Sunday


Science often (though not always or exclusively) involves a quantitative approach to phenomena. As the phenomena of the world are often (though not always or exclusively) continuous, the continuum of phenomena must be broken up into discrete chunks of experience, however imperfect the division. If we are to quantify knowledge, we must have distinctions, and distinctions must be interpolated at some particular point in a continuum.

The truncation principle is the principled justification of this practice, and the truncation fallacy is the claim that distinctions in the name of quantification are illegitimate. The claim of the illegitimacy of a given distinction is usually based on an ideal standard of distinctions having to be based on a sharply-bounded concept that marks an exhaustive division that admits of no exceptions. This is an unreasonable standard for human experience or its systematization in scientific knowledge.

One of my motivations (though not my only motivation) for formulating the truncation principle was the obvious application to historical periodization. Historians have always been forced to confront the truncation fallacy, though I am not aware that there has previously been any name for the conceptual problems involved in historical periodization, though it has been ever-present in the background of historical thought.

Here is an implicit exposition of the problems of the truncation principle by Marc Bloch, one of the most eminent members of the Annales school of historians (which also included Fernand Braudel, of whom I have written on many occasions), and who was killed by the Gestapo while working for the French resistance during the Second World War:

“…it is difficult to imagine that any of the sciences could treat time as a mere abstraction. Yet, for a great number of those who, for their own purposes, chop it up into arbitrary homogenous segments, time is nothing more than a measurement. In contrast, historical time is a concrete and living reality with an irreversible onward rush… this real time is, in essence, a continuum. It is also perpetual change. The great problems of historical inquiry derive from the antithesis of these two attributes. There is one problem especially, which raises the very raison d’être of our studies. Let us assume two consecutive periods taken out of the uninterrupted sequence of the ages. To what extent does the connection which the flow of time sets between them predominate, or fail to predominate, over the differences born out of the same flow?”

Marc Bloch, The Historian’s Craft, translated by Peter Putnam, New York: Vintage, 1953, Chapter I, sec. 3, “Historical Time,” pp. 27-29

Bloch, then, sees times itself, the structure of time, as the source both of historical continuity and historical discontinuity. For Bloch the historian, time is the truncation principle, as for some metaphysicians space (or time, for that matter) simply is the principle of individuation.

The truncation principle and the principle of individuation are closely related. What makes an individual an individual? When it is cut off from the rest of the world and designated as an individual. I haven’t thought about this yet, so I will reserve further remarks until I’ve made an effort to review the history of the principium individuationis.

The “two attributes” of continuity and change are both functions of time; both the connection and the differences between any two “arbitrary homogenous segments” are due to the action of time, according to Bloch.

The truncation principle, however, has a wider application than time. To express the truncation principle in terms of time invites a formulation (or an example) in terms of space, and there is an excellent example ready to hand: that of the color spectrum of visible light. There is a convention of dividing the color spectrum into red, orange, yellow, green, blue, indigo, and violet. But this is not the only convention. Because the word “indigo” is becoming almost archaic, one now sees the color spectrum decomposed into red, orange, yellow, green, blue, and purple.

Both decompositions of the color spectrum, and any others that might be proposed, constitute something like, “arbitrary homogenous segments.” The decomposition of the color spectrum is justified by the truncation principle, but the principle does not privilege any one decomposition over any other. All distinctions are equal, and if any one distinction is taken to be more equal than others, it is only because this distinction has the sanction of tradition.

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Saturday


Truncated sphere: can we appeal to any principle in our truncations?

We can make a distinction among distinctions between ad hoc and principled distinctions. The former category — ad hoc distinctions — may ultimately prove to be based on a principle, but that principle is unknown as long as the distinction remains an ad hoc distinction. This suggests a further distinction among distinctions between ad hoc distinctions that really are ad hoc, and which are based on no principle, and ad hoc distinctions that are really principled distinctions but the principle in question is not yet known, or not yet formulated, at the time the distinction is made. So there you have a principled distinction between distinctions.

A perfect evocation of ad hoc distinctions is to be found in the opening paragraph of the Preface to Foucault’s The Order of Things:

This book first arose out of a passage in Borges, out of the laughter that shattered, as I read the passage, all the familiar landmarks of my thought — our thought, the thought that bears the stamp of our age and our geography — breaking up all the ordered surfaces and all the planes with which we are accustomed to tame the wild profusion of existing things, and continuing long afterwards to disturb and threaten with collapse our age-old distinction between the Same and the Other. This passage quotes a ‘certain Chinese encyclopedia’ in which it is written that ‘animals are divided into: (a) belonging to the Emperor, (b) embalmed, (c) tame, (d) sucking pigs, (e) sirens, (f) fabulous, (g) stray dogs, (h) included in the present classification, (i) frenzied, (j) innumerable, (k) drawn with a very fine camelhair brush, (1) et cetera, (m) having just broken the water pitcher, (n) that from a long way off look like flies’. In the wonderment of this taxonomy, the thing we apprehend in one great leap, the thing that, by means of the fable, is demonstrated as the exotic charm of another system of thought, is the limitation of our own, the stark impossibility of thinking that.

