Physics explains color in terms of frequencies — or, equivalently, in terms of wave-lengths — of light. And the colors of substances are explained in terms of what frequencies these substances absorb and what frequencies they radiate or allow to pass.

An object of some color is so because it radiates or allows to pass light of frequencies participating in that color, but absorbs all others. When the object absorbs light of other frequencies, the energy of that light is absorbed. The reason that everyday objects don't then heat-up indefinitely is that they radiate the energy as light, but in the frequencies natural to their substances, rather than simply in the frequencies that were absorbed. A blue sweater converts white light to blue light. Moreover, under stable conditions, substances radiate light in the same set of frequencies. The blue sweater stays blue.

If the Earth did not radiate back into outer-space a significant amount of the light energy in which the Earth is bathed, then it would be dramatically hotter. But, as the surface of the Earth radiates light into the atmosphere, some of that light is absorbed by the atmosphere and then radiated back towards the surface; and, as some of the light radiated by lower parts of the atmosphere is radiated upward, some of it is absorbed by higher parts of the atmosphere and then radiated downward.

What are called greenhouse gasses have their effect by absorbing light energy and then returning a share of it downward, instead of allowing it to escape into outer-space. (Actual greenhouses work by a different principle.)

The theory of anthropogenic global climate change says that release of greenhouse gasses into the atmosphere by human activity causes significant changes to climate by increasing the amount of light energy radiated back downward, instead of being allowed to escape into outer-space.

Climate surely changes, and indeed has warmed and cooled, but that much was true before the first humans appeared, and well before humans released nearly so much of greenhouses gasses since the onset of the Industrial Age. So, if we seem to observe change now, or even warming now, then we want a means of determining whether the human contribution is significant or the change is about what it would be without us.

When the theory of anthropogenic global climate change first came into fashion, its models assumed a linear effect. That is to say that doubling the amount of a greenhouse gas would double the amount of light energy trapped, and so forth. Proper economists were naturally doubtful; we are used to marginal effects beyond some point diminishing. Indeed, the particles of a greenhouse gas would have to be arranged in some remarkable configuration in the sky to have linear effect. We could also imagine remarkable configurations in which marginal effects became immeasurably tiny. It should be no surprise that the linear models failed miserably.

As it happens, the effects of the greenhouse gasses are empirically measurable. Carbon dioxide, methane, and the other greenhouse gasses are like other chemicals, including the dye of that blue sweater that I mentioned. The greenhouse gasses don't simply radiate light energy back towards the surface of the Earth; they radiate it in characteristic frequencies. If you had the resources for strategically placed sensors, then for the last ten, twenty, or thirty years you could have measured the light energy in the relevant frequencies, and could have compared these measurements against the changing levels of the gasses — if you wanted to know.

Many of the various national states have the resources, and every reason to support such a study if its results conform to the theory of anthropogenic global climate change. But the results of a study effecting such measurements on a meaningful scale have not been reported. If that silence is because the results are being kept secret, then plainly they are inconvenient to whomever has kept them secret. If the silence is because such a study has not been undertaken, then plainly that is because the results are expected to be inconvenient.

If you're not autistic, then you recognize the significance for the theory that results have not been reported.

Famously, the Euclidean axiomata for space seemed necessary to many, so that various philosophers concluded or argued that some knowledge or something playing a rôle like that of knowledge derived from something other than experience. Yet there were doubters of one of these axiomata — that parallel lines would never intersect — and eventually physicists concluded that the universe would be better described were this axiom regarded as incorrect. Once one axiom was abandoned, the presumption of necessity of the others evaporated.

I think that our concept of space is built upon an experience of an object sometimes affecting another in ways that it sometimes does not, with the first being classified as near when it does and not near when it does; which ways are associated in the concept of near-ness are selected by experience. The concept of distance — variability of near-ness — develops from the variability of how one object affects another; and it is experience that selects which variabilities are associated with distance. Our concept of space is that of potential (realized or not) of near-ness.

The axiomata of Euclid were, implicitly, an attempted codification of observed properties of distance; in the adoption of this codification or of another, one might revise which variabilities one associated with distance. One might, in fact, hold onto those axiomata exactly by revising which variabilities are associated with distance. In saying that space is non-Euclidean, one ought to mean that the Euclidean axiomata are not the best suited to physics.

Just as the axiomata of Euclid become ill-suited to physics when distances become very large, they may be ill-suited when distances become very small.

Space might not even be divisible without limit. The mathematical construct of continuity may not apply to the physical world. At least some physical quantities that were once imagined potentially to have measures corresponding to any real number are now regarded as having measures corresponding only to integer multiples of quanta; perhaps distance cannot be reduced below some minimum.

