From 9c0214b92acd6816b93ee4edca995257914630ae Mon Sep 17 00:00:00 2001 From: phoebe jenkins Date: Wed, 21 Aug 2024 23:48:18 -0400 Subject: confuse myself around some weird invisible characters --- extra/apprehension_of_plurality.tex | 1199 ----------------------------------- 1 file changed, 1199 deletions(-) delete mode 100644 extra/apprehension_of_plurality.tex (limited to 'extra/apprehension_of_plurality.tex') diff --git a/extra/apprehension_of_plurality.tex b/extra/apprehension_of_plurality.tex deleted file mode 100644 index eee4290..0000000 --- a/extra/apprehension_of_plurality.tex +++ /dev/null @@ -1,1199 +0,0 @@ -\chapter{The Apprehension of Plurality (1987)} - -% if we end up needing to add it: -% https://henryflynt.org/studies_sci/reqmath.html - -{\centering\itshape -(An instruction manual for 1987 concept art)\par} - -\section{Original Stroke-Numerals} - -Stroke-numerals were introduced in foundations of mathematics -by the German mathematician David Hilbert early in the twentieth -century. Instead of a given Arabic numeral such as `6', for example, one -has the expression consisting of six concatenated occurrences of the -stroke, e.g. `$||||||$'. - -To explain the use of stroke-numerals, and to provide a background -for my innovations, some historical remarks about the philosophy -of mathematics are necessary. Traditional mathematics had -treated positive whole-number arithmetic as if the positive whole -numbers (and geometrical figures also) were objective intangible -beings. Plato is usually named as the originator of this view. Actually, -there is a scholarly controversy over the degree to which Plato espoused -the doctrine of Forms---over whether Aristotle's \booktitle{Metaphysics} put -words in Plato's mouth---but that is not important for my purposes. -For an intimation of the objective intangible reality of mathematical -objects in Plato's own words, see the remarks about "divine" geometric -figures in Plato's \essaytitle{Philebus.} Aristotle's \booktitle{Metaphysics}, -1.6, says that mathematical entities -\begin{quotation} -are intermediate, differing from things perceived in being eternal and -unchanging, and differing from the Forms in that they exist in copies, -whereas each Form is unique. -\end{quotation} - -For early modern philosophers such as Hume and Mill, any such -"Platonic" view was not credible and could not be defended seriously. -Thus, attempts were made to explain number and arithmetic in ways -which did not require a realm of objective intangible beings. In fact, -Hume said that arithmetic consisted of tautologies; Mill that it -consisted of truths of experience. - -Following upon subsequent developments---the philosophical -climate at the end of the nineteenth century, and specifically -mathematical developments such as non-Euclidian geometry---Hilbert proposed -that mathematics should be understood as a game played with meaningless marks. -So, for example, arithmetic concerns nothing but formal -terms---numerals---in a network of rules. Actually, what made arithmetic -problematic for mathematicians was its infinitary character---as -expressed, for example, by the principle of complete induction. Thus, -the principal concern for Hilbert was that this formal game should not, -as a result of being infinitary, allow the deduction of both a proposition -and its negation, or of such a proposition as $0=1$. - -But at the same time (without delving into Hilbert's distinction -between mathematics and metamathematics), the stroke-numerals -replace the traditional answer to the question of what a number is. The -stroke-numeral '||||||' is a concrete semantics for the sign `6', and at the -same time can serve as a sign in place of `6'. The problem of positive -whole numbers as abstract beings is supposedly avoided by inventing -e.g. a number-sign, a numeral, for six, which is identically a concrete -semantics for six. Let me elaborate a little further. A string of six copies -of a token having no internal structure is used as the numeral `6', the -sign for six. Thus the numeral is itself a collection which supposedly -demands a count of six, thereby showing its meaning. Hans Freudenthal -calls this device an "ostensive numeral." - -So traditionally, there is a question as to what domain of beings -the propositions of arithmetic refer to, a question as to what the -referents of number-words are. \emph{Correlative to this, mathematicians' -intentions require numerous presuppositions about content, and -require extensive competancies---which the rationalizations for math- -ematics today are unable to acknowledge, much less to defend.} - -For example, if mathematics rests on concrete signs, as Hilbert -proposed, then, since concrete signs are objects of perception, the -reliability of mathematics would depend on the reliability of percep- -tion. Given the script numeral -{\plainbreak{1}\centering\includegraphics[width=1in]{img/oneortwo}\plainbreak{1}} -which is ambiguous between one and two, conventional mathematics -would have to guarantee the exclusion of any such ambiguity as this. -Yet foundations of mathematics excludes perception and the reliability -of concrete signs as topics---much as Plato divorced mathematics from -these topics. (Roughly, modern mathematicians would say that reliability -of concrete signs does not interact with any advanced mathematical -results. So this precondition can simply be transferred from the requisites -of cognition in general. But it would not be sincere for Hilbert to -give this answer. Moreover, my purpose is to investigate the possibility -of reconstructing our intuitions of quantity beyond the limits of the -present culture. In this connection, I need to activate the role of -perception of signs.) - -But the most characteristic repressed presuppositions of mathematics -run in the opposite, supra-terrestrial direction. Mathematicians' -intentions require a realm of abstract beings. Again, it is academically -taboo today to expose such presuppositions. -\footnote{G\"{o}del and Quine admit the need to assume the non-spatial, abstract -existence of classes. But they cannot elaborate this admission; they cannot -provide a supporting metaphysics.} But to recur to the -purpose of this investigation, concept art is about reconstructing our -intuitions of quantity beyond the limits of the present culture. This -project demands an account of these repressed presuppositions. To -compile such an account is a substantial task; I focus on it ina collateral -manuscript entitled "The Repressed Content-Requirements of Math- -ematics." To uncover the repressed presuppositions, a combination of -approaches is required.\footnote{One anthroplogist has written about \enquote{the locus of -mathematical reality}---but, being an academic, he merely reproduces a stock answer outside -his field (namely that the shape of mathematics is dictated by the physiology of -the brain).} I will not dwell further on the matter here--- -but a suitable sample of my results is the section "The Reality-Character -of Pure Whole Numbers and Euclidian Figures" in \emph{The Repressed Content-Requirements.