Dear Reader,
I have excerpted, below, parts of the forthcoming F.E.D. Vignette #11, by F.E.D. Public Liaison Officer Aoristos Dyosphanthos, entitled '''Number''' Theory, Ancient vs. Modern vs. 'Trans-Modern', below, for your reading and learning pleasure.
I will add further excerpts as they become extant.
The original text can be accessed via the following URLs --
http://www.dialectics.info/dialectics/Welcome.html
http://www.dialectics.info/dialectics/Vignettes.html
http://www.dialectics.info/dialectics/Vignettes_files/Aoristos_Dyosphainthos,v.1.0,F.E.D._Vignette_11,%27%27%27Number%27%27%27_Theory,_Ancient_vs._Modern_vs._%27Trans-Modern%27,28MAR2013.pdf
Regards,
Miguel
The original text can be accessed via the following URLs --
http://www.dialectics.info/dialectics/Welcome.html
http://www.dialectics.info/dialectics/Vignettes.html
http://www.dialectics.info/dialectics/Vignettes_files/Aoristos_Dyosphainthos,v.1.0,F.E.D._Vignette_11,%27%27%27Number%27%27%27_Theory,_Ancient_vs._Modern_vs._%27Trans-Modern%27,28MAR2013.pdf
Regards,
Miguel
P.S. I have sometimes had to render, below, Greek letters, and other Ancient Greek symbols, in ways that could be accommodated within the available typography, or even as blank character-strings -- _______ -- when I found no more adequate way to render them.
F.E.D. Vignette #11 --
‘‘‘Number’’’
Theory,
Ancient vs.
Modern vs.
‘Trans-Modern’.
by
Aoristos Dyosphainthos
Author’s Preface. The purpose of F.E.D. Vignette #11 is to present
F.E.D.’s «Arithmos»
Theory -- a psychohistorical dialectical synthesis of Ancient «Arithmos» Theory &/with Modern ‘‘‘Number Theory’’’ -- without recourse to numbers.
A Note about the On-Line
Availability of Definitions of F.E.D. Key Technical Terms. Definitions
of Encyclopedia Dialectica technical terms and ‘neologia’ are available
on-line via the following URLs --
-- by clicking on the links associated with each
such term, listed, alphabetically, on the web-pages linked above.
The Encyclopedia Dialectica special terms
most fundamental to this vignette are indicated below, together with links to
their E.D. definitions
--
«arithmos»
and «arithmoi»
«aufheben»
‘cumulum’
«monad»
NQ dialectical arithmetic/algebra
‘self-meta-monad-ization’ or ‘self-meta-individual-ization’
-- we plan to expand these definitions resources as
the F.E.D. Encyclopedia Project unfolds.
q1 <---] I. Ancient «Arithmos»
Theories.
Not generally realized by we
Moderns is the psychohistorical fact that the Ancients’ concept of ‘‘‘Number’’’
-- named, in ancient Greek, by the word «Arithmos», the ancient Greek word from which the
modern English word “Arithmetic”
descends -- as proven by the
‘psychoartefacts’ that the Ancients left behind, and that have come down to us,
still extant, was qualitatively, ‘ideo-ontologically’ different from our Modern concept of “Number”.
Summarily we can say that
Ancients defined an «Arithmos» as an ‘‘‘assemblage of qualitative -- of multiple-qualities-exhibiting -- units/things of a given, single
kind’’’. «Arithmoi» are thus both ‘quanto-qualitative’,
and sensuous and ideational, phenomena, not “purely quantitative”,
‘quantifier-only’ ideo-phenomena, such as, e.g., 2, 3, 4, 5, 6, and 7, ... .
Herein we will take, as
representative of the Ancients’ concept of ‘‘‘Number’’’, in its philosophical
and/or mathematically technical form, the still-extant recorded thoughts of
Euclid and Aristotle.
For Euclid [third Century
B.C.E.]: “Euclid defines in the Elements, VII,
2, a number as “the multitude made up of units” having previously (Elements,
VII, 1) said that a unit is “that by virtue of which each of existing things is
called one.” As a unit is not composed
of units, neither EUCLID nor ARISTOTLE regard a unit as a number, but rather as
“the basis of counting, or as the origin [i.e., as the «arché» -- A.D.] of number.” There is an echo of this Euclidean definition
in CANTOR’s definition of the cardinal number as a set composed of nothing but
units ... .” [H. Hermes, et al., Numbers,
Springer Verlag, [NY: 1991], p. 12].
Note that this --
“self-evident?” -- claim that “a unit is not composed of units” posits a
radical duality between multiplicity / number on the one hand, and unity on the
other, i.e., between «arithmos» & «monad». The Ancients excepted a single «monad» from their category of ‘‘‘number’’’
simply because a single «monad»
is not an assemblage of «monads»
-- is not plural.
