The MDS data used to produce the idea maps that will be discussed in this section were generated by means of the Triadic Method. A set of forty-six polar ideas ("diverse examples or sources of beauty") were intuitively and holistically characterized on the basis of a recurrent set of fifty virtually intra-set scaling dimensions and via 2,300 binary decisions (eg polar "examples or sources of beauty" x 50 recurrent scaling dyads = 2,300 decisions involving 4,600 monadic choices).
The scaling dimensions are described as "virtually intra-set" for two reasons. First, although the poles were usually drawn from the larger set of dyadized monads, the truncated nature of the exercise left some. of the latter unpoled and hence unmapped. Second, a handful of poles that are not represented in the scaling-dyad set were chosen from a table titled "141 Examples and Sources of Beauty". (The former table, listing all of the fifty scaling dyads, as well as the latter table are reproduced here as figures)
The scaling dyads and their monads were effectively chosen at random, but the poled and mapped items were selected mainly for interest and irredundancy (although the culling was rather crude).
The original 141-item table was simply a product of my own mind. I did my best to think of things that could serve as maximally diverse and far-ranging and minimally redundant examples of beauty or of things that for human beings are sources of beauty.-, I continued to add to the table until I essentially ran out of ideas, which is to say, until the new ideas I had were of things that upon analysis proved to be mainly aesthetic variants of the old, either in a generic (higher-level) or spetific (lower-level) sense. It only made sense for me to stop there, for it was not my purpose to scout reality and imagination for all possible instances of the different types and divergent sources of beauty.
Instead the idea was that the cases tabulated would suggest by their implicit matrix of mutual interrelations and implications the present spectrum of human aesthetic feelings. Which, however, should by no means be read as an assertion that the preliminary and probably somewhat personal table is free-of internal redundancy.or perfect in any other way. The only realistic way to check the table in these respects would be by attempting to [define, categorize, cluster, group, classify, hierarchize, condense, justify, and-Complete] its contents.
The important thing to emphasize about the table is that, despite its comprehensive ambitions, it is only in the end a listing of particular examples and sources of beauty, rather than an identification or naming of abstract types or taxons of beauty. At best it is the necessary preliminary that should subsequently enable the preparation of such a nomothetic, typological, and taxological scheme (and of a suitable table representing it) .
It is hard, in fact it is impossible, to know at the outset whether the beauties of human progress and an industrious ant colony, or of athletic competition and cooperative endeavor generically, are largely the same or largely different or merely related aesthetically; or whether they should be represented as being such in some interpretive scheme that does or does not pretend to be fundamental, universal, and ultimate.
Actually the cartographic techniques I am about to describe could play a major role in helping us to answer such questions or confront such problems.
| Academic graduation ceremony | Wonderfully fortuitous event |
| Acquisition of first home | Wedding |
| Apocalyptic psychostasia | Waterfall, river rapids, or freshet |
| Athletic competition | Vista from mountain top or canyon rim |
| Attractive voice | Visionary statemanship or sea-green incorruptible |
| Aurora | Victory in war |
| Beatitude (consummate bliss) | Unity of natural laws |
| Bees' pollinating flowers | Unflagging loyalty of spouse |
| Biological evolution of ontogeny | Twilight mystery |
| Birdsong | Tragic love |
| Birth of one's child | Tiny humans at foot of mountain |
| Bold architecture or elegant bridge | Time-lapse film of sky, anthesis, or child's growth |
| Brilliant attire | Supreme amical moment or act |
| Bubble bath | Sudden unexpected resolution of crisis |
| Butterfl wing color patterns | Sublime wickedness |
| Calligraphy | Sublime dream |
| Cancer patient's indomitable will to live | Springtime |
| Catahrsis | Spaceship launch |
| Cavern decorated with speleothems | Snowfall |
| Charitable act or self-giving | Skyful of soaring thunderheads |
| Charmed banter, falliance, or dazzling laughter | Skydiving or gliding |
| Child's toys | Singualr humility or simplicity |
| Coitus | Sights and aroma of meal being prepared |
| Concept of infinity or Apeiron | Seashore (surf, immense beach, or towering cliff) |
| Cooperative endeavor | Saintly kindness |
| Coral reef | Rose |
| Coruscant stars across night sky | Revelatory insight or epiphany |
| Cosmos-atom size ratio (mental juxtraposition) | Reunion with childhood friend |
| Jewel | Industious ant colony |
| Jupiter's surface or Saturn's ring system | Individual or panhuman wisdom |
| Justice, or villainy receiving its due | Impressionist painting |
| Lavish banquet | Hydrologic cycle |
| Lush and surreal rain forest | Human progress |
| Machinery of the mind and brain | Human ideals |
| Magnificent battleship | Harp's sound |
| Magnificent body (figure or physique) | Handsome face |
| Marine islet or archipelago | Glorious pagentry or pomp |
| Massive or lofty tree | Glamorous scientific laboratory |
| Maternal or filial devotion | Geometric proof |
| Melody, song, or chorus | General prosperity |
| Meteor shower or radiant, or spectacular comet | Forest fire |
| Microscope, telescope, or other instruments | Foreign travel |
| Old-age reminiscence and reverie | Exuberant elfin child |
| Palatial estate | Extraordinary wealth or good fortune |
| Patterns of frost or dew | Epic interpersonal tableau (eg diplomatic) |
| Personal grace or magnanimity | Elevatory metanoia |
| Probity and zeal for truth | Drifting bubble |
| Profound metaphor | Dramatic chess > game, position, or tactic |
| Rainbow or green flash | Destiny or time's river |
| Rescue from misery or horror | Democratic processes |
In figure -2 I present the analytic regions countermap that I constructed after I spent some time trying to make sense of the primary map. As it indicates, it depicts various regions, directions, poles, concepts, etc that are visibly or conjecturably present, emerged, or confused in the plot that the computer calculated and rendered.
