Regarding the works of M. C. Escher

January 20, 2009

At some point over the past week (I oddly cannot remember any specifics), I encountered two of M. C. Escher’s drawings, and it struck me that he is truly doing nothing more than combining well-formed units into not-well-formed wholes,1 a process that should be very familiar to us in a different context.

In the English language (and, mutatis mutandis, any using a phonography or syllabary, although not a logography), the fundamental unit of writing is a letter. Letters are composited into words; words are composited into sentences (this three-level hierarchy is somewhat arbitrary; in an inflected language, one could say that letters are composited into roots and endings and these into words, while in most languages one could say that words are composited into subjects and predicates and these into sentences. However, three levels are all we need here). One could print letters at random, but the result would make no sense. Thus, we may distinguish a written word (a sequence of letters) as well-formed if it corresponds to a conceptual word and ill-formed otherwise. Any sentence (a sequence of words) that involves one or more ill-formed word is inherently meaningless; among sentences of well-formed words, we may distinguish between well-formed sentences where the sequence of words corresponds to a thought and ill-formed sentences where it does not. Hence:
“Adw” is an ill-formed word
“Adw oin wfe.” is a sequence of ill-formed words
“Cat” is a well-formed word
“The cat sat on the rug.” is a well-formed sentence
“The cat rug sat the.” is an ill-formed sentence

What does this have to do with Escher? Escher is not, strictly speaking, an abstract painter: no aspects of their paintings are well-formed, and thus the lack of proper formation does not seem incongruous (whatever its artistic merits, or lack thereof). Escher, on the other hand, clearly does use well-formed elements; most sufficiently small sections of his paintings correspond very well to what we see in reality. The painting as a whole, however, is not well-formed; while each flight of stairs corresponds to something that might exist in reality, the staircase as a whole could not.2

What causes our reactions to Escher’s paintings to differ from our reactions to ill-formed sentences? I cannot say for certain. But I cannot escape noting that while language deals primarily with the constructed, sight deals primarily with the natural. All writing is directed by the mind with no structural barrier to the production of ill-formed sentences; only the fact that ill-formed sentences do not serve the purposes for which we use language cause their rarity (at least among the mature). Sight, on the other hand, primarily looks upon what actually exists. Only through some hindrance or through some artificial object of sight (artificial here meaning not man-made, but a means of presenting the eye with an image intended to be taken as something other than the physical object beheld) can sight yield an ill-formed image. When the sight is hindered, as by water or a deficiency in the eye, one tends to recognize it quickly (and, moreover, typically all levels of objects perceived are equally ill-formed, e.g. as out of focus for a near-sighted man. Thus, hindrance of the sight tends to produce ill-formed images, not ill-formed composites of well-formed images). In other instances, such as the infamous case of the oar in the water, while the hindrance causes the image as actually received to be false (here thinking of the image before the mind compensates for hindrances), it does not cause an ill-formed image. Only in the case of drawings such as those of Escher does the eye behold without deficiency an ill-formed composition of well-formed objects. Thus, what seems the product simple lack of skill when it occurs in language, a medium to which ill-formed figures are natural, may seem quite tantalizing when met in a medium to which ill-formed figures are foreign.

1 Well-formed: a composition of symbols that possesses meaning, without regard to whether the meaning is true or false (true meaning “corresponding to reality”; a well-formed image corresponds to what could exist in reality, a true to what does exist).

2 Note that here I distinguish between images and diagrams. A well-formed image produces an image in the eye that corresponds to the image formed by a potential reality; a diagram is meant to communicate a concept without such correspondence. Escher’s diagrams must be classified with images because they move the mind to attempt to imagine the object that would produce the corresponding image, even though the attempt cannot succeed.

