WHAT ARE “... OVERSIGHTS”?

{This started as a sub-section of Newton’s Great... Oversight, but is now being transformed into a freestanding paper/article. It’s not very far along: }

SECTIONS

HUMOROUS... INSIGHTS

PERENNIAL... OVERSIGHTS

Let us start with some humorous insights, and some serious ones.

When thinking of serious... oversights in science and mathematics there is a chess aphorism that comes to mind:

• When a beginner gives away his queen,
     it’s a blunder.

• When a grandmaster gives away his queen,
     it’s an... oversight.

Humor has found use since time immemorial as a mnemonic and teaching device, and perhaps even more as comic relief. Shakespeare is a constant reminder of how important that last is. Humor has been found by the wise to be not only helpful, but essential to aiding the process(es) of learning, growth and evolution toward wisdom, and in science and mathematics there are so many... oversights and of such a serious nature that comic relief will eventually be welcomed wholeheartedly. (Here, the ellipsis... has been added to the old aphorism to further all those.)

Perhaps humor will help ease the current situation, since scientists and mathematicians generally do not — or would not, if questioned — credit that science and especially mathematics could be guilty of any truly serious or prolonged... oversights. But we have Newton’s Great... Oversight as a clear counter-example in the “king” of sciences, physics, that for over 300 years scientists have held that lighter and heavier bodies — a la Galileo — fall at the same rate, when Newton’s own theory clearly predicts that they will fall at different rates (except at Lagrangian points L4 or L5).

The falling rate difference is crucial from the standpoint of the psychology and philosophy of science, but not from the standpoint of changing theory itself. Newton’s Great... Oversight  should provoke embarrassment, much soul and theory searching (looking for other... oversights), and renewed appreciation of science itself, but no upheaval in physics theory that will e.g. promote migration toward Einstein’s relativity.

But we have other clear counter-examples — much more devastating ones —  not only in “the queen of sciences”, mathematics, but in its flagship, Set Theory. We can peer into the psychology of mathematicians concerning one of the more important of these Fundamental... Oversights (which see in Mathematics; reiterates for overall emphasis but expands on some of what follows in this section):

• ALL mathematicians believe that it is theoretically possible that set theory is inconsistent, but NO mathematicians believe it is actually possible that set theory is inconsistent.

Relatedly, and more seriously:

• If mathematicians informally — or “formally”  — refrain from allowing/accepting as valid any derivations that yield contradictions, then it is literally impossible for any theory to be (seen as/shown to be) inconsistent.

But mathematicians do systematically reject such derivations, such as subtracting 1 element at a time from the set of all natural numbers Á 4 {1,2,3...}. The reason this otherwise standard derivation from the standard axioms and rules of inference is rejected is that the sets cardinality remains a₀ as each element is removed, even when all the elements have been subtracted out and all that remains is the empty set!

PERENNIAL... OVERSIGHTS
     IN SCIENCE and MATHEMATICS

It is more than intriguing psychologically that Sir Isaac Newton, of all people, not to mention every scientist since, overlooked that Galileo’s lighter and heavier bodies generally fall at different rates (with the fascinating exception of... well, you can read about it in Newton’s Great... Oversight); it is even somewhat terrifying.

Newton’s... oversight was certainly important enough scientifically in its day. Newton and his contemporaries should have applied the new “law” of gravity to the same-falling-rate hypothesis of Galileo. It is very important and somewhat terrifying that they did not do so. And, extremely important and even more terrifying, this oversight has continued for over 300 years, even into the beginning of the 21^st Century.

But the far more grave source of terror that we all should feel is... anger. Seriously, in just the last few years (here nameless) leading scientists literally have become overtly angry at the questioning of established scientific belief, especially such a basic belief, when the case for the difference in falling rates was first presented to them. This anger is far more terrifying than even great oversights. Only grudgingly did some of them later permit the equations and analyses to be presented and studied. Only grudgingly did they then finally admit that the falling rate difference did actually exist, but they generally disparaged the result saying things like “it’s too small to be important.” When it was pointed out that physics is proud of detecting the “infinitesimal” advance in the perihelion of the orbit of Mercury... all further conversation came to an abrupt halt.