This claim ignores the fact
that reality is rife with ‘assemblages of metan+1-units, each of which is composes of a sub-assemblage of metan-units’ -- e.g., as a population
with molecules as its units is one of each of whose units is composed of atoms as its sub-units, as a population with
atomic nuclei as its units is one each of whose units is composed of “sub-atomic particles” [e.g., protons] as its sub-units, and as a population of “sub-atomic particles” is one each of whose units is composed of “pre-sub-atomic particles” [e.g., quarks and gluons] as
its sub-units.
For Aristotle [circa 335 B.C.E]: “Apart from this definition of number, which
is oriented towards the idea of counting, one can find in ARISTOTLE the
following statement: that which is
divisible into discrete parts is called [A.D.:
«plethos»] _____ (multitude), and the bounded
(finite) multiplicity is called the number (ARISTOTLE [1], 1020a, 7.14). The [A.D.:
Ancient] Greeks thus regarded as numbers, only the natural numbers,
excluding unity; fractions were treated as ratios of [A.D.: “natural”] numbers,
and irrational numbers as relationships between incommensurable magnitudes in
geometry ... . [Ibidem].
Actually, the statement
above is an anachronism, a ‘moderno-morphism’, and a ‘retro-projection’ of
the Modern meme of “number” back upon the Ancient one: the Ancients did not hold the modern
conception of the “natural” number, as “pure, unqualified quantifier”. On the contrary, as we shall show herein, via
Diophantus’s circa 250 C.E. treatise The
Arithmetica, the ancient meme of ‘‘‘number’’’ was a hybrid, ‘quanto-qualitative’ one.
Attending closely to the
qualitative, ‘ideo-ontological’ distinction of the Ancient concept of
‘‘‘number’’’ from the Modern can enable one to solve -- with both speed and
clarity -- mysteries that still baffle many scholars of philosophy, e.g. --
“arithmos: number; arithmêtikê: the science of number. Zero was unknown as a number and one also was
not counted as a number, the first number being duas [A.D.: or ‘dyos’] -- two. From the Pythagoreans, ton arithmon
nomizontes arkhên einai -- who consider number to be the first principle
(Ar. Met. 986a15) -- number played a great part in metaphysics, especially in
Plato’s unwritten doctrines, involving obscure distinctions of e.g. sumblêtoi
and asumblêtoi -- addible and non-addible numbers.” [J. O. Urmson, The Greek Philosophical Vocabularly, Duckworth [London: 2001], pp. 31-32].
The «Arithmoi Eide-tikoi» of Plato’s static, eternal dialectic, or ‘ideo-taxonomy’, were, in his conception, «arithmoi» of «Eide-Monads» -- Assemblages of «Idea»-Units -- for Plato’s reified, deified «Ideas», which he supposed to be the immutable, perfect,
Parmenidean Causes behind the imperfect copies of them which somehow constituted
and conducted the dynamic flux of our sensuous world.
Per Plato, for each such Causal «Idea», call it ‘I1’, any perfect
copy of It was redundant in terms of philosophical
logic, and could not exist: I1 + I1 ~= ‘2I1’;
instead, I1 + I1 = I1 [an
algebraic property which Modern mathematics names “additive idempotency”].
Moreover, for any two -- heterogeneous,
qualitatively/ontologically distinct -- such «Ideas», call them I1 and I2, their very “apples versus oranges” heterogeneity makes them “non-amalgamative” [cf. Dr.
Charles Musès] if added together:
I1 + I2 ~= ‘2I1’,
and I1 + I2 ~= ‘2I2’;
instead,
I1 + I2 = I1 + I2, without
further possibility of reduction within this language;
“apples plus oranges” equals “apples plus oranges”,
irreducibly so, at this level.
Thus, in both ‘self-addition’ and ‘other-addition’,
Plato’s «Eide»-Units
are “unaddible”, ‘unsum-able’
-- «asumblêtoi».
Also, given that the original
Pythagoreans held that «arithmoi» -- ‘‘‘assemblages of qualitative,
multiple-qualities-exhibiting units/things of various single kinds’’’, i.e.,
‘‘‘populations of individual things [including of physical, sensuous
things]’’’ -- constitute reality, it is no longer sure that the original
Pythagoreans were raving idealist mystics, as is so often presumed,
based upon the ‘retro-projection’ of the modern meme of ‘‘‘number’’’ upon their
Ancient «arithmos»
idea.
Ancient Alexandria’s ‘Proto-Renaissance’, & Diophantus’s ‘Qualifier-Quantifier Proto-Algebra’, at Dark
Ages’ Door.
The first known ‘protoic’ emergence of “symbolical algebra”,
as distinct from the already ancient ‘prose algebra’ -- or “rhetorical algebra”
-- and of an algebra “symbolical” in the specific sense of ‘ideogramic symbols’, not exclusively of either ‘pictogramic symbols’ and/or of ‘phonogramic [“phonetic”] symbols’ [which,
after all, would simply mean ‘prose algebra’, or “rhetorical algebra” again] --
was in a circa 250 C.E. work by Diophantus,
entitled The Arithmetica.
