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 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.

[This post was delayed from yester-day, as my hosting service had a technical failure, and it took me rather a long time to persuade them of such.]

I read

This past week it was reported that the hacktivist collective known as Anonymous claimed credit for taking offline over 40 websites used for sharing pedophilia — and for exposing the names and identifying information of more than 1500 alleged pedophiles that had been using the sites.

But the actual list is of user aliases, not of personal names.

Not only are pædophiles not being exposed here, but non-pædophiles who've had the misfortune of pædophiles' using the same aliases (by chance or from malice) are going to come under suspicion by those who think that they recognize them on this list.

Further, if agents of law enforcement were themselves working to track-down the actual legal identities of the pædophiles, their investigation has now been severely compromised, possibly fatally so.

Once again, Anonymous has done less good than they have led the gullible to believe, and have caused more damage than they have acknowledged.

One claim about Libertarians that won't withstand any real scrutiny — yet is very common amongst journalists and educators — is that Libertarians don't believe in doing anything to address the immediate needs of the poor. If asked to defend the claim, those who make it will either note Libertarian opposition to various state programmes, and with a crude induction draw the inference that Libertarians don't believe in doing anything to achieve the ostensible goals of those programmes, or they'll note the Libertarian objection in principle to any state programme with such goals, and treat this as QED.

Well, let's lay the form of that out:

L does not believe that X should be done by S,
therefore
L does not believe that X should be done.

Oooops! That isn't really very logical, is it? I mean that we can find plenty of X and S where this won't work, when we make ourselves L.

Libertarians don't believe that the state should do a lot of things, including farming, financial intermediation, and managing roads. Genuine anarchists go further, to claim that the state shouldn't do anything. That hardly means that they don't think that these things should be done by someone. It doesn't even mean that they won't agree that they should be those who do these things. (Indeed, people who rely upon the state are most likely to say that it ought to do whatever it does at the expense of someone else, as when they call for higher taxes on those who make more money.)

This point of logic ought to be obvious. Well, many journalists and educators are such damn'd fools that they truly don't see it, and an awful lot are knaves, who see it but don't want it to be seen by others.

One way that I see the eristicism effected is by the specious society-state equation — by treating the state as if it is society, which is to say as if it is us. Formally, this would be

L does not believe that X should be done by the state,
which is to say that
L does not believe that X should be done by society,
which is to say that
L does not believe that X should be done by any of us.

except that it's not explicitly expanded in this way, else the jig would be up. One place you'll see this eristic equation employed is in many quizzes that purport to tell the taker what his or her political classification is. If he or she answers affirmatively to a claim such as that society should help the poor then the typical quiz will score that towards state socialism and away from classical liberalism (of which Libertarianism is the extreme).

(Actually, one needs to be very careful whenever encountering the word society. In practice, it is often used to mean everyone else. Sometimes it's used to refer to some hypothetical entity which is somehow more than a group of people and their system of interaction; this latter notion tends to operationalize, again, as everyone else. Equating society with the state, and coupling this with demands for the state to make greater demands on other people is a popular way of making society mean everyone else.)

The fact is that one simply cannot tell, one way or another, from the datum that a person is a Libertarian whether he or she thinks that some goal ought to be pursued, unless the goal involves what a Libertarian would label coercion; because Libertarianism itself is no more than a belief that one ought not to initiate the class of behaviors to which they apply this label. A person can be a Libertarian and be all for voluntary redistribution, or that person might indeed be someone who embraced some of the more callous proclamations of Ayn Rand, or the Libertarian might hold some intermediate postion. Libertarianism itself is neutral.

(Within the Randian camp, there has been a willful confusion of the fact that Libertarianism itself has limited scope with the proposition that any given person who is a Libertarian must somehow have no view about matters not within that scope, or with the claim that a Libertarian must think that anything not prohibitable is good.)

Parallels can be found here with the claim that atheists do not believe in morality of any sort. Not only is the underlying fallacy very similar, but the implication in each case is that, should the persons in question believe that something ought to be done, they are more likely to see themselves as the someone who ought 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.)

(Actually, there is some evidence that a quantitative theory of probability developed and then disappeared in ancient India.[10] But Franklin's essential point here is none-the-less well-taken.)