Perhaps it seems a bit odd that I should later find it to be necessary to confront the consequences of my own data in this way, or to explain to myself what a graphical representation of that data means or could mean. But is it really so odd?
One often notes one's reactions to things without properly, or at all, understanding those reactions. This is true even when the reactions are sophisticated, descriptive, and confident.
The beauty of nonmetric MDS is that it allows the basic dimensions and structures of one's reactions to and ideas about things to surface, or to be broken free of their customary masking matrix and mutual entanglements in the mind. Where A had always been confused with B, suddenly their mutual independence or degrees of freedom are made visible. Suddenly clues can be gotten to the nature and interrelation of A, B, C, and D by the exact mapping of the distances, angles, orders, and patterns they have to one another in our passive and active intelligence. (There is more, but I will get on to that later.)
In crafting the analytic regions countermap I used different colors to distinguish and imply connectedness of different loci and entries in the graph. Where I have actually written my words is not necessarily where the words accurately belong. There was-only so much room, after all, and many of the loci alluded to overlap or are identical, are complex or diffusely distributed, are ambiguous or uncertain, are discontinuous, are multivalent, are derivative or hierarchical, etc.
Because of the annular symmetry of the original map (which gives it a one-dimensional quality), the azimuthal locations I have indicated are usually far more accurate, and real, than the radial positions (or distances from the origin of the graph), which often mean nothing (or are wholly artifactual).
The arrows merely signify a general direction away from the center or the direction opposite from the center; the circular sections that enclose heads of arrows bound an azimuthal range.
Part of the reason for the imprecision of the loci indicated lies in the insufficiency of the spatial samples represented by the mere forty-six beauties that are graphed; a great density of the examples would have enabled a far more reliable and authoritative countermap.
Before I use the countermap to analyze and synthesize the conceptual structure of the primary map, let me make a few observations about the possible dot structure of the latter.
The pattern of dots in the primary map, in the case of this particular MDS ideomap, strongly suggests, not a simple circle or annulus, but a spiral, perhaps of 11 turns (540 degrees). The spiral might have some angularities, cusps, branchlets, anastomoses, and/or clumps.
But is it a real spiral, or just a ring with a cluster or chainiet of points enclosed but free near its center?
Be it spiral or annular, the structure resembles a square with rounded corners and balanced on an end vertically.
Hinted simple (and complex) geometric (and topological) patterns like this are a common sight in MDS plots, certainly MDS plots of idea-sets. They are extraordinarily tantalizing. But it is possible they are simply the product of random noise, of the idiosyncrasies of the mathematical or computational procedures, of optical illusions or mental fantasies, or of irrelevant psycho-neural phenomena that play a role in the weighting (Triadic decision) process.
Since the mathematics of nonmetric multidimensional scaling treats the so-called variables or stimuli (or, in our case, the ideas) in the manner of a gravitational system of mutually rotating and interacting particles, patterns mimicing those seen in the cosmic zoo of galaxies are not to be wondered at; and this includes rings, rectilinear and curvilinear lines, spirals, branches, loops, trees, bunches, and sub-whorls.
One frequently sees what appear to be radiations, decussations, webs and reticles, parallel rows, triangles, etc. (See figure)
Patterns seen can seem to make great semantic sense, given the nature and placement of the ideas the dots represent.
I have succeeded in generating some of these patterns using random numbers submitted to the MDS programs in lieu of actual idea-set weightings. Yet even if all of the types of patterns.,could be produced in this way, or would be expected stochastically or as a consequence of the mathematical processes, this would not necessarily mean that the patterns are ideonomically unimportant or valueless. Thus the mathematic may package the useful information in the form of such inevitable patterns.
It is obvious that one has to be careful here and avoid drawing premature conclusions.
I will now undertake to apply and explain the countermap. Because half a year has-passed since I created the primary and counter maps, I will actually be joining the reader in trying to reconstruct what they mean. This may make what I have to say more interesting and understandable, if only because I will not be complacent about my results and analyses.
The first question concerns what the nature of the horizontal axis, or of Dimension 1, might be.
The countermap suggests that the biggest conceptual factor underlying the dimension and organizing the axis is an antipolarity between beauty or beauties on the right that represent the positive side of the dimension and that are or appertain to what is fun or sensual, and the opposite beauty or beauties upon the left that represent the negative side of the dimension and that are or appertain to what is serious or impressive.
Thus at the right limb one finds situated the beauties of charmed banter, dalliance, or dazzling laughter, of lavish banquet, of kaleidoscope, of butterfly wing color patterns, of birdsong, etc, whereas these are confronted on the left limb by the beauties of or that are associated with justice (as in villainy getting its due), with microscope, telescope, or other instruments, with probity and zeal for truth, with athletic competition, with maternal or filial devotion, etc.
This Dimension 1 is reminiscent, then, albeit in reverse, of the Dimension 1 that so conspicuously figured in our analysis of the lowest two-dimensional map of the interanalogousness of 36 emotions (figure )
It is always important, and practically always enlightening, to compare the equivalent-dimensionality MDS maps (both primary and counter) of different and disparate idea-sets. This is a subject I will discuss at some length elsewhere, for it is full of surprises and promise.
Inevitably the second question to be asked is about the meaning of the vertical axis or Dimension 2. What obvious aesthetic concept or qualities cause the forty-six 'beauties' to be ordered vertically or to be elevated or depressed in the absolute and relative way and degree they are with respect to the zero horizon?
Why is Dimension 2 less important than Dimension 1 and progressively more important than Dimensions 3, 4, 5, ... (none of which are explicitly expressed in this lowest-two-dimensional map)?
I am going to postpone for the moment any attempt to answer these questions, since doing so would be hard and probably premature.
My analysis will concentrate instead upon various local relationships of two or several named 'beauties'. I will try to discover, describe, and-where possible-explain these microscale and mesoscale contiguities, proximities, clumps, collinear chains, spurs, corners, junctions, and polygons.
