The Tetrast

Plausibility, verisimilitude, novelty, nontriviality, versus optima, probabilities, information, n-ary givens

First posted on Tuesday, December 24, 2013—

—Recentest significant edit: February 7, 2014.

This post needs a lot of work, but I want to get it started.

The merits in inference in Table (B) are one-to-one reminiscent of, but NOT equatable to, the deductive topics in Table (A):

(A) Quasi-modal deductive topics
1. Optima & feasibles.
with directions.

2. Probabilities.

 X  3. Information, 'news'.

4. Givens (data, etc.) as
n-ary complexuses.
Bases, roots.
(B) Merits in inference
1. Cogency,
natural simplicity

in surmise.

2. Verisimilitude (in
C.S. Peirce's sense)

in induction.
 X  3. New aspect

4. Nontriviality
in equipollential

Both tables have the following pattern, with notable oppositions along the diagonals:

(C) Pattern in common
1. Simple, doable, compelling.

2. Likely, consistent.
 X  3. New, distinctive.

4. Complex, structured.

As to the ordering "optima, probabilities, information, givens [i.e., logic]," I didn't reach it by association with the pattern "difference, ratio, logarithm, ...." Instead I reached the ordering as part of some broad correlations (see Table (D)).
(D) Some correlations
  • Optima & feasibles.
  • Decision processes.
  • Motion, forces.
  • Probabilities.
  • Stochastic processes.
  • Matter.
  • Information.
  • Communicational and
    control processes.
  • Life.
  • Givens, data,
    basēs (for further conclusions).
  • Logic, learning processes.
  • Mind.

    All of the topics in Table (A) are the main topics of abstract and significant mathematically deductive areas concerned with structures of alternatives in timelike or (quasi-)modal perspectives. All the inference merits listed in Table (B) have familiar forms and some intellectual history.

    1. An explanatory hypothesis's simplicity is a familiar idea. One version of it is parsimony or Ockham's Razor; C.S. Peirce held that logical simplicity is secondary and that at its extreme it would add no explanation to a surprising observation; he explored simplicity of explanation as plausibility, facility, naturalness, instinctual attunement, and Galileo's 'natural light of reason' (see the linked passage from "A Neglected Argument," 1908).

    2. An inductive conclusion's verisimilitude, in Peirce's sense, is an idea familiar in the form of that which people mean in speaking of an induction as an inductive generalization, taking a sample (preferably a fair one) for a likeness of the whole.

    3. New or novel aspect of conclusion of a 'forward-only' deduction such as a categorical syllogism has been noted by various people including Peirce, and they have typically seen it as pertinent to deduction generally. It is familiar in the sense that it is considered perspectivally redundant, for example, to conclude a deduction merely by restating a premiss in unchanged form.

    4. Nontriviality or, in stronger form, 'depth' of a mathematical conclusion is a familiar idea among mathematicians, and has been an element in the formation of ideas of complexity.

    Each fruitful aspect counterbalances, quasi-ironically, the essential structure of the entailment relations between premisses and conclusions that defines the inference mode to which the aspect pertains. So, the collectively systematic character of the definitive entailment relations is reflected in the collectively systematic character of the fruitful aspects.

    1. A surmise's conclusion (with 'surmise' as defined below) is complex in the sense of both adding to, and subtracting from, what the premisses claim, and is of interest when it nonetheless brings a simple perspective.

    2. An induction's conclusion is novel in the sense of adding to, but not subtracting from, what the premisses claim, and is of interest when it nonetheless brings a 'conservative' or 'frugal' (verisimilitudinous or 'likely') perspective.

    3. A 'forward-only' deduction's conclusion retrenches in the sense of subtracting from, but not adding to, what the premisses claim, and is of interest when it nonetheless brings a novel perspective.

    4. An equipollential deduction's conclusion is simple in the sense of neither adding to, nor subtracting from, what the premisses claim, and is of interest when it nonetheless brings a complex or nontrivial perspective.

    Deduction is so defined that its conclusions are true if its premisses are true, while other modes of inference lack that character; this has been seen as a problem for the other modes. In fact any inference depends on being correctable in its premisses and procedure through larger inquiry and the testability of conclusions. Certainly surmise to an explanation is the least secure mode of inference, but also the most expeditious, as Peirce pointed out. Now a valid deduction secures its conclusion's truth if its premisses are true, while other inference modes only suggest, with more or less strength, their conclusions' truth, and do so by the perspective they bring and other considerations (methods of sampling, weakness of alternate explanations, etc.) rather than only by general definitive entailment structure. But a deduction's conditional assurance of a true conclusion is counterbalanced by the perspective of novelty or nontriviality that a worthwhile deduction brings; dubitability is natural, and right in a way, in deduction. The deduction's conclusional perspectives (novelty, nontriviality) that, with varying strength, incline one to check one's reasonings and claims for falsity, inconsistency, etc., are the same ones that make deduction worthwhile.

    1. A surmise's natural plausibility, and
    2. an induction's Peircean verisimilitude,
    suggest that a conclusion might be true although non-deductive, and offer the hope of helping inquiry toward eventually finding (respectively) the real natural simplicities and the real reliabilities in things.

    3. A 'forward-only' deduction's novelty of aspect, and
    4. an equipollential deduction's nontriviality,
    suggest that a conclusion might be false despite being or seeming deductive (involving an error perhaps in the reasoning or perhaps in the premisses — indeed, the main purpose of some deductions is to expose the premisses to testing), and offer the hope of helping inquiry toward eventually finding (respectively) the real surprising ramifications and the real deep structures in things.

