Real Science Debates Are Not Rare

Guest Post by Dr. Robert G. Brown

The following is an “elevated comment” appearing originally in the comments to “A Rare Debate on the ‘Settled Science’ of Climate Change”, a guest essay by Steve Goreham. It is RG Brown’s reply to the Steven Mosher comment partially quoted at the beginning of the essay. This essay has been lightly edited by occasional WUWT contributor Kip Hansen with the author’s permission and subsequently slightly modified with a postscript by RGB.

rgbatduke

October 3, 2014 at 8:41 am

“…debates are rare because science is not a debate, or more specifically, science does not proceed or advance by verbal debates in front of audiences. You can win a debate and be wrong about the science. Debates prove one thing. Folks who engage in them don’t get it, folks who demand them don’t get it and folks who attend them don’t get it”.

Steven Mosher – comment

Um, Steven [Steven Mosher], it is pretty clear that you’ve never been to a major physics meeting that had a section presenting some unsettled science where the organizers had set up two or more scientists with entirely opposing views to give invited talks and participate in a panel just like the one presented. This isn’t “rare”, it is very nearly standard operating procedure to avoid giving the impression that the organizers are favoring one side or the other of the debate. I have not only attended meetings of this sort, I’ve been one of the two parties directly on the firing line (the topic of discussion was a bit esoteric — whether or not a particular expansion of the Green’s function for the Helmholtz or time-independent Schrodinger equation, which comes with a restriction that one argument must be strictly greater than the other in order for the expansion to converge, could be used to integrate over cells that de facto required the expansion to be used out of order). Sounds a bit, err, “mathy”, right, but would you believe that the debate grew so heated that we were almost (most cordially 🙂 shouting at each other by the end? And not just the primary participants — members of the packed-room audience were up, gesticulating, making pithy observations, validating parts of the math.

You’re right that you can “win the debate and be wrong about the science”, however, for two reasons. One is that in science, we profoundly believe that there is an independent objective standard of truth, and that is nature itself, the world around us. We attempt to build a mathematical-conceptual map to describe the real terrain, but (as any general semantician would tell you) the map is not the terrain, it is at best a representation of the terrain, almost certainly an imperfect one. Many of the maps developed in physics are truly excellent. Others are perhaps flawed, but are “good enough” — they might not lead someone to your cufflinks in the upstairs left dresser drawer, but they can at least get someone to your house. Others simply lead you to the wrong house, in the wrong neighborhood, or lead you out into the middle of the desert to die horribly (metaphorically speaking). In the end, scientific truth is determined by correspondence with real-world data — indeed, real world future data — nothing more, nothing less. There’s a pithy Einstein quote somewhere that makes the point more ably than I can (now there was a debate — one totally unknown patent clerk against an entire scientific establishment vested in Newtonian-Galilean physics 🙂 but I am too lazy to look it up.

Second, human language is often the language of debates and comes with all of the emotionalism and opportunity for logical fallacy inherent in an imprecise, multivalued symbol set. Science, however, ultimately is usually about mathematics, logic and requires a kind of logical-mathematical consistency to be a candidate for a possible scientific truth in the sense of correspondence with data. It may be that somebody armed with a dowsing rod can show an extraordinary ability to find your house and your cufflinks when tested some limited number of times with no map at all, but unless they can explain how the dowsing rod works and unless others can replicate their results it doesn’t become anything more than an anecdotal footnote that might — or might not — one day lead to a startling discovery of cuff-linked ley lines with a sound physical basis that fit consistently into a larger schema than we have today. Or it could be that the dowser is a con artist who secretly memorizes a map and whose wife covertly learned where you keep your cufflinks at the hairdresser. Either way, for a theory to be a candidate truth, it cannot contain logical or mathematical contradictions. And even though you would think that this is not really a matter for debate, as mathematics is cut and dried pure (axiomatically contingent) truth — like I said, a room full of theoretical physicists almost shouting over whether or not the Green’s function expansion could converge out of order — even after I presented both the absolutely clear mathematical argument and direct numerical evidence from a trivial computation that it does not.