Such distinctions are comic, though Foucault recognizes that our laughter is uneasy: even as we immediately recognize the ad hoc character of these distinctions, we realize that the principled distinctions we routinely employ may not be so principled as we supposed.

Foucault continues this theme for several pages, and then gives another formulation — perhaps, given his interest in mental illness, an illustration that is closer to reality than Borges’ Chinese dictionary:

“It appears that certain aphasiacs, when shown various differently coloured skeins of wool on a table top, are consistently unable to arrange them into any coherent pattern; as though that simple rectangle were unable to serve in their case as a homogeneous and neutral space in which things could be placed so as to display at the same time the continuous order of their identities or differences as well as the semantic field of their denomination. Within this simple space in which things are normally arranged and given names, the aphasiac will create a multiplicity of tiny, fragmented regions in which nameless resemblances agglutinate things into unconnected islets; in one corner, they will place the lightest-coloured skeins, in another the red ones, somewhere else those that are softest in texture, in yet another place the longest, or those that have a tinge of purple or those that have been wound up into a ball. But no sooner have they been adumbrated than all these groupings dissolve again, for the field of identity that sustains them, however limited it may be, is still too wide not to be unstable; and so the sick mind continues to infinity, creating groups then dispersing them again, heaping up diverse similarities, destroying those that seem clearest, splitting up things that are identical, superimposing different criteria, frenziedly beginning all over again, becoming more and more disturbed, and teetering finally on the brink of anxiety.”

Foucault here writes that, “the sick mind continues to infinity,” in other words, the process does not terminate in a definite state-of-affairs. This implies that the healthy mind does not continue to infinity: rational thought must make concessions to human finitude. While I find the use of the concept of the pathological in this context questionable, and I have to wonder if Foucault was unwittingly drawn into the continental anti-Cantorian tradition (Brouwerian intuitionism and the like, though I will leave this aside for now), there is some value to the idea that a scientific process (such as classification) must terminate in a finite state-of-affairs, even if only tentatively. I will try to show, moreover, that there is an implicit principle in this attitude, and that it is in fact a principle that I have discussed previously.

The quantification of continuous data requires certain compromises. Two of these compromises include finite precision errors (also called rounding errors) and finite dimension errors (also called truncation). Rounding errors should be pretty obvious: finite parameters cannot abide infinite decimal expansions, and so we set a limit of six decimal places, or twenty, or more — but we must set a limit. The difference between actual figures and limited decimal expansions of the same figure is called a finite precision error. Finite dimension errors result from the need to arbitrarily introduce gradations into a continuum. Using the real number system, any continuum can be faithfully represented, but this representation would require infinite decimal expansions, so we see that there is a deep consonance between finite precision errors and finite dimension errors. Thus, for example, we measure temperature by degrees, and the arbitrariness of this measure is driven home to us by the different scales we can use for this measurement. And if we could specify temperature using real numbers (including transcendental numbers) we would not have to compromise. But engineering and computers and even human minds need to break things up into manageable finite quantities, so we speak of 3 degrees C, or even 3.14 degrees C, but we don’t try to work with pi degrees C. Thus the increments of temperature, or of any another measurement, involve both finite precision errors and finite dimension errors.

In so far as quantification is necessary to the scientific method, finite dimension errors are necessary to the scientific method. In several posts (e.g., Axioms and Postulates in Strategy) I have cited Carnap’s tripartite distinction among scientific concepts, the three being classificatory, comparative, and quantitative concepts. Carnap characterizes the emergence of quantitative scientific concepts as the most sophisticated form of scientific thought, but in reviewing Carnap’s scientific concepts in the light of finite precision errors and finite dimension errors, it is immediately obvious that classificatory concepts and comparative concepts do not necessarily involve finite precision errors and finite dimension errors. It is only with the introduction of quantitative concepts that science becomes sufficiently precise that its precision forces compromises upon us. However, I should point out that classificatory concepts routinely force us to accept finite dimension errors, although they do not involve finite precision errors. The example given by Foucault, quoted above, illustrates the inherent tension in classificatory concepts.

We accept finite precision errors and finite dimension errors as the price of doing science, and indeed as the price of engaging in rational thought. As Foucault implied in the above quote, the healthy and sane mind must draw lines and define limits and call a halt to things. Sometimes these limits are close to being arbitrary. We retain the ambition of “carving nature at the joints,” but we accept that we can’t always locate the joint but at times must cleave the carcass of nature regardless.

For this willingness to draw lines and establish limits and to call a halt to proceedings I will give the name The Truncation Principle, since it is in virtue to cutting off some portion of the world and treating it as though it were a unified whole that we are able to reason about the world.

As I mentioned above, I have discussed this problem previously, and in my discussion I noted that I wanted to give an exposition of a principle and a fallacy, but that I did not have a name for it yet, so I called it An Unnamed Principle and an Unnamed Fallacy. Now I have a name for it, and I will use this name, i.e., the truncation principle, from now on.

