The Tetrast
Sketcher of various interrelated fourfolds.

What of these other fours?

July 14, 2005.
(Recentest significant change: a badly needed correction in the section on source, encoding, decoding, destination, on April 7, 2012). This post is increasingly superseded by my adjunct blog What of these other fours?.

Here I talk about correlations and discorrelations between my tetrastic structures and some of the fours —fourfolds, tetrachotomies, tetrads — which I've encountered in reading.

The Four Causes

(efficient, material, final, formal) and their Principles (agent, patient, act) of Aristotle and the Scholastics — they're part of what got me into fourfolds.

Important update Oct. 29, 2013: I recently got over a major brain glitch about the beginning-means-ending-check four-fold. I had long thought of my fourth stage (entelechy, "check") as involving verification that one has, so to speak, hit a target. So the prospect of verifiability could influence one's choice of targets. That was just not a strong enough conception of entelechy's part in the four-fold, and it kept nagging me though at length I drifted to other concerns. Then something that Dewey said finally set me on a better path . It's not just a matter of verifying one's "hits," but of whether the goals were good ideas in the first place. The prospective entelechy helps us consider unintended consequences, develop general values, and deal with conflicts among values. Thus entelechy guides ends (not to mention means and beginnings). Hedonism focuses only on end, telos, culmination, etc., and ignores entelechy. That's it in a nutshell. See "Telos, entelechy, Aristotle's Four Causes, pleasure, & happiness" at The Tetrast. End of update.

I discuss them quite a bit, especially in my posts: "Compare to Aristotle, Aquinas, & Peirce"; "The Four Causes" at The Tetrast4 - What of these other fours? (I made a whole blog out of the post that you're reading); and "The Four Causes, their principles, special relativity, Thomistic beauty, and I touch on them elsewhere, especially in "Why tetrastic?".

Basically, I think that Aristotle and the Scholastics missed the potential symmetry in the causal principles (agent, patient, act). Aquinas subdivided act into action and form, so that he had something like four causal principles for four causes, but it still isn't regular enough. The principles must be re-conceived, somewhat, as the agent, the bearer, the act, and, furthermore, the borne — which seems too neat and simple to be true, surely philosophers thought of it and, apparently, dismissed it for whatever reason — but for my part (I'm no expert on the ancients or the Scholastics) I don't know of any ancient or Scholastic philosopher's considering it (and dismissing it or otherwise), and it works. The borne (or "borneness"), as a cause, is the more-or-less stable balancement of forces, the form or structure, the standing finished, the settlement, establishment, entelechy.
Tetrastic version of the Four Causes & their Principles.
Beginning, efficient cause.
Source of change or rest.
Compare versus net momentum,
impulse, force.
Middle, means, material, process.
Mediation of change or rest.
Compare versus rest mass,
rest energy, internal work & power.
End(ing), teleiosis.
Culmination of change or rest.
Compare versus linear energy,
work, power.
Check, structure, entelechy.
Settlement/resolution of change or rest.
Compare versus internally balanced
momenta (potential & kinetic), impulses, forces.

Note: Momentum, force, etc., do not "cause" energy, work, power, as "effects."
Instead the quantities were originally conceived of in the attempt
to quantify "causativeness" and effect.
Tetrastic (elaborated)
Existence (consistently extreme version).Efficient cause.Sustainer.Consumer, exhauster.Assimilator / suppressor.
Causes as turns of becoming.Beginning.Middle, means.End (-ing), teleiosis.Check, entelechy, standing finished.
Causes as stages.Impetus.Development, process.Culmination.Settlement, establishment.
Static or quasi-static causes.Essential tensions, pressures.Composition, material.Differentiation, diversification (into parts, organs).Unitary structure.
Human causal principles.Will, conation.
Character. Virtues, vices, etc.
Ability, dealing.
Competence. Métiers, etc.
Sensibility. Values, etc.
Intelligence. Knowledgeability, etc.

Light cone.

Einstein's four zones of communication and cause & effect (the light cone)

— the future light cone's surface, the future light cone's inside, the past light cone's surface, the past light cone's inside. I incorporate a generalization of that ubiquitous structure into my tetrastics. My best discussion is probably "Special relativity's light cone & the mind's temporal perspectives."
Tetrastic 4x4 of modes of the psyche.Will, conation:Dealing, ability:Affectivity:Cognition:
(Like future light cone's surface.) For almost now:Trying.Testing, devising.Desire.Fancy, "impression."
(Like future light cone's inside.) For later:Seeking.Preparing, approach.Hope, confid.Expectation, anticip.
(Like past light cone's surface.) For just now:Taking, picking.Achieving.Pleasure, satisf.Noticing, discernment.
(Like past light cone's inside) For earlier:Adherence, habit.Maintaining, skill.Attachment.Memory.

Claude Shannon's communication-theoretic scenario when cast as four stages 

Communication theory
1. Source.
2. Encoding. 
2½. Channel.
3. Decoding. 
4. Destination.
Tetrastic semiotic
1. Object.
2. Sign. 
3. Interpretant. 
4. Recognizant, veri­ficant, establishment.
which are source, encoding (sender), decoding (reciever), and destination. (However, the communication channel is often included as a stage, on a par with the others and between encoding and decoding. Solutions to the challenge of the channel and its noise shape a lot of communication theory; but the challenge is to minimize the noise and avoid information loss; the generation or modification of signals is desirable only in the other stages.) My augmentation of C.S. Peirce's semiotic process (a.k.a. semiosis) to include a fourth and (dis)verificatory/(dis)confirmatory stage brings semiosis into alignment and correlation with the fourfold version of Shannon's scenario. The field of experience in or against which a decoding (and ultimately the encoding and source as well) is tested is, first of all, that of the destination. Note: semiosis differs in that it is not code-bound like info-theoretic communication; the continual renovation and occasional redesign of a communication system is a kind of "evolution" (pace biologists!) whereby semiosis is arguably definable, and which, I argue, comes about through such testing. As for channels and noise, I'm unsure of precisely what the semiotic analogs to them would be.

E.J. Lowe's four ontological categories

. E.J. Lowe, also known as Jonathan Lowe, proposes four ontological categories — Kinds, attributes, objects, and modes (property-particulars, a.k.a. tropes, e.g., this redness). There's a kind of double half-way correlation to my fours.
E.J. Lowe's four-category ontology
(a.k.a. generals)

Kinds are
characterized by

Kinds are
Attributes are
Attributes are
(a.k.a. singulars,

Objects are
characterized by

a.k.a. tropes,
e.g., this redness)
Tetrastic version of logical quantities, categories, & classes of research, at a glance
Conjunctively compounded logical quantity:Category with affinity for the logical quantity in the first column:Class of research with corresponding subject matter (objectives may be broader):

••  ••
••  ••     ••  ••

Correspondences / variances
("another than," "double of," "inverted order of," "sum of," "anti­derivative of," etc.).
'Pure' mathematics.
Pie chart. Photobucket(e.g., total popula­tion & para­meters).
Modes of attributability
("indeed," "not," "if," "novelly," "with a probability of 37%," "feasibly," "optimally," etc.).
Applied yet mathema­tically deep/nontrivial mathematics
(maths of optimization, probability, information, logic).
(i.e., neither singular nor universal).
# # # # # …
* * * * *
# # # # # …
Accidents, attributes, properties, etc.
("strong," "weak," "dependable," "healthy," "brave," etc.).
General ('domain-independent') studies of positive phenomena
(inverse optimization, statistics, inductive & descriptive info theory, philosophy).
A. B-C. D. E. F. G. …
(primary substance: "this man," "this horse," etc.).
'Special' sciences
(sciences of forces, matter, life, mind).

Basically, Lowe considers, as comprising his first categorial dichotomy, the particular (the singular or individual) and the universal (by which he means that which has more than one instance, and which I instead call the general). Logically there are four such quantities, not just two, arising simply and naturally. But Lowe is eclectic and doesn't consider as categorial divisions (1) the sweepingly universal in the sense of that which is true monadically or polyadically of everything ("one," "two," etc.) and is therefore extremely formalizable, or (2) the special in the sense of that which lacks such extremely formalizable universality ("blue," "elastic," "Jack," etc.).

