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.