A
COURSE IN DIALECTICS,
Part 1 –
The
Centrality of ‘Ordinality’ in Dialectics,
and
‘Ordinal Spectral Color-Coding’.
Dear Reader,
The five “Peano Postulates” form the recognized core of the
axioms for the “counting numbers”, “cardinal numbers”, or “Natural” numbers.
Apart from set theory, if taken to be the foundation of all of
mathematics, these “Natural” numbers – 1, 2, 3, … -- form the foundation of all
of arithmetic and algebra – of all of the richer systems of arithmetic that
have developed subsequent to the establishment of the ideography of counting.
The first four of the five Peano Postulates are expressions in
the mode of “first order” logic – they make assertions only about individual “Natural”
numbers.
It is worth considering, here, these first four, first order, postulated
propositions about “Natural” numbers, and to scrutinize precisely what they are
about – what, on the whole, they assert about “Natural” number in general. Here they are, as per their earliest incarnation
in Peano’s work –
a. 1 is a [“Natural”]
number.
b. The successor of any [“Natural”]
number is [also] a [“Natural”] number.
c. No two [distinct “Natural”]
numbers have the same successor.
d. 1 is not the successor
of any [“Natural”] number.
It may be surprising to note that there is nothing explicitly
quantitative, or counting-related, in these four statements. These statements begin with the abstract
unit, 1, and 1 is the only “Natural” number explicitly mentioned in them.
As a whole, they assert a strict consecutive order
among these “Natural” numbers – something more about “ordinal number” than
about “cardinal number”, than about counting, or than about quantity in general.
In this course, we will call the quality that these first four
Peano “axioms” describe by the term ‘ordinality’.
It took me awhile to notice that these postulates do not even
assert a strictly ‘quantitative ordinality’ – as in the concept of ‘ordinal
number’.
As ‘ordinal quantities’, the “Natural” numbers represent the
values 1st, 2nd, 3rd, and so on.
But I eventually noticed that, as they stand, they express a “primitive
undifferentiated unity” of ‘quantitative ordinality’ and its relative opposite,
which I came to call ‘qualitative ordinality’.
The ‘qualitative ordinality’, I came to see, can be expressed in
the following sequence, or ‘consecuum’ – the quality of ‘first-ness’, the
quality of ‘second-ness’, the quality of ‘third-ness’, and so on.
By the ‘quality of first-ness’, I mean the abstract quality
shared by all categories that come first in their native
categorial progressions.
Such categorial progressions come in at least two kinds.
Historical categorial progressions model the sequence in which the kind of
units which each category in such a progression gathers under its heading, in
the temporal “order of appearance” of each kind of units in the history of the
domain described by each given such categorial progression.
Systematic categorial progressions model the sequence in which the kind of
units which each category in such a progression gathers under its heading, when
the categories of a given domain are presented in the strict order of rising
complexity, from the least complex category, to the next more complex category,
…, and finally to the most complex category of the domain.
By the ‘quality of second-ness’, I mean the abstract quality
shared by all categories that come second in their native
categorial progressions.
And so on.
Extending the notion of the ‘Peanic’ “Natural” numbers from their
‘quantitative ordinality’ interpretation, leads to the progression of the “Standard
Arithmetics” – Whole Numbers, Integers, Rational Numbers, Real Number – whose numerals
all represent abstract “pure” quantities, “purely” quantitative numbers,
unqualified by any ‘arithmeticized qualifiers’, not even by ‘metrological
qualifiers”, such as the standard “syncopations” -- “cm.”, “gm.”, “sec.”, “°F.”,
etc.
Extending the notion of ‘Peanic’ “Natural” number from their ‘qualitative
ordinality’ interpretation, I found, led me to a new progression of higher, ‘“Non-Standard”’
arithmetics, all of which, I discovered, were capable of modeling dialectic,
with increasing richness.
The first of these ‘arithmetics for dialectics’ is a ‘‘‘Non-Standard
Model’’’ of the Peano “Natural” numbers, an arithmetic of ontological categories
that are represented by ‘unquantifiable ontological qualifiers’. Using the standard symbol N to denote space of the “Standard
Natural numbers”, I represent the space of these “Non-Standard Natural Numbers ‘pre-subscript-N next to script-level Q’, i.e., NQ.
This ‘first arithmetic for dialectics’, and its many ‘clarificational’
and ‘discoverential’ uses, as a dialectical ‘algorithmic heuristic’, will be
the focus of the first ~half of this course on dialectics.
The illustration below attempts to depict the course, and the
logic, of my discovery of the NQ from the first-order ‘Peanic’ N.
We will explicate this discovery with greater amplitude in subsequent
parts of this course.
Because of the centrality of the principle of ‘qualitative
ordinality’ in the foundation of this first, NQ, ‘dialectical arithmetic’, and its sequel
of ever-richer ‘arithmetics for dialectics’, we have adopted a
conventionalization of the qualitatively-different colors of the visible light
spectrum, that we use in ‘ordinal spectral color-coding’ for the successive
categories of dialectical categorial progressions.
This color-coding reminds – both subliminally and liminally – of
the order-place of each category in such a dialectical progression of
categories, and of the qualitative -- not quantitative – differences
distinguishing each category-symbol from every other category-symbol in such a
categorial progression.
It is this qualitative distinguishment which prevents these – thus
mutually heterogeneous – category-symbols from amalgamating and
collapsing, at their level of discourse, into any single category-symbol value.
The three images below summarize this ‘ordinal spectral
color-coding’ convention that we will apply throughout this course. These images to will be further explicated in
subsequent parts of this course.
Dialectically yours,
Karl Seldon