Perhaps the biggest impediment in taking inventory of logical quantities has been that we don't usually consider both the monadic singular and the polyadic singular (or polyad of singulars) as being, both of them, singular, just as we consider both the monadic general and the polyadic general as being general. Another impediment has been a common initial veering into regarding the sweepingly universal only as a highest genus, strictly monadic, trivial for most purposes, and confined to a narrowly gabled attic, so to speak, of the house of logical quantities, an attic with room for just one such universal, logically equivalent to every such universal. Perhaps a third impediment has been some sort of neglect about defining logical quantity for terms through some same question or questions asked in all cases.

Given a term "H" true of something (call it "x"), the question of its logical quantity then depends on quantification over the rest of the universe of discourse:
Is there something which isn't that thing x and of which the term "H" is also true?
If no, then "H" is singular. If yes, then let us call "H" general.
- and -
Is there something which isn't that thing x and of which the term "H" is instead false?
If no, then let us call "H" universal. If yes, then let us call "H" special.

The twin questions stand mutually independent and resolve into four answers, conjoinable in four ways (see the table "Tetrastic versions..." above), notwithstanding issues of term purport which multiply relevant options. For the polyadic case, incorporate criteria requiring one-to-one correspondences as needed and slackening as needed to compensate for sequence variety.

One may think at first glance that one of the conjunctions, the universal-cum-singular, enframes a nearly blind window, looking out only on the case of a one-object universe. But let us practice consistency of conception, avoiding special wrinkles and complications, and classify the singular and the singulars-in-a-polyad together as singular in logical quantity, just as we class both the monadic general and the polyadic general as general. Then the monadic-or-polyadic singular-cum-universal comes forth naturally as a logical quantity corresponding to a gamut, a total population and its parameters, a universe of discourse, etc., supporting for example a collective predicate such as "30% (are) blue." (Those collective predicates are pretty hard to get without polyads.) Usually when we think of the universal, we think of something like a law, with many, even indefinitely many instances. That is actually a compound quantification; such a universal is also non-singular, i.e., also general.

Of course, in the sense that two are not three, "two" is not universal. But "two," "three," etc., are universal in the sense of being true of anything in some polyad or other; the qualities, the particular attributes, of the counted objects don't matter; only each object's being other than the others matters. Then we abstract the numbers and think of them as singulars. Thanks to its imaginative apparati, mathematics can re-create the world's logical quantitative diversities and variegation on abstract levels.

Now, since a universe-encompassing polyadic subject fully spelt out in sequenced monadics is sometimes daunting, consider a universe-encompassing relative or collective predicate; consider a universe-encompassing expressionally streamlined polyadic subject; and consider also a predicate-formative functor such as "with a (frequency) probability of 35%."

The question of variety among exhaustive sequences of the same total population's members is not a vexatious complication (raising the question of whether the total population is really "singular" or "multi-singular" or whatever you wish to call it) but instead a good complication and part of the solution to the question of what might be interesting about the monadic-or-polyadic singular-cum-universal logical quantity as a perspective. (In any case it's not limited to a one-object universe.) It goes to show that one should check to see whether one has defined parametric options in a consistent manner, especially in order to avoid jumping to conclusions about seemingly trivial or seemingly near-empty compounds of parametric options.

If one defines logical term quantities such as the universal, the general, and the special such that the terms 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 term quantity. If one is proceeding exploratorily, then one’s logic should not be given special wrinkles in order to prejudge such questions as whether there’s any point to defining a monadically-or-polyadically-singular quantity. Such an anti-pre-judicial consistency, in the exploration of logical quantities, matters especially when one is interested in grasping logical quantities in a general way (general like statisticality and information) as perspectives characteristically emerging, even without formal articulate ado, as scopes in research and intelligent decision-making, performance, affectivity, cognition, etc., of whatever kind.

Because of the common philosophical failure to differentiate the singular as non-general sharply enough from the single as monadic, Lowe doesn't notice that a polyadic version of the singular could be a whole universe, and therefore sweepingly universal (in its universe of discourse), without being general and non-unique (again, in its universe of discourse) and is not a trivial almost-blind window onto a mere one-object universe. In other words, in missing some of the simple quantities, Lowe misses their conjunctive compounds. Thus he also misses the fact that the singular as usually understood actually involves a conjunctive compound of singular and special (or non-universal) in the sense that the singular, as usually understood, is not a total population. If "H" is a singular predicate, one usually assumes not only that there is some x of which "H" is true such that there exists nothing else of which "H" is true, but also that there exist, distinct from x, things of which "H" is false. That assumption is actually an option with a significance that becomes clear if by "singular" one means only "decidedly non-general" and not also "decidedly monadic." One will do that for conceptual consistency in considering logical quantity and term "adicity" or "valence" as separate dimensions.

I didn't start out hoping to find some way to include "total population" or "universe of discourse" as a logical quantity on a par with "general," "singular," etc. I hadn't given any thought to the idea of totality or universe as logical quantity. I simply followed the logic out consistently and tried to understand where it led. It leads to an old philosophical desideratum, a correlation of logical quantities to major classes of research subject matters. It even makes the old nominalist-realist wrangle seem less interesting, because now one is not confronting the same old stark dichotomy again and again. A universal such as "two" is as different from a non-universal general like "healthy," as either of them is from a non-universal singular like "Jack."

So I recognize four quantificational divisions, conjunctively compoundable in four ways, where E.J. Lowe recognizes only two logical quantities. Mine are logically more systematic and turn out to correlate to the subject matter perspectives of the major classes of research. If, for instance, one considers singulars polyadically as well as monadically, it is more natural to regard the 'special' sciences as being about singulars in a larger world. Before crying "there is no science of singulars," one should also remember that the subject matter of a science can differ from the object or objective of a science, and that the special sciences seek, as their objective, to discover laws, populations of elements, kinds, and individual histories of the subject matter, concrete singulars sometimes individually and sometimes in their multitudes. Even laws in physics take on the singular aspects of giant events, for instance the signal speed limit, which may have changed over time in relation to other fundamental physical quantities.

Insofar as more traditional categories such as substance and property are arrangeable in a pattern of nonbinding affinities with logical quantities, there again, I have four where E.J. Lowe has two. (Skip tables )
Simple & conjoined logical quantities for terms (subject, predicate, etc.)  Skip
General:1. Universal-cum-general.3. Special-cum-general
(neither singular nor universal)
(Multi-)singular (monadic, polyadic, etc.):2. (Monadic, polyadic, etc.) Universal-cum-singular
(gamut, universe of discourse, total population & its parameters)
4. (Monadic, polyadic, etc.)
special-cum-singular (monadic, polyadized, etc., singulars in a larger world).

Tetrastic categories  Skip
Positive phenomena:
1. Correspondences/ variances
(another than, sum of, inverted order of, anti-derivative of, etc.).
3. Attributes, properties, accidents etc.
(firm, unsound, well, ill, steady, irregualar, strong, weak, etc.)

2. Modes of attributability
(indeed, not, if, novelly, probably, feasibly, optimally, etc.)
4. Substances
(primary substances: this man, this horse.).

Tetrastic classifications of research  Skip
Typically, the premisses deductively imply the conclusions:Typically, the premisses don't deductively imply the conclusions.¹
Typically, the conclusions deductively imply the premisses:1. 'Pure' mathematics:
equations, topology, graphs, integration, measure, enumeration, functions & derivatives, algebra, limits, order
3. General ('domain-independent') studies of positive phenomena:
Inverse or multi-objective optimization, statistics, inductive & descriptive areas of info theory (& their math formalisms), philosophy.
Typically, the conclusions don't deductively imply the premisses¹′:2. Applied yet mathematically deep deductive theories of:
4. 'Special' sciences:
sciences of
motion & forces,
mind, intelligence, intelligent life

¹, ¹′ Notwithstanding inferences within imported formalisms.
² Or, more generally, uncertainty theory.
³ Mathematics of information overlaps significantly into pure math, especially abstract algebra.