It is extremely unlikely that anyone will say that the discovery of this... oversight of Newton — and unfortunately also of every scientist since Newton to the beginning of the 21^st Century — terrifying though it is, is the apocalypse. The “infinitesimal” advance in the perihelion of the orbit of Mercury could spell the difference between Newton and Einstein, but the falling-rate-difference question is just... Newton. It is fascinating from a historical perspective, essential from a psychology of science and scientists perspective (see below). It may be hoped — greatly — that it will help science give up its unspoken intent to be the new one-true-religion (by one or another variant of “killing the father and marrying the mother”), but it will provoke no revolution or shift of standard scientific paradigm, in the sense of Thomas Kuhn.

Actually, that last needs qualification. Realizing that lighter and heavier bodies fall at different rates will not really improve e.g. rocket science. The falling rate differences are not critical there, and are largely taken into account implicitly in the equations and calculations that the computers carry out. We already knew by way of Lagrange about Trojan asteroids, and about the possibility of putting space stations at Trojan points. Astronomy will be affected rather more, because there is a good chance that this result will yield inspiration to reexamine the Lagrangian concept of Trojan bodies and their stability. New possibilities will be looked at and for, such as ternary star systems, currently thought to be unstable per Lagrange’s analysis. (See also Figure 6a and Figure 6b.) But, still, this does not constitute a “standard” paradigm shift (if there is such a thing). The real paradigm shift — a true apocalypse for some — will be a “meta-paradigm” shift, a conjoining of the “science wars” with “scientific revolution”.

The “science wars” do not currently take science to task for failing on its own terms. But, here, we will take a quick look at the systemic nature of error and failure in science.

5.2         Approximation, Extrapolation, and Error in Science

If the angular deviation of 2 straight lines from parallel is non-zero but very small, few scientists will say that they will not diverge eventually, if — and this is essential — if the problem is stated in these terms. If we have an electronic device, few scientists will say that it will operate correctly under all conditions, again, if the problem is stated in these terms.

It must be emphasized that:

• Just because we can’t pragmatically measure a deviation within a limited experimental context does not mean that outside of that context the deviation will not become very great, even if it remains pragmatically unmeasurable because it is outside of the experimental context. Science produces beautiful flowers, but all too often they are hot-house flowers.

When the position is put forward that lighter and heavier bodies must interact gravitationally, and that there is an asymmetry that means that the falling rates might in fact not be precisely equal, many scientists respond that the rate difference is “so small...”, some even that it is “so small it doesn’t really make any difference”. This is not the most “scientific” attitude one can imagine. Science is supposed to be interested in such small quantities if in fact they are real, as the falling rate difference is. Scientists neglect that when this rate difference is given enough time we can actually begin to measure it, or at least notice its effects. Science should also be interested in small differences even — especially — if they are methodological artifacts; these, too, need to be studied and accounted for.

• Even “small” theoretical oversights are important, if one wishes to become and remain truly competent.

We have just seen that standard Newtonian gravity (the theory) is enough to at least derive equations that show that, according to that same theory, the falling rate difference is — generally, but not always — not zero. In conjunction, the astronomically known existence of Trojan asteroids is visible, scientific proof that the falling rate difference is in fact — generally — not zero. By Newton’s theory of gravity:

• If Galileo had held a 1 kg mass in one hand and a 2 kg in the other, 1 meter apart, and dropped them from the Tower of Pisa, at the instant of release the lighter body and the Earth would have accelerated together faster by approximately m/sec²  (ignoring, of course, all the usual: “gravitational anomalies”, wind, buoyancy, electro-magnetic effects, etc.)

When such bodies have an extended period of time to fall, as they do in orbit, the difference shows up; the bodies behave as though the approximation of “fall at the same rate” were incorrect, which it is if extrapolated sufficiently far.

Approximation is a necessary convenience to science, but scientists — with disappointing frequency — confuse their scientific approximations with “Laws of Science”, which by definition do not allow for any exceptions. They also confuse their “Laws of Science” with “laws of Nature”.

• Perhaps the greatest... oversight in modern science is that the only possible
  “One True Law Of Science”
  is:
  “Nature ALWAYS does as She darn well pleases!”

E.g. we may have a scientific inverse square force law, but this does not mean that all forces in Nature are inverse square, or even that gravity is always inverse square. Whether or not the “Laws of Science” are considered “universal”, approximations — by definition — only hold in limited contexts. The limits of these contexts are all too often ignored by scientists.

