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DOGMATIC THEORIES AND AXIOMATIC MODELS
in the light of the ERN logic
Foundations and definitions
Epistemological impact of the ERN logic concerns mainly
-foundations of logic,
-definitions and distinction of "Theory" and "Model",
-definitions and distinction of "Axiom" and "Dogma".
Foundations of logic.
We have postulated that Logical Systems may be evaluated and
justified exclusively by their capacity to simulate Mind's
intrinsic, ER based Logic. ERN is the first Logical System
founded in Mind's intrinsic Logic, rather than in noumenal
linguistic expressions. It seems to simulate it efficiently,
which has been verified by its several practical applications.
Theory and Model.
Contemporary Epistemology sees falsifiability as a necessary
quality of scientific structures. ERN embodies it rigorously
in its two complementary aspects:
1.Conceptual, deductive Theory,
2.Experimental, inductively falsifiable Model.
Axiom.
Full-fledged model structure supporting both, necessary
deduction and fuzzy factual induction will be called
"axiomatic" and its top arbitrary presumptions - "Axioms".
Axioms and thence deduced Theory are falsifiable and
refutable by inconclusive induction from factual experiences.
Dogma.
A Theory lacking bottom factual Theorems and thus unable to
support the falsifiable induction will be called "Dogmatic".
and its top arbitrary presumptions - "Dogma". Unlike Axioms,
Dogma are not falsifiable, cannot be refuted and repose in
unshakable faith in transcendental "Truth".
LOCAL AND EXTERNAL FOUNDATIONS
Besides being founded in their own, "local" axioms, models
may be founded in other "founding" models. By definition,
a model or a discipline is "founded" in a "founding" one,
when it accepts the latter's axioms and theorems as its own
axioms.
Thus, physics is founded in mathematics and does not derive
the principles of calculus, of vectors, tensors, etc. but
considers their mathematical formulations as axioms of its
own models.
This foundation hierarchies culminate in the ontological
intuition of continuum, the primary aspect of the Polarity
Continuum/Discreteness (CD), the fundamental construct of
the physical and human reality ("NATURAL MODEL"). Hence,
rational axiomatic models are ultimately founded in continuum.
In "SET THEORY" we saw the fallacies resulting from attempting
to found mathematics in discreteness. In Tome 2 "FOUNDATIONS
OF QUANTUM PHYSICS" we discuss the controversy between
quantizing the fundamental continuum of field and attempts
to found quantum physics in sheer discreteness without
considering SPACE or field continuum.
POSSIBLE AMBIGUITIES
Of all branches of Science Physics has been most perniciously
afflicted by the dogmatic reaction to the rationality of
the Fist Enlightenment, namely by the Dogma of Aether.
Aether is discussed in some detail in "SECOND SCIENTIFIC
REVOLUTION". Here we shall concentrate on its interest for
Epistemology, as illustration of the conflict of dogmatic
and axiomatic attitude.
The question indeed arises if Aether was Dogma or Axiom.
Before attempting to answer, let's recall its context and
essential features. It has been founded in the mechanistic,
"billiard ball" view of the underlying "reality" plus the
additional postulate that light is a wave and, by analogy
with known wave phenomena such as sound, must be supported
by oscillating particles of some fluid, some cosmic gas or
liquid: the Aether.
So far so good, at the outset Aether looked like an Axiom.
True, from the very beginning it raised unusual amount of
exceptionally tough empiric problems: it had to behave like
a solid with respect to light, like no-interaction vacuum
with respect to "matter" of stellar bodies, while interacting
with "matter" which it permeates like glass or water. Yet,
for each new problem falsifying a current version of Aether
a new, pertinent version was duly created, at the expenditure
of effort and ingenuity hardly ever matched in the history of
Science. Physicists were certainly not lazy, but if they were
less busy adjusting Aether, they might have heard Ockham
whispering that its exceptional complexity called for some
simpler Postulate. Still, Aether could pass so far for a
particularly complex Axiom.
