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RELATIVISTIC DIALECTIC
Polarity Continuum/Discreteness
Few terms are more confused than "dialectic" and we shall not
embark on comparative study of historic partial or total dialectic
failures, to mention the tutorial Socratic, the speculative Medieval,
the fatuous Hegelian, the deadly Marx-Engelsian and the ill-founded
Sartrian.
We shall mean by "dialectic" the Relativistic Dialectic (RD) defined
as reasoning based on Einstein's Physical Reality ("NATURAL MODEL")
and in particular on the Continuum/Discreteness (CD) Polarity and on
the foundational preponderance of its continuous aspect.
We have defined the CD polarity as the basic structure of all human
experiences ("TIME, AWARENESS AND EVENTS"). Inference pertaining to
RD is embodied in the ERN Logic ("ERN LOGIC").
Apparently contradictory aspects determine conjointly the polarity
intuited as a physical phenomenon or as a human situation.
Thus, apparently contradictory aspects, the discrete photon and the
continuous wave determine conjointly the phenomenon "light".
We shall consider them as "phenomenally equivalent".
Thus, apparent contradictions "friend" and "enemy", attributed to
the same entity and considered dialectically determine conjointly
a situation. We shall call them "dialectically complementary".
On the contrary, taken as absolute "logical" contradictions, they
misrepresent the situation as may be illustrated with the following
historic example.
Traditional naive pseudo-logic uses sort of +/-1 arithmetic. Friend
of my friend is my friend (1*1=1). Enemy of my enemy is my friend
(-1*-1=1).
Thus USSR, the enemy of my Nazi enemy was necessarily my friend.
Enemy of my friend is my enemy (-1*1=-1).
Thus Finland, fighting against USSR was necessarily my enemy.
And, logically, France and Britain committed one of the most shameful
acts in their history - declared war on a tiny Finland defending her
existence against the monstrous Soviet giant.
But ERN dialectic would show Finland as both friend AND enemy with
the respective plausibilities about 95% and 5%.
On condition of remembering that the WWII has been fought in defense
of freedom, against totalitarian enslavement and setting accordingly
the axioms. Besides reality conform deductive/inductive interplay
and fuzzy algorithms, the principal merit of ERN consists in boosting
pertinent axiomatics.
One may object that "friend" and "enemy" are both discrete
apperceptions and that their contradiction lacks a continuous
aspect. However, the network structure of ERN admits many-to-many
relations among discrete and continuous aspects. "Friend" and
"enemy" are indeed eventtual and discrete, but they have their
continuous counterpart(s) in intuitive belief(s) of (one's)
awareness embodied in the axiom(s).
Similarly, the positive and negative poles of the magnetic polarity
are discrete, eventtual aspects of the intuitive continuous magnetic
field.
FOUNDATIONS OF PHYSICS
RD being derived from Einstein's Physical Reality, and its basic
CD polarity with continuum preponderance, one expects it to be
supported and verified by concurrent physics. It amounts to presume
falsification and failure of discreteness based models.
And, indeed, two principal failures of recent physics concern two
famous discreteness based models, the Aether and the (rudimental)
Quantum Mechanics. We discuss them respectively in "SECOND SCIENTIFIC
REVOLUTION" and in "FOUNDATIONS OF QUANTUM PHYSICS".
One may set against them the vertiginous successes of continuum
based models such as the extended relativity an the quantum field
theory.
FOUNDATIONS OF MATHEMATICS
In "PREDICATE LOGIC" and "SET THEORY" we have seen that the
obstination to found mathematics in discreteness led to hundreds
inconclusive theories and that this proliferation leaves after
100 years the fundamental concepts of set and number totally
undefined.
We set against this ubiquitous failure Einstein's conviction that
mathematics are founded in continuum, which we introduced in the
"NATURAL MODEL" as follows:
***
mind's faculty of putting every body situated in any arbitrary
manner into contact with the quasi rigid continuation of a
chosen body of relation B0 is the basis of our intuition of space.
In pre-scientific thinking, the solid earth's crust plays the role
of B0. The very name geometry indicates that the idea of space is
mentally connected with the earth considered as the relation body.
We shall find the intuition of space founded by rigid bodies
at the base of Einstein's Covering Principle which requires
physical distance to be measured, also in mental experiments,
with physical rods complying with physical rules, such as
the Lorentz Contraction. The Covering Principle underlies
directly or indirectly the entire Extended Relativity.
Continuous space, its primacy and the discrete covering
measurement rods are aspects of the fundamental Polarity
Continuum/Discreteness (CD) defined in "TIME, AWARENESS
AND EVENTS" as the elementary structure of human Universe.
In mathematical terms geometry symbolizes the basic intuitive
image of continuous space and arithmetic - its discrete
covering measurements. The ultimate foundation of mathematics
is the continuum. Discrete concepts starting with that of
"number" are founded in continuum and symbolize its covering
measurements. Shortly, arithmetic is founded in geometry.
***
In "FOUNDATIONS OF MATHEMATICS" we extend the continuum
foundations over the totality of mathematics and conclude:
***
We believe to have shown the equivalence of foundations of
mathematics and physics in ontological continuum. In other
terms, mathematics does not have, nor need particular
foundations, different from the general foundations of
rational symbolic constructs.
Another confirmation of our view may be found in the
ultimate Frege's disappointment. In 1923 he came to the
conclusion that the aim he had set himself throughout most
of his career, namely to found arithmetic in (predicate)
logic, was wrong. He decided instead (like ourselves) that
one had to base the whole of mathematics on geometry.
He began to work on these ideas but had not progressed far
by the time of his death.
***
To finish with a poetic note, if anybody, then Frege was
worth of these lines:
If you can make one heap of all your winnings
And risk it on one turn of pitch-and-toss,
And lose, and start again at your beginnings
And never breathe a word about your loss
It's most gratifying to see his "start again" confirm
our ideas.