When dropping “higher order terms” from a Taylor series in a paper, for example, scientists rarely — or perhaps never — give any indication that they have studied the limits or thresholds beyond which those dropped terms would become large enough to be pragmatically interesting. That they might remain a very small percentage of order terms is not the essential issue. That the perihelion of the orbit of Mercury advances even if “infinitesimally”, that the continents and asteroids drift slooowly so that eventually our maps, predictions, and other extrapolations are “wrong”, these are essential issues of science, ones that are all too often ignored. The case of the difference in falling rates of lighter and heavier bodies and their relation to Trojan/Lagrangian points is an example of this.

Approximations are inextricably linked with abstract reasoning, and abstract reasoning is the sword that science lives by... and perishes by. Eventually we must realize that abstraction — by definition, by its own Nature — is only capable of reasoning about an infinitesimal part of the infinity of reality. It is a perennial error and failing of science that we do not universally recognize and realize this.

Ignored is the warning of, of all people, Aristotle. He warned that for logic to be and remain valid, every entity reasoned about by that logic must be and remain identically equal to itself, sometimes rendered as: A ≡ A. (He probably... overlooked... that this requirement also extends to what we today term logical operators, but, what the heck.) Since science and applied logic/applied mathematics are both/all in the position that an abstraction can never be the thing from which the abstraction was... abstracted, both/all are essentially invalid. They are also both/all formally invalid, since A 4 A is a formal requirement of the foundational logic of both mathematics and science, thanks to, of all people, Aristotle.

• Science can never truly be essentially — or even formally — valid on its own terms, since the map of abstraction — along with its abstract reasoning — can never really be the territory of reality, can never really be more than infinitesimally analogous to the territory, can never even map more than an infinitesimal portion of the territory of reality, or even of its Nature, who thankfully will keep on doing as She darn well pleases.

5.3         Newton’s... Oversight: How?! Why?!

How could such a simple, obvious result — the non-zero falling rate difference of lighter and heavier bodies and its essential relation to Trojan points — be overlooked by “the genius who discovered gravity”? The answer most likely lies in the character of the Renaissance which was flowering during Newton’s life. We will forgo descriptions of Newton’s genius and character, and of his role in this rebirth, to focus on what was happening spiritually and intellectually in his day.

Europe was beginning to throw off the suffocating rule and stultifying authority and power of religion as it had come to be practiced in the western world. Galileo had actually been in peril of his life from the Inquisition, and many others had lost their lives — and souls — to it. Aristotle, ironically a pagan Greek, and his teachings had come to solidly represent, and underpin, this ostensibly religious rule, this ostensibly spiritual authority and power.

The world was not then known for great wisdom, nor is it today. Because Aristotle and his science represented the power that many were liberating themselves from, including the budding European neo-intellectual and neo-scientific communities, they found it “necessary” to find fault with him, to dismiss him, even to depose him and his teachings. The work of Galileo concerning falling bodies flew in the face of some of Aristotle’s teachings, overthrew them, in fact, helping to hasten the waning of the rule and of the authority and power of the church. It was far too soon to think of questioning this new Golden Calf, Galileo and his new science, which had helped drive out that Sacred Cow, or rather Sacred Bull, Aristotle and his now old science. If Newton or his contemporaries had questioned Galileo, even on this one point, even with overpowering equations, to these newly self-established renaissance renegades, themselves among them, it would have been tantamount to turning traitor to their newfound salvation, traitor to their divinely inspired but still importantly self-achieved liberation from ignorance and error, traitor to this new life that was beginning to form in the world, life that they too were conceiving and giving birth to, and that they too were, themselves.

There is still the question of why no scientists since Newton have questioned the no-falling-rate-difference hypothesis of Galileo. Here we will merely note that, if by the time of Newton Galileo had become a demi-god, Newton himself — even today “generally recognized as the greatest scientist who ever lived” — soon became all but God Himself, almost within his lifetime.

• “Nature and Nature’s laws lay hid in Night:
  God said, Let Newton be, and all was Light”
  Alexander Pope (1688-1744)

If “He” didn’t question Galileo...