} - -Returning to the original stroke-numerals, they were meant -(among other things) to be part of an attempt to explain arithmetic -without requiring numbers as abstract beings. They were meant as -signs, for numbers, which are identically their own concrete semantics. -Whether I think Hilbert succeeded in dispensing with abstract entities is -not the point here. I am interested in how far the exercise of positing -stroke-numerals as primitives can be elaborated. My notions of the -original stroke-numerals are adapted from Hilbert, Weyl, Markov, -Kneebone, and Freudenthal. For example, how does one test two -stroke-numerals for equality? To give the answer that "you count the -strokes, first in one numeral and then in the other," is not in the spirit of -the exercise. For if that is the answer, then that means that you have a -competency, "counting," which must remain a complete mystery to -foundations of mathematics. What one wants to say, rather, is that you -test equality of stroke-numerals by "cross-tallying": by e.g. deleting -strokes alternately from the two numerals and finding if there is a -remainder from one of the numerals. This is also the test of whether one -numeral precedes the other. So, now, given an adult mastery of quality -and abstraction, you can identify stroke-numerals without being able -to "count." - -In the same vein, you add two stroke-numerals by copying the -second to the right of the first. You subtract a shorter numeral from a -longer numeral by using the shorter numeral to tally deletion of strokes -from the longer numeral. You multiply two stroke-numerals by copying the second as many times as there are strokes in the first: that is, by -using the strokes of the first to tally the copying of the second numeral. - -To say that all this is superfluous, because we already acquired -these "skills" as a child, misses the point. The child does not face the -question, posed in the Western tradition, of whether we can avoid -positing whole numbers as abstract beings. To weaken the requirements -of arithmetic to the point that somebody with an adult mastery -of quality and abstraction can do feasible arithmetic "blindly"---i.e. -without being able to "count," and without being able to see number-names -('five', 'seven', etc.) in concrete pluralities---is a notable exercise, -one that correlates culturally with positivism and with the machine age. - -To reiterate, the stroke-numeral is meant to replace numbers as -abstract beings by providing number-signs which are their own concrete -semantics. Freudenthal said that we should communicate positive -whole numbers to alien species by broadcasting stroke-numerals to -them (in the form of time-series of beeps). Still, Freudenthal said that -the aliens would have to resemble us psychologically to get the point.\footnote{\booktitle{Lincos}, pp. 14--15.} - -When Hilbert first announced stroke-numerals, certain difficulties -were pointed out immediately. It is not feasible to write the -stroke-numerals for very large integers. (And yet, if it is feasible to write the -stroke-numeral for the integer n, then there is no apparent reason why -it would not also be feasible to write the stroke-numeral for n+1. So -stroke-numerals are closed under succession, and yet are contained in a -finite segment of the classical natural number series.) Moreover, large -feasible stroke-numerals, such as that for 10,001, are not surveyable. - -But this is not a study of metamathematical stroke-numerals. And -I do not wish to go into Hilbert's question of the consistency of -arithmetic as an infinitary game here; "The Repressed Content-Requirements" -will have more to say on the consistency question. The -purpose of this manual, and of the artworks which it accompanies, is to -establish apprehensions of plurality beyond the limits of traditional -civilizations (beyond the limits of Freudenthal's "us"). Moreover, these -apprehensions of plurality are meant to violate the repressed presuppo- -sitions of mathematics. I refer back to original stroke-numerals because -certain devices which I will use in assembling my novelties cannot be -supposed to be intuitively comprehensible---certainly not to the -traditionally-indoctrinated reader---and will more likely be understood -if I mention that they are adaptations of features of original stroke- -numerals. Let me mention one point right away. In our culture, we -usually see numerals as positional notations---e.g. 111 is decimal -$1\times 10^2+1\times 10^1+1$ or binary $1\times 2^2+1\times 2^1+1$. But stroke-numerals -are not a positional notation (except trivially for base 1). Likewise, my -novelties will not be positional notations; I will even nullify the -reference to base 1. (Only much later in my investigations, when broad -scope becomes important, will I use positional notation.) So the -foregoing introduction to stroke-numerals has only the purpose of -motivating my novelties. And references to the academic canon are given -only for completeness. They cannot be norms for what I am "permitted" to posit. - -\section{Simple Necker-Cube Numerals} - -In my stroke-numerals, the printed figure, instead of being a -stroke, is a Necker cube. (Refer to the attached reproduction, "Stroke- -Numeral.") A Necker cube is a two-dimensional representation of a -cubical frame, formed without foreshortening so that its perspective is -perceptually equivocal or multistable. The Necker cube can be seen as -flat, as slanting down from a central facet like a gem, etc.; but for the -moment I am exclusively concerned with the two easiest variants in -which it is seen as an ordinary cube, either projecting up toward the -front or down toward the front. - -{\center\includegraphics[width=4in]{img/neckercube}\plainbreak{2} -\includegraphics[width=2in]{img/neckerkey}\par} - -Since I will use perceptually multistable figures as notations, I -need a terminology for distinctions which do not arise relative to -conventional notation. I call the ink-shape on paper a \term{figure}. I call the -stable apparition which one sees in a moment---which has imputed -perspective---the \term{image}.\footnote{I may note, without wanting to be precious, that a bar does not count as a Hilbert stroke unless it is vertical relative to its reader.} -As you gaze at the figure, the image changes -from one orientation to the other, according to intricate subjective -circumstances. It changes spontaneously; also, you can change it -voluntarily. - -Strictly---and very importantly---it is the image which in this -context becomes the notation. Thus, I will work with notations which -are not ink-shapes and are not on a page. They arise as active interactions -of awareness with an "external" or "material" print-shape or -object. - -So far, then, we have images---partly subjective, pseudo-solid -shapes. I now stipulate an alphabetic role for the two orientations in -question. The up orientation is a \term{stroke}; the down orientation is called -"\term{vacant}," and acts as the proofreaders' symbol $\closure$, meaning "close up space." -(So that "vacant" is not "even" an alphabetic space.) Now the -two images in question are \term{signs}. The transition from image to sign can -be analogized to the stipulation that circles of a certain size are (occurances -of) the letter "o."\footnote{And---the shape, bar, positioned vertically relative to its reader, is the symbol, Hilbert stroke.