Archive for the ‘physical science’ Category

Thinking inside the Box

Sunday, 4 March 2012

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.


[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 doie 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.

Smoke Gets in My Eyes

Friday, 2 September 2011

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.)

Weighty Matters

Sunday, 26 September 2010

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

Why aren't you just using pounds for the whole lot?

and

Wait, now that I think of it, what happened to that metric stuff?

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

Degenerate Matter

Tuesday, 1 June 2010

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: Ray Palmer leaps over a wall in pursuit of a meteor seen in the distance, about to hit the Earth.Ray Palmer excavates a meteor composed of about 1000 cu cm of degenerate matter from a white dwarf star, buried about two feet in the earth. 'So heavy-- I can hardly lift it!'Palmer, holding the meteor, looks at in amazement. 'Puff!'Palmer carries the meteor back to his car. 'Puff!'

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.

D_mn'd Yanquis

Friday, 22 January 2010

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.

Ball-Parking

Friday, 7 August 2009

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).