5.4         Questioning Scientific Dogma

The importance of Newton’s... oversight, great or not, can be judged somewhat by the response of some (luckily not all) leading scientists to this formal falsification of Galileo’s fundamental result: anger. Some scientists’ first reaction is just that, overt anger, anger that science is being questioned, especially such accepted and fundamental results. One, at first quite angry (“you shouldn’t...”), later admitted, after it was insisted that going over the equations was in order, that there was indeed a difference in the falling rates, but he dismissed it contemptuously as “too small to be important”. When it was pointed out that the advance in the perihelion of the orbit of Mercury was also very small, but was considered scientifically important none-the-less... end of communication.

Another reaction was that of a science teacher whose students had shown her the equations that they found of the PAIAS web site. We exchanged several e-mails discussing the falling rate difference, and she admitted that the falling rate difference seemed to be scientifically correct. But she finally pleaded: “but is it all right if I tell my students that lighter and heavier bodies do fall at the same rate?” When I reiterated that although they fall at different rates, they do fall at approximately the same rate for our usual purposes here on Earth, that satisfied her enough that our correspondence ended.

These sorts of reactions should give us all pause — scientists and non-scientists alike. We are made aware, perhaps yet again, of the stultifying effect of insistence on not truly questioning beliefs, scientific or otherwise, even of implicit insistence. Science — like religion — has accepted too much on “faith”. By the way, the original (Biblical) meaning of “faith” in religion was not what it is today; it was never any kind of willful ignorance (“we won’t look through your telescope, because we have... faith”). It was rather a “let them who have eyes see” kind of thing, or “you will perceive well the truth of reality, and the truth will set you free”, from which we have sadly departed.

The very real falling rate difference was readily accessible to Newton’s contemporaries through his theory of gravity (at least post 1687), not to mention Newton himself, and it is incredible that not only did they all miss it, but every astronomer and physicist since has missed it, as well. Perhaps we should note again that the temper of Newton’s times was one which was still not intellectually free of the repressive affects of Aristotle’s... eccentricities, at least not free of the uses to which they were put by those in power. Even after Copernicus these remained quite powerful, even as they slowly waned. Today...

Guilt — or innocence — by association is a popular and deadly fallacy. There was a (subliminally?) felt need to reject all of Aristotle, even when he may have been right, since he was both a symbol of the earthly and spiritual rule, authority and power of e.g. the church, and a heavy club to beat down “heresies” that didn’t want to submit to that rule, authority and power. The struggle was very much one of the young Turks of scientific Theory versus the old Greeks of religious Theology. We can note with serious irony that Theory and Theology are actually spelled the same, and the “Theo” in each is the Greek word for “God”. And both of these tend to lead to internecine — and even intranecine — warfare. The War of the Roses may be over, but this “War of the Theoses” continues even today. A Sacred Cow may sometimes be attacked, but only if it is on the other side of the fence. Galileo was on Newton’s side — and today’s modern science’s side — of the fence, and therefore to be “protected” from questioning, to be protected from “when wrong to be put right.”

We are currently in need of a good dose of:

• “... when wrong, to be put right.”

5.5         Our Aristotles, Our Galileos

Galileo is hard to fault since he had no real chance of measuring a falling rate difference of   m/sec² from the top of the Tower of Pisa (or rolling balls down a inclined plane), and since he as yet had no theory from which to derive it. Newton, however... and every physicist and astronomer since, at least to the end of the 20^th Century... Well, because science is really still in its infancy, or more correctly, still being conceived, perhaps we should continue to grace the situation with the term... “oversight”. Blunder sounds “so... harsh”. If there are any “scientists”, however, who would dare to maintain anything like “the difference is so small, it’s scientifically negligible!”, we could negligibly quote Galileo:

• “Ma eppur...” (“But yet...”; note: some omit the “ma”)

Relatedly, Aristotle has historically come through this falling rate difference contretemps not all that well. His falling rate thesis... well... it is mainly humorous to point out that he was 2/3 right. Perhaps we should allow that Aristotle, too, did not blunder, that he, too... “oversaw”. And when we stop to think that we are still teaching in our schools a 300+ year old... oversight, it is good to remember that, of all people, Aristotle also said:

• “The fate of an Empire depends on the education of its youth.”

Perhaps Aristotle doesn’t come out looking so bad. After all, we really can’t afford to go on teaching science’s... oversights, nor can we afford to go on teaching our future scientists to be angry when people point out and question... The Emperors’ New Fall Line. And he was 2/3 right!