} I may say that one sees the image; one -apprehends the image as sign. - -When a few additional explanations are made, then the signs -become plurality-names or "numerals." First, figures, Necker cubes, -are concatenated. When this is done, a display results. So the -stroke-numeral in the artwork, as an assembly of marks on a surface, is a -display of nine Necker cubes. An image-row occurs when one looks at -the display and sees nine subjectively oriented cubes, for just so long as -the apparition is stable (no cube reverses orientation). I chose nine -Necker cubes as an extreme limit of what one can apprehend in a fixed -field of vision. (So one must view the painting from several meters -away, at least.) The reader is encouraged to make shorter displays for -practice. Incidentally, if one printed a stroke-numeral so long that one -could only apprehend it serially, by shifting one's visual field, it would -be doubtful that it was well-defined. (Or it would incorporate a feature -which I do not provide for.) The universe of pluralities which can be -represented by these stroke-numerals is "small." My first goal is to -establish "subjectified" stroke-numerals at all. They don't need to be -large. - -The concatenated signs which you apprehend in a moment of -looking at the display are now apprehended or judged as a -plurality-name, a numeral. At the level where you apprehend signs (which, -remember, are alphabetized, partly subjective images, not figures), the -apparition is disambiguated. Thus I can explain this step of judging the -signs as plurality-names by using fixed notation. For nine Necker cubes -with the assigned syntactical role, you might apprehend such -permutations of signs as -\begin{enumerate}[label=\alpha*.] - \item $|\closure\closure||\closure\closure\closure|$ - \item $|\closure\closure\closure\closure\closure|||$ - \item $||||\closure\closure\closure\closure\closure$ - \item $||||\closure\closure\closure\closure|$ - \item $\closure\closure\closure\closure\closure\closure\closure\closure\closure$ -\end{enumerate} - -My Necker-cube stroke-numerals are something new; but (a)--(e) are -not---they are just a redundant version of Hilbert stroke-numerals -(which nullifies the base 1 reference as I promised). The "close up -space" signs function as stated; and the numeral concluded from the -expression corresponds to the number of strokes; i.e. the net result is -the Hilbert stroke-numeral having the presented number of strokes. So -(a) and (b) and (c) all amount to $|||$. (d) amounts to $|||||$. - -As for (e), it has the alphabetic role of a blank. My initial interpretation -of this blank is "no numeral present." Later I may interpret the -blank as "zero," so that every possibility will be a numeral. Let me -explain further. Even when I will interpret the blank as "zero." it will -not come about from having nine zeros mapped to one zero (like a sum -of zeros). (e) has nine occurrences of "close up space," making a blank. -There is always only one way of getting "blank." (A two-place display -allows two ways of getting "one" and one way of getting "two"; etc.) -The notation is not positional. It is immaterial whether one "focuses" -starting at the left or at the right. - -Relative to the heuristic numerals (a)--(e), you may judge the -intended numerals by counting strokes, using your naive competency -in counting. (It is also possible to use such numerals as (a)--(e) "blindly" -as explained earlier. This might mean that there would be no recognition -of particular numbers as gestalts; identity of numbers would be -handled entirely by cross-tallying.) The Necker-cube numerals, however, -pertain to a realm which is in flux because it is coupled to -subjectivity. My numerals provide plurality-names and models of that -realm. Thus, the issue of what you do when you conclude a numeral -from a sign in perception is not simple. \emph{We have to consider different -hermeneutics for the numerals---and the ramifications of those hermeneutics.} -Here we begin to get a perspective of the mutability which my -devices render manageable. - -For one thing, given a (stable) image-row, and thus a sign-row, you -can indeed use your naive arithmetical competency to count strokes, -and so conclude the appropriate numeral. This is \term{bicultural hermeneutic}, -because you are using the old numbers to read a new notation for -which they were not intended. We use the same traditional counting, of -course, to speak of the number of figures in a display. - -(This prescription of a hermeneutic is not entirely straightforward. -The competency called counting is required in traditional mathematics. -But such counting is already paradoxical "phenomenologically." I -explain this in the section called "Phenomenology of Counting" in \essaytitle{The -Repressed Content-Requirements}. As for the Necker-cube numerals, -the elements counted are not intended in a way which supports the -being of numbers as eternally self-identical. So the Necker-cube -numerals might resonate with the phenomenological paradoxes of -ordinary counting. The meaning of ordinary numbering, invoked in -this context, might begin to dissolve. But I mention this only to hint at -later elaborations. At this stage, it is proper to recall one's inculcated -school-counting; and to suppose that e.g. the number of figures in a -display is fixed in the ordinary way.) - -Then, there is the \term{ostensive hermeneutic}. Recall that I explained -Hilbert stroke-numerals as signs which identically provide a concrete -semantics for themselves; and as an attempt to do arithmetic without -assuming that one already possesses arithmetic in the form of competency -in counting, or of seeing number-names in pluralities. My -intention was to prepare the reader for features to be explained now. -On the other hand, at present we drop the notion of handling identity of -numerals by cross-tallying.\footnote{Because this notion corresponds to a situation in which we are unable to appraise image-rows as numerals, as gestalts.} -For the ostensive hermeneutic, it is crucial -that the display is short enough to be apprehended in a fixed field of -vision. - -With respect to short Hilbert numerals, I ask that when you see -e.g. - -$$||$$ - -marked ona wall, you grasp it asa sign for a definite plurality, without -mediation---without translating to the word "two." A similar intention -is involved in recognizing -$$\sout{||||}$$ -as a definite plurality, as a gestalt, without translating to "five." - -Now I ask you to apply this sort of hermeneutic to Necker-cube -stroke-numerals. I ask you to grasp the sign-row as a numeral, as a -gestalt. (Without using ordinary counting to call off the strokes.) Fora -two-place display, you are to take such images as - -\newcommand{\neckup}{\includegraphics[width=1in]{img/neckerup}} -\newcommand{\neckdown}{\includegraphics[width=1in]{img/neckerdown}} - -{\centering\neckup\neckdown\par} -and -{\centering\neckup\neckup\par} -as plurality-names without translating into English words. (Similarly - -{\centering\neckdown\neckdown\par} - -in the case where I choose to read "blank" as "zero.") Perhaps it is -necessary to spend considerable time with this new symbolism before -recognition is achieved. Again, I encourage the reader to make short -displays for practice. I have set a display of nine figures as the upper -limit for which it might be possible to learn to grasp every sign-row as a -numeral, as a gestalt. - -The circumstance that the apprehended numeral may be different -the next moment is not a mistake; the apprehended numeral is supposed -to be in flux. So when you see image-rows, you take them as -identical signs/semantics for the appearing pluralities. - -But who wants such numerals---where are there any phenomena -for them to count? For one thing, they count the very image-rows which -constitute them. The realm of these image-rows is a realm of subjective -flux: its plurality is authentically represented by my numerals, and -cannot be authentically represented by traditional arithmetic. - -A further remark which may be helpful is that here numerals arise -only visually. So far, my numerals have no phonic or audio equivalent. -(Whereas Freudenthal in effect posited an audio version of Hilbert -numerals, using beeps.) - -To repeat, by the "ostensive hermeneutic" I mean grasping the -sign-row, without mediation, as a numeral. But there is, as well, the -point that the Necker-cube numerals are \term{ostensive numerals}. That is, -the (momentary) numeral for six would in fact be an image-row with -just six occurrences of the image "upward cube." (Compare e.g. -$|||\closure\closure||\closure|$) The numeral is a collection in which only the "copies" of -"upward cube" contribute positively, so to speak; and these copies -demand a count of six (bicuturally). This feature needs to be clear, -because later I will introduce numerals for