Narratology

Narratology (pronounced nar-uh-tol-uh-jee)

The study of narrative & narrative structure and the ways these affect human perception (with some mission creep over the years).

1967: The construct was narrate +‎ -ology, an Anglicization of the French narratologie, coined by Bulgarian-French historian, philosopher & structuralist literary critic Tzvetan Todorov (1939–2017), it first appeared in his book Grammaire du Décaméron (1967), a structural analysis of Decameron (The Decameron (1348-1353)) by the Italian writer Giovanni Boccaccio (1313–1375).  Although once thought an arcane appendage to literature and a mere academic abstraction, structuralism and narratology in the 1970s and 1980s became a very popular (and controversial) field and while postmodernism’s historic movement may have passed, the tools are an important part of the “learning” process used by generative AI (artificial intelligence) to produce meaning from the LLM (large language models.)

Title page from a 1620 printing of Decameron.

Boccaccio’s Decameron (literally “ten days”) was a collection of short stories, structured into a hundred tales of seven young women and three young men who had secluded themselves in a villa outside Florence, seeking to avoid the Black Death pandemic (1346-1353) then sweeping Europe.  Although not too much should be made of this comparison, the work in some aspects is not dissimilar to reality television, being a mash-up of erotic scenes, set-piece jokes, suspense and unrequited love.  Todorov’s Grammaire du Décaméron was a literary analysis of the work but “grammaire” must be understood as meaning “grammar” in the sense of the structural or narratological principles rather than as its used in its “everyday” sense.  Historians and literary scholars have for centuries regarded Decameron as a valuable document because, written in the Florentine vernacular of the era, although fictional, it’s a kind of “snapshot” of life in what was one of Europe’s many troubled times.  It was Boccaccio who dubbed Dante’s (Dante Alighieri (circa 1265–1321)) Divina Commedia (Divine Comedy (circa 1310-1321)) “divine” (in the sense of “very good” rather than “holy”).

Narrate (to relate a story or series of events (historically in speech or writing)) may for years (or even decades) have been in oral use in English before the first known use in print in 1656, etymologists noting that until the nineteenth century it was stigmatized as “Scottish” (long a slur among the more fastidious) although it’s thought it was derived from the “respectable” narration.  Narrative ((1) a story or account of events or (2) the art, process or technique or telling the story) was in use by the 1440s and was from the Middle French noun & adjective narrative, from the Late Latin narrātīvus (narration (noun) & suitable for narration (adjective)), the construct being narrāt(us) (related, told), past participle of narrāre (to relate, tell, say) + -īvus (the adjectival suffix).  Again, like “narrate”, narrative was once used exclusively of speech or writing but in recent decades the terms have been more widely applied and not restricted to describing the efforts of humans.

Since the nineteenth century, “-ologies” have proliferated.

The suffix -ology was formed from -o- (as an interconsonantal vowel) +‎ -logy.  The origin in English of the -logy suffix lies with loanwords from the Ancient Greek, usually via Latin and French, where the suffix (-λογία) is an integral part of the word loaned (eg astrology from astrologia) since the sixteenth century.  French picked up -logie from the Latin -logia, from the Ancient Greek -λογία (-logía).  Within Greek, the suffix is an -ία (-ía) abstract from λόγος (lógos) (account, explanation, narrative), and that a verbal noun from λέγω (légō) (I say, speak, converse, tell a story).  In English the suffix became extraordinarily productive, used notably to form names of sciences or disciplines of study, analogous to the names traditionally borrowed from the Latin (eg astrology from astrologia; geology from geologia) and by the late eighteenth century, the practice (despite the disapproval of the pedants) extended to terms with no connection to Greek or Latin such as those building on French or German bases (eg insectology (1766) after the French insectologie; terminology (1801) after the German Terminologie).  Within a few decades of the intrusion of modern languages, combinations emerged using English terms (eg undergroundology (1820); hatology (1837)).  In this evolution, the development may be though similar to the latter-day proliferation of “-isms” (fascism; feminism et al).

A narrative is a story and it can run to thousands of pages or appear in a few words on a restaurant menu describing their fish & chips: “Ethically sourced, line-caught Atlantic cod, liberated from the frigid depths, encased in a whisper-light, effervescent golden shroud of our signature micro-foamed artisanal lager batter and served with hand-sliced, elongated potato batons fried to a crisp perfection in sustainably produced vegetable oil.”  In the age of every customer being able to post from their phone a rating and review of a restaurant, wisely, some institutions include a footnote along the lines: “These narratives are a guide and because natural products vary greatly, there will be variation.”  That’s an aspect of narratology, a process which is not the reading and interpretation of individual texts but an attempt to study the nature of “story” itself, as a concept and as a cultural practice or construct.

Crooked Hillary Clinton's book tour (2017).

Narratologists know that what to a narrator can be a narrative, a naratee will receive as spin.  In What Happened (2017), a work of a few dozen pages somehow padded out to a two-inch thick wad of over 500 using the “how to write an Amazon best-seller” template, crooked Hillary Clinton (b 1947; US secretary of state 2009-2013) explained who was to blame for her loss in the 2016 US presidential election (spoiler alert: it was everybody except her).