This text, The
Arithmetica [«Arithmêtikê»], taught
the ‘‘‘art’’’, or ‘‘‘technology’’’ or ‘‘‘technique(s)’’’ [«tekhnê»], or ‘‘‘craft’’’, or ‘‘‘skill’’’, or ‘‘‘science’’’
of «Arithmoi» in general.
This text developed an intermediate stage between
“rhetorical” algebra and full-blown “symbolical”, ‘equational’ algebra, which
has often been termed “syncopated” [abbreviated] algebra, in which minimized
abbreviations [“syncopations”] of words served as ‘proto-ideogramic’ symbols
for arithmetical quantities,
or ‘‘‘quantifiers’’’,
and for arithmetical
qualities or ‘‘‘qualifiers’’’, including for equations involving unknown quantities which
Diophantus showed how to “solve” -- how to systematically render the unknown quantities known.
Diophantus’s particular style of “abbreviation” or
“syncopation” -- an unprecedented style as far as is known -- was, apparently,
to take the first letter of the Greek word to be abbreviated, and to place atop
that first letter the second Greek letter of that word. Thus, only two Greek letters -- the first two
letters -- of the Greek word “survived” his abbreviation process. Known numerical values were expressed using
single Greek letters, with a dash or a “ prime” atop each letter, in ordinal
correspondence [i.e., alpha = I, beta = II, gamma = III, etc.], in accord with longstanding Ancient
arithmetical tradition.
The context of this work by
Diophantus was the Ancient Egyptian city of Alexandria, after the zenith of the
extraordinary, unprecedented Human-Phenomic -- scientific, technological, and
institutional -- developments there, that Karl Seldon has described as the
Western “Proto-Renaissance”.
Diophantus’s revolution in mathematics was cut short, in part, because
it arose circa 250 C.E., just a few centuries
before the tidal wave of the fall of the Roman Empire, and the undertow
dragging Ancient Hellenistic civilization down into the hellish abyss of the European
Dark Ages, smashed into Alexandria, suppressing this progressive trend, and
delaying its resumption, continuation, and supersession for another ~ ten centuries.
Regarding the mathematical
aspect of this “Proto-Renaissance” in Ancient Alexandria, we find the following
from the historical record: “The
earliest attempt to found a university, as we understand the word, was made at Alexandria. ... It
was particularly fortunate in producing within the first century of its
existence three of the greatest mathematicians of antiquity -- Euclid, Archimedes, and
Apollonius. They laid down the lines on
which mathematics subsequently developed, and treated it as a subject distinct
from philosophy: hence the foundation of
the Alexandrian Schools is rightly taken as the commencement of a new era. Thenceforward, until the destruction of the
city by the Arabs in 641 A.D. [i.e., C.E.], the history of mathematics centers
more or less round that of Alexandria”.
[W. W. Rouse Ball, A
Short Account of the History of Mathematics, Dover
[New York: 1960], pp. 50-51].
Howard Eves describes, as follows, the lead-up to
the founding of Alexandria
--
“The period following the Peloponnesian War was one of political
disunity among the Greek states, rendering them easy prey for the now strong kingdom of Macedonia which lay to the north. King Philip of Macedonia was gradually extending
his power southward and Demosthenes thundered his unheeded warnings. The
Greeks rallied too late for a successful defense and, with the Athenian defeat
at Chaeronea in 338 B.C.[E.], Greece became a part of the
Macedonian empire. Two years after the
fall of the Greek states, ambitious Alexander the Great succeeded his father
Philip and set out upon his unparalleled career of conquest which added vast
portions of the civilized world to the growing Macedonian domains. Behind him, wherever he led his victorious army,
he created, at well-chosen places, a string of new cities. It was in this way, when Alexander entered Egypt, that the city of Alexandria was founded in 332 B.C.[E.]. ... It
is said that the choice of the site, the drawing of the ground plan, and the process
of colonization for Alexandria
were directed by Alexander himself. From
its inception, Alexandria
showed every sign of fulfilling a remarkable future. In an incredibly short time, largely
due to its very fortunate location at a natural intersection of
some important trade routes, it grew in
wealth, and became the most magnificent and cosmopolitan
center of the world. ..." [Howard Eves, An Introduction
to the History of Mathematics (3rd ed.), Holt, Rinehart &
Winston (NY: 1969),
pp. 112-113 emphasis added by A.D.].
-- and the institutional innovations which seeded its
unprecedented destiny --
“After Alexander the Great died in 323 B.C.[E.], his empire was
partitioned among some of his military leaders, resulting in the eventual
emergence of three empires, under separate rule, but nevertheless united by the
bonds of the Hellenistic civilization that had followed Alexander's conquests. Egypt fell to the lot of Ptolemy.
... He selected Alexandria
as his capital and, to attract learned men to his city,
immediately began the erection of the famed University of Alexandria.