which it does not hold. - -Let me add another proviso concerning the ostensive hermeneutic -which will be important later. I will illustrate the feature in question -with an example which, however, is only an analogy. Referring to -Arabic decimal-positional numerals, you can appraise the number-name of -$$1001$$ -(comma omitted) immediately. But consider -$$786493015201483492147$$ -Here you cannot appraise the number-name without mediation. That -is, if you are asked to read the number aloud, you don't know whether -to begin with "seven" or "seventy-eight" or "seven hundred eighty-six." -Lacking commas, you have to group this expression from the right, in -triples, to find what to call it. An act of analysis is required. - -In the case of Necker-cube numerals and the ostensive hermeneutic, -I don't want you to see traditional number-names in the pluralities. -However, I ask you to grasp a sign-row as a numeral, as a gestalt. I now -add that the gestalt appraisal is definitive. I rule out appraising image-rows -analytically (by procedures analogous to mentally grouping an -Arabic number in triples). (I established a display of nine figures as the -upper limit to support this.) - -The need for this proviso will be obscure now. It prepares for a -later device in which, even for short displays, gestalt appraisal and -appraisal by analysis give different answers, either of which could be -made binding. - -\breatk - -The bicultural hermeneutic is applied, in effect, in my uninterpreted -calculus \textsc{"Derivation,"} which serves as a simplified analogue of -my early concept art piece \textsc{"Illusions."} (Refer to the reproductions on -the next four pages.) Strictly, though, "Derivation" does not concern a -Necker-cube stroke-numeral. The individual figures are not Necker -cubes, but "Wedberg cubes," formed with some foreshortening to make -one of the two orientations more likely to be seen than the other. What -is of interest is not apprehension of image-rows as numerals, but rather -appraisal of lengths of the image-rows via ordinary counting. As for the -lessons of this piece, a few simple observations are made in the piece's -instructions. But to pursue the topic of concept art as uninterpreted -calculi, and derive substantial lessons from it, will require an entire -further study---taking off from earlier writings on post-formalism and -uncanny calculi, and from my current writings collateral to this essay. - - -1987 Concept Art --- Henry Flynt -"DERIVATION" (August 1987 corrected version) - - -Purpose: To provide a simplified analogue of my 1961 concept art piece "'IIlusions'' which is -discrete and non-''warping.''* Thereby certain features of "'Illusions'' become more -clearly discernible. - - -Given a perceptually multistable figure, the ""Wedberg cube," which can be seen in two -orientations: as a cube; as a prism (trapezohedron.) - -Call what is seen at an instant an /mage. - -Nine figures are concatenated to form the display. - - -An element is an image of the display for as long as that image remains constant (Thus, -elements include: the image from the first instant of a viewing until the image first -changes; an image for the duration between two changes; the image from the last -change you see in a viewing until the end of the viewing.) - - -The /ength of an element equals the number of prisms seen. Lengths from O through nine -are possible. Two different elements can have the same length. Length of element X -is written /(X). - - -Elements are seen in temporal order in the lived time of the spectator. | refer to this order by -words with prefix 'T'. T-first; T-next; etc. - - -Element Y succeeds element X if and only if -i) (X) = KY), and Y is T-next after X of all elements with this length; or -ii) ¥ is the T-earliest element you ever see with length /(X) + 1. -Note that (ii) permits Y to be T-earlier than X: the relationship is rather artificial. - - -The initial element A is the T-first element. (/(A) may be greater than O; but it is likely to be O -because the figure is biased.) - - -The conclusion C is the T-earliest element of length 9 (exclusive of Ain the unlikely case in -which /(A) = 9). - - -A derivation is a series of elements in lived time which contains A and C and in which every -element but A succeeds some other element. - - -Discussion - -To believe that you have seen a derivation, you need to keep track that you see each -possible length, and to force yourself to see lengths which do not occur spontane- -ously. - - -You may know that you have seen a derivation, without being able to identify in memory the -particular successions. - - -"Derivation" is not isomorphic to "Illusions" for a number of reasons. ''Illusions" doesn't -require you to see individually every possible ratio between the T-first ratio and unity. -"Illusions" allows an element to succeed itself. The version of 'Derivation' pres- -ented here is a compromise between mimicking "'Illusions"' and avoiding a trivial or -cluttered structure. Any change such as allowing elements to succeed themselves -would require several definitions to be modified accordingly. - - -*In "Illusions," psychic coercion, which may be called "false seeing" or "warping," is -recommended to make yourself see the ration as unity. In ''Derivation," this warping is not -necessary; all that may be needed is that you see certain lengths willfully. - - -ABABA AAS - - -Concept Art Version of Mathematics System 3/26/6l(6/19/61) - -An "element"is the facing page (with the figure on it) so long -as the apparent, perceived, ratio of the length of the vertical -line to that of the horizontal line (the element's "associated -ratio") does not change. - -A "selection sequence" is asequence of elements of which the -first is the one having the greatest associated ratio, and -each of the others has the associated ratio next smallerthan -that of the preceding one. (To decrease the ratio, come to -see the vertical line as shorter, relative to the horizontal -line, one might try measuring the lines with a ruler to con- -vince oneself that the vertical one is not longer than the -other, and then trying to see the lines as equal in length; -constructing similar figures with a variety of real (measured) -ratios and practicing judging these ratios; and so forth.) -(Observe that the order of elements in a selection sequence -may not be the order in which one sees them.] - - -An elaboration of "Stroke-Numeral" should be mentioned here, -the piece called "an Impossible Constancy." (Refer to the facing page.) -As written, this piece presupposes the bicultural hermeneutic, and that -is probably the way it should be formulated. The point of this piece, -paradoxically, is that one seeks to annul the flux designed into the -apprehended numeral. Viewing of the Necker-cube numeral is placed -in the context of a lived experience which is interconfirmationally -weak: namely, memory of past moments within a dream (a single -dream). Presumably, appraisals of the numeral at different times could -come out the same because evidence to the contrary does not survive. -So inconstancy passes as constancy. Either hermeneutic can be -employed; but when I explained the hermetic hermeneutic, I encour- -aged you to follow the flux. Here you wouldn't do that---you wouldn't -stare at the display over a retentional interval. - - -As for the concept of equality with regard to Necker-cube numerals, -what can be said about it at this point? We have equality of numbers of -figures in displays, by ordinary counting. We have two hermeneutics -for identifying an apprehended numeral. In the course of expounding -them, I expounded equivalence of different permutations of "stroke" -and "vacant." Nevertheless, given that, for example, a display of two -figures can momentarily count the numeral