⁹ Foot-note in the original, citing Decretales Pseudo-Isidorianae, et Capitula Angilramni edited by Paul Hinschius, and recommending comparison with The Collection in Seventy-Four Titles: A Canon Law Manual of the Gregorian Reform edited by John Gilchrist.

[10] In The Story of Nala and Damayanti within the Mahābhārata, there is a character Rtuparna (aka Rituparna, and mistakenly as Rtupama and as Ritupama) who seems to have a marvelous understanding of sampling and is a master of dice-play. I learned about Rtuparna by way of Ian Hacking's outstanding The Emergence of Probability; Hacking seems to have learned of it by way of V.P. Godambe, who noted the apparent implication in A historical perspective of the recent developments in the theory of sampling from actual populations, Journal of the Indian Society of Agricultural Statistics v. 38 #1 (Apr 1976) pp 1-12.

A previous entry quotes a foot-note from Austrian Marginalism and Mathematical Economics by Karl Menger; that foot-note is tied to a sentence that I found particularly striking.

(musings on the relationship of mathematics to economics)

But to assume from the superiority of Galilean principles in the sciences of inanimate nature that they must provide the model for the sciences of animate behaviour is to make a speculative leap, not to enunciate a necessary conclusion.

Whether a decision as such is good or bad is never determined by its actual consequences as such.

Decisions are made before their consequences are reälized (made actual). Instead, decisions are made in the face of possible consequences. There may be an ordering of these consequences in terms of plausibility, in which case that ordering should be incorporated into the making of the decision. Most theories even presume that levels of plausibility may be meaningfully quantified, in which case (ex hypothesi) these quantifications should be incorporated into the process. But even in a case where there were only one outcome possible, while the decision could (and should) be made in response to that unique possibility, it still were possibility of the consequence that informed the decision, and not actuality. (Inevitability is not actuality.)

When the reälized consequences of a decision are undesirable, many people will assert or believe that whoever made the choice (perhaps they themselves) should have done something different. Well, it might be that a bad outcome illustrates that a decision were poor, but that will only be true if the inappropriateness of the decision could have been seen without the illustration. For example, if someone failed to see a possibility as such, then its reälization will show the possibility, but there had to have been some failure of reasoning for a possibility to have ever been deemed impossible. On the other hand, if someone deemed something to be highly unlikely, yet it occurred anyway, that doesn't prove that it were more likely than he or she had thought — in a world with an enormous number of events, many highly unlikely things happen. If an event were highly unlikely but its consequences were so dire that they should have been factored into the decision, and yet were not, the reälization of the event might bring that to one's attention; but, again, that could have been seen without the event actually occurring. The decision was good or bad before its consequences were reälized.

A painter whose canvas is improved by the hand of another is not a better painter for this, and one whose work is slashed by a madman (other than perhaps himself) is not a worse painter for that. Likewise, choosing well is simply not the same thing as being lucky in one's choice, and choosing badly not the same as being unlucky.

Sometimes people say that this-or-that should have been chosen simply as an expression of the wish that more information had been available; in other cases, they are really declaring a change in future policy based upon experience and its new information. In either case, the form of expression is misleading.

Some readers may be thinking that what I'm saying here is obvious (and some of these may have abandoned reading this entry). But people fail to take reasonable risks because they will or fear that they will be thought fools should they be unlucky; some have responded to me as if I were being absurd when I've referred to something as a good idea that didn't work; our culture treats people who attempt heinous acts but fail at them as somehow less wicked than those who succeed at them; and I was drawn to thinking about this matter to-day in considering the debate between those who defend a consequentialist ethics and those who defend a deöntological ethics, and the amount of confusion on this issue of the rôle of consequences in decision-making (especially on the side of the self-identified consequentialists) that underlies that debate.

When someone uses the word random, part of me immediately wants a definition.[1]

One notion of randomness is essentially that of lawlessness. For example, I was recently slogging through a book that rejects the proposition that quantum-level events are determined by hidden variables, and insists that the universe is instead irreducibly random. The problem that I have with such a claim is that it seems incoherent.

There is no being without being something; the idea of existence is no more or less than that of properties in the extreme abstract. And a property is no more or less than a law of behavior.