This was the first institution of its kind.
... Report has it that it was highly endowed and that its
attractive and elaborate plan contained lecture rooms,
laboratories, gardens, museums, library facilities,
and living quarters. The core
of the institution was the great library,
which for a long time was the largest repository of
learned works to be found anywhere in
the world, boasting, within forty
years of its founding, over 600,000 papyrus rolls.
It was about 300 B.C.[E.]
that the university opened its doors and Alexandria became, and
remained for close to a thousand years,
the intellectual metropolis of the Greek race [and not of the Greek “race”
alone, but of the Occidental Afro/Euro/Near-Asian hemisphere of humanity
entire! -- A.D.]." [Ibid., page 113, emphasis added
by A.D.].
In summary: “No other city has been the center of
mathematical activity for so long a period as was Alexandria
from the days of Euclid
(ca. 300 B.C.[E.]) to
the time of Hypatia (A.D. 415 [C.E.]).
It was a very cosmopolitan center, and
the mathematics that resulted from Alexandrian scholarship was not all of the
same type. ...” [Carl Boyer, Uta Merzbach, A History of Mathematics (2nd edition), John Wiley &
Sons, Inc. (NY: 1991), p. 178, emphasis added by A.D.].
Morris Kline well-describes the
mathematical, technological, economic, and cultural momenta that converged into
the genesis of the Ancient Alexandrian “Proto-Renaissance” in the following
passages.
After the early death of
Alexander, the Ptolemaic emperors of Egypt carried forward with Alexander’s
plans: “After his death ... the empire was split into
three independent parts. ... Egypt,
ruled by the Greek Ptolemy dynasty, became the third empire. Antigonid Greece
and Macedonia gradually fell
under Roman domination and became unimportant as far as the development of
mathematics is concerned ... The major creations following the classical Greek
period were made in the Ptolemaic empire, primarily in Alexandria.”
“That the Ptolemaic empire
became the mathematical heir of classical Greece was not accidental. The kings of the empire ... pursued
Alexander’s plan to build a cultural center at Alexandria. ... These rulers therefore
brought to Alexandria
scholars from all the existing centers of civilization and supported them with
state funds.”
“About 290 B.C.[E.] Ptolemy
Soter built a center in which the scholars could study and teach. This building, dedicated to the muses, became known as the Museum, and it housed poets,
philosophers, philologists, astronomers, geographers, physicians, historians,
artists, and most of the famous mathematicians of the Alexandrian Greek
civilization.”
“Adjacent to the Museum,
Ptolemy built a library, not only for the preservation of important documents
but for the use of the general public.
This famous library was said at one time to contain 750,000 volumes,
including the personal library of Aristotle and his successor Theophrastus. Books, incidentally, were more readily
available in Alexandria than in classical Greece because
Egyptian papyrus was at hand. In fact, Alexandria became the
center of the book-copying trade of the ancient world.”
“The Ptolemies also pursued
Alexander’s plan of encouraging a mixture of peoples, so that Greeks, Persians,
Jews, Ethiopians, Arabs, Romans, Indians, and Negroes came unhindered to Alexandria and mingled
freely in the city. Aristocrat, citizen,
and slave jostled each other and, in fact, the class divisions of the older
Greek civilization broke down.” [Morris
Kline, Mathematical Thought from Ancient to Modern Times, Volume I, Oxford University Press [New York: 1972], pp. 101-102, emphases added by A.D.].
Ancient Alexandria’s
favorable locus, with respect to the concentration and centralization of
ancient commerce and wealth there, also contributed crucially to the consummation
of its peoples’ cultural ambitions: “The
civilization in Egypt
was influenced further by knowledge brought in by traders and by the special
expeditions organized by the scholars to learn more about other parts of the
world. Consequently, intellectual
horizons broadened. The long sea voyages
of the Alexandrians called for far better knowledge of geography, methods of
telling time, and navigational techniques, while commercial competition
generated interest in materials, in efficiency of production, and in
improvement of skills. Arts that had
been despised in the classical period were taken up with new zest and training
schools were established. Pure science
continued to be pursued but was also applied.”
[ibid., pp. 102-103].
Part of what resulted was an unprecedented flowering of engineering and technology, even though not supported
by strong incentives to apply this technology in production, given the still
predominantly pre-capitalist, peasant-/serf-, ‘artisanal-’, and slavery basis
of the prevailing social relations of production, especially after the Roman
conquest of Egypt, in 31 B.C.E.: “The mechanical devices created
by the Alexandrians were astonishing even by modern standards. Pumps
to bring up water from wells and cisterns, pulleys, wedges, tackles,
systems of gears, and a mileage measuring device no different
from what may be found in the modern automobile were used
commonly. Steam power was employed to drive a vehicle along the city streets
in the annual religious parade. Water
or air heated by fire in secret vessels of temple altars was used to make
statues move. ... Water power operated a musical organ and made figures on a
fountain move automatically while compressed air was used to operate a gun. New mechanical instruments, including an
improved sundial, were invented to refine astronomical measurements.” [ibid.;
pp. 103, emphases by
A.D.].
