(See updates further down.)
There is a perspective of differentiation, distinction, and the building a classification in such a way as could lead to singling an object out but oftener seems to lead, in putting like with like, to the general, operating from many to one, as for instance successive differentiations of a curve can lead up to where rate of change is constant and the next derivative is zero and a lot of curves thus end up sorted at some level into a common class in that manner. Think also of a table of a group or any table of addition or multiplication.
|a b c d|
|a b c d|
b a d c
c d a b
d c b a
Each combination of two values leads to just one value, but more than one combination can lead to the same value.
It's like enumerative combinatorics turned inside out. Likewise, differentiation and integration are considered inverse operations. Generally the differentiation/algebra/information/life perspective is the "inside-out" version of the perspective which cuts across mathematical integration, measure, enumerative combinatorics, probability, statistics, and matter.
This leading of more than one (set of) value(s) to one (set of) value(s) is sometimes called "many-to-one" but that phrase is sometimes employed to signify a non-injective function.
Likewise by "one-to-one" is sometimes meant only a bijective function, not also an injective function. I find no distinct names for the kinds of relation which I'm discussing here. What would be good names for them? Also -- one-to-many (e.g., antiderivatives) and many-to-many (e.g., conservation equations). Note, for instance, that by "many-to-many" I don't mean merely many variables to be valued on each side of the equation, rather I mean that for each set of values for the variables on either side of the equation, there are, for the variables on the other side of the equation, more than one set of values which will satisfy the equation.
Update Friday, March 30, 2007
I've played around with coinages like "epluribune" (from e pluribus unum) but that one leads to formations like "epluripplure" and so on. "Unadune, pluradune, unapplure, plurapplure" might not be so bad yet who besides myself would ever use them? My best bet is to work from mathematical terms. Since "one-to-one" is sometimes used to mean "bijective," I thought -- why not borrow the prefixes from "bijective," "surjective," and "injective"?
One-to-one correspondence -- Bispondence.
Many-to-one or one-to-one correspondence -- Surspondence.
One-to-many or one-to-one correspondence -- Inspondence.
Then, to finish the job, I still have to engage in more invention than I'd like:
Many-to-one correspondence -- Surcospondence.
One-to-many correspondence -- Incospondence.
Many-to-many correspondence -- Plexspondence (does somebody have a better idea?).
These terms suffer from the ambiguities of their definitions. For instance, by "many-to-one" do I mean "many-to-at-most-one"? That's probably the most convenient sense.
Second Update Friday, March 30, 2007
By coincidence I met a computer programmer turned math schoolteacher this afternoon after writing the update above. He told me that he's sure that there are technical terms, which he's temporarily forgotten, for "many-to-one" etc., from discrete mathematics, and suggested that I google around including "C++" as a search term. I thought also of including "discrete mathematics" in some searches, and of checking an introductory textbook of discrete mathematics which I have lying around.
I looked. No luck!
Sometimes ours is said to be an "innocence/guilt" culture rather than an "honor/shame" culture, but in fact ours remains more or less an honor/shame culture, too. It's just that the terms of honor and shame have been shifted from -- for example -- that of proving deities' will by winning a battle, shifted and concentrated into proving things in terms ultimately of a scientific or science-influenced worldview.
It's said that the Ancient Greeks thought of perfect things as the real things. These were the things which they honored as real. Honor is not just about awards and symbolic certificates. Honor is about reality such that reality may not be reactively compelling us in the situation. This question of what we will honor is the question of what we will allow to determine us as if it were reactively undergirding or compelling us to do so. What agreements will we honor? What imaginary objects will the mathematician honor? To behave as if one had "something to prove" is a kind of behavior for the sake of honor. (Update January 5, 2011: I discussed this a bit further in a December 28, 2008 post "William Vallicella's Mighty Tetrad" at my blog What of these other fours? End of update.)