And, at some sub-atomic level, any useable rules of distance might be more complex. On a larger scale, non-Euclidean spaces are sometimes imagined to have worm-holes, which is really to say that some spaces would have near-ness by peculiar paths. Perhaps worm-holes or some discontinuous analogue thereöf are pervasive at a sub-atomic level, making space into something of a rat's nest.

Reading a book first published in 1951, I am reminded that, at one time, the definition of humanities included sciences of human behavior within its scope. Now, one seldom encounters that inclusion in contemporary use, and the Merriam-Webster Dictionary explicitly excludes the study of social relations (though it says nothing explicit about that part of behavior outside of the social).

In the earlier period, there was a question of whether the study of human behavior were fundamentally different from the study of the properties of other things. Those who insisted upon such a difference would speak and write of science and the humanities as if of two separate things.

But the tools by which the physical, biological, and behavioral science were studied were increasingly shared. The physical and biological sciences took-up probability and statistics; the biological sciences have taken-up chemistry, mechanics, and game theory; the behavioral science have taken-up biological explanation and mathematical modelling. All have been affected by the same philosophic theories of method. A dichotomy of science and the humanities cannot prevail so long as the behavioral sciences are included amongst what are called humanities.

Apparently that dichotomy was so dear to some of those who insisted upon it that they attempted its preservation by implicitly changing what they intended with humanities in order to hold fast to it. Of course, the newer definition doesn't maintain the original dichotomy; but replaces it with a new one.

I recently finished reading A Budget of Paradoxes (1872) by Augustus de Morgan.

Now-a-days, we are most likely to encounter the word paradox as referring to apparent truth that seems to fly in the face of reason, but its original sense, not so radical, was of a tenet opposed to received opinion. De Morgan uses it more specifically for such tenets when they go beyond mere heterodoxy. Subscribers to paradox are those typically viewed as crackpot, though de Morgan occasionally takes pains to explain that, in some cases, the paradoxical pot is quite sound, and it is the orthodox pot that will not hold water. None-the-less, most of the paradoxers, as he calls them, proceed on an unsound basis (and he sometimes rhetorically loses sight of the exceptions).

A recurring topic in his book is attempt at quadrature of the circle. Most of us have heard of squaring the circle, though far fewer know to just what it refers.

I guess that most students are now taught to think about geometry in terms of Cartesian coördinates,[1] but there's an approach, called constructive, which concerns itself with what might be accomplished using nothing but a stylus, drawing surface, straight-edge, and compass. The equipment is assumed to be perfect: the stylus to have infinitesimal width; the surface to be perfectly planar, the straight-edge to be perfectly linear, and the pivot of the compass to stay exactly where placed. The user is assumed able to place the pivot of the compass exactly at any marked point and to open it to any other marked point; likewise, the user is assumed to be able to place the straight-edge exactly touching any one or two marked points. A marked point may be randomly placed, or constructed as the intersection of a line with a line, of an arc with an arc, or of an arc with a line. A line may be constructed by drawing along the straight-edge. An arc may be constructed by placing the compass on a marked point, opening it to touch another marked point, and then turning it. (Conceptually, these processes can be generalized into n dimensions.)

A classic problem of constructive geometry was to construct a square whose area was equal to that of a given circle. Now, if you think about it, you'll reälize that this problem is equivalent to arriving at the value of π; with a little more thought, you might see that to construct this square in a finite number of steps would be equivalent to finding a rational value for π. So, assuming that one is restricted to a finite number of steps, the problem is insoluable. It was shown to be so in the middle of the 18th Century, when it was demonstrated that π were irrational.

The demonstration not-with-standing, people continued to try to square the circle into de Morgan's day, and some of them fought in print with de Morgan. (One of them, a successful merchant, was able to self-publish repeatedly.) De Morgan tended to deal with them the way that I often deal with people who are not merely wrong but are arguing foolishly — he critiqued the argument as such, rather than attempting to walk them through a proper argument to some conclusion. I think that he did so for a number of reasons. First, bad argumentation is a deeper problem that mistaken conclusions, and de Morgan had greater concern to attack the former than the latter, in a manner that exhibited the defects to his readers. Second, some of these would-be squarers of the circle had been furnished with proper argumentation, but had just plowed-on, without attending to it. (Indeed, de Morgan notes that most paradoxers will not bother to familiarize themselves with the arguments for the systems that they seek to overthrow, let alone master those arguments.) Third, the standard proof that π is not rational is tedious to mount, and tedious to read.

