Dear Reader,
On this 19th anniversary
of Seldon’s April 7th,
1996
breakthrough -- his sudden discovery of the NQ_
‘First Dialectical
Arithmetic for contra-Boolean Algebra’,
after years of [re-]searching for, and of slow progress toward finding, a “mathematics of dialectics” -- the F.E.D. General Council has cleared,
for public dissemination, its eight specifications sheets defining the ‘meta-number’ value that we call ‘Full Zero’, ‘.’ -- as distinct from ordinary zero, 0, which
we, in this context, call ‘Empty Zero’,
0 -- and elaborating upon the candidate
postulate(s) to govern the use of this new 'ideo-ontology', this new dialectical-ideographical
symbol, within the Seldonian seventh, or 'Mu', dialectical arithmetic.
I have, in my dissertation-contribution to the Foundation, for my induction-into-membership
in the Foundation,
entitled “The Gödelian Dialectic of the Standard
Arithmetics” [which is accessible via the following links, on
the Vignettes Page, as Vignette #4, Parts 0, I, and II, at --
-- described the internal inadequacies/‘self-incompletenesses’
of each successive axioms-system of the standard arithmetics -- how each standard arithmetic is marked by algebraic,
“diophantine equations” which, grounding an ‘‘‘immanent critique’’’, or ‘‘‘self-critique’’’, and an ‘ideo-intra-duality’,
or ‘ideo-self-duality’, of each such system by itself, are -- syntactically -- well-formed within that arithmetic,
but for which, semantically, no ‘semantification’ of the unknown, x, of that algebraic
equation, i.e., no solution(s) of that equation, are available/expressible within
that standard arithmetic’s
axioms-system,
i.e., within the, partially tacit, ‘‘‘ideo-ontological commitments/presumptions/self-limitations’’’
of that system.
Thus, the equation
x +
1 = 1 is unsolvable within the system of arithmetic of the so-called “Natural” numbers,
N,
wherein N =
{1, 2, 3, ...}, and indeed this equation asserts a psychohistorical paradox for the concept of addition native to that system.
Thus, in a sense, within the limitations of the N system, x = ., although this equation of x and/to . must be considered a 'meta-arithmetical', '''meta-mathematical''' assertion, because . is not an element of -- is not a "number" within -- N.
this equation, x +
1 = 1, marks the presentational transition from the N system to the W system, the axioms-system
of the so-called “Whole” numbers, W = {0, 1, 2, 3, ...},
wherein that equation
is readily solvable:
x =
0.
However, the equation x + 1 = 0
is unsolvable
within the W system,
and indeed asserts a psychohistorical paradox
for the concept of addition
native to that
system.
Thus, within the limitations of W system, in a sense, x = . [again, as a 'meta-arithmetical' assertion, because . is not an element of -- is not a "number" within -- W] and this equation marks the presentational transition from the W system to the Z system, the axioms-system
of the so-called “integers”
--
Z = {..., -3,
-2, -1, ±0, +1, +2, +3, ...}
-- wherein that equation is readily
solvable: x = -1.
However, the equation 2x = 1
is unsolvable
within the Z system,
and indeed asserts a psychohistorical paradox
for the concept of multiplication
native to that system.
Thus, within the limitations of the Z system, in a sense, x = . [again, as a 'meta-arithmetical' assertion, because . is not an element of -- is not a "number" within -- Z], and this equation marks the presentational transition from the Z system to the Q system, the axioms-system
of the so-called “rational numbers” --
Q = {....-3/2...-2/1...-1/2...±0/1...+1/2...+2/1...+3/2....}
-- wherein that equation is readily
solvable: x = +1/2.
However, the equation x2 = 2 is unsolvable within the Q system, and indeed implies a
psychohistorical paradox
for the concept of exponentiation
native to that system
-- that x
must be either both
odd and even, or neither
odd nor even.
Thus, within the limitations of the Q system, in a sense, x =
. [again, as a 'meta-arithmetical' assertion, because . is not an element of -- is not a "number" within -- Q], and this equation marks the presentational transition from the Q system to the R system, the axioms-system
of the so-called “Real numbers”
--
R = {.....-pi....-e....-\/2....±0.000.......+\/2....+e....+pi.....}
-- wherein that equation is readily solvable:
x =
±\/2.
