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| author | grr <grr@lo2.org> | 2024-05-22 21:40:19 -0400 |
|---|---|---|
| committer | grr <grr@lo2.org> | 2024-05-22 21:40:19 -0400 |
| commit | b805073879bb9de0dda93b84518dcc676026811e (patch) | |
| tree | 4f2d51cf3ce24aef73c1152bf13126d63618fd89 | |
| parent | 46b67e2911488da5ccd6e4f20ac6b0547ff33acb (diff) | |
| download | blueprint-b805073879bb9de0dda93b84518dcc676026811e.tar.gz | |
fixes in appendix and images
| -rw-r--r-- | extra/anthology_non_philosophical.tex | 17 | ||||
| -rw-r--r-- | extra/apprehension_of_pluraility.tex | 1265 | ||||
| -rw-r--r-- | extra/philosophy_of_concept_art.tex | 296 | ||||
| -rw-r--r-- | extra/structure_art_pure_mathematics.tex | 4 | ||||
| -rw-r--r-- | img/ManipulatingReality.pdf | bin | 0 -> 1939608 bytes | |||
| -rw-r--r-- | img/illusions.png | bin | 8184 -> 14328 bytes | |||
| -rw-r--r-- | img/strokes.gif | bin | 0 -> 2090 bytes | |||
| -rw-r--r-- | img/strokes.png | bin | 1621 -> 11289 bytes | |||
| -rw-r--r-- | img/structure_art.png | bin | 167164 -> 156501 bytes | |||
| -rw-r--r-- | img/terry_flynt_name.png | bin | 3895 -> 16067 bytes |
10 files changed, 154 insertions, 1428 deletions
diff --git a/extra/anthology_non_philosophical.tex b/extra/anthology_non_philosophical.tex index b48f385..9bd29c7 100644 --- a/extra/anthology_non_philosophical.tex +++ b/extra/anthology_non_philosophical.tex @@ -1,17 +1,7 @@ \chapter{Anthology of Non-Philosophical Cultural Works (1961)} - - - - - - - - - \section{Introduction} - I cannot include here my essays which discuss at length the purposes of lingart, audart, strange culture description, concept art, and so forth, tell what they aro good for. However, @@ -29,6 +19,8 @@ or having acadmic certification, or having a standard label (such as 'music' or 'mathematics') to tell him. These works stand by themselves. +\clearpage + \section{Lingart: Poem 1 (early 1960 / August 1961)} {\vskip 2em \centering [Instructions \vskip 2em} @@ -93,6 +85,7 @@ Empty cream quiets moons.\\ Monsters prayed in screaming vinegar.\\ Bitter moons were carbon,\\ +\clearpage \section{Audart Composition (May 1961)} To experience this composition, one must be alone in a @@ -114,6 +107,7 @@ array of bright lights before starting, the way to appreciate it is to seize on images as soon as they appear and concentrate to bring them cut. If done properly this should be a very strange experience, +\clearpage \section{Audart: A way of enjoying a Non-Controlled Acoustical Environment (July 1961)} Let me distinguish what I will say, for want of better terms, are "highly select sounds", such as popular music and @@ -146,6 +140,7 @@ head". The music is made from the environmental sounds with imaginings, and seems to be part of the non-mental environmental sounds, come from the non-mental environment. +\clearpage \section{Strange Culture Description: Instructions Accompanying Two Identity Structure Standards (April--May 1961)} These standards are for determining the type of identity @@ -188,10 +183,12 @@ a gold leaf by nuclear bombardment, it is \uline{ruined}. It is advised that persons who do not know all this or are careless or destructive not be allowed to handle the standards. +\clearpage \section{Concept Art: Work Such That No One Knows What's Going On (July 1961)} [Ono just has to guess whether this work exists and if it does what it is like.] +\clearpage \section{Concept Art: Innperseqs (May--July 1961)} A "halpoint" iff whatever is at any point in space, in the fading rainbow halo which appears to surround a small bright light when one looks at it through glasses fogged by having been breathed on, for as long as the point is in the halo. diff --git a/extra/apprehension_of_pluraility.tex b/extra/apprehension_of_pluraility.tex deleted file mode 100644 index c09bcbd..0000000 --- a/extra/apprehension_of_pluraility.tex +++ /dev/null @@ -1,1265 +0,0 @@ -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 - - diff --git a/extra/philosophy_of_concept_art.tex b/extra/philosophy_of_concept_art.tex index f2431ef..1c8c71b 100644 --- a/extra/philosophy_of_concept_art.tex +++ b/extra/philosophy_of_concept_art.tex @@ -1,7 +1,7 @@ \newcommand{\action}[1]{[\textit{#1}]} -\newcommand{\speaker}[1]{\vskip 0.2em \textsc{#1}: } -\newcommand{\speakermod}[2]{\vskip 0.2em \textsc{#1} \textit{(#2)}: } +\newcommand{\spk}[1]{\vskip 0.4em \textsc{#1}: } +\newcommand{\spkmod}[2]{\vskip 0.4em \textsc{#1} \textit{(#2)}: } \chapter{Philosophy of Concept Art (1987)} @@ -11,7 +11,7 @@ by Christer Hennix \\ Dec. 6, 1987 \par } -\speaker{FLYNT} I'm going to give a summary of how I originated Concept Art +\spk{Flynt} I'm going to give a summary of how I originated Concept Art in order to bring it up to the point where it's understandable why I speak of you (Catherine Christer Hennix) as my only successor in the genre. Summarizing briefly, I see two things coming together. One of them @@ -30,16 +30,16 @@ intellectual dimension in Stockhausen---Stockhausen's theoretical journal \journaltitle{die Reihe}---the impression that they were doing science actually---for example Stockhausen had a long essay on how the duration of the notes had to correspond to the twelve pitches of the -chromatic scale \ldots +chromatic scale\ldots -\speaker{HENNIX} "\ldots\ how time passes\ldots"\footnote{\journaltitle{die Reihe 3}} +\spk{Hennix} \enquote{\ldots\ \textit{how time passes}\ldots}\footnote{\journaltitle{die Reihe 3}} -\speaker{FLYNT} Yes, and what is more, the other rhythms had to correspond to +\spk{Flynt} Yes, and what is more, the other rhythms had to correspond to the overtone structure above those frequencies as fundamentals. -\speaker{HENNIX} Yes, I'm quite familiar with that. +\spk{Hennix} Yes, I'm quite familiar with that. -\speaker{FLYNT} Yes, I would expect you would be. I remember Bo +\spk{Flynt} Yes, I would expect you would be. I remember Bo Nilson---you will like this---in 