Now, such quantificational divisions are no more to be eschewed for parsimony than corners of the Square of Opposition; they are so systematic that it takes more information to eclectically select a few than to take them all. Occam doesn't raze exactly one or two corners of the Square of Opposition. To fail to recognize this leads to arguments over how few angels can dance on the head of a pin. If instead one listens to that which the logical structure is "trying to tell" one, then one may avoid the excessive foreshortening of the world's divisions that is echoed by the classic Saul Steinberg cartoon. For another instance, logical connectives can all be done in terms of negative alternation, and can all be done in terms of negative conjunction. But this means that the negative alternative and negative conjunctive are particularly versatile logical connectives; it does not mean that one or the other of them is really the only logical connective. Now, Lowe intends his ontology for the natural sciences, but I don't know what the natural sciences gain by minimizing or ignoring the difference between "blue" (true of this and of that, but not of some third or of some fourth thing) and a numberish universal like "three" (true of everything in one or another polyad xyz where xyz are all of them distinct objects and in a universe with more than two objects). I suspect that the real issue is an avoidance of evoking or suggesting further mountain ranges of those entities (beyond the generals or "universals" which Lowe already countenances) which nominalists dislike. Thus does the addictive battle between realism and its opponents distract from other interesting issues, distort and crop straightforward logical formalisms and their potential applications, and prevent philosophers from doing justice to the ideas to which they commit themselves in adopting a logical formalism such as that of logical quantity. For my part, I generally take the involvement of questions of a subject matter's ontological status in questions of math and science classification as an intrusion signifying that the classification is either deficient in firm and fertile constraints or just plain nebulous.

John Boyd's OODA loop

Correlations, not flat equations.
1. Observation (data intake).
2. Orientation.
3. Decision.
4. Action.
Tetrastic modes
of the psyche:
3. Affectivity.
4. Cognition.
1. Will, conation.
2. Dealing, ability.
Tetrastic stages in
a generalized loop.
1. Adopt.
2. Apply.
3. Take in.
4. Digest.
— Observation, orientation, decision, action. There seems not too bad a correlation, except for a few things. In my non-reordered, "default" version the loop would begin with decision — DAOO. There my short answer is that a reordering can be perfectly "okay, philosophically" as long as it is regular, relationship-preservative. The biggest difference seems instead to be that in my version, "observation" (the intake of data) would have, at its core, affective evaluation — i.e., one is confronted by good or by bad or by an irresistable challenge — etc. In battle, of course, it's important to keep cool under fire, and meanwhile Boyd emphasizes the intake of data via the senses. Next, Boyd portrays orientation as a cognitively digestive stage, yet "orientation" remains the right word for that which he's discussing, cognition with pertinence to one's immediate situation.
But why would I want to order even the generalized loop differently? If I think that Boyd's ordering is just fine, then why don't I make it the standard for more general orderings? Where the action is a kind of means, the decision to it is a beginning, an undertaking. From one's ensuing action springs a result, an effect or lack thereof, which one observes (or tries to observe, anyway) and evaluates, especially for its likeness or unlikeness to one's intent, and one considers it carefully, checking it against various things including one's experience and expectations. Beginning — middle/means — end — check. Thence one may loop back to the decision stage again, as indeed one may have already done in getting into the current go-round. (Discussion of beginnings, means, ends, and checks.)

Marshall McLuhan's tetrad

— a tetrachotomy of enhancement, erosion, retrieval, reversal — I’ve tried but haven’t yet found a correlation.

Karl Popper's tetrad

— (a sequential tetrachotomy? or a genuine tetrad? of) problem, tentative theory, (attempted) error-elimination (especially by way of critical discussion), new problem(s): P 1 » TT » EE » P2 . ” I’ve tried but haven’t yet found a correlation. I can see that it could be argued that it’s a triad beginning to cycle.

Buckminster Fuller's tetrahedra

— are not about specific foursomes of philosophically relevant conceptions, as far as I’ve been able to tell.

Walker Percy's tetrad

of Symbol (or sign), Object, Organism1 (I), and Organism2 (Thou). Symbol and Object comprise a relation of quasi-identity or meaning. Organism1 (I), and Organism2 (Thou) comprise a relation of intersubjectivity. The two relations are sometimes shown intersecting across a diamond-shaped diagram.
Walker Percy's Tetrad
Organism1 (I)

Relation ofQuasi-

Organism2 (Thou)

Walker Percy jumbles functionally-defined semiotic elements with their bearers, sets up an ungainly tetrad of two organisms (I and Thou) and two things (object and symbol)), and that tetrad does not lend itself to generalization in terms of correlations to philosophical categories. Percy’s tetrad does not strike me as really philosophical.

Irrespectively, there might be some correlation with my foursomes, a correlation to that extent to which one may interpret one of the Organisms as the interpretant and the other Organism as the collaterally based recognition (which I call the "recognizant" or the "establishment"), or perhaps each of the Organisms as an interpretant and their supportings, checkings, & balancings of each other (in regard to symbol and object) as the recognizant — but these semiotic functions should be embodied by separate dedicated terms in the tetrad just as object and symbol are. And to some extent a single cognizant organism or at any rate a mind acts as its own cognitive support, check, & balance, even though it has learned much of how to do so from collaboration, strife, etc., with other organisms. (The recognizant is the verificatory/disconfirmatory element whereby I augment the Peircean triad to a tetrad). However, I wouldn’t tie interpretant and recognizant to being the fundamental semiotic “I” and “Thou” in whatever order (I would hold that the semiotic object is addressed to its sign, the sign to its interpretant, and the interpretant to its recognizant. I discuss semiotics in “Why Tetrastic?” under “Semiotics” and in “Semiotics: collaterally based recognition, the proxy, and counting-as.”)

Kant's basic four categories

1 of Quantity:

2 Quality
Substance and accident 
Cause and effect 
Action and reaction 

4 Modality
Possibility -- Impossibility
Existence -- Non-existence
Necessity -- Contingency
Correspondence /

(another than,
double of, sum of,
antiderivative of, etc.)

Mode of

(indeed, not, if, possibly,
novelly, probably,
optimally, feasibly, etc.)
attribute, accident

(firm, unsound, well, ill,
steady, irregular,
strong, weak, etc.)


(this man,
this horse,
— quantity, quality, relation, modality. I don't see any special resemblance to my fours.

Martin Heidegger's fourfold

— earth, sky, mortals, and divinities — I’ve tried but haven’t yet found a correlation.

Carl Jung's four functions of consciousness

— sensation & intuition, and thinking & feeling. There isn't a correlation. Despite their opposition, sensation and intuition are closer to each other than thinking and feeling are to each other. I don't want to get complicated here but, for various reasons, it doesn't seem a good, systematic fourfold. However I'm disinclined to judge it as a philosophical product; it is the work of a psychologist of homo sapiens, a psychologist whom I haven't read in decades. In any case, it is hardly without value. I don't go along with the ESP stuff, but the difference between intuition (even minus actual ESP) and sensation is significant both on its own account and as an instance possibly of a pattern.

Ken Wilber's Four Quadrants

— Interior-Individual, Exterior-Individual, Interior-Collective, Exterior-Collective.
Ken Wilber's Four Quadrants
(Source: Wikipedia
Quadrant (UL)

e.g. Freud
Quadrant (UR)

e.g. Skinner
Quadrant (LL)

e.g. Gadamer
Quadrant (LR)

e.g. Marx [sic]
As far as I can tell, his foursome of quadrants doesn't correlate with any of my fours. It's interesting, though, and it consists of four combinations of values of paired two-valued parameters. Maybe I'll find a way to adapt it, though I'd be likelier to include Smith or Hayek than Marx as an example.

On the other hand, the way in which Wilber divides stages of moral development does seem to correlate, somewhat, with the way that I treat logical quantity, which involves conjoined quantifications, four conjunctions of answers to two twinned but mutually independent quantity questions (see my post "E.J. Lowe's four-category ontology" or the longer "Logical quantity & the problem of universals").
Correlations, not equations
Wilber's moral development stages
(source: Wikipedia
Tetrastic logical quantities
(Also see my "Logical quantity & the problem of universals."
Note: Wilber apparently assigns meanings to colors. My use of colors unrelated to his. Note, however, that the hues of color which I use (habitually) for the logical quantities are systematically opposite in feeling to the correlated Wilberian moral development stage.
Egocentric (similar to
Carol Gilligan's 'Selfish' stage).
Singular, or singulars taken as in a polyad, in a larger world.A. B-C. D. E. F. G. …
Ethnocentric or Sociocentric
(Gilligan's 'Care' stage).
Special-cum-general, i.e.,
neither universal (e.g., mathematical) nor singular (like you & me)
# # # # # …
* * * * *
# # # # # …
(Gilligan's 'Universal Care' stage).
(total population & its parameters, universe of discourse, gamut).
Pie chart. Photobucket
(Gilligan's 'Integrated' stage)
Universal but not a universe, i.e.,
there's more than one instantiation of it in its universe

••  ••
••  ••     ••  ••

The most questionable correlation is that between Wilber's "Being-centric" stage and the repeatedly instantiated universal. Wilber associates the "Being-centric" stage with a final and mystical stage of moral development. The immediate problem isn't the mysticism since, on my side of the correlation, there are merely logical quantities. The immediate question is whether by "Being" he means something that correlates with the repeatedly (indeed sometimes endlessly repeatably) instantiated universal. Of course, since being is that which everything has (though never in the same way twice), he probably does mean something similar to the universal that is not the universe or world.