However, the equation
x2
+ 1 =
0 is unsolvable
within the R
system, and
indeed implies a psychohistorical paradox
for the concept of inverse values native to that system -- for that equation implies that,
for that x,
its additive inverse value
and its multiplicative inverse value
must be equal:
-x = +1/+x.
Thus, within the limitations of the R system, in a sense, x = . [again, as a 'meta-arithmetical' assertion, because . is not an element of -- is not a "number" within -- R], and this equation marks the presentational transition from the R system
to the C system, the axioms-system
of the so-called “Complex
numbers”
--
C = {R + Ri}
-- wherein that equation is readily solvable:
x = ±i. And so on . . ..
However, notice also that, in NONE of these systems -- [not in N], not in W, not in Z, not in Q, not in R, not in C,
... -- is division by ZERO
workable; is an equation of the form x = c/0 solvable
[in the N
system, such an equation is not even ‘‘‘well-formed’’’,
because the number 0
is not even part of the ‘ideo-ontology’
-- of either the syntax, or the semantics -- of that system].
This internal, immanent inadequacy and ‘‘‘incompleteness’’’
of ALL systems in the progression of the systems of the standard arithmetics is evidently of a far deeper sort
than the inadequacies and ‘‘‘incompleteness’’’ that drive that progression, and that
were progressively solved
in that progression,
as outlined above.
The [implicitly-dialectical] first-order-logic,
Peano axioms system
of the “Natural”
numbers, which Encyclopedia Dialectica
denotes by N_, and sees as being standardly interpreted as a “purely”-quantitative arithmetic, is the first, «arché» category/system of arithmetic in the Seldonian progression of non-standard,
dialectical arithmetics.
The Seldonian ‘First [explicitly-]Dialectical Arithmetic’,
which Encyclopedia Dialectica
denotes by NQ_, and interprets as a “purely”-qualitative ordinal arithmetic, is the second category/system of arithmetic in the Seldonian progression of non-standard,
dialectical arithmetics, the first ‘contra-category’/‘ contra-system’,
in that progression.
The seventh
system of dialectical arithmetic in that
Seldonian progression,
which Encyclopedia Dialectica
connotes by Rq_MQN = Rq_MU = Rm_ -- the second uni-category system of dialectical arithmetic -- arises
naturally as the first non-“syncopated”,
fully-ideographic, fully-algorithmic arithmetic for “dimensional analysis”.
I that seventh
system, questions
leading to the ‘Full Zero meta-number’
concept also arise naturally, and yield, at long last,
an arithmetic in
which division by zero
appears to become non-problematic.
As a result of that ‘‘‘rectification’’’ and ‘‘‘regularization’’’
of division by zero, dynamical
“singularities”,
presently manifesting as ‘infinity residuals’, i.e., as infinite errors, in the predictions
of [especially nonlinear] dynamical differential equations,
including of those which represent this humanity’s presently most advanced scientific-consensus
expressions
of the “laws” of Nature,
can be ‘semantified’ by correct solution-values, under intuitively satisfying new
axioms, which can be stated, briefly, as:
“ ‘Empty Zero’ “times” a metrological unit qualifier yields ‘Full Zero’
”.
-- and --
“ ‘Full Zero’
is operationally dominant, in multiplication and division, with respect to all
other ‘[meta-]number’ values in this Rm_ system, that is,
multiplication and division operations if they involve ‘Full Zero’, yield only ‘Full Zero’ ”.
-- or --
0mo = ..
-- and --
[ for
all mo in Rm_ ][ [.xmo = mox. = .] & [./mo = mo/. = .] ].
Unlike in the cases of the systems of arithmetic -- of the N, W, Z, Q, R, and C, ..., systems of arithmetic -- considered earlier, above, in this case, the present case, the case of the system of arithmetic, Rm_, . IS, finally, an element of the set -- is, at last, a number within the '''number-space''' -- Rm. Thus --
0mo = ..
-- is no longer a '''meta-mathematical''' assertion.
The classic published rendition of an earlier version of
this theory is available via --
-- on pages A-7 through A-21 of the latter.
I have posted the eight sheets of the new ‘Full
Zero’ specification, below,
for your convenience.
May you much enjoy this deeper glimpse into the world-historical
fruition of these arithmetics
of dialectic!
Regards,
Miguel