Decisive blow came with the MM (Michelson-Morley) experiment.
Galilean additive Transformation assumed CE, speed of light
measured at the earth, as the sum of speeds C of light and
V of earth both with respect to Aether: CE = C+V (similarly
to somebody walking within a moving train). Yet, MM
experiment has shown that C was invariant, independent of
the speed of source and Observer. Aether got falsified beyond
repair and from this moment the superhuman efforts to save it
at any price and at the expense of facts glaringly reveal its
dogmatic nature.
HISTORIC OVERVIEW
In the chapter "NATURAL MODEL" we asserted that the intuition
of continuous and infinite space stems from the imaginary,
quasi rigid continuation of a relation body B0.
In pre-scientific thinking, the solid earth's crust played
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.
Science, to wit the Euclidean geometry, based its axiomatics
upon this natural, intuitive view and considered it as
"self-evident". However, Euclidean geometry or "art of earth
measuring" was indeed a natural science and its axioms were
factually falsifiable, even if Euclid did not state it
explicitly.
Self-evidence stayed as the official characteristic of axioms
until the 19th Century. Yet, the factual falsifiability was
always implied and allowed to distinguish rational models
from dogmatic theories. Galileo founded his Relativity in the
deductive/inductive method implying an axiomatic model in our
sense, emerging from the background of traditional purely
deductive, speculative methods, misrepresented as "axiomatic",
whose "self-evident axioms" were indeed camouflaged dogma,
a case in point being the famous Spinoza's "axiomatic" Ethics.
At the end of the 19th Century, epistemology has officially
adapted the principle of factual falsifiability as the
cornerstone of axiomatic models.
It appears incredible, but the common "wisdom" did not notice
it and stays 2300 years behind concurrent rationality, as can
be seen in English dictionaries defining axiom as:
-generally accepted truth.
-a statement or proposition that needs no proof because its
truth is obvious, or one that is accepted as true without
proof.
-an obvious or generally accepted principle.
-self-evident or universally recognized truth.
-self-evident and necessary truth, or a proposition whose
truth is so evident as first sight that no reasoning or
demonstration can make it plainer; a proposition which it
is necessary to take for granted.
-self evident truth, or a proposition whose truth is so
evident at first sight, that no process of reasoning or
demonstration can make it plainer.
-necessary and accepted truth; basic and universal principle.
None would even hint that axiom is not "axiomatic" by itself,
but by virtue of the role it plays in a theory, viz. the
deductively founding and inductively falsifiable presumption.
That the same assumption may be an axiom in one theory and
a dogma or a theorem in another.
ONE OF CONCURRENT FALLACIES
In "Goedel's Proof" by Peter Suber, Philosophy Department,
Earlham College,we read:
***
Suppose we added G to the axiom set of S. Then G would become
provable, since all axioms are provable by definition.
***
The mind boggles. Since 2300 years all scientific theories
are axiomatic and their founding axioms are unanimously
considered as "by definition" true and unprovable. So Suber's
"definition" barring the whole scientific history seems
a bit exotic and one would associate it rather with some
exclusive loony bin than with science, mathematics, logic
or anything rational.
And Mr Suber continues:
***
Goedel proved that if we do this (add G to the axiom set of S),
creating S', then we could always construct another undecidable
wff, G', which asserted that it could not be proved in S'.
Of course we could then add G' to S', creating S'', but then
we could construct G'', and so on ad infinitum.
***
Very exclusive must indeed be the loony bin where Peter Suber
may venerate his co-certified master Goedel confusing mathematics
with russian dolls.
This Goedel's betise may be added to all those shown in the
chapter "PREDICATE LOGIC".
POSTFACE
We have defined axiom and dogma as deductively founding
presumptions of a theory. The difference consists in axiom
being in addition inductively and factually falsifiable.
Now, mathematical axioms seem to lack factual falsifiability.
Would therefore mathematics be dogmatic?
We shall examine it in the chapter "FOUNDATIONS OF MATHEMATICS".