Jacques Lacan's Four Discourses

— Master, University, Hysteric, and Analyst. I wouldn't go along with a presumptive attitude of authority = bad, resistance = good. Yet, I do discern a certain weak but unmistakable echo of some of my fourfolds. In order to resist verbosity, maybe I can get away with bit of connect-the-dots.
Lacan (Source: Veryard Projects at Zizek's example
from the opera Don Giovanni
(Source: Veryard Projects at
Tetrastic echoes, not equivalents (mine). Imagine the items in this column as if they had been qualified or altered a little in order to apply in particular to academic knowledge and discourse. Note: I don't share Lacan's & Zizek's antipathy toward non-socialist power and wealth. Update: I've learned that Zizek is quite the enfant terrible, profitably praising Hitler & Stalin (snifter clink to David Thompson).
Discourse of the MasterStruggle for mastery / domination / penetration. Based on Hegel's Master/Slave paradox.Don Ottavio inauthentic,
Power. Ruling/governing arts.
Discourse of the UniversityProvision and worship of "objective" knowledge - usually in the unacknowledged service of some external master discourse.Leporelloinauthentic,
Wealth, means. Productive arts.
Discourse of the HystericSymptoms embodying and revealing resistance to the prevailing master discourse.Donna Elviraauthentic,
Splendor, glamour, "wattage," etc. The affective, expressive, "consumptual" arts.
Discourse of the AnalystDeliberate subversion of the prevailing master discourse.Donna Annaauthentic,
Honor, standing, legitimacy. The "ruminative arts" -- maths & sciences.

Alain Badiou's Four Discourses

or “truth procedures” — Art, Love, Politics, and Science — (Skip table )

Tetrastic 4×4 classification of aspects of humanity
which lend themselves to social compartmentalization,
containing at least 3 of Badiou's "4 Discourses."

SECOND level
(the ROWS),


Beginnings, leadings, decision-makings
Middles, means, abilities, resources, dealings
Ends, endings,
teleioses, satisfactions
Checks, ente­lechies, establi­shings, knowledge
Subhead (sector)
Subhead (sector)
Subhead (sector)
Subhead (sector)

Subhead (inter-behavior)
Vyings, arenas:
Affairs of power, freedom.
(Badiou's "Politics.")
Business, trade,
finance, wealth.
Show, games, sports, fashion, “wattage.”Case-building, validation, standing,
Subhead (inter-behavior)
Practices, cooperations, tolerances, occupational spheres & concourses:
Administration, management, compliance, adjustment.
Labor, work, collaboration.
Leisure, hobbies, repasts, celebrations, observances, recreations, pastimes.
Study, investigation, review, discussion, reporting.
Ends, endings,
Subhead (inter-behavior)
Valuings, distinctive unitings, communities:
Ruling / governing valuings.
Tastes & valuings about feelings. 
(Badiou's "Love"?)
Valuings about cognition & legitimacy.
Subhead (inter-behavior)
Checks & balances, supports, disciplines:
Ruling / governing arts.
Know-how, productive arts/sciences.
Affective arts.

(Badiou's "Art.")
Maths & sciences.
(Badiou's "Science.")
“...the four generic 'conditions' of philosophy itself.... These are the only four fields in which a pure subjective commitment is possible, i.e. one indifferent to procedures of interpretation, representation or verification.” (See Badiou’s EGS faculty biography.) The Wikipedia article adds the caveat that “Badiou consistently maintains throughout his work that philosophy must avoid the temptation to attach its own truth to that of any of the discourses, a process he terms a philosophical ‘disaster’.” I see no obvious correlation here with my fourfolds. Three or possibly all four of Badiou’s Four Discourses appear in my 4x4 classification of aspects of humanity which lend themselves to social compartmentalization. There’s hardly any pattern in the resultant distribution. So, right off the bat, I see hardly any correlation. I haven’t studied in any detail Badiou’s justifications for singling exactly those four things out, but the justifications sound unpromising. If Badiou makes a four-way distinction among object (subject matter), representation, interpretation, and verification, then, at least there, I’ll agree with him.

The Andean Tetralectics of Jorge Emilio Molina, Javier Amaru Ruiz Garcia, Jorge Miranda Luizaga, & others.

Their old sites are gone, just bits remain here on the WayBack Machine (at (More info on their Tetralectics can be found by searching on Tetralectica OR Tetraléctica). At the old sites I had found talk of the stone Gate of The Sun in Tiahuanaco, prime numbers, string theory. I’ve no idea as to whether there’s any correlation to my fours.

The postmodernist Tetralectics

of R. Hargitai, Ö. Farkas, L+. Ropolyi, G. Veress & Gy. Vankó — there is necessarily a correlation to my Tetrastics insofar as their Tetralectics assumes the Aristotelian four causes but, beyond that, I don’t quite know what to make of it. The orientation seems systems-theoretic. They display conceptual oppositions assembled into a tetrahedron, discuss symmetry transformations, various kinds of oppositions (vertex/face, edge/edge) etc. They discuss three levels of description — (1) standard scientific theories, (2) metatheories, and (3) tetralectics. The philosopher and information theorist John Collier has given a favorable talk "Tetralectics: Ancient and modern precursors" (at the Symmetry Festival 2003 on Culture and Science). In a March 31, 2002 peirce-l message (lost from the archive and unavailable online), he links to the Hungarian group's paper and adds, "In my own work, which is unpublished, I find over and over that there is an underlying triadic symmetry to classic tetrads. The symmetry group is that of a tetrahedron. This has been a hobby of mine for over thirty years now." (He and the Hungarian group arrived at their ideas independently of each other. See Collier's Nov. 14, 2005 peirce-l mssage).
Anyway, I've looked at the “Tetralectics” paper a number of times over the past few years. It would be nice if some of the more detailed discussion which they mention became available, in particular in regard to their reinterpretation of Aristotle’s Four Causes, their mapping of theory families in physics, the discrete/continuous/global/local division, its alignment with infinite/infinite/infinite/finite, and so forth.
Tetralectics Table gathered from “Tetralectics” (authors R. Hargitai, Ö. Farkas, L+. Ropolyi, G. Veress & Gy. Vankó) at
Central Concept /
in a tetralectics of
natural sciences
Theory families
in physics
The Central Concepts'
Properties connected to
Central Concepts/
Properties connected to
Central Concepts/
Properties connected to
Central Concepts / Metatheories
(opposition of vertex to
the opposite triangular face)
MatterMatter (M)Corpuscularsubstrate --&-- structurestatic, closed, individualdiscrete, stochastic, disorderedequipositionalqualityrealityinfinite
FormSpace-time (S)Fieldspace --&-- timestatic, open, collectivecontinuous, homogeneous, causalhierarchicalquantityrealityinfinite
EfficiencyAction (A)Variation principlesaction --&-- interactiondynamic, open, individualglobal, deterministic, inhomogeneoushierarchicalqualitypossibilityinfinite
AimChange (C)Conservation lawstransformation --&-- equilibriumdynamic, closed, collectivelocal, teleological, orderedhierarchicalqualityrealityfinite

The Quadralectics of Marten Kuilman.

He seems to be particularly interested in a four-way version of dialogical structure, and in bringing it to light in old systems of thought, but his main interest right now is in applying his ideas to architecture. I have had some contact with the genial Kuilman during the past year, and he sent me some material. I’ve read some parts a number of times, explored other parts, but I really do have trouble understanding it, particularly the mathematics. I also suspect that his very European philosophical context makes his conceptions (e.g., his conception of “visibility”) harder for me to understand. In such situations, sometimes it is best to plow ahead, in hopes of catching broad gists and then working one’s way down to some of the more difficult stuff. That hasn’t worked yet for me in this case.

The Quadralectics of Kent D. Palmer

, which are part of a larger system of his. It's well beyond my understanding. Palmer left the following comment on June 2, 2010 at another post at The Tetrast:
Emergent design : explorations in systems phenomenology in relation to ontology, hermeneutics and the meta-dialectics of design by Kent Palmer at talks about Quadralectics.
He brings together continental philosophy with systems engineering and builds on the work of Metapattern author Pieter Wisse.

