CATEGORIES
ARE
NUMBERS:
Hence
Their NQ Calculus is an ‘‘‘Arithmetic’’’.
Dear Reader,
The space denoted by NQ, the mathematical, analytical-geometric space of our standard N Q_ axiomatic system for the modeling of
dialectics, is a space of generic, strictly-ordered, ‘ordinal
quality’ categories, that can be used to model specific
ontological categories, for specific Domains-of-discourse, and to calculate their
dialectical categorial progression interconnexions, e.g., via the various
‘‘‘Seldon Functions’’’.
¿But why do we call this ‘categorial combinatoric calculus’ an
‘‘‘arithmetic’’’?
We do so because this ‘calculus of categories’ is about ‘‘‘numbers’’’,
i.e., is about «arithmoi», partially in the ancient Greek sense
of that word.
Yes, this means that
ontological categories are ‘‘‘numbers’’’.
¿But how can this be?
By number, in this context, we mean a number of units of some kind, an assemblage of
at least two units. One unit by itself
is not a number in this sense, but just one unit of some kind – of the quality
that defines its kind – hence just one qualitative unit. Number, in this sense, begins with two units.
This concept of a number, as
an assemblage of at least two individuals of a single kind, was the predominant
meaning of the word “number” – e.g., of the word “«arithmos»” – in the
ancient, western, Mediterranean world, e.g., in the ancient Hellenistic world.
It is from the word “«arithmos»”
that the modern word “arithmetic” is derived.
The history of this word – its etymology – begins, at latest, with the
famous proto-algebraic, proto-symbolic-algebra, proto-ideogramic scroll of
Diophantus, in the proto-renaissance city of ancient Alexandria in ancient
Egypt. That text was titled «Arithmêtikê»,
or, in English transliteration, “[the] Arithmetica”.
The ancient Hellenistic
concept of number was a more concrete, more sensuous one than our own, in
keeping with the psychohistorical character of that ancient world, more
grounded in wealth as use-value rather than as exchange-value, as money-wealth.
Our modern concept of numbers
is far more abstract, in keeping with the psychohistorical character of our
modern world.
Our world is a world ever-more
permeated by the modernly – far more incessant than anciently – praxis of
money-mediated exchange, with money-capital as the core form of
wealth.
The constant equation of
qualitatively different, heterogeneous goods and services by prices; by
quantities of money, i.e., apparently by pure, abstract quantity alone –
unconsciously inculcates an “abstract quantity” intuition of number.
It does so because money
prices subliminally appear to us to denote “pure”, unqualified quantities,
given predominant ignorance of the real, commensurable unit of modern, world-market-competition-formed
economic values of general goods and services – namely, the generic human labor
hour time-cost of the present competitive reproduction
of the good or service in question.
For the ancient «mentalité»
– e.g. for Diophantus’s Arithmetica, even the qualitative
aspect of ‘“number”’ – even the qualitative unit of each «arithmos» – was
abstracted, and was represented in “syncopated”, abbreviated, generic
form, together with and next to its generic arithmetical quantifier.
That is, for example, the Arithmetica
represented the “number” 2 by Mob¢.
Note the ancient “priority”
in relation to our own: the ‘arithmetical qualifier’, e.g., Mo, came first, the ‘arithmetical quantifier’, b¢, qualifier.
Today, we usually write, e.g., “4 grams”, “4 gm.”, not “grams 4” or
“gm.4”, although, perhaps tellingly, e.g., “$4” is an exception. We usually don’t write “4$”, except on
checks: “4 and no/100s dollars”. But the currency sign, e.g., the dollar sign, [usually] comes first. Money
comes first.
That “syncopated” and
gematric symbol, “Mob¢”, for Diophantus, stood for two generic «Monads», two Mo, or two units, using b¢, the second letter in the Greek alphabet, with
an overbar or a “prime”/apostrophe, as the numeral for the arithmetical quantifier
“two”, and not just the, gematric, b¢ by itself for the number 2.
Let us next review some
comments, by scholars of arithmetic and of number, which help to illuminate
this psychohistorical difference between the ancient and the modern «mentalités»
and ‘human phenomes’.
For example, J. O. Urmson’s
dictionary The Greek Philosophical Vocabulary notes the
following about the ancient Greek view of “number” –
“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 the duas
– two. … number played a great part in metaphysics, especially in Plato’s
unwritten doctrines of e.g. sumblêtoi and asumblêtoi – addible
and non-addible numbers.” [pp. 31-32].
Note that certain modern cases of “hypernumber”
involvement – such as 1 +
i in the “Complex” arithmetic, 1 + i + j + k in Quaternion arithmetic, the orthonormal basis
vectors e1 + e2 + e3… in “Linear Algebra”, and cm.1 + cm.2
+ cm.3 in the, Jacob Klein’s magisterial Greek
Mathematical Thought and the Origin of Algebra states as follows:
“The Greek word arithmos
(a’riqmo'V) is rendered in the German text as Anzahl: “a
number of things,” to distinguish it from our modern Zahl: “number.” Since English approximations to Anzahl are
either obsolescent (e.g., “tale”) or awkward (e.g., “counting-number,”
“numbered assemblage”), Anzahl, like Zahl, has been rendered
simply as “number,” although it is a chief object of this study to show that
Greek “arithmos” and modern “number” do not mean the same thing,
that they differ in their intentionality, for the former intends things
[M.D.: for which “Mo” was Diophantus’s generic symbol], i.e., a number of them, while the latter intends a concept,
i.e., that of quantity…” [p. vii].
