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Subject: fixes in appendix and images

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-The Apprehension of Plurality 
-
-
-Henry Flynt 
-
-
-(An instruction manual 
-for 1987 concept art) 
-
-
-I. 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. ‘III’. 
-
-To explain the use of stroke-numerals, and to provide a back- 
-ground for my innovations, some historical remarks about the philo- 
-sophy 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 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 “Philebus.” Aristotle’s Metaphysics, 1.6, says that 
-mathematical entities 
-
-
-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. 
-
-
-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 con- 
-sisted of truths of experience. 
-
-Following upon subsequent developments—the philosophical 
-climate at the end of the nineteenth century, and specifically mathema- 
-tical developments suchas non-Euclidian geometry—Hilbert proposed 
-that mathematics should be understood as a game played with mean- 
-ingless marks. So, for example, arithmetic concerns nothing but formal 
-terms—numerals—in a network of rules. Actually, what made arith- 
-metic 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 botha 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 ‘IIIIII’ 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 Freud- 
-enthal 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. 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 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 reliabil- 
-ity of concrete signs does not interact with any advanced mathematical 
-results. So this precondition can simply be transferred from the requi- 
-sites 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 mathe- 
-matics 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.* 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.** 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 “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 
-
-
-*Godel 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. 
-
-**One anthroplogist has written about “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). 
-
-
-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 copy- 
-ing 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 require- 
-ments 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 con- 
-crete 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. 
-(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 ina 
-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 1 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 
-1X 102+ 1X 10!+1 or binary | X 22+ 1 X 2!+ 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 refer- 
-ence to base 1. (Only much later in my investigations, when broad 
-scope becomes important, will I use positional notation.) So the fore- 
-going introduction to stroke-numerals has only the purpose of moti- 
-vating my novelties. And references to the academic canon are given 
-only for completeness. They cannot be norms for what I am “per- 
-mitted” to posit. 
-
-
-IT. 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. 
-
-
-Biel] Bie] bie/ bie) in] Bie bia) 
-
-
-STROKE-NUMERAL 
-
-
-STROKE 
-
-
-Q) 
-O VACANT 
-
-
-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 figure. I call the 
-stable apparition which one sees in a moment—which has imputed 
-perspective—the image.* 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 interac- 
-tions 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 stroke; the down orientation is called 
-“vacant,” and acts as the proofreaders’ symbol © , meaning “close up 
-space.” (So that “vacant” is not “even” an alphabetic space.) Now the 
-two images in question are signs. The transition from image to sign can 
-be analogized to the stipulation that circles of a certain size are (occu- 
-rances of) the letter “o."**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 
-
-
-*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. 
-** And—the shape, bar, positioned vertically relative to its reader, is the 
-symbol, Hilbert stroke. 
-
-
-the apparition is stable (no cube reverses orientation). I chose nine 
-Necker cubes as an extreme limit of what one can apprehend ina 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 permuta- 
-tions of signs as 
-
-
-a) ISCHOOCSI 
-by ISTSoC SH 
-c) IIIS DOCS 
-d) HINO CCTI 
-
-
-RNA ARRANN 
-OC) vuevvuvvuves 
-
-
-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 | 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 III. (d) amounts to IIIII. 
-
-As for (e), it has the alphabetic role of a blank. My initial interpre- 
-tation 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 recogni- 
-tion of particular numbers as gestalts; identity of numbers would uv 
-handled entirely by cross-tallying.) The Necker-cube numerals, how- 
-ever, 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. We have to consider different 
-hermeneutics for the numerals—and the ramifications of those herme- 
-neutics. 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 bicultural hermeneu- 
-tic, 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 “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 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 com- 
-
-
-petency 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.* 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. 
-
-Il 
-
-
-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 
-
-
-THLE 
-
-
-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 
-
-
-ae 
-
-
-as plurality-names without translating into English words. (Similarly 
-
-
-BR 
-
-
-in the case where I choose to read “blank” as “zero.”) Perhaps it is 
-necessary to spend considerable time with this new symbolism before 
-
-
-and 
-
-
-*Because this notion corresponds to a situation in which we are unable to 
-appraise image-rows as numerals, as gestalts. 
-
-
-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 sup- 
-posed 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 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. 
-I 2111) 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 hermeneu- 
-tic, 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. | 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. 
-
-
-The bicultural hermeneutic is applied, in effect, in my uninter- 
-preted calculus “Derivation,” which serves as a simplified analogue of 
-my early concept art piece “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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