The disparaging
squeamishness and ‘needlessness’ of classical Greek “aristocratic”
slave-holders with regard to ‘‘‘hands-dirtying work’’’ [‘‘‘fit only for
slaves’’’] -- and with regard to practical and commercial applications of the
fruits of intellectual labor -- was overcome in Ancient Alexandria:
“Proclus, who drew material from Germinus of Rhodes (1st cent.
B.C.[E.]), cites the latter on the divisions of mathematics...: arithmetic (our theory of numbers), geometry,
mechanics, astronomy, optics, geodesy, canonic (science of musical harmony),
and logistics (applied arithmetic).
According to Proclus, Germinus says:
The entire mathematics was separated into two main divisions with the
following distinction: one part
concerned itself with the intellectual concepts and the other with material
concepts.” Arithmetic and geometry were
intellectual. The other division was
material. However, the distinction was
gradually lost sight of ... One can say, as a broad generalization, that the
mathematicians of the Alexandrian period severed their relation with philosophy
and allied themselves with engineering.” [ibid., pp. 104-105, emphases by A.D.].
Hero[n] of Alexandria, and his teacher, Ctesibius
[who may have been responsible for the “Antikythera Mechanism”], incarnate this
mathematico-technological momenta of the Ancient Alexandrian
‘Proto-Renaissance’: “Proclus refers
to Heron as mechanicus, which might mean a mechanical
engineer today, and discusses him in connection with the inventor Ctesibius,
his teacher. Heron
was also a good surveyor. ... The striking fact about Heron’s work is his commingling of
rigorous mathematics and the approximate procedures and formulas of the
Egyptians. On the one hand, he wrote a
commentary on Euclid,
used the exact results of Archimedes (indeed he refers
to him often), and in original works proved a number of new theorems of
Euclidean geometry. On the other hand,
he was concerned with applied geometry and mechanics and gave all sorts of
approximate results without apology. He
used Egyptian formulas freely and much of his geometry was also Egyptian in
character. ...”
“His applied works include Mechanics,
The Construction of Catapults, Measurements, The Design of
Guns, Pneumatica (the theory and use of air pressure), and On The
Art of Construction of Automata. He gives designs for water clocks, measuring
instruments, automatic machines, weight lifting
machines, and war engines.” [ibid., pp. 116-117, emphases added by
A.D.].
Factors in the demise of
this Ancient Alexandrian “Proto-Renaissance” are described, by Howard Eves, as
follows --
“The city of Alexandria enjoyed many
advantages, not the least of which was long-lasting peace. During the reign of the Ptolemies, which
lasted for almost 300 years, the city, although on occasion beset with internal
power struggles, remained free from external strife. This was ended by a short period of conflict
when Egypt became part of
the Roman empire ... The closing period of
ancient times was dominated by Rome. ... The economic structure ... was
essentially based on agriculture, with a spreading use of slave labor. The eventual decline of the slave market, with its disastrous effect on Roman
economy, found
science reduced to a mediocre level. The
Alexandrian school gradually faded, along with the breakup of ancient society. [op.
cit., p. 164, emphases added by A.D.].
-- and --
“Greek science reached its
pinnacle at Alexandria
... The decline was caused by a combination of technological, political,
economic, and social factors. ... The Romans used slave labor to an almost
unprecedented degree, especially after the founding of the Empire by Augustus in 31
B.C.[E.]. More than half of
the Empire’s inhabitants were slaves. With slaves to do most of the backbreaking
work, there was little perceived need for labor-saving devices, such as the
pulleys and levers invented by Archimedes ... hence, scientists had little
incentive to invent them.” [op. cit.,
pp. 137-138, emphases added by
A.D.].
-- and by Morris Kline
thusly --
“The fate of Hypatia, an
Alexandrian mathematician of note and the daughter of Theon of Alexandria [the redactor of Euclid's
Elements -- A.D.], symbolizes the end of the era. Because she refused to abandon the Greek religion, Christian fanatics seized her in the
streets of Alexandria
and tore her to pieces.
... From the standpoint of the history of
mathematics, the rise of Christianity had unfortunate consequences. Though the Christian leaders adopted many
Greek and Oriental myths and customs with the intent of making Christianity
more acceptable to converts, they opposed pagan learning and ridiculed mathematics, astronomy,
and physical science; Christians were forbidden to contaminate themselves with
Greek learning. Despite cruel persecution by the Romans,
Christianity spread and became so powerful that the emperor Constantine (272-337 [C.E.]) was obliged to consign
it a privileged position in the Roman Empire. The Christians were now able to effect even greater destruction of
Greek culture. The emperor
Theodosius proscribed the pagan religions and, in 392 [C.E.] ordered
that the Greek temples be destroyed. Pagans
were attacked and murdered throughout the empire. Greek books were burned by the thousands. In that year Theodosius
banned the pagan religions, the Christians destroyed the temple of Serapis [in Alexandria
-- A.D.], which still housed the only extensive collection of Greek works. It is estimated that 300,000 manuscripts were destroyed. Many other works written on parchment were
expunged by the Christians so that they could use the parchment for their own
writings ... In 529 [C.E.], the Eastern Roman emperor
Justinian closed all the Greek schools of philosophy, including Plato’s Academy. ... The final blow to Alexandria
was the conquest of Egypt
by the upsurging Moslems in ... 640 [C.E.].