We suspect that the
“obsolescent” English words that correspond to the meanings of the German word
«Anzahl» and the ancient Greek word «arithmos» are obsolescent
precisely because they represent an ancient, mostly now forgotten, «mentalité»
regarding numbers, one that is psychohistorically incommensurable with the
modern, abstract, unqualified, exchange-value-inculcated meaning.
In their book Numbers,
H. Hermes et al., describe the Hellenistic notion of “number” in these
terms:
“ 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 [M.D.:
absent the Seldonian core concept of ‘meta-unit-ization’], neither EUCLID or ARISTOTLE regard a unit as a
number, but rather as “the basis of counting, or as the origin [M.D.: «arché»] of number.” ”
[p. 12].
Again, we see that our modern
“0” and “1”
did not fit the ancient «arithmos» concept of
“number”, and we now know why they did not fit it.
OK, so «arithmoi»
are composed of units, of two or more things of the same quality – of
the same kind; of the same ontological type.
¿But what have «arithmoi» to do with
dialectical, ontological categories?
Let’s see what Tony Smith’s The
Logic of Marx’s Capital, a book about Marx’s dialectical
logic, has to say about such categories:
“…we first have to consider
what a category is. It is a principle (a
universal) for unifying a manifold of some sort or other (different
individuals, or particulars). A category
thus articulates a structure with two poles, a pole of unity and a pole of [M.D.: individual]
differences.” [p. 5].
Note that talk about
categories today, dwells on a category as a unit[y], and as a “unit” in its own
right, and typically omits any mention of the multiplicity of [two or more] individual
units or « Monads», all
of the same kind or quality, that a category implicitly “contains” and “stands
for”.
So a category, too, is “an
assemblage of units”, typically named for the quality that all of its units
share. Such a category-name today should
be, but so often is not, a plural noun, to signify
the multiplicity of individuals for which it stands. E.g., the general category “Money” ought to
be “Monies” instead, but the modern, hyper-abstract «mentalité»
tends to favor the singular, i.e., a word that names the abstraction.
So, a category is an
“assemblage of units”, a “numbered assemblage”.
A category is an «arithmos»,
made up of «monads».
A category is/represents a
“number” of individuals, all of the same kind.
A category is a ‘“number”’ [of units, treated as a “unit[y]” in its own right].
However, in typical
discourse, the exact multiplicity of a typical category’s units is typically not
known.
We do not
usually have at hand a “census” of, e.g., how many “Money” units exist in the
world, now or at some former time; of how many “Commodity” units exist[ed] at
some given time, of how many units of a given kind of “Good” existed during
some past period.
¿What remains, explicitly, for such ‘un-census-ed’
ontological categories, of their ‘«arithmos»-ness’, of their
‘“number”’-ness?
All that remains, explicitly,
is the key quality shared by all units of their kind, and
defining their category.
This is that which the,
purely-qualitative, and thus unquantifiable arithmetical qualifers, {qn}, of the NQ arithmetic for modeling
categorial-progression dialectics, can be assigned; can capture ‘ideogram-ically’.
Of course, the
‘ideo-ontological’, dialectical categorial progression of the categories of
arithmetics for modeling dialectics does not end with NQ.
Albeit that there are no more
“purely-quantitative” categories of arithmetics in that progression, after its first, its «arché»,
category, that of the first-order-logic “Natural Numbers” axioms-system, denoted by N_ [double-underscored], and assigned to the NQ ordinal-first
‘meta-number’, q1.
The N Q category of
axiomatic arithmetics –
of axiomatic, ‘ideogram-ic’ languages
– is assigned to q2 in this dialectical progression of
arithmetical axioms-systems;
N Q is the
“purely-qualitative” ‘antithesis-category’ of arithmetical systems/-languages
relative to “purely”-quantitative ‘thesis-category’ of arithmetical
axioms-systems/languages, denoted by N_.
However – there emerges, starting
with the third category of
arithmetics-systems/languages in this progression, the
‘dialectical synthesis category’ and Unity of “purely”-quantitative N_ with “purely”-qualitative N_Q_, ‘‘‘named’’’ N_U_ , or, N_q QN, or N_q U, and assigned to q3 –
the quantifiable {un} ‘arithmetical unit qualifiers’ {uon}: {un ´ uon}.
Thereafter, in that
dialectical categorial progression of ‘«arithmos»-etical ideo-ontology’,
other, higher such ‘qualo-quantitative’ arithmetics/languages alternate with higher
“purely”-qualitative arithmetics/languages,
all for the modeling of dialectics – of dialectical ontological-categorial
progressions – with ever richer determinateness and expressive power.
For more
information regarding these
Seldonian insights, and to read and/or download, free
of charge, PDFs and/or JPGs of Foundation books, other texts, and images, please see:
www.dialectics.info
For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights – specimens of ‘dialectical art’ – as well as dialectically-illustrated books
published by
the F.E.D. Press, see:
https://www.etsy.com/shop/DialecticsMATH
¡ENJOY!
Regards,
Miguel
Detonacciones,
Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];
Elected Member, F.E.D. General Council;
Participant, F.E.D. Special Council for Public Liaison;
Officer, F.E.D. Office of Public Liaison.
YOU are invited to post
your comments on this blog-entry below!
