From 0bfdf11360f1b5e3d93a4527acfb33e71640e216 Mon Sep 17 00:00:00 2001
From: grr <grr@lo2.org>
Date: Mon, 24 Jun 2024 19:25:03 -0400
Subject: throw in the extra tex files for essays im not sure i end up using

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+\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 
+
+
-- 
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