throw in the extra tex files for essays im not sure i end up using

4 files changed, 1615 insertions, 0 deletions

diff --git a/extra/apprehension_of_plurality.tex b/extra/apprehension_of_plurality.tex
new file mode 100644
index 0000000..eee4290
--- /dev/null
+++ b/extra/apprehension_of_plurality.tex
@@ -0,0 +1,1199 @@
+\chapter{The Apprehension of Plurality (1987)}
+
+% if we end up needing to add it:
+% https://henryflynt.org/studies_sci/reqmath.html
+
+{\centering\itshape
+(An instruction manual for 1987 concept art)\par}
+
+\section{Original Stroke-Numerals}
+
+Stroke-numerals were introduced in foundations of mathematics 
+by the German mathematician David Hilbert early in the twentieth 
+century. Instead of a given Arabic numeral such as `6', for example, one 
+has the expression consisting of six concatenated occurrences of the 
+stroke, e.g. `$||||||$'. 
+
+To explain the use of stroke-numerals, and to provide a background 
+for my innovations, some historical remarks about the philosophy 
+of mathematics are necessary. Traditional mathematics had 
+treated positive whole-number arithmetic as if the positive whole 
+numbers (and geometrical figures also) were objective intangible 
+beings. Plato is usually named as the originator of this view. Actually, 
+there is a scholarly controversy over the degree to which Plato espoused 
+the doctrine of Forms---over whether Aristotle's \booktitle{Metaphysics} put 
+words in Plato's mouth---but that is not important for my purposes. 
+For an intimation of the objective intangible reality of mathematical 
+objects in Plato's own words, see the remarks about "divine" geometric 
+figures in Plato's \essaytitle{Philebus.} Aristotle's \booktitle{Metaphysics}, 
+1.6, says that mathematical entities 
+\begin{quotation}
+are intermediate, differing from things perceived in being eternal and 
+unchanging, and differing from the Forms in that they exist in copies, 
+whereas each Form is unique. 
+\end{quotation}
+
+For early modern philosophers such as Hume and Mill, any such 
+"Platonic" view was not credible and could not be defended seriously. 
+Thus, attempts were made to explain number and arithmetic in ways 
+which did not require a realm of objective intangible beings. In fact, 
+Hume said that arithmetic consisted of tautologies; Mill that it 
+consisted of truths of experience. 
+
+Following upon subsequent developments---the philosophical 
+climate at the end of the nineteenth century, and specifically 
+mathematical developments such as non-Euclidian geometry---Hilbert proposed 
+that mathematics should be understood as a game played with meaningless marks. 
+So, for example, arithmetic concerns nothing but formal 
+terms---numerals---in a network of rules. Actually, what made arithmetic 
+problematic for mathematicians was its infinitary character---as 
+expressed, for example, by the principle of complete induction. Thus, 
+the principal concern for Hilbert was that this formal game should not, 
+as a result of being infinitary, allow the deduction of both a proposition 
+and its negation, or of such a proposition as $0=1$. 
+
+But at the same time (without delving into Hilbert's distinction 
+between mathematics and metamathematics), the stroke-numerals 
+replace the traditional answer to the question of what a number is. The 
+stroke-numeral '||||||' is a concrete semantics for the sign `6', and at the 
+same time can serve as a sign in place of `6'. The problem of positive 
+whole numbers as abstract beings is supposedly avoided by inventing 
+e.g. a number-sign, a numeral, for six, which is identically a concrete 
+semantics for six. Let me elaborate a little further. A string of six copies 
+of a token having no internal structure is used as the numeral `6', the 
+sign for six. Thus the numeral is itself a collection which supposedly 
+demands a count of six, thereby showing its meaning. Hans Freudenthal 
+calls this device an "ostensive numeral." 
+
+So traditionally, there is a question as to what domain of beings 
+the propositions of arithmetic refer to, a question as to what the 
+referents of number-words are. \emph{Correlative to this, mathematicians' 
+intentions require numerous presuppositions about content, and 
+require extensive competancies---which the rationalizations for math- 
+ematics today are unable to acknowledge, much less to defend.}
+
+For example, if mathematics rests on concrete signs, as Hilbert 
+proposed, then, since concrete signs are objects of perception, the 
+reliability of mathematics would depend on the reliability of percep- 
+tion. Given the script numeral 
+{\plainbreak{1}\centering\includegraphics[width=1in]{img/oneortwo}\plainbreak{1}}
+which is ambiguous between one and two, conventional mathematics 
+would have to guarantee the exclusion of any such ambiguity as this. 
+Yet foundations of mathematics excludes perception and the reliability 
+of concrete signs as topics---much as Plato divorced mathematics from 
+these topics. (Roughly, modern mathematicians would say that reliability 
+of concrete signs does not interact with any advanced mathematical 
+results. So this precondition can simply be transferred from the requisites 
+of cognition in general. But it would not be sincere for Hilbert to 
+give this answer. Moreover, my purpose is to investigate the possibility 
+of reconstructing our intuitions of quantity beyond the limits of the 
+present culture. In this connection, I need to activate the role of 
+perception of signs.) 
+
+But the most characteristic repressed presuppositions of mathematics 
+run in the opposite, supra-terrestrial direction. Mathematicians' 
+intentions require a realm of abstract beings. Again, it is academically 
+taboo today to expose such presuppositions.
+\footnote{G\"{o}del and Quine admit the need to assume the non-spatial, abstract 
+existence of classes. But they cannot elaborate this admission; they cannot 
+provide a supporting metaphysics.} But to recur to the 
+purpose of this investigation, concept art is about reconstructing our 
+intuitions of quantity beyond the limits of the present culture. This 
+project demands an account of these repressed presuppositions. To 
+compile such an account is a substantial task; I focus on it ina collateral 
+manuscript entitled "The Repressed Content-Requirements of Math- 
+ematics." To uncover the repressed presuppositions, a combination of 
+approaches is required.\footnote{One anthroplogist has written about \enquote{the locus of 
+mathematical reality}---but, being an academic, he merely reproduces a stock answer outside 
+his field (namely that the shape of mathematics is dictated by the physiology of 
+the brain).} I will not dwell further on the matter here---
+but a suitable sample of my results is the section "The Reality-Character 
+of Pure Whole Numbers and Euclidian Figures" in \emph{The Repressed Content-Requirements.}
+
+Returning to the original stroke-numerals, they were meant 
+(among other things) to be part of an attempt to explain arithmetic 
+without requiring numbers as abstract beings. They were meant as 
+signs, for numbers, which are identically their own concrete semantics. 
+Whether I think Hilbert succeeded in dispensing with abstract entities is 
+not the point here. I am interested in how far the exercise of positing 
+stroke-numerals as primitives can be elaborated. My notions of the 
+original stroke-numerals are adapted from Hilbert, Weyl, Markov, 
+Kneebone, and Freudenthal. For example, how does one test two 
+stroke-numerals for equality? To give the answer that "you count the 
+strokes, first in one numeral and then in the other," is not in the spirit of 
+the exercise. For if that is the answer, then that means that you have a 
+competency, "counting," which must remain a complete mystery to 
+foundations of mathematics. What one wants to say, rather, is that you 
+test equality of stroke-numerals by "cross-tallying": by e.g. deleting 
+strokes alternately from the two numerals and finding if there is a 
+remainder from one of the numerals. This is also the test of whether one 
+numeral precedes the other. So, now, given an adult mastery of quality 
+and abstraction, you can identify stroke-numerals without being able 
+to "count." 
+
+In the same vein, you add two stroke-numerals by copying the 
+second to the right of the first. You subtract a shorter numeral from a 
+longer numeral by using the shorter numeral to tally deletion of strokes 
+from the longer numeral. You multiply two stroke-numerals by copying the second as many times as there are strokes in the first: that is, by 
+using the strokes of the first to tally the copying of the second numeral. 
+
+To say that all this is superfluous, because we already acquired 
+these "skills" as a child, misses the point. The child does not face the 
+question, posed in the Western tradition, of whether we can avoid 
+positing whole numbers as abstract beings. To weaken the requirements 
+of arithmetic to the point that somebody with an adult mastery 
+of quality and abstraction can do feasible arithmetic "blindly"---i.e. 
+without being able to "count," and without being able to see number-names 
+('five', 'seven', etc.) in concrete pluralities---is a notable exercise, 
+one that correlates culturally with positivism and with the machine age. 
+
+To reiterate, the stroke-numeral is meant to replace numbers as 
+abstract beings by providing number-signs which are their own concrete 
+semantics. Freudenthal said that we should communicate positive 
+whole numbers to alien species by broadcasting stroke-numerals to 
+them (in the form of time-series of beeps). Still, Freudenthal said that 
+the aliens would have to resemble us psychologically to get the point.\footnote{\booktitle{Lincos}, pp. 14--15.} 
+
+When Hilbert first announced stroke-numerals, certain difficulties 
+were pointed out immediately. It is not feasible to write the 
+stroke-numerals for very large integers. (And yet, if it is feasible to write the 
+stroke-numeral for the integer n, then there is no apparent reason why 
+it would not also be feasible to write the stroke-numeral for n+1. So 
+stroke-numerals are closed under succession, and yet are contained in a
+finite segment of the classical natural number series.) Moreover, large 
+feasible stroke-numerals, such as that for 10,001, are not surveyable. 
+
+But this is not a study of metamathematical stroke-numerals. And 
+I do not wish to go into Hilbert's question of the consistency of 
+arithmetic as an infinitary game here; "The Repressed Content-Requirements" 
+will have more to say on the consistency question. The 
+purpose of this manual, and of the artworks which it accompanies, is to 
+establish apprehensions of plurality beyond the limits of traditional 
+civilizations (beyond the limits of Freudenthal's "us"). Moreover, these 
+apprehensions of plurality are meant to violate the repressed presuppo- 
+sitions of mathematics. I refer back to original stroke-numerals because 
+certain devices which I will use in assembling my novelties cannot be 
+supposed to be intuitively comprehensible---certainly not to the 
+traditionally-indoctrinated reader---and will more likely be understood 
+if I mention that they are adaptations of features of original stroke- 
+numerals. Let me mention one point right away. In our culture, we 
+usually see numerals as positional notations---e.g. 111 is decimal 
+$1\times 10^2+1\times 10^1+1$ or binary $1\times 2^2+1\times 2^1+1$. But stroke-numerals 
+are not a positional notation (except trivially for base 1). Likewise, my 
+novelties will not be positional notations; I will even nullify the 
+reference to base 1. (Only much later in my investigations, when broad 
+scope becomes important, will I use positional notation.) So the 
+foregoing introduction to stroke-numerals has only the purpose of 
+motivating my novelties. And references to the academic canon are given 
+only for completeness. They cannot be norms for what I am "permitted" to posit. 
+
+\section{Simple Necker-Cube Numerals}
+
+In my stroke-numerals, the printed figure, instead of being a 
+stroke, is a Necker cube. (Refer to the attached reproduction, "Stroke- 
+Numeral.") A Necker cube is a two-dimensional representation of a 
+cubical frame, formed without foreshortening so that its perspective is 
+perceptually equivocal or multistable. The Necker cube can be seen as 
+flat, as slanting down from a central facet like a gem, etc.; but for the 
+moment I am exclusively concerned with the two easiest variants in 
+which it is seen as an ordinary cube, either projecting up toward the 
+front or down toward the front. 
+
+{\center\includegraphics[width=4in]{img/neckercube}\plainbreak{2}
+\includegraphics[width=2in]{img/neckerkey}\par}
+
+Since I will use perceptually multistable figures as notations, I 
+need a terminology for distinctions which do not arise relative to 
+conventional notation. I call the ink-shape on paper a \term{figure}. I call the 
+stable apparition which one sees in a moment---which has imputed 
+perspective---the \term{image}.\footnote{I may note, without wanting to be precious, that a bar does not count as a Hilbert stroke unless it is vertical relative to its reader.}
+As you gaze at the figure, the image changes 
+from one orientation to the other, according to intricate subjective 
+circumstances. It changes spontaneously; also, you can change it 
+voluntarily. 