From this smallest, lowest, richest, local, and most empirical level I will then try to ascend to the biggest, highest, most universal, most meaningful, most comprehensive, simplest, most abstract, most necessary or lawful, and most dimensionalized level.
Much of the beauty of the left half of the map's entries seems to have to do with things like achievement, endeavor, ambition, accomplishment, labor, strength, struggle, action, and the like; while the right half of the mapped items speak of the beauty of existence in a passive, observational, or experiential sense. Thus at the western edge of the countermap one finds inscribed the words "Bold, Exertive, Aspiring"; and near the center of the diagram an arc of arrows pointing leftward indicate "Seeking".
Probity and zeal for truth (@160 degrees), in these terms, is probably a matter of aspiration and of the virtuous accomplishment honesty is apt to represent.
The beauty of microscope, telescope, and other instruments may be situated so far west because of the association of instruments with labor and because of the fine or even miraculous accomplishment a scientific instrument represents.
A massive or lofty tree is a grand accomplishment of Nature. The beauty of magnificent battleship relates to the potential for military accomplishment, the strength symbolized by and embodied in the object, the admirable cooperation of the sailors who man that intricate engine of war, etc. Tiny humans at foot of mountain is a testament to the embodied strength and titanic challenge of a massif, and perhaps as well to the unexpected power of the dwarfed men ironically symbolized by'the presence of the mountain.
Why then is the beauty of cancer patient's indomitable will to live only just left of center in the ideomap? Perhaps because it is as apt as not to be a tragic beauty, the beauty of a doomed struggle or futile will to live.
The beauties of evolution or ontogeny, of the machinery of the mind and brain, and of a forest fire are presumably located on the left but only barely because they are peculiar hybrids of action and observation or contemplation. Biological evolution is a perpetual and nearly infinite achievement but it is ancient, slow, nonhuman, and unobserved; imagining it is like contemplating a scene in Nature. The architecture and industry of the mind and brain are sublime and yet inaccessible and abstract.
What conceptual patterns are to be found in the southeast quadrant of the ideomap of beauty? Precisely ten 'beauties' reside here, their clockwise roster being: charmed banter, dalliance, or dazzling laughter (M), marine isle or archipelago (h), ocean swim (o), birdsong (D), mountain meadow (n), drifting bubble (U), desert silence, openness, and luminosity (T), first love (X), coitus P), and jewel (Z). (Letter-names are those assigned to the so-called variables in the MDS plot.) The azimuthal range of the ten 'beauties' is ~ 7O degrees (circle/5.1). It might also be noted that 10 variables 46 variables = 0.22 (as opposed to 0.25, so that the quadrant is slightly underpopulated) .
Mountain meadow (n) and,drifting bubble (U) being near the azimuthal center, an obvious theme here is "Quiet, Stilill", as the countermap avers.
Such a theme clearly extends to desert silence ... (T), mutely symbolic (Z), and ocean swim (o). But it also has aspectual, metaphorical, or connotative (mental-associational) pertinency to the tranquil and isolated marine isle or archipelago (h), the silential shyness of first love (X), the post-orgasmic phase of coitus (P), and the contrastive and refluent forestal quiet of birdsong (D). The only member of the quadrant to which the theme apparently does not apply (unless it be by analogy to what was said about birdsong) is charmed banter, dalliance, or dazzling laughte_r.(M); but there is no reason why every member of an arbitrary like a quadrant should embody or owe its position to a single trait, and the organizational complexity of ideonomic MDS projections is so great, in any case, that such homogeneity would be neither expected nor desirable.
If one has an identified or postulated a countermap theme such as "Quiet, Still", an attempt should be made to elaborate upon, internally differentiate, and extend the theme, which might be thought of as a seed from which a tree of analogous, related, and divergent themes or thematic ideas can or must naturally grow. For example, the present theme might be enlarged-both in theory and by consulting the actual composition and content of the ideomap-to "Quiet, Still, Peaceful".
Are these subthemes equally or at all applicable, instantially, outside the arbitrary quadrant or in its complement? They do seem pertinent to the beauties of lush, surreal rain forest (d) and patterns of frost or dew (p), which are vicinal, and to the beauty of coruscant stars over night sky (S). In this particular ideomap the latter is remote from the quadrant, but I see that in another map created from the same dataset (figure ) this variable, and this variable alone, is radically displaced in the plot structure, so that it then resides at nearly the same azimuth as drifting bubble (U) (note asterisk). (This figure belongs to a pair of figures that I will explain later.)
Adjacent to the quadrant in the other direction, or clockwise, are found several examples and sources of beauty to which the quadrant theme could conceivably be extended aspectually: catharsis (K), birth of one's child (E), rescue from misery or horror (r), and academic graduation ceremony (A). Quiet, stillness, and/or peace attend or follow upon all of them.
Some of the beauties collocated in the quadrant are of interest because of their metaphorical value inter se, or their natural tendency to combine in the completion, aboration, or intensification of mental images. The beauties of jewel (Z), coitus (P), and first love (X) share a line and practically an azimuth, and the analytic countermap imagines "Beloved" as a possible lineal, azimuthal, or regional theme of minor or faint character. In any case, jewel (Z) and first love (X) are almost reciprocal metaphors.
Desert silence, openness, and luminosity (T) can be used to suggest some of the aesthetic qualities of coitus (P). Drifting bubble (U) is a marvelous metaphor or symbol for first love (X), and resonates aesthetically with the miraculous levity, tension, extension, fragility, perfection, concentricity, and ephemerality that are aspects or elements of coitus (P). So also do ocean swim (o), birdsong (D), and mountain meadow (n) have the power to add to the aesthetic meaning of coitus (P) and first love (X).
