Part 02: ‘Seldon’s Dialectic Algebras’ Series.
The Encyclopedia
Dialectica first arithmetic/algebra for modeling Dialectic, N Q , is a NON-STANDARD MODEL of the first-order Peano “Natural Numbers”, N .
Dear Reader,
It
is my pleasure,
and my honor, as an elected member
of the Foundation Encyclopedia
Dialectica [F.E.D.] General Council, and as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder,
Karl Seldon.
This 2nd release in this
new such series
is posted below [Some E.D.
standard edits have been applied, in the version presented below, by the editors
of the F.E.D. Special Council for the Encyclopedia,
to the direct transcript of our co-founder’s
discourse].
Seldon
–
“We have elsewhere
described the Encyclopedia Dialectica NQ arithmetic/algebra for modeling dialectics as an
algebraic model of logic – as a ‘contra-Boolean’
arithmetic/algebra modeling dialectical logic.”
“To
summarize that description, the NQ arithmetic/algebra opposes Boolean algebraic
formal logic’s
“fundamental law of thought”, or “law of duality”, which Boole stated
algebraically as x2 = x.”
“This
Boolean law, when its 0 = x – x2
is factored into x(1 -
x) =
0, with (1 - x) representing the Boolean negation of class/category x – thus asserting that any class x, when it “multiplies” the everything class [=
‘1’] less [“minus”]
that x itself, equal 0 – represents the Aristotelian “law of
non-contradiction” in its Boolean form.”
“Translation:
Class x AND class not-x is NOTHING [‘0’].
Also, the “addition”, or ‘ORing’, of these two “factors” – of x and/plus (1 - x) – asserts the Boolean form of the Aristotelian “law
of the excluded middle”:
x + (1 –
x) = 1.
Translation: Class x OR class not-x is EVERYTHING [‘1’].”
“The NQ arithmetic/algebra, it turns out, by contradicting
the Boolean “law” x2 = x, constitutes,
in particular, an algebraic dialectical logic, whose fundamental theorem is a
strong contrary to that Boolean “fundamental law”, and can be stated
algebraically as –
x2 ~<=> x,
because x2 = x + Dx,
and Dx ~<=> x
–
wherein the relation-sign ‘~<=>’
signifies a relationship of non-quantitative, i.e.,
of qualitative – of ontological – inequality. That is, the ‘self-product’, x2, produces a net ‘onto-dynamasis’, a new class
or kind-category, denoted by Dx.”
“This
inequation turns out to mean also that the value of x2, i.e., x + Dx, ‘‘‘diagonally transcends’’’ the “space” in which
both x and Dx inhere, when x denotes a singleton element of that “space”, i.e., a
single ‘dialectical meta-number’, so that x2 thus escapes the would-be “closure” of that “number-space”,
even though each term within x2 does, separately, inhere in that “space”, and
is part of the “closure” of that “space” – both x = x1, and Dx = (Dx)1.”
“The
focus of this text, however, is a description of another relation of
contrariety involving the NQ arithmetic/algebra.”
“In
this case, the relation of contrariety is to the “Standard Natural Numbers”, as
based upon the four, “first-order logic”, “Peano Postulates”, stated below –
the standard core axioms for the first-order “Natural” numbers.”
“The
phrase “first order logic” means logical assertions which address only the
features of individual elements of the axioms-system, i.e., of individual
“Natural” numbers, but that do not make any assertions about features
of groups of such elements, such as about the qualities of all even
numbers, of all odd numbers, of all prime numbers, etc.”
“John
W. Dawson, Jr., in his 1997 book entitled Logical Dilemmas:
The Life and Work of Kurt Gödel, lays out, with careful clarity,
the way in which Gödel’s theorems – his Completeness Theorem for first-order
logic, and his First Incompleteness Theorem, which applies for first-order logic systems, and also applies for second and higher
order logic axiomatic systems of arithmetic – predict the necessary “co-existence”
of “non-standard models” of the “Natural” numbers, given the
“existence” of the standard “Natural” numbers themselves. He does so as follows –
“Most
discussions of Gödel’s proof, … focus on its quasi-paradoxical nature
[K.S.: in that the ‘Gödel Sentence’ “says of itself”
that ‘I am not deductively provable from the axioms of the axioms-system within which I have been well-formed’.
Thus, if that assertion is true, then that axioms-system is
“incomplete” – contains internal truths which cannot be deduced from its axioms. If that assertion is false, then that
axioms-system in “inconsistent”, or self-contradictory, because a false “theorem”
can be deduced – “proven” – from its axions. ].”
“It
is illuminating, however, to ignore the proof and ponder the implications of
the theorems themselves.”
“It
is particularly enlightening to consider together both the completeness and
incompleteness theorems and to clarify the terminology, since the names of the
two theorems might wrongly be taken to imply their incompatibility.”
“The
confusion arises from the two different senses in which the term “complete” is
used within logic.”
“In
the semantic sense, “complete” means “capable of proving
whatever is valid [K.S.: i.e., of proving
all propositions that are true within the theory]”, whereas in the syntactic sense it means
capable of proving or refuting each sentence [K.S.: i.e., each sentence that is ‘well-formable’ within the theory's syntactic rules] of the theory.”