    Correlation between quasi-modal deductive topics and inference modes' valuable aspects (plausibility, verisimilitude, novelty, nontriviality)

    Quasi-modal topics of so-called 'applied' but abstract and significant mathematically deductive areas (below). Aspects with merit in inference (below).
      Valuable, fruitful aspects or perspectives into which conclusions put premisses are sorted below by inference mode.
    1. Optima & feasibles.
    • Differences (distances) with directions.
    • Mathematics of optimization (longer known as linear & nonlinear programming) applies extremization, calculus of variations, Morse theory.
    Naturalness, simplicity, plausibility, viability, cogency.
    Suggested technical name: viatility.
    • Pertains to surmise, explanatory hypothesis, that which Peirce called "abductive inference." I propose a (non-Peircean) definition of surmise as inference that automatically preserves neither truth nor falsity (the premisses don't deductively imply the conclusions, and the conclusions don't deductively imply the premisses). Even if one excludes cases where the premiss set is inconsistent with the conclusion set (e.g., "there is a horse, so there isn't a horse"), natural simplicity, cogency, etc., remain an important consideration in the surmises that remain. The inference "There is a horse here, so there is a squid here" is a surmise without natural simplicity or cogency. Now, one could restate a cogent surmise to make it seem, technically, an induction, its premisses implied by but not implying its full conclusions: "Normally after rain the lawn is wet and this morning the lawn is wet. So, normally after rain the lawn is wet, this morning the lawn is wet, and last night it rained." But this is like turning a traditional syllogism into an equipollential deduction by restating all the premisses in the conclusion. Such representation of the inference does not faithfully enough reflect the guiding inquiry interest of the inference and instead counterproductively obscures it with redundancies.
    • Studies of concrete phenomena (that is, sciences and studies physical, material, biological, and human / social) are for drawing conclusions that typically are (cogent) surmises (but they do apply the other modes of inference along the way). As Peirce points out in discussing verisimilitude in induction (Collected Papers Vol. 2, in Paragraph 633): "Strictly speaking, matters of fact never can be demonstrably proved, since it will always remain conceivable that there should be some mistake about it. [....] Indeed, I cannot specify any date on which any certain person informed me I had been born there; and it certainly would have been easy to deceive me in the matter had there been any serious reason for doing so; and how can I be so sure as I surely am that no such reason did exist? It would be a theory without plausibility; that is all." Hence, even a verisimilar and high-confidence induction, if it is from concrete individual cases, rests in the end on premisses which are affirmed because their falsity is implausible, or so taken, and which amount in the end to cogent surmises. Hence the ultimate standing of conclusions sought by studies of concrete phenomena is that of cogent surmise. Now, I've cited Peirce saying something that I believe helps imply my conclusion; but in or around the same year (1910), he held that all determination of probabilities of actual objects (such as actual dice) rests on verisimilitudes (Collected Papers Vol. 8, in Paragraph 224) and did not mention a dependence of those verisimilitudes on plausibilities (or cogencies).
    2. Probabilities.
    • Ratios (such that 0 ≤ the ratio ≤ 1).
    • Mathematics of probability applies enumerative combinatorics, general measure theory.
    Verisimilitude, likelihood, in C.S. Peirce's sense.
    Suggested technical name: veteratility.
    • Pertains to induction, inference that automatically preserves falsity but not truth (the premisses don't deductively imply the conclusions, but the conclusions deductively imply the premisses).
        Verisimilitude in Peirce's sense consists in that, if pertinent further data were to continue, until complete, to have the same character as the data supporting the conclusion, the conclusion would be proven true. The phrase "inductive generalization" is an approximate way of saying "induction with some verisimilitude." The inference "there is a horse here, so there are a horse and a squid here" is an induction without verisimilitude. I would add that the idea of verisimilitude should also involve the idea of not being too influenced by a premiss that seems to express an outlier or random fluctuation.
    • Studies of positive phenomena in general (studies such as inverse optimization, statistics, information theory's induction-oriented areas, and maybe philosophy) are for drawing conclusions typically inductive.
    3. Information, 'news'.
    • Logarithms.
    • Mathematics of information applies abstract algebra (i.e., abstract theory of calculation).
    New aspect, elucidativeness, significance in a sense.
    Suggested technical name: novatility.
    • Pertains to 'forward-only' deduction, inference that automatically preserves truth but not falsity (the premisses deductively imply the conclusions, but the conclusions do not deductively imply the premisses). The inference "There are a horse and a squid here, so there is a horse here" is a 'forward-only' deduction without novelty of aspect. The inference "All A is B, and all B is C, so all A is C" is a 'forward-only' deduction with some novelty of aspect.
        Novelty of aspect of the kind sought in 'forward-only' deduction involves some tightening of focus, which is lacking in such a 'forward-only' deduction as "Socrates is yonder, so Socrates is here or yonder." (Still, it is important sometimes to remember how "surprisingly" broad alternatives are deductively implied — especially on those occasions when one is surprised by a subtle role of such implications in one's reasoning.)
    • 'Applied' but abstract and significant mathematically deductive areas — mathematics of optimization, of probability, of information, and of logic — are for drawing conclusions that typically are 'forward-only' deductions. I don't say that equivalences, e.g., that between p and T→p, are unimportant in those fields. But their general aim at 'forward-only' deduction is why they are correlated with areas of inductive research as their "inverses" or "reverses." Reverse pure mathematics, on the other hand, is, if not pure mathematics per se, still mathematical logic about pure mathematics; it would not be the first research program to originate in mathematical logic and then grow into a part of pure mathematics, if some old accounts that I recall about non-standard analysis are correct. My understanding is that probability theory is also often applied in pure mathematics, and not always trivially. These so-called 'applied' areas need another generic name.
    4. Givens (data, etc.) as nullary, unary, binary, n-ary, complexuses. (See Arity.)
    • Roots, bases.
    • Mathematics of logic applies order theory.
    Nontriviality, depth, complexity, 'lessonfulness'.
    Suggested technical name: basatility.
    • Pertains to equipollential deduction, inference that automatically preserves truth and falsity alike (the premisses deductively imply the conclusions and vice versa). The inference "There are a horse and squid here, so there are a horse and a squid here" is an equipollential deduction without nontriviality. The inference "3 × 5 = 15, so 3 ∕ 15 = 5" is an equipollential deduction with at least a jot of nontriviality. When the well-orderedness of the pertinent set is a standing given, then the mathematical-induction step, from the ancestral case conjoined with the hereditary case, to the conclusion, is an equipollential deduction.
    • 'Pure' mathematics is for drawing conclusions typically deductive through equivalences and equipollencies. 'Forward-only' deduction sometimes occurs, especially when greater-than or less-than statements are involved. I've read that, because of this, in a mathematical induction a premiss (such as the ancestral case) is not always to be proven by simply reversing the order of one's scratch work; sometimes instead one must find some other way back.