Presumably not comparing what they’re doing with making “fish & chips” sound like something expensive, politicians and their operatives will often describe something they offer as a “narrative” although were mush the same stuff to come from their opponents it might be dismissed as “spin”.  A political narrative functions as a cognitive schema intended to simplify complexity, motivate support and legitimizes particular courses of action.  The concept has a long history but in recent decades the emphasis has been on “simplicity”, something illustrated by comparing a narrative like The Federalist Papers (1878-1788; a collection of several dozen essays advocating the ratification of the Constitution of the United States) with how things are now done (mostly fleshed-out, three-word slogans endlessly repeated).  That descent doesn’t mean both are not narratives in that both are crafted interpretive frame rather than objective descriptions although the extent of the deception obviously had tended to change.  Political spin can also be a narrative and should be thought a parallel stream rather than a tributary; variations on a theme as it were.  Although the purpose may differ (a narrative a storyline intended to set and define and agenda whereas spin is a “damage control” story designed to re-shape perceptions.  Given that, a narrative can be thought of a “macro-management” and spin “micro-management”, both providing fine case-studies for narratologists.

Narratology is a noun; the noun plural is narratologies.  The derived forms are the noun antenarratology (the study of antenarratives and their interplay with narratives and stories), the noun antenarrative (the process by which a retrospective narrative is linked to a living story (the word unrelated to the noun antinarrative (a narrative, as of a play or novel, that deliberately avoids the typical conventions of the narrative, such as a coherent plot and resolution)), the noun  econarratology (an approach to literary criticism combining aspects of ecocriticism (the interdisciplinary study of literature and ecology) and narratology), the noun narratologist (one who (1) studies or (2) practices narratology), the adjective narratological (of or pertaining to narratology) and the adverb narratologically (in terms of narratology).  Remarkably (given the literary theory industry), the adjective narratologistic seems never to have appeared; it can be only a matter of time.

Tzvetan Todorov on the rooftop of Casa Milà (La Pedrera), Barcelona, Spain, November 2014.

Although not a lineal descendent, what Todorov did in Grammaire du Décaméron was in the tradition of Aristotle’s (384-322 BC) work, especially ποιητικῆς (Peri poietikês (De Poetica De Poe in the Latin and traditionally rendered in English as Poetics).  Poetics is notable as the earliest known study of the structure of Greek drama and remains the oldest known text written exclusively in the form of what now would be called literary theory.  To a modern audience the word “poetics” can mislead because the author’s focus was ποιητική (literally “the poetic art”, from ποιητής (poet, author, writer) and his scope encompassed verse drama (comedy, tragedy, and the satyr play), lyric poetry, and the epic.  For centuries, Poetics loomed over the Western understanding of Greek theatre; it was revered by scholars of the late Medieval period and especially the Renaissance and their influence endured.  As far as is known, the Greeks were the first of the tragedians and it’s through the surviving texts of Aristotle that later understandings were filtered but all of his conclusions were based only on the tragedies and such was his historic and intellectual authority that for centuries those theories came to be misapplied and misused, either by mapping them on to all forms of tragedy or using them as exclusionary, dismissing from the canon those works which couldn’t be made to fit his descriptions.  However, as well as being an invaluable historic text explain how Greek theatre handled mimesis (imitation of life, fiction, allegory etc), Poetics genuinely can be read as proto-critical theory and in it lies a framework for structuralism.

Paintings of Claude Lévi-Strauss: Portrait de Claude Lévi-Strauss, 1991 (1991), oil on panel by Bengt Lindström (1925-2008) (left) and Claude Lévi-Strauss (undated), oil on other by Cal Lekie (b 1999).

Narratology as a distinct fork of structuralism does pre-date Todorov’s use of the word in 1967, the seminal work in the parameters of the discipline by Russian folklorist & literary historian of the formalist school Vladimir Propp (1895-1970) who doubtlessly never anticipated “formalism” would come to be weaponized by comrade Stalin (1878-1953; Soviet leader 1924-1953).  Indeed, by the late 1920s the school of formalism had become unfashionable (something which in the Soviet Union could be dangerous for authors) and their works essentially “disappeared” until being re-discovered by structuralists in the 1950s.  In the West, the idea of narratology as the “theory, discourse or critique of narrative or narration” owes a debt to Belgian-born French anthropologist & ethnologist Claude Lévi-Strauss (1908–2009) who defined the structural analysis by narrative as its now understood.  His landmark text Anthropologie structurale (Structural Anthropology (1958)) suggested myths are variations on basic themes and that in their totality (which runs to thousands) their narratives contain certain constant, basic and universal structures by which any one myth can be explained.  In that way, myths (collectively) exist as a kind “language” which can be deconstructed into units or “mythemes” (by analogy with phonemes (an indivisible unit of sound in a given language)).  Although he didn’t pursue the notion of the comparison with mathematics, others did and that (inherently more segmented) field perhaps better illustrates “structural roles” within language in elements which, although individually standing as minimal contrastive units, can be combined or manipulated according to rules to produce meaningful expressions.  As in formal language theory, in mathematical logic, the smallest units are the primitive symbols of a language which can be quantifiers, variables, logical connectives, relation symbols, function symbols or punctuation.  Broken into the individual parts, these need have no (or only minimal) semantic meaning but gain much meaning when assembled or otherwise handled through syntactic combination governed by a recognized grammar (ie although conceptual primitives rather than “building blocks”, complex meaning can be attained by applying axioms and rules).

Azerbaijani folk art, following Layla and Majnun (1188), a narrative poem by the Persian poet Nizami Ganjavi (circa 1141–1209), printed in Morphology of the Folk Tale (1928) by Vladimir Propp.  In something of a Russian tradition, there are no known photographs of Propp smiling.