The remaining books
were destroyed on the
ground given by Omar, the Arab conqueror: “Either the books contain what is in the
Koran, in which case we do not have to read them, or they contain the opposite
of what is in the Koran, in which case we must not read them.” And so for six months the baths of Alexandria were heated by burning rolls of parchment. After
the capture of Alexandria by the
Mohammedans, the majority of the scholars migrated to Constantinople, which had
become the capital of the Eastern Roman Empire.
Though no activity along the lines
of Greek thought could flourish in the unfriendly Christian atmosphere of Byzantium, this flux of scholars and their works to comparative safety increased
the treasury of knowledge that was to reach Europe
eight hundred years later. It is perhaps pointless to contemplate what
might have been. But one
cannot help observe that the Alexandrian Greek civilization ended its active
scientific life on the threshold of the modern age. It had the unusual combination of theoretical
and practical interests that proved so fertile a thousand years later. Until the last few centuries of its existence,
it enjoyed freedom of thought, which is also essential to a
flourishing culture. And
it tackled and made major advances in several fields that were to become all-important
in the Renaissance: quantitative
plane and solid geometry; trigonometry; algebra; calculus; and astronomy.” [op. cit.,
pp. 180-181, emphases added by A.D.].
It is in the above-described
‘‘‘psychohistorical’’’ context that the work of
Diophantus of Alexandria can be comprehended -- as a hybrid product of waning Hellenistic memes, and of
a ‘protoic’, precocious, prevenient partial prefigurement of core components of
the as yet unborn Human Phenome of Modernity.
Morris Kline assesses the
work of Diophantus in the following terms:
“The highest point of Alexandrian Greek
algebra is reached with Diophantus. ... His work towers above that of his contemporaries; unfortunately, it came too late to be highly influential
in his time because a destructive tide was already engulfing the civilization. Diophantus
wrote several books that are lost in their entirety. ... His great work is the Arithmetica
which, Diophantus says, comprises thirteen books. We have six [6 surviving in Greek, that is; 4 more
were recently found, in Arabic, possibly translations into Arabic of Hypatia’s
Greek commentaries on books 4 through 7, rather
than of Diophantus' originals -- A.D.] ... One of Diophantus’
major steps is the introduction of symbolism
[i.e., of proto-ideography
-- A.D.] in algebra. ...
The appearance of such symbolism
is of course remarkable but the use of powers higher than three
is even more extraordinary. The
classical Greeks could not and would not consider a product of more than three
factors because such a product had no [then-recognized -- A.D.] geometrical
significance [i.e., given the apparently 3-and-no-more/no-less-dimensional
physical space of our world -- A.D.]. On
a purely arithmetical basis, however, such products do have a meaning;
and this is precisely the basis Diophantus adopts." [op. cit., pp. 138-139, emphases added by A.D.].
Diophantus symbolized a[ny],
generic number, in a dual format, as a juxtaposition
-- a ‘‘‘product’’’, in effect -- of two semantic ‘‘‘co-factors’’’, called, by Karl Seldon, an
‘‘‘arithmetical qualifier’’’, and an ‘‘‘arithmetical quantifier’’’, viz. --
Ms
-- with the “syncopated” unit qualifier symbol M signifying
the «Mo-nad», the generic, abstract [and ‘quantifiable’] “unit”, or ‘‘‘one-ness”’, standing
generically and indifferently for any specific kind of unit -- e.g., for an ontological unit, or for a metrical unit, or even for an
undifferentiated combination of the two.
Examples include a unit of the “kind of
thing” category --
or ‘‘‘ontological
category’’’ -- of the quality of “apple-ness”, i.e., an apple unit, or an orange unit, or a pound
unit as “unit of
measure” or “metrical
unit”, or the combined, undifferentiated unity of a metrical and an ontological
quality unit, e.g., “a
pound of apples”, or “a pound of oranges”.
The symbol s, is the generic quantifier
symbol, often used by Diophantus to represent the unknown, and
to-be-solved-for, value in one of Diophantus’s ‘proto-algebraic
proto-equations’.
This number symbol is drawn, as was typical in Ancient Greek ‘‘‘logistics’’’ [practical arithmetic], from the Greek alphabet. It is the version of the Greek letter sigma, _, that is used when sigma is the final letter of a Greek word, e.g., in particular, s is the last letter of the Greek word «arithmos» ... .