+
+Strictly---and very importantly---it is the image which in this 
+context becomes the notation. Thus, I will work with notations which 
+are not ink-shapes and are not on a page. They arise as active interactions 
+of awareness with an "external" or "material" print-shape or 
+object. 
+
+So far, then, we have images---partly subjective, pseudo-solid 
+shapes. I now stipulate an alphabetic role for the two orientations in 
+question. The up orientation is a \term{stroke}; the down orientation is called 
+"\term{vacant}," and acts as the proofreaders' symbol $\closure$, meaning "close up space." 
+(So that "vacant" is not "even" an alphabetic space.) Now the 
+two images in question are \term{signs}. The transition from image to sign can 
+be analogized to the stipulation that circles of a certain size are (occurances 
+of) the letter "o."\footnote{And---the shape, bar, positioned vertically relative to its reader, is the symbol, Hilbert stroke.} I may say that one sees the image; one 
+apprehends the image as sign. 
+
+When a few additional explanations are made, then the signs 
+become plurality-names or "numerals." First, figures, Necker cubes, 
+are concatenated. When this is done, a display results. So the 
+stroke-numeral in the artwork, as an assembly of marks on a surface, is a 
+display of nine Necker cubes. An image-row occurs when one looks at 
+the display and sees nine subjectively oriented cubes, for just so long as 
+the apparition is stable (no cube reverses orientation). I chose nine 
+Necker cubes as an extreme limit of what one can apprehend in a fixed 
+field of vision. (So one must view the painting from several meters 
+away, at least.) The reader is encouraged to make shorter displays for 
+practice. Incidentally, if one printed a stroke-numeral so long that one 
+could only apprehend it serially, by shifting one's visual field, it would 
+be doubtful that it was well-defined. (Or it would incorporate a feature 
+which I do not provide for.) The universe of pluralities which can be 
+represented by these stroke-numerals is "small." My first goal is to 
+establish "subjectified" stroke-numerals at all. They don't need to be 
+large. 
+
+The concatenated signs which you apprehend in a moment of 
+looking at the display are now apprehended or judged as a 
+plurality-name, a numeral. At the level where you apprehend signs (which, 
+remember, are alphabetized, partly subjective images, not figures), the 
+apparition is disambiguated. Thus I can explain this step of judging the 
+signs as plurality-names by using fixed notation. For nine Necker cubes 
+with the assigned syntactical role, you might apprehend such 
+permutations of signs as 
+\begin{enumerate}[label=\alpha*.]
+	\item $|\closure\closure||\closure\closure\closure|$
+	\item $|\closure\closure\closure\closure\closure|||$
+	\item $||||\closure\closure\closure\closure\closure$
+	\item $||||\closure\closure\closure\closure|$
+	\item $\closure\closure\closure\closure\closure\closure\closure\closure\closure$
+\end{enumerate}
+
+My Necker-cube stroke-numerals are something new; but (a)--(e) are 
+not---they are just a redundant version of Hilbert stroke-numerals 
+(which nullifies the base 1 reference as I promised). The "close up 
+space" signs function as stated; and the numeral concluded from the 
+expression corresponds to the number of strokes; i.e. the net result is 
+the Hilbert stroke-numeral having the presented number of strokes. So 
+(a) and (b) and (c) all amount to $|||$. (d) amounts to $|||||$.
+
+As for (e), it has the alphabetic role of a blank. My initial interpretation 
+of this blank is "no numeral present." Later I may interpret the 
+blank as "zero," so that every possibility will be a numeral. Let me 
+explain further. Even when I will interpret the blank as "zero." it will 
+not come about from having nine zeros mapped to one zero (like a sum 
+of zeros). (e) has nine occurrences of "close up space," making a blank. 
+There is always only one way of getting "blank." (A two-place display 
+allows two ways of getting "one" and one way of getting "two"; etc.) 
+The notation is not positional. It is immaterial whether one "focuses" 
+starting at the left or at the right. 
+
+Relative to the heuristic numerals (a)--(e), you may judge the 
+intended numerals by counting strokes, using your naive competency 
+in counting. (It is also possible to use such numerals as (a)--(e) "blindly" 
+as explained earlier. This might mean that there would be no recognition 
+of particular numbers as gestalts; identity of numbers would be
+handled entirely by cross-tallying.) The Necker-cube numerals, however, 
+pertain to a realm which is in flux because it is coupled to 
+subjectivity. My numerals provide plurality-names and models of that 
+realm. Thus, the issue of what you do when you conclude a numeral 
+from a sign in perception is not simple. \emph{We have to consider different 
+hermeneutics for the numerals---and the ramifications of those hermeneutics.}
+Here we begin to get a perspective of the mutability which my 
+devices render manageable. 
+
+For one thing, given a (stable) image-row, and thus a sign-row, you 
+can indeed use your naive arithmetical competency to count strokes, 
+and so conclude the appropriate numeral. This is \term{bicultural hermeneutic}, 
+because you are using the old numbers to read a new notation for 
+which they were not intended. We use the same traditional counting, of 
+course, to speak of the number of figures in a display. 
+
+(This prescription of a hermeneutic is not entirely straightforward. 
+The competency called counting is required in traditional mathematics. 
+But such counting is already paradoxical "phenomenologically." I 
+explain this in the section called "Phenomenology of Counting" in \essaytitle{The 
+Repressed Content-Requirements}. As for the Necker-cube numerals, 
+the elements counted are not intended in a way which supports the 
+being of numbers as eternally self-identical. So the Necker-cube 
+numerals might resonate with the phenomenological paradoxes of 
+ordinary counting. The meaning of ordinary numbering, invoked in 
+this context, might begin to dissolve. But I mention this only to hint at 
+later elaborations. At this stage, it is proper to recall one's inculcated 
+school-counting; and to suppose that e.g. the number of figures in a 
+display is fixed in the ordinary way.) 
+
+Then, there is the \term{ostensive hermeneutic}. Recall that I explained 
+Hilbert stroke-numerals as signs which identically provide a concrete 
+semantics for themselves; and as an attempt to do arithmetic without 
+assuming that one already possesses arithmetic in the form of competency 
+in counting, or of seeing number-names in pluralities. My 
+intention was to prepare the reader for features to be explained now. 
+On the other hand, at present we drop the notion of handling identity of 
+numerals by cross-tallying.\footnote{Because this notion corresponds to a situation in which we are unable to appraise image-rows as numerals, as gestalts.}
+For the ostensive hermeneutic, it is crucial 
+that the display is short enough to be apprehended in a fixed field of 
+vision. 
+
+With respect to short Hilbert numerals, I ask that when you see 
+e.g. 
+
+$$||$$
+
+marked ona wall, you grasp it asa sign for a definite plurality, without 
+mediation---without translating to the word "two." A similar intention 
+is involved in recognizing 
+$$\sout{||||}$$
+as a definite plurality, as a gestalt, without translating to "five." 
+
+Now I ask you to apply this sort of hermeneutic to Necker-cube 
+stroke-numerals. I ask you to grasp the sign-row as a numeral, as a 
+gestalt. (Without using ordinary counting to call off the strokes.) Fora 
+two-place display, you are to take such images as 
+
+\newcommand{\neckup}{\includegraphics[width=1in]{img/neckerup}}
+\newcommand{\neckdown}{\includegraphics[width=1in]{img/neckerdown}}
+
+{\centering\neckup\neckdown\par}
+and
+{\centering\neckup\neckup\par}
+as plurality-names without translating into English words. (Similarly 
+
+{\centering\neckdown\neckdown\par}
+
+in the case where I choose to read "blank" as "zero.") Perhaps it is 
+necessary to spend considerable time with this new symbolism before 
+recognition is achieved. Again, I encourage the reader to make short 
+displays for practice. I have set a display of nine figures as the upper 
+limit for which it might be possible to learn to grasp every sign-row as a 
+numeral, as a gestalt. 
+
+The circumstance that the apprehended numeral may be different 
+the next moment is not a mistake; the apprehended numeral is supposed 
+to be in flux. So when you see image-rows, you take them as 
+identical signs/semantics for the appearing pluralities. 
+
+But who wants such numerals---where are there any phenomena 
+for them to count? For one thing, they count the very image-rows which 
+constitute them. The realm of these image-rows is a realm of subjective 
+flux: its plurality is authentically represented by my numerals, and 
+cannot be authentically represented by traditional arithmetic. 
+
+A further remark which may be helpful is that here numerals arise 
+only visually. So far, my numerals have no phonic or audio equivalent. 
+(Whereas Freudenthal in effect posited an audio version of Hilbert 
+numerals, using beeps.) 
+
+To repeat, by the "ostensive hermeneutic" I mean grasping the 
+sign-row, without mediation, as a numeral. But there is, as well, the 
+point that the Necker-cube numerals are \term{ostensive numerals}. That is, 
+the (momentary) numeral for six would in fact be an image-row with 
+just six occurrences of the image "upward cube." (Compare e.g. 
+$|||\closure\closure||\closure|$) The numeral is a collection in which only the "copies" of 
+"upward cube" contribute positively, so to speak; and these copies 
+demand a count of six (bicuturally). This feature needs to be clear, 
+because later I will introduce numerals for which it does not hold. 
+
+Let me add another proviso concerning the ostensive hermeneutic 
+which will be important later. I will illustrate the feature in question 
+with an example which, however, is only an analogy. Referring to 
+Arabic decimal-positional numerals, you can appraise the number-name of 
+$$1001$$
+(comma omitted) immediately. But consider 
+$$786493015201483492147$$
+Here you cannot appraise the number-name without mediation. That 
+is, if you are asked to read the number aloud, you don't know whether 
+to begin with "seven" or "seventy-eight" or "seven hundred eighty-six." 
+Lacking commas, you have to group this expression from the right, in 
+triples, to find what to call it. An act of analysis is required. 
+
+In the case of Necker-cube numerals and the ostensive hermeneutic, 
+I don't want you to see traditional number-names in the pluralities. 
+However, I ask you to grasp a sign-row as a numeral, as a gestalt. I now 
+add that the gestalt appraisal is definitive. I rule out appraising image-rows 
+analytically (by procedures analogous to mentally grouping an 
+Arabic number in triples). (I established a display of nine figures as the 
+upper limit to support this.) 
+
+The need for this proviso will be obscure now. It prepares for a 
+later device in which, even for short displays, gestalt appraisal and 
+appraisal by analysis give different answers, either of which could be 
+made binding. 
+
+\breatk
+
+The bicultural hermeneutic is applied, in effect, in my uninterpreted 
+calculus \textsc{"Derivation,"} which serves as a simplified analogue of 
+my early concept art piece \textsc{"Illusions."} (Refer to the reproductions on 
+the next four pages.) Strictly, though, "Derivation" does not concern a 
+Necker-cube stroke-numeral. The individual figures are not Necker 
+cubes, but "Wedberg cubes," formed with some foreshortening to make 
+one of the two orientations more likely to be seen than the other. What 
+is of interest is not apprehension of image-rows as numerals, but rather 
+appraisal of lengths of the image-rows via ordinary counting. As for the 
+lessons of this piece, a few simple observations are made in the piece's 
+instructions. But to pursue the topic of concept art as uninterpreted 
+calculi, and derive substantial lessons from it, will require an entire 
+further study---taking off from earlier writings on post-formalism and 
+uncanny calculi, and from my current writings collateral to this essay. 
+
+
+1987 Concept Art --- Henry Flynt 
+"DERIVATION" (August 1987 corrected version) 
+
+
+Purpose: To provide a simplified analogue of my 1961 concept art piece "'IIlusions'' which is 
+discrete and non-''warping.''* Thereby certain features of "'Illusions'' become more 
+clearly discernible. 
+
+
+Given a perceptually multistable figure, the ""Wedberg cube," which can be seen in two 
+orientations: as a cube; as a prism (trapezohedron.) 
+
+Call what is seen at an instant an /mage. 
+
+Nine figures are concatenated to form the display. 