Levi-Strauss’s contribution was that myths can be read in relation to each other rather than as reflecting a particular version, thus the his concept of a kind of “grammar” (the set of relations lying beneath the narrative’s surface), thus the general principle of the “collective existence of myths”, independent of individual thought.  That was of course interesting but the startling aspect was the implication myths as related to other myths rather than truth and reality; they are, in a sense, “outside” decentred, and possess their own truth and logic which, when contemplated in a “traditional” way, may be judged neither truthful nor logical.  In that, Levi-Strauss applied something of the method of Propp who, in Morphology of the Folk Tale (1928), “reduced” all folk tales to seven “spheres of action” and 31 fixed elements or “functions” of narrative.  In Propp’s scheme, the function was the basic unit of the narrative “language’ and denoted or referred to the actions which constitute the narrative while the functions tend to follow a logical sequence.  The concept would have been familiar to engineers and shipbuilders but genuinely there was some novelty when applied to literature

Lithuanian semiotician A. J “Julien” Greimas (1917–1992) was among the many academics working in France who found Propp’s reductionism compelling and in Sémantique Structurale Recherche de méthode (Structural Semantics: An Attempt at a Method (1966)) he further atomized things, apparently seeking something like a “universal macro language”, a grammar of narrative which could be derived from a semantic analysis of sentence structure.  That was as ambitious as it sounds and to replace Propp’s “spheres of action” he suggested the “actant” (or role): a structural unit which is neither character or narrative.  To handle the mechanics of this approach he posited three pairs of binary oppositions which included six actants: subject/object; sender/receiver; helper/opponent.  The interactions of these binary oppositions served to account for or describe the three basic patterns which are to be found in narrative: (1) desire, search or aim (subject/object), (2) communication (sender/receiver) and (3) auxiliary support or hindrance (helper/opponent).

An eleven-volume first edition of Marcel Proust’s À la recherche du temps perdu (published originally in seven (1913-1927); in the the original French it contained some 1.267 million words.  By comparison, Leo Tolstoy's (1828-1910) War and Peace (1898) ran ran (depending on the edition) to 560-590 thousand.

While Greimas didn’t explicitly claim his model successfully could be mapped on to “any and every” narrative, he does appear to have built his model as a general theory and while not all critics were convinced, it seems generally to have been acknowledged his toolbox would work on a much wider range than that of Propp which did break down as narrative complexity increased.  Another French literary theorist associated with the structural movement was Gérard Genette (1930–2018) and in choosing a case study for his model he described in Discours du récit est un essai de méthode (Narrative Discourse: An essay in method (1972)) he selected Marcel Proust’s (1871-1922) À la recherche du temps perdu (1913–1927) (originally translated in English as “Remembrance of Things Past” and of late as “In Search of Lost Time”) which spans many volumes and narrative streams.  This time the critics seemed more convinced and seem to have concluded Genette’s approach was “more accessible” (these things are relative).  Noting the distinctions made in Russian Formalism between fabula (story) & syuzhet (plot), Genette distinguished between récit (the chronological sequence of a narrative’s events), historie (the sequence in which the event actually occurred and narration (the act of narrating itself); atop that framework, he built a complex discussion.  Being a French structuralist, he of course added to the field some new jargon to delight the academy, concluding there were three basic kinds of narrator: (1) the heterodiegetic' (where the narrator is absent from his own narrative), (2) the homodiegetic (the narrator is inside his narrative, as in a story told in the first person) and the autodiegetic (the narrator is inside the narrative and also the main character).  Genene’s approach was thus relational, envisaging narrative as a product or consequence of the interplay of its different components, meaning all and all aspects of narrative can be seen as dependent units (or, debatably, layers).

Narrator & protagonist: Lindsay Lohan as Cady Heron in Mean Girls (2004).  What in literary theory is known as homodiegetic narration is in film production usually called “subjective narration” or “first-person narration”, realized usually in a “voice-over narration by the protagonist”.

In formulating his three categories Genene nodded to Aristotle and Plato (circa 427-348 BC), the ancient worthies distinguishing three basic kinds of narrator: (1) the speaker or writer using their own voice, (2) (b) one who assumes the voice of another or others and (3) one who uses both their own voice and that of others.  These categories need not be exclusive for a story may begin in the voice of a narrator who may then introduce another narrator who proceeds to tell the story of characters who usually have their own voices and one or more of them may turn to narration.  Structurally (and even logically), there’s no reason why such a progression (or regression) cannot be infinite.  Although it’s obvious the term “narrate” denotes the person to whom a narrative is addressed, just because there is a narrative, it need not be axiomatic a narratee is present or ever existed, T. S. Eliot (1888–1965) in The Three Voices of Poetry (1953-1954) discerning three modes (voices) of poetic expression: (1) the poet speaking to himself, a personal, often obscure meditation, (2) the poet addressing an audience, aiming to teach, persuade, or amuse and (3) the poet creating a dramatic character, as in verse drama, something demanding complex communication between imagined characters.  Eliot argued that “good” poetry often was a blend of these voices and distinguishing them helps in understanding a poem's social and artistic purpose, beyond its mere self-expression.  However, Eliot did note that in “talking to himself”, the writer could also be “talking to nobody”.  He was at pains also to point out that when speaking in the third voice, the poet is saying not what he would say in his own person, but only what he can say within the limits of one imaginary character addressing another imaginary character.  More than many, Eliot knew narrative was not always reliable but the techniques of narratology (and structuralism generally) exist for purposes other than determining truth.