Humans become both emotionally and financially attached to their theories, in other words. Emotionally because scientists don’t like being proven wrong any more than anybody else, and are no more noble than the average Joe at admitting it when they are wrong, even after they come to realize in their heart of hearts that it is so. That is, some do and apologize handsomely and actively change their public point of view, but plenty do not — many scientists went to their graves never accepting either the relativistic or quantum revolutions in physics. Financially, we’ve created a world of short-term public funding of science that rewards the short-run winners and punishes — badly — the short-run losers. Grants are typically from 1 to 3 years, and then you have to write all over again. I quit research in physics primarily because I was sick and tired of participating in this rat race — spending almost a quarter of your grant-funded time writing your next grant proposal, with your ass hanging out over a hollow because if you lose your funding your career is likely enough to be over — you have a very few years (tenure or not) to find new funding in a new field before you get moved into a broom closet and end up teaching junk classes (if tenured) or have to leave to proverbially work at Walmart (without tenure).

Since roughly six people in the room where I was presenting were actively using a broken theory to do computations of crystal band structure, my assertion that the theory they were using was broken was not met with the joy one might expect even though the theory I had developed permitted them to do almost the same computation and end up with a systematically and properly convergent result. I was threatening to pull the bread from the mouths of their children, metaphorically speaking (and vice versa!).

At this point, the forces that give rise to this sort of defensive science are thoroughly entrenched. The tenure system that was intended to prevent this sort of thing has been transformed into a money pump for Universities that can no longer survive without the constant influx of soft and indirect cost money farmed every year by their current tenured faculty, especially those in the sciences. Because in most cases that support comes from the federal government, that is to say our taxes, there is constant pressure to keep the research “relevant” to public interests. There is little money to fund research into (say) the formation of fractal crystal patterns by matter that is slowly condensing into a solid (like a snowflake) unless you can argue that your research will result in improved catalysis, or a way of building new nano-materials, or that condensed matter of this sort might form the basis for a new drug, or…

Or today, of course, that by studying this, you will help promote the understanding of the tiny ice crystals that make up clouds, and thereby promote our understanding of a critical part of the water cycle and albedo feedback in Climate Science and thereby do your bit to stave off the coming Climate Apocalypse.

I mean, seriously. Just go to any of the major search engines and enter “climate” along with anything you like as part of the search string. You would be literally amazed at how many disparate branches of utterly disconnected research manage to sneak some sort of climate connection into their proposals, and then (by necessity) into their abstracts and/or paper text. One cannot study poison dart frogs in the Amazon rainforest any more just because they are pretty, or pretty cool, or even because we might find therapeutically useful substances mixed into the chemical poisons that they generate (medical therapy being a Public Good even more powerful that Climate Science, quite frankly, and everything I say here goes double for dubious connections between biology research and medicine) — one has to argue somewhere that Climate Change might be dooming the poor frogs to extinction before we even have a chance to properly explore them for the next cure to cancer. Studying the frogs just because they are damn interesting, knowledge for its own sake? Forget it. Nobody’s buying.

In this sense, Climate Science is the ultimate save. Let’s face it, lots of poison dart frogs probably don’t produce anything we don’t already know about (if only from studying the first few species decades ago) and the odds of finding a really valuable therapy are slender, however much of a patent-producing home run it might be to succeed. The poor biologists who have made frogs their life work need a Plan B. And here Climate is absolutely perfect! Anybody can do an old fashioned data dredge and find some population of frogs that they are studying that is changing, because ecology and the environment is not static. One subpopulation of frogs is thriving — boo, hiss, cannot use you — but another is decreasing! Oh My Gosh! We’ve discovered a subpopulation of frogs that is succumbing to Climate Change! Their next grant is now a sure thing. They are socially relevant. Their grant reviewers will feel ennobled by renewing them, as they will be protecting Poison Dart Frogs from the ravages of a human-caused changing climate by funding further research into precisely how it is human activity that is causing this subpopulation to diminish.

This isn’t in any sense a metaphor, nor is it only poison dart frogs. Think polar bears — the total population is if anything rapidly rising, but one can always find some part of the Arctic where it is diminishing and blame it on the climate. Think coral reefs — many of them are thriving, some of them are not, those that are not may not be thriving for many reasons, some of those reasons may well be human (e.g. dumping vast amounts of sewage into the water that feeds them, agricultural silt overwashing them, or sure — maybe even climate change. But scientists seeking to write grants to study coral reefs have to have some reason in the public interest to be funded to travel all over the world to really amazing locations and spend their workdays doing what many a tourist pays big money to do once in a lifetime — scuba or snorkel over a tropical coral reef. Since there is literally no change to a coral reef that cannot somehow be attributed to a changing environment (because we refuse to believe that things can just change in and of themselves in a chaotic evolution too complex to linearize and reduce to simple causes), climate change is once again the ultimate save, one where they don’t even have to state that it is occurring now, they can just claim to be studying what will happen when eventually it does because everybody knows that the models have long since proven that climate change is inevitable. And Oh My! If they discover that a coral reef is bleaching, that some patch of coral, growing somewhere in a marginal environment somewhere in the world (as opposed to on one of the near infinity of perfectly healthy coral reefs) then their funding is once again ensured for decades, baby-sitting that particular reef and trying to find more like it so that they can assert that the danger to our reefs is growing.

