## Archive for the ‘epistemology’ Category

### The Paradox of Shadows

Friday, 5 March 2021For many years, one of the projects on a back-burner in my mind has been the writing of a novel, A Paradox of Shadows, in which the principal character is attempting to reconstruct or to otherwise recover an ancient work, the title of which might have been περὶ τοῦ ἀτόπου τοῦ τῶν σκιῶν

, or De [Anomalia de] Obumbratio

, or perhaps something else.

Everything about the work is a matter of doubt or of conjecture, including its author and the era and language in which it were originally composed; even that there ever were such a work is uncertain. Its existence is primarily inferred from how parts of it seem to be esoterically embedded in other works; sometimes these passages can be made to fit together like bits of a jigsaw, but different ways of fitting are possible, especially allowing for lacunæ, interpolations, and unintended errors in translation or in transcription. In ancient art and literature are found what may be other references to the work, but these apparent references are subject to alternate interpretation, especially as many of them would be quite oblique if indeed they refer to the work. The search is largely a matter of poring over old manuscripts and documents.

No rational person would look at any one piece of evidence known to the main character and conclude that the work must have existed or just probably existed. Few would take the evidence as jointly establishing such a probability. There is both too much and too little information, so that bold intellectual leaps must be made in chaos or in darkness. A searcher may encounter unscalable cliffs or unbridgable chasms; and, if forced to stop at any point, one is likely to look pathetic. But the evidence, taken jointly, associates a relative plausibility of recovery with each of various possibilities as to the nature of the work. In that context, the possible profundity is enough to drive the search by the principal character, even with likely failure.

### Tiny Spaces

Wednesday, 20 January 2021Famously, 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.

### Humpty Dumpty and Commerce

Thursday, 7 January 2021Fairly inexpensive hair combs made of hard rubber — rubber vulcanized to a state in which it is as about firm as a modern plastic — could be found in most American drugstores at least into the mid-'90s. Now-a-days, they have become something of a premium item. I was looking at listing on Amazon *supposedly* of hard rubber combs and discovered, to my annoyance, that a careful reading of the descriptions showed that most of the combs explicitly described as hard rubber

were made of *plastic*. To me, the situation seemed to be of pervasive fraud, as it will to many others.

But then I realized that it is more likely to be something else. Fraud, after all, involves *deliberate* misrepresentation. Whereäs we live in a world in which a great many people believe that *no use of a word or phrase is objectively improper* — that if they think that hard rubber

means a rubbery plastic or a plastic that looks like another substance called hard rubber

, then it indeed means just that. (Of course, we cannot trust any *verbal* explanation from them of these idiosyncratic meanings, as they may be assigning different meanings to any words with which they define other words.)

My defense of linguistic prescriptivism has for the most part been driven by concerns other than those immediate to commercial transactions. And, when I've seen things such on eBay as items described with mint condition for its age

or with draped nude

, my inclination has been merely to groan or to laugh. But it seems to me that the effects of ignoring or of rejecting linguistic prescription have found their way into commercial transactions beyond the casual.

Well, those who are not prescriptivists are hypocrites if they complain, and they're getting no worse than they deserve.

### Perverted Locusts

Wednesday, 9 December 2020Those who support locking-down in response to SARS-CoV-2 are like weird locusts. Instead of eating the crops; these locusts prevent growth and harvest. That is to say that they prevent economic activity, which is an implicit consumption of an especially perverse sort. In any case, they leave despair and literal starvation in their wake.

### Transcription Error

Monday, 23 November 2020To my chagrin, I find that I made a transcription error for an axiom in Formal Qualitative Probability

. More specifically, I placed a quantification in the wrong place. Axiom (A6) should read I've corrected this error in the working version.

### Lack of Infrastructure

Saturday, 26 September 2020### Libertine Bayesianism

Thursday, 24 September 2020As repeatedly noted by me and by many others, there are multiple theories about the *fundamental* notion of probability, including (though not restricted to) the notion of probabilities as objective, logical relationships amongst propositions and that of probabilities as degrees of belief.

Though those two notions are distinct, subscribers to each typically agree with subscribers to the other upon a great deal of the axiomatic structure of the logic of probability. Further, *in practice* the *main-stream* of the first group and that of the second group both arrive at their estimates of *measures* of probability by adjusting initial values through repeated application, as observations accumulate, of a principle known as Bayes' theorem

. Indeed, the main-stream of one group are called objective Bayesian

and the mainstream of the other are often called subjective Bayesian

.[1] Where the two main-streams differ in practice is *in the source of those initial values*.

The objective Bayesians believe that, in the absence of information, one begins with what are called non-informative priors

. This notion is evolved from the classical idea of a principle of insufficient reason, which said that one should assign *equal probabilities* to events or to propositions, in the absence of a reason for assigning different probabilities. (For example, begin by assume that a die is fair

.) The objective Bayesians attempt to be more shrewd than the classical theorists, but will often admit that in some cases non-informative priors cannot be found because of a lack of understanding of how to divide the possibilities (in some cases because of complexity).

The subjective Bayesians believe that one may use as a prior whatever initial degree of belief one has, measured on an interval from 0 through 1. As measures of probability are taken to *be* degrees of belief, any application of Bayes' theorem that results in a new value is supposed to result in a new degree of belief.

I want to suggest what I think to be *a new school of thought*, with a Bayesian sub-school, not-withstanding that I have no intention of *joining* this school.

If a set of things is completely ranked, it's possible to *proxy* that ranking with a *quantification*, such that if one thing has a higher rank than another then it is assigned a greater quantification, and that if two things have the same rank then they are assigned the same quantification. If *all* that we have is a ranking, with no further stipulations, then there will be infinitely many possible quantifications that will work as proxies. Often, we may want to *tighten-up* the rules of quantification (for example, by requiring that all quantities be in the interval from 0 through 1), and yet *still* it may be the case that infinitely many quantifications would work equally well as proxies.

