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diff --git a/extra/apprehension_of_pluraility.tex b/extra/apprehension_of_pluraility.tex new file mode 100644 index 0000000..c09bcbd --- /dev/null +++ b/extra/apprehension_of_pluraility.tex @@ -0,0 +1,1265 @@ +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 + + |