Our ordinary discourse does not distinguish between claims about a thing and claims about the idea of a thing. Thus, we can seem to talk about unicorns when we are really talking about the idea of unicorns. When we say that unicorns do not exist, we are really talking about the idea of unicorns, which is how unicorns can be this-or-that without unicorns really being anything.

When it is claimed that a behavior is random in the sense of being without law, it seems to me that the behavior and the idea of the behavior have been confused; that, supposedly, there's no property in some dimension, yet it's going to express itself in that dimension.

Another idea of randomness is one of complexity, especially of hopeless complexity. In this case, there's no denial of underlying lawfulness; there's just a throwing-up of the hands at the difficulty in finding a law or in applying a law once found.

This complexity notion makes awfully good sense to me, but it's not quite the notion that I want to present here. What unites the notion of lawlessness with that of complexity is that of practical unpredictability. But I think that we can usefully look at things from a different perspective.

After the recognition that space could be usefully conceptualized within a framework of three orthogonal, arithmetic dimensions, there came a recognition that time could be considered as a fourth arithmetic dimension, orthogonal to the other three. But, as an analogy was sensed amongst these four dimensions, a puzzle presented itself. That puzzle is the arrow of time. If time were just like the other dimensions, why cannot we reverse ourselves along that dimension just as along the other three. I don't propose to offer a solution to that puzzle, but I propose to take a critical look at a class of ostensible solutions, reject them, and then pull something from the ashes.

Some authors propose to find the arrow of time in disorder; as they would have it, for a system to move into the future is no more or less than for it to become more disorderly.

One of the implications of this proposition is that time would be macroscopic; in sufficiently small systems, there is no increase nor decrease in order, so time would be said neither to more forward nor backward. And, as some of these authors note, because the propensity of macroscopic systems to become more disorderly is statistical, rather than specifically absolute, it would be possible for time to be reversed, if a macroscopic system happened to become more orderly.

But I immediately want to ask what it would even mean to be reversed here. Reversal is always relative. The universe cannot be pointed in a different direction, unless by universe one means something other than everything. Perhaps we could have a local system become more orderly, and thus be reversed in time relative to some other, except, then, that the local system doesn't seem to be closed. And, since the propensity to disorder is statistical, it's possible for it to be reversed for the universe as a whole, even if the odds are not only against that but astronomically against it. What are we to make of a distinction between a universe flying into reverse and a universe just coming to an end? And what are we to make of a universe in which over-all order increases for some time less than the universe has already existed? Couldn't this be, and yet how could it be if the arrow of time were a consequence of disorder?

But I also have a big problem with notions of disorder. In fact, this heads us back in the direction of notions of randomness.

If I take a deck of cards that has been shuffled, hand it to someone, and ask him or her to put it in order, there are multiple ways that he or she might do so. Numbers could be ascending or descending within suits, suits could be separated or interleaved, &c. There are as many possible orderings as there are possible rules for ordering, and for any sequence, there is some rule to fit it. In a very important sense, the cards are always ordered. To describe anything is to fit a rule to it, to find an order for it. That someone whom I asked to put the cards in order would be perfectly correct to just hand them right back to me, unless I'd specified some order other than that in which they already were.

Time's arrow is not found in real disorder generally, because there is always order. One could focus on specific classes of order, but, for reasons noted earlier, I don't see the explanation of time in, say, thermodynamic entropy.

But, return to decks of cards. I could present two decks of card, with the individual cards still seeming to be in mint state, with one deck ordered familiarly and with the other in unfamiliar order. Most people would classify the deck in familiar order as ordered and the other as random; and most people would think the ordered deck as more likely straight from the pack than the random deck. Unfamiliar orderings of some things are often the same thing as complex orderings, but the familiar orderings of decks of cards are actually conventional. It's only if we use a mapping from a familiar ordering to an unfamiliar ordering that the unfamiliar ordering seems complex. Yet even people who know this are going to think of the deck in less familiar order as likely having gone through something more than the deck with more familiar order. Perhaps it is less fundamentally complexity than experience of the evolution of orderings that causes us to see the unfamiliar orderings as random. (Note that, in fact, many people insist that unfamiliar things are complicated even when they're quite simple, or that familiar things are simple even when they're quite complex.)