apprehended from a dis- -play of three figures,* we are in unexplored territory. Cross-tallying, -suitable for judging equality of Hilbert numerals, seems maladapted to -Necker-cube numerals; in fact, I dismissed it when introducing the -ostensive hermeneutic. - -If the "impossible constancy" from the paragraph before last were -manageable, then one might consider restricting the ultimate definition -of equality to impossible constancies. That is, with respect to a single -display, if one wanted to investigate the intention of constancy (self- -equivalence of the apprehended numeral), one might start with the -impossible constancy. Appraisals of a given display become constant -(the numeral becomes self-equivalent) in the dream. Then two displays -which are copies might become constantly equivalent to each other, in -the dream. - -Such is a possibility. To elaborate the basics and give an incisive -notion of equality is really an open problem, though. Other avenues -might require additional devices such as the use of figures with distinc- -tions of appearance. - - -*that it is not assured that copies of a numeral will be apprehended or -appraised correlatively - - -1987 Concept Art --- Henry Flynt -Necker-Cube Stroke-Numeral: AN IMPOSSIBLE CONSTANCY - - -The purpose of this treatment is to say how a Necker-cube stroke numeral may be -judged (from the standpoint of private subjectivity) to have the same value at different -times; even though the conventional belief-system says that the value is likely to change -frequently. - - -This is accomplished by selecting a juncture in an available mode of illusion, namely -dreaming, which annuls any distinction between an objective circumstance, and the -circumstance which exists according to your subjective judgment. In the first instance, | -don't ask you to change your epistemology. Instead, to repeat, | select an available juncture -in lived experience at which the conventional epistomology gets collapsed. - - -You have to occupy yourself with the stroke-numeral to the point that you induce -yourself to dream about it. - -When, in apprehending a stroke-numeral, you "judge" the value of the numeral, the -number, this refers to the image you see and to the number-word which you may conclude -from the image. - -Suppose that in a single dreamed episode, you judge the value of the numeral at two -different moments. Suppose that at the second moment, you do not register any discre- -pancy between the value at the second moment and what the value was at the first -moment. Then you are permitted to disregard fallibility of memory, and to conclude that the -values were the same at both moments: because if your memory has changed the past, it -has done so tracelessly. A tracelessly-altered past may be accepted as the genuine past. - - -Refinements. The foregoing dream-construct may be "'lifted" to waking experience, as -per the lengthy explanations in ""An Epistemic Calculus."' Now you are asked to alter your -epistemology, selectively to suspend a norm of realism. - -Now that we are concerned with waking experience, a supporting refinement is -possible. Suppose | make an expectation (which may be unverbalized) that the value of the -numeral at a future moment will be the same that it is now. This expectation cannot be -proved false, if: the undetermined time-reference 'future moment" is applied only at those -later moments when the value is the same as at the moment the expectation was made. -(Any later moment when the value is not the same is set aside as not pertinent, or forgotten -at still later moments when the value is the same.) - - -As a postscript, there is another respect in which testing a fact requires trust in a -comparable fact. Suppose | make a verbalized expectation that the value of the numeral in -the future will be the same as at present. Then to test this expectation in the future depends -on my memory of my verbalization. My expectation cannot be belied unless | have a sound - -"memory that the number | verbalized in my expectation is different from the number | -conclude from the image now. - - -HT. Inconsistently-Valued Numerals - - -As the "Wedberg cube" illustrates, a cubical frame can be formed -in different ways, altering the likelihood that one or another image is -seen. With respect to the initial uses of the Necker-cube stroke-numeral -a figure is wanted which lends itself to the image of a cube projecting -up, or of a cube projecting down, with an approximately equal likeli- -hood for the two images---and which makes other images unlikely. -Now let a Necker cube be drawn large, with heavy line-segments, with -all segments equally long, with rhomboid front and back faces; and -display it below eye level. - - -As you look for the up and down orientations, there should be -moments when paradoxically you see the figure taking on both of these -mutually-exclusive orientations at once---yielding an apparition which -is a logical/ geometric impossibility. The sense-content in this case is -dizzying. - -That we have perceptions of the logically impossible when we -suffer illusions has been mentioned by academic authors. (Negative -afterimages of motion---the waterfall illusion.) Evidently, though, these -phenomenaare so distasteful to sciences which are still firmly Aristote- -lian that the relations of perception, habituation, language, and logic -manifested in these phenomena have never been assessed academically. -For me to treat the paradoxical image thoroughly here would be too -much of a digression from our subject, the apprehension of plurality. -However, a sketchy treatment of the features of the impossible image is -necessary here. - -To begin with, the paradoxical image of the Necker cube is not the -same phenomenon as the "impossible figures" shown in visual percep- -tion textbooks. The latter figures employ "puns" in perspective coding -such that parts of a figure are unambiguous, but the entire figure - - -cannot be grasped as a gestalt coherently. Then, the paradoxical Necker- -cube image is not an inconsistently oriented object (as the reader may -have noted). It is an apparitional depiction of an inconsistently oriented -object. But this is itself remarkable. For since a dually-oriented cube (in -Euclidean 3-space) is self-contradictory by geometric standards, a -picture of it amounts to a non-vacuous semantics for an inconsistency. -Another way of saying the same thing is that the paradoxically- -oriented image is real as an apparition. - -If one is serious about wanting a "logic of contradictions"---a logic -which admits inconsistencies, without a void semantics and without -entailing everything---then one will not attempt to get it by a contorted -weakening of received academic logic. One will start from a concrete -phenomenon which demands a logic of contradictions for its authentic -representation---and will let the contours of the phenomenon shape the -logic. - -In this connection, the paradoxically-oriented Necker-cube image -provides a lesson which I must explain here. Consider states or proper- -ties which are mutually exclusive, such as "married" and "bachelor." -Their conjunction---in English, the compound noun "married -bachelor"---is inconsistent.