In modern English, it coincides with the final s, i.e., with English letter suffix that signifies plurality.
This number symbol is drawn, as was typical in Ancient Greek ‘‘‘logistics’’’ [practical arithmetic], from the Greek alphabet. It is the version of the Greek letter sigma, _, that is used when sigma is the final letter of a Greek word, e.g., in particular, s is the last letter of the Greek word «arithmos» ... .
In modern English, it coincides with the final s, i.e., with English letter suffix that signifies plurality.
That is, Diophantus, in
keeping -- for the most part -- with Ancient «Arithmos»
Theory, does not
symbolize number in general simply as --
s
-- i.e., as an abstract,
“pure” quantifier, without qualification, as would be the
case if Diophantus had followed -- i.e., if he had anticipated -- the meme of European Renaissance humanity, after the world-historic ‘Elision of the Qualifiers’.
This world-historic ‘Elision’ was brought about, ‘‘‘psychohistorically’’’, we hold, in the post-Dark-Ages
European Human Phenome -- which was also the point-of-origin of the [psycho]historically-specific Capitalist Phenome, or «mentalité»,
by the intensive practice of the capital-relation by so much of the
population: of the monies-[capitals-]mediated
exchanges of commodities[-capitals], that emerged, in the lead-up to the
Western European Renaissance, as a far more intensive such praxis than was ever
reached within the socio-economic limitations of Ancient Mediterranean times,
and of their substantially slavery-based mode of social production.
This ‘capital-praxis’ was
captured, in its purest, simplest essence -- abstracting from its more concrete
determinations, involving mediation by money [price] and by production
processes, outside of the process of circulation of capitals, by Marx’s «arché» for «Das Kapital» as a whole, “The Elementary or Accidental Form of
Value”, set forth by Marx from the beginning of that
work, in Vol. I, Part I, Chapter I., Section 3.A. of, as the systematic-dialectical ‘seed cell’ of that entire work, and
expressed by Marx, in his ‘algebraic/rhetorical’
notation, in the form of the ‘‘‘exchange-equations’’’ --
"x commodity A = y commodity B"
or,
e.g., as:
"20 yards of linen = 1 coat"
"20 yards of linen = 1 coat"
-- and, later, by Seldon, as
--
{ cjCj = ckCk }
-- with commodity quantifiers cj < or = or > ck, despite the fact that, for the Commodity qualifiers, Cj is qualitatively unequal to Ck.
.
. .
q2 <---] II. Modern “Number Theories”.
[Forthcoming]
q3 <---] III. F.E.D.’s Seldonian, Trans-Modern -- Modern/Ancient Hybrid -- «Arithmos» Theories.
We of F.E.D.use the word ‘‘‘number’’’ in a far more concrete sense than has become habitual in the Modern World, and, in certain ways, with a sense much more like it had in the Ancient World.
In our ‘‘‘Number Theory’’’, as a modernization of the ancient ‘«Arithmos»-Theory’, or ‘«Monads»-Theory’, ‘‘‘number’’’ means not an abstract, “pure” quantity as such, as per those “number” conceptions so central to the Modern [unconsciously, experientially law-of-capital-value-inculcated] «mentalité».
On the contrary, in our usage, ‘‘‘number’’’ means something far closer to sensuous ‘empiricality’.
It refers to a specific multiplicity of units/individuals/monads, akin to a plural but finite “population” of the individuals of the same kind, such that each individual is a concrete, determinate, ‘multi-qualitative’ [‘multi-quality’], attributes-rich [ev]entity, not a distilled, rarefied mental abstraction of “pure, unqualified quantity”.
In such a usage, ‘‘‘numbers’’’ thus no longer differ only quantitatively: such ‘‘‘numbers’’’ have different “kinds”.
And, Old ‘‘‘numbers’’’ create New ‘‘‘numbers’’’: they not only expand themselves quantitatively, as populations of their units, but qualitatively, ontologically as well.
That is, Old kinds of ‘‘‘numbers’’’ create New kinds of ‘‘‘numbers’’’ by means of ‘self-meta-monad-ization’, that is, via ‘self-meta-unit-ization’, or ‘self-meta-individual-ization’.
The process of self-meta-monad-ization’ is a self-«aufheben» process, which is to say, a dialectical process.
