Logical quantities, categories of research, and categories.

(Latest significant change: Thursday, July 31, 2006)

Note: "Classifications of Research" (in "Compare to Aristotle, Aquinas, & Peirce") is a more recent discussion of the subject addressed in this post here. In particular, there I address Peirce's science-classificational trichotomy of (1) universal elements or laws, (2) classes, (3) descriptions. I most recently discussed related matters in "Logical quantity & the problem of universals" at my blog The Tetrast2.

What a strange, weed-hidden yard is the subject of logical quantities such as “singular” and “general.” Considering the importance that the conceptions of such quantities have had in philosophy, it’s something of a shock to find the subject cultivated by philosophers with such nicety in certain of its corners but otherwise wild. Its terminology has fared worse, a sad thing of paper clips and rubber bands. Particulars, singulars, individuals, and related distinctions have been developed in various ways, distinguishing a pointlike atomic individual from an everyday singular like you or me (at times the terms have exchanged meanings). The subject matter as a whole has no good name. The subject matters to questions of scopes of denotation, pertinence, interest, etc., as typifying various perspectives cognitive and otherwise. Well, I suppose the only thing more dime-a-dozen than a professional philosopher dissatisfied with philosophy’s current condition is an amateur philosopher like me dissatisfied with philosophy’s current condition. So I have spoken merely the feelings common to almost every philosophical heart, though the philosophical topic occasioning the feelings may vary contentiously. I won’t propose a terminological reform but I will stipulate or define a few terms for the purpose of the present discussion. I will try to keep it brief and will get a bit breezy in passing through some of the technicalities.


Take the singular. When we hold a term to be true of only one thing, we usually hold it also to be false of something (indeed, of many a thing) — this is a second element in that which we usually mean by “singular”, and, to put it another way, we think of an everyday singular thing like you or me as being decidedly not alone in or as its own universe, but instead as being among still more singulars, indeed embedded among them in the world. But logically we need to become explicit about this, for the “road less taken” (in this case the one-object universe) in logic turns out, for its part, also to point, as occasionally happens, to a quite active and populous area of research. Now, if a singular thing were so alone, then it would also be a universal — the universal. It would be a universal-cum-singular and would be the universe, a one-object universe. It seems an almost blind window. But a singular is monadic, while a general or a universal-cum-nonsingular can be either monadic or polyadic. The singular is defined in opposition not only to the general but also to the polyadic such as xyz or John, Paul, George, Ringo. In that regard the singular is over-defined for the purpose of comparison to the general or universal. One needs a broader conception than that of the singular in order to see that the singular-cum-universal does not represent merely an almost blind window. Instead what peeks out there is a broad conception of a universe, a gamut cdefgab, a total population straight out of conceiving the conjunctive compounds of elementary logical quantities that characterize terms.

I wish I could rewrite this discussion of the polyadic singular or multisingular into much briefer form — “singular (monadic or polyadic, just like the general or the universal)” seems to sum this idea up. But I start to think of what a reader might say, and doubtlessly I’m not even managing to anticipate every objection. However, if the conception in question is already acceptable to you, I suggest that you breeze through the next four paragraphs.

Breeze on.
Importantly, it is of singulars in a sequence or in a congeries, be it a total population or only a sample, that one can affirm a relative or collective predicate such as “seven” or “35% are blue.” One may instead, for instance, affix a subscript “7” to an existence functor, but this turns out to need to be rephrasable back into forms which lead sooner or later to something like “xyzwusr”. Such singulars form into something like a plural, but “plural” is a term with specialized meanings in logic and, in any case, I’m pursuing a conception of a logical quantity, a denotative scope, of terms either monadic or polyadic just like the general and the universal. Now, a name like “mono- or poly-singular” is too lengthy for this.

So for the purpose of the present discussion I will call “transingular” (a) that subject term which is a singular monadic term or (“multi-singular”) polyad-of-singulars term and is either sequenced or unsequenced, and (b) that predicate true of just the singular or true polyadically of just the singulars whereof it is truly predicated; i.e., “transingular” means “singular” only as opposed to “general,” not as opposed to “gathered” or “sequenced.” (Note: in contemporary logic, “general” is said of a predicate term which does not purport as to logical quantity: however, herein, “general” is used simply as opposed to “transingular,” whether or not the issue of term purport arises.) One can think of “yzw” as a sequential polyadic singular subject term spelt out in sequenced monadic singular subject terms; one might instead devise an unspeltout version “x^3” etc. But we don’t need to fuss too much — the object of this discussion is not to make terms work our every whim with logical quantities, but to see the logical quantities as perspectives of all research.