“Gödel’s
completeness theorem states that every…first-order theory, whatever its
nonlogical axioms may be, is complete in the former sense: Its theorems
coincide with the statements true in all models of its axioms.”
“The
incompleteness theorems, on the other hand, show that if formal number theory
is consistent [K.S.: if no propositional
contradiction is deducible from its axioms],
it fails to be complete in the second sense [K.S.: i.e., the theory is "incomplete" because its formal arithmetic cannot deductively “decide” -- prove or
refute -- every sentence that is ‘well-formable’ within its first-order logic
axioms-system, by means of deduction from its axioms].”
“The
incompleteness theorems hold also for higher-order formalizations [K.S: i.e., axiomatizations] of number theory.”
“If
only first-order formalizations are considered, then the completeness theorem
applies as well, and together they yield not a contradiction, but an
interesting conclusion: Any sentence of arithmetic that is undecidable [K.S: i.e., is not deducible from the first-order
axioms as either true or false, a truth or a falsity within that theory that is
discernible as such by means other than formal deduction] must be true in some models of
Peano’s axioms (lest it be formally refutable [K.S.: i.e., refutable in all models of the first-order
axioms]) and false in [K.S.: some of the] others (lest it be formally provable [K.S.: i.e., provable in all models
of the first-order axioms]).”
“In
particular, there must be [K.S.: some] models of first-order Peano arithmetic whose
elements [K.S.: i.e., whose “numbers”] do not “behave” the same as the [K.S: “standard”]
natural numbers.”
“Such
nonstandard models were unforeseen
and unintended, but they cannot be ignored, for their
existence implies that no first-order axiomatization of number theory can be
adequate to the task of deriving as theorems exactly those statements
that are true of the [K.S.: standard] natural numbers.
[J. W. Dawson, Logical Dilemmas: The
Life and Work of Kurt Gödel, A. K. Peters, Wellesley, MA., 1997, pp.
67-68, bold & underscored & italicized emphases
added by K.S.].”
“The
sentences of the original “Natural” Numbers first-order axioms, known as the ‘“Peano-Dedekind Postulates”’, are essentially as follows –
P1. ‘1’ is a “Natural Number”.
Or: ‘1 Î N’.
P2. The successor of any “Natural Number” is also a
“Natural
Number”.
Or: ‘"n Î N, s(n)
Î N’. [s(n) º n + 1].
P3. No two
distinct “Natural
Numbers” have the
same successor.
Or: ‘"n, "m Î N, n ¹ m
Þ
s(n)
¹ s(m).
P4. There is no
“Natural
Number” that is
the predecessor of ‘1’.
Or: ¬$x Î N
| s(x)
= 1.”
“We
may tend typically to think of the “Natural” numbers as cardinal numbers – as
representing “pure”, unqualified cardinal quantities: “counts
of identical, abstract unit[ie]s”.”
“And,
indeed, the full, first and second-order axioms for the “Natural”
numbers do encompass both the cardinal aspects and the ordinal
aspects of the standard “Natural” numbers.”
“However,
by inspection of the four “Peano” sentences above, one can see that the first-order
axioms for these “numbers” implicitly emphasize their ordinal number,
or ‘order-number’, character; their aspects as “ordinal quantities”.”
“This
means that, as “ordinal numbers”, the “Naturals” represent first, second,
third, fourth,…, rather than “counts” such as one unit, two units, three units,
four units,… .”
“The NQ non-standard,
dialectical ‘meta-numbers’, it turns out, pick up on another possible aspect of
the ‘‘‘ordinality’’’ emphasized by the first-order Peano-Dedekind axioms: what
we call ‘ordinal quality’.”
“Thus,
the NQ
sequence of ‘meta-numbers’ represent ‘the quality of first-ness’,
then ‘the quality of second-ness’, then ‘the quality of third-ness’,
and so on. We will see how these
‘dialectical meta-numbers’ do so below.”
“The
chart posted below contrasts the first-order axioms of the NQ versus the N – of the non-standard
versus “standard” “Naturals”, and their syntax and symbology – as well as their
distinct ‘‘‘analytical geometries’’’.”
“This chart thus shows what a “non-standard” model of the
first-order Peano axioms means, in a specific case.”
“The
NQ
non-standard model of the “Natural” Numbers is
implicit in, immanent in, potential in, and hidden in the first-order
Peano-Dedekind axioms, as is the N model, the standard model [albeit less hiddenly so,
for powerful psychohistorical reasons].
But NQ
and N
stand, nonetheless, as two different
interpretations of the same four, first-order Peano axioms.”
“But
these two interpretations are not just “different”; they are qualitatively,
‘ideo-ontologically’, dialectically opposed, forming a dialectical, ‘categorial
antithesis’. This is because their
elements, their ‘number-kinds’, are opposite in their qualities. The N kind of numbers are ‘unqualified
“pure” quantifiers’.
The NQ
kind are ‘unquantified “pure” qualifiers. To be more precise, the NQ
‘meta-numbers’ kind are, at root, ‘unquantifiable
“pure”, ordinal arithmetical qualifiers’.”
“With
further interpretation, the NQ
‘meta-numbers’ can serve to model ontological categories, kinds-of-things
categories, and categorial progressions – progressions or series of such
categories, that model categorial dialectics – as instantiated so abundantly and
for so long, elsewhere in this blog.”
For more
information regarding these
Seldonian insights, 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.
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