The last pair of variables (P,X) may also be mentally and aesthetically illuminated by other qualities of the variables lying to the left of the quadrant. The beauties of catharsis (K), birth of one's child (E), rescue from misery or horror (r), and academic graduation ceremony (A) are celebrative of the bliss of escape or delivery, and the beauties of coitus (P) and first love (X) are easily related to the same transitional feelings: e.g. the coital beauty-may be triumphant or successive to struggle, and the beauty of first love (X) may be heightened or defined by the contrast to or delivery from the special or aimless agonies, or unnatural emotional individuality, of childhood before the metanoiac psychobiological discoveries of puberty, (The azimuthal distance from (P,X) to those other variables is about 30-65 degrees, or 0.17-0.36 semicircles.)
Charitable act or self-giving (L), at +52 degrees (0.29 semicircles), also resonates with the donative, self-denying, and self-sacrificial aspects of coitus (P) and first love (X).
The countermap likewise ascribes the property of "Soft" to some region in or near the southeast quadrant; although of course in this particular countermap, or this particular version of it, no effort has been made to specify or pictorially depict any precise peripheral or intercurrent boundaries of regions, any precise axes or underlying structures, any gradients, or the like.
From eight to ten of the ten quadrant variables seem to partake of this property, in or through one or more ways, elements, or respects: Drifting bubbles because of the gentle motion, humble raison dletre, vague boundaries, soft surface, and fragility of such a bubble. Desert because of the describable softness of its sound and quiet, its paroxical gentleness and its surprisingly delicate and charming minutiae, the soft contours of its dunes, the 'soft' sparsity of its vegetation, its 'soft' sweet mysterious fragrances, its lack of temporal or existential sharpness or hardness, its 'softly' distant horizon, the weakness and irresistance of its sand, the 'softness' of its colors (their want of brilliance and of chromatic and spatial diversity and differentiation), the 'softness' of the featureless sky and shimmery air of the desert, the soft spectacular grandeur and negativity of the desert's night, etc. Mountain meadow because of the obvious softness of its still quiet, aspect of a garden, marshy springiness, utter innocence, air of secrecy and of protective maternity, and seductive loveliness. Birdson because of its aural softness and delicacy. Ocean swim bcause of the irresistance of the seals supportive waters and the gentle pleasures of such a swim and of the sights and minutes that border it. First love because of its delicate shades and graceful nuances of feeling and meaning and its transcendent benignity. Coitus because of its wafting ecstasies, soft warm contacts, merger and extinction of identities, liquid penetration, hallucinatory levitation, resistless and unresisted progression and consummation, and hedonic omneity. Marine isle or archipelago,- because of the softly paradisiacal aspects and symbolism of same (a la nearby ocean swim).
The two doubtful variables are charmed banter, dalliance, or dazzling Taught and jewel,. Yet the first has the describably 'soft' lightness, delicacy, pleasantness, and/or association with quiet of its companions in the quadrant (of birdsona-, etc); and the second has a 'softness' about it by virtue of its prysical and symbolic beauty and its amorous, sentimental, royal, and other mental associations.
Realistically the full azimuthal representation, or conceptual field, of, "Soft", could be interpreted as overspreading a semicircle or even 230 degrees (0.64-circles) by reaching counterclockwise from the nearer maternal or filial devotion to the farther spider's web. Thus there is a literal or metaphoric softness about, or an element-related or pertinent to softness in: devotion (j), post-exertional graduation (A), charity (L), salvation (r), childbirth (E), being freed or purged (K), Jewelry (Z), love (P,X), desert (T), bubble (U), meadow (n), birdsong (D), natation (o), Oceanian isles (h), dalliance (M), bubble bath (G), delicate calligraphy (l), butterflies (H), rain forest (d), banquet (c), dew (p) , some toys (N), music (k), pollinating bees (c), dollhouse (V) , night sky (S), migrating geese (m), and the web of the spider (s).
This means that "Soft" has thematic relevance to at least 29 of the 30 variables in the-230 degree sector.
Whatever might specifically or in general be meant by "Soft", no diametrically opposite antonymous (anti-semous) property of 'hard' is explicitly named or localized by the countermap. Either this implies explicitly that the cartographic region is naturally unipolar or else the antipolar part of a properly bipolar region was overlooked in the construction of the countermap or was unrepresented by the chance subset of variables ('beauties') that were graphed in the primary map.
Can I discover a 'hard' region in the sixteen-variable gap (of 130 degrees 0.36 circles) between the arms of the "Soft" region?. I can check this in two different ways: by selecting a few variables in the gap at random to see if the imagined theme of 'hard' pertains to them, or instead by examining all of the variables in the gap (to once again see if they 'are' exclusively, mainly, or equally 'hard' or 'soft').
Use of a random number generator picks the beauty of evolution or ontogeny (W). In my opinion this 'beauty' is not 'soft, (at least not in the sense that the twenty-nine variables in the "Soft" region are soft), but I would also hesitate to associate 'hardness' with it. Of course, it does lie close to the edge of the hypothetical "Soft" region, so perhaps there are two neutral or ambiguous sectors between the soft and hard regions, and the variables lying in these sectors can be either soft, hard, soft-and-hard, or neither-soft-nor-hard.
In any case, if I look in the gap for 'beauties' that 'are' mainly or wholly 'soft' I come up without any bone in my mouth. Perhaps partial or minor elements of 'soft' beauty are suggested by machinery of the mind and brain (e), because of the operational subtlety of the machinery; by clockwork (0), which in loudness can be soft; by evolution or ontogeny (W, again, by analogy to the mind and brain; by tiny humans at foot of mountain (t), because of the 'soft' silence of such a scene; by massive or lofty tree (i) owing to the softness of the foliage and arboreal quiet, peace, mystery, and stillness; by cooperative endeavor (R)^a. and by magnificent body (g) because that is as apt as not to be the soft and rounded body of a woman. (This list includes 7/16 = 44% of the sixteen gap variables.)