As I said in “Why tetrastic?”: If one defines logical quantities such as the universal, the general, and the special such that they may be either monadic or polyadic, then one should likewise define the singular, even if it means giving the singular another name, so as to keep the parameter of monadicity/polyadicity consistently independent of the parameter of logical quantity; if one is proceeding exploratorily, then one’s logic should not prejudge which philosophical views one should adopt in such matters as whether there’s any point to defining a monadically-or-polyadically-singular logical quantity. Such an anti-pre-judicial consistency, in the logical exploration of logical quantity, matters especially when one is interested in grasping logical quantities in a general way (general like statisticality and information) as mental perspectives, scopes of reference, interest, etc., characteristically emerging even without formal articulate ado in research and intelligent decision-making, performance, affectivity, cognition, etc., of whatever kind. The singular is a case in point (and, by now, obviously a case on my mind), understood, as it usually is, as being in two oppositions, one versus the general, and the other versus the polyadic.

The conception of the transingular lets us treat logical quantities independently of monadicity and polyadicity, distinguishing logical quantity from adicity in thought, the better to recombine them as becomes certainly necessary. (Part of grammar’s power is in not being rigidly tied to its correlated categorial elements and this is not to be scorned by logic.) The phrase “polyadic singular” is no more a denial that relations lead to generality than the phrase “monadic universal” is either a claim or denial that monadicity is a thing of singularity. You see the importance of generality to relation pretty clearly in a term sequence like “xyzw”, which is pretty much bare of relational meaning but stands ready to receive relational meaning from predicates and other grammatical forms. Symmetrically enough, a monadic universal true of everything whatsoever has almost no meaning. Whatever juice of meaning such forms have is whatever juice of meaning can be squeezed out as ramifications of their syntactical structure, to which one may add that which arises in consideraton of philosophical questions whose outcomes are not necessarily built into the structure of one’s first-order logic. I should at least note, however, that sequenced monadic singulars, or a single-term polyadic subject “x^3” are one thing. A found cluster of indices of singular things is another. There, relationality and generality are already at play.
/Breeze Off.

Now, singulars, in any number, which were collectively all that are, would be, collectively, universal, the universe. They would become the denizens of an abstract conception. For instance, take the musical notes cdefgab as a universe — in such a universe they are the gamut of that which exists. That “X” which is true of them collectively is true of them such that there is nothing else of which “X” is true and nothing else of which “X” is false — at least there is no sequence which contains something y such that y is not among cdefgab. This perspective is the one found in an active and populous family of research areas — abstract studies such as deductive logic and the deductive maths of information and probability, where a universe of discourse or a total population is given such that “ceteris paribus” becomes “ceteris non existentibus” — “the rest [held] equal” becomes “the rest being non-existent.” This is not the everday perspective — the everyday perspective is that of a singular or bunch of singulars among still more singulars. The everday singular or bunch of singulars is not universal, is not the universe. For the purpose of the present discussion, I will call such everyday singular or bunch of singulars (or transingulars) “special” because it is decidedly not everything — in sum it is special-cum-transingular. For the same reason a general, like red, which is not universal, I will call “special” as well as “general” — in sum, “special-cum-general.”

Now, when faced with possible combinations of basic logical options, one should not just accept uncritically a philosophically traditional eclectic selection of combinations of those options. The point in particular here is that, when we call a term singular or “transingular,” it’s in negative answer to the question “is there anything besides this (or these) such that the term is true of it?” — if the answer were instead yes, then the term would be general (as herein already defined) — yet there is another question: “is there anything besides this (or these) such that the term is false of it?” If the answer is no, then the term is universal; if the answer is yes, then the term is special (in the sense which I have already stipulated). And, as I’ve suggested, a mind usually works with implicit answers to both questions. The two logical-quantity questions are independent parameters. It’s very important to realize that. Therefore, out of the four elementary logical quaantities — transingular, general, special, and universal — there will be four conjunctively compound elementary logical quantities. They in turn are, so to speak, the four corners of this floor, this level of logical-quantity logic. There is to examine them systematically and, when it’s not already obvious, to look for what it is to which they might apply. There really ought to be a technical term for each. I will provide “working” terms for the purpose of the present discussion. People may find them so funny that I will regret having ever mentioned them. I will try, at least, to make them evocative of their meanings but reasonably brief.