    The complexity theorist Cosma Shalizi (in his notes "Complexity" and "Complexity Measures") has said that complexity is ill-defined and its proposed general measures not actually useful. Now, the idea of complexity seems based on the idea of nontriviality in mathematics ("deep" in mathematics means "very nontrivial," I've been told), and the effort at quantifying complexity seems aimed at finding an information-like quantity. Yet, the idea of information as a quantity is rather simple. Now, one might argue that the analogous idea for complexity will be complex because it's about COMPLEXITY, of course. But that is to say that it is not a quantity on a mathematical par with information in the sense that probability is. Complexity, at least in that which seems the idea's core sense of nontriviality and depth as in mathematics, seems akin in various ways, including its importance and tantalizing character, to aspectual novelty, verisimilitude (in C. S. Peirce's sense), and plausibility (natural simplicity). So classing it, one ends up with four kindred aspects in inference, aspects that resist computation-friendly formulations but are vital as forms of value or merit in inference. They form a set that seems complete in its way, such that in their collective absence, no mind would ever bother to draw a conclusion and there would not be mind as we know it. Data or, more generally, givens, in their character of adicity or arity, with its definabilities and its own share of importance, seem a better "complexity" counterpart to information. For logic, there are, first of all, the structures of alternatives (such as that represented by the variable x) and of compounds conjunctive, conditional, etc., and such is the turn of interest that groups mathematics of logic with those of information, probability, and optimization. Now, when each element of such a compound or relation is itself a relation with the same arity, the arity is like a root or base raised to successive powers, though its sheer size is not the only point. Anyway, logic gains stature as a serious subject, as Quine pointed out, when it comes to the study of relative terms (in polyadic quantification), which have arities of their own; relative terms are a complicating factor in logic. Now, such a relation as the dyadic '__discussing__' is itself not a compound conjunctive, disjunctive, or otherwise; nor are relations such as pure mathematical operations, functions, etc.; yet the forms of such relations are what is applied to represent the forms of alternatives and other such compounds which, for their part, are in a sense (be it literal or figurative) relations among worlds.

    Logical quantity & research scopes - universal, general, special, particular, individual, singular

    First posted on Monday, November 18, 2013—

    —Recentest significant edit: June 28, 2014.

    Some logical quantities, such as the general and the singular, pertain mainly to terms or their objects, rather than mainly to propositions, and are the occasion for the perennial argument between nominalists and realists over the problem of universals. My treatment of such logical quantities differs from tradition. I've discussed these matters in older posts. Here I will adopt some different terminology. I will discuss uniformity for logical quantities, definitions, the conjunctive compounds and correlated areas of research, philosophical tradition, the arts and still other areas of knowledge, and C. S. Peirce.

    Uniformity for logical quantities

    Now, a singular is usually taken as monadic, that is, Socrates is singular, but the dyadic Socrates, Aristotle are taken as singular separately, not also polyadically in the sense of singular, singular, polyadically such that one might want to call them "multi-singular" or some such (even if the polyad is just Socrates, Socrates). So, I coin the word "solular" to refer alike to the monadic singular and to singulars taken polyadically. Unlike the idea of the singular, the idea of the solular is on the same footing as the idea of the general in having and evoking polyadic versions as well as monadic versions. (An example of a polyadic general is the two-place (i.e., dyadic) general "__discussing__.") Such sameness of footing is desirable when one seeks to be systematic, and this pays off for example in the case of the consideration of a universe or total population and its more-or-less collective description (its parameter set) as solulars that are also fully universal (in that universe of discourse). Traditional singularity conjoined with such full universality evokes merely a one-object universe, but that conjoint vista seems mostly barren only because the window is narrowed to the monadicity of the traditional singular. Broaden the singular into the solular, and said conjoint vista broadens itself into at least a rudimentary version of the populous subject matter proper to deductive mathematics of optimization, of probability, of information, and of logic.


    It seems best for the elementary definitions to be in a monadic and de facto perspective where formal considerations to accommodate polyads don't come into play (although I'll still use the term "solular" which applies to monads as well as polyads). Additionally, I coin the word "omnial" to take the place of the informal and potentially misleading phrase "fully universal."

    Suppose that there is something glad.
    Question A: Is there something else glad? If yes, then I call 'glad' general. If no, then I call 'glad' solular.
    Question B: Is there also something non-glad? If yes, then I call 'glad' special. If no, then I call 'glad' omnial.

    The above supposition that there is something glad (or whatever) is not axiomatic but merely a hypothetical condition, so such a thing as is glad (or whatever) may completely lack instances and thus be both a de facto non-general and a de facto non-solular, even though the general and the solular seem each the other's negative. The same goes for the special and the omnial.

    Questions A and B may seem excessively simple. For example, one thinks of the general term not just as actually true of something else, but as potentially or purportively true of something else, indeed of various other things, perhaps indefinitely many. Still, it seems best to keep the elementary definitions crude but refinable to suit the occasion, and to remember them in that light.

    Now, Question A ("Is there something else glad?") and Question B ("Is there also something non-glad?") do not depend on each other at all. The four answers can be conjoined without contradiction in four ways.

    Simple & conjoined positive logical quantities for terms or objects
    General: 1. General-cum-omnial.
    Simple example: two (things among many).
    3. General-cum-special.
    Examples in practical contexts: blue, resilient, melodious, etc.

    Solular (monadic singular, polyadized singulars, etc.):2. Solular-cum-omnial.
    Gamut, universe of discourse, total population, its parameters.

    4. Solular-cum-special. Monadic, polyadized, etc., singular(s) in a larger world.

    The solular-cum-omnial, when monadic, is the object in a single-object universe, but is polyadic for a larger universe and can be much less boring then.

    The conjunctive compounds and correlated areas of research

    Now, I have coined some further terms for brevity. I hope that I haven't erred in making trade-offs between conventionality and evocativeness of the coinages.

    1. The general-cum-omnial, or etceteral.

    The general-cum-omnial is that omnial which, given a monadic or polyadic instance, is also instantiated by further monads or polyads that don't share all the same members. Roughly speaking, it's the fully-universal that is not the whole universe at once. Consider in a first-order logical sense the idea of two such that "two" is true collectively of any x⁠y such that x is not y, and consider the case where, besides x⁠y, there are also z, w,..., etc., that are not x or y and are distinct from one another. (This conception of a number won't get us to Peano arithmetic but, analogously, conceptions of the logical individual don't get us to empirical science.) The term "two" will be true of everything in that universe, each thing not monadically but instead in some combination or other, and indeed in every dyad of distinct things and in every polyad of just a one and an other, be they mentioned soever many times under soever many designations (unless it is mentions or designations that are being counted as objects themselves). It does not depend on particular positive qualities or characters of things, or on distributions of such qualities, and it does not depend on the positive thisness or haecceities of things (e.g., it doesn't matter if one is talking about Socrates and Aristotle, only that one is talking about two distinct objects). For any given non-zero whole number, this works in any sufficiently large universe. The perspective is that of two things (or three things, etc.) such that there is still another two of things (or three of things, etc.), or indeed indefinitely many twos, threes, etc. It's an idea of universality combined with an idea of further instances, even unto "infinity, or the miraculous jar of mathematics." Hence my coinage "etceteral." Particularly natural expressions of the general-cum-omnial, a.k.a. the etceteral, are mathematical operations and (lambda) functions, as well as mathematical one-to-many relations and many-to-many relations. Now, one often thinks of The Number Two, etc., as abstract singulars rather than as etceterals. By abstraction and imaginational machineries such as set theory one revives the logical-quantitative variegation of overall experience, and gets numbers like Zero besides.