I remember watching a PBS history show in which they discussed ancient kings' desire for prestige, the desire to be counted among, say, the six big kings in some land. I don't think that there's any getting away from the interplay of ideas of power and authority/legitimacy etc. There's a similar interplay between ideas of resources, wealth, etc., and ideas of glory, splendor, radiance; and I don't think that that's merely because one allows you to buy the other -- unless one takes out the "merely" -- there's that which is prized for its own sake, and that which is prized because it helps you get that which you prize for its own sake. Power determines one by its own agency, but legitimacy and authoritativeness determine one because one grants it (as it may deserve) the status of that which determines one by its own agency; it is a stand-in for some determinant; one honors it as real. (One's slack for discretion in this matter, however, is not unlimited, and instead may, in the given case, very possibly be only for the short term; there are prices to pay for errors, even when the payment is deferred.)
Power (political etc.,
forcelike & directional).
Middle, means, approach
Note: the parallelisms of mechanical quantities with terms like "agency" and "act" should not be taken to mean that momentum, impulse, or force cause energy, work, or power as effects. They don't. These quantities were originally conceived in the effort to quantify amounts of "causativeness," "causedness," etc., but are not conceptions of causes and effects per se, much less are they conceptions of things related directly to each other as cause and effect.
Design and architecture seem to tend toward customized or tailored practical application. Questions of character and decision-making, ethics in its original broader-than-morality sense, seem more specially relevant to design and architecture than do questions of beauty and delight. There isn't a word generalizing from "ethics" to encompass both individuals and polities, so I'll just speak of "ethics" in a broad sense, like C.S. Peirce does, as including politics.
Since much of architecture is concerned with use by many people at a time, the politics seems relevant. I can think of a personal experience, a minor example, as might be expected from my more general lack of focus on architecture. I worked for some years in a building which, from afar, looked at once tall and hunched and not especially beautiful. I noticed a number of nice things about working in it: the floors aren't very large, so you didn't feel like you were working in a hive. This was a virtue made of necessity, since the company did not have a larger area on which to build (but they have finally gotten sufficient neighboring land and are preparing to build a second building there next to the first). Second, I noticed that its floors were so shaped that, at each expected corner, there were three corners, allowing not just four but up to twelve corner offices on each floor. (On the majority of floors, the "middle" corner in each of the four trios of corners had no office but had chairs, a table, full-length windows, and was reached by a walk-path turning the corner there -- quite a nice amenity for everybody). This was not a necessity but a design choice, one that definitely rendered less extreme and claustrophobic and sunlight-blocking the consequences of office hierarchy. An excellent design choice in political terms. (I didn't like the smaller upper floors, they were too small, a person (me) needed to use the elevators altogether too often, while the stairways were kept off limits by lawsuit frenzy and some folks' desire to deny a conspicuous advantage to the physically fit.)
It is in dealing with such issues, from the trivial to the important, that design and architecture are particularly distinct from purely aesthetic arts and it is in virtue of such dealings that design and architecture were traditionally included among the so-called "ruling arts." One doesn't have to be bored as I am with royalty in general in order to feel that "ruling" is a silly word for this sort of thing -- heck, architects have trouble from the start, just in getting the builders to build according to specifications. None of us give up our power to ruling artists so freely, and that's probably a good thing (except perhaps in the case of architects and builders, if you're an architect or builder reading this :-) ). Anyway I'd prefer to call them "governing arts" -- arts of governing, and of being governed too (I'm always reminded in this connection of the title of Wyndham Lewis's book The Art of Being Ruled). Such an art would also be an art of self-government. This perspective, that of the governing art (please let me call it, I can't stand the phrase "ruling art") seems the broadest one whereby to distinguish architecture and design from purely aesthetic arts, and furthermore it allies architecture and design with at least a few other fields, fields which are arguably sibs of architecture and design and with which they are particularly prone to overlap -- community planning, and whatever less formal arts there are for designing the frameworks in which we move and act and make decisions, and where the old architectural grand prize of being "a wonder" comes down to earth in the common good of freeing and empowering generally the people who use, inhabit, or otherwise interact with the work -- indeed, freeing and empowering them to govern themselves better . If there is an art of the education or development or evolution of personal and social character, it will fall under this classification. I suppose that civil engineering sometimes exercises a governing-art function. Bottom line, I would associate ethics -- in its broadest philosophically traditional sense, as including politics among other things -- with the governing arts especially and pre-eminently among the arts and sciences .