1958 at the same time I saw Stockhausen's score---he went even one step further than Stockhausen because he used fractional amplitude specifications---so this is even more than @@ -55,11 +55,9 @@ is on one axis and the length of the note is on another axis. What he would do was to superimpose that on some picture like from a star catalogue--- +\spk{Hennix} \opustitle{Atlas Eclipticalis}--- -\speaker{HENNIX} \opustitle{Atlas Eclipticalis}--- - - -\speaker{FLYNT} Yeah, well, that's the particular piece. I'm making up a +\spk{Flynt} Yeah, well, that's the particular piece. I'm making up a composite of his compositional techniques but the result is that when you break up a sequential event in that way, it's not like a pitch-time graph where there's an intuitive recognition of the way the process unfolds. @@ -102,12 +100,10 @@ the attitude, they were rewriting the history of music, trying to show that all previous important figures were essentially preoccupied with structure, that they had been complete structuralists. - -\speaker{HENNIX} Really? I thought it was only Webern that was given that +\spk{Hennix} Really? I thought it was only Webern that was given that treatment. - -\speaker{FLYNT} Well, they were digging up all these composers from the +\spk{Flynt} Well, they were digging up all these composers from the Middle Ages, the isorhythmic motet and everything like that---they were sort of dredging that up because that was the previous period---the medieval scores in the form of a circle and the use of insertion @@ -123,25 +119,25 @@ quite an excursion. At any rate there is in music, there is this preoccupation with---it may be a kind of quasi-Pythagoreanism, I don't know\ldots -\speaker{HENNIX} The way I looked at it was that they saw in Webern, first of +\spk{Hennix} The way I looked at it was that they saw in Webern, first of all the harmony was going away. And they saw in Webern a way of determining the note more and more precisely, in terms of all of its parameters, pitch, duration, timbre and all that. What was left was that timbre was not serialized yet. And that, as far I see it, was what the Darmstadt school did---they added--- -\speaker{FLYNT} Stockhausen's \opustitle{Kontra-Punkte}--- +\spk{Flynt} Stockhausen's \opustitle{Kontra-Punkte}--- -\speaker{HENNIX} Yeah. And they all considered Webern the god of the new +\spk{Hennix} Yeah. And they all considered Webern the god of the new music--- -\speaker{FLYNT} Yes--- +\spk{Flynt} Yes--- -\speaker{HENNIX} ---and also a little bit Messiaen--- +\spk{Hennix} ---and also a little bit Messiaen--- -\speaker{FLYNT} Yes. +\spk{Flynt} Yes. -\speaker{HENNIX} It was Webern and Messaien that determined the entire +\spk{Hennix} It was Webern and Messaien that determined the entire fifties in Darmstadt. In other words, they were saying that Cage was no good. He was just looking in \booktitle{I Ching}---it was a random thing. And you cannot recover the structure, it's hidden, as you said. The problem was @@ -154,15 +150,13 @@ had to fake it. Because---you find that yourself when you do serial music---the music moves too slowly. So you change the numbers to get the music up a little bit. -\speaker{FLYNT} Yes. We're taking longer on this than I meant to\ldots +\spk{Flynt} Yes. We're taking longer on this than I meant to\ldots -\speaker{HENNIX} But I wanted to say this. The completely deterministic com- -position technique and the completely random, aleatoric technique, +\spk{Hennix} But I wanted to say this. The completely deterministic composition technique and the completely random, aleatoric technique, gave exactly the same results. And that was the complete breakdown of -the Darmstadt school. That's when they started to improvise in Darm- -stadt. Not before that was there improvisation in Darmstadt. +the Darmstadt school. That's when they started to improvise in Darmstadt. Not before that was there improvisation in Darmstadt. -\speaker{FLYNT} When they first tried to serialize duration, they tried to pick a +\spk{Flynt} When they first tried to serialize duration, they tried to pick a fundamental unit and use multiples of it; in other words, that's not the way you serialize pitch. You don't take one cycle per second and then use two cycles per second, up to twelve. That's not what you do. But @@ -210,9 +204,7 @@ Carnap. And I was a schoolmate of Kripke, Solovay, Goodman \etc\ my conversations with them were insignificant as far as the philosophy of mathematics was concerned, there was no discussion between me and them on any of that but it will locate the time frame that I'm talking -about.* - -\footnote{I'm being too diffident. I had quite significant discussions with Kripke and Goodman in 1961. [H.F,, note added]} +about.\footnote{I'm being too diffident. I had quite significant discussions with Kripke and Goodman in 1961. [H.F,, note added]} But observing what was going on at that time, I picked up the idea that the most plausible explanation of what mathematics is, is that it is @@ -259,7 +251,7 @@ mathematician was to study structures which do not have any reality. And that from time to time you will give an interpretation to one or the other of these structures, like a physical interpretation, and then it may be found to be true or false in reality or not. Meanwhile, you have -another sense of the word "interpretation" which has to do with relative +another sense of the word \enquote{interpretation} which has to do with relative consistency proofs by something having a model. This is now a completely open question for me, what they thought @@ -268,11 +260,11 @@ doing---he interpreted one or another non-Euclidean geometry---what was the interpretation that he used? It was a denumerable domain of algebraic numbers.