+
+
+An element is an image of the display for as long as that image remains constant (Thus, 
+elements include: the image from the first instant of a viewing until the image first 
+changes; an image for the duration between two changes; the image from the last 
+change you see in a viewing until the end of the viewing.) 
+
+
+The /ength of an element equals the number of prisms seen. Lengths from O through nine 
+are possible. Two different elements can have the same length. Length of element X 
+is written /(X). 
+
+
+Elements are seen in temporal order in the lived time of the spectator. | refer to this order by 
+words with prefix 'T'. T-first; T-next; etc. 
+
+
+Element Y succeeds element X if and only if 
+i) (X) = KY), and Y is T-next after X of all elements with this length; or 
+ii) ¥ is the T-earliest element you ever see with length /(X) + 1. 
+Note that (ii) permits Y to be T-earlier than X: the relationship is rather artificial. 
+
+
+The initial element A is the T-first element. (/(A) may be greater than O; but it is likely to be O 
+because the figure is biased.) 
+
+
+The conclusion C is the T-earliest element of length 9 (exclusive of Ain the unlikely case in 
+which /(A) = 9). 
+
+
+A derivation is a series of elements in lived time which contains A and C and in which every 
+element but A succeeds some other element. 
+
+
+Discussion 
+
+To believe that you have seen a derivation, you need to keep track that you see each 
+possible length, and to force yourself to see lengths which do not occur spontane- 
+ously. 
+
+
+You may know that you have seen a derivation, without being able to identify in memory the 
+particular successions. 
+
+
+"Derivation" is not isomorphic to "Illusions" for a number of reasons. ''Illusions" doesn't 
+require you to see individually every possible ratio between the T-first ratio and unity. 
+"Illusions" allows an element to succeed itself. The version of 'Derivation' pres- 
+ented here is a compromise between mimicking "'Illusions"' and avoiding a trivial or 
+cluttered structure. Any change such as allowing elements to succeed themselves 
+would require several definitions to be modified accordingly. 
+
+
+*In "Illusions," psychic coercion, which may be called "false seeing" or "warping," is 
+recommended to make yourself see the ration as unity. In ''Derivation," this warping is not 
+necessary; all that may be needed is that you see certain lengths willfully. 
+
+
+ABABA AAS 
+
+
+Concept Art Version of Mathematics System 3/26/6l(6/19/61) 
+
+An "element"is the facing page (with the figure on 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 asequence of elements of which the 
+first is the one having the greatest associated ratio, and 
+each of the others has the associated ratio next smallerthan 
+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 con- 
+vince 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.] 
+
+
+An elaboration of "Stroke-Numeral" should be mentioned here, 
+the piece called "an Impossible Constancy." (Refer to the facing page.) 
+As written, this piece presupposes the bicultural hermeneutic, and that 
+is probably the way it should be formulated. The point of this piece, 
+paradoxically, is that one seeks to annul the flux designed into the 
+apprehended numeral. Viewing of the Necker-cube numeral is placed 
+in the context of a lived experience which is interconfirmationally 
+weak: namely, memory of past moments within a dream (a single 
+dream). Presumably, appraisals of the numeral at different times could 
+come out the same because evidence to the contrary does not survive. 
+So inconstancy passes as constancy. Either hermeneutic can be 
+employed; but when I explained the hermetic hermeneutic, I encour- 
+aged you to follow the flux. Here you wouldn't do that---you wouldn't 
+stare at the display over a retentional interval. 
+
+
+As for the concept of equality with regard to Necker-cube numerals, 
+what can be said about it at this point? We have equality of numbers of 
+figures in displays, by ordinary counting. We have two hermeneutics 
+for identifying an apprehended numeral. In the course of expounding 
+them, I expounded equivalence of different permutations of "stroke" 
+and "vacant." Nevertheless, given that, for example, a display of two 
+figures can momentarily count the numeral apprehended from a dis- 
+play of three figures,* we are in unexplored territory. Cross-tallying, 
+suitable for judging equality of Hilbert numerals, seems maladapted to 
+Necker-cube numerals; in fact, I dismissed it when introducing the 
+ostensive hermeneutic. 
+
+If the "impossible constancy" from the paragraph before last were 
+manageable, then one might consider restricting the ultimate definition 
+of equality to impossible constancies. That is, with respect to a single 
+display, if one wanted to investigate the intention of constancy (self- 
+equivalence of the apprehended numeral), one might start with the 
+impossible constancy. Appraisals of a given display become constant 
+(the numeral becomes self-equivalent) in the dream. Then two displays 
+which are copies might become constantly equivalent to each other, in 
+the dream. 
+
+Such is a possibility. To elaborate the basics and give an incisive 
+notion of equality is really an open problem, though. Other avenues 
+might require additional devices such as the use of figures with distinc- 
+tions of appearance. 
+
+
+*that it is not assured that copies of a numeral will be apprehended or 
+appraised correlatively 
+
+
+1987 Concept Art --- Henry Flynt 
+Necker-Cube Stroke-Numeral: AN IMPOSSIBLE CONSTANCY 
+
+
+The purpose of this treatment is to say how a Necker-cube stroke numeral may be 
+judged (from the standpoint of private subjectivity) to have the same value at different 
+times; even though the conventional belief-system says that the value is likely to change 
+frequently. 
+
+
+This is accomplished by selecting a juncture in an available mode of illusion, namely 
+dreaming, which annuls any distinction between an objective circumstance, and the 
+circumstance which exists according to your subjective judgment. In the first instance, | 
+don't ask you to change your epistemology. Instead, to repeat, | select an available juncture 
+in lived experience at which the conventional epistomology gets collapsed. 
+
+
+You have to occupy yourself with the stroke-numeral to the point that you induce 
+yourself to dream about it. 
+
+When, in apprehending a stroke-numeral, you "judge" the value of the numeral, the 
+number, this refers to the image you see and to the number-word which you may conclude 
+from the image. 
+
+Suppose that in a single dreamed episode, you judge the value of the numeral at two 
+different moments. Suppose that at the second moment, you do not register any discre- 
+pancy between the value at the second moment and what the value was at the first 
+moment. Then you are permitted to disregard fallibility of memory, and to conclude that the 
+values were the same at both moments: because if your memory has changed the past, it 
+has done so tracelessly. A tracelessly-altered past may be accepted as the genuine past. 
+
+
+Refinements. The foregoing dream-construct may be "'lifted" to waking experience, as 
+per the lengthy explanations in ""An Epistemic Calculus."' Now you are asked to alter your 
+epistemology, selectively to suspend a norm of realism. 
+
+Now that we are concerned with waking experience, a supporting refinement is 
+possible. Suppose | make an expectation (which may be unverbalized) that the value of the 
+numeral at a future moment will be the same that it is now. This expectation cannot be 
+proved false, if: the undetermined time-reference 'future moment" is applied only at those 
+later moments when the value is the same as at the moment the expectation was made. 
+(Any later moment when the value is not the same is set aside as not pertinent, or forgotten 
+at still later moments when the value is the same.) 
+
+
+As a postscript, there is another respect in which testing a fact requires trust in a 
+comparable fact. Suppose | make a verbalized expectation that the value of the numeral in 
+the future will be the same as at present. Then to test this expectation in the future depends 
+on my memory of my verbalization. My expectation cannot be belied unless | have a sound 
+
+"memory that the number | verbalized in my expectation is different from the number | 
+conclude from the image now. 
+
+
+HT. Inconsistently-Valued Numerals 
+
+
+As the "Wedberg cube" illustrates, a cubical frame can be formed 
+in different ways, altering the likelihood that one or another image is 
+seen. With respect to the initial uses of the Necker-cube stroke-numeral 
+a figure is wanted which lends itself to the image of a cube projecting 
+up, or of a cube projecting down, with an approximately equal likeli- 
+hood for the two images---and which makes other images unlikely. 
+Now let a Necker cube be drawn large, with heavy line-segments, with 
+all segments equally long, with rhomboid front and back faces; and 
+display it below eye level. 
+
+
+As you look for the up and down orientations, there should be 
+moments when paradoxically you see the figure taking on both of these 
+mutually-exclusive orientations at once---yielding an apparition which 
+is a logical/ geometric impossibility. The sense-content in this case is 
+dizzying. 
+
+That we have perceptions of the logically impossible when we 
+suffer illusions has been mentioned by academic authors. (Negative 
+afterimages of motion---the waterfall illusion.) Evidently, though, these 
+phenomenaare so distasteful to sciences which are still firmly Aristote- 
+lian that the relations of perception, habituation, language, and logic 
+manifested in these phenomena have never been assessed academically. 
+For me to treat the paradoxical image thoroughly here would be too 
+much of a digression from our subject, the apprehension of plurality. 
+However, a sketchy treatment of the features of the impossible image is 
+necessary here. 
+
+To begin with, the paradoxical image of the Necker cube is not the 
+same phenomenon as the "impossible figures" shown in visual percep- 
+tion textbooks. The latter figures employ "puns" in perspective coding 
+such that parts of a figure are unambiguous, but the entire figure 
+
+
+cannot be grasped as a gestalt coherently. Then, the paradoxical Necker- 
+cube image is not an inconsistently oriented object (as the reader may 
+have noted). It is an apparitional depiction of an inconsistently oriented 
+object. But this is itself remarkable. For since a dually-oriented cube (in 
+Euclidean 3-space) is self-contradictory by geometric standards, a 
+picture of it amounts to a non-vacuous semantics for an inconsistency. 
+Another way of saying the same thing is that the paradoxically- 
+oriented image is real as an apparition. 
+
+If one is serious about wanting a "logic of contradictions"---a logic 
+which admits inconsistencies, without a void semantics and without 
+entailing everything---then one will not attempt to get it by a contorted 
+weakening of received academic logic. One will start from a concrete 
+phenomenon which demands a logic of contradictions for its authentic 
+representation---and will let the contours of the phenomenon shape the 
+logic. 
+
+In this connection, the paradoxically-oriented Necker-cube image 
+provides a lesson which I must explain here. Consider states or proper- 
+ties which are mutually exclusive, such as "married" and "bachelor." 
+Their conjunction---in English, the compound noun "married 
+bachelor"---is inconsistent.* On the other hand, the joint denial 
+"unmarried nonbachelor" is perfectly consistent and is satisfied by 
+nonpersons: a table is an unmarried nonbachelor. "Married" and 
+"bachelor" are mutually exclusive, but not exhaustive, properties. Only 
+when the domain of possibility, or intensional domain, is restricted to 
+persons, so "married" and "bachelor" become exhaustive properties. ** 
+Then, by classical logic, "married bachelor" and "unmarried nonbache- 
+lor" both have the same semantics: they are both inconsistent, and thus 
+vacuous, and thus indistinguishable. For exhaustive opposites, joint 
+affirmation and joint denial are identically vacuous. 
+
+But the paradoxically-oriented Necker-cube image provides a 
+concrete phenomenon which combines mutually exclusive states---as 
+an apparition. We can ascertain whether a concrete case behaves as the 
+tenets of logic prescribe. As I have said, various images can be seen ina 
+Necker cube, including a flat image. Thus, the "up" and "down" cubes 
+
+
+*If I must show that it is academically permitted to posit notions such as 
+these, then let me mention that Jan Mycielski calls "triangular circle" incon- 
+sistent in The Journal of Symbolic logic, Vol. 46, p. 625. 
+
+**] invoke this device so that I may proceed to the main point quickly. If it 
+is felt to be too artificial, perhaps it can be eliminated later. 
+
+
+are analogous to "married" and "bachelor" in that they are not exhaus- 
+tive of a domain unless the domain is produced by restriction. Then 
+"neither up nor down" is made inconsistent. (It is very helpful if you 
+haven't learned to see any stable images other than "up" and "down.") 