I do not intend to imply by the above that all science is corrupt, or that scientists are in any sense ill-intentioned or evil. Not at all. Most scientists are quite honest, and most of them are reasonably fair in their assessment of facts and doubt. But scientists have to eat, and for better or worse we have created a world where they are in thrall to their funding. The human brain is a tricky thing, and it is not at all difficult to find a perfectly honest way to present one’s work that nevertheless contains nearly obligatory references to at least the possibility that it is relevant, and the more publicly important that relevance is, the better. I’ve been there myself, and done it myself. You have to. Otherwise you simply won’t get funded, unless you are a lucky recipient of a grant to do e.g. pure mathematics or win a no-strings fellowship or the Nobel Prize and are hence nearly guaranteed a lifetime of renewed grants no matter how they are written.

This is the really sad thing, Steve [Steven Mosher]. Science is supposed to be a debate. What many don’t realize is that peer review is not about the debate. When I review a paper, I’m not passing a judgment as a participant on whether or not its conclusion is correct politically or otherwise (or I shouldn’t be — that is gatekeeping, unless my opinion is directly solicited by an editor as the paper is e.g. critical of my own previous work). I am supposed to be determining whether or not the paper is clear, whether its arguments contain any logical or mathematical inconsistencies, whether it is well enough done to pass muster as “reasonable”, if it is worthy of publication, now not whether or not it is right or even convincing beyond not being obviously wrong or in direct contradiction of known facts. I might even judge the writing and English to some extent, at least to the point where I make suggestions for the authors to fix.

In climate science, however, the ClimateGate letters openly revealed that it has long since become covertly corrupted, with most of the refereeing being done by a small, closed, cabal of researchers who accept one another’s papers and reject as referees (well, technically only “recommend” rejection as referees) any paper that seriously challenges their conclusions. Furthermore, they revealed that this group of researchers was perfectly willing to ruin academic careers and pressure journals to fire any editor that dared to cross them. They corrupted the peer review process itself — articles are no longer judged on the basis of whether or not the science is well presented and moderately sound, they have twisted it so that the very science being challenged by those papers is used as the basis for asserting that they are unsound.

Here’s the logic:

a) We know that human caused climate change is a fact. (We heard this repeatedly asserted in the “debate” above, did we not? It is a fact that CO₂ is a radiatively coupled gas, completely ignoring the actual logarithmic curve Goreham presented, it is a fact that our models show that that more CO₂ must lead to more warming, it is a fact that all sorts of climate changes are soundly observed, occurred when CO₂ was rising so it is a fact that CO₂ is the cause, count the logical and scientific fallacies at your leisure).

b) This paper that I’m reviewing asserts that human caused climate change is not a fact. It therefore contradicts “known science”, because human caused climate change is a fact. Indeed, I can cite hundreds of peer reviewed publications that conclude that it is a fact, so it must be so.

c) Therefore, I recommend rejecting this paper.

It is a good thing that Einstein’s results didn’t occur in Climate Science. He had a hard enough time getting published in physics journals, but physicists more often than not follow the rules and accept a properly written paper without judging whether or not its conclusions are true, with the clear understanding that debate in the literature is precisely where and how this sort of thing should be cleared up, and that if that debate is stifled by gatekeeping, one more or less guarantees that no great scientific revolutions can occur because radical new ideas even when correct are, well, radical. In one stroke they can render the conclusions of entire decades of learned publications by the world’s savants pointless and wrong. This means that physics is just a little bit tolerant of the (possible) crackpot. All too often the crackpot has proven not only to be right, but so right that their names are learned by each succeeding generation of physicist with great reverence.

Maybe that is what is missing in climate science — the lack of any sort of tradition of the maverick being righter than the entire body of established work, a tradition of big mistakes that work amazingly well — until they don’t and demand explanations that prove revolutionary. Once upon a time we celebrated this sort of thing throughout science, but now science itself is one vast bureaucracy, one that actively repels the very mavericks that we rely on to set things right when we go badly astray.

