From e83b8620bc4cd159e6e34db2bc92160afbc7e87c Mon Sep 17 00:00:00 2001
From: grr <grr@lo2.org>
Date: Wed, 8 May 2024 19:43:11 -0400
Subject: addition of several extra essays

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+\chapter{Structure Art and Pure Mathematics (1960)}
+
+In some art---music, visual art, poetry, and the rest---there is a tendency for 
+"structure" to predominate. When structure tends to predominate in art, then \emph{if} the 
+artist wants the interest of the structure to predominate, wants to communicate the 
+interest of the structure, I will say that the art is "structure art." Much structure 
+art is a vestige of Serious Art; for exemple, of medieval music, which was conceived 
+to be a metaphysical science. Now consider, for example, a piece of structure music, 
+a serial piece. The "structure" of the piece is \emph{not} (in) the sounds in (a performance 
+of) the piece. It is a categorization of the sounds, that represented by the score 
+together with that typically given in the first instance by the composer in an "analysis" 
+of the "piece" (actually the analysis is more a part of the piece). Thus if I speak 
+of the "intended structure" of a piece it will be the composer's categorization; and I 
+will speak of others' categorizations, the audiences' categorizations, as "associated 
+structures" of the piece. (To some extent the composer can work to the audience's 
+background so that one association is more probable than another.) Many structure 
+artists do claim "that the structure (particulerly the intended structure) is in the 
+sounds" in that, for example, there is an objective relation between the categorization 
+and the sounds. This claim is unjustifiable; I will return to it later. There is an 
+important division of structure art into two kinds, exemplified by the fugue and total 
+serial music, according to how the structure is "appreciated." In the case of a fugue, 
+one is aware of its structure in listening to it; one mentally imposes "reletionships," 
+a categorization (hopefully the intended one) on the sounds while listening to them 
+that is, there is an "associated artistic structuring by oneself." In the case of total 
+serial music, the structure is such that this cannot be done; one just has to read an 
+"analysis" of the music, a specification of relationships. Incidentally, there is 
+another, less important kind of art in which the important thing is categorization; 
+the art involving conceptual cleverness, play with the concepts of the art-form such as, 
+in music, "the score," "performer versus listener," "playing a composition." In 
+structure poetry, there is a lack of concern with syntacticel structure. The poetry 
+is mere phonemes or graphemes with an artistic structure. 
+
+The following is an attempt at a formal definition of "artistic structure." 
+The artistic structure of a production is a division or segmentation of the raw work 
+(the body of material), a grouping of the segments, and a "weighting" of the subgroupings 
+in this grouping (according to their"structural importance"); that is, it is a system 
+of definitions. When structure is regarded as the most important aspect of the 
+production, the production is merely a diagram illustrating the description of its 
+structure. Certain pieces of music are merely acoustical diagrams of their structures. 
+Such a production consists of the production proper together with a concept poem, a 
+body of definitions. Here is a canonical method of specifying such structures. 
+Given the raw work, the informal description of its structure is as follows. The 
+segments are blocks of color; the first two are grouped together, and each of the 
+others is grouped separately; the weights of the successive groups are $5, 2, 4, 2, 4$
+(2 is the weight of \eg\ a bridge passage in music). The formal specification is 
+$(AB)_5(C)_2(D)_4(E)_2(F)_4$;  that is, the production is structurally a "$(AB)_5(C)_2(D)_4(E)_2(F)_4$."
+
+
+The method, then, is that the terminology for a certain structure is formed from 
+letters corresponding to segments, parentheses to indicate grouping, and numerical 
+subscripts to indicate weights. (Does the method need to be elaborated to take into 
+account relations between segments?) It can be seen that this kind of structure is 
+definitional, stipulational, like logical syntax; it is not intuitive and statistical 
+like an individual's use of inflection in speech. I now turn to the analyses of 
+the structure of a production made by critics, what I call "associated definitions of the 
+structure" (in line with the terminology of the previous paragraph), Consider the 
+following examples. 
+
+\includegraphics[width=4in]{img/structure_art}
+
+In each example, the actual sounds, the body of material, is exactly the same. 
+The difference is in the different structures defined on the material. The examples 
+substantiate my contentions thet the structure is not in the sounds; that the composer's 
+analysis of the piece is really a definition and a part of the piece; and that the 
+critics' analyses of the structure are definitions attached to the piece, not discoveries 
+of intrinsic properties of the sounds. As another example, consider the difference 
+between hearing the "Sanctus," \opustitle{Missa Prolationum} of Ockhegem, in no meter (by 
+a non-European listener), in one meter (by a lay European listener), and in four 
+meters (the intended structure). Arguments such as the one over whether the structure 
+of Webern's music is "really" motivic or serial are absurd, since Webern himself did not 
+define this point. Many academic structural analyses of art have been irrelevant 
+to the aesthetics of the works. 