+The great lesson here is that given "both up and down" and "neither up 
+nor down" as inconsistent, their concrete reference is quite different. To 
+see a cube which manifests both orientations at the same time is one 
+paradoxical condition, which we know how to realize. To see a cube 
+which has no orientation (absence of "stroke" and absence of "vacant" 
+both) would be a different paradoxical condition, which we do not 
+know how to realize and which may not be realizable from the Necker- 
+cube figure. I don't claim that this is fully worked out; but it intimates a 
+violation of classical logic so important that I had to mention it. When 
+concept art reaches the level of reconstructing our inferential intuitions 
+as well as our quantitative intuitions, such anomalies as these will surely 
+be important. 
+
+Referring back to the Necker cube of page 210, let us now intend it 
+as a stroke-numeral (display of one figure). Let me modify the previous 
+assignments and stipulate that "blank" means "zero," rather than "no 
+numeral present." (It is more convenient if every sign yields a numeral.) 
+When you see the paradoxical image, you are genuinely seeing "a" 
+numeral which is the simultaneous presence of two mutually exclusive 
+numerals "one" and "zero" ---because it is the simultaneous presence of 
+images which are mutually exclusive geometrically.*** 
+
+It's not the same thing as 
+
+
+| 
+
+
+---because these are merely ambiguous scripts. In the Necker-cube case, 
+two determinate images which by logic preclude each other are present 
+at once; and as these images are different numerals, we have a genuine 
+
+
+---or as an alternative, 
+
+
+*For brevity, I may compress the three levels image, sign, numeral in 
+exposition. 
+
+
+inconsistently-valued numeral. 
+
+This situation changes features of the Necker-cube numerals in 
+important ways, however. Lessons from above become crucial. We 
+transfer the ostensive hermeneutic to the new situation, and find an 
+inconsistent-valued numeral. But this is no longer an ostensive 
+numeral. We have a name which is one and zero simultaneously, but 
+this is because of the impossible shape (orientation) of the notation- 
+token. What we do not have is a collection of images of a single kind 
+(the stroke) which paradoxically requires a count of one and a count of 
+zero. "Stroke" is positively present, while "vacant" is positively present 
+in the same place. We will find that a display with two figures can be 
+inconsistent as zero and two; but it is not an ostensive numeral, because 
+the number of strokes present is two uniquely.* Here the numerals are 
+not identically their semantics: for the anomaly is not an anomaly of 
+counting. The ambiguous script numeral is a proper analogy in this 
+respect. To give an anomaly of counting which serves as a concrete 
+semantics for the inconsistently-valued numerals, I will turn to an 
+entirely different modality. 
+
+From work with the paradoxical image, we learn that the Necker 
+cube allows some apprehensions which are not as commonas others--- 
+but which can be fostered by the way the figure is made and by 
+indicating what is to be seen. These rare apprehensions then become 
+intersubjectively determinate. If one observes Necker-cube displays for 
+a long time, one may well observe subtle, transient effects. For exam- 
+ple, you might see the "up" and "down" orientations at the same time, 
+but see one as dominating the other. In fact, there are too many such 
+effects and their interpersonal replicability is dubious. If we accepted 
+such effects as determining numerals, the interpersonal replicability of 
+the symbols would be eroded. Also the concrete definiteness of my 
+anomalous, paradoxical effects would be eroded. So I must stipulate 
+that every subtle transient effect which I do not acknowledge explicitly 
+is not definitive, and is unwanted, when the display is intended as a 
+symbolism. 
+
+Let me continue the explanation, for the inconsistently-valued 
+
+
+*Referring to my "person-world analysis" and to the dichotomy of 
+Paradigm | and Paradigm 2 expounded in "Personhood III," this token which 
+is two mutually exclusive numerals because its shape is inconsistent is outside 
+that dichotomy: because established signs acquire a complication which is 
+more or less self-explanatory, but the meanings do not follow suit. 
+
+
+numerals, for displays of more than one figure. When the display 
+consists of two Necker cubes, and the paradoxical images are admitted, 
+what are the variations? In the first place, one figure might be seen (ina 
+moment) as a paradoxical image and the other as a unary image. 
+Actually, if it is important to obtain this variant, we can compel it, by 
+drawing one of the cubes in a way which hampers the double image. 
+(Thin lines, square front and back faces, the four side segments much 
+shorter than the front and back segments.) Then we stipulate that the 
+differently-formed cubes continue to have the same assigned interpre- 
+tation. 
+
+
+Reading the two-figure display, then, the paradoxical and unary 
+images concatenate so that the resulting numeral is in one case one and 
+two at the same time; and in the other case zero and one at the same 
+time. Of course, it is only ina moment that either of these two cases will 
+be realized. At other moments, one may have only unary images, so 
+that the numeral is noncontradictorily zero, one, or two as the case may 
+be. (If it is important to know that we can obtain a numeral which is 
+both one and two at the same time without using dissimilar figures, 
+then, of course, we can use a single figure and redefine the signs as "one" 
+and "two.") 
+
+Now let us consider a display of two copies of the cube which lends 
+itself to the paradoxical image. Suppose that two paradoxical images 
+are seen; what is the numeral? Here is where I need the proviso which I 
+introduced earlier. Every sign-row is capable of being grasped as a 
+numeral, as a gestalt; and the appraisal of image-rows as numerals, 
+analytically, is ruled out. Let me explain how this proviso applies when 
+two paradoxical images are seen. 
+
+Indeed, let me begin with the case of a pair of ambiguous 
+
+
+script-numerals: ] ] 
+
+
+When these numerals are formed as exact copies, and I appraise the 
+expression as a numeral, as a gestalt, then I see 11 or I see 22. ("Conca- 
+tenating in parallel") I do not see 21 or 12---although these variants are 
+possible to an analytical appraisal of the expression. In the gestalt, it is 
+unlikely to intend the left and right figures differently. This case is 
+helpful heuristically, because it provides a situation in which the percep- 
+tual modification is only a matter of emphasis (as opposed to imputa- 
+tion of depth). To this degree, the juncture at issue is externalized; and it 
+is easier to argue a particular outcome. On the other hand, the mechan- 
+ics differ essentially in the script case and the Necker-cube case. 
+
+In the Necker-cube case, one sees both the left and the right image 
+determinately both ways at once. This case may be represented as 
+
+
+stroke stroke 
+vacant vacant 
+
+
+Analytically, then, four variants are available here, 
+
+
+stroke-stroke 
+
+stroke-vacant 
+vacant-stroke 
+vacant-vacant 
+
+
+However, to complete the present explanation, only two of these 
+variants appear as gestalts, 
+
+
+stroke-stroke 
+vacant-vacant 
+
+
+I chose to rule out the three-valued numeral which would be obtained 
+by analytically inventorying the permutations of the signs afforded in 
+the perception. The two-valued numeral arising when the sign-row is 
+grasped as a gestalt is definitive. 
+
+Let me summarize informally what I have established. Relative to 
+a two-figure display with paradoxical images admitted, we have a 
+numeral which is inconsistenly two and zero. We can also have a 
+numeral which is inconsistently one and zero, and a numeral which is 
+inconsistently two and one. (In fact, these variants occur in several 
+ways.) But we don't have a numeral which is inconsistently zero, one, 
+and two---even though such a variant is available in an analytical 
+appraisal---because such a numeral does not appear, in perception, asa 
+gestalt. 
+
+Academic logic would never imagine that there is a situation 
+which demands just this configuration as its representation. Certain 
+
+
+definite positive inconsistencies are available in perception. Other defi- 
+nite positive inconsistencies, very near to them, are not available. Once 
+again, if one wants a vital "logic of contradictions," one has to develop 
+it as a representation of concrete phenomena; not as an unmotivated 
+contortion of received academic logics. 
+
+
+But what is the use of inconsistently-valued numerals? I shall now 
+provide the promised concrete semantics for them. This semantics 
+utilizes another experience of a logical impossibility in perception. This 
+time the sensory modality is touch; and the experienced contradiction 
+is one of enumeration. Aristotle's illusion is well known in whicha rod, 
+placed between the tips of crossed fingers, is felt as two rods. (Actually, 
+the greater oddity is that when the rod is held between uncrossed 
+fingers, it is felt as one even though it makes two contacts with the 
+hand.) I now replace the rod with a finger of the other hand: the same 
+finger is felt as one finger in one hand, as two fingers by the other hand. 
+So the same entity is apprehended as being of different pluralities, in 
+one sensory modality. 
+
+Let me introduce some notation to make it easier to elaborate. 
+Abbreviate "left-hand" as L and "right-hand" as R. Denote the first, 
+middle, ring, and little fingers, respectively, as 1, 2,3, and 4. Now cross 
+L2 and L3, and touch R3 between the tips of L2 and L3. One feels R3 as 
+one finger in the right hand, and as two fingers with the left hand. As 
+apparition, R3 gets a count of both one and two, apprehended in the 
+same sensory modality at the same time. Here is a phenomenon 
+authentically signified by a Necker-cube numeral which is both "1" and 
+"> 
+
+The crossed-finger device is obviously unwieldy. The possibilities 
+can, however, be enlarged somewhat, to make a further useful point. 
+For example, touch L1 and R3, while touching crossed L2 and L3 with 
+R4. Here we have a plurality, concatenated from one unary and one 
+paradoxical constituent, which numbers two and three at the same 
+time. 
+
+Then, we may cross L1 and L2 and touch R3, while crossing L3 
+and L4 and touching R4. Now we have a plurality which is two and 
+four at the same time. In terms of perceptual structure, it is analogous 
+to the numeral concatenated from two paradoxical images. As gestalt, 
+we concatenate in parallel. In the case of the fingers, we do not find a 
+plurality of three unless we appraise the perception analytically (block- 
+
+
+ing concatenation in parallel). 
+
+If one wants the inconsistently-valued numerals to be ostensive 
+numerals, then one can use finger-apparitions to constitute stroke- 
+numerals. Referring back to the first example, if we specify that the 
+stroke(s) is your R3-perception, or the apparition R3, then we obtaina 
+stroke which is single and double at the same time. Now the 
+inconsistently-valued numeral is identically its semantics: it authenti- 
+cally names the token-plurality which constitutes it. 
+
+I choose not to rely heavily on this device because it is so unwieldy. 
+The visual device is superior in that considerably longer constellations 
+are in the grasp of one person. Of course, if one chose to define fingers 
+as the tokens of ordinary counting, one might keep track of numbers 
+larger than ten by calling upon more than one person. The analogous 
+device could be posited with respect to the inconsistently-valued 
+numbers; but then postulates about intersubjectivity would have to be 
+stated formally. I do not wish to pursue this approach. 
+
+It is worth mentioning that if you hold a rod vertically in the near 
+center of your visual field, hold a mirror beyond it, and focus your gaze 
+on the rod, then you will see the rod reflected double in the mirror. This 
+is probably not an inconsistent perception, because the inconsistent 
+counts don't apply to the same apparition. (But if we add Kant's 
+postulate that a reflection exactly copies spacial relations among parts 
+of the object, then the illusion does bring us close to inconsistency.) The 
+illusion illustrates, though, that there is a rich domain of phenomena 
+which support mutable and inconsistent enumeration. 
+
+
+IV. Magnitude A rithmatic 
+
+
+I will end this stage of the work with an entirely different approach 
+to subjectively variable numerals and quantities. I use the horizontal- 
+vertical illusion, the same that appeared in "Ilusions," to form numer- 
+als. The numeral called "one" is now the standard horizontal-vertical 
+illusion with a measured ratio of one between the segments. The 
+numeral called "two" becomes a horizontal-vertical figure such that the 
+vertical has a measured ratio of two to the horizontal segment. Etc. If 
+"zero" is wanted, it consists of the horizontal segment only. 
+
+The meaning of each numeral is defined as the apparent, perceived 
+length-ratio of the vertical to the horizontal segment. Thus, for exam- 
+ple, the meaning of the numeral called "one" admits subjective varia- 
+tion above the measured magnitude. For brevity, I call this approach 
+magnitude arithmetic---although the important thing is how the mag- 
+nitudes are realized. 