* On the other hand, the joint denial -"unmarried nonbachelor" is perfectly consistent and is satisfied by -nonpersons: a table is an unmarried nonbachelor. "Married" and -"bachelor" are mutually exclusive, but not exhaustive, properties. Only -when the domain of possibility, or intensional domain, is restricted to -persons, so "married" and "bachelor" become exhaustive properties. ** -Then, by classical logic, "married bachelor" and "unmarried nonbache- -lor" both have the same semantics: they are both inconsistent, and thus -vacuous, and thus indistinguishable. For exhaustive opposites, joint -affirmation and joint denial are identically vacuous. - -But the paradoxically-oriented Necker-cube image provides a -concrete phenomenon which combines mutually exclusive states---as -an apparition. We can ascertain whether a concrete case behaves as the -tenets of logic prescribe. As I have said, various images can be seen ina -Necker cube, including a flat image. Thus, the "up" and "down" cubes - - -*If I must show that it is academically permitted to posit notions such as -these, then let me mention that Jan Mycielski calls "triangular circle" incon- -sistent in The Journal of Symbolic logic, Vol. 46, p. 625. - -**] invoke this device so that I may proceed to the main point quickly. If it -is felt to be too artificial, perhaps it can be eliminated later. - - -are analogous to "married" and "bachelor" in that they are not exhaus- -tive of a domain unless the domain is produced by restriction. Then -"neither up nor down" is made inconsistent. (It is very helpful if you -haven't learned to see any stable images other than "up" and "down.") -The great lesson here is that given "both up and down" and "neither up -nor down" as inconsistent, their concrete reference is quite different. To -see a cube which manifests both orientations at the same time is one -paradoxical condition, which we know how to realize. To see a cube -which has no orientation (absence of "stroke" and absence of "vacant" -both) would be a different paradoxical condition, which we do not -know how to realize and which may not be realizable from the Necker- -cube figure. I don't claim that this is fully worked out; but it intimates a -violation of classical logic so important that I had to mention it. When -concept art reaches the level of reconstructing our inferential intuitions -as well as our quantitative intuitions, such anomalies as these will surely -be important. - -Referring back to the Necker cube of page 210, let us now intend it -as a stroke-numeral (display of one figure). Let me modify the previous -assignments and stipulate that "blank" means "zero," rather than "no -numeral present." (It is more convenient if every sign yields a numeral.) -When you see the paradoxical image, you are genuinely seeing "a" -numeral which is the simultaneous presence of two mutually exclusive -numerals "one" and "zero" ---because it is the simultaneous presence of -images which are mutually exclusive geometrically.*** - -It's not the same thing as - - -| - - ----because these are merely ambiguous scripts. In the Necker-cube case, -two determinate images which by logic preclude each other are present -at once; and as these images are different numerals, we have a genuine - - ----or as an alternative, - - -*For brevity, I may compress the three levels image, sign, numeral in -exposition. - - -inconsistently-valued numeral. - -This situation changes features of the Necker-cube numerals in -important ways, however. Lessons from above become crucial. We -transfer the ostensive hermeneutic to the new situation, and find an -inconsistent-valued numeral. But this is no longer an ostensive -numeral. We have a name which is one and zero simultaneously, but -this is because of the impossible shape (orientation) of the notation- -token. What we do not have is a collection of images of a single kind -(the stroke) which paradoxically requires a count of one and a count of -zero. "Stroke" is positively present, while "vacant" is positively present -in the same place. We will find that a display with two figures can be -inconsistent as zero and two; but it is not an ostensive numeral, because -the number of strokes present is two uniquely.* Here the numerals are -not identically their semantics: for the anomaly is not an anomaly of -counting. The ambiguous script numeral is a proper analogy in this -respect. To give an anomaly of counting which serves as a concrete -semantics for the inconsistently-valued numerals, I will turn to an -entirely different modality. - -From work with the paradoxical image, we learn that the Necker -cube allows some apprehensions which are not as commonas others--- -but which can be fostered by the way the figure is made and by -indicating what is to be seen. These rare apprehensions then become -intersubjectively determinate. If one observes Necker-cube displays for -a long time, one may well observe subtle, transient effects. For exam- -ple, you might see the "up" and "down" orientations at the same time, -but see one as dominating the other. In fact, there are too many such -effects and their interpersonal replicability is dubious. If we accepted -such effects as determining numerals, the interpersonal replicability of -the symbols would be eroded. Also the concrete definiteness of my -anomalous, paradoxical effects would be eroded. So I must stipulate -that every subtle transient effect which I do not acknowledge explicitly -is not definitive, and is unwanted, when the display is intended as a -symbolism. - -Let me continue the explanation, for the inconsistently-valued - - -*Referring to my "person-world analysis" and to the dichotomy of -Paradigm | and Paradigm 2 expounded in "Personhood III," this token which -is two mutually exclusive numerals because its shape is inconsistent is outside -that dichotomy: because established signs acquire a complication which is -more or less self-explanatory, but the meanings do not follow suit. - - -numerals, for displays of more than one figure. When the display -consists of two Necker cubes, and the paradoxical images are admitted, -what are the variations? In the first place, one figure might be seen (ina -moment) as a paradoxical image and the other as a unary image. -Actually, if it is important to obtain this variant, we can compel it, by -drawing one of the cubes in a way which hampers the double image. -(Thin lines, square front and back faces, the four side segments much -shorter than the front and back segments.) Then we stipulate that the -differently-formed cubes continue to have the same assigned interpre- -tation. - - -Reading the two-figure display, then, the paradoxical and unary -images concatenate so that the resulting numeral is in one case one and -two at the same time; and in the other case zero and one at the same -time. Of course, it is only ina moment that either of these two cases will -be realized. At other moments, one may have only unary images, so -that the numeral is noncontradictorily zero, one, or two as the case may -be. (If it is important to know that we can obtain a numeral which is -both one and two at the same time without using dissimilar figures, -then, of course, we can use a single figure and redefine the signs as "one" -and "two.") - -Now let us consider a display of two copies of the cube which lends -itself to the paradoxical image. Suppose that two paradoxical images -are seen; what is the numeral? Here is where I need the proviso which I -introduced earlier. Every sign-row is capable of being grasped as a -numeral, as a gestalt; and the appraisal of image-rows as numerals, -analytically, is ruled out. Let me explain how this proviso applies when -two paradoxical images are seen. - -Indeed, let me begin with the case of a pair of ambiguous - - -script-numerals: ] ] - - -When these numerals are formed as exact copies, and I appraise the -expression as a numeral, as a gestalt, then I see 11 or I see 22. ("Conca- -tenating in parallel") I do not see 21 or 12---although these variants are -possible to an analytical appraisal of the expression. In the gestalt, it is -unlikely to intend the left and right figures differently. This case is -helpful heuristically, because it provides a situation in which the percep- -tual modification is only a matter of emphasis (as opposed to imputa- -tion of depth). To this degree, the juncture at issue