At the moment, I’m reading Gleick’s lovely book on Chaos [Chaos: The Making of a New Science], which outlines both the science and early history of the concept. In it, he repeatedly points out that all of the things above are part of a well-known flaw in science and the scientific method. We (as scientists) are all too often literally blinded by our knowledge. We teach physics by idealizing it from day one, linearizing it on day two, and forcing students to solve problem after problem of linearized, idealized, contrived stuff literally engineered to teach basic principles. In the process we end up with students that are very well trained and skilled and knowledgeable about those principles, but the price we pay is that they all too often find phenomena that fall outside of their linearized and idealized understanding literally inconceivable. This was the barrier that Chaos theory (one of the latest in the long line of revolutions in physics) had to overcome.

And it still hasn’t fully succeeded. The climate is a highly nonlinear chaotic system. Worse, chaos was discovered by Lorenz [Edward Norton Lorenz] in the very first computational climate models. Chaos, right down to apparent period doubling, is clearly visible (IMO) in the 5 million year climate record. Chaotic systems, in a chaotic regime, are nearly uncomputable even for very, simple, toy problems — that is the essence of Lorenz’s discovery as his first weather model was crude in the extreme, little more than a toy. What nobody is acknowledging is that current climate models, for all of their computational complexity and enormous size and expense, are still no more than toys, countless orders of magnitude away from the integration scale where we might have some reasonable hope of success. They are being used with gay abandon to generate countless climate trajectories, none of which particularly resemble the climate, and then they are averaged in ways that are an absolute statistical obscenity as if the linearized average of a Feigenbaum tree of chaotic behavior is somehow a good predictor of the behavior of a chaotic system!

This isn’t just dumb, it is beyond dumb. It is literally betraying the roots of the entire discipline for manna.

One of the most interesting papers I have to date looked at that was posted on WUWT was the one a year or three ago in which four prominent climate models were applied to a toy “water world” planet, one with no continents, no axial tilt, literally “nothing interesting” happening, with fixed atmospheric chemistry.

The four models — not at all unsurprisingly — converged to four completely different steady state descriptions of the planetary weather.

And — trust me! — there isn’t any good reason to think that if those models were run a million times each that any one of them would generate the same probability distribution of outcomes as any other, or that any of those distributions are in any sense “correct” representations of the actual probability distribution of “planetary climates” or their time evolution trajectories. There are wonderful reasons to think exactly the opposite, since the models are solving the problem at a scale that we know is orders of magnitude to [too] coarse to succeed in the general realm of integrating chaotic nonlinear coupled systems of PDEs in fluid dynamics.

Metaphor fails me. It’s not like we are ignorant (any more) about general properties of chaotic systems. There is a wealth of knowledge to draw on at this point. We know about period doubling, period three to chaos, we know about fractal dimension, we know about the dangers of projecting dynamics in a very high dimensional space into lower dimensions, linearizing it, and then solving it. It would be a miracle if climate models worked for even ten years, let alone thirty, or fifty, or a hundred.

Here’s the climate model argument in a nutshell. CO₂ is a greenhouse gas. Increasing it will without any reasonable doubt cause some warming all things being equal (that is, linearizing the model in our minds before we even begin to write the computation!) The Earth’s climate is clearly at least locally pretty stable, so we’ll start by making this a fundamental principle (stated clearly in the talk above) — The Earth’s Climate is Stable By Default. This requires minimizing or blinding ourselves to any evidence to the contrary, hence the MWP and LIA must go away. Check. This also removes the pesky problem of multiple attractors and the disappearance and appearance of old/new attractors (Lorenz, along with Poincaré [Jules Henri Poincaré], coined the very notion of attractors). Hurst-Kolmogorov statistics, punctuated equilibrium, and all the rest is nonlinear and non-deterministic, it has to go away. Check. None of the models therefore exhibit it (but the climate does!). They have been carefully written so that they cannot exhibit it!

Fine, so now we’re down to a single attractor, and it has to both be stable when nothing changes and change, linearly, when underlying driving parameters change. This requires linearizing all of the forcings and trivially coupling all of the feedbacks and then searching hard — as pointed out in the talk, very hard indeed! — for some forlorn and non-robust combination of the forcing parameters, some balance of CO₂forcing, aerosol anti-forcing, water vapor feedback, and luck that balances this teetering pen of a system on a metaphorical point and tracks a training set climate for at least some small but carefully selected reference period, naturally, the single period where the balance they discover actually works and one where the climate is actively warming. Since they know that CO₂ is the cause, the parameter sets they search around are all centered on “CO₂ is the cause” (fixed) plus tweaking the feedbacks until this sort of works.