Aesthetics is important in almost any governing art, as important at least as scientific truth is in technology, engineering, medicine, and other practical/productive sciences/arts. In a way, an ethics is an applied aesthetics, the novel which is intended as a manual, from which characters really are intended to walk from off the page, as Frankenstein's monsters perhaps (let's hope not), whereas purely aesthetic art may purge and drain, like Dracula perhaps (well, I couldn't resist playing out the metaphor). The incorporation of more purely aesthetic values into the ethical/political view I would see as a function of the ethical/political values themselves. There are, for instance, beauties and graces fitting for emulation. So, anyway, I would see aesthetics as important in the governing arts, but not as their essential difference.
Now, Peirce has an even broader sense of "ethics" -- it includes the norms not only of decision-making but of carrying out.
While obviously there is in performance and execution a dimension which we in everyday language would call ethical, really there is also a distinguishable and very commonly distinguished dimension particularly for norms and values of performance and execution, a dimension of norms and values of competence, hicanoteta, as distinguished from character, ethos. In terms of semiotic resemblances, character is like the semiotic object, deciding, determining, and sometimes bumping and clashing with other things. Competence is like the sign, carrying forth. However, experience teaches that competence is no simply understood sign of character itself. Anyway, most of us are perfectionists in one kind of thing or another. Most of us are "merely-adequate-ists" about many other kinds of things. Certainly no know-how can come into existence without that which might be called care-how. Questions of "hicanotetic" or "competential" values and norms, questions of values and norms of performance and execution, as opposed to those of decision-making, are questions of care-how, and, pace Peirce, are not directly questions of ethics and character as usually and commonly understood. One sometimes takes pains to explain that one is criticizing a person's competence, not the person's character, or vice versa, quite in the manner of speaking of two different things, just as one sometimes takes pains to explain that one is criticizing a person's sensibility or temper, not the person's intelligence or assertions, or vice versa. If competence were even a species or subclass of character, it would not be considered cruel or unusual to treat a flaw in competence as, in and of itself, a flaw in character. We sometimes try to treat and remedy flaws in character as a special class of flaws in competence, but it's an uphill battle as we all know. It's a lot like trying to treat and remedy a flaw in sensibility as a flaw in intelligence and knowledge. A flaw of one kind may result from a flaw of another kind, and the "emotional IQ," as it is called, appears subject to improvement. But that doesn't logically mean that flaws switch or merge kinds.
The norms and values of competence pertain to all kinds of work, of course, but, among the arts and sciences, they pertain pre-eminently to engineering, technology, medicine, etc.. Medical doctors do frequently run into ethical issues, but I don't think that it's the same thing. Lawyers more frequently and naturally run into challenging questions of how to design and tailor the structure of a situation than do medical doctors -- not that trial lawyers in the USA respond to such challenges as constructively as they used to respond under erstwhile healthier ethical standards, practices, and attitudes no longer even taught, but that's another story. But this points to something which seems to be another distinguishing characteristic of the governing art.-- the tailoring.
A governing art seems to be typically a cognitive, conceptive discipline focused toward work which is "occasional" in that sense in which some poems are said to be "occasional" -- devoted to a concrete singular occasion at which the poem is read aloud. More technically focused fields such as engineering and medicine, while obviously involved in individual solutions to individual problems, are focused on the development of general, though still somewhat specialized, methods or techniques of dealing reliably with one situation after another after another after etc. Design is not so concerned with efficiently churning things out, though it can be concerned with designing a kind of thing which may be churned out. The only reasons that paintings and sculptures are so seldom reproduced (as compared, say, with novels) are technical and not because a painting or sculpture is the kind of thing which people see little point in reproducing -- except to the extent that a painting or, notably often, a sculpture, is part of an architectural work. Now, Georgia O'Keeffe's paintings tend not even to be photographable adequately -- yet this is a merely technical problem of reproduction . On the other hand, the Parthenon just isn't the same when it's not on the Acropolis, though in some places it may still be quite good. And I'm sure it's already not nearly as wonderful as it was when it was at the center of Athenian life. The point isn't the spectacle as re-creatable in one's mind. The point is the actuality of the thing in people's lives, our lives -- not only when we inhabit or use it, but even when it is remote from us or from some of us. It just seems natural to leave a good or great work of architecture in place in the concrete singular geographic and historical fabric in which it was made, rather than moving it to New Mexico, though sometimes that has to suffice.