Roland Barthes (2015), oil and acrylic on canvas by Benoit Erwann Boucherot (b 1983).

Layers in narrative structure were identified by the French philosopher & literary theorist Roland Barthes (1915–1980) and his work had great appeal, something of an academic cult once surrounded him and, almost half a century after his death, he retains a following.  In Introduction à l'analyse structurale des récits (Introduction to Structural Analysis of Narrative (1966)), Barthes presumed a hierarchy of levels existed within narrative, suggesting that, up to a point, they can be discussed separately.  Narrative (at least for this purpose), he conceived as a “long sentence”, just as every constative (in linguistics, pertaining to an utterance relaying information and likely to be regarded as true or false) sentence can be the “rough outline” of a short narrative.  Barthes’ model was more building block-like in that he selects basic units of narrative (such as “function” & “index”, functions constituting a chain of acts while indices are a kind of metadata containing information about characters.

François Mitterrand (1984), acrylic on canvas by Bryan Organ (b 1935).

On X (formerly known as Twitter), one tweeter analysed the images on Barthes which exists and the indexed web, finding in 72% he was smoking a cigarette or cigar.  The statistical risks associated with routinely inhaling a known carcinogen have for decades been well-known but Barthes didn’t live long enter the age of “peak statistical risk”.  In February, 1980, having just taken lunch with François Mitterrand (1916–1996; President of France 1981-1995) in a restaurant on Paris’s Rue des Blancs-Manteaux, Barthes was using a zebra crossing on the Rue des Ecoles when knocked down by a laundry van; never recovering from his injuries, he died a month later.  The van’s driver was one Yvan Delahov, of Bulgarian nationality who tested positive for alcohol, but his reading of 0.6 fell below the legal maximum of 0.8; admitting he was late delivering his shirts, he claimed he’d not exceeded 60 km/h (37.3) mph.  At the time, Barthes was carrying no identity documents but was identified his colleague, the philosopher Michel Foucault (1926–1984).

Northrop Frye, Anatomy of Criticism (first edition, 1957).

Finally must be acknowledged the contribution of Canadian literary critic & literary theorist Northrop Frye (1912–1991) whose Anatomy of Criticism (1957) is regarded still as one of the more “remarkable and original” (in the words of the English historian and critic J.A. Cuddon (1928-1966)) works of literary theory in the English-speaking world.  In the narrow technical sense, Frye's theory is not structuralist (something which doubtless burnished its reputation among many) but it certainly contains strands which can be seen as structuralist.  Frye positioned literature as an “autonomous verbal structure”' unrelated to anything beyond itself, a world which contains “life and reality in a system of verbal relationships”.  In this “self-contained literary universe”, there were four radical “mythoi” (plot forms and basic organizing structural principles) which corresponded to the four seasons of the natural order and constitute the four main genres of comedy romance, tragedy and satire.  For those non-postmodernists who still long for l'art pour l'art (art for art's sake), Frye’s mythois are there to be used and he proved their utility in a wide range of texts, including the Bible.

Axiomatic

Axiomatic (pronounced ak-see-uh-mat-ik)

(1) Of or pertaining to the nature of an axiom.

(2) That which is self-evident or unquestionable; the obvious.

(3) Containing maxims; aphoristic.

(4) In formal logic (or any logical system), as axiomatic system, a set of axioms from which theorems can be derived by the application of by transformation rules.

(5) In mathematics, relating to or containing axioms (now less common).

1797: From the Ancient Greek ἀξιωματικός (axiōmatikós), from ἀξίωμα ((axíōma), genitive axiomatos) (a self-evident principle), the construct being axiōmat (stem of axíōma) + the suffix ikos (and the related ic).  The now less common form axiomatical was known as early as the 1580s.  The ikos suffix was from κός (kós) with an added i, from i-stems such as φυσι-κός (phusi-kós) (natural), through the same process by which ῑ́της (ī́tēs) developed from της (tēs), occurring in some original case and later used freely.  It was cognate with the Latin icus and the Proto-Germanic igaz, from which came Old English iġ (which in Modern English ultimately was resolved as y), the Old High German ig and the Gothic eigs.  The ic suffix forms adjectives from other parts of speech.  It occurred originally in Greek and Latin loanwords (metallic; poetic; archaic; public etc) and, on this model, was used as an adjective-forming suffix with the particular sense of “having some characteristics of”, as opposed to the simple attributive use of the base noun (balletic; sophomoric etc), “in the style of” (Byronic; Miltonic etc), or “pertaining to a family of peoples or languages” (Finnic; Semitic; Turkic).  The -ic suffix was from the Middle English -ik, from the Old French -ique, from the Latin -icus, from the primitive Indo-European -kos & -ḱos, formed with the i-stem suffix -i- and the adjectival suffix -kos & -ḱos.  The form existed also in the Ancient Greek as -ικός (-ikós), in Sanskrit as -इक (-ika) and the Old Church Slavonic as -ъкъ (-ŭkŭ); A doublet of -y.  In European languages, adding -kos to noun stems carried the meaning "characteristic of, like, typical, pertaining to" while on adjectival stems it acted emphatically; in English it's always been used to form adjectives from nouns with the meaning “of or pertaining to”.  A precise technical use exists in physical chemistry where it's used to denote certain chemical compounds in which a specified chemical element has a higher oxidation number than in the equivalent compound whose name ends in the suffix -ous; (eg sulphuric acid (H₂SO₄) has more oxygen atoms per molecule than sulphurous acid (H₂SO₃).  Axiomatic & axiomatical are adjectives, axiomatize & axiomatize are verbs and axiomatically is an adverb.  Clumsy forms (sometimes hyphenated) like nonaxiomatic & unaxiomatic are created as required.