Now they crank up CO₂, and because CO₂ is the cause of more warming, they have successfully built a linearized, single attractor system that does not easily admit nonlinear jumps or appearances and disappearances of attractors so that the attractor itself must move monotonically to warmer when CO₂ is increasing. They run the model and — gasp! — increasing CO₂ makes the whole system warmer!

Now, they haven’t really gotten rid of the pesky attractor problem. They discover when they run the models that in spite of their best efforts they are still chaotic! The models jump all over the place, started with only tiny changes in parametric settings or initial conditions. Sometimes a run just plain cools, in spite of all the additional CO₂. Sometimes they heat up and boil over, making Venus Earth and melting the polar caps. The variance they obtain is utterly incorrect, because after all, they balanced the parameter space on a point with opposing forcings in order to reproduce the data in the reference period and one of many prices they have to pay is that the forcings in opposition have the wrong time constants and autocorrelation and the climate attractors are far too shallow, allowing for vast excursions around the old slowly varying attractor instead of selecting a new attractor from the near-infinity of possibilities (one that might well be more efficient at dissipating energy) and favoring its growth at the expense of a far narrower old attractor. But even so, new attractors appear and disappear and instead of getting a prediction of the Earth’s climate they get an irrelevantly wide shotgun blast of possible future climates (that is, as noted above, probably not even distributed correctly, or at least we haven’t the slightest reason to think that it would be). Anyone who looked at an actual computed trajectory would instantly reject it as being a reasonable approximation to the actual climate — variance as much as an order of magnitude too large, wrong time constants, oversensitive to small changes in forcings or discrete events like volcanoes.

So they bring on the final trick. They average over all of these climates. Say what? Each climate is the result of a physics computation. One with horrible and probably wrong approximations galore in the “physics” determining (for example) what clouds do in a cell from one timestep to the next, but at least one can argue that the computation is in fact modeling an actual climate trajectory in a Universe where that physics and scale turned out to be adequate. The average of the many climates is nothing at all. In the short run, this trick is useful in weather forecasting as long as one doesn’t try to use it much longer than the time required for the set of possible trajectories to smear out and cover the phase space to where the mean is no longer meaningful. This is governed by e.g. the Lyupanov exponents of the chaotic processes. For a while, the trajectories form a predictive bundle, and then they diverge and don’t. Bigger better computers, finer grained computations, can extend the time before divergence slowly, but we’re talking at most weeks, even with the best of modern tools.

In the long run, there isn’t the slightest reason — no, not even a fond hope — that this averaging will in any way be predictive of the weather or climate. There is indeed a near certainty that it will not be, as it isn’t in any other chaotic system studied so why should it be so in this one? But hey! The overlarge variance goes away! Now the variance of the average of the trajectories looks to the eye like it isn’t insanely out of scale with the observed variance of the climate, neatly hiding the fact that the individual trajectories are obviously wrong and that you aren’t comparing the output of your model to the real climate at all, you are comparing the average of the output of your model to the real climate when the two are not the same thing!

Incidentally, at this point the assertion that the results of the climate models are determined by physics becomes laughable. If I average over the trajectories observed in a chaotic oscillator, does the result converge to the actual trajectory? Seriously dudes, get a grip!

Oh, sorry, it isn’t quite the final trick. They actually average internally over climate runs, which at least is sort of justifiable as an almost certainly non-convergent sort of Monte Carlo computation of the set of accessible/probable trajectories, even though averaging over the set when the set doesn’t have the right probability distribution of outcomes or variance or internal autocorrelation is a bit pointless, but they end up finding that some of the models actually come out, after all of this, far too close to the actual climate, which sadly is not warming and hence which then makes it all too easy for the public to enquire why, exactly, we’re dropping a few trillion dollars per decade solving a problem that doesn’t exist.

So they then average over all of the average trajectories! That’s right folks, they take some 36 climate models (not the “twenty” erroneously cited in the presentation, I mean come on, get your facts right even if the estimate for the number of independent models in CMIP5 is more like seven). Some of these run absurdly hot, so hot that if you saw even the average model trajectory by itself you would ask why it is being included at all. Others as noted are dangerously close to a reality that — if proven — means that you lose your funding (and then, Walmart looms). So they average them together, and present the resulting line as if that is a “physics based” “projection” of the future climate. Because they keep the absurdly hot, they balance the nearly realistically cool and hide them under a safely rapidly warming “central estimate”, and get the double bonus that by forming the envelope of all of the models they can create a lower bound (and completely, utterly unfounded) “error estimate” that is barely large enough to reach the actual climate trajectory, so far.