    2. The solular-cum-omnial, or totular.

    The solular-cum-omnial is the logical quantity of a single object in, and only in, a one-object universe. A more populous case would be that of the universe of a plinker's notes c⁠d⁠e⁠f⁠g⁠a⁠b, with an idea of ceteris paribus, "the rest staying the same," the rest coarse-grained out, summed over, to the extreme of ceteris non existentibus, i.e., the rest (of one's world) not existing. (I got this idea of total populations and universes of discourse as just various ways of coarse-graining the same grand world from somewhere in The Quark and the Jaguar by Murray Gell-Mann. The solular-cum-omnial generally seems to me the logical quantity with a tinge of The Twilight Zone — which in turn is a reminder that a universe of discourse can't be a mere coarse-graining of the real if that universe harbors fictional elements.) So this is the logical quantity for a total population's or universe-of-discourse's members taken polyadically and more-or-less collectively, and quite collectively when one considers probabilities. Hence my coinage "totular." One could consider the solular-cum-omnial, a.k.a. the totular, that specifies sequence, the kind that does not (e.g., "The Three" in a three-object universe), and mixed cases. A frequential distribution (such as '30% of the total population') of a characteristic across a total population is a totular. Natural expressions of the totular include information as a quantity, probabilities, and feasibility and optimality as mathematically studied. An abstract total population's members will not usually be spelt out with singular constant designations, but, if spelt out at all, then with singular dummy letters perhaps regarded as singular veiled constants, or with variables; one could have a solular term that looks monadic but is to be construed as polyadic, but probably people prefer to express these things with sets. The totular is a total population and also parameters of attribution or distribution of characters, said parameters as belonging to the total population. It lends itself to abstraction and formalization such that the particular qualities distributed do not matter, but only their samenesses and differences, as well as the samenesses and distinctnesses among individuals, and not their 'thisness' or haecceities, their individual "identities" in the sense of everyday English.

    3. The general-cum-special, or transcernal.

    This is the logical quantity most natural for monadic & polyadic positive qualities and characters of things, which are such that we expect further and even indefinitely many instances and also some and even indefinitely many counter-instances. Hence my coinage "transcernal," evoking an idea of sifting through. This is the perspective of inductive fields like inverse optimization, statistics, information theory's inductive areas, and (I think) philosophy, concerned with positive phenomena in general but not, except in applications, with individual positive phenomena in their thisness or haecceities.

    4. The solular-cum-special, or obstular.

    This is the logical quantity most natural for concrete individuals taken monadically and polyadically, but not as a total population in the abstract with ceteris non existentibus. This is the usual sense about individuals and singulars — that they're not only individual or singular, but also not absolutely alone, as if each one or each handful were a universe unto itself. Instead they're individuals in a larger world. Hence my coinage "obstular." It's the perspective of sciences and studies of concrete phenomena, what C. S. Peirce called "idioscopy, or the special sciences." Aristotle said that there is no epistêmê (often translated as "science") of the individual, but by epistêmê he meant something deductive, or nearly so, and not including concrete experimentation. The subject matter of idioscopy is concrete individuals in the larger concrete world, but the objective is to learn about their individual connections and tapestries, their positive qualities and characters, their parameters and laws, and their applicable mathematics.

    Simple & conjoined positive logical quantities for terms or objects.
    Researches correlated thereto by scope.
    General:1. General-cum-omnial, i.e.:
    Subject matter of pure mathematics — fields for deductive conclusions typically through equivalences & equipollencies.
    3. General-cum-special, i.e.:
    Subject matter of studies of positive phenomena in general: inverse optimization, statistics, information theory's inductive areas, and (I think) philosophy — fields for inductive conclusions from parts or samples to larger wholes.

    Solular (monadic singular, polyadized singulars, etc.):2. Solular-cum-omnial, i.e.:
    Subject matter of so-called "applied" yet abstract and significant maths drawing from ideas of total populations, universes of discourse, etc.: deductive mathematics of optimization, of probability, of information, and of logic — fields for deductive conclusions typically "forward-only," from wholes to parts or particular cases.

    4. Solular-cum-special, i.e.:
    Subject matter of sciences & studies of concrete phenomena: sciences & studies of forces & motion, matter, life, and mind — fields for (soever cogent) hypothetical conclusions.

    In discussing pure mathematics, I said, "By abstraction and imaginational machineries such as set theory one revives the logical-quantitative variegation of overall experience...." I'll go out on a limb here (I'm no mathematician) to give examples, not of how, for example, some mathematical ideas are, in their way, obviously more general than others, but of how logical-quantitative properties in mathematics are associable with systematic purposes analogous to those outside mathematics. Ordinals are like singular obstulars for systematically capturing, if not concrete thisness, still a kind of "whichness." Think of series and summability, theory of limits, structures of order, conditions for mathematical induction. A function's derivative, and an arithmetical calculation's result, are like transcernals for systematically classifying together the various functions or various sets of numbers or letters that result in them. Combinatorial enumeration and mathematical integration are like totulars for systematically analyzing a number or an area into its constituents. Topological forms and graph-theoretical graphs are like etceterals for systematically determining traversals and transformations. Well, that last assertion is a bit too vague, but I hope to improve it in time.

    Philosophical tradition

    (The sections on tradition were originally near this post's start but I reorganized the post so as better to 'cut to the chase'.)