\footnote{Foundations of Geometry, pp. 27--30} -\speaker{HENNIX} I think his ideas go back to Klein's models---which are +\spk{Hennix} I think his ideas go back to Klein's models---which are Euclidean in the center of the circle and then at the periphery they have turned non-Euclidean (in the complex plane). -\speaker{FLYNT} You had to have an explanation of how mathematics could be +\spk{Flynt} You had to have an explanation of how mathematics could be true in any sense whatsoever even though any claim of a connection with the real world had been completely severed, and it was being pursued in some kind of vacuum. What does mathematics mean in that @@ -286,28 +278,28 @@ infinitary games. By the way, I completely overlooked that aspect at that time. You know, I can only see it now, kind of like two superimposed pictures, because I see what I know now and compare it with what I knew then. -\speaker{HENNIX} Yeah, the same for myself. I didn't know that this idea of +\spk{Hennix} Yeah, the same for myself. I didn't know that this idea of Hilbert's was forced by Frege until later. Frege was the one who said that either the parallel axiom is true, or it's not. Which way do you want it? And so he caused the big stir in the foundations of geometry in the end of the nineteenth century and that's why he became enemies with Hilbert. They were life enemies. -\speaker{FLYNT} The reason I see it like two superimposed transparencies--- +\spk{Flynt} The reason I see it like two superimposed transparencies--- -\speaker{HENNIX} But even today this debate with Frege---you have to go to a +\spk{Hennix} But even today this debate with Frege---you have to go to a single volume in Frege's posthumous writings---it is not mentioned in any textbook---no lecture mentions it, and, so far, nobody has explained it properly.\footnote{\booktitle{Nachgelassene Schriften und Wissenschaftlicher Briefwechsel}, vol. 2, Felix Meiner, Hamburg: 1976. (Gottlob Frege, The Philosophical and Mathematical Correspondence, University of Chicago Press: 1980)} -\speaker{FLYNT} Yes, yes, yes. You're talking about an obscure origin of something +\spk{Flynt} Yes, yes, yes. You're talking about an obscure origin of something and what I'm talking about is a kind of consensus that had grown up, since everybody agreed that mathematics should study unreal structures. -\speaker{HENNIX} But that consensus was forced on us, that that was what we +\spk{Hennix} But that consensus was forced on us, that that was what we were supposed to do. -\speaker{FLYNT} The problem then---I thought mathematics was like chess. +\spk{Flynt} The problem then---I thought mathematics was like chess. What I understand now is that even a good formalist would not agree with that. A good formalist would say that when you have a finite game like chess, the problems of validity and soundness become transparent @@ -321,11 +313,11 @@ problem to stand for mathematics. Or that the reliability of a finite game was sufficiently complicated to stand for mathematics so I basically focused just on a finite game. -\speaker{HENNIX} By the way, this was exactly the late Wittgenstein's view of +\spk{Hennix} By the way, this was exactly the late Wittgenstein's view of the philosophy of mathematics---it's not a complete misunderstanding, that is to say, other people thought of it that way too. -\speaker{FLYNT} The question then arose of even the soundness, the reliability, +\spk{Flynt} The question then arose of even the soundness, the reliability, the consistency of a finite game---this then is the problem for example whether it is possible to follow a very simple rule correctly or not. The other thing that was feeding into everything that was going on was that @@ -334,23 +326,23 @@ the Harvard Bookstore when I walked in as a freshman my very first day there---so in other words I was looking at Wittgenstein's Remarks on The Foundations of Mathematics from 1957--- -\speaker{HENNIX} Ten years before me--- +\spk{Hennix} Ten years before me--- -\speaker{FLYNT} ---but very cursorily. Because I had a philosophical +\spk{Flynt} ---but very cursorily. Because I had a philosophical agenda---I passed over this material in a very cursory way because I had a philosophical agenda. I was not involved in the distinction between a finite and an infinite structure. I was not involved in that. -\speaker{HENNIX} You thought there was no such distinction? +\spk{Hennix} You thought there was no such distinction? -\speaker{FLYNT} Well no, I thought that---it didn't seem that there was very +\spk{Flynt} Well no, I thought that---it didn't seem that there was very much point in worrying about that when there were much more extreme problems to be worried about. But Wittgenstein wrote a lot about the possibility of following very simple rules. And I assumed that if there were epistemological questions for mathematics that this game interpretation---this chess interpretation---had displaced the question of the soundness and reliability of the mathematics to the possibility of -understanding a very simple rule like writing the series "plus 2". +understanding a very simple rule like writing the series \enquote{plus 2}. And having gathered that this was the way that I should picture mathematics---I mean we understood very well that there were other @@ -359,9 +351,9 @@ obsolete. In other words the person who believed that mathematics was a description of a real supra-terrestrial structure, and certainly there were people like that--- -\speaker{HENNIX} Still today. +\spk{Hennix} Still today. -\speaker{FLYNT} ---we thought that this was a philosophy that had been +\spk{Flynt} ---we thought that this was a philosophy that had been exposed as superstitious by Positivism and possibly even by Ockham several centuries earlier. So it was not that we didn't know about that. I drew a personal conclusion that that position could not be defended by @@ -372,8 +364,8 @@ meant to me. In my philosophy I was not concerned with the specifics of mathematics; I was concerned with the problem of how I knowa world beyond my immediate sensations. That was actually the question that I -began with---the question of propositions of material fact, like "it is raining" -or "the \textsc{Empire State Building} is at Fifth Avenue and 34th Street." +began with---the question of propositions of material fact, like \enquote{it is raining} +or \enquote{the \textsc{Empire State Building} is at Fifth Avenue and 34th Street.