+
+
+In all of the work with stroke-numerals, numbers were determina- 
+tions of plurality. An ostensive numeral was a numeral formed from a 
+quantity of simple tokens, which quantity was named by the expres- 
+sion. The issue in perception was the ability to make gestalt judgments 
+of assemblies of copies of a simple token. 
+
+The magnitude numerals establish a different situation. Magni- 
+tude numerals pertain to quantity as magnitude. They relate to plural- 
+ity only in the sense that in fact, measured vertical segments are integer 
+multiples of a unit length; and e.g. the apprehended meaning of "two" 
+will be a magnitude always between the apprehended meanings of 
+"one" and "three"---etc. 
+
+Once again we can distinguish a bicultural and an ostensive 
+hermeneutic. The bicultural hermeneutic involves judging meanings of 
+the numerals with estimates in terms of the conventional assignment of 
+fractions to lengths (as on a ruler). I find, for example, that the 
+magnitude numeral "two" may have a meaning which is almost 3. 
+(Larger numerals become completely unwieldly, of course. The point of 
+the device is to establish a principle, and I'm not required to provide for 
+large numerals.) 
+
+Then there must be an ostensive hermeneutic, a "magnitude- 
+ostensive" hermeneutic. Here the subjective variations of magnitude do 
+not receive number-names. They are apprehended (and retentionally 
+remembered) ostensively. 
+
+As I pointed out, above, the concept of equality with regard to 
+Necker-cube numerals is at present an open problem. To write an 
+equality between two Necker-cube displays of the same length is not 
+obviously cogent; in fat, it is distinctly implausible. For magnitude 
+numerals, however, it is entirely plausible to set numbers equal to 
+themselves---e.g. 
+
+
+The point is that it is highly likely that copies of a magnitude numeral 
+will be apprehended or appraised correlatively. This was by no means 
+guaranteed for copies of a Necker-cube numeral displayed in proximity. 
+
+
+Upon being convinced that these simplest of equations are mean- 
+ingful, we may stipulate a simple addition, "one" plus "one" equals 
+"two." (It was not possible to do anything this straightforward with 
+Necker-cube numerals.) Continuing, we may write a subtraction with 
+these numerals. There may now appear a complication in the rationale 
+of combination of these quantities. The "two" in the subtraction may 
+appear shorter than the "two" in the addition. A dependence of percep- 
+tions of these numbers on context may be involved. 
+
+We find, further, that "readings" of these equations according to 
+the bicutural hermeneutic yield propositions which are false when 
+referred back to school-arithmetic---e.g. the addition might be read as 
+
+
+I'/s + 1's = 24/s 
+
+
+So the effect of inventing a context in which a relationship called "one 
+plus one equals two" is appraised as 1!/5 + 1!/; = 24/5 (where there is a 
+palpable motivation for doing this) is to erode school-arithmetic. 
+
+Another approach to the same problem is to ask whether magni- 
+tude arithmetic authentically describes any palpable phenomenon. The 
+answer is that it does, but that the phenomenon in question is the 
+illusion, or rationale of the illusion. The significant phenomenon arises 
+from having both a measured ratio and a visually-apparent ratio, which 
+diverge. This is very different from claiming equations among non- 
+integral magnitudes without any motivation for doing so. Indeed, given 
+that the divergence is the phenomenon, the numerals are not really 
+ostensive in a straightforward way. 
+
+One way of illustrating the power of the phenomenon which 
+models magnitude arithmetic is to display ruler grids flush with the 
+segments of a horizontal-vertical figure. 
+
+
+What we find is that the illusion visually captures the ruler grids: it 
+withstands objective measurement and overcomes it. We have a non- 
+trivial, systematic divergence between two overlapping modalities for 
+appraising length-ratios---one modality being considered by this cul- 
+ture to be subjective, and the other not. 
+
+
+In "Derivation" I used multistable cube figures to give a simplified, 
+discrete analogue of the potentially continuous "vocabulary" in "Illu- 
+sions." I could try something similar for magnitude numerals. Take as 
+the magnitude unit a black bar representing an objective unit of twenty 
+20ths, concatenated with a row of five Necker cubes. Each cube seen in 
+the "up" orientation adds another 20th to the judged magnitude of the 
+subjective unit, so that the unit's subjective magnitude can range to 14. 
+When, however, we write the basic equality between units, it becomes 
+clear that this device does not function as it is meant to. In particular, 
+the claim of equality applied to the Necker-cube tails is not plausible, 
+because it is not guaranteed that these tails will be apprehended or 
+appraised correlatively. I have included this case as another illutration 
+of the sort of inventiveness which this work requires; and also to 
+illustrate how a device may be inadequate. 
+
+
+* * * 
+
+
+This completes the present stage of the work. Let me emphasize 
+that this manual does little more than define certain devices developed 
+in the summer of 1987. These devices can surely give rise to substantial 
+lessons and substantial applications. 
+
+There is my pending project in a priori neurocybernetics. Given 
+that mechanistic neurophysiology arrives at a mind-reading machine--- 
+called, in neurophysiological theory, an autocerebroscope---devise a 
+text for the human subject such that reading it will place the machine in 
+an impossible state (or short-circuit it). Such a problem is treated 
+facetiously in Raymond Smullyan's 5000 B.C.; and more seriously by 
+Gordon G. Globus' "Mind, Structure, and Contradiction," in Con- 
+sciousness and the Brain, ed. Gordon Globus et al. (New York, 1976), p. 
+283 in particular. But I imagine that my Necker-cube notations will be 
+the key to the first profound, extra-cultural solution. 
+
+In any case, this essay is only the beginning of an enterprise which 
+requires collateral studies and persistence far into the future to be 
+fulfilled. (I may say that I first envisioned the possibility of the present 
+results about twenty-five years ago.) 
+
+
+Background References 
+
+
+David Hilbert, three papers in From Frege to Godel, ed. Jean van Heijenoort 
+(1967) 
+
+David Hilbert, "Neubegrundung der Mathematik" (1922) 
+
+David Hilbert and P. Bernays, Grundlagen der Mathematik I (Berlin, 1968), 
+pp. 20-25 
+
+Plato, "Philebus" 
+
+Aristotle, Metaphysics, 1.6 
+
+Proclus, A Commentary on the First Book of Euclid's Elements, tr. Glenn 
+Morrow (Princeton, 1970), 54-55 
+
+Hans Freudenthal, Lincos: Design of a Language for Cosmic Intercourse 
+(Amsterdam, 1960), pp. 14-5, 17, 21, 45-6 
+
+Kurt Godel in The Philosophy of Bertrand Russell, ed. Paul Schilpp (1944), p. 
+137 
+
+W.V.O. Quine, Mathematical Logic (revised), pp. 121-2 
+
+Paul Benacerraf, "What numbers could not be," in Philosophy of Mathemat- 
+ics (2nd edition), ed. Paul Beneacerraf and Hilary Putnam (1983) 
+
+Leslie A. White, "The Locus of Mathematical Reality: An Anthropological 
+Footnote," in The World of Mathematics, ed. J.R. Newman, Vol. 4, pp. 
+2348-2364 
+
+Herman Weyl, Philosophy of Mathematics and Natural Science (Princeton, 
+1949), pp. 34-7, 55-66 
+
+Andrei Markov, Theory of Algorithms (Jerusalem, 1961) 
+
+G.T. Kneebone, Mathematical Logic and the Foundations of Mathematics 
+(London, 1963), p. 204ff. 
+
+Michael Resnik, Frege and the Philosophy of Mathematics (Ithaca, 1980), pp. 
+82, 99 
+
+Ludwig Wittgenstein, Wittgenstein's Lectures on the Foundations of Mathe- 
+matics (1976), p. 24; but p. 273 
+
+Ludwig Wittgenstein, Philosophical Grammer (Oxford, 1974), pp. 330-331 
+
+Steven M. Rosen in Physics and the Ultimate Significance of Time, ed. David 
+R. Griffin (1986), pp. 225-7 
+
+Edgar Rubin, "Visual Figures Apparently Incompatible with Geometry," 
+Acta Psychologica, Vol. 7 (1950), pp. 365-87 
+
+E.T. Rasmussen, "On Perspectoid Distances," Acta Pschologica, Vol. Il 
+(1955), pp. 297-302 
+
+N.C.A. da Costa, "On the Theory of Inconsistent Formal Systems," Notre 
+Dame Journal of Formal Logic, Vol. 15, pp. 497-510 
+
+FG. Asenjo and J. Tamburino, "Logic of Antinomies," Notre Dame Journal 
+of Formal Logic, Vol. 16, pp. 17-44 
+
+
+Richard Routley and R.K. Meyer, "Dialectical Logic, Classical Logic, and the 
+Consistency of the World," Studies in Soviet Thought, Vol. 16, pp. 1-25 
+
+Nicolas Goodman, "The Logic of Contradiction," Zeitschr. f. math. Logik und 
+Grundlagen d. Math., Vol. 27, pp. 119-126 
+
+Hristo Smolenov, "Paraconsistency, Paracompleteness and Intentional Con- 
+tradictions," in Epistemology and Philosophy of Science (1982) 
+
+J.B. Rosser and A.R. Turquette, Many-valued Logics (1952), pp. 1-9 
+
+Gordon G. Globus, "Mind, Structure, and Contradiction," in Conciousness 
+and the Brain, ed. Gordon Globus et al. (New York, 1976), p. 283 
+
+
diff --git a/extra/challenge_conceptual_artists.tex b/extra/challenge_conceptual_artists.tex
new file mode 100644
index 0000000..dd5f71c
--- /dev/null
+++ b/extra/challenge_conceptual_artists.tex
@@ -0,0 +1,3 @@
+\chapter[Challenge for Conceptual Artists (1990)][Challenge for Conceptual Artists (1990)]{}
+
+{\centering\includegraphics[width=4in]{img/challenge}\par}
diff --git a/extra/preface.tex b/extra/preface.tex
new file mode 100644
index 0000000..dd260f1
--- /dev/null
+++ b/extra/preface.tex
@@ -0,0 +1,4 @@
+\chapter{2024 Editorial Preface}
+
+Just a few comments for some things that may not entirely speak for themselves:
+
diff --git a/extra/the_art_connection.tex b/extra/the_art_connection.tex
new file mode 100644
index 0000000..90dc709
--- /dev/null
+++ b/extra/the_art_connection.tex
@@ -0,0 +1,409 @@
+\newcommand{\topnote}[1]{{\vskip 1em \raggedleft \parbox{2.5in}{\raggedleft \textit{#1}} \vskip 1em}}
+\newcommand{\Pb}{\plainbreak{2}}
+\newcommand{\Db}{\fancybreak{{$\diamond$}}}
+\newcommand{\arttitle}[1]{\textit{#1}}
+\newcommand{\sidenote}[1]{\vskip 1em {\raggedleft \parbox{2.75in}{\raggedright\itshape #1}\vskip 1em}}
+\newcommand{\asidenote}[1]{{\footnotesize{#1}}}
+
+\newcommand{\lilsection}[2]{{\Large \textbf{{#1}.} \hskip 2em \textit{#2} \hspace\fill}}
+\newcommand{\lilsubsection}[2]{\vskip 0.75em {\large\textbf{{#1}.} \hskip 2em \textit{#2} \hspace\fill} \vskip 0.75em}
+
+\chapter[\textsc{The Art Connection:  My endeavor's intersections with art (2005)}][\textsc{The Art Connection}]{The Art Connection: My endeavor's intersections with art}
+
+\topnote{26 October 2005}
+
+If you ask me to review the art world or the institution of art, it's not an interesting question if I am asked to speak like an art critic.  On the other hand, I have a great deal to say about the art institution if I am allowed to view it as a piece of a civilization, and to practice censorious sociological aesthetics.  Beginning in 1963 especially, I made appraisals of fine art as:  an elite institution which distinguishes modern European civilization from other civilizations. 
+
+But that critique of the status quo is less interesting to me right now than autobiography.