Even if we do not explain the arrow of time with disorder, we associate randomness with the effects of physical processes, which processes take time. Perhaps we could invert the explanation. Perhaps we could operationalize our conception of randomness in terms of what we expect from a class of processes (specifically, those not guided by intelligence) over time.

(Someone might now object that I'm begging the question of the arrow of time, but I didn't propose to explain it, and my readers all have the experience of that arrow; it's not a rabbit pulled from a hat.)

[1] Other words that cause the same reäction are probability and capitalism.

Sometimes I've simplistically said that invocation of consensus is not a scientific method. A more accurate claim would be that its use is a way of approximating the results of more rigorous methods — a way of approximation that should never be mistaken for the more rigorous methods, and that is often unacceptable as science.

Calling upon consensus is a generalization of calling upon an expert. Using an expert can be analogous to using an electronic calculator. In some sense, using a calculator could be said to be scientific; there are sound empirical reasons for trusting a calculator to give one the right answer — at least for some classes of problems.

But note that, while possibly scientific, the use of the calculator is, itself, not scientifically expert in answering the question actually asked of the calculator (though some scientific expertise may have gone into answering the questions of whether to use a calculator, and of which calculator to use). Likewise, calling upon opinion from a human expert is not itself scientifically expert in answering the question actually asked. That distinction might not matter much, if ultimately scientific expertise from someone (or from some thing) ultimately went into the answer.

The generalization of invoking consensus proceeds in at least one direction, and perhaps in two. First, using consensus generalizes from using one expert to using n experts. But, second, invoking consensus often generalizes from invoking the views of experts to invoking the views of those who are less expert, or even not expert at all.

Individual human experts, like individual electronic calculators, may not be perfectly reliable for answers to some sorts of questions. One response to this problem is the generalization of getting an answer from more than one, and, using a sort of probabilistic reasoning, going with the answer given by a majority of the respondents, or with some weighted sum of the answers. However, this approach goes astray when a common error prevails amongst most of the experts. If one returns to the analogy of digital calculators, various limitations and defects are typical, but not universal; a minority of calculators will answer some questions correctly, even as the majority agree on an incorrect answer. Likewise with human experts. That's not to say that being in the minority somehow proves a calculator or a human being to be correct, but it does indicate that one should be careful in how one responds to minority views as such. (In particular, mocking an answer for being unpopular amongst experts is like mocking an answer for being unpopular amongst calculators.) Counting the votes is a poor substitute for doing the math.

A hugely important special case of the problem of common design flaws obtains when most specialists form their opinions by reference to the opinions of other specialists. In this case, the expert opinion is not itself scientifically expert. Its foundation might be in perfectly sound work by some scientists, or it might be in unsound work, in misreading, in intuïtion and in guess-work, or in wishful thinking; but, in any case, what is taken to be the scientifically expert opinion of n experts proves instead to be that of some smaller number, or of none at all! In such cases, consensus may be little better, or nothing other, than a leap-of-faith. It isn't made more scientific by being a consensus.

In a world in which expert opinion were always scientifically expert, broadening the pool to include those less expert would typically be seeking the center of opinion in less reliable opinion. However, as noted above, a field of expertise isn't necessarily dominated by scientific experts, in which case, people less expert but more scientific may move the center of opinion to a better approximation of a scientific opinion.

Additionally, for an outsider in seeking the opinion of experts, there is the problem of identifying who counts as an expert. The relevant knowledge and the relevant focus do not necessarily reside in the same people. As well as experts failing to behave like scientists, there are often people instead focussed on other matters who yet have as much relevant knowledge as any of those focussed on the subject in question.

So a case can be made for sometimes looking at the opinions of more than those most specialized around the questions. None-the-less, as the pool is broadened, the ultimate tendency is for the consensus to be ever less reliable as an approximation of scientific opinion. One should become wary of a consensus of broadly defined groups, and one should especially be wary if evidence can be shown of consensus shopping, where different pools were examined until a pool was found that gave an optimal threshold of conviction for whatever proposition is being advocated.

What I've really been trying to convey when I've said that invocation of consensus is not a scientific method is that a scientist, acting as a scientist, would never treat invocation of consensus — not even the consensus of bona fide experts — within his or her own area of expertise as scientific method, and that everyone else needs to see consensus for no more than what it is: a second-hand approximation that can fail grotesquely, sometimes even by design.