Thoughts on Translation

November 28, 2008

As with many issues of language, I believe that multitudinous difficulties of translation categorize well as lexical and syntactical. Lexical difficulties stem from the fact that there is not a one-to-one correspondence between words in sundry languages. The most obvious case is translating between two languages, one of an extensive vocabulary and the other a scarce, but with a functional correspondence of words in the former to those of the latter.1 Translating from the former to the latter, the translator typically faces little interpretive challenge, as only one word in the final language matches a word in the source language. However, if he translates word-for-word in this manner, he fails to convey the original precision. Thus, a conceptually accurate translation must add words, and one stylistically accurate must add ambiguity. The reverse, translating from the poorer to the richer, encounters the opposite problem. A given word in the source language will, in different contexts, translate to a different word in the final. Thus, to maintain the original ambiguity of meaning, he must add words to indicate the ambiguity; to maintain the style, he must act as interpreter of meaning, adding his own opinion of the precise meaning of the original. And this is in an optimal situation, where the correspondence of words is functional in one direction. In real translation, meanings will overlap, so that the translator risks adding ambiguity in some respects and removing it in others. For example, in English words may be vague in exact meaning, and specific in context; in Greek, the converse is true. L&S defines ŒªœåŒ≥ŒøœÇ as “Computation, reckoning, relation, correspondence, proportion, explanation, debate, continuous statement, narrative, verbal expression or utterance, a particular utterance, saying, thing spoken of, subject-matter, speech”. To an English speaker, this seems to be a very broad range of meanings. At the same time, it conveys a fairly specific general meaning, applied in a variety of contexts. In English, we use a different word for each context, but frequently use a word for a variety of meanings within one context for which Greek uses multiple words. English better specifies context; Greek meaning. Thus, word-for-word translation from Greek to English specifies the context more strongly and the meaning less; the converse, the inverse. The only means of circumventing these difficulties is to attempt to force a correspondence, as the scholastics did in creating a technical terminology within Latin to parallel the nuances of Aristotle’s Greek.

Moreover, lexical difficulties may pale in comparison to syntactical. Syntactical difficulties further resolve into conceptual, accentual, and stylistic. Conceptual difficulties are very similar to lexical; certain forms in one language are more precise than comparable forms in another, e.g. the careful distinction in Greek between natural and artificial result, difficult to express except by convention in English. Accentual difficulties arise particularly when translating from a highly inflected language into a weakly; highly inflected languages tend to use word-order to express accent, while weakly syntactical role. Thus, maintaining the original word order introduces odd word orders where the original was natural and even grammatical errors, while transposing words changes emphasis. Going the other way, maintenance of the original word order implies an emphasis not in the original; transposition an interpretation of the intended emphasis. Yet more challenging are stylistic difficulties. Style, more so than any other aspect of a language, is relative to context; what may seem informal and idiomatic, when literally translated, may appear formal and stilted. Thus, preserving the original style may require unidiomatic translation.

1. A function associating each member of the range to one member of the domain, but not necessarily the converse.

The Importance of Experimental Error

November 17, 2008

You have probably heard Galileo’s experiment wherein he verified that acceleration due to gravity does not depend on mass by releasing two cannonballs of different mass from the tower of Pisa and observed that they hit the ground at the same time. The problem is that it never happened. At least, I hope that it never happened.

The problem arises from uncontrolled variables. Acceleration due to gravity only matches actual acceleration when all forces other than gravity sum to zero. On the surface of the earth, however, one other force has significance in most problems: friction. Thus, when Galileo released the cannonballs and they gained speed, both gravity and friction acted upon them. FG is directly proportional to mass; friction approximately with surface area (if I recall correctly, it does not vary directly with surface area; however, my purpose here is to show the inequality between the actual rates of acceleration of the cannonballs, not the exact relationship). Mass varies with volume, or with r3; surface area varies with r2. Thus, the net force on the cannonball is not directly proportional to mass when at non-zero speed, and the larger cannonball falls faster. However, as FF would have been small relative to FG for a metal cannonball, the cannonballs would probably have appeared to hit the ground at the same time.

Yet if this had been the observed result, the experiment would have either (depending on whether Galileo attributed the difference to friction or to the antiquated theories) been inconclusive or even supported the theory that heavier objects have greater acceleration due to gravity. Only because of the error in collecting data did the experiment come to the correct conclusion.

This makes me wonder: how many conclusions of experimental science are due to such slight inaccuracies which may have produced an incorrect conclusion? As one obvious example, take Newtonian physics. Newtonian physics is not actually true, ignoring as it does relativity. However, because the effects of relativity are small, most experiments seem to verify Newtonian physics. Only after relativity was first hypothesized were sufficiently precise experiments designed. Thus, it seems to me that even the physical sciences are not, at their core, inductive, conducting experiments and then looking for relationships. Great changes in scientific theory arise from purely theoretical hypotheses later experimentally verified. Why, therefore, expect inductive reasoning, insufficient even in the physical sciences, to work in the social sciences?