is externalized; and it -is easier to argue a particular outcome. On the other hand, the mechan- -ics differ essentially in the script case and the Necker-cube case. - -In the Necker-cube case, one sees both the left and the right image -determinately both ways at once. This case may be represented as - - -stroke stroke -vacant vacant - - -Analytically, then, four variants are available here, - - -stroke-stroke - -stroke-vacant -vacant-stroke -vacant-vacant - - -However, to complete the present explanation, only two of these -variants appear as gestalts, - - -stroke-stroke -vacant-vacant - - -I chose to rule out the three-valued numeral which would be obtained -by analytically inventorying the permutations of the signs afforded in -the perception. The two-valued numeral arising when the sign-row is -grasped as a gestalt is definitive. - -Let me summarize informally what I have established. Relative to -a two-figure display with paradoxical images admitted, we have a -numeral which is inconsistenly two and zero. We can also have a -numeral which is inconsistently one and zero, and a numeral which is -inconsistently two and one. (In fact, these variants occur in several -ways.) But we don't have a numeral which is inconsistently zero, one, -and two---even though such a variant is available in an analytical -appraisal---because such a numeral does not appear, in perception, asa -gestalt. - -Academic logic would never imagine that there is a situation -which demands just this configuration as its representation. Certain - - -definite positive inconsistencies are available in perception. Other defi- -nite positive inconsistencies, very near to them, are not available. Once -again, if one wants a vital "logic of contradictions," one has to develop -it as a representation of concrete phenomena; not as an unmotivated -contortion of received academic logics. - - -But what is the use of inconsistently-valued numerals? I shall now -provide the promised concrete semantics for them. This semantics -utilizes another experience of a logical impossibility in perception. This -time the sensory modality is touch; and the experienced contradiction -is one of enumeration. Aristotle's illusion is well known in whicha rod, -placed between the tips of crossed fingers, is felt as two rods. (Actually, -the greater oddity is that when the rod is held between uncrossed -fingers, it is felt as one even though it makes two contacts with the -hand.) I now replace the rod with a finger of the other hand: the same -finger is felt as one finger in one hand, as two fingers by the other hand. -So the same entity is apprehended as being of different pluralities, in -one sensory modality. - -Let me introduce some notation to make it easier to elaborate. -Abbreviate "left-hand" as L and "right-hand" as R. Denote the first, -middle, ring, and little fingers, respectively, as 1, 2,3, and 4. Now cross -L2 and L3, and touch R3 between the tips of L2 and L3. One feels R3 as -one finger in the right hand, and as two fingers with the left hand. As -apparition, R3 gets a count of both one and two, apprehended in the -same sensory modality at the same time. Here is a phenomenon -authentically signified by a Necker-cube numeral which is both "1" and -"> - -The crossed-finger device is obviously unwieldy. The possibilities -can, however, be enlarged somewhat, to make a further useful point. -For example, touch L1 and R3, while touching crossed L2 and L3 with -R4. Here we have a plurality, concatenated from one unary and one -paradoxical constituent, which numbers two and three at the same -time. - -Then, we may cross L1 and L2 and touch R3, while crossing L3 -and L4 and touching R4. Now we have a plurality which is two and -four at the same time. In terms of perceptual structure, it is analogous -to the numeral concatenated from two paradoxical images. As gestalt, -we concatenate in parallel. In the case of the fingers, we do not find a -plurality of three unless we appraise the perception analytically (block- - - -ing concatenation in parallel). - -If one wants the inconsistently-valued numerals to be ostensive -numerals, then one can use finger-apparitions to constitute stroke- -numerals. Referring back to the first example, if we specify that the -stroke(s) is your R3-perception, or the apparition R3, then we obtaina -stroke which is single and double at the same time. Now the -inconsistently-valued numeral is identically its semantics: it authenti- -cally names the token-plurality which constitutes it. - -I choose not to rely heavily on this device because it is so unwieldy. -The visual device is superior in that considerably longer constellations -are in the grasp of one person. Of course, if one chose to define fingers -as the tokens of ordinary counting, one might keep track of numbers -larger than ten by calling upon more than one person. The analogous -device could be posited with respect to the inconsistently-valued -numbers; but then postulates about intersubjectivity would have to be -stated formally. I do not wish to pursue this approach. - -It is worth mentioning that if you hold a rod vertically in the near -center of your visual field, hold a mirror beyond it, and focus your gaze -on the rod, then you will see the rod reflected double in the mirror. This -is probably not an inconsistent perception, because the inconsistent -counts don't apply to the same apparition. (But if we add Kant's -postulate that a reflection exactly copies spacial relations among parts -of the object, then the illusion does bring us close to inconsistency.) The -illusion illustrates, though, that there is a rich domain of phenomena -which support mutable and inconsistent enumeration. - - -IV. Magnitude A rithmatic - - -I will end this stage of the work with an entirely different approach -to subjectively variable numerals and quantities. I use the horizontal- -vertical illusion, the same that appeared in "Ilusions," to form numer- -als. The numeral called "one" is now the standard horizontal-vertical -illusion with a measured ratio of one between the segments. The -numeral called "two" becomes a horizontal-vertical figure such that the -vertical has a measured ratio of two to the horizontal segment. Etc. If -"zero" is wanted, it consists of the horizontal segment only. - -The meaning of each numeral is defined as the apparent, perceived -length-ratio of the vertical to the horizontal segment. Thus, for exam- -ple, the meaning of the numeral called "one" admits subjective varia- -tion above the measured magnitude. For brevity, I call this approach -magnitude arithmetic---although the important thing is how the mag- -nitudes are realized. - - -In all of the work with stroke-numerals, numbers were determina- -tions of plurality. An ostensive numeral was a numeral formed from a -quantity of simple tokens, which quantity was named by the expres- -sion. The issue in perception was the ability to make gestalt judgments -of assemblies of copies of a simple token. - -The magnitude numerals establish a different situation. Magni- -tude numerals pertain to quantity as magnitude. They relate to plural- -ity only in the sense that in fact, measured vertical segments are integer -multiples of a unit length; and e.g. the apprehended meaning of "two" -will be a magnitude always between the apprehended meanings of -"one" and "three"---etc. - -Once again we can distinguish a bicultural and an ostensive -hermeneutic. The bicultural hermeneutic involves judging meanings of -the numerals with estimates in terms of the conventional assignment of -fractions to lengths (as on a ruler). I find, for example, that the -magnitude numeral "two" may have a meaning which is almost 3. -(Larger numerals become completely unwieldly, of course. The point of -the device is to establish a principle, and