In mathematics (notably in including geometry, algebra, and set theory), an axiomatic system (“deductive system” or “formal system” seem to be the more fashionable terms) is a set of axioms or postulates, which, coupled with rules of inference, can be used to derive theorems or statements from those axioms.  In mathematics, there are collections of equations which can be used to document the processes but in any form of applied logic these systems provide a rigorous foundation for reasoning and proof, using what can be reduced to a mathematical process.  In axiomatic systems, axioms are assumed to be true without proof and the rules of inference are used to derive new statements from the axioms; theorems derived from the axioms are then considered to be true, based on the validity of the axioms and the rules of inference.

Axiomatic: If crooked Hillary Clinton is using a cell phone, she will be deleting something.

In the discipline of philosophy, even those parts which are not inherently mathematical (such as formal logic), the axiomatic system works in a similar way in that a statement, proposition, or principle that is considered self-evident or universally accepted without needing to be proven.  Axioms are thus often used as the starting point for logical reasoning or the foundation upon which a system of thought or theory is built, assumed to be true and are not subject to further analysis or questioning within the context of the system they are part of.  There are a number of highly technical rules which define whether a axiomatic system can be described as “consistent” but that means that within its own terms it contains nothing contradictory.  In other words, from the elements of any axiomatic system, it’s not possible to be either proven or disproven. This differs from one labeled “independent” in that that status is defined by them not being proven or disproven from other axioms in the system.  An axiomatic system is labeled “complete” if for every statement, either itself or its negation is derivable from the system's axioms (implicit in which is that every statement is capable of being proven true or false).

Not axiomatic: The real Lindsay Lohan and Take Two Interactive’s alleged doppelganger in Grand Theft Auto V (GTA5).

Lawyers too like the word “axiomatic”, possibly because concepts like “foreseeability” and “causation” are such an essential part of their training.  The use though exists within different parameters to that of mathematics.  In Lindsay Lohan vs Take Two Interactive Software Inc et al (APL-2017—00027 and APL-2017-00028 (November 2017)), the New York Court of Appeals held that a certain section of an act “categorically excludes works of fiction, a protected category of expression beyond the narrow scope of the statutory phrases advertising and trade”, noting the US Supreme Court (USSC) had “reversed course to recognize First Amendment protections for fiction”.  The Court of Appeal explained that after the USSC “limited Section 51 claims for fictionalization” to factual works that place persons in a false light, subsequent case law both isolated the commercial interest protected by the right of publicity and recognized “the right of publicity does not attach” where “it is evident to the public that the events so depicted are fictitious.”  The judgment noted with approval the decision of the California Supreme Court which “famously” recognized fiction writers may “more persuasively be able to more accurately express themselves by weaving into the tale persons or events familiar to their readers”, adding “correctly”, that “the choice is theirs”.  “This categorical protection is now axiomatic”.  Once can see what the judges meant and of course they were correct but what can be held to be “axiomatic” in law can differ from the same thing in mathematics because in the world of numbers, there is no superior court able to rule 2+2=5.  Their position is more akin to the philosophers who for centuries until 1697 could regard as inviolate the axiom to “all swans are white and all non-white birds are not swans”.

Asymptote

Asymptote (pronounced as-im-toht)

(1) In mathematics, a straight line which a curve approaches arbitrarily closely as it extends to infinity; the limit of the curve; its tangent “at an imaginary representation of infinity”.

(2) By extension, figuratively, that which comes near to but never meets something else (used in philosophy, politics, conflict resolution etc).

1650–1660: From the Greek asýmptōtos (not falling together).  The Ancient Greek ἀσύμπτωτη (asúmptōtē) was the feminine of Apollonius Pergaeus' Ἀπολλώνιος ὁ Περγαῖος Apollṓnios ho Pergaîos (Apollonius of Perga (Apollonius Pergaeus (circa 240-190 BC)), an astronomer whose most noted contribution to mathematics were his equations exploring quadratic curves.  The construct of the Ancient Greek adjective ἀσύμπτωτος (asúmptōtos) (not falling together) was a- (not) + sýmptōtos (falling together (the construct being +‎ συν (-sym-) (together) + πτωτός (ptōtós) (falling; fallen inclined to fall), the construct being ptō- (a variant stem of píptein (to fall) (from the primitive Indo-European root pet (to rush; to fly)) + -tos (the verbid suffix).  The adjective asymptotic (having the characteristics of an asymptote) dates only from the 1970s.  Asymptote is a noun & verb, asymptotia & asymptoter are nouns, asymptotic & asymptotical are adjectives, asymptoted & asymptoting are verbs and asymptotically is an adverb; the noun plural is asymptotes.

Lines, curves & infinity

The noun asymptote describes a straight line continually approaching but never meeting a curve, even if extending to infinity.  This means that although the distance between line and curve may tend towards zero, it can never reach that point, which is hard to visualize but explained by the notion of the line only ever able to move half the distance required to achieve intersection.  At some point such a thing becomes impossible usefully to represent graphically and even exactly to define the asymptotic using integer mathematics would be unmanageable, thus the use of the infinity symbol (∞).

