspect is designed to provide the corresponding holistic qualitative aspect of mathematical proof (based on a circular type logical approach).
The Type 3 aspect is then designed to integrate both aspects of proof in a dynamic interactive manner.
The possibilities here are limitless! However ultimately I suspect that when propositions are proved in an appropriate manner (in accordance with the Type 3 approach) that two two aspects of proof should match in a synchronous manner. Then one should simultaneously see the relationship of the rational Type 1 connections to the holistic Type intuitive type connections (indirectly translated in a rational circular manner).
This would then open up immensely creative possibilities for quickly moving from a (recognised) Type 1 to an (as yet unrecognised) Type 2 aspect of proof; or alternatively for moving from a (recognised Type 2) to an (as yet unrecognised) Type 1 aspect.
So one of my present goals is to prove some mathematical proposition, which I am not presently aware has been achieved in Type 1 terms, from a corresponding proof of the Type 2 aspect (that I can intuitively "see" first in holistic terms).
I did succeed (at least to my own satisfaction) in one of my earliest mathematical goals, which was to resolve - what I term - the Pythagorean Dilemma.
The Pythagoreans did indeed recognise that Mathematics possessed both quantitative and qualitative aspects. It is only the subsequent specialisation in a merely quantitative direction, that has led to such a significant loss of the qualitative dimension!
So in their discovery that the square root of 2 was irrational, they were able apparently to provide a (Type 1) quantitative proof. However their dilemma was that they were not able to provide the corresponding qualitative explanation that would make this new finding intuitively accessible. So the all important holistic appreciation of the true nature of an irrational number was missing. In other words they lacked the Type 2 aspect of proof!
Therefore, after some time developing holistic mathematical notions, I set the task of providing this missing Type 2 aspect using my own insights and terminology to resolve the issue.
Recently, I briefly explored this issue again in Type 3 terms and to a considerable degree was able to establish the mutual correspondence as between both aspects of proof.
So in this context, the Type 1 aspect relates to proof of why - or rather how - the square root of 2 is irrational (in a specific quantitative manner).
The Type 2 aspect relates to proof of why (philosophically) the square of root of 2 is irrational (in a holistic qualitative manner).
The Type 3 aspect relates to establishing a mutual correspondence of both quantitative and qualitative aspects through matching explanations so that the linear can be readily adopted to circular, and the circular to linear appreciation respectively. In this way the two aspects are seen as reflecting the "same" interactive truth reality (containing both quantitative and qualitative aspects) in a complementary fashion.