If I look for 'beauties' in the gap that are mainly or wholly 'hard' I find: magnificent battleship (f), athletic competition (B), microscope L telescope, or other instruments (1), justice (e.g. villainy getting its due (a), bold architecture or elegant bridge (F), cancer patient's indomitable will to live (J), clockwork (O) probity and zeal for truth (q), concept of infinity or the Apeiron (Q), tiny humans at foot of mountain (t); and perhaps forest fire (Y) and evolution or ontogeny (W). (This includes 12/16 = 75% of the sixteen gap variables.)
Apart from this last category, perhaps partial or minor elements-of 'hard' beauty are suggested by magnificent body (g), massive or lofty tree (i), cooperative endeavor and machinery of the mind and brain (e). (Iluded here are 4/16 = 25% of the variables.)
To summarize my findings, then, respecting the sixteen gap 'beauties':
(1) MAINLY OR WHOLLY 'HARD': 75% (12/16);
(2) 'HARD' IN MINOR WAY: 25% (4/16);
(3) 'SOFT' IN MINOR WAY: 44% (7/16);
(4) MAINLY OR WHOLLY 'SOFT': 0% (o/16).
The evidence would therefore suggest that the analytic-regions countermap of 46 examples and sources of beauty is azimuthally divisible by a diagonal southeast-northwest axis into two opposite but highly unequal sectoral regions of 'soft beauties' (centered toward the southeast) and of 'hard beauties' (centered northwest), with both the azimuthal and populational ratios of the two regions (see figure
Once again it would be highly desirable to attempt to define, supplement, and explain countermap properties such as 'soft and hard beauty'. But properly this is a task that should be undertaken at a stage later than even that of the analytic-regions countermap, and that should produce a countermap of a higher order, a meta-countermap that might be referred to as "synthetic or justificatory countermap"
The purpose of this third MDS map would be, then: (1) to logically [interconnect and unify] the different [regions, structures, axes, poles, and concepts] identified in the analytic countermap; and (2) to [supply or speculate upon] the fundamental raisons d'etre of both the [synthetic and the earlier analytic] regions and patterns.
Perhaps one should really visualize a multistep, hierarchical process possessed of at least four distinct stages: 1 Primary Map.-Stage, 2 analytic map stage, 3 synthetic map stage, and 4 justificatory map stage.
In addition there are other types of countermaps that it would commonly be appropriate to produce when investigating ideonomic MDS data or primary maps, but that would not fall into the same sequence, or at least not unilineally.
An example of such would be a countermap that would note the various important or expected things-such as primary map variables, or regions in the analytic countermap-that are actually found to be absent in the, maps that are studied. Anontological maps of naughts of this sort may turn out to be quite valuable in the future statistical ideonomy.
There are two separate arrows that point directly east in the analytic regions countermap of 'beauties'. The one named "Fun, Sensual" I have already dealt with.
The arrow directly below it, titled "Delicate" is crudely indicated to refer to the entire eastern half of the countermap. This property of aesthetic delicacy has been mentioned several times already in connection with various variables, but I will now discuss it centrally and systematically.
Delicacy is a complex and ultimately somewhat obscure concept that is broadly applicable, both literally and as a metaphor, and that fairly cries out for refinement and generalization. The reader should keep this in mind as I proceed with my topographic analysis. The dictionary defines delicacy as "fineness or daintiness of form, texture, or constitution", but so many other senses of delicacy and delicate are also provided that, even though they are relevant, I will have to pass them by.
Delicacy most obviously contributes to the beauty of the following things in the primary map, beginning from the right side of the map and,working leftward, with the horizontal (Dimension 1) coordinate-of each 'beauty' indicated: +1.19 charmed banter, dalliance... (M), via the verbal, intonational, behavioral, and psychic nuances; +1.17 lavish banquet (c), via delicate morsels and flavors; +1.1 kaleidoscope (b), +-9-butterfly wing color patterns (H), via the subtle colorations and textural minutiae, +.88 lush and surreal rain forest (d), via the intricacy of its sights and life and their exotic variations; +.88 birdsong (D), via the delicacy of its warblings; +.86 child's to via their delicacy of construction; +.81 bubble bath (G), via the delicacy of the foam and individual bubbles; +.79 patterns of frost or dew (p), via the feathery ice and fragile half-held droplets; +.79 mountain meadow (n), via the delicacy of the ecology and of the patternings of innumerable flowers; +.54 bees pollinating flowers (c), via the anatomic delicacy of bees and flowers, and the bees' behavioral and 'sonic' delicacy; +.52 desert silence, openness, luminosity (T) via.the associations delicacy of the desert's scant ecosystem and the sensory delicacy of its immense stillness, quiet, and clarity; +.49 calligraphy (1), via the kinetic and visual delicacy of marking; +.47 enchanting centurial dollhouse (V), via the delicacy of all miniaturization; +.44 drifting bubble (U), via the kinetic, structural, and visual delicacy, and a bubble's delicate sensitivity to human and zephyrean movements; +.47 first love (X), via the delicate web of subjective and intersubjective interdependent and protean meanings; +.41 melody, song, chorus (k), via e.g. the hierarchic fragilitiet of music; +.2 coruscant stars overnight sky (S), via the delicate atmospheric scintillations of 10,000 all-but-imperceptible beacons; +.11 migrating flock of geese (m), via the delicacy of the birds flight formation and of the distant silent flapping of their wings; +.07 jewel (Z), via the symbolically enlarged importance of the microscopically perfect facets; +.05 spider's web (s), via tht flutter of the gossamer net, -.13 birth of one's child (E), via the fragility of embryogeny, the newborn, and the mother-child relationship; -.41 machinery of the mind and brain (e), via the unfathomable delicacy of their loom and of the cloth they weave from instant to instant; -.42 clockwork (0), most obviously via the dynamic intricacy of the innards of a watch; -.55 evolution or ontogeny (W), via their labyrinthal complexity that bridges and exactly balances nature's greates and least scales; -.99 tiny humans of mountain (t), via the symbolic delicacy of the juxtaposed opposites; and -1.18 microscope, telescope, other instruments (l), via the delicacy of their adjustment and of what they are capable of resolving or detecting.