Note: I was tempted to call this principle the “baby retention principle” or even the “hang on to your baby principle” since it is all about the commonsense notion of not throwing out the baby with the bathwater.

In An Unnamed Principle and an Unnamed Fallacy I initially formulated the principle as follows:

The principle is simply this: for any distinction that is made, there will be cases in which the distinction is problematic, but there will also be cases when the distinction is not problematic. The correlative unnamed fallacy is the failure to recognize this principle.

What I most want to highlight is that when someone points out there are gray areas that seem to elude classification by any clear cut distinction, this is sometimes used as a skeptical argument intended to undercut the possibility of making any distinctions whatsoever. The point is that the existence of gray areas and problematic cases does not address the other cases (possibly even the majority of the cases) for which the distinction isn’t in the least problematic.

A distinction that that admits of problematic cases not clearly falling on one side of the distinction or the other, may yet have other cases that are clearly decided by the distinction in question. This might seem too obvious to mention, but distinctions that admit of problematic instances are often impugned and rejected for this reason. Admitting of no exceptions whatsoever is an unrealistic standard for a distinction.

I hope to be able to elaborate on this formulation as I continue to think about the truncation principle and its applications in philosophical, formal, and scientific thought.

Usually when we hear “truncation” we immediately think of the geometrical exercise of regularly cutting away parts of the regular (Platonic) solids, yielding truncated polyhedra and converging on rectified polyhedra. This is truncation in space. Truncation in time, on the other hand, is what is more commonly known as historical periodization. How exactly one historical period is to be cut off from another is always problematic, not least due to the complexity of history and the sheer number of outliers that seem to falsify any attempt at periodization. And yet, we need to break history up into comprehensible chunks. When we do so, we engage in temporal truncation.

All the problems of philosophical logic that present themselves to the subtle and perceptive mind when contemplating a spatial truncation, as, for example, in defining the Pacific Ocean — where exactly does it end in relation to the Indian Ocean? — occur in spades in making a temporal truncation. Yet if rational inquiry is to begin (and here we do not even raise the question of where rational inquiry ends) we must make such truncations, and our initial truncations are crude and mostly ad hoc concessions to human finitude. Thus I introduce the truncation principle as an explicit justification of truncations as we employ them throughout reasoning.

And, as if we hadn’t already laid up enough principles and distinctions for today, here is a principle of principles of distinctions: every principled distinction implies a fallacy that takes the form of neglecting this distinction. With an ad hoc distinction there is no question of fallacy, because there is no principle to violate. Where there is a principle involved, however, the violation of the principle constitutes a fallacy.

Contrariwise, every fallacy implies a principled distinction that ought to have been made. If we observe the appropriate principled distinctions, we avoid fallacies, and if we avoid fallacies we appropriately distinguish that which ought to be distinguished.

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Tuesday


In earlier posts to this forum I have discussed the dissatisfaction that comes from introducing an idea before one has the right name for it. An appropriate name will immediately communicate the intuitive content of the idea to the reader, as when I wrote about the civilization of the hand in contradistinction to the civilization of the mind, after having already sketched the idea in a previous post.

Again I find myself in the position of wanting to write about something for which I don’t yet have the perfect intuitive name, and I have even had to name this post “an unnamed principle and an unnamed fallacy” because I can’t even think of a mediocre name for the principle and its related fallacy.

In yesterday’s Defunct Ideas I argued that new ideas are always emerging in history (though they aren’t always being lost), and it isn’t too difficult to come up with a new idea if one has the knack for it. But most new ideas are pretty run-of-the-mill. One can always build on past ideas and add another brick to the growing structure of human knowledge.

That being said, it is only occasionally, in the midst of a lot of ideas of the middling sort, that one comes up with a really good idea. It is even more rare when one comes up with a truly fundamental idea. Formulating a logical fallacy that has not been noticed to date, despite at least twenty-five hundred years of cataloging fallacies would constitute a somewhat fundamental idea. As this is unlikely in the present context, the principle and the associated fallacy below have probably already been noticed and named by others long ago. If not, they should have been.

The principle is simply this: for any distinction that is made, there will be cases in which the distinction is problematic, but there will also be cases when the distinction is not problematic. The correlative unnamed fallacy is the failure to recognize this principle.

This unnamed principle is not the same as the principle of bivalence or the law of the excluded middle (tertium non datur), though any clear distinction depends, to a certain extent upon them. This unnamed principle is also not to be confused with a simple denial of clear cut distinctions. What I most want to highlight is that when someone points out there are gray areas that seem to elude classification by any clear cut distinction, this is sometimes used as a skeptical argument intended to undercut the possibility of making any distinctions whatsoever. The point is that the existence of gray areas and problematic cases does not address the other cases (possibly even the majority of the cases) for which the distinction isn’t in the least problematic.

Again: a distinction that that admits of problematic cases not clearly falling on one side of the distinction or the other, may yet have other cases that are clearly decided by the distinction in question. This might seem too obvious to mention, but distinctions that admit of problematic instances are often impugned and rejected for this reason. Admitting of no exceptions whatsoever is an unrealistic standard for a distinction.

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