    Of the logical quantities exhibited by things as represented by terms, only two sorts have been regarded as noteworthy by most philosophers, and the two have gone under at least two pairs of labels: universals & particulars; and generals & singulars. (C. S. Peirce emphasized three such logical quantities.) By "a universal" or "a general" is usually meant, by philosophers, a thing of which there are AT LEAST TWO INSTANCES (actually or, for some philosophers, at least potentially), and often indefinitely many instances. By "a singular" is meant a term that has or is defined to have just one object, or a thing that has accidentally or oftener intrinsically JUST ONE INSTANCE (if any at all). Sometimes such a thing is called "an individual" or "a particular" (for example by E. J. Lowe, who classes as particulars not only individual substantial objects but also individual monadic and relational property-instances, a.k.a. tropes).

    Despite perennial philosophical attention to the problem of universals — the question, disputed between realists and nominalists, of whether universals (a.k.a. generals) are real or merely verbal ("nominal") — philosophers have mostly ignored the structure of such logical quantities. The terminology is threadbare.

    The tradition's edge and beyond

    The idea of something universal to simply everything is involved in the idea of being itself, also in such tautologous ideas as known-or-unknown, and in the Scholastic idea of the transcendentals of being (unity, truth, goodness). The Aristotelian categories are sometimes regarded as summa genera, highest genera.

    Now, a quality such as blue is typically regarded as a universal but not as being fully universal, universal to everything; one wouldn't expect any positive quality to belong to everything. Rudimentary ideas of one and two do seem fully universal to everything monadically or polyadically, in the sense that anything x is one, and any x⁠y such that x is not y are two, and so on. Of course "three" is not true of two things per se but it is true of them in combination with any still other thing. I think that that is a viable idea of reasonably full universality, even if it is not the fullest imaginable universality, and that it is more fruitful in that, unlike the fullest imaginable, it is populous, indeed infinitely so, with non-equivalent examples. Such ideas as two and three do not depend on things' positive qualities, much less on things' being any individual this such as Socrates or Bucephalus, but only on their selfsamenesses and distinctnesses, which are abstractibles that pour themselves into formalization in ways that individual and qualitative positive phenomena do not. So one has notions of the (reasonably) fully universal and of the not-fully universal but special like blue and Socrates.

    What about the affective arts (those of music, dance, sculpture, drawing, painting, language, story, theater, cinema, etc.), and still other kinds of knowledge?

    Here I seem to have used up the logical quantities as perspectives, scopes of subject matter, just to map, so to speak, the main classes of more-or-less theoretical research. So maybe one should do likewise for each of the other knowledge disciplines (such as the affective arts), but that seems a daunting task. Still, I'd say that there seems:

    A. across theoretically-oriented research, an overall tendency, a kind of overall urge, toward etceteralities where individual cases or facts harbor enhanced lessons.

    Maybe the other disciplines of (fallibilistic) knowledge share the overall tendency toward the etceteral, yet vary insofar as they are cognitive disciplines not of cognitive bases but of decisional impetuses, competential means, and affective effects. I am unsure about this. But at any rate, there seem:

    B. in the affective arts, some overall tendency toward the totular, the selective composing, the selective combining, of things, modes, etc., into worlds in which qualities harbor enhanced values;
    C. in the productive arts/sciences (know-how — engineering, etc.), some overall tendency toward transcernalities where domains harbor enhanced reliabilities, norms; and
    D. in the so-called ruling arts (design, architecture, community planning, education of personal character, etc.), some overall tendency toward obstularities where regimes (rules and constraints, in some sense) harbor enhanced optima, with the obstulars concretely (multi-)individualized and customized into the larger world or even, in some sense, imposed upon or against the larger world.

    But I probably should save such talk for my Speculation Lounge blog.

    Let me note a still bigger picture, where questions of correlations to logical quantities may arise. The above are areas with a kind of upper or second-order level on which the prevailing element — not the only element, but the prevailing one — is that of

    A. cognitively assessing (i.e., as to lessons) — and (fallibilistic) knowledge (to wit: ruling art is knowing from what impetuses one decides things; know-how is knowing by what means one achieves things; affective art is knowing in what effects one affectively feels things; math, science, etc., is knowing on what bases one knows things; I use "knowing" vaguely here; often it's better to say and think "(cognitively) learning" instead, and "coming to feel" instead of "feeling," etc.);

    but the aforementioned bigger picture also includes areas with upper prevailing (though not exclusive) elements of:

    B. valuing, appraising — and sensibility, appreciation, devotion (including but not limited to religion);
    C. practice — and skill; and
    D. struggle / striving — and (volitional / conational) strength, dedication (in conflict, competition, rivalry, dispute).

    For a big table, see "A periodic table of aspects of humanity." For associated methods of learning (cognitively and otherwise), see "Methods of learning" (at The Tetrast2: Speculation Lounge).

    C. S. Peirce

    The structure of such logical quantities as the singular and the general has been barely studied in philosophy except, as far as I know, by C. S. Peirce. Yet philosophy perennially pursues the problem of universals, the question of what sort of being or reality belongs or can belong to that which is not a concrete individual object (where, again, 'universal' refers to that which characterizes more than one thing, at least two things, and, in some contexts, possibly indefinitely many things). The terminology has varied: "universals and particulars," "generals and singulars," and so on. Among major philosophers as far as I know, only Peirce has introduced a more-than-two-way distinction, for which I don't know where to send the reader for a brief sketch, so I will supply one here. He made a three-way distinction, a trichotomy, of:
      (1) the vague, the indefinite, such as a quality as contemplated without reaction or reflection,
      (2) the individual, determinate, and
      (3) the general.
    Said trichotomy
    (A) is based by him in his three respective phenomenological categories:
      (1) Firstness, quality of feeling (more as quality of a sensation than of an affect such as pleasure or pain), essentially monadic,
      (2) Secondness, reaction/resistance, essentially dyadic (individuals, brute facts, etc.), and
      (3) Thirdness, representation/mediation, essentially triadic (rules, habits, norms, dispositions, etc.); 
    (B) reflects three traditional affirmative logical quantities for propositions, respectively:
      (1) the existential particular (Some G is H),
      (2) the singular (This G is H), and
      (3) the hypothetical universal (All G is H). This hypotheticality (as in "each thing is, IF glad, THEN hearty") is important in Peirce, since he usually treated Thirdness as involving conditional necessities, conditional rules, etc.

    Peirce made a distinction to which he did not always adhere terminologically:
    Singular individuals, or singulars for short, "occupy neither time nor space, but can only be at one point and can only be at one date" (i.e., point-instants).
    General individuals, or individuals for short, do occupy time and space and "can only be in one place at one time."
    (See "Questions on Reality," 1868.)

