CATEGORIES ARE NUMBERS ... .

  

 

 

 

 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 M^ob¢.

 

Note the ancient “priority” in relation to our own: the ‘arithmetical qualifier’, e.g., M^o, 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, “M^ob¢”, for Diophantus, stood for two generic «Monads»,  two M^o, 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 “M^o” 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 « M^onads», 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, {q_n}, 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 {u_n} ‘arithmetical unit qualifiers’ {u^o_n}: {u_n ´ u^o_n}.

 

 

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!