} I had read a very simplified exposition---it was actually some lectures that Carnap gave in England in the 1930s on what Positivism @@ -403,29 +395,29 @@ text which in effect gave my own empiricist constructions of what it means to say that A causes B and so forth, to give empiricist constructive definitions of those---which is, I suppose, in the spirit of Carnap's program, even though I hadn't actually seen what he had written, and if -I had it would have confused me---no, I wouldn't say "confused"; I +I had it would have confused me---no, I wouldn't say \enquote{confused}; I would say it would have discredited him completely. I wouldn't say -"confused" because that's too modest. +\enquote{confused} because that's too modest. -\speaker{HENNIX} No, I wouldn't think "confused," I would think it would +\spk{Hennix} No, I wouldn't think \enquote{confused,} I would think it would have upset you\ldots -\speaker{FLYNT} No, I wouldn't say "confused." I would say he had been +\spk{Flynt} No, I wouldn't say \enquote{confused.} I would say he had been discredited. I very quickly passed to the position that the propositions of natural science were meaningless metaphysics. -\speaker{HENNIX} On what basis? Can you pin that down? A little bit, only. +\spk{Hennix} On what basis? Can you pin that down? A little bit, only. -\speaker{FLYNT} This is something I want to compress---it says a little bit about +\spk{Flynt} This is something I want to compress---it says a little bit about this in \booktitle{Blueprint for a Higher Civilization}\footnote{H. Flynt, Blueprint for a Higher Civilization (Milan, 1975). Recently reissued and an expanded and corrected edition by \textsc{Salitter Workings}}---like -the proposition, "this key is made of iron" or something like that, I comment on that in the +the proposition, \enquote{this key is made of iron} or something like that, I comment on that in the essay \essaytitle{Philosophical Aspects of Walking Through Walls}. -\speaker{HENNIX} I didn't recall the example actually. +\spk{Hennix} I didn't recall the example actually. -\speakermod{FLYNT}{reading} "The natural sciences must certainly be dismantled. +\spkmod{Flynt}{reading} \textquote{The natural sciences must certainly be dismantled. In this connection it is appropriate to make a criticism about the logic of science as Carnap rationalized it. Carnap considered a proposition meaningful if it had any empirically verifiable proposition as an @@ -441,7 +433,7 @@ fabrication which amalgamates a few trivially-testable meanings with an infinite number of untestable meanings and inveigles us to accept the whole conglomeration at once. It is apparent at the very beginning of \booktitle{Philosophy and Logical Syntax} that Carnap recognized this quite -clearly; but it did not occur to him to do anything about it." +clearly; but it did not occur to him to do anything about it.} The only point that I'm trying to make here is that I began to move very quickly when I was still very young towards a position of extreme @@ -450,7 +442,7 @@ was not a slow process. I just immediately took Carnap's critique of metaphysics, decided that it applied directly to natural science---you dismiss natural science as meaningless. The problem: is there an object that is beyond my experience, is there a glass which is beyond what they -would call the "scopic" glass, the "tactile" glass \action{gestures toward the +would call the \enquote{scopic} glass, the \enquote{tactile} glass \action{gestures toward the glass from which he has been drinking}---is there a glass other than those glasses---when you first think about it, that question seems to have exactly the status of the propositions about God, freedom, and @@ -478,16 +470,16 @@ Kant makes the same point. In order to ask the question whether there is a glass beyond my sense impression of it---I cannot ask that question\ldots -\speaker{HENNIX} Oh you mean the \term{ding an sich} question. +\spk{Hennix} Oh you mean the \term{ding an sich} question. -\speaker{FLYNT} Well that's what Kant would have been talking about but I +\spk{Flynt} Well that's what Kant would have been talking about but I don't want to fit that narrowly into Kant's controlling the terms of the discussion. I'm trying to ask it as someone who has embraced Logical Positivism and is now turning around to question Logical Positivism---you see the point that I was just making there---when you say that this key is made of iron, which is Carnap's favorite example---and then a protocol sentence, for example -"if I hold a magnet near this key, the key will be attracted to the magnet"---it +\enquote{if I hold a magnet near this key, the key will be attracted to the magnet}---it is not clear where Carnap stands on the question whether only my sense impressions are real---just talking about this situation---only my sense impressions are real---or is there @@ -540,26 +532,26 @@ forces a yes answer. This does not mean that a proof of the existence of the external world has been given. It meant that the proposition of the existence of the external world would verify itself even if it were false! -\speaker{HENNIX} I find this extremely interesting and rewarding, what you are +\spk{Hennix} I find this extremely interesting and rewarding, what you are saying now, because I never heard you say it this way before. I just want to ask you one question before you go on: namely, I see something for the first time which I hadn't seen before---but before you go on I just want to ask you one leading question: the simple existential statement, -"there is a glass on the table." You include that also in what will be -doubtable here. In other words not just "there is a glass on the table" -but "there exists a glass," the existential statement. I guess I wasn't very +\enquote{there is a glass on the table.