Stipulations (for the purpose of the present discussion) regarding elementary logical quantities of terms. Understand “true of these” etc. to mean “true of these in a polyad” etc.
“universal” = true of this and false of no thing that is not this (or true of these and false of no things whereof not all are these).
“general” = true of this and true of some thing that is not this (or true of these and true of some things whereof not all are these).
“special” = true of this and false of some thing that is not this (or true of these and false of some things whereof not all are these).
“transingular” = true of this and true of no thing that is not this (or true of these and true of no things whereof not all are these).
Incorporate criteria requiring one-to-one correspondences as needed or slackening as needed to compensate for sequence variety.

Basic conjunctively compound logical quantities:

~ ~ ~ ~ ~ ~ ~ ~ universal ~ ~ ~ ~ ~ special
general ~ ~ ~ ~ 1. etceterant ~ ~ 3. pluscernant
transingular~ ~ 2. solipsant ~ ~ 4. obsingulant

1. The etceterant: the universal but not transingular, a universal which is not the universe — i.e., the universal-cum-general.
The universal-cum-general is the perspective of that which is universal but is not the universe. This I will call the etceterant when said of a term, a predicate, etc., and the etceterate when said of things — e.g., “three” is etceterantly true of you, me, and the lamppost, while you, I, and the lamppost are etceterately three. We are three such that there are no mutually other xyz that are not three and there are mutually other xyz which contain something other than at least one of us. We mutually other xyz are etceterately three in that we are three just as are any of also the rest of the mutually other x y z ’ s . “Five” likewise is universal even though five things are not three. The point is that it does not matter what sorts of things are five, as long as each of the five is other than each of the others. Their attributes are irrelevant. Their being other than each other is not, for them, an “attribute.” My not being that there lamppost is not an attribute of me (or of that there lamppost or of us both). It might be an attribute relative to the term “that there lamppost.” But let’s not lose sight of substance. Likewise any three things can have the sequence schemata AABC, ABBC, etc. One has to qualify one’s statements carefully in regard to exact sequences, but one can do so. (One should not grasp at complications inevitable to any system of logical quantities in order to avoid unfolding the full elementary ramifications of elementary conceptions — one should not insist on scooting up and down ladders without having even explored a whole floor — some philosophers may have erred most unfortunately in this.) Now, the universality of mathematics is often not that of a number that can be expressed as a predicate affirmed collectively of some subjects. The grammatical form most germane to pure-mathematical universals appears to be the subject-forming functor, e.g., “inverted order of,” “double of,” “sum of,” “antiderivative of,” etc. It is on a kind of higher level that one says that some mathematical conceptions are more general than others. At such higher levels, the world in its wild variegation may be re-created.

2. The solipsant: the universal-cum-transingular. This is the perspective of that which is the universe. This I will call the solipsant and the solipsate. “Gamut” is solipsantly true of the notes cdefgab in the universe of distinctly lettered notes, and in that universe the notes cdefgab are solipsately the gamut.

3. The pluscernant: that which is neither universal nor transingular but “in between” — special-cum-general or, if you prefer, the non-universal general. This I will call the pluscernant and the pluscernate. “Red” is pluscernantly true of this ruby (there’s something else that’s red and also there’s something that’s not red) and this ruby is pluscernately red.

4. The obsingulant: the transingular which is not universal but instead is among more transingulars — i.e., the transingular-cum-special. This I will call the obsingulant and the obsingulate. “C.S. Peirce” (or, if you prefer, “= C.S. Peirce”) is obsingulantly true of C.S. Peirce (or, again if you prefer, “C.S. Peirce” obsingulantly corresponds to him), and C.S. Peirce is obsingulately C.S. Peirce.


The logical-quantity perspectives, the categories of research, and their typical inferential modes of conclusions drawn:

1. The perspective of pure mathematics is etceterant. Pure mathematics deals with universals such that there are always more things for the universals to be true of. Ever extending its universalities through structures of equivalances, it tends to draw reversibly deductive conclusions. “Reciprocation of premisses and conclusion is more frequent in mathematics, because mathematics takes definitions, but never an accident, for its premisses”—Aristotle, Posterior Analytics, Bk. 1, Ch. 12. Given the existence of the well ordered set to which it is applied, a mathematical induction involves a reversible deduction from the minimal case and the heredity, to the conclusion. The first and every succeeding domino falls if and only if every domino falls. (However, the proofs of the minimal case and of the heredity — I mean, of course, proofs which usefully don’t assume the induction’s conclusion — are not always reversibly deductive, especially when inequalities or greater-than or less-than statements are involved.) If there is no mathematical induction to prove a given statement about all members of a well-ordered set, then there is no proof at all for that statement about all members of the set (except perhaps via a richer mathematical level). Though mathematical induction is used throughout mathematics, it does not exhaust mathematics’ kinds of inference through equivalences. Reversibly deductive inference tends to be just the thing to logically gird and sustain the metamorphic imagination for the building of bridges across subtle differences, bustling diversities, stupendous gulfs, among appearances.