    Now, Peirce defined the real as the object (the topic or subject matter, not necessarily a concrete thing) of a true proposition (whether actually expressed or not), such that anything real and any truth are what they are irrespectively of the opinions of particular minds and particular communities (fallibilism) and would be discovered by investigation if such investigation were to be pursued sufficiently (cognizabilism, opposition to radical skepticism). Thus he held that there are real generals, the objects of true general propositions. He held that this is a logical presupposition which, in its turn, metaphysics, which he held to be based on logic, not vice versa, fleshes out as robust and nontrivial. Thus Peirce was that which is called a realist in metaphysics. He was quite anti-nominalist. By "actual" on the other hand, Peirce meant the individual, the this, i.e., the concrete individual objects that nominalists take as the only real things. Thus Peirce held that rules, qualities, and individuals can be real, but rules and qualities can't be actual (strictly speaking) since they are not individual things. In particular, Peirce held that indeterminacy is real and that there is spontaneity, absolute chance.

    I'll probably add to this post later.

    Telos, entelechy, Aristotle's Four Causes, pleasure, & happiness

    First posted on Sunday, June 23, 2013—

    (Recentest significant edit: March 6, 2015).

    My ideas on entelechy and the Four Causes diverge in some ways from tradition. General references:

    If, in some everyday sense, ends govern beginnings and means, then in what such sense, if any, do entelechies, i.e., forms in their finality or stability, govern ends (as culminations), means, and beginnings?

    In short: I conceive of entelechy as more distinct from end than do Aristotle, Peirce, etc. As the good, the well, etc., has, in Aquinas's phrase, "the rational character of an end," so (pace Aquinas) the true, the sound, the (fallibilistically) wise, has the rational character of an entelechy; the entelechy pertains to confirmation, disconfirmation, etc.—but of what?—merely of hitting, so to speak, a chosen target? Finally a remark by John Dewey led me to see that one observes, imagines, etc., an entelechy, an actual or potential settled state or legacy, in order to check not only whether one has achieved or would achieve a given aim, but also whether the end, targeted as good, really has been or would be good, or bad instead, or better or worse, etc., in light of unintended consequences, conflicts of values, etc. From that standpoint, an entelechy is a structure supporting, checking, and balancing the involved impetuses, means, ends, and sub-entelechies too. Hedonism focuses on the end and ignores the entelechy. ■

    In old posts on this blog I have discussed an expansion of the means-end dichotomy to a tetrachotomy:
    1. Beginning, impetus.
    2. Middle, means, resource, development.
    3. End, telos, culmination, actualization.
    4. "Check," standing-finished, entelechy. (Not absolute finishedness or retirement from evolution or learning.)

    They have a nice regularity and completeness, non-eclectically paralleling the four possible turns of being or becoming: commencing, continuing, ceasing, and refraining. (Still, it's just a simplistic link-up of ideas requiring flexibility and, in some sense, recursiveness. For biological example and as Stephen Jay Gould argued somewhere, not every organism's final stage should be regarded as its only adult stage.)

    Said parallel was what guided me, in thinking also of Aristotle's Four Causes, to the idea of a fourth stage beyond beginning, middle, end, a "checkedness" stage (long before I started my blogs). Eventually I correlated it to the idea of form and, in particular, form as structure, a more or less stable (and thus relatively "final") balance of forces or motions; not form as appearance or look (original meaning of Latin species and Greek eidos). After a while longer I learned that my idea of that stage might correspond more or less to Aristotle's idea of entelechy. Pronounced en-TEL-eh-kee.

    I correlated the four stages one-to-one with Aristotle's Four Causes (after some dithering on my part) and eventually with such things as modalities (in a broadened sense) and processes; levels of concrete phenomena; and human causal capacities:

    Summary of some analogies and correlations
    # Teleological term: Aristotelian
    (with somewhat
    revised principles — agent, patient, act, borne):
    Correlated modality; and processes: Analogous level of
    concrete phenomena:
    Analogous causal principle of the psyche,
    —along with some notable modes, & excellences:
    1. Beginning, impetus. Efficient cause
    (more or less forceful agent as cause).
    Optima & feasibles. Decision processes. Forces, motion. Volition, conation.
    —Struggle / striving; & strength (of body or character).
    (Psyche as agent).
    2. Middle, means, resource, development. Matter, material
    (more or less enduring patient, bearer, allower, fosterer, as cause).
    Probabilities. Stochastic processes. Physical matter,
    physical material process.
    Ability, handling, etc.
    —Practice; & skill.
    (Psyche as patient, bearer, allower, fosterer; like Latin patiens insofar as it diverges from Greek páschonta (from páschô) which it usually translates).
    3. End, culmination, actualization, etc. End, telos
    (more or less vigorous act as cause).
    Information, news. Communicational & control / cybernetic processes. Life. Affectivity (in the broad sense of any feeling of good or bad, i.e., sentiment, emotion, pleasure, pain, etc.)
    —Valuing; & sensibility, appreciation, devotion.
    (Psyche as acted-on, undergoing).
    4. Entelechy, standing-finished. Form
    (more or less stable borneness, balance, as cause).
    Bas(is)es, givens, data, etc. Inference processes. Mind. Cognition.
    —Assessing as to lessons; & (fallibilistic) knowledge ('ruling' art, productive art/science, affective art, and theoretical science, math, etc.).
    (Psyche as borne, balanced).

    Life at the vegetable-organismic level is distinguished by its being governed by species-level standards of functionality, a kind of telos. (I don't mean retroactive causation by an end; I just mean the ways in which an organism's character depends on biological functions or "purposes," dependences that help explain the organism's character to us.) But what about smart living things? I have thought that, in a living mind, cognition of an expected or imagined final form or structure, like a legacy, is involved — one has an idea of how one will verify (in the legacy of side effects, after-effects, etc.) that one has achieved one's goal. The issue of verifiability of having hit a chosen target tends to influence one's choice of target. Eventually I came to see this as a matter of not just of verification, but of learning lessons with potentially broader applications (potential learnings as influencing the choice of target).

    Still, I got stuck in trying to think of how these understandings influence goals themselves in general, not only via desire for lessonfully verifiable outcomes but as goals per se — not only knowledge goals, and not only one's means for reaching one's goals (i.e., not only verifiability that it was by the intended means that one hit the target). How would final forms, final states, come more fully into their own as ways to explain things about living minds likewise as ends, functions, actualizations, help explain things about organisms generally? This was my biggest brain glitch ever; I missed the key staring me in the face.

