Mathematical Landscapes by Zarko D. Mijajlovich |

The crucial significance of both Simplification and Idealisation in Mankind's attempts to, first describe, and then understand, Reality, must be understood for how they have enabled progress in this demanding endeavour, while at the same time guaranteeing both mistakes and even impasses - some of which seem totally impossible to transcend, and have remained so for extremely long periods in Man's relatively short history. Indeed, it is this contradictory status that these "gains" most certainly possess, which makes their relationships to what actually pertains in Reality-as-is, so difficult to grasp.

Mankind's frequent solution to such conundrums, has always been their quality of remarkable flexibility, which allowed them to use the pragmatic stance embodied in, "If it works, it is right!", to actually step-around such impasses, merely on the basis of experience - without necessarily understanding

*why*, and in what circumstances, a particular assumption works. Such "successful-steps" appear everywhere in the panoply of Man's ideas of his World, and deliver, therefore, many such universal, "sticking-plaster-solutions" throughout that constructed-view of Reality.

In contrast, with a purely holist view of Reality, such means may seem wholly wrong, but that clearly isn't the case. Such eclectic methods can, and indeed do (to an extent) reflect Reality, especially in certain situations.

The seeming contradiction between "Everything affects everything else" and fixed, Idealised Forms and Relationships, is NOT a mutually-exclusive, and contradictory pair of stances.

Indeed, because of simplifying in various persisting natural situations, occurring at particular times - which we call Stable Situations (or natural Stabilities), these assumptions can indeed approximate to what pertains there.

It is, of course, due to an arrived-at "balance" of contradictory factors, which can be, for an extended period, self-maintaining, and, consequently and quite-naturally, simplifies the situation, in an overall way, delivering an extractable combined relation.

We don't know why this occurs, but we clearly see it, and can extract and use it, for as long as the Stability persists!

Now, situations can naturally occur, which are close to being stable, and these allow glimpses of such simple relationships, which observers can latch on to as the key-producing-parts of such complex situations, and, if they are extractable, they can become the assumed-to-be producing "idealised components" of Reality. So, such Idealisations can indeed reflect such situations, and can be successfully used to predict what will happen under any particular, non-dissociating changes, within a Stability.

But, even so, the question, "Why?" is never addressed.

So, disentangling such naturally-complex, yet reasonably-stable situations is often impossible: and there can be no doubt that multiple, individual factors are involved, and also that, in specially arranged-for circumstances, a particular Single Factor can actually dominate, and its individual contribution in those circumstances, can be extracted. But, it will not be the same as it would be in other non-dominant situations.

That assumption - that it will always be the same, is the flaw in Idealisation!

The extracted Form of the individual contributing factor, taken from the dominant situation, is merely the Idealised Version of that factor, and the assumption that it is always exactly like that in all situations, that it is eternal, is quite definitely incorrect!

The assumption (that it is fixed) depends upon the universally-adopted Principle of Plurality, which underpins the whole of the usual methodology of Science. So clearly, such a mistake is exceedingly important.

The much more truthful, but currently "technologically-unusable" stance, is that delivered by the totally-opposite Principle of Holism!

Andy Goldsworthy |

For multiple simultaneous factors, acting together are neither eternal, nor do they merely add-together, in varying amounts, to produce all possible situations. Indeed, every single one of them is different in different situations: for they most certainly affect one another!

They can, however, be organised to approach Plurality, for a given individual, targeted factor, by the careful-farming, and then sustained-maintenance of the exact same conditions, under which that particular simplified and idealised factor was made overt and then extracted.

Now, perhaps surprisingly, these transforming pluralist assumptions do not prohibit effective Use! As long as the appropriate, farmed conditions are provided for a given idealised factor, that will deliver what its extracted Form predicts.

But, to get anywhere near what the original, unfettered, many-factor, complex situation actually produced (though really, even then, only something similar) would always need a new application for each-and-every extracted factor, each in its own farmed environment, and carried through as a complete sequence over time!

[See all production processes in Industry for proof of this!]