Horizontal (left), vertical (centre) and oblique asymptotes (right).

There are (1) horizontal asymptotes (as x goes to infinity (in either direction (ie also negative (-) infinity)), the curve approaches b which has a constant value), (2) vertical asymptotes (as x (from any direction) approaches c (which has a constant value), the curve proceeds towards infinity (or -infinity) and (3) oblique asymptotes (as x proceeds towards infinity (or -infinity), the curve goes towards a line y=mx+b (m is not 0 as that is a horizontal asymptote).

The logarithmic spiral and the asymptote.

Although usually depicted on a flat plane, a curve may intersect the asymptote an infinite amount of times.  A spiral with a radius is a logarithmic spiral, distinguished by the property of the angle between the tangent and the radius vector being constant (hence the more popular names “equiangular spiral” or “growth spiral”, the latter favored by laissez faire economists.  The shape appears often in the natural environment in objects and phenomenon as otherwise dissimilar as sea-shells, hurricanes and galaxies near (in cosmic terms) and far.  This diagram was posted on X (formerly known as Twitter) by Dr Cliff Pickover (@pickover) who writes the most elegant explanations which help draw the eye to the often otherwise hidden beauty of mathematics.

Zeno of Elea (Ζήνων ὁ Ἐλεᾱ́της (circa  490–430 BC)) was a Greek philosopher of the Eleatic school, an ever-shifting aggregation of pre-Socratic thinkers based in the lands around the old colony of Ἐλέα (Elea, in the present day southern Italian region of Campania, then called Magna Graecia).  Among his surviving thoughts were nine musings (now called Zeno's paradoxes) on the nature of reality, the details of which survived only in the writings of others which has led to some speculation perhaps not all came originally from the quill of Zeno.  Although most of the paradoxes revolve around the notion movement is illusory (and thus effortlessly & instantly resolved by every student in their first Philosophy 101 lecture), they are all less about physics than language and mathematics, the most intriguing of them one of the underlying structures of the argument about whether “now” does or can exist, the “ultras” of one faction asserting “now cannot exist” the other that “only now can exist”.  In that spirit, there’s much to suggest Zeno was aware of the absurdity of many of “his” paradoxes and created them as (1) tools of intellectual training for his students and (2) devices to illustrate how ridiculous can be the result if abstraction is pursued far beyond the possibilities of reality (ie not all arguments pursued to their “logical conclusion” produce a “logical” result).  One of Zeno’s paradoxes contains an explanation of why a curve might never reach a straight line, even if that line stretches to infinity: If the curve can at any time move closer to the line only by half the distance required to intersect, then the curve can only ever tend towards the line.  The two will never touch.

Christian von Wolff (circa 1740), mezzotint by Johann Jacob Haid (1704-1767).

The German philosopher Baron Christian von Wolff (1679-1754) was an author whose writings cover an extraordinary range in formal philosophy, metaphysics, ethics and mathematics and were it not for the way in which Immanuel Kant’s (1724-1804) work has tended to be an intellectual steamroller flattening the history of German Enlightenment rationality, he probably now be better remembered beyond the profession.

What most historians agree is the paradoxes were written to provide some framework supporting Parmenides' (Parmenides of Elea (Παρμενίδης ὁ Ἐλεάτης (circa 515-570 BC)) was a teacher of the younger Zeno) doctrine of monism (that all that exists is one and cannot be changed, separable only descriptively for purposes of explanation).  The word “monism” was coined by Christian von Wolff and first used in English in 1862; it was from the New Latin monismus, from the Ancient Greek μόνος (mónos) (alone).  Spending years contemplating things like monism may be one of the reasons why so many German philosophers went mad.  So the doctrine of monism is one of the oneness and unity of reality, despite the appearance of what seems a most diverse universe.  That “one-thingism” (that one of philosophy’s great contributions to language) attracted political thinkers along the spectrum but most appealed to those who hold there must be a single source of political authority, expressed frequently as the need for the church to be subordinate to the state or vice versa although the differences may be less apparent than defined: the systems imposed by the ayatollahs in the Islamic Republic of Iran and the Chinese Communist Party (CCP) in the People’s Republic of China structurally more similar than divergent.  Winston Churchill (1875-1965; UK prime-minister 1940-1945 & 1951-1955) once observed that while to political scientists fascism & communism seemed polar opposites, to many living under either the difference may have been something like comparing the North & South Poles, one frozen wilderness much the same as any other.  Arctic geographers would quibble over the details of that but his point was well-understood.

Lindsay Lohan and her lawyer in court, Los Angeles, December 2011.

Because of the self-contained, internal beauty, Monism has attracted long attracted political philosophers with axes to grind.  According to Sir Isaiah Berlin (1909-1997), “value monism” holds there are discoverable, axiomatic ethical principles from which all ethical knowledge may be derived, that ethical reasoning is algorithmic and mechanical, and that it seeks permanent, “final solutions” (no historical baggage in the phrase) to all ethical conflicts.  Berlin had his agenda and that was to warn monism tends to support political despotism, rejecting Immanuel Kant’s (1724–1804) argument “asymptotic monism” is not merely compatible with liberty and liberal toleration but actually a prerequisite for these values.  Although the phrase “Kant’s asymptotic monism” appears often, the phrase was never in his writings and is an encapsulation used by later philosophers to describe positions identifiably Kantesque.  His own philosophy has often been called “a form of transcendental idealism” which holds that the mind plays an active role in shaping our experience of the world, one’s individual’s experience of things not a direct reflection of what is but a construct shaped by the categories and concepts one’s minds impose on one’s experience.  Implicit in Kant is there is certainly one, ultimate, objective reality but experience of reality is limited and shaped by one’s cognitive capacities: because one’s experience of reality is always incomplete and imperfect, it can only ever approach a complete understanding of reality.  One’s cognitive capacities might improve but can only ever tend toward and never attain perfection.  Reality is the asymptote, one’s cognitive capacity the curve.