2. The perspective of those areas of mathematics which are called “applied” yet are regarded as mathematically deep or nontrivial is solipsant. These include the deductive mathematical theories of optimization, probability, information, and logic. They deal with totalities and structures of alternatives in those totalities. They tend to move from totalities to parts and to draw non-reversibly (a.k.a. strictly) deductive conclusions, which can also be fairly termed precisively deductive. (Deductive maths of information theory, however, independently developed, as laws of information, some of group theory’s principles, so some information theorists may really be essentially pure mathematicians in research interest and in tending to draw reversibly deductive conclusions.) By simplifying or closing many questions as details or complicative wrinkles, the mind intellectually abstracts manageable total structures of alternatives among cases; then by precisive deduction, it throws light from the whole onto the particular cases which began the overall question. Both moves seem reductive, but the first one, in implicitly or explictly closing questions, implicitly or explicitly adds information.

3. The perspective of those areas of research which are called “abstract” and are not dedicated to any special class of positive phenomena yet are regarded as deep in their treatment of positive phenomena is pluscernant. These include the fields of inverse optimization problems, statistical theory, cybernetics and much of information theory, and philosophy. Each treats of a problem which is the “inverse” of the problem treated by a correlated deductive mathematical field. The pluscernant fields (which I would argue fit Peirce’s definition of cenoscopy reasonably well), when not merely descriptive, tend to draw ampliatively inductive conclusions — working from samples or some similar pluscernant sort of thing, to make inferences to larger or total populations or some similar sort of thing.

4. The perspective of those areas of research which are called “idioscopy” or “the special sciences” is obsingulant. These include researches physical, material and chemical, biological, and human (or intelligent-biological). The objects or objectives of such studies are often general, but the subject matter is obsingulate things in those sought-after regularities and collectivities which distinguish them from each other and at least from things at least general enough to be treated more abstractly, as subjects of inverse-optimizational or statistical or cybernetic or philosophical theory or some mathematical theory. The idioscopic fields, when not merely descriptive, tend to move toward explanatory entities and laws (which are not mere inductive generalizations) and to draw abductive or “surmisive” conclusions — less compelling than straighforward perceptual judgments which are, as Peirce says in the Harvard Lectures on Pragmatism, CP 5.181, 1903, here and here, an extreme case of abductive inference, but more compelling than sheer guesses soever educated. Since these are surmises to laws, entities, etc., they may be overturned by just a few experiments, whereas the inductive inference to a trend might well be merely adjusted by a few negative results. A surmise may be so cogent that one reexamines, re-tests, and seeks to revise, for a better fit, the assumptions which occasioned its imaginative leap. However, a promisingly cogent surmise and, even moreso, a well established surmise on which much has been built, when coming into conflict with new experience, will tend (often reasonably enough) to be adjusted with various further devices of surmise instead of being swiftly discarded; much depends on whether an alternate surmise is available which explains both the experience previously explained by the now-challenged surmise and the new and unexpected experience. It should be added that the “inductive syllogism” whereby empirical verification is often considered to be achieved is, taken as a whole, an inference which both adds and removes information and is in that sense a surmise — a given kind of test has always worked in the past, ergo it has worked this time — and it is to be remembered that the removal of information in the course of inference is not the very same thing as the denial of said information but instead a shift or tightening of focus. Whether a given scientific inference about positive phenomena is best represented in the form of a surmise or an inductive generalization (or whatever else) is itself a matter of assessing what is the end or guiding research interest of the inference.

I discuss the division of research into families also in my post “What of these other fours?”. There it’s in regard to Max Tegmark’s theory of a four-level multiverse.

There is an overall tendency in the sciences and maths toward the etceterant, despite their diversity of scopes at other levels. Likewise there is an overall tendency in the affective arts toward the solipsant, toward universes, worlds, modalities, etc., in which various qualities and relations take on special and vibrant meanings. In the practical/productive arts/sciences, the overall tendency is to the pluscernant, the special-cum-general — not so universalizing as the theoretically oriented sciences and maths, but still interested in general methods which can be reliably reapplied. In the ruling/governing arts there is an overall tendency to the obsingulant, the custom-tailored so to speak.