+
+I can legitimately say that taking art as a the thematic axis for a chronicle of my work is not fair to the work.  In the first place, from the beginning I was interested in the correlation of arts.  I quickly graduated to \enquote{interdisciplinary projects} such as concept art, which had art as a precedent, but stemmed from my iconoclastic philosophy of 1960, and had outgrown art.  To force my projects back into the art mold made it impossible to understand them.
+
+That bears directly on this recitation.  From the beginning, I was committed to correlation of the arts and of all culture.  I used compositional techniques from \enquote{new music} in visual works, and I used \enquote{abstract expressionist} drawings as pitch-time graphs, etc.  The early story cannot be fairly told unless unless I can move freely among the disciplines.  Concept art was assembled from works formerly called music and works formerly called mathematics.
+
+As I turned to new ethnic music, my musical practice became more separable---and I will keep references to it to a minimum.  But that is still artificial, because various artists I have known have been musicians, Conrad, Hennix, and my exchanges with them are not compartmentalized into disciplines.
+
+In the late Eighties, I revived concept art for tactical reasons.  Then I learned all over that one can't force the public to \enquote{get} concept art.  As long as its premise is unacceptable---outside the civilization---it will be received as \enquote{art junk.}
+
+My engagement with art had many distinctly different periods.  Because I am unprecedented or iconoclastic, that's what I want to review.  I'm going to be autobiographical.
+
+\Pb
+
+During 1957--60, I fell in with an (American) crowd whose trailblazers made a mystique out of newness.  (Definable as \enquote{baffling \'{a} la Dada.})  I had probably dismissed opera, dance, and even fiction as corny in my teens.  I went through a rapid evolution in which I talked myself out of the branches into which culture was divided such as painting, sculpture, poetry, music.  I felt invited, by Young and his entourage, to charge ahead into the bafflingly unprecedented and the iconoclastic.  By late 1962, I had rejected art altogether, along with entertainment and competitive pastimes such as bridge and checkers.  In fact, in spring 1962 I called for \textbf{the civilization in one mind.}  By early 1963, I had convinced myself that Marx's social critique and social utopianism had to be integrated into the perspective of culture; the relatively apolitical postures of Cage and Young were incomplete at best and collaborationist at worst. 
+
+\Pb
+
+I took positions at the beginning of my twenties which I may still admire.  But I periodically have retreated from my severe positions for tactical reasons.  As to the Marxism, I disagree with what I formerly talked myself into assenting to.  However, it had great plausibility to be a Leftist in the early and mid-Sixties---I can't regret it.
+
+At certain junctures in my life, it seemed that I was hopelessly isolated, that I was limited to mere survival while I piled up work that nobody knew or cared about.  I changed direction, sometimes retreating from my severe positions to do so. 
+
+In some cases I'm comfortable with these retreats. 
+
+In other cases, I feel that I was imposed on, that there was a social pressure which pigeon-holed me in ways that were unfair to my work.  However, there were redeeming features to the pigeon-holing.  My relationships with the impresarios, Maciunas and Harvey, were much richer than the usual artist-dealer relationship.  They were my support systems and confidants.   (Not that any \enquote{big} dealer ever recruited me or ever followed through after we had met.)
+
+\Pb
+ 
+\newcommand{\Ed}[1]{\textit{#1} --- } % "enumDate"
+\begin{enumerate}[label=\Alph*]
+\item \Ed{1958 or 1959 through 1960}  College and the first year out of college.  My modern art pilot projects, so to speak.  Correlation of the modern arts---formalism.
+
+\item \Ed{1961} Approximate date of my first in-person meeting with La Monte Young.  The period of \essaytitle{Philosophy Proper} and my \enquote{iconoclastic interdisciplinary projects,} particularly concept art.
+
+\item \Ed{1962}  Continuing the \enquote{iconoclastic interdisciplinary projects}; beginning of my anti-art crusade.  First printing of \booktitle{An Anthology}.
+
+\item \Ed{1963--4}  Affiliation with Marxism.  Adding a censorious sociological aesthetics as preface to the anti-art theory.  Accusation of political imperialism in the way musics were ranked in high culture.
+
+\item \Ed{1965--6}  Bearing down on a number of agendas simultaneously.  I reconfigured my critique of culture as a Communist program in culture.  The rock songs with Walter De Maria.  I begin aggressively to recast my \enquote{interdisciplinary projects.}
+
+\item \Ed{1967--8}  Return to anti-art utopianism.  Down with participation.  The absolutization of subjectivity.  Publication of two more \enquote{interdisciplinary works} from 1961--2 (in \journaltitle{Ikon} magazine).
+
+\item \Ed{1970--1984}  Inactivity in \enquote{visual art,} except for photographing the SAMO\scalebox{1.2}{\textcopyright} Graffiti and except for the \enquote{archeology} embodied in my 1982 Backworks show.
+
+\item \Ed{1985--1989}  I begin to revive concept art for tactical reasons.
+
+\item \Ed{1989--1993}  Having joined Emily Harvey's gallery, I become a career artist, defining what I do as concept art, modern art, and fantasy.
+
+\item \Ed{1994--1999}  My art career on hold.  Occasional pieces in group shows. 
+
+\item \Ed{2000--2005}  Development of my attack on \textbf{modern art} as a turn in European civilization which crystallized at the beginning of the twentieth century.  \enquote{Baffling without substance, cult of the lurid, impoverishment chic, making the collector pay to be scammed.}
+
+\item \Ed{2005}  Commence making abstract cinemas as an extension of abstract painting:  pilot projects.  Only a small venture so far.
+\end{enumerate}
+ 
+\Db
+ 
+\lilsection{A}{1958 or 1959 through 1960.  College and the first year out of college.  My modern art pilot projects, so to speak.  Correlation of the modern arts---formalism.}
+\lilsubsection{A}{Starting my sophomore year in college and lasting through my first year out of college.}
+
+Roughly, works surviving, or reconstructed, as \arttitle{Grey Planes}, \arttitle{Ugly Drawing}, \arttitle{Poems 1--4}, \arttitle{Aleatoric Painting}, \arttitle{Spirit-World Paintings} (originally a single cartoon with multiple titles).  \enquote{An abstract expressionist} drawings for music, namely November 1960 No. 2 \scalebox{0.9}{[survives].}
+
+Aleatoric Painting was oils on canvas board.  The others were drawings on bond paper or notebook paper, physically diminutive.
+
+Retrieval in this period depends on my memory, and remakes of various pieces from surviving fragments.  (a random number table for Grey Planes; the \enquote{English translation} of \arttitle{Poem 4}.)  A few works survive intact in the collections of others.  Ugly Drawing.  Musical scores from late 1960.
+
+These works, whch were carefuly thought through before I executed them, had three sources.  My sincere preference for abstract painting, especially Pollock.  My willingness to entertain modern poetry, which fell short of actual love for it.  My affiliation with the so-called new music.  I had arrived at college as a Bartok fan, was told that that was passe.  To be respected by the people whose respect I wanted, I had to get on the \enquote{new music} bandwagon.  I was very enthusiastic about Pollock, but knew it only from reproductions.  (I did not stand in the presence of a Pollock until I was 65.)
+
+{\vskip 1em \raggedleft \parbox{3in}{\itshape \textsc{Note}: The credibility of ethnic music for me, which means that I find the potential for a complete musical experience in musical languages learned by ear: \textit{I do not find an analogy in visual art.}  Genre painting, if that is what it is called, folky painting, is kitsch to me.  As a visual artist, the prospect of trying to \textit{be Southern} never interested me in the least.  I am a critic of modern art, but I am not even slightly an aesthetic conservative, a classicist.  I was never able to commit to visual art based on rendering.
+
+To me, iconic painting amounts to a handmade photograph with elements of cartooning.  I could not commit to it as a goal.  Maybe just cartooning.
+
+I respect rendering in ancient Egypt, but nowhere else.  I have no interest in duplicating that achievement which united their language and their mythology. It would be utterly anachronistic.
+
+In fact, I think that abstract cinema is a better medium than painting because painting is a handcraft medium.  (The problem with abstract cinema is that its practitioners did not know what they ought to be doing.  Also abstract cinema has needed the \textsc{dvd} to come into its own.)}\vskip 1em}
+
+      From the outset, say 1958, the division of art, culture, into categories which became separate professions meant nothing to me.  I simply disrregarded the compartmentation of culture, and assumed that I should pass freely among philosophy, exact science, linguistics, poetry, painting, music, whatever.  Using each to illuminate the others and transferring methods from one to the other.  My first \enquote{flat visual works} were precisely translations from serial music and so-called chance music.  My poems also.
+
+      While enrolled at Harvard, I began familiarizing myself with jazz recordings.  I had begun a conversion away from modern music.
+
+There had been an exchange about the latest jazz via Young's correspondence with Conrad.
+
+      The direction of everything I did was given by investigations in \enquote{epistemology} or whatever you call it which were unique to me.  I espoused positivism in high school and while enrolled in college, but quickly moved far beyond it.  Chomsky told me in 1960 that Philosophy Proper, Version 1 (it must have been Version 1) was worthless.
+
+ 
+\lilsubsection{B}{My attitude toward success, 1958--60.}
+
+      Those I fraternized with at Harvard who were trying their luck at the creative pursuits were not mercenary.  Wilder, who introduced us to the new music, did not become a career composer.  Conrad showed no interest in specializing in music.  Christian Wolff was satisfied to be a succes d'estime and did not intend to specialize in music.
+
+      At Harvard I proceeded as if art were pursued for its own sake as a branch of worthy human possibility.  I took it for granted that I should \enquote{be creative,} and I approached my encounters with $<$ the culture in play in my milieu $>$ $<$ as the early Marx assumes it will be in a pure Communist society>:  the exploration of human possibility for its own sake. 
+
+      I was aware that there was grumbing that Universal Edition was manipulatively promoting International Style.  Obviously Stockhausen, a product of apprenticeships with famous composition teachers, had to be a master at securing funding.  But I didn't take any of that as a signal of what to do.
+
+      All in my circle in e.g. 1959--61 assumed that the arts were a phase of worthy human possibility.  As far as I saw at the time, we were non-mercenary.  De Maria and Morris may have known that they would quickly begin to earn.  Young soon became a career composer but the affluence did not come until later.  Cale would become commercially successful by crossing to Pop.  I didn't see that.  I assumed that our efforts were labors of love.  We were knowledgeable and competitive but we were not earning.
+
+      The commercially successful artists I was aware of were i) Johns, ii) the Pop Artists, who would be a smash in the early Sixties.  The Pop Artists meant nothing to me as role models.
+
+      I left Harvard on probation for low grades to write my first philosophical monograph, Philosophy Proper Version 1.  The orientation for everything else I ever did.  The warrant for my iconoclasm toward cognitive claims for art, laws of art; the warrant for my transfer of \enquote{intederminacy} etc. into the exact sciences.
+
+\Pb
+ 
+
+      Probably 1960, I attended a John Tate lecture which began with algebraic varieties and passed to ringed space.  \textit{[Tate's specialty, class field theory]} It inspired me to write a one-page text as a parody.  I seem to have anticipated the proliferating diagrams of category theory, and to have used color and non-alphanumeric symbols.  Cf. Wette for color.
+
+Bafflingly out there, and I could not substantiate its cogency.  Like double-talk.
+
+Mathematicians would have said then, and now, that it was worthless, but that's too easy.  Composers were producing scores that amounted to notational double-talk.  Indeterminacy had become a feature of ordained structure in \enquote{serious music.}  It had to translate to the \enquote{exact sciences} (and the very label would become a misnomer).
+
+Increasingly I would come to see a legitimate role for indeterminate mathematics.  Mathematics waiting for its content to be focused. 
+
+\sidenote{[That actually happens in mathematics, Miles Tierney cleaned up topos theory. Then Hennix blurred it again.]}
+
+${}^\circ$ It's the idea that a text can be indeterminate, can be ahead of its substantiation.  comes back in full strength in \essaytitle{1966 Mathematical Studies}, then in the proposal for \journaltitle{Journal of Indeterminate Mathematical Investigations.}\editornote{Referred to in \essaytitle{Creep}}
+
+ 
+
+\lilsection{B}{\Ed{1961} Approximate date of my first in-person meeting with La Monte Young.  The period of Philosophy Proper and my \enquote{iconoclastic interdisciplinary projects,} particularly concept art.}
+
+ I met Young in person and met his entourage (as I thought of them).