Joel Miller's Tetrology and the Tetrastic System

(once there, scroll down), unpublished. In the meantime, he recommends From DNA to ABC and is willing to send it to me for a $20 bill in the mail to him in Sweden; I trust him but I wonder whether he understands about the advanced state of thievery within the U.S. postal system. I have to get around to ordering his book in some more usual way; after all, he used the word “tetrastic” before I even thought of it! (Update: He now has PayPal.) (His “Majority English: the dialect of the non-native speaker,” which you’ll see if you click on From DNA to ABC above, is the good-humored title of what seems a pleasant Website for those interested in the ins and outs of English-language usage.) Anyway, his division of the concrete world into atomic, chemical, biological, and human systems may fit well with my four-way division; I just need to learn why he does it.

Robert Worstell's tetrads

of ideas (e.g., Ability as analyzed into “Purpose, Confront, Responsibility and Understanding”. He has several blogs on various subjects. I haven’t grasped the pattern in his tetrads, and I’m starting to realize that it may be that he’s not primarily seeking a particularized, family-resemblance kind of pattern in the sense that I do, but instead has other criteria, involving: interaction and mutual enhancement of a tetrad's four points; a kind of tetrahedron-like philosophical stability of such tetrad; and productive results following soon upon application in practice. I see that he has linked to this post. By way of update, I add that he says there that at his current blog(s), “tetrads have been updated to ‘four-way thunks,’ but I do have a Lulu book which contains the original doctorate thesis.”

Hyatt Carter's Meta-Fours

Book cover of _Thinking Is the Best Way to Travel_. May the Fours Be With You. Image hosted by Photobucket — pun intended. Updated December 31, 2010. He has redone his Website and moved it to a new domain. In "Meta-Fours" he discusses a number of four-fold structures, along with their pursuers, such as David Spooner, Ken Wilber, Michael Denton, C.J. Jung, Wolgang Pauli, and John Sanford. Even I appear there! The cheerful Carter's interests include religion, spirituality, process philosophy, thinkers such as Alfred North Whitehead, Zen Master Dogen, and Charles Hartshorne, Joyce's fiction, and other things. He's written two books, Thinking Is the Best Way to Travel: Essays along the Journey and Some Little Night Musings: 137 HyC Adventures. There's no comparison for me to make between his fours and mine, since Carter is working not so much on a particular four-fold pattern of ideas (as far as I can tell) as exploring four-fold patterns of ideas in general. As he puts it, I'm less eclectic than he is. We've been emailing each other for some years now, and I can vouch for the focus on tesserophiliac puns that he reports. In fact, we've been guiltier than he reports, and he's the guiltiest and best at it of all.

X of crossing diagonals. Each diagonal itself is a narrow X. Particularly interesting are his essays on chiasmus in poems and in the New Testament. My four-folds always have a chiastic structure, so I find this sort of thing interesting in an almost eerie way. Well not very eerie, but you know what I mean. Or if you don't know what I mean, you can try reading these essays of his:
  • "Chiasmus"
  • "A Miracle of Composition"
  • "Amazing Literary Grace: The Structure of Paul’s Hymn to Love"

  • David Spooner's fours

    . Insect metamorphosis — (A) the earlier-evolved triadic and gradual hemimetabolic metamorphosis (ovum, nymph, completed imago, e.g. the grasshopper) and (B) the tetradic and abrupt holometabolic metamorphosis, (1) from ovum, egg, (2) to larva, grub, caterpillar, (3) then pupa or chrysalis, and (4) finally the imago - butterfly, bee, moth, wasp or beetle, — are processes which Spooner seeks in his books to show
    are more crucial to natural selection than evolutionary theorists have accepted. While not disputing Darwin, I work from A.R. Wallace`s insights. If we start from the greatest works of human consciousness (Beethoven, Mozart, Melville, Shakespeare), then humanity owes as much obliquely to the insect as the ape.

    — Spooner on the homepage of his Website at the Authors' Guild.
    At that Website, on a discussion page, I asked Spooner a question about the Helmholz-Poincaré picture of the creative process (which I find well-correlated with my tetrastic structure): "How would you relate (or not relate) the four-stage process of egg, larva, pupa, imago to the four-stage creative process of saturation, incubation, illumination, and verification as discussed by Murray Gell-Mann on pp. 264-265 of _The Quark and the Jaguar_? The passage can be viewed at Google Books as linked via a Google search on: each-found-a-contradiction Gell-Mann" — and Spooner responded:
    Response from David Spooner to Ben UdeIΙ:
    `chitinous` (so to say) 4s are zooming about the universe. They determine the structure of much of reality, thought and experimental processes. You have many that are new to me, especially in regard to that latter category, and Hyatt Carter has many more ( The Gell-Mann endorsement of the Helmholz-Poincaré tetradic system of discovery/experimentation is a compacted version of the philological and evolutionary cluster ova-larva-pupa-imago.

    However, this latter organic process has the virtue of ultimately opening out and fusing the human species, in its long-term evolution and maturation, more profoundly with the living world of nature. {as I explain in my books from the 1995 Metaphysics to the 2005 Insect-Populated}. It draws the realm of abstract thought back towards the earth, and thereby supplements the human relation to the great apes with one to the insect world in its evolution.
    It will take me a while to absorb his basic ideas.

    Dr. Stephen R. Palmquist's diagrammatic four-folds

    . Palmquist has written books on Kant and is interested in the diagrammatic representation of structures among ideas, including ideas related by twinned two-valued parameters (my lingo, not his), which is where we cross paths. He has said that a good place to start in understanding his diagrams is Chapter 5 of his book The Tree of Philosophy.

    Richard McKeon

    , the pluralist philosopher, developed some four-fold classifications of philosophical issues, approaches, etc. McKeon was just recently brought to my attention, so I've barely had a chance to read him, but off the top of my head I'd say that his fourfolds differ from mine. There are at least two philosophers, Walter Watson and David A. Dilworth, who have used McKeon's fourfolds (and maybe his threefolds too), and I may attempt some comments on Watson and Dilworth eventually. Anyway, below is my rendition of a table in McKeon's 17-page paper "Philosophic Semantics and Philosophic Inquiry" which is available from a former student of his at There's a link there to a photoimage of the table, which I used. I can't render the slanting lines in html, so I made do with the slash characters. The horizontal spacing is the same, but I changed the vertical spacing to single-spaced wherever I could.

    Modes of Being     
    Modes of Thought  
    That which is 
      Modes of Fact
     Modes of Simplicity
    Being and Becoming-----------Assimilation and Exemplification--
    Reality and Approximation--Categories of Thought
    (Ideas and presentations)
    Phenomena and Projections----
    Discrimination and Postulation----
    Process and Frame----------Categories of Language and
    action (Symbols and rules)
    Elements and Composites------
    Construction and Decomposition----
    Object and Immpression-----Categories of Things
    (Cognition and Emotion)
    Actuality and Potentiality---
    Resolution and Question-----------
    Substance and Accident-----Categories of Terms
       Principles    Methods   InterpretationsSelections
       Comprehensive————————————Dialectical——————————————————Ontological————————————————————Hierarchy (transcendental)
       Reflexive—————\/—————————Operational———————\/—————————Entitative—————————————————————Matter (reductive)
    Meroscopic       /\Particular            /\Phenomenal
       Simple———————/——\————————Logistic—————————/  \————————Existentialist—————————————————Types (perspective)
       Actional————/    \———————Problematic——————————————————Essentialist———————————————————Kinds (functional)
                           BASIC DIVISIONS OF PHILOSOPHY
          Theoretic                    Physics                      Philosophy                 Logic
                          BASIC PROBLEMS

    Pythagoras’s Tetractys

    • •
    • • •
    • • • •
    — its numerological imagery has seemed little correlated to my fours and, in some ways, quite discorrelated, except perhaps in certain of its outlines such as turn up in an account of traditional four-symbolism by Penelope Merritt (see below).