4. Primary substance in the philosophical sense of this man, this horse, is obsingulate, most naturally represented by a subject term.

3. Attribute (unhypostatized) is pluscernate, most naturally represented by a predicate term.

2. Whetherhood (once known as “anity” from Latin an, “whether”), information, probability, optimality are (when unhypostatized) solipsate, most naturally represented by a kind of adverb, a kind of predicate-genic functor such as “not,” “probably,” “with a probability of 7/8,” “novelly,” “...if and only if p,” etc.

1. Otherness/identity and further pure-mathematical relationships and mappings, (operations, functions, antiderivatives, equations such as “x^2 + y^2 =1”, etc., are (when unhypostatized) etceterate, most naturally represented by a kind of subject-genic functor such as “another than” or “inverted order of” or “triple of” or “multiplicative product of” or etc.

The four basic conjunctively compound logical quantities — etceterant, solipsant, pluscernant, obsingulant — seem to characterize the perspectives of four quadrants of research not too badly. The characterization of those quadrants in terms of inferential mode of conclusions typically drawn helps fill the picture more. All the same, one is tempted to try to do it in terms of the concrete and various kinds of abstraction (if indeed they all should be called abstraction), for which the above outlining of categories seems suggestive.

1. There is a level where one abstracts various properties and relations, idealizes them to an “extreme of precision” as objects, and re-defines them in each other’s terms — i.e., pulls the ladder up after oneself, so to speak. Mathematics does this with especial rigor. Geometry and analysis are arithmeticized. In the course of such imaginative abstraction, one removes some information and adds other information. At the same time, just as proofs are typically by info-preservative inference, the mathematical characterization which one forms applicably to some situation (mathematical, concrete, whatever) involves neither purposefully increasing nor purposefully decreasing any uncertainty due to extraneous factors. Three oranges, five other oranges, 3+5=8. This is notwithstanding difficulties in obtaining exact information in order to state the problem in the first place, difficulties which present themselves across the board in research and not just in pure mathematics. What I mean is, that in distinction from all this, in a field like probability theory one typically purposely rules out things like the effect of flies on a horse race -- one does a lot of purposeful "summing over" and details-elimination, perhaps precisely in order to expose the involvement of a hidden factor by contrast with the probability calculation. Meanwhile in statistical theory one typically purposely includes the possibility of such an effect even if, outside the context of the study, one has reason to think that the flies aren’t a factor. Yes, I need a better way to say all this.

2. There is a level where one “abstracts” or at any rate info-increasingly “attributes” or “imputes” (or perhaps some other word is needed_ whetherhoods, universes, totalities, and structures thereof. At the same time, one reduces uncertainty due to extraneous factors, uncertainties which, be they details or not, would, if unclosed, complicate the resultant abstraction and which, if unclosed while one is pursuing a question of essential or underlying conditions, might make a difference and distract one toward idiosyncratic or superficial answers. One may want to show those closed-out factors by contrast, but also at some point in the one may cease to care about such questions at all — “the rest being held equal” becomes “the rest being non-existent” — and instead one starts to pursue deductive study of the abstract structures of alternatives and conditionalities brought into relief. E.g., in probability theory you start with a total population’s parameters as given and deduce probabilities for particular outcomes.

3. There is a level where we deal with samples, samplings, surfaces, qualities abstracted as samples, tastings, not in their own little universes, but as signs and tastings of a larger and somewhat opaque world. One may even ignore some facts which one knows or thinks oneself to know, in pursuing such signs’ patterns, without closing off the questions to which those facts are answers. This involves increasing the uncertainty of “outside” factors. A larger world is regarded as still there and is the objective of one’s inquiry. However, at some point in the abstraction one may cease to care about any particular case and inductively pursue, almost as samples, these inductive studies of samples — e.g., in the scientific case, one may pursue statistical theory as such.

4. The concrete things around us, the things as concrete, are taken as they are, with their information intact, “preserved,” free of abstraction. This is not to say that one is in possession of all that information. The fact that one is not in such possession, is one of the motives for abstraction — it’s in order to make the most of that which one does have. As regards uncertainties due to further factors, at this level one characteristically gets into both increases and decreases of such uncertainties (and sometimes physically enforcing the increases and decreases in controlled experiments), to isolate certain things in order to learn other things, some of which are supposedly “already known,” while holding other factors equal.