+
+{ \vskip 1em
+\begin{enumerate}[label=\alph*)]
+\item Young challenged the \enquote{infinitely new and radical} claims of International Style and Cage.  his response:  word pieces, monotony (minimalism)
+
+\item Young absolutized the mystique of the new.  New was the definition of best.  If I had listened, I would have realized that his associates did not subscribe to this.
+\end{enumerate}
+\vskip 1em}
+
+ I interpreted all this as an exploration of human potential that had become engrossed in being enigmatic.  art was not an arena in which it was the greatest achievement to discover a new beauty.  The greatest achievement was to baffle and frustrate the audience.
+
+I was told that the Dadaists were the greatest precedents because they had been the most extreme, and the most extreme always won.  Young's entourage made sure that I knew about \booktitle{The Dada Painters and Poets, ed. Robert Motherwell} (1951).  Young's greatest coups had to do with dismantling the art process logically and with baffling the audience. (E.g. \opustitle{Compositions 1961}, identical pieces performed before they were composed.)
+
+\Pb
+
+Young and I rejected opera and dance as intrinsically corny.
+Young was also the first jazz musician with whom I truly fraternized socially.  December 1960.  That confirmed the conversion I had been undergoing. Young also confirmed my already estabished attraction to Hindustani music.  He was already knowledgeable about the vocal music, and introduced me to many phases of Hindustani music.  (Conrad had recordings of some prominent Carnatic musicians.)
+
+After being presented Young's word pieces in December 1960, I did numerous pieces directly prompted by them.
+
+\Pb
+
+\begin{itemize}
+\item My first formalist system (mathematical logic), January 1961, had what amounted to an abstract color drawing as the \enquote{system's base,} with plastic overlays a la Cage.
+\item Möbius strip piece.  \opustitle{Piece No. 2 (2/3/61)}.  \enquote{The instructions for this composition are on the other side of this strip}   
+
+the fact I did these pieces is now proved by surviving works and their references to no longer existing pieces and by correspondence.\editornote{They're also viewable in MOMA's online collection.}
+\item 1961.  Each point on this line is a composition.
+\item Music of the future. \asidenote{[composition ordained that some music a thousand years from now was the composition's content]}
+\item Rationale of concert held in my mind.
+\end{itemize}
+
+\Pb
+
+At some point in the first half of 1961, I completely lost interest in the \enquote{art professions,} music, painting, sculpture, poetry.  My works at this time were \enquote{interdisciplinary projects} or out-of-category projects---which were shaped by my philosophical perspective, which I continued to refine throughout the spring.  When I mailed \essaytitle{Philosophy Proper, Version 3} to Carnap in 1961, he didn't reply.  \asidenote{[if not \essaytitle{Version 1} in 1960]}
+
+At that time I saw a unity between intellectual innovations and artistic embodiment:  concept art.  But I would continuously re-examine my own premises for rationality.
+
+\Pb
+
+As I arrive at concept art, June 1961, I part company with the syllabus of art school---because everything I did stands or falls on the intellectual premises.  I'm not going to browbeat you about logic or philosophical linguistics.  You are the products of specializations which I happen to regard as derelict and irresponsible and retarded.
+
+\Pb
+
+People who are actually comfortable with this culture love the idea of the artist as a raging creature of instinct, who vomits emotion onto a canvas.  Who hops around like a jungle shaman.  As a matter of fact, post-war art music was not like that:  it was pseudo-scientific.  In fact, as everyone knows, many successful visual artists intimidated or baffled by being clinical and sterile.  However, their \enquote{blank wall} art was all posture---the equivalent of looking cool by wearing dark glasses---it made no intellectual contribution. 
+
+The larger public still loves the mystique of artists as a caste which channels irrationalism and superstition and instinct.  Of course the instinctual artist is a fraud, since what they turn out is dictated far more by an evolving stylistic consensus. 
+
+The assumption that infantile incompetence better qualifies one to \enquote{get feeling into the object} was a notion of European painting as classic painting disintegrated in the $19^{th}$ century.  Regression, insanity, etc.  Stockhausen's \opustitle{Aus den Sieben Tagen}.  Totally contemptible.
+
+[[I accept that an artist would be intuitive and would draw on alternative consciousness.  I don't like the caste role assigned to the artist as a raging savage.  Artists supposed to be intellectually vacant.  Not to mention the art theory produced by critics and curators:  pseudo-intellectual trash. (\#)
+
+\Pb
+
+back to me:  I began to do \enquote{interdisciplinary} projects: \\
+\begin{itemize}
+\item \essaytitle{Concept Art}, which became \essaytitle{1966 Mathematical Studies}, \essaytitle{The Apprehension of Plurality}, etc.
+\item \essaytitle{Mock Risk Games} \asidenote{[its publication history in \journaltitle{Ikon}]}\editornote{That is: published 1961 as \essaytitle{Exercise Awareness States}, disavowed and/or lost, re-created from memory as \essaytitle{Mock Risk Games}, then anthologized as \essaytitle{Exercise Awareness States} after all.}
+\item \essaytitle{Energy Cube Organism}, which ended as \essaytitle{Choice Chronology Project}
+\item \essaytitle{Perception-Dissociator} which became \essaytitle{Superseding} \asidenote{[the unfinished work of 1962 redone and published in 1968]}
+\end{itemize}
+
+\Pb
+\textbullet WSTNOKWGO\\
+another step in indeterminate mathematics (or specification of structure) \\
+---instructions for performance in letter to Young, late 1961 \\
+structural black box: \\
+---extra-terrestrial broadcasts that earth theoretically capable of intercepting. \\
+---actually existing verbalizations, texts, that are irrecoverably lost (aside from my destroyed work) \\
+
+\Pb
+
+When I met Maciunas in connection with appearing in his gallery in 1961, he made some favorable remarks about the Soviet Union.  in the same evening, Jackson MacLow was making caustic remarks along non-Communist Left lines.  (MacLow had been a rigorous anarchist-apacifist in the Forties.)  Dick Higgins seemed to be making gestures in the direction of the CPUSA.  I began to convert myself to Marxism.
+
+\Pb
+
+As of 1960 I was spoiled because so many people seemed to be admired for being iconoclastic and intransigent.  Cage and Ornette Coleman were just now famous.  In 1961 Young's entourage told me that Dada was the best because it had been the \enquote{most radical.}
+
+The artists whom I met through Young did not seem to be full-time artists.  Only De Maria was already a \enquote{power artist}; I didn't register it because in person he was affable and generous and because I had signed off on the art machine almost before I knew what it was. Morris, a student of Lippold at Hunter, explicitly condemned wanting to get rich and famous in a letter to Young. In other words, Morris nominally rejected the actual purpose of major public art, which is professional success. (The golden paintbrush.)
+
+\textbf{In the second half of 1961,} after half a year of active association with Young and his entourage, I began to imagine that I could gain recognition and earn my living by playing improvised music in clubs with Young.  I thought we could follow in the wake of Ornette Coleman.  (I did not register that Coleman got disgusted with club performance and quit.)
+
+\Pb
+
+That being said, I did nothing shrewd to pour myself into the mold of success.  I only cared about doing what had my enthusiasm.  I experienced the avant-garde as a crisis situation; I cared deeply about whether activity in the crisis situation was rational.  I was never shrewd about cultivating the exhibition system or giving the audience what it wanted.  I was happy to be naive and heedless.
+
+I would write \enquote{The Exploitation of Cultural Revolutionaries in Present Society} in fall 1961.  I would take cases like Galois, Abel, and Van Gogh, and make an issue of them.  The cultural judges couldn't get it right because cultural revolution upset the standards of excellence themselves.  At the same time, there was a legitimate role for arcane work which could not be asked to support itself commercially.  I experienced this as an appalling trap:  for the cultural revolutionary, life was a death march.  The only real solution would be a Communist utopia as envisioned by the early Marx. \textbf{I converted to Communism out of self-interest, because of a \textit{rare} personal problem, before I began to worry about the masses and to endorse institutional Communism.}
+
+\lilsection{C}{\Ed{1962} Continuing the \enquote{iconoclastic interdisciplinary projects}; beginning of my anti-art crusade.  \enquote{First printing} of \booktitle{An Anthology}.}
+
+Begins 1961 but takes shape in 1962.  I realized that I was involved in a knock-down drag-out competition---the neo-Dada avant-garde---and I began to wonder if it was not like being a college student.  Being forced to compete on demeaning terms, being immersed in a demeaning subject-matter.
+
+      I become more and more uncomfortable that artists were offering things that inrinsically weren't worth doing, whose only payoff was to leave the audience feeling baffled and frustrated.  They were competing for social approval on that basis.  They were making careers out of bluff, posture, hoodwinking the experts into giving approval for what nobody would do without the social context. 
+
+      Already at the time of the loft appearances, Feb. 1961, when I had the ability to garner respect from the people whose opinion I most valued\slash whose approval mattered to me\slash I was bothered by the game we were playing. \\
+
+the Harvard concert of 1961:  the reason it was \enquote{possibly Henry Flynt} and I didn't perform.
+
+{\itshape \footnotesize
+[Despite all the talk about new new new, the artistic fraternity could only deal with painting this, sculpture that.  their inovation consisted in brandishing postures at each other for social effect.  by definition they were intellectually vacant.
+
+{\scriptsize [it's all posture, a game being played inside an elite institution, the self-important overpriviledged cognosenti, posture game or intimadation game.  this cognoscenti does not have anything to say about philosophy, science, economics, government that I respect in the least.] }
+
+they were not seeking interdisciplinary or out-of-category innovation.  Thus, the works which I poured myself into developing went utterly over their heads.  drew a blank.
+
+Substantial innovation, e.g. concept art, went over their heads.]}
+
+I passed from the mystique of the avant-garde to the conviction that art had a flawed premise.  Cage had already said it, with a different rationale.  But he didn't mean it.  {\itshape \footnotesize [Later, Ben Vautier would deliberately use anti-art as a ploy, the collectors paid him to scam them.]}
+
+\Pb 
+ 
+All the while, the \enquote{first printing} of An Anthology, with Concept Art, was 1962.  Definitive printing 1963.
+
+\Pb 
+ 
+\begin{itemize}
+\item My New Concept of General Acognitive Culture was first presented publicly with the creep lecture, Harvard, May 15, 1962
+
+\item New York, June 5, 1962  acognitive culture talk.  Young was mainly responsible for assembling the audience.  Almost all were people who were or would be famous in the \enquote{creative world.}  Like the February 1963.  I was attacking art to an audience of fledgling famous artists.
+
+\item lecture \essaytitle{Pure Recreation}, Harvard, August 7, 1962
+
+\item publication in \journaltitle{decollage No. 3}
+\end{itemize}
+
+ \lilsection{D}{\Ed{1963--4}  Affiliation with Marxism.  Adding a censorious sociological aesthetics as preface to the anti-art theory.  Accusation of political imperialism in the way musics were ranked in high culture.}
+
+In 1962--63 I became deeply interested in artistic culture as a phase of modern European civilization:  for the purpose of a destructive critique, a hostile or censorious sociological aesthetic.  I spoke to audiences of culture professionals who would go on to become icons.  I told them to destroy their work and renounce art. 
+
+I converted myself to Marxism and began to require that everything I was doing be squared with a basic Marxist premise:  that market democracy was intolerable, it had to be opposed.  it was unacceptable to be at peace with it.  Mills' \booktitle{Listen Yankee}, apartheid in South Africa, what become the Vietnam war, the history of bourgeois-democratic intervention in primary producer nations.  It seemed that bourgeois democracy was incapable of supporting the bourgeois-democratic revolution and in fact saw every liberation as a threat.
