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+\chapter{Concept Art}
+{ \raggedleft (1961) \par }
+
+
+Concept art is first of all an art of which the material is concepts, as the 
+material of e.g. music is sound. Since concepts are closely bound up with 
+language, concept art is a kind of art of which the material is language. That 
+is, unlike e.g. a work of music, in which the music proper (as opposed to 
+notation, analysis, etc.) is just sound, concept art proper will involve 
+language. From the philosophy of language, we learn that a concept may as 
+well be thought of as the intension of a name; this is the relation between 
+concepts and language.\footnote{The extension of the word 'table' is all 
+existing tables; the intension of 'table' is all possible instances of a table.}
+The notion of a concept is a vestige of the notion of 
+a platonic form (the thing which e.g. all tables have in common: tableness), 
+which notion is replaced by the notion of a name objectively, metaphysically 
+related to its intension (so that all tables now have in common their 
+objective relation to table). Now the claim that there can be an objective 
+relation between a name and its intension is wrong, and (the word) concept, 
+as commonly used now, can be discredited (see my book, Philosophy 
+Proper). If, however, it is enough for one that there be a subjective relation 
+between a name and its intension, namely the unhesitant decision as to the 
+way one wants to use the name, the unhesitant decisions to affirm the names 
+of some things but not others, then concept is valid language, and concept 
+art has a philosophically valid basis. 
+
+Now what is artistic, aesthetic, about a work which is a body of 
+concepts? This question can best be answered by telling where concept art 
+came from; I developed it in an attempt to straighten out certain traditional 
+activities generally regarded as aesthetic. The first of these is structure art, 
+music, visual art, etc., in which the important thing is "structure." My 
+definitive discussion of structure art is in my unpublished essay \essaytitle{Structure 
+Art and Pure Mathematics}; here I will just summarize that discussion. Much 
+structure art is a vestige of the time when \eg music was believed to be 
+knowledge, a science, which had important things to say in astronomy \etc
+Contemporary structure artists, on the other hand, tend to claim the kind of 
+cognitive value for their art that conventional contemporary mathematicians 
+claim for mathematics. Modern examples of structure art are the fugue and 
+total serial music. These examples illustrate the important division of 
+structure art into two kinds according to how the structure is appreciated. In 
+the case of a fugue, one is aware of its structure in listening to it; one 
+imposes relationships, a categorization (hopefully that intended by the 
+composer) on the sounds while listening to them, that is, has an (associated) 
+artistic structure experience. 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, definition of the relationships. Now there are two things wrong with 
+structure art. First, its cognitive pretensions are utterly wrong. Secondly, by 
+trying to be music or whatever (which has nothing to do with knowledge), 
+and knowledge represented by structure, structure art both fails, is 
+completely boring, as music, and doesn't begin to explore the aesthetic 
+possibilities structure can have when freed from trying to be music or 
+whatever.The first step in straightening out e.g. structure music is to stop 
+calling it music, and start saying that the sound is used only to carry the 
+structure and that the real point is the structure--and then you will see how 
+limited, impoverished, the structure is. Incidentally, anyone who says that 
+works of structure music do occasionally have musical value just doesn't 
+know how good real music (the Goli Dance of the Baoule; Cans on Windows 
+by La Monte Young; the contemporary American hit song Sweets for My 
+Sweets, by the Drifters) can get. When you make the change, then since 
+structures are concepts, you have concept art. Incidentally, there is another, 
+less important kind of art which when straightened out becomes concept art: 
+art involving play with the concepts of the art such as, in music, the score, 
+performer vs. listener, playing a work. The second criticism of structure art 
+applies, with the necessary changes, to this art. 
+
+The second main antecedent of structure art is mathematics. This is the 
+result of my revolution in mathematics, presented in my 1966 \essaytitle{Mathematical 
+Studies}; here I will only summarize. The revolution occured first because for 
+reasons of taste I wanted to deemphasize discovery in mathematics, 
+mathematics as discovering theorems and proofs. I wasn't good at such 
+discovery, and it bored me. The first way I thought of to de-emphasize 
+discovery came not later than Summer, 1960; it was that since the value of 
+pure mathematics is now regarded as aesthetic rather than cognitive, why not 
+try to make up aesthetic theorems, without considering whether they are 
+true. The second way, which came at about the same time, was to find, as a 
+philosopher, that the conventional claim that theorems and proofs are 
+discovered is wrong, for the same reason I have already given that 'concept' 
+can be discredited. The third way, which came in the fall-winter of 1960, 
+was to work in unexplored regions of formalist mathematics. The resulting 
+mathematics still had statements, theorems, proofs, but the latter weren't 
+discovered in the way they traditionally were. Now exploration of the wider 
+possibilities of mathematics as revolutionized by me tends to lead beyond 
+what it makes sense to call mathematics; the category of mathematics, a 
+vestige of Platonism, is an unnatural, bad one. My work in mathematics leads 
+to the new category of concept art, of which straightened out traditional 
+mathematics (mathematics as discovery) is an untypical, small but 
+intensively developed part. 
+
+I can now return to the question of why concept art is art. Why isn't it an 
+absolutely new, or at least a non-artistic, non-aesthetic activity? The answer 
+is that the antecedents of concept art are commonly regarded as artistic, 
+aesthetic activities; on a deeper level, interesting concepts, concepts 
+enjoyable in themselves, especially as they occur in mathematics, are 
+commonly said to have beauty. By calling my activity art, therefore, I am 
+simply recognizing this common usage, and the origin of the activity in 
+structure art and mathematics. However: it is confusing to call things as 
+irrelevant as the emotional enjoyment of (real) music, and the intellectual 
+enjoyment of concepts, the same kind of enjoyment. Since concept art 
+includes almost everything ever said to be music, at least, which is not music 
+for the emotions, perhaps it would be better to restrict art to apply to art for 
+the emotions, and recognize my activity as an independent, new activity, 
+irrelevant to art (and knowledge). 