But de Morgan, towards justifying attending as much as he does specifically to those who would square the circle, expresses a concern that they might gain a foothold within the social structure that allowed them to demand positions amongst the learnèd, and that they might thus undermine the advancement of useful knowledge.[2] And, with this concern in-mind, I wonder why I didn't, to my recollection, encounter de Morgan once mentioning that constructive quadrature of the circle would take an infinite number of operations; he certainly didn't emphasize this point. It seems to me that the vast majority of would-be squarers of the circle (and trisectors of the angle) simply don't see how many steps it would take; that their intuïtion fails them exactly there. And their intuïtion is an essential aspect of the problem; a large part of why the typical paradoxer will not expend the effort to learn the orthodox system is that he or she is convinced that his or her intuïtion has found a way around any need to do so. But sometimes a lynch-pin in the intuïtion may be pulled, causing the machine to be arrested, and the paradoxer to pause. Granted that this may not be as potentially edifying to the audience, but if one has real fear of the effects of paradoxers on scientific pursuit, then it is perhaps best to reduce their number by a low-cost conversion.

De Morgan's concern for the effect of these géomètres manqués might seem odd these days, though I presume that it was quite sincere. I've not even heard of an attempt in my life-time actually to square the circle[3] (though I'm sure that some could be found). I think that attempts have gone out of fashion for two reasons. First, a greater share of the population is exposed to the idea that π is irrational almost as soon as its very existence is reported to them. Second, technology, founded upon science, has got notably further along, and largely by using and thereby vindicating the mathematical notions that de Morgan was so concerned to protect because of their importance. To insist now that π is, say 3 1/8, as did some of the would-be circle-squarers of de Morgan's day, would be to insist that so much of what we do use is unusable.

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[1] Cartesian coördinates are named for René Descartes (31 March 1596 – 11 February 1650) because they were invented by Nicole Oresme (c 1320 – 11 July 1382).

[2] Somewhat similarly, many people to-day are concerned that paradoxers not be allowed to influence palæobiology, climatology, or economics. But, whereäs de Morgan proposed to keep the foolish paradoxers of his day in-check by exhibiting the problems with their modes of reasoning, most of those concerned to protect to-day's orthodoxies in alleged science want to do so by methods of ostensibly wise censorship that in-practice excludes views for being unorthodox rather than for being genuinely unreasonable. When jurists and journalists propose to operationalize the definition of science with the formula that science is what scientists do — ie that science may be identified by the activity of those acknowledged by some social class to be scientists — actual science is being displaced by orthodoxy as such.

[3] Trisection of the angle is another matter. As a university undergraduate, I had a roommate who believed that one of his high-school classmates had worked-out how to do it.

If one wanted to know the solution to particular mathematical problem, and found that different groups gave different answers, then it might be interesting to hear or to read what each group said about the motives of rival groups, but one really ought to chose which answer or answers were correct based upon principles of mathematics, rather than based upon which groups seemed most noble. If one lacked the competence to decide the issue based upon principles of mathematics, then it would probably be best to resist coming to any decision if at all possible.

Likewise, if one wanted to know the solution to a particular problem of the natural sciences, but found that different groups gave different answers, then it might be interesting to hear or to read what each group said about the motives of rival groups, but one really ought to chose which answer or answers were correct based upon principles of science, rather than based upon which group seemed most noble. If one lacked the competence to decide the issue based upon principles of science, then it would probably be best to resist coming to any decision if at all possible.

And if one wanted to know what sort of social policy ought to be applied to some case, but found that different groups gave one different answers, then it might be interesting to hear or to read what each group said about the motives of rival groups, but one really ought to chose which answer or answers were correct based upon principles of science in combination with rational criteria for evaluating ethical philosophies (if, indeed, those criteria are not themselves scientific). And if one lacked the competence to decide the issue based upon such principles, then it would probably be best to resist coming to any decision if at all possible.

Now, all of that ought to be obvious; but consider how much pundits and the major media focus on personalities and theories of motive when it comes both to policy and to science applicable to policy, and how little real science and how little careful dissection of philosophical case is presented. If one party wants one thing, and another wants something different, then we are given some tale of the nobility or at least the level-headedness of one group, and of the knavery or foolishness of the other; accompanying this narrative will be cartoon physics, cartoon biology, or cartoon economics. If ethics are relevant, then one might get cartoon philosophy of ethics, or some ethical philosophy might be implicitly imposed, as if no rival philosophy were conceivable. (If something is treated as good, there generally ought to be an explanation somewhere of what makes it good. If something is treated as bad, there likewise ought to be an explanation of what makes it bad.)

This practice is so prevalent because so many listeners and readers unthinkingly accept it. And I'm not just talking about low-brow or middle-brow people. The self-supposed high-brow folk, more educated and ostensibly more thoughtful, accept this practice. Most of the people who would, if they read them, say that the previous four paragraphs were trivially obvious accept this practice. I don't simply mean that they don't cancel subscriptions or write angry letters to the editor; I mean that they allow their own beliefs to be shaped by some group engaging in the practice. They fall into attending to one narration of this sort, and let it guide them until and unless some crisis causes them to turn their backs on it, at which point they almost always begin to be guided by a narration using the same basic practice to advance some different set of policies.