Here at the beginning of the new millennium, as we try to put it right, let us hope that science will soon give up its tacitly-implicitly promoted position of “all-but-omniscience” and “all-but-infallibility” (both in the “objective” “practical” sense, e.g. one that “justifies” “ordering” people’s lives), and its more explicitly public stance of science-versus-the-subjective-world that can be poetically rendered as:

• “If it ain’t dreamt of in our philosophy
  (there’s no need for more than one),
  it   just   ain’t!”

Questioning Sacred Cows is still considered heresy, even in today’s science, so we can guess how the scientists of Newton’s time felt, both as heretics themselves, and as a new establishment. But the Inquisition won to this extent, that those who later came to question the new scientific dogma did so in imposed silence, and all too often still do.

5.6         “Science Wars” and “Ablative Shielding”
(and Science’s Implicit Infallibility)

In the current “science wars”, the critics of science tend to recount the philosophical, social-sociological, ethical, and even political failings of science, all rejected by science as unscientific bases for criticism (of science, at any rate). Scientists, in turn, tend to recount the “scientific” failings of science’s critics (academics in the “soft” social sciences and cultural studies, feminists, etc). But still overlooked is the fundamental fact that science has a long history of failing on its own terms, and on its own turf. Aristotle was a modern scientist of his day, as was Ptolemy, and as of course were many others. As these theorists and/or their theories fall into scientific disfavor, the worker-bees of science kick these now unneeded drones out of the hive into the cold of the winter of their discontent — or some such. Like footsteps of the past in that journey of a thousand miles, they have served their purpose, that of advancing science from whatever it was before to whatever it is that is also about to be left behind. They are now an embarrassment, cut from the team by a subtle variant of Occam’s Razor. Thus (the implicit infallibility of) science is maintained.

The space program has endowed us with a special term for this type of process. The early space capsules had heat shields of a special design that would protect the capsule from the extremes of re-entry temperatures and heat. These heat shields were made of a special material composed with a large number of special potential “flakes”. Upon re-entry, as the surface of the heat shield is heated by friction with the atmosphere, these “flakes”, heated to extreme temperatures, will “take the fall” and “take the heat”, away from the heat shield before it gets too hot, and... away from the astronauts.

• “Ablative Shielding”: a “heat” shield that functions by letting “flakes” “take the heat” and “take the fall”

If this sounds social-political as well as scientific, it should.

When science changes so much that the old paradigms and their champions are too much of an embarrassment — people might start questioning (the infallibility of) science as they did the infallibility of religion — science, like so many others, practices a variant of “ablative shielding” by casting out “flakes”, “fall guys”, who, “falling from grace”, also “take the heat”... away from science. Thus, Aristotle, Ptolemy, and many others were ejected from the pantheon of science, even though they were leading “modern scientists” in their day. Kepler, on the other hand, is still a close enough approximation to keep in the pantheon. (Kepler (1571-1630) is lucky that the center of mass of the Sun is still very close to the center of mass of the Solar System, or he’d be outa here.)

Science deals with its critics in very much the same way, and all too often with great, “righteous” anger. They are afraid of any kind of shift that might... disenfranchise... them from their preeminent position of power, a position and a power which the warnings of e.g. the Bible unambiguously describe as a damnation (“... when He shall have put down all rule and all authority and power”; check out 1^st Corinthians 15:24). Those who truly love science often feel an all too actual terror at the current state of science’s... soul.

5.7         “Science Wars” and “Scientific Revolution”

There is an impending shift in the relationship between science and the world, somewhat like the “paradigm shifts” that Thomas Kuhn wrote about in his “The Structure of Scientific Revolutions” (University of Chicago Press, 1962, 3^rd Edition in 1996). The real shift — and it will be a true apocalypse for some — will be a “meta-paradigm” shift. There will be a conjoining of the “science wars” with “scientific revolution”. The real revolution that this falling rate result will encourage has to do with the terrors mentioned above. It is not a source of terror that Newton’s law of gravity will be displaced and replaced, and in any case it won’t be, at least not due to the result presented here.

• One source of the terror that will give impetus to a new and different kind of scientific revolution is the one repeated so often here, that, not only did Newton fail to note the rather obvious falling rate difference that his own theory predicts, but every scientist since has failed to, as well, including many brilliant ones. Another source of the terror is the “righteous” anger that scientists express at those who try to point out such... oversights. There is a pattern of... oversights, and there is a corresponding pattern of anger.

