F.E.D. Dialectics: A COURSE IN DIALECTICS, Part 1 – The Centrality of ‘Ordinality’ in Dialectics, and ‘Ordinal Spectral Color-Coding’.

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 1^st, 2^nd, 3^rd, 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.