Meh. Just Meh. This is actively insulting, an open abuse of the principles of science, logic, and computer modeling all three. The average of failed models is not a successful model. The average of deterministic microtrajectories is not a deterministic microtrajectory. A microtrajectory numerically generated at a scale inadequate to solve a nonlinear chaotic problem is most unlikely to represent anything like the actual microtrajectory of the actual system. And finally, the system itself realizes at most one of the possible future trajectories available to it from initial conditions subject to the butterfly effect that we cannot even accurately measure at the granularity needed to initialize the computation at the inadequate computational scale we can afford to use.

That’s what Goreham didn’t point out in his talk this time — but should. The GCMs are the ultimate shell game, hiding the pea under an avalanche of misapplied statistical reasoning that nobody but some mathematicians and maverick physicists understand well enough to challenge, and they just don’t seem to give a, uh, “flip”. With a few very notable exceptions, of course.

Rgb

Postscript (from a related slashdot post):

1° C is what one expects from CO₂ forcing at all, with no net feedbacks. It is what one expects as the null hypothesis from the very unbelievably simplest of linearized physical models — one where the current temperature is the result of a crossover in feedback so that any warming produces net cooling, any cooling produces net warming. This sort of crossover is key to stabilizing a linearized physical model (like a harmonic oscillator) — small perturbations have to push one back towards equilibrium, and the net displacement from equilibrium is strictly due to the linear response to the additional driving force. We use this all of the time in introductory physics to show how the only effect of solving a vertical harmonic oscillator in external, uniform gravitational field is to shift the equilibrium down by Δy = mg/k. Precisely the same sort of computation, applied to the climate, suggests that ΔT ≈ 1° C at 600 ppm relative to 300 ppm. The null hypothesis for the climate is that it is similarly locally linearly stable, so that perturbing the climate away from equilibrium either way causes negative feedbacks that push it back to equilibrium. We have no empirical foundation for assuming positive feedbacks in the vicinity of the local equilibrium — that’s what linearization is all about!

That’s right folks. Climate is what happens over 30+ years of weather, but Hansen and indeed the entire climate research establishment never bothered to falsify the null hypothesis of simple linear response before building enormously complex and unwieldy climate models, building strong positive feedback into those models from the beginning, working tirelessly to “explain” the single stretch of only 20 years in the second half of the 20th century, badly, by balancing the strong feedbacks with a term that was and remains poorly known (aerosols), and asserting that this would be a reliable predictor of future climate.

I personally would argue that historical climate data manifestly a) fail to falsify the null hypothesis; b) strongly support the assertion that the climate is highly naturally variable as a chaotic nonlinear highly multivariate system is expected to be; and c) that at this point, we have extremely excellent reason to believe that the climate problem is non-computable, quite probably non-computable with any reasonable allocation of computational resources the human species is likely to be able to engineer or afford, even with Moore’s Law, anytime in the next few decades, if Moore’s Law itself doesn’t fail in the meantime. 30 orders of magnitude is 100 doublings — at least half a century. Even then we will face the difficulty if initializing the computation as we are not going to be able to afford to measure the Earth’s microstate on this scale, and we will need theorems in the theory of nonlinear ODEs that I do not believe have yet been proven to have any good reason to think that we will succeed in the meantime with some sort of interpolatory approximation scheme.

rgb

Author: Dr. Robert G. Brown is a Lecturer in Physics at Duke University where he teaches undergraduate introductory physics, undergraduate quantum theory, graduate classical electrodynamics, and graduate mathematical methods of physics. In addition Brown has taught independent study courses in computer science, programming, genetic algorithms, quantum mechanics, information theory, and neural network.

Moderation and Author’s Replies Note: This elevated comment has been posted at the request of several commenters here. It was edited by occasional WUWT contributor Kip Hansen with the author’s approval. Anything added to the comment was denoted in [square brackets]. There are only a few corrections of typos shown by strikeout [correction]. When in doubt, refer to the original comment here. RGB is currently teaching at Duke University with a very heavy teaching schedule and may not have time to interact or answer your questions.

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