There are a number of things that could be done to yield clues as to the nature of aesthetic delicacy (beauty-via-delicacy).
For example, one could try to think of as many examples and types of this form of beauty as possible, and then subject just this special set of 'beauties' to MDS. Dimensions and other patterns found in this way might then be imposed upon or used to clarify the general set of forty-six 'beauties' that have been under discussion here.
One might also attempt to imagine things whose beauty exactly combines "delicacy" with each of the other themes identified in the analytic-regions countermap, one combination at a time. For example, what things best synthesize beauty-via-delicacy with beauty-via-bigness, beauty-via-multitude, or beauty-via-height (which may comprige over 100,000 discrete rings), and Venezuela's Angel Falls (which-drops 0.98 kilometers), perhaps?
If one has mapped via MDS a set of ideas, but then takes a fractional subset of those ideas and has MDS create another map, or set of co-plots, based on the smaller set, geometric and topologic patterns will result that are somewhat different from those corresponding to the interrelations of the same subset of ideas in the original exercise.
Such changes need not be viewed as evidence against nonmetric MDS, or that limits the accuracy, reliability, utility, objectivity, or meaning of the ideic dimensions, rank-orderings, plots, structures, relations, regions, or behavior it purports to discover.
The question that must be addressed, before drawing conclusions of that highly critical sort, is whether the alterations induced possess [additional, their own 'transformationally' conservative] semantic patterns, or are instead simply manifestations of noise, error, or randomness. The latter would indeed be damning; the former would merely represent an adaptational challenge or opportunity for accomplishing something greater.
Careful experimental and theoretical attention to this matter has persuaded me that the more optimistic possibility is apt to be closer to the truth. In particular cartographic transformations caused by subset restrictions impress me as being both meaningful and irredundant - an enlarged opportunity associated with MDS.
I did in fact map such a subset of the 'beauties', but before I discuss what resulted I should introduce and expatiate upon the subject of paramaps.
The number of regions, so-called, that are identified in the analytic-regions countermap (or at least in the version given as figure ) is 33, and these include 61, or about twice as many, discrete terms representing more or less related concepts (related, that is, to the region identified and to one another).
Many though not all of these 33 regions and 61 terms have been separately mapped, for their major and minor positive and negative distribution over the primary MDS map of the mutual analogousness of the forty-six examples and sources of beauty, in figures-,- through . This suite of 18 related countermaps I will henceforth refer to as monothematic paramaps, since all of them are functionally equivalent and yet each concerns the cartographic distribution of a single, unipolar or bipolar, theme over the identical ideospace or primary MDS map of the ideospace. As will be seen shortly, these maps are ultimately intended to be analyzed in connection with one another, or for the codistributional patterns they exhibit when virtually or actually superimposed, two or more at a time. (Here "codistributional" does not imply that such spatial and amplitude patterns as may come to light when the different paramaps are superimposed will necessarily be wholly or even partially identical, but only that they should have a tendency to be consignificant, or mutually meaningful.
Figures (with the exception of figure), which I will distinguish in a moment) have all been color-coded in such a way that each of the forty-six 'beauties' they share have been assigned six alternative valuations in each paramap according to whether the 'beauty' in that paramap was intuitively judged to be related in a major, minor, or opposite way to the special aesthetic theme of each map. The six-color scaling was (1) RED, "Mainly or wholly related to the positive theme", (2) ORANGE, "Related to the positive theme in a minor way, but 'not related' to the negative theme (except of course negatively)," (3) CONCENTRIC COLORS, "Simultaneously related to the theme both positively and negatively (but in degrees indicated by the colors chosen)," (4) GREEN, "Related to the negative theme in a minor way, but not related to the positive theme (except of course inversely) (5) BLUE, "Mainly or wholly related to the negative theme", (6) ACHROMATIC, "Not related to the theme, either positively or negatively, even in a minor way."
The exception, figure is a 25-degree map that complements figure (as discussed below)
As I have already hinted, the purpose of creating all of these paramaps is to enable them to be subsequently combined or compared in intelligent and revealing ways. I should point out at once, by the way, that even with small sets of paramaps the number of such comparisons that are possible is stupendous: 176 dyadic comparisons for 17, and 528 dyadic comparisons for 33 paramaps (in the present instance). The number of possible decisionally-chained combinations of the same number of paramaps is, moreover, in infinite (depending only) upon the amount of ferment of the pertinent ideas, able to give rise to diverse predictive hypotheses, in the mind that makes repeated use of the finite set of basic combinations).
When different random or intentional, pair or larger, combinations of the present set of 17 paramaps are made, for exploratory purpose, one discovers that the patterns of simple and complex : overlap (intersection), avoidance, co-avoidance, concentration, intensification ('thematic , etc : of the different thematic distributions are ideonomically, extraordinarily meaningful and important.
They permit, indeed invite, the framing and predictive checking of innumerable theories and hypotheses with a bearing on the causes, effects, interrelationships, meaning, etc of the concepts that are (primarily,. secondarily, and tertiarily) mapped, and hence they afford means for attaining a far deeper understanding of the general nature and possibilities of beauty.
Of course all that has been said here applies no less to any set of ideonomic paramaps whatever that might be constructed and treated in these ways.
I must strongly emphasize that as, for example, the number of different MDS countermap regions or themes-and hence the quantity of monothematic paramaps-goes up, the richness of the associated ideonomic possibilities and opportunities will eventually, in effect, exceed a cognitive threshold or critical mass, and subsequently grow explosively. A veritable universe of investigatory possibilities will be created in connection with the original set of concepts that were mapped by means of nonmetric MDS.
All of this inevitably suggests marvelous possibilities for the design of ideonomic graphical software, both commercial and scientific.