    Mikhail Epstein on tetrads in Soviet rhetoric

    Four disks against black background, 2x2 arrangement: upper left & lower right, two hammer-&-sickles, red against green star against red background, & green against red star against green background, & lower left & upper right, two Statues of Liberties, gold on storm-blue & vice versa, all traversed horizontally by a steeply zigzagging waveform suggesting troubled signal reception and maybe something going haywire, all of it against black background shaped like old Soviet-style squarish TV screen. In his “Relativistic Patterns in Totalitarian Thinking: an Inquiry into the Language of Soviet Ideology,” Epstein (also spelt "Epshtein") discusses the Soviet ideolinguistic use of tetradic (I would say “tetrachotomical”) structures arising from pairs of two-value parameters. (In a few places, Epstein refers to it as the Soviet ideologists’ “tetralectics,” apparently only in order to suggest a four-pole version of “dialectics” and not in reference or allusion to any of the other particular brands of “Tetralectics” which I’ve mentioned in this post.) Epstein offers a fascinating account of the malign use of conceptual tetrachotomies — not particularly strong tetrachotomies philosophically, in my view, but there they are, they do their jobs. Basically, two opposite actions by an ally receive laudatory labels from the ideologist, and the same two opposite actions, when carried out by an enemy, receive denunciatory labels from the ideologist. For a simple example, the Soviet ideologist might systematically call Soviet boldness “brave” and Soviet caution “prudent,” and just as systematically call U.S. boldness “rash” and U.S. caution “cowardly.” (I discuss these particular concepts in “Tetrachotomies of future-oriented virtues and vices.”) One could imagine that the pattern could be found in political rhetoric more generally, though political rhetoric is not always shaped in full awareness of such inconsistency or hypocrisy as it harbors. When, in respect of the same behavior, one applies more favorable standards to one’s allies and less favorable standards to one’s adversaries, one may fall into such patterns. There is something of that aspect of the Soviet ideologist in each of us. Mikhail Epstein analyzes the malign Soviet extreme of weaponization of language and its prolongation into an insistent quadru-venomous stream of propaganda.

    Penelope Merritt’s account of traditional four-symbolism

    Red drop-shape, storm-blue lips-shape, vivid-gold solid triangle, at jazzy tilts, and hard-green solid square, all against a black infinity sign against a white pentagon against a black hexagaon, etc., i.e. a background of higher polygons crowding up behind one another, alternating black & white.A Few Thoughts On the Number Four” at is one of the very few which I’ve read which reminds me at all of my fours. Incidentally thereto (at least I think it's a coincidence), it’s one of the few accounts of number symbolism which don’t make me sleepy. Most such accounts that I’ve seen, even the brief ones, soon amble into vague numerological mazes. But this Penelope weaves plain and clear. (She is with the Community Center for the Performing Arts, Eugene, Oregon.)

    Now, I’m interested rather more in recurrent logical patterns, than in number symbolism and elaborate games of artificially synesthetic apprehensions of small positive integers (and I don’t believe in synchronicity or believe that numbers have magic powers). But logic and reason involve fourfolds which do get reflected in common ideas, whence traditional number symbolism draws.

    After Penelope’s initial discussion, she goes on to discuss the number five, which represents things like expansion, destabilization, catalysis. This is like a new beginning, a new first stage, that works upon the stabilization which is the fourth stage.

    Then Penelope discusses the four Gospels, the four elements, the four humors, and there the correlations with my tetrastic structures seem weak, so I will focus on her initial discussion.

    “One represents the male principle, the ‘yang’. It is raw energy, positive, original and creative. In the creative process it is the original spark of an idea.”

    Here, at a beginning, I think of forces, movements, directional and opposable, roving and wandering, more than I think of energy.

    “Two is the feminine principle, the ‘yin’. It is the gestational period in which things begin to form, the earth into which the seed of one’s idea is planted. In the creative process there is almost always a similar period when an original impulse ‘cooks’ for a time, even if only in sleep or distraction.

    Here, at a middle, I similarly think of gestation, processing, producing, adaptation. Here I also think particularly of rhythm, regularity, constancy, homeostasis, patience, endurance, dependability, perseverance, etc.
    “Three is the synthesis of one and two. It is ideation and self-expression, the creation itself, the finished idea.

    Here, at an end or culmination, I think of those things and of vibrancy, claritas or radiance, energy, vigor, and also selectiveness, perfectiveness, etc.

    “Four is the material manifestation of three, the actual physical realisation, order and systematisation of the idea. It is the making real of the dream represented by three.

    Here, at a check or checking, similarly I think of stability, firmness, solidification, confirmation, entelechy.
    Penelope goes on to say, “Four has come to be considered the number of labour and stability” I don’t associate stage four (my “check” or “checking”) with labor except (as often happens) insofar as labor bears out and verifies, or disconfirms, that which is discovered in stage three (my “end” or “culmination”). Instead I would associate, most of all, stage two (my “middle,” “means,” “mediation”) with processing, production, labor, adaptation, etc. Penelope elsewhere in her essay says that four is associated with both dependability and stability; I think, for "four," less about dependability across time and more about balance and stability across space, structuring and stabilization (of opposed forces and movements), etc., rumination, digestion, assimilation, integration, concrete embodiment. Staunchness and solidity.
    In terms of various kinds of strength, one might do it this way:
    1 {beginnings}. Might, dynamism.
    2 {middles}. Endurance, patience.
    3 {ends}. Vigor, vibrance.
    4 {checks}. Firmness, solidity.

    I have long been somewhat aware of yin-yang ideas, seed and soil, etc., but I know little of any further number symbolism. Yet I didn’t pick my four out of a hat. Above, note the diagonal oppositions between 1. might, dynamism, & 4. firmness, solidity, (will travel vs. won’t travel) whereof the familiar fantastic extremes are the irresistable force and the immovable object, and between 2. endurance, patience, & 3. vigor, vibrance, (will be patient vs. won’t be patient), whereof the respective fantastic extremes are the unflappable and the undampable. These are ideas in abstract balance. And they are anything but an arbitrary pairing of dyads. Note that 1. might, dynamism, & 4. firmness, solidity, (will travel vs. won’t travel) are space or distance ideas, while 2. endurance, patience, & 3. vigor, vibrance, (will be patient vs. won’t be patient), are time ideas. They have distinguishable physical meanings reflected in a system’s
    1. Momentum, impulse, force.
    2. Rest mass, rest energy, internal work & power
    3. Energy, work, power.
    4. Internally balanced momenta (kinetic & potential), impulses, forces.

    Physics quantities. Momentum, mass, energy, etc.They also correlate pretty well with Aristotle’s Four Causes:
    1. {might} efficient,
    2. {endurance} material,
    3. {vigor} final,
    4. {firmness} formal.
    Worthy of note is the correlation of Aristotle’s four causes with the systematically interrelated kinetic & mechanical conceptions above (remembering that kinetic and related mechanical conceptions arose from attempts to quantify cause and effect, but are not conceptions of causes and effects per se, much less conceptions of things related to each other as cause and effect, e.g., momentum and force are not considered to “cause” energy, work, or power as “effects”).
    — In comparing with Aristotle’s causes, one may wish to think not just of momentum and energy but also of impulse and work, and of force and power. Force, for instance, involves change (or rigidity, opposition to change) of a system’s motion, shape, state, or condition. And thinking of internal force and power makes us think of a material system rather than, say, merely a cloud of variously traveling photons (which as a whole travels slower than light and so has the kinetic values which some given material system might have).
    — “Power” here means rate of work done or energy transported, such that “wattage” would be the least bad word for it in everyday metaphors, because the quantity called “power” in physics is decidedly unlike political-style power, which is instead forcelike, directional and opposable, winner of a contest among those who would lead and be first; wattage-style power is comparatively more suggestive of a different prize, that of being that which wins the contest among ends and perfections: “vibes,” charisma, radiance, popularity, glamour, show, etc., though one should think of horsepower, vigor, whatever kind of vitality, and not only of candlepower. To be sure, I don’t think for a moment that social and poetic forces determine theoretical physics; however I like some kinds of common metaphors and I think that it’s interesting to see how far they can be taken and to see whether underlying logical similarities between systematic sets (especially foursomes) of conceptions can be brought to light).