    For example, identifying entelechy with form, especially final form (as is more or less traditional), I fretted about the dictum that form follows function, i.e., that form is a means to the end of function, in which cases form is just another means or resource and not fully a cause in its own right, whereas my guiding idea has been that for each of the Four Causes there is a concrete level or sphere where it comes into its own as an explanatory cause. The same problem arises if one classifies final form simply as a further telos, i.e., a further or more detailed culmination, end, or goal. (Note that the distinction between final and pre-final forms is no more over-nuanced than the distinction between the matter out of which a thing is made and the matter in which a thing consists.) Anyway, eventually my focus drifted to other aspects of my four-ist "project."

    Now, one can think of a structure as an organization, i.e., a division (and coordination) of labor. So far, that seems only a guidance of means. Finally an idea of John Dewey's helped things fall into place for me. Dewey is quoted in the following passage from "Limited Horizons: The Habitual Basis of the Imagination" by Jason Hills in Transactions v. 48, n. 1:

    In Dewey's words, structure is "a character of events…. A set of traits is called a structure because of its limiting function in relation to other traits of events" (LW 1:64). A significance of a structure is that it determines the telos of an activity or complex of activities.

    At Hills's blog Immanent Transcendence I said in a comment:

    [....] The imagined entelechy, the imagined potential legacy-structure, pertains furthermore to questions of whether the good that one seeks would really turn out to be good at all.

    That's the key. Misusing the word "teleiosis" to emphasize telos as culmination, I went on to say:

    Therefore as organization the entelechical structure guides the ends, the means, and the beginnings wherein one decided to pursue the ends in the first place. Now, insofar as the traditional idea of _telos_ seems closer to teleiosis, a culminal action, than to entelechy, hedonism seems a ditch hard to escape. But in fact hedonism focuses on only one aspect of _telos_. It focuses on teleiosis and ignores entelechy. [....]

    Think. In other words, as Aretha Franklin sings, "Think!" Note that the freedom about which she sings (read the lyrics) is not just the freedom, for cinematic example, of her man to quit their restaurant and go play with the Blues Brothers, but the freedom to THINK and make a thoughtful choice rather than acting automatically on impulse. (In the movie he walks off the job anyway, with comically contrarian directness.)

    The entelechy, the final form, is a structure of supports, checks, and balances. It also guides or influences the (already telos-limiting) structure in which one is already operating.

    Note, I'm not saying that thinking and recognizing are the key to everything. There are also to fight, work, and love. For every key—a season and harmonies. Usually like dimensions, all at once in varying proportions:
    1. Struggle/striving & strength (of body or character).—Conflict, competition, etc. ("So be it?!").
    2. Practice & skill—Facilitation, co-operation, etc. ("So be it.").
    3. Valuing & sensibility, appreciation, devotion.—Community, distinctively shared values ("Is it?!").
    4. Cognitively assessing & (fallibilistic) knowledge.—Systems (such as knowledge disciplines) of checks & balances and (structural) supports ("It is.").

    By learning and imagination of entelechies, one can consider unintended consequences, develop general values, and deal with conflicts among values — those affairs about which Plato's Socrates dialogued so much. Pace the manipulative Rousseau, psychologists and philosophers aren't the only critters interested in the question of what is well-being or, in that sense, happiness (a further question is that of happiness as one's feeling or appreciation of well-being). One addresses questions of well-being in all learning's basic forms:
    1. Struggle (trial & error, and variation).
    2. Practice & repetition.
    3. Valuation & emulation (better known as identification & imitation).
    4. Assessment (as to lessons) & testing (the reasoning process, inquiry).

    Moreover, all causation that works through the imagined, expected, believed, known, etc. (e.g., as in economic affairs), is entelechiacal causation channeled through cognition. Even biological evolution, as a quasi-learning process of trial and error, involves an entelechiacal influence, as stands out for example in convergent evolution. Again, I use the word "causation" in an Aristotelian sense, NOT in the sense of retroactive efficient causation. Biological evolution has arranged for us to be able to let ourselves be determined by what will be, what would be, etc. I feel diffident about discussing things like "emergent properties." I'll just say that, if you believe that all causation is mechanical, then you believe that mechanical causation can arrange for ink marks on paper to depend actually on an expansion of π to a huge number of digits; and you believe that your mechanical causation can arrange for you to be determined, by real things, real possibilities, and even real indeterminacies, to truths about them and their possibilities and indeterminacies. That determination to truth is what I'm talking about.

    I went on in that comment at Hills's blog to say that, thanks to entelechy, an intelligent eudaemonism beats a silly hedonism, but I didn't know that "eudaemonism" generally rejects the relevance of feeling, affectivity. However, I don't see how one can have well-being if one doesn't properly "feel" and appreciate the good; it's not that well-being consists purely in feeling good (it doesn't), and it's not only that, indeed, well-being devoid of any kind of good feeling would starve the spirit and thus not be well-being; it's also that affectivity helps guide and spur one as to good and bad. If one simply doesn't care about anything, one will not gain or keep well-being, except maybe in some overly narrow sense and by hardly believable luck. On the other hand, by intelligence one can cultivate improved affective appreciations. Thus with good logic may one self-sacrifice for others; in one sense it is a sacrifice; in the sense of love it may involve a trade-off, yet finally not a sacrifice — I mean, not the kind of sacrifice that some have a way of demanding when they contrive to rob you blind or worse.

    Hedonism's tunnelvisionary focus on culminal pleasure is bound to divorce the good from the pleasure itself. It concentrates on the feeling of good; it concentrates on sensory pleasure as a reward drug, which it partly is. Hedonism increases one's vulnerability to manipulation by oneself or others, just as belief that only the perceptually familiar is real increases one's vulnerability to deception by oneself or others. Nothing is more typical of affectivity than sensory affectivity, especially sensory pleasure and pain, so that is where hedonism, focused on telos as actualization, culmination, leads. We have these affective capacities built into us to keep us focused on survival, reproduction, etc. But we also have the inborn capacity to improve on these, to consider consequences, and to look to final forms, end states, entelechies — supposed, expected, noticed, remembered — to help determine, guide, influence (I don't mean create absolutely and ex nihilo) our impetuses, decisions, and rules, our means and methods, and our goals, appreciations, and values.