With regard to the semiotic tetrad of objectification, representation, interpretation, and recognition (recognition as inference to an acknowledgement, concluding judgment, etc.), there is a sense in which:
1. every objectification is (intended as) info-preservative,
2. every representation is abstractive,
3. every interpretant is attributive, and
4. every recognition is adductive (as adducing a judgment to serve as conclusion).

1. Etceterant (universal-cum-general, for the universal that isn’t the universe)	Pure mathematics	Object(s)-to-object(s) mappings, operations, functions, antiderivatives, equations such as “x^2 + y^2 =1”, etc.	Subject-genic functor	Info-adding-&-removing adductive ‘abstraction,’ imaginative	4. Reversible deduction (preserves information)
2. Solipsant (universal-cum-transingular, for the universe, totality, gamut)	Applied-yet-deep mathematics	Whetherhoods, probabilities, info (qua novelty, newsiness), logical conditionings & compoundings, etc.	Predicate-genic functor	Info-increasing attributive ‘abstraction,’ intellectual	3. Non-reversible deduction (decreases information)
3. Pluscernant (special-cum-general, for the non-universal general)	Abstract-yet-deep phenomenal sciences	Attributes	Predicate	Info-decreasing abstraction, sensory/intuitive	2. Ampliative induction (increases information)
4. Obsingulant (special-cum-transingular, for the transingular that isn’t the universe.)	Idioscopy, the special sciences	Substances	Subject	Concrete (info preserved), commonsense-perceptual	1. Abduction, surmise (adds and removes information)

Note the curious “inverse” relationships between the modes of abstraction/concreteness and the modes of inference of conclusions typically drawn.

As regards the trichotomy of Peircean categories (firstness, secondness, thirdness, often characterized respectively as quality of feeling, reaction/resistance, and representation), they don’t completely disappear in my system, but they are so revised that they aren’t really the three Peircean categories any more. Peirce in “On a New List of Categories” presented the three categories as a trichotomy of categories of accident (or as it is now commonly called, modification), while in Peirce’s later work his non-accident categories (being and substance) are more or less unseen and perhaps eventually absent in his categorial picture (this pertains to his development of the logic of relations). I do see categories (or perhaps I should call them subcategories) of attribute (Peirce himself seemed to employ the term “accident” as a category’ name somewhat reluctantly), and they are revisions or reconceptions of the Peircean three, plus a fourth. All of them pertain to feeling in the sense of sensation and (in the contemporary sense) intuition — and the third of them also especially to affectivity. My views on this tetrachotomy of acccidents are neither tested in argument nor long held by me, so they will seem much more intuitive than argued, struggling to amount to informal qualitative inductions.

First, reactivities, resistances, instabilities, sensitivities. They connect concrete singulars, tend to obey variational a.k.a. optimizational rules, and are such sensitivities that each case is unique in its way, even though they are attributes. Now, according to remarks which I’ve encountered on the Internet, inverse optimization is a young research field, maybe nine or ten years old as of the present writing. Usefulness is seen for it for sensitivity studies. I would expect that it will turn out to be illuminating in the study of chaos and catastrophe; I imagine that an inverse-variational or inverse-optimizational process would involve the emergence and evolution of extremizational rules in phenomena; at any rate, I expect inverse optimization to become eventually a major discipline not unlike statistics, for it deals with the inverse of the problem in deductive maths of optimization, just as statistics deals with the inverse of the probability problem. In terms of pure maths, the difference between inverse optimization and statistics can be related to the difference between two wings of combinatorics — graph theory, which deals with many-to-many relationships and is a basis for deductive maths of optimization, and enumerative combinatorics, which deals with one-to-many relationships and is a basis for probability theory. A reactivity/resistance is, in a sense, an index stripped of its garb (or perhaps a generalization of the index, a subindex or something like that). Reactivity and resistance, as a kind of insistence, are echoed in human conceptions of the good as strength and arete.

Second, qualities, rhythms, norms. A quality is already a kind of norm, a homogeneity, a likeness across some space or time. Likeness holds not only across discrete things but across expanses of stuff. Statistics exploits likenesses and unlikenesses in order to infer to a total population from representative samples, representative in elementary cases such that the totality will be like the sample. Qualities are echoed in human conceptions of the good as assets and resources. A quality, rhythm, norm, is, in a sense, an icon stripped of its garb. The raw quale has a persistent, somewhat arbitrary suchness, just as a brute concrete force has an insistent arbitrary thisness or haecceity. The Peircean conception of the quality seems to involve conceiving the quality as something as exact and singular in its way as an atomic pointlike individual. I speak of qualities in a less abstract way, just as I usually mean singulars and individuals in the everyday sense. For the time being, I’ll keep these conceptions of exact things separate from the others, at least until I understand them better. I think it’s important to remember that qualities have some relativity to the mind sensing or intuiting them, and that one of the reasons for values, meanings, etc., is, so to speak, to try to translate qualities across modalities, universes of discourse, perspectives, etc.