} You include that also in what will be +doubtable here. In other words not just \enquote{there is a glass on the table} +but \enquote{there exists a glass,} the existential statement. I guess I wasn't very clear now. -\speaker{FLYNT} No, the thing is, the approach that I'm taking doesn't break it +\spk{Flynt} No, the thing is, the approach that I'm taking doesn't break it down the way that you're talking about. Let me tell you. You may not be \emph{sympatico} with empiricism. When you are trying to deal with philosophy at all---you have to make some allowance for the fact---you have to understand that the philosopher may be carving up problems in a way that is temperamentally alien to you. -\speaker{HENNIX} Yeah\ldots +\spk{Hennix} Yeah\ldots -\speaker{FLYNT} You have to understand that. This is why somebody like +\spk{Flynt} You have to understand that. This is why somebody like Carnap would read Hegel and say it's not saying anything. Actually, Hegel is saying something. In fact, you might go so far as to make a case that Hegel is actually rebutting Carnap, becaue if you understand what @@ -574,17 +566,17 @@ solved problem. I'm trying to give you a sense of misunderstandings between philosophers that are the results of temperamental incompatibilities. -\speaker{HENNIX} What you are giving me is a two-step way to skepticism. You +\spk{Hennix} What you are giving me is a two-step way to skepticism. You ask a certain question---is there something beyond this perception of -the glass? And you say the answer "yes" is forced on me, but then you +the glass? And you say the answer \enquote{yes} is forced on me, but then you realize this was a meaningless question. -\speaker{FLYNT} No, it's the other way around. +\spk{Flynt} No, it's the other way around. -\speaker{HENNIX} Oh, okay, but here's where you have to explain in detail +\spk{Hennix} Oh, okay, but here's where you have to explain in detail because here's where I miss you. -\speaker{FLYNT} Let me go through the series of steps again. The series of steps +\spk{Flynt} Let me go through the series of steps again. The series of steps was\ldots\ I'll have to doit all at the same time. You have to understand---I don't think that you even understand what an empiricist is. It's a peculiar attitude. And one of the reasons why you have very little @@ -598,9 +590,9 @@ because in addition to having the doctrine of the construction of the world from sense impressions, they also want to have things like science--- -\speaker{HENNIX} Ethics\ldots +\spk{Hennix} Ethics\ldots -\speaker{FLYNT} No, not ethics---one of the characteristics of the twentieth- +\spk{Flynt} No, not ethics---one of the characteristics of the twentieth- century philosopher was the appearance of the tough-guy philosopher who rejects all of ethics as meaningless, which Carnap certainly did and people who are close to him like A.J. Ayer---no, they did not want @@ -625,9 +617,9 @@ stupidity, because the man did not realize that his answers were not adequate, did not realize how preposterous his constructions of the world were--- -\speaker{HENNIX} I would say vulgar. +\spk{Hennix} I would say vulgar. -\speaker{FLYNT} Yes, yes. And\ldots\ what is even worse about empiricism is, in the +\spk{Flynt} Yes, yes. And\ldots\ what is even worse about empiricism is, in the case of somebody like Mach, not only does he want to have his sense impressions and does he want to have his science, but he wants to have science explain sense impressions! And nevertheless it was supposed to @@ -646,12 +638,12 @@ authentic empiricist. You ask does a glass exist; an authentic empiricist would have to say that he already has a problem with that---that he has to regard that as an undefined question or statement. It's undefined, because if you are asking me if at this moment I quote unquote -have---interesting word there, "have"---that is what our ordinary +have---interesting word there, \enquote{have}---that is what our ordinary language gives us as the idiom for this. -\speaker{HENNIX} Or "suffer!" +\spk{Hennix} Or \enquote{suffer!} -\speaker{FLYNT} Yes, "have" or "suffer," that's right. I have or I suffer a scopic +\spk{Flynt} Yes, \enquote{have} or \enquote{suffer,} that's right. I have or I suffer a scopic glass or visual glass apparition---then that is identically true. That is identically true. If you express any surprise at that, we have a problem here. I have a scopic glass. If I say I have an apparitional glass, would @@ -671,77 +663,77 @@ an objectivity to compare it to. And a bona fide empiricist would not agree that my sense impression is subjective---subjective in comparison to \emph{what}? -\speaker{HENNIX} So an empiricist would be a person who would not doubt +\spk{Hennix} So an empiricist would be a person who would not doubt whether he had a toothache or not. In other words, if he had a toothache\ldots -\speaker{FLYNT} You would regard it as being a mistake to do what? I'm not -sure about the word "toothache"---if you mean that he would not +\spk{Flynt} You would regard it as being a mistake to do what? I'm not +sure about the word \enquote{toothache}---if you mean that he would not doubt whether he had a toothache sensation. Whether there is an organic---in the language of medicine---whether there is an organic substrate for the toothache impression---this in a medical sense is a question of what is called hysteria or something like that\ldots -\speaker{HENNIX} Suppose I have a toothache. But now I'm an empiricist so I +\spk{Hennix} Suppose I have a toothache. But now I'm an empiricist so I say I'm doubting this impression. I probably don't have a toothache. -\speaker{FLYNT} No, no\ldots +\spk{Flynt} No, no\ldots -\speaker{HENNIX} I have to accept the toothache? +\spk{Hennix} I have to accept the toothache? -\speaker{FLYNT} No, you don't have--- +\spk{Flynt} No, you don't have--- -\speaker{HENNIX} The glass you said was---I couldn't doubt the perception of +\spk{Hennix} The glass you said was---I couldn't doubt the perception of the glass. You said that was beyond doubt, in some sense, for the empiricist. -\speaker{FLYNT} It would be some kind of logical mistake to think that there +\spk{Flynt} It would be some kind of logical mistake to think that there was anything there to be doubted. -\speaker{HENNIX} Okay. And the same with the toothache. +\spk{Hennix} Okay. And the same with the toothache. -\speaker{FLYNT} Yes, yes. I mean the point is not so much that we have come +\spk{Flynt} Yes, yes. I mean the point is not so much that we have come into an area in which the empiricist is prepared to have faith---that would be completely missing the point. No faith is required---that's the point. The point is that it would be some kind of logical error. Once you understand what a sense impression is, the terminology of doubt does not apply to that level. -\speaker{HENNIX} I see. Just that was my question. +\spk{Hennix} I see. Just that was my question. -\speaker{FLYNT} The terminology of doubt does not apply to apparitions. It +\spk{Flynt} The terminology of doubt does not apply to apparitions. It doesn't make sense to doubt subjective apparitions. The empiricist is already nervous when you ask does a glass exist. If you are asking -whether I have a "scopic" glass, it's identically true. Wait, wait. There +whether I have a \enquote{scopic} glass, it's identically true. Wait, wait. There are already problems there. I'll come back to them. But when you say---it sounds like what you're asking me is whether the fact that I see a glass is sufficient to prove an objective glass---that sounds like \ldots -\speaker{HENNIX} No, no, that's not what--- +\spk{Hennix} No, no, that's not what--- -\speaker{FLYNT} Well, ok. Most people when they say: -"do you concede that there is a glass on the table---I'm sitting here looking at it," what they -mean is: "do you concede that from your visual glass apparition you should conclude an objective glass, a substantial glass?" I'm taking it for +\spk{Flynt} Well, ok. Most people when they say: +\enquote{do you concede that there is a glass on the table---I'm sitting here looking at it,} what they +mean is: \enquote{do you concede that from your visual glass apparition you should conclude an objective glass, a substantial glass?} I'm taking it for granted that you know enough about philosophy to have a sense of the -full weight those two words "substantial" and "objective" have in +full weight those two words \enquote{substantial} and \enquote{objective} have in philosophy. -\speaker{HENNIX} Yes. +\spk{Hennix} Yes. -\speaker{FLYNT} That at great length is my reaction to your question about -doubting "there is a glass on the table" versus doubting "there exists a glass." -A bona fide empiricist would say, "Why are you asking me this?" +\spk{Flynt} That at great length is my reaction to your question about +doubting \enquote{there is a glass on the table} versus doubting \enquote{there exists a glass.} +A bona fide empiricist would say, \enquote{Why are you asking me this?} The scopic glass is simply here for me. As far as concluding that an objective glass exists from the existence of that apparition---the traditional problem of concluding whether the apparition is a symptom of -some transcendent world---I think the word "transcendent" is sometimes +some transcendent world---I think the word \enquote{transcendent} is sometimes used in that sense in philosophy---the world beyond any sense impression--- -\speaker{HENNIX} This is why I used the example of the pain---because it +\spk{Hennix} This is why I used the example of the pain---because it would be senseless for me to claim that \emph{I} can have \emph{your} toothache! -\speaker{FLYNT} Now just a minute. An empiricist---what you're really getting +\spk{Flynt} Now just a minute. An empiricist---what you're really getting at what you're sort of squeezing out of me here---I'm glad to have it squeezed out of me---I have no embarrassment about this---is that with empiricism either you must be prepared immediately to depart @@ -758,7 +750,7 @@ For the empiricist, nothing remotely like that question has arisen yet, because I haven't got outside of my own quote unquote head yet. Maybe you're just squeezing more and more. Either the empiricist -must be a "madman" or else he must be insincere. I took the alternative +must be a \enquote{madman} or else he must be insincere. I took the alternative of the madman. This is important not for me but for the general public to be told---something which the general public has never been told---and I know why they have never been told---maybe it is necessary to @@ -776,14 +768,14 @@ the scientist needed. And, at the same time, empiricism was supposed to be---in the case of Neurath---he wanted to make some kind of unification of empiricism with Marxism and make it like a complete demythified view of society. -\speaker{HENNIX} There was even an attempt to bring ethics into it. +\spk{Hennix} There was even an attempt to bring ethics into it. -\speaker{FLYNT} Well, in Neurath's case, yes. +\spk{Flynt} Well, in Neurath's case, yes. -\speaker{HENNIX} Schlick too, I think---Schlick, I recall, did something in -ethics.\bootnote{\booktitle{Fragen der Ethik}, Vienna, 1930.} +\spk{Hennix} Schlick too, I think---Schlick, I recall, did something in +ethics.\footnote{\booktitle{Fragen der Ethik}, Vienna, 1930.