• This is science failing at its most fundamental levels.

• This is science failing on its own terms.

The current heating up in the last few decades of the millennia old “science wars” (a modern term for it) is still based mainly on finding that science fails on philosophical-moral-ethical-social grounds, e.g. regarding epistemology or “how we know what we know”, which of course it does even though science rejects such grounds as... “unscientific”. Add to this the terror we should all — especially scientists — feel at the thought of people — especially ostensibly competent scientists — getting angry when science is questioned, even when questioned about such an accepted fundamental result.

The real revolution will be in world-community awareness of this serious failure of science, and relatedly in our science education, and we could say in meta-science, how we all — not just scientists — do science. The questions are inescapable: If science has overlooked something so theoretically obvious for over 300 years, and so astronomically obvious for almost 100 years, what else is science... overlooking?! What (Greek?!) fatal flaws is science burdened with, and burdening our world community with?! Talk about “science wars”: science failing on its own terms and its own turf...

There will be a meta-paradigm shift e.g. in our communally accepting the actual, even the inherent, fallibility and fragility of science. A subtly related idea is expressed in a recently published book (this gets a bit complicated): In Chet Raymo’s review in Scientific American (September, 2000) of Steve Fuller’s new book Thomas Kuhn: A Philosophical History for Our Times (University of Chicago Press, 2000), he has:

• “The paradigms of normal science are not the ideal form of science, he [Fuller] says, but rather ‘an arrested social movement in which the natural spread of knowledge is captured by a community that gains relative advantage by forcing other communities to rely on its expertise to get what they want.’”
  [Chet Raymo, quoting Steve Fuller]

It would be difficult for science, as religion has also found, to maintain its status, to promote its implicit “practically omniscient, practically infallible” image and its explicitly “the only real way to go” image, if it admitted to... oversights.

Community recognition of the falling-rate-difference... oversight could very well help stimulate the already budding revolution of how and why we do science in the larger sense. When we hold:

• “Science Right or Wrong;”

we will be far more likely to insist on:

• “when right, to be kept right;

•   when wrong, to be put right.”

We will find ourselves less inclined to teach scientific jingoism or chauvinism to our children. We will find that hypocrisy will make it impossible for anyone to win the current “science wars”.

• We will start to seek out and publicize discovery of... oversights as whole-heartedly as we seek out and publicize discovery of... “truth”. We will become far less accepting of science as a means or a basis for the control of our daily lives.

Science — and mathematics — still have many more undiscovered... oversights. Discovery of these... oversights, or rather their careful examination and the soul searching that will go with it, need not make us sadder, but will help make us — both science and society — wiser. If we don’t keep questioning accepted scientific beliefs, science will literally die out, far more quickly and completely than the dinosaurs or passenger pigeons, and perhaps take everyone else with it.

5.8         The Future of Science and Its... Oversights

The science wars — and someday science itself — will soon take a new turn. Not only critics, but lovers of science, as well, will start taking the stance that...

• “How can we trust science, even scientifically?
    After all, if they can’t even get that right...”

Science must learn to come to terms with its... oversights. It turns out that there are many, and some of them are far more important than this albeit fascinating one, somewhat theatrically referred to as “Newton’s Great... Oversight”. Science must also learn to come to terms with its penchant for making... oversights and taking sanctuary in denial, and even in “righteous” anger. This is accepted by the community because of science’s reputation, a reputation that does not include the possibility of any serious... oversights. Science must learn that it is fundamentally fallible, that it can decidedly fail, on its own terms, flagrantly, even in its fundamental tenets, and for hundreds of years at a stretch, without noticing, and with anger when there are attempts to question or “put right”. It is of telling significance that... “oversights” will be easier for scientists to deal with psychologically than “blunders”.

For Newton it can be considered a “great”... oversight, but perhaps for other, lesser scientists it can be considered merely a “normal”... oversight, an even more terrifying concept. In any case, dealing with this oversight — the non-zero falling rate difference of lighter and heavier bodies, and its consequences — will be useful in this regard, since it is so obvious that no scientist since the time of Newton should have missed it. It is easy to demonstrate both the falling rate difference and its consequences in popular terms, without abstruse-arcane or difficult mathematics. This will ease the way for the meta-paradigm shift to take place, perhaps in less than a generation, a shift toward transmuting... oversights into... insights.