Club

Club (pronounced kluhb)

(1) A heavy stick, usually thicker at one end than at the other, suitable for use as a weapon; a cudgel.

(2) A group of persons organized for a social, literary, athletic, political, or other purpose.

(3) The building or rooms occupied by such a group.

(4) An organization that offers its subscribers certain benefits, as discounts, bonuses, or interest, in return for regular purchases or payments.

(5) In sport, a stick or bat used to drive a ball in various games, as golf.

(6) A nightclub, especially one in which people dance to popular music, drink, and socialize.

(7) A black trefoil-shaped figure on a playing card.

(8) To beat with or as with a club.

(9) To gather or form into a club-like mass.

(10) To contribute as one's share toward a joint expense; make up by joint contribution (often followed by up or together).

(11) To defray by proportional shares.

(12) To combine or join together, as for a common purpose.

(13) In nautical, use, to drift in a current with an anchor, usually rigged with a spring, dragging or dangling to reduce speed.

(14) In casual military use, in the maneuvering of troops, blunders in command whereby troops get into a position from which they cannot extricate themselves by ordinary tactics.

(15) In zoological anatomy, a body part near the tail of some dinosaurs and mammals

(16) In mathematical logic and set theory, a subset of a limit ordinal which is closed under the order topology, and is unbounded relative to the limit ordinal.

(17) In axiomatic set theory, a set of combinatorial principles that are a weaker version of the corresponding diamond principle.

(18) A birth defect where one or both feet are rotated inwards and downward.

1175-1225: From the Middle English clubbe, derived from the Old Norse klubba (club or cudgel) akin to clump, from Old English clympre (lump of metal) related to the Middle High German klumpe (group of trees).  The Proto-Germanic klumbon was also related to clump.  Old English words for this were sagol and cycgel.  The Danish klőver and Dutch klaver (a club at cards) is literally "a clover."  Ultimate root is the classical Latin globus or glomus (forming into a globe or ball), a later influence the Middle Low German kolve (bulb) and German Kolben (butt, bulb, club).  The sense of a "bat used in games" is from mid-fifteenth century; the club suit in the deck of cards is from the 1560s although the pattern adopted on English cards is the French trefoil.  The social club emerged in the 1660s, apparently an organic evolution from the verbal sense "gather in a club-like mass", first noted in the 1620s, then, as a noun, the "association of people", dating from the 1640s.  The Club Sandwich was probably first offered in 1899, the unrelated club soda in 1877, originally as the proprietary name Club Soda.  Club, clubbishness & clubbing are nouns & verbs, clubber is a noun, clubbed is a verb & adjective, clubby & clubbish are adjectives and clubbily is an adverb; the noun plural is clubs.

On her Only Fans page, Tash Petersen shows her club membership.

Something of a local legend in the world of vegan activism, Tash Peterson (b circa 1995) is an animal rights activist based in Perth, Australia.  Not actually part of the the militant extreme of the movement which engages in actual physical attacks on the personnel, plant & equipment of the industries associated with animal slaughter, Ms Peterson's form of direct action is the set-piece event, staged to produce images and video with cross-platform appeal, the footage she posts on social media freely available for re-distribution by the legacy media, her Instagram feed providing a sample of her work in various contexts.  The accessories used include blood (reputedly from slaughterhouses) and very fetching figure-fitting costumes styled to resemble various animals including cows, her favored locations including the meat section of supermarkets, cafés and restaurants serving animal flesh, processing facilities associated with the slaughter industry and any events celebrating the carnivorous.  Ms Peterson's other club membership is that of the vegansexuals (vegans who chooses to have sex or pursue sexual relationships only with other vegans).

The Club Sandwich

A majority of historians of food suggest the club sandwich, as an item able to be ordered, first appeared on a menu in 1899 at the Union Club of New York City.  It was however made with two toasted slices of bread with a layer of turkey or chicken and ham between them, served warm, not the three slices with which it’s now associated; at that point the "club" was merely self referential of the institution at which it was served.  Others suggest it originated in 1894 at an exclusive gambling club in New York’s Saratoga Springs.  The former is more accepted because there’s documentary evidence while the latter based on references in secondary sources.  It’s a mere etymological point; as a recipe, what’s now thought of as a club sandwich had doubtless been eaten for decades or centuries before the words Club Sandwich appeared on a menu.

The notion that club is actually an acronym for "chicken and lettuce under bacon" appears to be a modern pop-culture invention, derived from a British TV sitcom, Peter Kay’s Car Share (episode 5: Unscripted, 7 May 2018) in what’s claimed to be an un-scripted take, although, on television, very little is really ad-lib and "reality" has a specific technical meaning.  On the show, the discussion was about the difference between a BLT (bacon, lettuce & tomato) and a club sandwich.  It spread quickly on the internet but was fake news.

Seemingly sceptical: Lindsay Lohan contemplates club.