} -\speaker{FLYNT} I was talking about why empiricism is not portrayed honestly +\spk{Flynt} I was talking about why empiricism is not portrayed honestly in the general picture that exists of philosophy---the public picture of philosophy---it was brought in to solve the problem of what is a base for science---namely, sense impressions are going to be taken as @@ -791,20 +783,20 @@ elemental. Science is going to arise from sense impressions by construction. Nevertheless it is required that both scientific knowledge and the common-sense social world be produced by this approach--- -\speaker{HENNIX} Neurath, you mean. +\spk{Hennix} Neurath, you mean. -\speaker{FLYNT} No, no. Well, Carnap did not deny the existence of other +\spk{Flynt} No, no. Well, Carnap did not deny the existence of other people. All of the positivists\ldots -\speaker{HENNIX} Rather, he had nothing to say about it. +\spk{Hennix} Rather, he had nothing to say about it. -\speaker{FLYNT} I didn't say ethics---I said the common-sense social world. I +\spk{Flynt} I didn't say ethics---I said the common-sense social world. I wasn't talking about anything ethical\ldots -\speaker{HENNIX} The existence of tables and cars and--- +\spk{Hennix} The existence of tables and cars and--- -\speaker{FLYNT} Well, what I'm saying is that the existence of other people is on +\spk{Flynt} Well, what I'm saying is that the existence of other people is on the same level as the existence of tables and automobiles. And what is even worse than that is that the ones who were scientists in fact wanted to see perception itself as the product of the abstract and quantified @@ -827,14 +819,14 @@ take that \emph{seriously} anymore. As a matter of fact Hume wrote two philosophical works and in the first work\footnote{\booktitle{Treatise on Human Nature}} there is the notorious passage in which he himself -understands what it means to be a genuine empiricist.\footnote{Book I, Part IV, VII "Conclusion"} +understands what it means to be a genuine empiricist.\footnote{Book I, Part IV, VII \enquote{Conclusion}} He says, \textquote{I feel that I am an outcast from the human race,} and so forth in this famous passage---he says, -"I do not know if the glass continues to exist after I've looked away from it." +\enquote{I do not know if the glass continues to exist after I've looked away from it.} That line in Hume should have told you whatever you wanted to know about the existence of the glass. You should be able to ascertain the appropriate answer to your question. -Hume says: "I do not know if the glass exists when I look away from it." +Hume says: \enquote{I do not know if the glass exists when I look away from it.} Hume's second book\footnote{\booktitle{An Enquiry Concerning Human Understanding}}, when he was trying to vindicate himself, @@ -867,7 +859,7 @@ how dreams would affect the validity of the proof. He touches on all of those in a way which is just awful. It's a disgraceful performance. But he had the issue there, actually. -Well, your first reaction is, "I have no way of answering this." Your +Well, your first reaction is, \enquote{I have no way of answering this.} Your second reaction is, that \emph{if I understand the question}, then there must be an external world. So it would seem that I have actually proved the external world---that's what Kant actually said. Or he came very near @@ -875,11 +867,11 @@ to saying something like that. The third step is the realization that the statement would validate itself not only if it's true---but if it's false it validates itself equally well! -\speaker{HENNIX} Given this method of understanding the question. And the +\spk{Hennix} Given this method of understanding the question. And the method remained unspecified so far---as far as I know nobody has been able to do very well at specifying it. -\speaker{FLYNT} What? Do you mean if somebody asks whether there is an +\spk{Flynt} What? Do you mean if somebody asks whether there is an external world---my last remark is a comment about semantics---the genuine semantic issue, as I said, and it's very different from the sort of thing that Tarski is going on about which I think is just ridiculous. @@ -899,8 +891,8 @@ then there is no guarantee of the continuity of the position of the pieces in the absence of moves. What happens is that people treat those basic questions as if they are so basic that it's sort of preposterous to make an issue of them. Kripke said very clearly in his book on Wittgenstein that -once the question, "Does language exist?" has been asked, not to give -an affirmative answer is "insane and intolerable."\footnote{S. Kripke, +once the question, \enquote{Does language exist?} has been asked, not to give +an affirmative answer is \enquote{insane and intolerable.}\footnote{S. Kripke, \booktitle{Wittgenstein on Rules and Private Language}, p.60} It's the same reaction as there is to solipsism---that solipsism is the philosophy of the man in the lunatic asylum. @@ -909,23 +901,23 @@ The thing that may come before all the discussion so far is the question of \emph{what is my position on being classified as insane} is the beginning This of philosophy for me. -\speaker{HENNIX} Well, this is the classical beginning of philosophy. +\spk{Hennix} Well, this is the classical beginning of philosophy. -\speaker{FLYNT} Because if you're not willing to face up to being classified as +\spk{Flynt} Because if you're not willing to face up to being classified as insane---if you want to avoid that confrontation---you can't be a philosopher. That confrontation is at the center of bona fide philosophy. -\speaker{HENNIX} Or was\ldots +\spk{Hennix} Or was\ldots -\speaker{FLYNT} Yes. At any rate, I had reached this point in something like -1961. I had not yet done \essaytitlte{The "Is There Language?" Trap}. But I had reached +\spk{Flynt} Yes. At any rate, I had reached this point in something like +1961. I had not yet done \essaytitle{The \enquote{Is There Language?