I don't mean that the governing arts are "about" the concrete singular any more than I would say that the discovery sciences are flatly about the universal. The discovery sciences have an exhaustive diversity of typical viewpoint-scope. Still, on another level, they have nevertheless a kind of overall tendency to seek to discover universals, a tendency to seek to be logically determined by them in that sense. It seems to me that likewise in some sense, on another level, the governing arts have a kind of overall tendency toward to seek to design or to "architect" singulars, a tendency to seek to be logically determined by them in that sense, though some governing arts may be more general and abstract than others. (And one should not imagine resultant singular work as represented by a point dwindling to an infinitesimal "singularity." Instead, the work might instead be even a whole culture or civilization.) How to distinguish and relate these two "levels" of scope is a discussion perhaps for another time. But I'd add here that there is a further complexity in this as well.
Despite how my talk of universals and singulars may make it sound, I wouldn't regard the discovery sciences and the governing arts as being simply each other's symmetrical opposites. Discovery science aims to know or learn things in or on whose light or basis it will learn or know more things. Knowledge which leads to knowledge. With these at-least-two cognitive phases of the goal, both of them cognitive, discovery science is in a sense a pure type. But a governing art is, in that same sense, a mixed type. It aims to know or at least conceive things by whose power or liberativeness one may move, act, self-govern, decide things etc. (and "one," in that phrase, may represent the governing artist or the client or the material -- as some teachers regard their students), which conceptions are embodied first in designs, plans. Here we have, in some primary or simplistic sense, not a goal consisting in cognition in or on light or basis of cognition as in discovery science, but a goal consisting in will by a power or liberation cognitively conceived (though of course not necessarily everybody's power or liberation, e.g., jails are designed, too).
(Just a concluding note on the foregoing paragraph: The pure type involving will, conation, etc.would be decision-making with regard to decision-making -- deciding who or what gets to decide -- deciding by what power or freedom things will be decided -- and this seems to encompass political and military/martial affairs, enforcement, and even, vis-a-vis nature, fire-fighting and hunting where the hunt is not a routinely conducted harvest -- and generally, an affair wherein control is unstable and actually at stake at least somewhat dependently on choices and actions of people in the affair. Seemingly simple acts take on a political or more generally a "power" significance to the extent that they affect, e.g., set a precedent for, how further things could get decided.)
(Note March 3, 2007: I've edited this post after posting it earlier today.)
A number of years ago, before I read Peirce, I had scoured Roget's Thesaurus and come up with four time orientations of affectivity:
(The fourth one I called not "attachment" but "fondness" in the same sense.)
To these I correlated, also scouring Roget's Thesaurus:
(By "fancy" I meant something like imagination but possibly focused on one's actual setting or on some part of it.)
The times were ones which I which I came to align with special relativity's light cone:
1. One's direct
And to those sequences I correlated (again with RT's help):
1. Concept, percept, etc.
2. Belief, Suspicion etc.
Later, I read Peirce and, after some time, came to have some grasp of his object-sign-interpretant triad and his views on collateral experience.