To give substance and plausibility to these abstract assertions, I will now present the reader with a few of the actual comparisons of the 17 paramaps that I have made to date, where there have been interesting and illustrative results, though as always my examination of the possibilities will perforce remain largely superficial and not at all representative of the 'best' possibilities that a more systematic and exhaustive study of the same material would uncover.
Figures and respectively represent the paramaps for the spatial occurrence of beauty-via-multitude and of beauty-via-peacefulness. By "multitude" is meant numerousness, as opposed to fewness or singleness. "Peaceful" is a synonym for calm or tranquil, but the paramap neither explicitly names nor explicitly maps its opposite or opposites.
If you look at the distribution of those 'beauties' that are judged to maximally embody (mainly or wholly relate to) the two themes in question, and mentally superimpose them, you will see that they mainly represent two orthogonal arcs or bars that tend to converge and become superimposed toward the center-right of the primary map (whose underlying structure and coordinate system the paramaps retain).
When I noticed this I predicted to myself that if I referred over to an additional paramap, depicting the spatial distribution of the theme of beauty-via-charm, I would observe a tendency for beauty-via-charm to flourish in and near this region of overlap. And this is exactly what I did observe (compare figure)
I made the prediction because I intuitively sensed that a peaceful multitude ought to be charming almost generically.
Certainly one can imagine a scientist framing an hypothesis of this sort and then checking it out, quantitatively or qualitatively. Were there a contrary result, such a result would be puzzling, challenging, and useful, a prompt to new hypotheses, theories, and experiments.
Of the forty-six 'beauties', there were six that had previously been judged to be "maximally or wholly related" to both the "multitude" and "peaceful" themes: namely bubble bath (G), lush, surreal rain forest (d), patterns of frost and dew (p), bees pollinating flowers (C), coruscant stars over night sky (S), and massive or lofty tree (i).
Four of these six had also been judged equally related to the "charming" theme. The exceptions were bubble bath (G) and massive or lofty tree (i). The first was thought to be wholly unrelated and the second was deemed related in a minor way. "Charming" was a very difficult concept to cognize and valuate, however, and in retrospect I would think it obvious that charm must greatly contribute to or help explain the human beauty of both things. Yet I have no wish to be in a position of manufacturing excuses or of exonerating troublesome results.
Comparisons of paramaps can also be used to discover all, or at least diverse, ways in which the themes in analytic-regions countermaps differ. Thus if one compares the paramaps of the closely related aesthetic themes of "charming" and "delicate" one finds six discrepancies (respecting the valuation "mainly or wholly related"): migrating flock of geese (m) and birth of one's child (E) are judged to be that related to "charming" but only weakly related to "delicate", machinary of mind and brain (e) and melody, song, chorus (k), quite ihe contrary, are judged to be that related to "delicate" but only "weakly" related to "charming"; and judged equally related to "delicate" but NOT AT ALL related to charming are lavish banquet (C) and bubble bath (G). Of course an excuse has already been made for bubble bath (and could be as easily made for melody, song, chorus) but one also has to consider what the statistical noise level ought to be here (and elsewhere)
Can the nature of such beauty as is associated with theme "moral, right" be clarified by comparing the paramap of the latter with the paramap of some other theme? Examining the 16 other paramaps that currently exist, I find the distribution.of "singular, simple", in the paramap on the theme "diverse, complex : singular, simple" (where the intermediate colon signifies versus), especially analogous and appropriate cognitively. Truth presumably, and morality ideally, should be fundamentally singular, unified, and simple; at least, it is largely because we impute such properties to them that we think of them as also being beautiful. When confronting questions of truth and propriety we are forced to make a unique and definite choice, whereas at other times we are free to admire and affirm the infinite diversity, richness, and complexity of what is and can be.
When the monothematic paramaps upon the aesthetic themes of "delicate" and of "diverse/complex vs. singular/simpie" are compared, one finds that the regional distributions of beauty-via-delicacy and beauty-via-singular/simple are almost totally opposite and nonoverlapping in space, with four interesting exceptions (in terms of the valuation "wholly or mainly related").
This throws out a challenge to imagine what such a conceptual (thematic) combination might be like in the abstract (that is, to mentally demonstrate its feasibility or non-self-contradictoriness), and then to instantiate it: 'What beauty might arise from something by virtue of at once its delicacy and its singularity/simplicity?' Perhaps the nub of the problem is, can there be 'delicate singularity'? The paradox is wonderfully ideogenic.
The four interesting exceptions alluded to were the 'beauties' desert silence, openness, luminosity (T), jewel (Z), drifting bubble (U), and birdsong (D). All are at once delicate and singular/simple. Notice that first love (X) is near to being a fifth exception to the aesthetic rule that these properties are immiscible in the beauty things.
The "Moral/Right/True" southwestern region (in figure) interlinks (as shown in figure ) the northwestern "Mensural/Symmetric/Logical/ Functional" region to the southeastern Existential/Emotional/Atmospheric/ Subtle" region.
That the first region should occur where the other two regions abut is highly appropriate. Morality relates to the measurement, symmetries, logic, and functions of human existence and emotions. Similarly truth has to do with the collision of abstract patterns (of symmetries, measurables, etc) with real existence, feelings, and meanings.
Notice that the northeastern "Delicate" region (of the paramap "Delicate vs. Undelicate", presented as figure ) also interlinks the northwestern "Mensural/Symmetric/Logical/Functional" region to the southeastern "Existential/Emotional/Atmospheric/Subtle" region (in figure ), but from the opposite direction that the "Moral/Right/True" region does: in other words, from the northeast rather than the southwest.
This dual situation could be interpreted in the following way. The southwestern "'Moral/Right/True" interlinkage relates to the measurement, ETC of the emotions, ETC; whereas the opposite "Delicate'' interlinkage relates to the emotional enjoyment and artistic appreciation of mensural, symmetric, logical, and functional relationships. Such an interpretation would explain why the beauties probity and zeal for truth (q) and maternal or filial devotion (j) are located at the southwestern interregional convergence, whereas one finds the beauties melody, song, chorus (k), enchanting centurial dollhouse (V), kaleidoscope (b), and calligraphy (I) at the northeastern interregional convergence. Again, the former beauties have more to do with the measurement of joys, the latter with the joys of measurements. (To be honest, however,it could be argued that music (k) has an equal amount to do with both functions.)