+
+\section*{Concept Art Version of Mathematics System 3/26/61 (6/19/61)}
+
+An element is the adjacent area (with the figure in it) so long as the 
+apparent, perceived, ratio of the length of the vertical line to that of the 
+horizontal line (the element's associated ratio) does not change. 
+
+A selection sequence is a sequence of elements of which the first is the one 
+having the greatest associated ratio, and each of the others has the associated 
+ratio next smaller than that of the preceding one. (To decrease the ratio, 
+come to see the vertical line as shorter, relative to the horizontal line, one 
+might try measuring the lines with a ruler to convince oneself that the 
+vertical one is not longer than the other, and then trying to see the lines as 
+equal in length; constructing similar figures with a variety of real (measured) 
+ratios and practicing judging these ratios; and so forth.) 
+
+[Observe that the order of elements in a selection sequence may not be the 
+order in which one sees them.] 
+
+
+\img{implications}
+
+\section*{Implications---Concept Art Version of Colored Sheet Music No. 1 3/14/61 (10/11/61)}
+
+[This is a mathematical system without general concepts of statement, 
+implication, axiom, and proof. Instead, you make the object, and stipulate 
+by ostension that it is an axiom, theorem, or whatever. My thesis is that 
+since there is no objective relation between name and intension, all 
+mathematics is this arbitrary. Originally, the successive statements, or sheets, 
+were to be played on an optical audiorecorder.]
+
+\begin{sysrules}
+The axiom: a sheet of cheap, thin white typewriter paper 
+
+The axiom implies statement 2: soak the axiom in inflammable liquid which 
+does not leave solid residue when burned; then burn it on horizontal 
+rectangular white fireproof surface---statement 2 is ashes (on surface) 
+
+Statement 2 implies s.3: make black and white photograph of s.2 in white 
+light (image of ashes' rectangle with respect to white surface (that is, of the 
+region (of surface, with the ashes on it) with bounding edges parallel to the 
+edges of the surface and intersecting the four points in the ashes nearest the 
+four edges of the surface) must exactly cover the film); develop film---s.3 is 
+the negative.
+
+s.2 and s.3 imply s.4: melt s.3 and cool in mold to form plastic doubly 
+convex lens with small curvature; take color photograph of ashes' rectangle 
+in yellow light using this lens; develop film---s.4 is color negative.
+
+s.2 and s.4 imply s.5: repeat last step with s.4 (instead of 3), using red 
+light---s.5 is second color negative 
+
+s.2 and s.5 imply s.6: repeat last step with s.5, using blue light---s.6 is third 
+color negative 
+
+s.2 and s.6 imply s.7: make lens from s.6 mixed with the ashes which have 
+been being photographed; make black and white photograph, in white fight, 
+of that part of the white surface where the ashes' rectangle was; develop film 
+--- s.7 is second black and white negative 
+
+s.2, s.6, and s.7 imply the theorem: melt, mold, and cool lens used in last 
+step to form negative, and make lens from s.7; using negative and lens in an 
+enlarger, make two prints, an enlargement and a reduction--enlargement and 
+reduction together constitute the theorem. 
+\end{sysrules}
+
+\section*{Concept Art: Innpersegs (May--July 1961)}
+
+\begin{sysrules}
+A "halpoint" iff whatever is at any point in space, in the fading rainbow halo 
+which appears to surround a small bright light when one looks at it through 
+glasses fogged by having been breathed on, for as long as the point is in the 
+halo. 
+
+An "init`point" iff a halpoint in the initial vague outer ring of its halo. 
+
+
+An "inn`perseq" iff a sequence of sequences of halpoints such that all the 
+halpoints are on one (initial) radius of a halo; the members of the first 
+sequence are initpoints; for each of the other sequences, the first member (a 
+consequent) is got from the non-first members of the preceding sequence 
+(the antecedents) by being the inner endpoint of the radial segment in the 
+vague outer ring when they are on the segment, and the other members (if 
+any) are initpoints or first members of preceding sequences; all first members 
+of sequences other than the last [two] appear as non-first members, and 
+halpoints appear only once as non-first members; and the last sequence has 
+one member. 
+\end{sysrules}
+
+\section*{Indeterminacy}
+
+\begin{sysrules}
+A $\ulcorner$totally determinate innperseq' iff an innperseq$\urcorner$ in which one is aware of 
+(specifies) all halpoints. 
+
+An $\ulcorner$antecedentally indeterminate innperseq' iff an innperseq$\urcorner$ in which one is 
+aware of (specifies) only each consequent and the radial seqment beyond it. 
+
+A $\ulcorner$halpointally indeterminate innperseq' iff an innperseq$\urcorner$ in which one is 
+aware of (specifies) only the radial segment in the vague outer ring, and its 
+inner endpoint, as it progresses inward. 
+\end{sysrules}
+
+\subsection*{Innperseqs Diagram}
+
+In the diagram, different positions of the vague outer ring at different times 
+are suggested by different shadings. The radial segment in the vague outer 
+ring moves down the page. The figure is by no means an innperseq, but is 
+supposed to help explain the definition. 
+
+\img{innperseqsdiagram}
+