So then it seemed natural to me to correlate as follows:
4. Recognition, verification, corroboration, legitimation, etc.
1. Conception, percept-formation, etc.
3. Inference to a concept, percept, etc.
4. Inference to a judgment.
I added the proviso that "knowledge" should be understood fallibilistically, and one should think in terms of "knowing something as..." just as one sometimes says "I remember him as being such-&-such." Now I wonder whether those phrasings are a bit of cheating. Anyway, I would say technically that knowledge is that which is sufficiently well confirmed that it can be fairly regarded as hardly worth further special efforts at confirmation. This is somewhat subtly yet decidedly different from treating "knowledge" as an achievement word meaning that p is true and that one believes p.
I've suspected that one of the reasons that some misunderstand the Pragmatic Maxim to mean that a conception's meaning consists in the actual consequences of its object (or even of the conception itself), is that there isn't room made in the object-sign-interpretant triad for a verificational stage. So the interpretant gets regarded in that way, obscuring its clarificatory rather than verificational character.
Verification, corroboration, validation, etc., involve experience of the object (or something verifiably counting as the object), and so are not interpretation or construal but instead are collateral to interpretation and representation in respect of the object, yet still for that very reason have a determinational role in logical processes, or at least so I claim. The two most common responses from others have been (1) that verification etc. can be reduced to relations of object, sign, interpretant, and, contrarily, (2) that verification etc. aren't semiotic per se and don't have an essential determinational role in logical processes. The first response depends on shifting the given logical frame of reference during examples. The second response leads me to ask, if logic (deductive and non-deductive) isn't about verification and arranging for verifiability, then what is it about? It would be like trying to make beef stew without beef. If confirmation and disconfirmation determine nothing in a supposedly inferential process, then said process is incapable of learning and is non-inferential. I've passed more than a smidgen of time in such arguments. Not wasted time, though; at least, not yet.
|Timeward modes of affectivity.||Desire.||Hope.||Delight.||Attachment.|
|Timeward modes of cognition.||Fancy.||Expectation, Anticipation.||Noticing.||Memory.|
|Times.||One's almost-now||One's unambiguously later.||One's just-now.||One's unambiguously earlier.|
|Logical psychological states.||Concept, percept, etc.||Belief, suspicion, etc.||Understanding.||Knowledge.|
|Logical psychological acts.||Conception, percept-formation, etc.||Judgment.||Inference to a concept, percept, etc.||Inference to a judgment.|
(Note that the aforementioned logical quantities are correlated, not equated, with levels of concrete phenomena such as physics, intelligent life, etc., and also can be correlated with divisions within each level.)
Thinkings about matter involve prominent use of symbols, while thinkings about life involve prominent use of likenesses, similarities of appearance. As if, respectively, (symbols) to represent night things and (likenesses) dawn things.
When matter and vegetable life get conflated (by philosophers) as co-occupying, vaguely or otherwise, a realm of classes in between universals and singulars, then, since matter is less attractive than life to many tastes, the inertial stuff ends up regarded as merely the night-enduring underside of the dawn-impassioned vigorous stuff. Indeed matter and life do evoke the distinction between proper time and time minus proper time, the distinction between mass and energy. But that's also a big distinction.
These two conflated opposed sides of "the" classificatory realm stand related as each other's complementaries or inverses, matter and life interlocking like probability and information, and distinguishable like the totalistic (gamut of elements, gamuts of combinations) and the mixed general-special (which really is in between the universal and the singular). This distinction between logical quantities is quite on a par with the distinction between laws or universals with many applications, and singulars (monadic or polyadized) among still more singulars. (The aforementioned four logical quantities exhaust the options at their level of analysis, but there are no technical terms for them.)
|Research level of concrete subject matter||Typical focus||Signs most prominently used in thinking on the subject|
|Physics.||Laws, universals with many applications.||Formal proxies, mathematical arrays and diagrams. "Dusk."|
|Matter science.||Totalities, gamuts, full complements of divisions.||Symbols. "Night"|
|Biology.||Classes differentiated and generalized.||Likenesses, similar appearances. "Dawn"|
|Intelligent life.||Singulars, sequences and congeries of singulars, among more singulars.||Indices. "Day."|
I obviously disagree with C.S. Peirce about some things, but not about the importance of lockstep trains of ideas.