For several months, just two manuscripts were in the listing of manuscripts accepted by The Review of Symbolic Logic, one on an application of modal logic to set theory which article had been listed on 4 October and mine on probability which had been listed on 20 February. On 25 June, another was added, followed by nine more on 29 June; on 2 July, four more.

I don't know why the oldest two on the list have still not been typeset and moved to the FirstView list. And I don't think that my article will find many readers before it has, further, been assigned to a specific issue. It seems unlikely that the last will happen before 2021.

I have been working on a paper concerned with De Morgan's contribution to an area of probability theory. I had wanted to mention that contribution in the introduction to my probability paper, but saw no way of doing it that would be succinct without seeming occult. I began the new paper thinking that I would finish it in a very few days, but as I engaged in some of the requisite research I found that the task of properly explaining things was going to be still more challenging than I had anticipated.

In the end, the paper on De Morgan will be seen as a minor contribution to the history of thought. Had I known at the beginning how troublesome would be the task of writing it, I would have postponed the undertaking.

On Sunday, 27 May, I received a pair of e.mail messages announcing formal acceptance for publication of my paper on indecision, and I ceased being braced for rejection. From 15 June, Elsevier had a version for sale on-line (first the uncorrected proof, then the corrected proof, now the version found in the journal). The issue itself (J Math Econ v48 #4) was made available on-line on 3 August. (I assume that the print copies will be received by subscribers soon.)

Reader may recall that, not very long ago, I was reading A Budget of Paradoxes by Augustus de Morgan, and that when de Morgan used the term paradox he did not use in in the sense of an apparent truth which seems to fly in the face of reason, but in the older sense of a tenet opposed to received opinion. De Morgan was especially concerned with cases of heterodoxy to which no credibility would be ascribed by the established mainstream.

Some paradoxes would later move from heterodoxy to orthodoxy, as when the Earth came to be viewed as closely approximated by a sphere, and with no particular claim to being the center of the universe. But most paradoxes are unreasonable, and have little chance of ever becoming orthodoxy.

I began reading de Morgan's Budget largely because I have at least a passing interest in cranky ideas. But reading it at the time that I did was not conducive to my mental health.

Under ideal circumstances, one would not use a weight of opinion — whether the opinion were popular or that of experts — to approximate most sorts of truth. But circumstances are seldom ideal, and social norms are often less than optimal whatever the circumstances. When confronted with work that is heterodox about foundational matters, the vast majority of people judge work to be crackpot if it is not treated with respect by some ostensibly relevant population.

In cases where respect is used as the measure of authority, there can be a problem of whose respect is itself taken to have some authority; often a layering obtains. The topology of that layering can be conceptualized in at least three ways, but the point is that the layers run from those considered to have little authority beyond that to declare who has more authority, to those who are considered to actually do the most respected research, with respected popularizers usually in one of the layers in-between. In such structures, absurdities can obtain, such as presumptions that popularizers have themselves done important research, or that the more famous authorities are the better authorities.

As I was reading de Morgan's book, my paper was waiting for a response from the seventh journal to which it had been offered. The first rejection had been preëmptory; no reason was given for it, though there was some assurance that this need not be taken as indicating that the paper were incompetent or unimportant. The next three rejections (2nd, 3rd, 4th) were less worrisome, as they seemed to be about the paper being too specialized, and two of them made a point of suggesting what the editor or reviewer thought to be more suitable journals. But then came the awful experience of my paper being held by Theory and Decision for more than a year-and-a half, with editor Mohammed Abdellaoui refusing to communicate with me about what the Hell were happening. And this was followed by a perverse rejection at the next journal from a reviewer with a conflict of interest. Six rejections[1] might not seem like a lot, but there really aren't that many academically respected journals which might have published my paper (especially as I vowed never again to submit anything to a Springer journal); I was running-out of possibilities.

I didn't produce my work with my reputation in mind, and I wouldn't see damage to my reputation as the worst consequence of my work being rejected; but de Morgan's book drew my attention to the grim fact that my work, which is heterodox and foundational, was in danger of being classified as crackpot, and I along with it.

Crackpots, finding their work dismissed, often vent about the injustice of that rejection. That venting is taken by some as confirmation that the crackpots are crackpots. It's not; it's a natural reäction to a rejection that is perceived to be unjust, whether the perception is correct or not. The psychological effect can be profoundly injurious; crackpots may collapse or snap, but so may people who were perfectly reasonable in their heterodoxy. (Society will be inclined to see a collapse or break as confirmation that the person were a crackpot, until and unless the ostensible authorities reverse themselves, at which point the person may be seen as a martyr.)

As things went from bad to worse for my paper, I dealt with how I felt by compartmentalization and dissociation. When the paper was first given conditional acceptance, my reäction was not one of happiness nor of relief; rather, with some greater prospect that the paper would be published, the structure of compartmentalization came largely undone, and I felt traumatized.

Meanwhile, some other things in my life were going or just-plain went wrong, at least one of which I'll note in some later entry. In any case, the recent quietude of this 'blog hasn't been because I'd lost interest in it, but because properly to continue the 'blog this entry was needed, and I've not been in a good frame-of-mind to write it.

[1] Actually five rejections joined with the behavior of Abdellaoui, which was something far worse than a rejection.

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.

Secondly, I mourn to think that when the New Zealander picks up his old copy of this book, and reads it by the associations of his own day, he may, in spite of the many assurances I have received that my Athenæum Budget was amusing, feel me to be as heavy as I feel James Gregory and Sanders.[1] But he will see that I knew what was coming, which Gregory did not.

[1] These two (principally Gregory) composed a bit of heavy jocosity, published pseudonymously in 1672. The Budget began as a series in a weekly, Athenæum; the assembled collection was published posthumously in 1872.

I don't know how New Zealanders reäct; but, in 2012, I don't find the jocosity of the Budget the least bit heavy. I laugh aloud at some passages and am at least amused by a larger share. However, I admit that much of the content seems to me a tedious enumeration.

Dutch would be clear English if it were properly spelt. For example, learn-master would be seen at once to be teacher; but they will spell it leermeester. Of these they write as van deze; window they make weduwe. All this is plain to me, who never saw a Dutch dictionary in my life; but many of their misspellings are quite unconquerable.