Sets of *measures* of probability may be considered as proxies for underlying *rankings* of propositions or of events by probability. The principles to which most theorists agree when they consider probability rankings as such *constrain* the sets of possible measures, but so long as only a finite set of propositions or of events is under consideration, there are infinitely many sets of measures that will work as proxies.

A subjectivist feels free to use his or her degrees of belief so long as they fit the constraints, even though someone else may have a different set of degrees of belief that also fit the constraints. However, the argument for the *admissibility* of the subjectivist's own set of degrees of belief is *not* that it is *believed*; the argument is that one's own set of degrees of belief *fits the constraints*. *Belief as such is irrelevant.* It might be that one's own belief is colored by private information, but then the argument is not that one *believes* the private information, but that the information as such is relevant (as indeed it might be); and there would always be some *other* sets of measures that *also* conformed to the private information.

Perhaps one might *as well* use one's own set of degrees of belief, but one *also* might *every bit as well* use *any* conforming set of measures.

So what I now suggest is what I call a libertine

school, which regards measures of probability as proxies for probability rankings and which accepts *any* set of measures that conform to what is *known* of the probability ranking of propositions or of events, regardless of whether these measures are thought to be the degrees of belief of *anyone*, and without any concern that these should *become* the degrees of belief of anyone; and in particular I suggest libertine Bayesianism

, which accepts the analytic principles common to the objective Bayesians and to the subjective Bayesians, but which will allow *any* set of priors that conforms to those principles.

[1] So great a share of subjectivists subscribe to a Bayesian principle of updating that often the subjective Bayesians are simply called subjectivists

as if there were no need to distinguish amongst subjectivists. And, until relatively recently, so little recognition was given to the objective Bayesians that Bayesian

was often taken as synonymous with subjectivist

.

### Voigt and Value

Friday, 10 April 2020In a previous entry, I asserted that Voigt's Zahl und Mass in der Ökonomik contain[s] more error than insight

. Here, I'll discuss one of the more egregious errors. In section V, Voigt writes

An die Spitze der Erörterung dieses vielberufenen Begriffes sollte gestellt werden, dass es Einheiten des Wertes giebt, dass man also untersuchen kann, wievielmal so gross ein Wert als ein anderer ist und Güter gleichen Wertes durch einander ersetzen kann, dass also der Wert ein eigentliches in einer Kardinalzahl ausdrückbares Mass hat.

which may be translated as

At the forefront of discussion of this much used concept should be placed that there are units of value that one thus can investigate how many time as large a value is as another and can replace goods of the same value with each other, that thus the value has a real measure expressible in a cardinal number.

I'll deal first with the point that it seems that one can investigate how many times as large a value is as another.

Numbers are used in many ways. Depending upon the use, what is revealed by arithmetic may be a great deal or very little. Sometimes numbers are ascribed with so little meaning that we may as well consider them strings of *numerals*, the *characters* that we use for numbers, and not numbers at all. Sometimes numbers do nothing but provide an *arbitrary* order, good for something such as a look-up table but nothing else. Sometimes they provide a *meaningful* order, but one in which the results of *most* arithmetic operations are meaningless, as when items produced at irregular intervals are given sequential serial numbers. (The *difference* between any two such numbers tells one which was produced before the other, but little else.) Sometimes the differences *between* the differences are meaningful, as when items are produce at *regular* intervals and given sequential serial numbers. And so forth.

Monetary prices are *quantities*, but they are more specifically quantities *of money*; that does not make them quantities *of value* nor *proxies* of quantities of value. One would have to show that the results of *every* arithmetic operation on such a quantity of money said something about value for it to be shown that value were itself a quantity.

The second part of Voigt's claim is that one Güter gleichen Wertes durch einander ersetzen kann

[can replace goods of the same value with each other

]. But an *equivalence* between things corresponding to the same numbers doesn't make results of the application of arithmetic to those numbers meaningful. (Consider *lots* of items produced at irregular intervals, with each item in the lot given the same serial number, unique to the lot but otherwise *random*.) And we should ask ourselves under just what circumstances we can and cannot ersetzen one set of commodities of a given price with another of the same price.

Nor does somehow combining the use of quantities of money for prices with a property of equivalence imply that value is a quantity.

Voigt is unusual not in making this unwarranted inference, but in so clearly expressing himself as he does. From the observation that prices are usually quantities of *something*, which quantities increase as value increases, *most* people, and even most *economists* blithely infer that value itself behaves as a quantity.

### A Translation of Voigt's Zahl und Mass in der Ökonomik

Wednesday, 8 April 2020In early 2013, I made freely available a transcription of Zahl und Mass in der Ökonomik: Eine kritische Untersuchung der mathematischen Methode und der mathematischen Preistheorie (1893) by Andreas Heinrich Voigt. I have to-day completed **a first pass** of a translation of this as Number and Measure in Economics: A Critical Examination of Mathematical Method and of Mathematical Price Theory. Although I believe that there are many errors to be corrected in that translation, I am making it available. I do not plan to use a different URI for corrected versions.

I have been very disappointed by my reading of Voigt's article. I regard it as containing more error than insight.

In the course of translation, I found and corrected extremely minor errors in the transcription of the original. A name was at one point misspelled by me, and I failed to capitalize a word beginning a sentence. I also marked a die die

as questionable which I've since concluded was deliberate. I do not believe that anyone could have been led to a mistaken reading as a result of those errors, but I have naturally corrected them.

I may change the URI for the transcription, moving it from another domain to place it amongst the uploads for this 'blog. If so, then I will edit entries to reflect that change.