The "Vertical/High vs. Flat/Low" and the "Big vs. Small" and the "Strong vs. Weak/Fragile" dipolar regions are, it should be noted, roughly coextensive and the signs of their trends are semantically correct (so that, for example, ''Vertical", "Big", and "Strong" coincide azimuthally) .
As already mentioned, one of the paramaps (figure) is of a different type from the others. It rank-orders the forty-six 'beauties' on a truncated 25-degree scale of beauty-via-delicacy. That is, it takes the subset of twenty-four 'beauties' that (in figure ) were previously valuated "mainly or wholly related to delicate or subtle" and estimates their beauty-via-delicacy rank-order from 1 to 24 via the scaling question, "What is the absolute contribution of a thing's delicacy to its beauty (or the absolute delicacy-related beauty of each thing)?" These finer degrees of aesthetic delicacy are color-coded in the figure.
The same sensitive treatment could have been given to all the regions in all of the other paramaps. Had this been done, it would have brought to light semantic patterns of a far more complex, subtle, and cogent (or testing) nature. That this is so is underscoreaby a comparison of the 6-degree and 25-degree paramaps (figures and ). The 'beauties' rated highest for beauty-via-delicacy by the latter are not concentrated, as one would have expected, in the center of the "Delicate" region of the former, but rather in its northwestern extreme. Hence simple reliance on the 6-degree paramap could be highly misleading.
I will provide just one example of the potential value of such a highly rank-ordered monothematic paramap, by means of the 25-degree delicacy map.
Comparing figures and it will immediately be seen that the strongest (most intense) beauty-via-delicacy occurs-as was by myself first hypothesized and then verified to be the case-where the "Beauty-Via-Small-With-Big Region" overlaps the "Beauty-Via-Delicacy Region". Which is to say that those 'beauties' that are rank-ordered -in the 24-rank delicacy paramap and simultaneously valuated as being maximally or wholly related to the theme of beauty-via-small-with-big, are almost all rated especially high in (placed in ranks 1-12 rather than 13-24 of) beauty-via-delicacy: patterns of frost or dew (p) is rank-ordered as #4, child's toys (N) as #19-it represents the one failure or exception, enchanting centurial dollhouse (V) as #6, coruscant stars over the night sky (S) as #3, machinery of the mind and brain (e) as #I, and microscope, telescope, other instruments (1) as #9. Of course, not all of the 'beauties' that were valuated highest for the theme "Small-With-Big" were also valuated "mainly or wholly related to delicacy": the two-of-nine-possible exceptions were evolution or ontogeny (W) and tiny humans at foot of mountain (t).
This convergence appears to suggest something about the causation-of delicacy-related beauty and, more generally, about the mental bases of the concept of delicacy itself. Time does not permit me to speculate in writing upon what.
Now that I have touched upon monothematic paramaps, let me discuss what happened when I mapped a monothematic subset of the 'beauties'. It was in fact this same set of twenty-four 'beauties' especially'related to delicacy that I independently mapped.
I did not do this mapping by employing the Triadic Method to produce a new set of data, but rather by exploiting the relevant subset of data in the original 46-beauty table. Partitionings of this sort do give rise to the changed maps, or sub-maps spoken of earlier. The smaller 24-beauty data set incorporated only 1,200 of the original 2,300 binary Triadic Method decisions.
Figure shows the reduced ideomap. Readers should be warned that the named 'beauties' are not coded by the same letters and colors that they are in the 46-beauty maps.
If the 24-beauty and 46-beauty dimensionality-two maps are compared (figures and ), it will be observed that the relative, and even the absolute, positions of the twenty-four 'beauties' are not so very different in the two cases (despite the distortion inevitably induced by the deletion of twenty-three other 'beauties'), except perhaps in the case of the beauties of coruscant stars over the night sky, of jewel and of first love.
Actually this degree of similarity is ordinarily to be expected where that remapped is fully half of the original set of ideas, rather than something like a mere sixth, say.
The twenty-four 'beauties', moreover, represent a very homogeneous region in the 46-beauty map.
So less conservative behavior would have been the real surprise.
It was not only the dimensionality-two map of the twenty-four beauties that I plotted and studied, but the entire set of ten dimensionality two - four co-plots (though these have not been reproduced here).
Logic would strongly suggest that the 'beauties' jewel and kaleidoscope share both appearance-related (phenological) and beauty-based (kalological) analogies and that they should therefore come together in space in at least some of the twenty D = 2-4 MDS co-plots of 'the twenty-four delicate related beauties', and yet inspection of the co-plots reveals that they never do so even once, and in fact they always remain stubbornly distant -almost opposite-so that at their closest approach in any of the co-plots (in the D1 x D3 co-Plot in D = 3) there are no less than seven 'beauties' more proximal to than kaleidoscope [to wit, in proximodistal order, calligraphy; melody, song, chorus; drifting bubble, clockwork; spider's. web; coruscant stars over night sky; and patterns of frost or and seventeen 'beauties' nearer to kaleidoscope than jewel.
What might this mean? Was the original 'classification' via the Triadic Method distorted, guilty of an oversight, superficial, or insufficient (e.g. were the scaling dyads used too few, positively or negatively biased, misappreciated, underappreciated, or perhaps mutually contradictory)? Or does nonmetric MDS itself suffer from some quantitative or qualitative limitation, defect, or quirk (whether slight, moderate, or severe)? Or am I just mistaken in seeing similarities between the two 'beauties'?
The existence or nonexistence of other logical or expected patterns such as this should be checked for systematically to determine what the : efficiencies, shortcomings, irregularities, mechanisms, and habits of ideonomic MDS are in general.