+
+How could I espouse a closed and \asidenote{[doctrinaire]} ideology like Marxism?  At that time, as far as I knew, the progressive forces had become embodied in vast institutions.  One was honor-bound to support them, as opposed to remaining on the sidelines. \asidenote{[There is not a single person in this audence who would recommend remaining on the sidelines during the Second World War, who would be neutral between the Allies and the Axis.]}
+
+Marxism had several versions, and had already redefined itself several times, as the locus of insurgency moved.  (Not the Western proletariat, but the colonial subject, then the college student.)  It was the only tradition that touched the essential bases and that aligned itself institutionally with liberation struggles.  Namely:
+\begin{enumerate}[label=\roman*)]
+\item \textbf{[market democracy was wage slavery and ruthless imperialism]} \\
+economic individualism said it was the greatest good for the greatest number \\
+why wasn't it the greatest good for the least number?
+\item \textbf{a just social order could and should be built, based on economic collectivism.} \\{}
+[at that time, I assumed that Cuba was what its supporters said it was] \\
+\end{enumerate}
+I agreed with Marxism that there was no future for the reversion to the simple life.  (In the Seventies, there would in fact be rural comune experiments.  They all failed, of course.)  The new society would be planned, would cultivate and extend technology.  Its economic laws would be counter-intutive at first:  not entrepreneurs producing for profit.  The general population would have more disposible time.  It would not mean becoming indolent and taking longer vacations; it would mean the exploration of human possibility without regard for market signals.  The life-style I spent my life fighting to have, I wanted to make it possible for many people.
+
+[\textbf{I admit to self-deception out of desperation.}  my interests isolated me so profoundly that the only thing I had to look forward to was mere physical survival, maximizing the length of life.  I had to find a way to break out.
+
+\Pb
+
+Precepts (i)--(ii), found in Marx, became deeply interwoven in my sociological aesthetic. 
+
+my rejection of European art music for black and East Indian music.  \asidenote{[or for that matter, Rumanian gypsy music, the one music on the continent of Europe which had my enthusiasm]}
+
+the official aesthetic dispensation, e.g. the musicology departments, which put the best music at the back of the bus---I tied this to colonialism.  the victorious empire could impose a culture which was intimidating, which incorporated technology and massed forces, but which was otherwise poison.
+
+\Pb 
+
+{\itshape\footnotesize[$\diamond$ I continued musical activity right through the anti-art period, even though I unfortunately destroyed my earliest \enquote{new ethnic music} recordings.
+
+Why wasn't I attracted to Southern ethnic music when I lived there as a child?  As a child I did not have the confidence or \enquote{creativity} to imagine the world for myself.  I saw, accurately, that Southern ethnic music was a truncated cultural experience.  It was associated with a brawling life-style and with obscurantist Protestantism. 
+
+the idea that there could be a refined Southern music which would engage the whole person.  I single-handedly invented that.  The role model I needed at 15:  I had to become that role model, and it did not crystallize until the Seventies at the earliest.]}
+
+\Pb 
+ 
+Another key issue was competition among artists.  If art was gratuitous self-expression, how could one artist be competitively superior to another?  My peers tried to define it, and gave answers such as new or the most extreme or the best ideas.
+
+Meanwhile, I looked back on my sojurn in classical music as an episode of competiting in a demeaning contest.  The horror of classical music to the competitive student---how I felt is indicated by a July 26, 2005 news story:
+
+Bryan O'Lone was scheduled to play at a competitive recital in Carnegie Hall.  He prepared Chopin.  When he arrived, they told him he must play something he hadn't prepared.  When he played the Chopin anyway, the music teacher who sponsored the recital walked onto the stage and shut the keyboard cover on his fingers.  He sued her for a million dollars.  This quirky story, to me, is emblematic of the entire experience of being a competitive classical music student.  The National Music Camp and Bloody Friday.
+
+\Pb 
+
+The demonstrations against Stockhausen in 1964.
+
+\lilsection{E}{\Ed{1965--6} Finds me bearing down on a number of agendas simultaneously. I reconfigured my critique of culture as a Communist program in culture.  The rock songs with Walter De Maria. I begin aggressively to recast my \enquote{interdisciplinary projects.}}
+
+Communism was an actual constellation of nations and insurgent organizations, stemming from the early Soviet Union, claiming an inspiration from the doctrine of Marx. 
+
+I give my sociological aesthetic the guise of a cultural program for this Communism.  Perhaps the exercise sounds dully compliant and subservient, but it was nothing of the sort.  At that time we supported their social objectives, or thought we did, but I and Maciunas ripped the consensus Marxist cultural program to pieces.  Our argument was that the entire Marxist tradition had gotten culture all wrong.
+
+I reconfigured \essaytitle{From Culture to Brend} (1963--4?) to an avowedly Communist program for what culture should be.  E.g. protest rock.  I give my sociological aesthetic the guise of a cultural program for Communism.  This was iconoclastic because it was a break with the official Soviet view of culture unheard-of on the hard Left.  I was the first person who said that rock ought to be politically radical.
+
+The perspective on music was my most obvious innovation in Left cultural appraisal.  But we re-examined the entire range of Communist cultural policy---to some extent converging with early Soviet figures like Rodchenko, although I did not know of him at the time.
+
+I was the first person in the Marxist Left to say that black music, and rock, were not American decadence, but rather musical languages [created] collectively by the very people our movement claimed to speak for.  The idea that Beethoven was wealth which transcended class, about which the Party ought to instruct the proletariat (like a Metropolitan Museum art appreciation course) was one of Communism's immense mistakes. 
+
+I was also an innovator relative to rock practice, because rock had been configured as saleable entertainment, and had not been concerned to find a political interest against patriotism and the existing economic order. In 1966 I made my recommendations concrete by recording what was released forty years later as \opustitle{I Don't Wanna.}
+
+{\itshape\footnotesize [I was more comfortable with music because I admired traditional musical languages and the possibility of renewing or broadening them through innovation.  Visual art:  I find traditional languages corny (after ancient Egypt).  I adhered to the modernist principle that visual art needs to create the language as it goes, not to work with a long-practiced language.]\par}
+
+\Pb 
+
+Even during the years of ostensible Marxism, I never surrendered my intellectual gains.  Primary Study was published in 1964. \essaytitle{Mathematical Studies} came in 1966.  As I saw it:  I found the necessary elements of a complete perspective embedded in various hidebound traditions or disciplines.  What I wanted was uniformity, to bring each necessary element up to the level of the other.  So that you didn't have to switch ideologies on different days of the week. 
+
+At the same time, I was transferring concept art and the other \enquote{interdisciplinary projects} out of art altogether---and into exact science formats or foundations of science formats. \\
+1966 Math. Studies \\
+Perception-Dissociation of Physics \\
+
+What began as concept art etc. now became insane extremism in mathematical logic and foundations of science.
+
+\lilsection{F}{\Ed{1967--8}  Return to anti-art utopianism.  Down with participation.  The absolutization of subjectivity.  Publication of two more \enquote{interdisciplinary works} from 1961--2 (in \journaltitle{Ikon}).}
+
+I left the organized Left because I was now willing to bet my life that the Communist regimes could not become the utopia of the future via \enquote{regeneration.}  I decided that the Stalinist regimes were a new exploiting class.  If they aren't, it doesn't vindicate Trotsky; it means that the regime he headed and unconditionally defended was a crackpot version of capitalism which gained its credibility from an \enquote{oppressed people} ideology. 
+
+I became willing to forego \enquote{participation.}  I revived the utopianism of my cultural position.  What ought to be was so far from what was socially feasible that there was no bridge between them.  I chose to again emphasize what ought to be.  It was a drop-out stance which combined utopian social speculation with solitary self-realization.
+
+My interest in technical economics began at this point, probably beginning 1968.  I began to realize how badly I had been served by the Left's economic illiteracy, or rather, denial.  It became one of my projects to substantiate the utopia in technical economcs.
+
+I left the organized Left because I became convinced that Leninism could never deliver the utopia Marx had promised, it was as far from that utopia as capitalism was.  To characterize the Soviet Union is still an unsolved problem, but those who called it state capitalism were on the right track.  The prospect of being a cultural administrator for official socialism lost all interest for me. 
+
+I resumed the pure rejection of art.
+
+I returned to the [devotion to] subjectivity and individuality as the necessary  outcome of avant-garde positions.  I was prepared to forego the participatory dimension of culture.  In fact, I rarely enjoyed attending concerts.  I almost always preferred to listen alone to recordings.  At concerts, other members of the audience became the message.
+
+\journaltitle{Journal of Indeterminate Mathematical Investigations}, November 1967.  Another reconfiguration of the interdisciplinary projects of 1961.
+
+HF, \opustitle{Lecture on Brend}, was Film-Makers' Cinematheque, Feb. 14, 1968
+
+I was prepared to forego the use of culture as symbols of a social tendency.
+
+\Pb
+
+\lilsection{G}{\Ed{1968--1984} Inactivity in \enquote{visual art,} except for photographing the SAMO\scalebox{1.2}{©} Graffiti and except for the \enquote{archeology} embodied in my 1982 Backworks show.}
+\begin{itemize}
+\item \Ed{1975} publication of \booktitle{Blueprint}
+
+\item \Ed{1977} Hennix arrives in New York and we quarrel because Hennix chose to show the Algebras in an art gallery.  Hennix wants the art world to subsidize radical thought.  I object that it just demeans the work.
+
+\item \Ed{1979} I photographed SAMO©
+
+\item \Ed{1982} Backworks: \opustitle{Fragments of a Destroyed Oeuvre}
+\end{itemize}
+
+\lilsection{H.}{\Ed{1985--1989} I begin to revive concept art for tactical reasons.}
+
+	I am asked to remake a 1961 concept art piece for a show in 1985.  1987, Avenue B Gallery, I make new work.  revival of concept art to publicize the joint work of myself and Hennix, to ask the belated reward for our pioneering, to unfold concept art to a much greater extent than I had in the beginning. \opustitle{Stroke Numeral}, \opustitle{Tritone Monochord}.  I kept a concept art journal.  I sought to unfold concept art and showcase it.  There was an entire series of concept art signs and only a few got made.
+\begin{itemize}
+\item Self-Validating Falsehood
+\item Short Additive Semigroup
+\item One True Sentence
+\item Two Honest Texts
+\end{itemize}
+
+\lilsection{I}{\Ed{1989--1993} Having joined Emily Harvey's gallery, I become a career artist, defining what I do as \textbf{concept art, modern art, and fantasy.}}
+\begin{itemize}
+\item \Ed{1987} new concept art works.  $\diamond$ the art career, the attempt to become successful by asking to be rewarded for my pioneering.  I became an art soldier. $\diamond$
+
+\item First one-person show 1989
+
+\item  \opustitle{One True Sentence}.  the wallpaper.  a piercing self-reference anomaly.  muted yellows, kitchen wallpaper, a mild tone.  If somebody actually used it for the intended purpose, it would be like residing next to one of the civilization's intellectual fault-lines.
+
+\item \opustitle{Logically Impossible Space}
+
+\item last one-person show 1993
+
+\item SAMO\scalebox{1.2}{©} at the Lyon Biennale.
+\end{itemize}
+
+it was not a waste of time, but I did not enjoy a \enquote{success.}   I had to submit to being pre-classified as a Fluxus artist although that didn't make any sense.
+
+\Pb
+
+\lilsection{J}{\Ed{1994--1999}  I let my life as a career artist lapse.  Occasional pieces in group shows.}
+
+The joint appearance of Flynt and Hennix at MELA was 18 May 1997.
+
+\lilsection{K}{\Ed{2000--2005} Development of the attack on modern art as a turn in European civilization which crystallized at the beginning of the twentieth century.}
+
+\enquote{Baffling without substance, cult of the lurid, impoverishment chic, making the collector pay to be scammed.} 
+
+$<$ All the while, my practice embraces one or two tenets of modern art, such as the move from rendering to abstraction. $>$
+
+ 
+\lilsection{L}{\Ed{2005} Commence making abstract cinemas as an extension of abstract painting:  pilot projects.  Only a small venture so far.}
+