Third, values, meanings, importances. I’m unsure here whether to think primarily of the valuedness of a thing, or primarily of the values which hold for a living thing, or to somehow slough over the distinction. A value or meaning is, in a sense, a symbol stripped of its garb, and this category corresponds in some ways but differs in others from Peirce’s category of representation. While information theory does not, at least as yet, treat of meaning in the human-linguistic sense, it does treat of importantness, seldomness, dearness. Inductively inferential information theory in taking on cybernetic and biological orientations will tend to deal with values and importances for the given system. Values, meanings, importances, pertain especially to affectivity, just as qualities, rhythms, norms pertain especially to ability and competence and skill, and just as reactivities and resistances pertain especially to will and character.

Fourth, legitimacies, validities, authorities. I’m unsure here whether to think primarily of the legitimacy attributed to a thing, or primarily of the legitimacies or legitimacy standards held by an intelligent thing, or to somehow slough over the distinction. A legitimacy is, in a sense, the garb-stripped form of a sign which I call a “proxy” defined by the recognition which it would merit as a sign counting as the object for an observer for purposes in interaction, observation, experiment, etc. (this is discussed in the previous post, below the current one). Philosophy tends to deal with questions of what is legitimate, valid, authoritative, etc., in a given system. Legitimacies, validities, authorities pertain especially to cognition and intelligence.

The neatness of the boundaries shown below among research fields in the chart does not reflect any neatness of division among the research disciplines themselves. At the very least I would rather have the colors blend into each other at the places where boundaries are shown, but a blog has its technical limitations; I wish to run up neither expenses nor file size. For instance, I’ve pared the html code for these tables down to a minimum. (Update: I’ve given in and incorporated plenty of graphics, but have found that there remain limits.) A table like this cannot reflect the kinds of overlaps and cross-applications found in reality. The relative WIDTHS of the columns for the research fields are NOT meant to reflect anything about the relative volume or importance of work in the given field.

Pure mathematics. Etceterancies. Tends to reversibly deductive conclusions.	Applied-yet-deep mathematics. Solipsancies. Tends to non-reversibly deductive conclusions.	Abstract-yet-deep phenomenal sciences. Pluscernancies. Tends to ampliatively inductive conclusions.	Idioscopy. Obsingulancies. Tends to abductive conclusions.		Categorially correlated human power.	Categorially correlated sign.
Many-to-many relationships.	Deductive maths of optimization.	Inverse optimization problems. Reactivities, resistances, sensitivities.	Forces.		Will and character. Strength, arete (virtus, virtue).	Indices (&/or subindices?).
One-to-many relationships.	Deductive maths of probability.	Statistics. Qualities, rhythms, norms.	Matter.		Ability and competence. Assets, resources, stamina, reliability.	Icons.
Many-to-one relationships.	Deductive maths of information.	Descriptive & inductively inferential information theory. Meanings, values, importances (if as studied by info theory, then not in the human-linguistic sense)	Life.		Affectivity and sensibility. Wellness, vigor, vibrancy.	Symbols.
One-to-one relationships.	Deductive maths of logic.	Philosophy. Legitimacies, validities, authorities.	Intelligent life.		Cognition and intelligence. Firmness, solidity, integrity.	Proxies.

One will notice that the way in which these fields naturally line up leads to an odd-looking situation. Those who hold logic to be basic among maths are often those who hold physics to be basic among the special sciences. Yet logic is at the maths column’s base, while physics sits atop the special sciences column. It doesn’t take long to see what is happening. To put logic first is to put proof first. That’s to put things in the order of knowledge, as tradition called it. To put physics first among the special sciences, on the other hand, is to put things in the order of what one takes to be fundamental “out there.” It’s to put things in the order of being, as tradition called it. Those who do both are not being quite consistent. I guess it’s because physics is so mathematical in the structure of a given theory, and what they really like is deductiveness.