Sometimes, one must make a decision, with nothing upon which to go except the discernible motives of conflicting parties. In those cases, one should bear in mind that, except to the extent that they are reporting brute fact (rather than interpretation), one typically learns more about the narrators themselves from what they say (and avoid saying) of their opponents, than one learns about their opponents. (And one should not allow the emotional appeal of a narrative to lead one to pretend that one must make a decision that one can in fact defer.)

The metric system has some points of genuine superiority to those of the English (aka American) system, but that superiority tends to be exaggerated. For example, the every-day English measures for volume tend to be implicitly binary, allowing easy halving or doubling. (If base 10 were everywhere superior to base 2, then our computers would be designed differently.)

One of the things that I was told as a child was that the metric system were superior because it measured in terms of mass, rather than weight, with the former being invariant while the latter would change in the face of a gravitational field. Well, actually, the English system has a unit of mass; it's the slug, 1 lb·sec²/ft, which is about 14.6 kg.

Meanwhile, I observe that, in countries where the metric system ostensibly prevails, people typically use its names of units of mass (gram and kilogram) for units of weight; they even refer to what is measured as a weight. Now, the real metric system does have a unit for weight, because weight is a force; weight can be measured by the newton (or by the dyne, which is a hundred-thousandth of a newton). But people aren't doing that; they're using kilogram as if it means about 9.807 N.

Much as it may be claimed that America is the only industrialized nation not on the metric system, really nobody's on it.

I notice that the Beeb most often wants to speak and write of weight, rather than of mass, but in the most ghastly unit of all, the stone (pronounced /stɛun/, with at least one pinkie extended). The stone is 14 pounds (divisible by 2 and, uh, 7). When weights don't divide into integer multiples of 14 pounds, tradition is to represent weight in terms of a combination of stone and pounds, as in Me mum weighs 19 stone and 12. Of course, if the Beeb were using pounds at all, there'd be the two obvious questions of

and So the Beeb feels compelled just to round everything up or down to an integral number of stone, and somebody's mum either gains two pounds or loses twelve.

At Kingdom Kane (a 'blog focussed upon the art of Gil Kane), Mykal Banta has reproduced The Birth of the Atom. a story which contains what I have long regarded as an epitomal sequence of what I call comic-book science:

As I noted to Mykal, a white dwarf star has a density of about 1 million grams per cc, and the meteor appears to be about 1000 cc, so the whole thing should mass at about 1 million kilograms.

It's not apparent why 1 million kilograms should stay compressed into such a small volume. In the case of a dwarf star itself, the gravitational mass of the star as a whole creätes sufficient force, but this is just a fractional piece of such a star. It ought to fly apart as a terrible burst of radiation. But let's assume that this somehow doesn't happen, that the meteor just stays together in a nifty one-liter piece.

The meteor that creäted Meteor Crater in Arizona was under 30,000 kilograms. Ray wouldn't be excavating the meteor at all; he would have been killed by the shock waves from the impact. Those who later did excavate the meteor wouldn't find it buried just a couple of feet deep.

At the surface of the Earth (which itself masses about 5.97 × 10²⁴ kilograms), this meteor would weigh about 11 hundred tons, but Ray picks it up! He subvocalizes a few puffs, but he manages to carry the thing back to his car! Now-a-days, they don't make cars that can carry 11 hundred tons. I don't think that any grad students can lift 11 hundred tons. And, really, Ray ought to be sinking into the ground, as even if he has big feet and has both feet on the ground he is applying over 7000 kPa of pressure to the soil.

It might be suggested that the meteor, while perhaps of material that were once compressed to a density of about 1 million grams per cc, were subsequently uncompressed, and that what Palmer recovered were only, say, 100 kilograms of material. But I don't know how, then, it would be recognizable as originating from a white dwarf star. For example, the core of the sun compresses matter to a greater density than 100 grams per cc.

Readers of this 'blog might recall the Decimator. Well, according to Hugo Chávez, the United States has one.

I'm just hoping that it doesn't fall into the wrong hands, and get directed at the Amsterdam Fault. Meanwhile, maybe I can become one of the Rocket Men. At my age, hopes of becoming a super-hero have dimmed, but I at least look younger than Jeff King.

The character is drawn about 33 pixels tall. If he's of average height, then that means that 1 pixel maps to about 2 1/8 in (5.39 cm). The pulley above him is drawn about 431 pixels above ground, which would then be about 76 ft (23.22 m) up.

At 14% of Earth's gravity, gravitational acceleration on Titan would be at about 4.5 ft (1.37 m) per squared second.

That all means that he hits the ground at about 17.8 mph (28.7 kph) or less. That would be the speed associated with a normal fall of about 10.5 ft (3.2 m).