    The volitional or conational characterizations which I made —
    1. wandering,
    2. perseverance,
    3. selectiveness,
    4. staunchness, unbudgingness.
    — are based on considerations about variability and constancy in light of the structure of logical quantity. As I said in “Why tetrastic?,” some fourfolds echo each other in ways for which I have not yet managed, at least to my satisfaction, to uncover the reasons, even when the fourfolds separately from each other have seemed clear enough. Turn a sign this way, then that, align it with others, the world seems to crack open, and the chase may be on. I disbelieve in a collective unconscious (Jungian, panpsychic, or otherwise) and I really have no precise idea why, for instance, there would be a correlation or analogy stretching from mechanical and kinetic concepts such as force, energy, mass, etc. (and related concepts of time and distance), to logical modes of constancy and variability, and even to, of all things, aspects of traditional number symbolism. I can only assume that it reflects some similarity in their respective logical structures, and guess, as I usually do, that broad conceptual structures elaborated so as to exhaust the logical possibilities in their respective realms sometimes end up with a family resemblance which sometimes spurs philosophical qualitative inductive generalizations but is seldom subjected to thematization and careful treatment and which may just as often spur a writer or artist as a philosopher. Penelope also, as shown, characterizes the numbers in terms of the creative process, which brings us to:

    Gell-Man's discussion of the creative process

    outlined by Helmholz and Poincaré. In The Quark and the Jaguar, theoretical physicist Murray Gell-Mann discusses the creative process in terms of Helmholz's three stages of saturation, incubation, and illumination, and the verification stage added thereto by Poincaré. This accords quite well with “my” foursome of beginning, middle, end, check. On pp. 264-265, Gell-Mann says that he and some physicists, biologists, painters, and poets compared experiences of discovery, & that their accounts were remarkably similar. The entire passage from which I've drawn excerpts is available through Google books (one needs to have a Google account in order to access it, I think) All ellipses below are mine.

    .... We had each found a contradiction between the established way of doing things and something we needed to accomplish: in art, the statement of a feeling, a thought, an insight; theoretical science, the explanation of some experimental facts in the face of an accepted “paradigm” that did not permit such an explanation.
      First, we had worked, for days or weeks or months, filling our minds with the difficulties of the problem in question and trying to overcome them. Second, there had come a time when further conscious thought was useless, even though we continued to carry the problem around with us. Third, suddenly, while we were cycling or shaving or cooking ..., the crucial idea had come. We had shaken loose from the rut we were in.
      We were all impressed with the congruence of our stories. Later on I learned that the insight about this act of creation was in fact rather old. Hermann Von Helmholtz ... described the three stages of conceiving an idea as saturation, incubation, and illumination, in perfect agreement with what the members of our group ... had discussed a century later.
      In 1908, Henri Poincaré added a fourth stage, important though rather obvious — verification. He described his own experience in developing a theory of a certain kind of mathematical function. He worked on the problem steadily for two weeks without success. One night, sleepless, it seemed to him that “ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination.” Still, he did not have the solution. But, a day or so later, he was boarding a bus .... “The idea came to me, without anything in my thoughts seeming to have paved the way for it, that the transformations I had used to define these functions were identical with those of non-Euclidean geometry. ... I felt a perfect certainty. On my return to Caen, for conscience’s sake, I verified the result.”
      The psychologist Graham Wallas formally described the process in 1926, and it has been standard ever since in the relevant branch of psychology, though I think none of us at the ... meeting had ever heard of it. I first came across it in a popular book by Morton Hunt entitled The Universe Within, from which the above translated quotations are drawn.

    (1) In saturation, one is taking hold of the problem, taking it on. That’s the beginning.
    (2) If this does not lead soon either to illumination or to dropping the problem, then there is incubation, in which, though the problem remains unsolved, one has gotten its elements sufficiently under control to process the problem without having to consciously think about it (though of course one still can so think). It may consist, as Gell-Mann points out in the passage's fuller version, in little more than unconscious stewing over established assumptions till one or another of them softens in the mind. Anyway, that’s the middle.
    (3) Illumination is the eureka, the ending, the climax.
    (4) Verification/falsification is the checking.
    I'd say that it's a very good match.

    Max Tegmark's Multiverse.

    There appears to be some structural correlation between my tetrastic classification of the fields of research and Max Tegmark’s theory of a four-level multiverse in which every possibility is actualized (“everything exists”) and in which mathematical existence is real existence. I’m not saying that I think that Tegmark’s four-level multiverse picture is true (or that any multiverse picture is true). Tegmark claims that it is at least testable. (I am not a physicist and feel unprepared to evaluate his claims of testability.) What I’m saying is that there is a correlation between my tetrastic classification of the fields of research and aspects of the structure of physics-relevant ideas built by Tegmark as the structure of his claimed four-level multiverse. As for the reality of Tegmark’s four-level multiverse, your guess is as good as and maybe better than mine.
    Tegmark’s MultiverseMy classification of research areas into familiesParticle possibilities, probabilities, detection, & experimentation as correlating to Tegmark levels
    (I haven't seen this correlation, or whatever it is, posed by anybody else, but it seems implicit.)
    Level IV: includes other mathematical structures, different fundamental physical equations.1. Pure mathematics (analytic equations, extremization, topology, graph theory, integration, measure, enumeration, differentiation, calculation, limits, order, etc.).(1st) IV. Feasibles & optima
    Variation across mathematical structures. The particle's (wave packet's) evolution involves contributions from classically absurd (but quantum-feasible) potential trajectories and counterfactual circumstances, which mostly cancel each other out.
    Do they correspond, as "would-have-beens," to potential variations of a Level-I experimental setup? Yet how meaningful would that correspondence be if they involve variations in shortest distances, spacetime metrics, and mathematical structure generally, well beyond variations that we could actually carry out in our Level I experimental setup? This raises a question of mathematical equivalences between seemingly dissimilar scenarios.
    Level III: includes alternate-outcome worlds (quantum branching); same features as Level II. (Needs unitarity of wave-function evolution in order to work.) 2. Applied yet mathematically deep mathematics (deductive maths of: optimization, probability, information, and logic).(2nd) III. Probabilities
    Variation across quantum branches.
    Level II: includes other post-inflation bubbles, same fundamental equations of physics, but possibly different particles, constants, and dimensionality.3. Abstract yet positive-phenomenally deep sciences/studies (inverse optimization, statistical theory, descriptive and ampliatively-inductive areas of information theory, and philosophy).(3rd) II. Information
    Variation along one quantum branch (repeated experiment with same setup).
    Level I: includes regions beyond our cosmic horizon, “universes” or Hubble volumes in a single given inflationary bubble.4. Concrete empirical sciences/studies (physical, material, biological, and behavioral/social/human).(4th) I. Logic
    Variation of experimental setup, an actual history (establishment of a hypothesis, theory, etc.)

    Tegmark correlates Level IV with mathematics; Tegmark takes mathematics as being the world from a “bird’s eye view”. Tegmark correlates Level I with the world at the level of our “frog’s eye view” (though it’s much too huge for us to observe most of it), the world of concrete empirical physics and chemistry as we know them. Ergo what about Levels III and II? If Tegmark’s picture were to prove true, then it would be exceedingly strange if there were a one-to-one correlation between research families and only two of four multiverse levels. Let me put it informally and as a question: we’re talking about the grand system of everything, right?

    In other words, one would expect that the “city of research,” in its evolved broad layout, would naturally come consistently — if it came at all — to resemble the “sky” of constellated multiverse structures “above” it. I mean that a resemblance that goes half-way and then simply quits seems rather unsatisfying. Another question is, of course, whether our civilization's “city of research” has evolved sufficiently for a systematic resemblance between it and multiverse structures to emerge. Whatever the case may be in that regard, I think I do see a correlation between the multiverse structures and the layout, as I see it, of research fields.

    However, in this correlation, fields such as deductive logic, which Tegmark associates with Level IV, are associated instead with Level III. Deductive logic is about the structures of alternatives among predicates or propositions which, according to the quantum Many Worlds view, are all actualized thanks to quantum branching into alternatives. Deductive logic is one of a family of fields, including also the deductive mathematics of optimization, probability, and information, studying such alternatives. They are considered mathematically deep, yet are not usually called “pure” mathematics, but “applied.” (One is stuck with their distinction’s being made with the terms “pure” and “applied”; one can see how it came about, but it’s neither the most illuminating way nor even true in every relevant sense. And as Dieudonné points out in his mathematics article in the Encyclopedia Britannica Fifteenth Edition, the rubric “applied” jumbles deep and trivial areas of math together. “Pure” does not.)

    Graphically recapitulates info from table of Tegmark's Multiverse Levels and the tetrastic division of research areas. Now, Tegmark follows tradition in regarding formal deductive logics as the most basic area in maths. I discuss issues of this kind at greater length in my post “Logical quantities, categories of research, and categories”. To summarize here, such deductive logics are about proof, and to put them as most basic within mathematics is to order the maths in the order of knowledge and of how we know things. Yet tradition also puts physics as more basic then chemistry, biology, etc., yet that is not in the order of knowledge but in the order of being. Tradition, on these points, is inconsistent, and the neat inter-family alignment of members of the research families tends to bear this out (see A periodic table of the maths, sciences, & areas intermediate between them in “Why Tetrastic?”). If Max Tegmark on some level liked an element of research-classificational traditionalism as “leavening” his cosmological radicalism, I’d say he should have been even more radical instead.