+
+The purpose throughout all this art is dual; structure or concept art tries to be, 
+first, music, visual art, or whatever (which suggests that it is to be listened to, or 
+looked at), \emph{and}, something else entirely, to be valuable for its structure or conceptual 
+cleverness. Then when the structure is "hidden," "unexperiencable," when it can only be 
+appreciated by reading the "analysis," why put emphasis on the body of sound, light, or 
+whatever, why listen to structure music, why look at structural visual art, why even call 
+them "music," "visual art"? Why not throw away the bodies of sound, light, or whatever, 
+and keep the "analyses" of the structure as the works of art? In general, logic, and 
+experience (with the results of the artists' efforts), show that the dual purpose of 
+structure art consists of irreconcilable objectives; that one can be attained only at the 
+other's expense. Which objective are the structure artists trying to attain?---they 
+obviously have no idea. Structure art represents obsolete, confused categories of 
+activities, categories which by now are obscurantist. Structure (or concept) music, 
+for example, needs straightening out, first, by ceasing to call it "music," and starting 
+to say thet the sound (or activity) is used only to carry the structure or conceptual 
+cleverness, and that the real point is the structure or conceptual cleverness---the 
+categorization---and then it will be seen how limited, impoverished the structure of 
+these productions trying to be music are. When you make the change, then you are led 
+to a far more consistent, integral activity, the same one arrived at below through 
+a consideration of pure mathematics. Games of intellectual skill such as chess fall 
+into this same category; since, after all, they can be regerded as formalist mathematics. 
+
+Neryt I will discuss pure mathematics. Originally, mathematics was a system of 
+beliefs, a doctrine, about the entities numbers, points, polygons, and so forth (Pythagoras, 
+Euclid, Platonic geometry). As mathematicians became skeptical, and thus less desirous 
+of resting the importance of mathematics on the validity of these beliefs, they changed 
+their minds about what the purpose of mathematics is. The purpose became for the theorems 
+to be true if the axioms are. In the nineteenth century, as a result of e.g. the ideas 
+of Riemann, they became unconcerned to claim that their axioms are true. They began 
+to say that the value of mathematics is "aesthetic." Here is when mathematics becomes 
+a subject for this essay; when it becomes pure mathematics, when its value is not claimed 
+to be that of technology or natural science, but rather more an aesthetic value, when it 
+becomes "adoctrinal culture." Mathematics becomes something to be considered alongside art. 
+When I became interested in contributing to pure mathematics, for reasons of taste I wanted 
+to de-emphasize discovery in mathematics, mathematics as discovering theorems and proofs. 
+(Such discovery bored me.) The first way I thought of to de-emphasize discovery was that 
+since the value of pure mathematics is now regarded as conceptual interest, aesthetic 
+rather than scientific value, why not try to make up aesthetic theorems, without considering 
+whether they are true. The second way was to find that the conventional claim that 
+theorems and proofs are discovered is unjustifiable; I will return to this point later. 
+In the twentieth century, as a result of the ideas of Hilbert, and then Carnap, 
+mathematicians became unconcerned to claim that mathematical "statements," the 
+mathematical object language, are (substantive) assertions having truth value (as are 
+English statements). Rather, they are "merely" series of signs formed according to 
+certain rules: formalist mathematics. Then my third way of de-emphasizing discovery was 
+to open up unexplored regions of formalist mathematics. The resulting mathematics still 
+had statements, theorems, proofs, but the latter weren't "discovered" the way they 
+traditionally were. 
+
+Now exploration of the wider possibilities of pure mathematics opened up by me 
+tends to lead beyond the form of "making statements," "proving," and the like, so thet 
+the term "pure mathematics" becomes completely incongruous. The category of pure 
+mathematics---a vestige ultimately of the old system of beliefs canonized by Plato 
+(hence the form of statements, proving, and the like)---is an obsolete category. My 
+contributions to pure mathematics lead to an integral, general activity of which the 
+point is categorizations (having the value of being "well-formed"); the contributions 
+need to be classified as such an activity rather than as pure mathematics to escape 
+confusion, Traditional mathematics (mathematics as discovery), reformulated, explicated 
+to take my findings into account, would be an untypical, small but intensively developed 
+part of such an activity. 
+
+The proponents of structure art, pure mathematics, and chess make similar claims 
+for them. I have mentioned the claims that structure is an objective property of things; 
+and that mathematical theorems and proofs are discovered; and there is a similar claim for 
+games of intellectual skill. Two important notions associated with these fundamentally 
+identical claims require comment. There is the notion that contribution to structure 
+art, pure mathematics, and chess requires high intelligence, the discovery of implications; 
+the notion of intelligence as the ability to discover implications. Then, there is the 
+notion that structure (as in mathematics pre-eminently) is an objective property of things, 
+capable of discovery, demonstration, rational cognition---with particular reference 
+to language, art, and the like---whereas meaning, expression, and emotion are not. 
+(These pretensions are traditionally an essential aspect of structure art, pure mathematics, 
+and chess.) Both notions come down to the belief that there can be an objective relation 
+between a name and its referents; for example, an objective relation between the 
+metamathematical term "true theorem" and certain theorems, or an objective relation 
+between "having serial structure" and a body of sound, or between "checkmate" and 
+checkmates. As I said, these notions are discreditable, as can be seen from my 
+\essaytitle{Philosophy Proper} and \essaytitle{Primary Paradox}. Thus the notion of intelligence, pretension 
+of intellectual superiority, as what mathematicians, chess players, and the like have; 
+and the prejudice in favor of structure; cannot be defended. It is about time that 
+these notions be discarded. 
+
+
-- 
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