    This is not to say that the good is generically an entelechy. Rather, the good has the rational character of an end as a fulfillment, an attaining of fulfillment, a culmination (which may be ongoing, not just a one-time event), as guided by entelechy (also possibly ongoing, evolving in its feasibility, probability, etc.). Happiness in the sense of well-being, including healthy appreciations, is, at least generically, an end (telos), not an entelechy. The true or valid or sound has the rational character of an entelechy. A kind of firmness or groundedness of one's being, including soundly based true cognitions, is, at least generically, an entelechy, not an end or culmination. In particular, a true-seeming cognition is not automatically a sound entelechy. This is likewise as a feeling or emotion that "feels good" is not automatically a good end or culmination.

    1. The strong, forceful, etc., has the rational character of a beginning, an impetus.

    2. The apt, suitable, resource-rich, etc., has the rational character of a middle, a means.
    ∞ 3. The good, well, healthy, vibrant, has the rational character of an end, a (possibly ongoing) culmination.

    4. The univocal / consistent, true, valid, sound, (fallibilistically) wise has the rational character of an entelechy, a having-in-completeness, a standing-finished (by which, again, I don't mean absolute finality or retirement from evolution or learning).

    Note: I cannibalized from this post in a 9/19/2013 message to peirce-l.

    Note: "Having-in-completeness" is one of Joe Sachs's translations (another is "being-at-an-end") of entelecheia, in the "Energeia and Entelecheia" section of his article "Aristotle (384-322 BCE): Motion and its Place in Nature" in the Internet Encyclopedia of Philosophy. A traditional interpretation of entelechy, written entirely or mostly by C. S. Peirce, appears in the Century Dictionary:

    entelechy (en-tel´e-ki), n. [ L. entelechia, Gr. ’εντελέχεια, actuality, ’εν τέλει ’έχειν, be complete (cf. ’εντελής, complete, full): ’εν, in; dat. of τέλος, end, completion; ’έχειν, have, hold, intr. be.] Realization: opposed to power or potentiality, and nearly the same as energy or act (actuality). The only difference is that entelechy implies a more perfect realization. The idea of entelechy is connected with that of form, the idea of power with that of matter. Thus, iron is potentially in its ore, which to be made iron must be worked; when this is done, the iron exists in entelechy. The development from being in posse or in germ to entelechy takes place, according to Aristotle, by means of a change, the imperfect action or energy, of which the perfected result is the entelechy. Entelechy is, however, either first or second. First entelechy is being in working order; second entelechy is being in action. [....]

    Joe Sachs holds that Aristotle holds that entelechy exists only in, through, and during the "activity" and "expenditure of effort" required to maintain it. One discerns at length that Sachs has Aristotle seeing even a static opposition of forces (a rock pushing downward against the seabed) as involving energeia, activity, whereas nowadays a static opposition of forces is seen, in simple cases at least, as involving potential energy rather than energy being expended, work being done. On the other hand, the example of the rock against the seabed is oddly complicated; maybe we're meant to understand that the rock has not yet come to rest and is still making a little downward progress against the seabed's resistance, with a corresponding expenditure of energy. In that case maybe Sachs is correct and Aristotle's conception of entelechy excludes static final structures and is hence less broad than it has traditionally seemed.

    A little more on ends as determinants of means. As C.S. Peirce points out, it is through the mind that prospective ends can work like agents through a person. At the vegetable level, it is through biological evolution that ends or functions help govern the means of organisms. At the material level, thermodynamic decay seems the general end (thanks to the probabilities involved) and, at the mechanical level (outside of General Relativity), conservation of certain quantities seems the general end (thanks to the extremal or optimizational principles involved), and there seems nothing retroactively agentlike about either of those two ends when sufficiently well understood (say physicists; the math is beyond me, but I'm quite willing to believe them on this).

    Questions. Does my way of distinguishing between end and entelechy correspond closely or only approximately to Aristotle's distinction between energeia and entelechia, and, likewise, to Aquinas's distinction between the agent cause's end (activity) and the matter's end (form)? Update September 5, 2013: I should add here that I mistakenly thought that I could use the word "teleiosis" to emphasize end as completion, culmination, etc., but have learned that Aristotle used that word to mean something more like development toward an end. The Ancient Greek word teleute won't suffice either. The word telos itself already meant, first of all, end in the sense of reaching an end, the attainment of fulfillment, according to the online dictionaries at Perseus-Tufts. Accordingly I've made corrections throughout this post. End of update.

    Peircean pragmatism. My idea of entelechy, especially the renovated version that is the subject of the present post, plainly crosses paths with C. S. Peirce's pragmatic maxim, first published in 1878:

    Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.

    By the phrase "crosses paths" I imply that my ideas still end up at a different place than Peirce's ideas do. To some extent that is true, but I'm not sure to what extent.

    A difference between Socratic and Peircean foresight and imagination as to consequences and legacies, is that Peirce focused more on potential goods than on the potential ills and conflicts that concern so much of the Socratic dialogues.

    A ball rolls, but a cube sits, on a slightly inclined plane. Is that not formal causation? It is mechanical causation "by forms" but not formal causation in Aristotle's sense (and it is quite traditional to see forms as also acting as agent causes into the future). In the Aristotelian sense, a thing's formal cause is, most simply put, the thing's form. (The word "cause" is redundant in speaking of material, end, and form; it isn't quite redundant in the phrase "efficient cause" but it is chiefly for linguistic reasons that we don't call it "the efficient" or "the effector"; on the other hand, the phrase "agent cause," synonymous with "efficient cause" or "effector," involves no redundancy; likewise "patient cause," which refers to the matter or ingredients).

    More generally, such things as balls rolling while cubes sit are sometimes taken as "geometrical" or "mathematical" causation; the form, or the mathematical structure, helps explain the phenomenon, even if it is not formal causation, or that which would be mathematical causation, in an Aristotelian sense. In fact, in mechanics, phenomena are always explained with the help of geometry, just ask Galileo. Motions and forces have directions, etc. Still, what about the mathematicality of form and structure? Is a thing's form simply its mathematicality (whatever that means?)? Aristotle said that the ratio of different kinds of ingredient in a thing is part of the form or formal cause. It is sometimes said that mathematics studies structures. It is also said to study relations in a general sense that includes operations and functions, not to mention relations, e.g. antiderivatives, that go from one set of values to many sets of values, and relations that go from many sets of values to many sets of values. So which is it, structures or some abstract sort of doings? If that choice comes down to splitting a hair, then splitting what hair?

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