    To put logic first among maths is an inclination of many people, usually anti-Platonistic, who regard the existence of mathematical objects as a fiction, at best a convenient fiction - for them, there is no order of being, but only order of knowledge, in mathematics. That's not a constraint which Tegmark needs to heed in his theory that mathematical existence is real existence.

    Now, two families of mathematics are regarded as deep, and one of them as pure and deep, and the other as applied yet (mathematically) deep. Pure mathematics includes such areas as simultaneous equations, topology, matrices, extremization, graph theory, integration, measure, enumeration, differentiation, calculation (algebra), groups, limits, and kinds of ordering e.g. well ordering. Conclusions drawn in these fields tend to be “reversibly” a.k.a. “equivalentially” deductive (in mathematical induction, the minimal case and the heredity, conjoined, are equivalent to the conclusion) and structures of equivalences are rife throughout pure mathematics. Applied yet mathematically deep mathematics consists of deductive mathematical theories of optimization, probability, information, and logic; conclusions in these fields tend to be non-reversibly deductive (though to the extent that deductive mathematical theory of information has “re-invented” group theory, it has developed pure-mathematical interests.) All of these applied yet deep mathematics are about structures of alternatives. They are about the structures of those alternatives which all are actualized across Tegmark’s Level III, the Many Worlds of quantum physics, and they deduce from totalities to parts.

    What about Level II? Now, Level III and Level II are each other’s “inverses,” Level III actualizing alternate outcomes across quantum branchings, and Level II actualizing alternate outcomes in various times and places along a single branch, so that the two levels come out the same in their features. Likewise is there a family of abstract yet positive-phenomenally deep areas of research, such as statistical theory, areas each of which deals with the inverse problem of a correlated area of applied yet deep mathematics, and each of which deals in a general way with gathering data from various actual places and times and drawing ampliatively-inductive conclusions from parts, samples, etc., to totalities. These areas pertain to phenomena in general rather than to any special class, any single sample of the concrete real (and thus are all cenoscopic in the Peircean sense). They include the young field of inverse optimization problems, statistical theory, descriptive and ampliatively inductive areas of information theory, and the descriptive and ampliatively inductive study of logic and intelligent processes — I mean philosophy, not AI or computer research. This family of research seem to stand to Level II as the deductive maths of optimization, probability, information, and logic stand to Level III.

    Finally of course, correlated to Level I, there are the concrete empirical or “special” sciences — physical, chemical, biological, behaviorial/social/human, which tend to conclude in surmises, as cogent as they can make them.

    I had kind of hoped to discuss some of this with the folks at the "everything" mailing list, but the arguments there tend to revolve around computationalism (most of the active participants are genial, e.g., Bruno Marchal, and they're all intelligent). Also, they don't think that much can be said about Tegmark’s Level IV. I suspect that this is because they haven’t yet been able to incorporate extremal principles into their work as they would like, but I don't think that I convinced them that there's any particular reason to think that there's a connection.

    Note: How to say “everything exists.” In standard first-order logic, the phrase “everything exists” would be taken to trivially mean “that, that is, is,” or the like. Is there a way to say it in Tegmark’s sense in first-order logic at all? Is it an idea that can be logically expressed at that basic level? What would it mean if it can’t? Well, there does appear to be a way to say it in a specially restricted kind of first-order logic, by use of a special kind of quantificational functor. As for whether this leads to a coherent logical idea in less restricted logic, you be the judge. The result is, at least, a kind of statement which seems to lead to an area of logical issues raised by Tegmark’s picture, in any case, with regard to saying that every “potential” particular definite individual is actualized somewhere and somewhen, or, on the other hand, that the world altogether lacks some particular definite individual(s). The objectual version of the formalism sharpens the problem by allowing the individual(s) in question to be unspecified and even unspecifiable.
    Now, in defining the existential particular quantification, one may start with a finite universe of objects named by constants a through h, and say “There is a such that...a...or there is b such that...b... ... ...or there is h such that...h....” and agree to write this as x ...x....” Then one drops the substitutionalist requirement that x ranges over only named objects a, b, c, etc. Then the variable x is no longer substitutional but instead is objectual. To get to our new special functor will be a matter of replacing the repeated “or” with a repeated “and”.
    Let’s define a functor Æ such that Æ x ...x....” is equivalent to “There is a such that...a...and there is b such that...b... ... ...and there is h such that...h....” In effect one is saying that every name names something. Now, what happens when the substitutionalist requirement is dropped? In considering just what it is that x now ranges over, and whether the objectual statement Æ x is contingently or formally true or contingently or formally false or formally or contingently undecidable or (despite its fraternal-twin relationship with the existential particular) just plain ill-defined, one is led to consider some of the logical problems which arise in any case in entertaining the general idea that “everything exists.” In other words, we seem to arrive at some of the right problematics.
    (Note: Æ x should NOT be called the “existential universal” which would instead be properly applied to whatever is equivalent to the conjunction or predicative combination of the existential particular and the hypothetical universal, where you say, e.g., “there’s some food that’s good, and any food is good” or “there’s some food that’s good such that any food is good” or “there’s food and any food is good” or x ( F x ) & x ( F x G x ) ” or x y ( [ F x ] & [ F y G y ] ) , ” etc. I suppose that Æ x could be called the “omniexistential.”


    Do you have these pages in pdf format?
    Sorry, no.
    Great stuff. I can see why someone would want to be able to print out a hardcopy version (via PDF).

    The point I use most is as you phrase it: “Why tetrastic?,” some fourfolds echo each other in ways for which I have not yet managed, at least to my satisfaction, to uncover the reasons, even when the fourfolds separately from each other have seemed clear enough. Turn a sign this way, then that, align it with others, the world seems to crack open, and the chase may be on.

    Certainly turns conventional thinking on its head. My use of four-valued/way thinking is in the metaphysical - the search for a single underlying system which tends to explain how we routinely operate. The only thing accomplished so far is to create an interactive set of tetradic figures which each interact with the others - forming a 16-valued set that speeds up digesting massive data incredibly. (Of course it opens the door to intuitive serendipitous inspiration, but that is a side benefit...)

    But do keep up the good work - as I can, I'll keep working at understanding all that you've posted here.
    Thanks for the encouragement.

    Occasionally one person or another has asked me "what's the point," "what's the key," etc., in what I'm doing, because it does seem hard to understand. I've no over-arching purpose except to trace out a pattern, a particular pattern, of fourfolds 'cuz I like 'em. I do think that items in a fourfold should have practical interrelations, and be, all of them, comparably significant (a fourfold with an ersatz member is much worse than a fourfold with "n/a" in one of its four cells, though one can't always rely on first impressions about what's ersatz). Anyway, aside from that, I'm not oriented mainly toward practice; i.e., practice helps inspire, confirm, etc., theory, but 'tis the fourfold pattern itself which motivates me, not its practical applications (there's a streak of the pedant in me; chaqu'un à son goût). If, nonetheless, it helps you along with your own applicational efforts, that's cool. What I've gleaned from your texts (e.g., here and here) is that your main criterion for a fourfold or "four-way" is that its four points are interrelated such that enhancement of one point enhances all four, and such that practical application soon leads to productive results in practical activities; and that there is stability of such structures which echoes that of the tetrahedron. The fourfold pattern, on the other hand, which I trace out, doesn't seem particularly tetrahedral; I don't think of it as "stable" so much as I think of it as the pattern of sets of four contraries co-exhaustive within a given context, co-exhaustive of a given dimension of classification, etc. Yet, the idea of stability itself recurs as being involved with the fourth point in my fourfolds. The four points often have an ordering, such that the first is an instability or destabilization, and the fourth is a stability or stabilization.
















    NUMBERS: 0, 1, 2, 3








    --- )
    You're welcome, thanks! Hyatt Carter and David Spooner (both of them linked above) also collect fours. At Wikipedia's article "4 (number)" I added a section "In logic and philosophy" (somebody else added the first bullet) and I included some fours - from Schopenhauer, Brentano, Peirce, and Weiss, that I haven't included here. I guess I should get around to that. Hegel has some fours too.
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