} Trap}. But I had reached the point of saying that to claim the existence of a world beyond experience is untenable. However I understood very well that it begins to create problems for me to say, \textquote{I have a visual glass apparition,} because there is a lot of structure in that sentence. And it's not clear what is supporting that structure after the world has been cut away. -Even the use of the idioms like "have" and "suffer." The use of the word -"I"---after the objective world has been cut away it's unclear what is the +Even the use of the idioms like \enquote{have} and \enquote{suffer.} The use of the word +\enquote{I}---after the objective world has been cut away it's unclear what is the basis for all of that. And this is the point I had reached in 1961 and this is the point when I did \essaytitle{Concept Art}. @@ -981,7 +973,7 @@ rule---but in each case they wish to express the violation, the failure of some traditional organizing principle of these uninterpreted calculi, For instance there is one where, among other things, the very notation itself has an undisplaced active interaction with the subjectivity of the -quote unquote reader.\footnote{dated 6/19/61---later titled "Illusions."} +quote unquote reader.\footnote{dated 6/19/61---later titled \enquote{Illusions.}} And that determines the structure of the derivation, the proof. It was pointed out to me many years later that it's not just that you don't get this in schoolbook mathematics---this is what @@ -994,15 +986,15 @@ In other words, for each step you are told, for that step only and for this moment only, what the transformation rule is. And by the time you are ready to take the next step, that rule is forgotten and inoperative. -\speaker{HENNIX} This is the \essaytitle{Energy Cube Organism}? +\spk{Hennix} This is the \essaytitle{Energy Cube Organism}? -\speaker{FLYNT} No, no. \essaytitle{The Energy Cube Organism} was not Concept Art at +\spk{Flynt} No, no. \essaytitle{The Energy Cube Organism} was not Concept Art at all. No, no. It was a different genre. That one was the piece called \essaytitle{Transformations}. \essaytitle{The Energy Cube Organism} and the \essaytitle{Perception-Dissociator} in my own classification are not Concept Art. Only the pieces labeled -"Concept Art" are Concept Art. And I only did four of them until 1987. +\enquote{Concept Art} are Concept Art. And I only did four of them until 1987. Three of them are in \booktitle{An Anthology}, and the fourth was published in \journaltitle{dimension 14} (1963). \essaytitle{The Energy Cube Organism} and the \essaytitle{Perception-Dissociator} were in other genres. I drew these distinctions of genre @@ -1024,18 +1016,18 @@ general rule written in Heaven. But in fact there isn't any general rule, and when you move the pawn to Bishop 3, you're just making up what you are doing right at that moment, and there isn't any general rule. -\speaker{HENNIX} You would label this ad hoc? +\spk{Hennix} You would label this ad hoc? -\speaker{FLYNT} That's right. That would be perhaps a better word for it. All +\spk{Flynt} That's right. That would be perhaps a better word for it. All transformation rules and probably even all formation rules are ad hoc, yes, yes. -I said "nominalistic" because they are only there individually. +I said \enquote{nominalistic} because they are only there individually. They do not add up to any general--- -\speaker{HENNIX} System of rules? +\spk{Hennix} System of rules? -\speaker{FLYNT} No---not that---they do not add up to any generality, to a +\spk{Flynt} No---not that---they do not add up to any generality, to a general rule that covers all cases of a certain class. What is inadequate about this---and I realized very quickly that @@ -1109,20 +1101,20 @@ third thing is that this does not claim to have objective truth. It is a construction for the world-hallucination or the world-apparition or even a construction for the private world-apparition. -\speaker{HENNIX} You are actually extending the world by new constructions. +\spk{Hennix} You are actually extending the world by new constructions. -\speaker{FLYNT} But it's the world-apparition. In a sense if I believed that these +\spk{Flynt} But it's the world-apparition. In a sense if I believed that these rules were objectively established, then it would almost indicate that I had not learned the lesson of the very piece which sits beside it on the -page!\footnote{\essaytitle{Innperseqs} versus \essaytitle{{Transformations,} second edition.} +page!\footnote{\essaytitle{Innperseqs} versus \essaytitle{Transformations,} second edition.} And what am I doing talking about a page and a text? So the answer is that I have abandoned the provision of truth as the purpose of this activity and I have moved to the provision of experiences where the possibility of these experiences is a surprise. -\speaker{HENNIX} And you don't have to be an empiricist to be surprised. +\spk{Hennix} And you don't have to be an empiricist to be surprised. -\speaker{FLYNT} Yes. Yes. But the truth claim that you would have from a +\spk{Flynt} Yes. Yes. But the truth claim that you would have from a Kripke or a Goodman has been dropped. The meaning of the text is the meaning that the reader associates to it. And the thing is, that in conventional intellectual work that's an unacceptable answer, because diff --git a/extra/structure_art_pure_mathematics.tex b/extra/structure_art_pure_mathematics.tex index dd8cb61..78b5557 100644 --- a/extra/structure_art_pure_mathematics.tex +++ b/extra/structure_art_pure_mathematics.tex @@ -56,7 +56,9 @@ the structure of a production made by critics, what I call "associated definitio structure" (in line with the terminology of the previous paragraph), Consider the following examples. -\includegraphics[width=4in]{img/structure_art} +{ \centering +\includegraphics[width=4in]{img/structure_art}\par +} In each example, the actual sounds, the body of material, is exactly the same. The difference is in the different structures defined on the material. 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