If we were to put the special sciences into the order of knowledge, then sciences of the human and social would come first. That seems natural enough if we wish to put things in the order of how we know things. We would want to know about minds and perception and society, especially those of the scientists among us. Between deductive logic and the sciences of intelligent life, which comes first in the order of knowledge? I suppose it depends on whether one means deductive proof or personal familiarity. I tend to think that it’s more in the spirit of these things as they appear on the chart of research fields, to regard humanity as first in the order of knowledge. I’ve also wondered whether one could illuminatively characterize two more orders: not only ABCD and DCBA, but also BADC and CDAB. (The Peircean Gary Richmond is pursuing the idea of six orderings or “vectors” of Peircean trichotomical and triadic elements.) It’s occurred to me that a test of the “majorness” of a reseach field may be its capacity to be conceived as first in one or another order before the rest.

By an intuitive constraint of lockstep patterns, if physics is so very “pure-math-like,” then the material sciences should be rather “applied-yet-deep-math-like,” while the biological sciences should be especially flavored with the abstract-yet-deep phenomenal sciences, and the sciences of intelligent life should be “especially” idioscopic. I wonder how well that pattern works.

Now, mathematics is notoriously difficult to subcategorize. The subcategorizations which have appeared most important to mathematicians are not even shown in the chart, since another kind of subcategorization turns out to line up best with the subcategorizations shown for the other quadrants of research.

1. Many-to-many relationships: graph theory, various areas in topology, simultaneous equations, extremization. I think that Morse Theory (the generalization of the calculus of variations) belongs here rather than in deductive maths of optimization. I would look to the mode of conclusions typically drawn: reversibly deductive, mathematically inductive (as with other pure maths)? Or non-reversibly deductive?

2. One-to-many relationships: enumerative combinatorics, measure theory, integration.

3. Many-to-one relationships: algebra (theory of calculation), derivative functions.

4. One-to-one relationships: order (well-ordering, etc.), relations, conditions for applicability of mathematical induction. I suspect that the theory of limits belongs here. Generally this subcategory seems to include mathematical foundations. It tends to be concerned with the study of sets in terms of conditions for mathematical induction.

As you can see, I have only the rudiments of a classification. I would not hope to place every area in pure maths neatly into one of those subcategories; rather I would hope that the four subcategories would prove useful in characterizing complicated overlaps and intra-applications of pure maths. Still, it would be nice if I could say more about geometry at this level.

Mathematicians consider especially discreteness versus continuity of the researched set or series or etc. in distinguishing research fields; such distinctions are especially important in foundations. One may consider such distinctions as constituting a here-unshown dimension of subcategorization and as running cross-wise to those shown, such that one would expect it to extend through all quadrants of research, though I don’t know what are the best analogous distinctions in the phenomenal research fields. There appear to be four subcategories of this kind which are significant in mathematics as commonly researched and which I will phrase in terms of sets:

1. Mathematics of sets larger than the continuum of points. The set of curves or functions is greater than the set of points in the continuum. So, functionals (functions of functions) and other mathematical relationships ranging over functions fall into this subcategory. It also includes, of course, non-Archimedean maths, with hyperreals or surreals. I don’t know whether the set of functions of finite numbers of real variables is considered to be the size of the set of hyperreals or of some larger set. I have heard that the hyperreals are nested within the surreals and that both lead to non-metric conceptions of space; I don’t know whether, as a rule, non-metric continua are non-Archimedean). As I understand or misunderstand it, global Lie groups (originally known as infinite continuous groups, infinite in the sense that each such group amounts to a continuous function with an infinite number of variables) represent, as functions, relationships of the kind which belong here. At least I think that global Lie groups and infinite continuous groups are the same thing. I surfed the Internet a good deal in order to learn that a few years ago and now I don’t know where my references went. Mathematics is like a maze of underwater caves. Whether the algebra (theory of calculation) of global Lie groups (their originative context) belongs here, I haven’t the soggiest idea.

2. Mathematics of continuum-sized sets. This includes a lot of mathematical analysis and point-set topology. Local Lie groups (not a very active field, from what I’ve gleaned on the Internet, and originally known as finite continuous groups, each amounting to a continuous function with a finite number of variables) represent functions which belong here (again, this is assuming that I remember my way around mathematics’ underwater caves). Maths of continuum-sized sets includes the calculus and analysis which one learns in high school. Theory of limits, derivative functions, integrals, extremization, etc.

3. Mathematics of the continuous but countable (e.g., the rationals). This includes a lot of non-analytic maths.

4. Mathematics of the discrete. - posted by The Tetrast @ 2:24 PM [top of post] [top of page] Copyright © BenjaminðA.ðUde11.

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