I'm not required to provide for -large numerals.) - -Then there must be an ostensive hermeneutic, a "magnitude- -ostensive" hermeneutic. Here the subjective variations of magnitude do -not receive number-names. They are apprehended (and retentionally -remembered) ostensively. - -As I pointed out, above, the concept of equality with regard to -Necker-cube numerals is at present an open problem. To write an -equality between two Necker-cube displays of the same length is not -obviously cogent; in fat, it is distinctly implausible. For magnitude -numerals, however, it is entirely plausible to set numbers equal to -themselves---e.g. - - -The point is that it is highly likely that copies of a magnitude numeral -will be apprehended or appraised correlatively. This was by no means -guaranteed for copies of a Necker-cube numeral displayed in proximity. - - -Upon being convinced that these simplest of equations are mean- -ingful, we may stipulate a simple addition, "one" plus "one" equals -"two." (It was not possible to do anything this straightforward with -Necker-cube numerals.) Continuing, we may write a subtraction with -these numerals. There may now appear a complication in the rationale -of combination of these quantities. The "two" in the subtraction may -appear shorter than the "two" in the addition. A dependence of percep- -tions of these numbers on context may be involved. - -We find, further, that "readings" of these equations according to -the bicutural hermeneutic yield propositions which are false when -referred back to school-arithmetic---e.g. the addition might be read as - - -I'/s + 1's = 24/s - - -So the effect of inventing a context in which a relationship called "one -plus one equals two" is appraised as 1!/5 + 1!/; = 24/5 (where there is a -palpable motivation for doing this) is to erode school-arithmetic. - -Another approach to the same problem is to ask whether magni- -tude arithmetic authentically describes any palpable phenomenon. The -answer is that it does, but that the phenomenon in question is the -illusion, or rationale of the illusion. The significant phenomenon arises -from having both a measured ratio and a visually-apparent ratio, which -diverge. This is very different from claiming equations among non- -integral magnitudes without any motivation for doing so. Indeed, given -that the divergence is the phenomenon, the numerals are not really -ostensive in a straightforward way. - -One way of illustrating the power of the phenomenon which -models magnitude arithmetic is to display ruler grids flush with the -segments of a horizontal-vertical figure. - - -What we find is that the illusion visually captures the ruler grids: it -withstands objective measurement and overcomes it. We have a non- -trivial, systematic divergence between two overlapping modalities for -appraising length-ratios---one modality being considered by this cul- -ture to be subjective, and the other not. - - -In "Derivation" I used multistable cube figures to give a simplified, -discrete analogue of the potentially continuous "vocabulary" in "Illu- -sions." I could try something similar for magnitude numerals. Take as -the magnitude unit a black bar representing an objective unit of twenty -20ths, concatenated with a row of five Necker cubes. Each cube seen in -the "up" orientation adds another 20th to the judged magnitude of the -subjective unit, so that the unit's subjective magnitude can range to 14. -When, however, we write the basic equality between units, it becomes -clear that this device does not function as it is meant to. In particular, -the claim of equality applied to the Necker-cube tails is not plausible, -because it is not guaranteed that these tails will be apprehended or -appraised correlatively. I have included this case as another illutration -of the sort of inventiveness which this work requires; and also to -illustrate how a device may be inadequate. - - -* * * - - -This completes the present stage of the work. Let me emphasize -that this manual does little more than define certain devices developed -in the summer of 1987. These devices can surely give rise to substantial -lessons and substantial applications. - -There is my pending project in a priori neurocybernetics. Given -that mechanistic neurophysiology arrives at a mind-reading machine--- -called, in neurophysiological theory, an autocerebroscope---devise a -text for the human subject such that reading it will place the machine in -an impossible state (or short-circuit it). Such a problem is treated -facetiously in Raymond Smullyan's 5000 B.C.; and more seriously by -Gordon G. Globus' "Mind, Structure, and Contradiction," in Con- -sciousness and the Brain, ed. Gordon Globus et al. (New York, 1976), p. -283 in particular. But I imagine that my Necker-cube notations will be -the key to the first profound, extra-cultural solution. - -In any case, this essay is only the beginning of an enterprise which -requires collateral studies and persistence far into the future to be -fulfilled. (I may say that I first envisioned the possibility of the present -results about twenty-five years ago.) - - -Background References - - -David Hilbert, three papers in From Frege to Godel, ed. Jean van Heijenoort -(1967) - -David Hilbert, "Neubegrundung der Mathematik" (1922) - -David Hilbert and P. Bernays, Grundlagen der Mathematik I (Berlin, 1968), -pp. 20-25 - -Plato, "Philebus" - -Aristotle, Metaphysics, 1.6 - -Proclus, A Commentary on the First Book of Euclid's Elements, tr. Glenn -Morrow (Princeton, 1970), 54-55 - -Hans Freudenthal, Lincos: Design of a Language for Cosmic Intercourse -(Amsterdam, 1960), pp. 14-5, 17, 21, 45-6 - -Kurt Godel in The Philosophy of Bertrand Russell, ed. Paul Schilpp (1944), p. -137 - -W.V.O. Quine, Mathematical Logic (revised), pp. 121-2 - -Paul Benacerraf, "What numbers could not be," in Philosophy of Mathemat- -ics (2nd edition), ed. Paul Beneacerraf and Hilary Putnam (1983) - -Leslie A. White, "The Locus of Mathematical Reality: An Anthropological -Footnote," in The World of Mathematics, ed. J.R. Newman, Vol. 4, pp. -2348-2364 - -Herman Weyl, Philosophy of Mathematics and Natural Science (Princeton, -1949), pp. 34-7, 55-66 - -Andrei Markov, Theory of Algorithms (Jerusalem, 1961) - -G.T. Kneebone, Mathematical Logic and the Foundations of Mathematics -(London, 1963), p. 204ff. - -Michael Resnik, Frege and the Philosophy of Mathematics (Ithaca, 1980), pp. -82, 99 - -Ludwig Wittgenstein, Wittgenstein's Lectures on the Foundations of Mathe- -matics (1976), p. 24; but p. 273 - -Ludwig Wittgenstein, Philosophical Grammer (Oxford, 1974), pp. 330-331 - -Steven M. Rosen in Physics and the Ultimate Significance of Time, ed. David -R. Griffin (1986), pp. 225-7 - -Edgar Rubin, "Visual Figures Apparently Incompatible with Geometry," -Acta Psychologica, Vol. 7 (1950), pp. 365-87 - -E.T. Rasmussen, "On Perspectoid Distances," Acta Pschologica, Vol. Il -(1955), pp. 297-302 - -N.C.A. da Costa, "On the Theory of Inconsistent Formal Systems," Notre -Dame Journal of Formal Logic, Vol. 15, pp. 497-510 - -FG. Asenjo and J. Tamburino, "Logic of Antinomies," Notre Dame Journal -of Formal Logic, Vol. 16, pp. 17-44 - - -Richard Routley and R.K. Meyer, "Dialectical Logic, Classical Logic, and the -Consistency of the World," Studies in Soviet Thought, Vol. 16, pp. 1-25 - -Nicolas Goodman, "The Logic of Contradiction," Zeitschr. f. math. Logik und -Grundlagen d. Math., Vol. 27, pp. 119-126 - -Hristo Smolenov, "Paraconsistency, Paracompleteness and Intentional Con- -tradictions," in Epistemology and Philosophy of Science (1982) - -J.B. Rosser and A.R. Turquette, Many-valued Logics (1952), pp. 1-9 - -Gordon G. Globus, "Mind, Structure, and Contradiction," in Conciousness -and the Brain, ed. Gordon Globus et al. (New York, 1976), p. 283 - - -- cgit v1.2.3