A theory of the progressive self-construction of our cosmos -- in the form of a single, recurrent, mounting, cumulative, helical ‘dialectic of nature’ -- can be constructed on the basis of noticing that, e.g. --
The [self-changing] ‘‘‘number’’’ [cosmological population] of pre-nuclear “particles” [e.g., of non-Hadronic, “non-composite” bosons and fermions, such as quarks] created the [dynamical, “fluent” [cf. Newton] self-changing] ‘‘‘number’’’ of sub-atomic “particles” [e.g., of primordial protons and neutrons], by their own ‘self-meta-monad-ization’;
The [self-changing] ‘‘‘number’’’ [cosmological population] of sub-atomic “particles” [e.g., non-Hadronic, “non-composite” bosons and fermions, such as, quarks] created the [self-changing/other-changed/other-changing] ‘‘‘number’’’ of [ionic] atomic nuclei [e.g., primordial Deuterium, Tritium, Helium, and Lithium], by their ‘self-meta-monad-ization’;
The [self-changing] ‘‘‘numbers’’’ [galactic populations] of atomic nuclei created the [dynamical, “fluent”, self-changing/other-changed/other-changing] ‘‘‘numbers’’’ of molecules [e.g., of galactic “inter-stellar medium” accumulating H2, O2, CN, H2O, CO2, CH4, etc.], by their own brand of such ‘self-meta-monad-ization’;
The [dynamical, “fluent”, self-changing] ‘‘‘numbers’’’ [cosmological populations] of molecules created the [self-changing/other-changed/other-changing] ‘‘‘numbers’’’ of ‘pre-eukaryotic’ living cells, by their own, natural-historically-specific «species» of ‘self-meta-monad-ization’;
etc.
Our Marxian, immanent critique of both the Modern and the Ancient conceptions of ‘‘‘Number’’’ find their foundation in the ‘‘‘psychohistorical’’’ insights, into both the Modern and the Ancient human ideologies -- into the Modern versus the Ancient ‘Human Phenomes’ -- embodied in Marx’s immanent, dialectical critique of capitalist political economy.
In his “Elementary Form of Value”, Marx discovered something much more momentous than even the ultimate “seed” category -- the «arché» category -- from which there ‘‘‘descends’’’, in an ‘ideo-meta-genealogical’, dialectical method-of-presentation sense, the rest of his entire, vast, comprehensive critique of the political economy of capital; of the capital social-relation-of-production; of the capitals social system of global, prehistoric humanity.
He also discovered the universal unconscious paradigm of ‘The Modern «mentalité»’, whose most characteristic symptom is the “purely quantitative” frame of mind, and, consequently, ‘The Elision of the Qualifiers’ from conception, from perception, and from mathematical -- starting especially with arithmetical -- symbolic expression.
Marx therein and thereby discovered the secret, not just of “The German Ideology”, but of the total, global, human “Modern Ideology” entire -- of the total‘Human Phenome’ of a planetary humanity that embodies and incarnates Capital [i.e., that incarnates the "Capital-relation-of-production' as the predominant social relation of social reproduction].
We of F.E.D. have found working with the NQ arithmetic/algebra, as with its successor systems, to be a worthwhile and cognitively healing practice for we F.E.D. monastics.
In working with the NQ, one is working with ‘‘‘numbers’’’ that are purely qualitative.
A given, generic qk is interpreted, or specified, as “standing for” an «arithmos», anumber, in part, in the Ancient sense: as “standing for”, in effect, an ontological category representing the special ‘common-kind-ness’ that unites all of the individuals; that all of the «monads» which inhere in that ontological category share, like the ‘‘‘in-tension’’’ of an ‘‘‘ex-tension’’’, i.e., of a “set of elements”.
The generic symbol qk, for a k in N, thus interpreted, means a number of indefinite/changing cardinality, creating a kind of Marxian version of the “intentional” variables of the original Boolean algebra.
The practice of the expression of experienced/experimented reality, using the language of the NQ numbers, is, we find, a liberating “spiritual practice” -- in the sense of a Marxian version of Hegel’s “Objective Spirit”: of ‘The Human Phenome’.
That is, this activity of ours is a healing modifier of our individual human phenomes, one that lifts us beyond the collective, ‘ideologized’ “Mind”, the typical «mentalité», of our time -- beyond the “Mind” of ‘The Modern Ideology’; beyond the ‘Money Mind’, beyond the one-sidedly, purely-quantitative «mentalité», the “Mind” of “The Elementary Form of [Commodity] Value” as unconscious universal paradigm -- in short, beyond ‘the capital-value «mentalité»’.
This practice thereby helps us to free our minds to see in new and wider ways -- to think beyond the blockages characteristic of ‘The Modern Ideology’, the ideology of capital-value as supreme value, or even as only-value.
If you believe that such seeing is a part of your life path, then we commend this practice also to you.
Links to definitions of additional Encyclopedia Dialectica special terms deployed in the discourse above --
«arché»
https://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Arche/Arche.htm
Boole’s Algebra
http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/BoolesAlgebra/BoolesAlgebra.htm
categorial
http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Categorial/Categorial.htm
category
http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Category/Category.htm
dialectical categorial progression
http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/CategorialProgression/CategorialProgression.htm
‘‘‘eventity’’’
https://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Eventity/Eventity.htm
ontological category
http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/CategoryOntological/CategoryOntological.htm
ontology
https://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Ontology/Ontology.htm
No comments:
Post a Comment