Anhydrous ammonia plant, ca. 1954 |

But, there was, still, a major fly-in-the-ointment: the pragmatic assumptions which did deliver-usefully in production, were always significantly-damaging in attempts to actually

Initially the Explanation was always attempted holistically, in terms of substances-and-their-properties, but though successful, these never gelled with the extracted equations.

So, the two approaches gradually changed in their roles. While the equations were considered reflections of underlying eternal Natural Laws, the holistic explanations became something of an apologetic-accompanying-narrative: a tale to tell to the uninitiated, who couldn't possibly appreciate the beauty and power of the abstracted eternal Natural Laws.

The only solution to this contradictory situation was to stress the Principle of Plurality, and insist that the unfettered phenomenon was merely an addition of the full set of eternal Natural Laws, in varying quantitative proportions.

*explain*phenomena. For, any individual equations produced by those pluralistic methods could not be brought together to explain the original unfettered situation.Initially the Explanation was always attempted holistically, in terms of substances-and-their-properties, but though successful, these never gelled with the extracted equations.

So, the two approaches gradually changed in their roles. While the equations were considered reflections of underlying eternal Natural Laws, the holistic explanations became something of an apologetic-accompanying-narrative: a tale to tell to the uninitiated, who couldn't possibly appreciate the beauty and power of the abstracted eternal Natural Laws.

The only solution to this contradictory situation was to stress the Principle of Plurality, and insist that the unfettered phenomenon was merely an addition of the full set of eternal Natural Laws, in varying quantitative proportions.

Theory in these circumstances had been abandoned for mere Productive Reliability. Pragmatism had re-established its old dominance, and if anyone asked for an explanation, they were now just given the equation.

Gradually, participants began to consider that "Theory" was just the skilful manipulation of just such equations, to fit all possible circumstances. And, of course, that wasn't ever correct or explanatory! It was a frig: and any consistency evident was that of Idealised Mathematical Form, and NOT of physically existing Reality.

Real Physical Theory was rapidly being abandoned, and the results would be the current Crisis in Physics, which has now existed ever since the decision at the Solvay Conference in 1927, when Bohr and Heisenberg defeated Einstein and SchrÃ¶dinger, with their Copenhagen Interpretation of Quantum Theory.

Mathematics is never the Essence of Reality, but only the study of idealised Forms, which are always just inviolate-able patterns or formal relations. To make them primary, as the "actual Drivers of Reality", instead of simplified and idealised forms derived from a carefully-tailored Reality, is clearly Idealism - a far cry from the avowed Materialist basis of Science!

Gradually, participants began to consider that "Theory" was just the skilful manipulation of just such equations, to fit all possible circumstances. And, of course, that wasn't ever correct or explanatory! It was a frig: and any consistency evident was that of Idealised Mathematical Form, and NOT of physically existing Reality.

Real Physical Theory was rapidly being abandoned, and the results would be the current Crisis in Physics, which has now existed ever since the decision at the Solvay Conference in 1927, when Bohr and Heisenberg defeated Einstein and SchrÃ¶dinger, with their Copenhagen Interpretation of Quantum Theory.

Mathematics is never the Essence of Reality, but only the study of idealised Forms, which are always just inviolate-able patterns or formal relations. To make them primary, as the "actual Drivers of Reality", instead of simplified and idealised forms derived from a carefully-tailored Reality, is clearly Idealism - a far cry from the avowed Materialist basis of Science!

Math Fantasy |

*explanations*!

When it came to understanding what was really going on, and "Why?", it was gravely flawed and limped across multiple impasses via old fashioned "suck-it-and-see" Pragmatism!

We have to be crystal-clear on all this!

Mathematics (like Formal Logic) is valid system of study for idealised Forms.

But, it is not the underlying driving basis of concrete Reality!

It was, and still is, a brilliant man-made simplification of aspects of Reality, which when used in appropriately simplified and maintained Domains - that actually bend a situation to something approaching that idealised form. But, it is, however, never appropriate in unfettered Reality, or in any attempt to understand and explain phenomena as they actually occur, naturally, in Reality-as-is!

Fundamentally, the pluralist approach and methodology, separates individual factors, artificially, from their natural joint occurrences, to use each one separately and sequentially, in tailored situations, to enable reliable predictions.

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