Ingredients

12 slices wholegrain or rye bread

12 rashers rindless, shortcut, peach-fed bacon

Extra-virgin olive oil

2 free-range eggs

1/2 cup whole egg mayonnaise

12 cos lettuce leaves

320g sliced lean turkey breast

4 ripe Paul Robeson heirloom tomatoes, sliced

A little freshly-chopped tarragon

Ground smoked sea salt & freshly cracked black peppercorns

Instructions

(1) Preheat a grill tray on medium.  Place half the bread under grill and cook until lightly toasted.  Repeat with remaining bread.

(2) Lightly brush both sides of bacon with oil.  Place under grill and cook for 2-4 minutes each side according to taste.  Once removed, place on a paper towel, turning over after one minute.

(3) Fry eggs, preferably leaving yokes soft and runny.  Fold tarragon into mayonnaise according to taste.   

(4) Spread 8 of the slices of toast with mayonnaise.  Arrange half of the lettuce, turkey and tomatoes over 4 slices.  Evenly distribute the fried eggs.

(5) Top with a second slice of toast with mayonnaise. Then, add remaining lettuce, bacon and tomato. Season well with salt and pepper. Top with remaining pieces of toast.

(6) Cut each sandwich in half or quarters according to preference, using toothpicks driven through centre to secure construction.

Variations

Chefs are a dictatorial lot and tend to insist a club sandwich must be a balanced construction with no predominant or overwhelming taste or texture.  Trick is to agree with everything they say and then make things to suit individual taste.  By varying the percentages of the ingredients, one can create things like a bacon club with extras and vegetarian creations are rendered by swapping bacon and turkey for aubergine and avocado.  A surprising number find tomato a mismatch, some add cheese or onion while many prefer butter to mayonnaise.  In commercial operations like cafés, tradition is to serve clubs with French fries but many now offer salads, often with a light vinaigrette dressing.  Served with soup, it’s a meal.

1946 Lincoln Club Coupe (body style 77).  When production of the V12 Lincoln Zephyr (1936-1942) resumed in 1946, the cars were sold simply as Lincolns with no model designation, differentiated by the style of coach-work (Sedan, Club Coupe & Convertible Coupe).  When production ended in 1948, it was the last of the American V12s.

The mysterious term “club coupe” emerged in the 1930s to distinguish the style from the “business coupe”, the latter a two door car with only a front seat, the rear compartment used to augment the space in the trunk (boot), the target market the numerous “travelling salesmen” who needed a vehicle with lots of secure storage for their wares.  What the term “club coupe” described was a two-door car with a rear passenger including a bench seat for two or three.  The use of the word “club” was an example of “aspirational branding”, a marketing flourish intended to suggest something more upscale than the utilitarian business coupe, the invocation that of the style and exclusivity of the “private club”.  Being a product of the marketing department, “club coupe” was never precisely defined and while the characteristics associated with the style were sometimes identifiable they were never consistent.

1951 Ford Custom Deluxe Club Coupe; long model names are nothing new.

The mid-century tendency was to use a body shorter than that of a sedan but retaining the convenience of a full-size back seat (unlike the single-seat business coupe) but as the “two door sedan” emerged as a descriptor things became fuzzy and by the time the two door hardtops appeared at scale in the 1950s, it wasn’t surprising “club coupe” fell from favour.  Ford in 1954 offered a club coupe but they were the next season renamed “Tudor sedan” (ie a two-door sedan) but made the use murkier still by calling the Customline Six two-door a "Tudor Sedan" and the new V8 Fairlane a “Club Sedan”, business coupes and club sedans lingering for years in the line-up but the club coupe vanished until 1966.  The 1960s revival was a use of the word to allude to the upmarket fittings once associated with the more luxurious club coupes of the pre-war years and like “landau”, “brougham” and such, was just another model designation, suggestive of some link to the past.

Promotional images used for 2015 Holden Commodore Clubsport R8 25th anniversary edition.  Note the bogan-themed tyre marks; Holden knew their target-market.

The meaning denoted was different in 1990 when Holden added the V8 HSV (Holden Special Vehicles) Clubsport to the VN range as a “de-contented” entry-level model, along the lines of the original Plymouth Road Runner (1968-1970), the message being: fewer fittings meant lower cost and higher performance.  The Clubsport would remain in the line-up until the end of Commodore production in 2017 although for various reasons, equipment levels steadily increased.

1992 Porsche 911 Club Coupé in Familiengrün.

Porsche also used the word and until the US use in the 1930s which was an allusion to the generic “private club”, the German like was to a literal club.  In 2012, Porsche celebrated the company’s 60th anniversary and part of the programme was the creation of 13 911 Club Coupés, the unusual production volume a tribute to the 13 fanboys who formed the world’s first “Porsche Club”: the Westfälischer Porsche Club Hohensyburg.  Although “fanboy” is understood to mean something like übertriebener Fan (excessively obsessive fan), blinder Anhänger (blind follower) or fanatischer Anhänger (fanatical follower), German has absorbed the English slang “fanboy” and uses it unmodified.  Based on the 991 series 911, the Club Coupé was bundled with the Sport Design package, X51 Powerkit, body-colored Sport Techno wheels, PCCB (Porsche Ceramic Composite Brakes which used a ceramic disk-rotor reinforced with carbon fibre), Club-themed door sills, and PASM (Porsche Active Suspension Management).  One marker of uniqueness was the color; although the factory listed the hue as Brewster Green, it was actually known internally as Familiengrün (Family Green), used for Wolfgang Porsche’s personal 911s.