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CBA.LOST PARADISE
NOTE: neologies and ambiguous terms clarified in GLOSSARY
are marked "[G]".
1.Cantorian Paradise
19th Century Physics and Logic were dominated by reaction of
Dogmatism against Rationality of the First Enlightenment.
Physics was founded in the dogmatic delusion of Aether.
Logical systems culminated in Predicate Logic (PL) founded
in linguistic structures reducible to "[subject] is
[property]", e.g. "my-car is green" or "my-car has (the
property of being) green" or "my-car has greenness" or,
generally "[entity] has [attribute]" ("attribute" being
synonym of "property").
This, rather surprising, ontological foundation in wrongly
interpreted structures of some selected natural languages
(BAC.LINGUISTIC INVERSION) sees Entity as "container of
Attributes", which clearly defines PL as a noumenal system.
Indeed, Noumenalism considers Entity and Attribute as
ontologically distinct constructs, Entity being a "Noumenon",
a "Container", which may receive and "have" Attributes but
exists as such, even empty and prior to receiving them. Kant
called Noumenon "Ding an sich" (thing in itself) and banned
it from cognition and rationality. Now, Noumenon is neither
observable nor factually verifiable, so that whatever we
assert about it is clearly a Dogma[G]. Thus, Noumenalism is
synonymous with Dogmatism and like the latter is nothing else
than Naive View[G] (also called "Naive Realism") dressed up in
highbrow verbiage.
Logic founded thus in Naive View may be called naive.
This noumenal, naive Logic was assumed to be the universal,
absolute foundation of transcendental "Reality" and of human
knowledge thereof. All areas of Science were assumed to be
recursively founded in Logic, although Mathematics stayed
for some time unfounded. Finally, in 1873 Cantor conceived
the naive Set Theory which supplied the missing link and
completed the hierarchy of "Cantorian Paradise":
1.(naive) Logic, the universal, absolute, olympian foundation,
2.(naive) Set Theory, founded in (naive) Logic,
3.Mathematics, founded in (naive) Set Theory,
4.Physics, founded in Mathematics and in the mechanistic
Dogma, itself founded in (naive) Logic.
5.Other "hard" Sciences, founded in Physics,
6."Soft" Sciences, founded in "hard" Sciences.
This "Paradise" represented the edifice of Science solid and
stable as Cheops pyramid, bestowing upon scientists the
gratifying conviction that Logic, THEIR Logic, explains and
determines the "Real Universe".
2.Breaches in the pyramid
At the end of the 19th century two breaches appeared in the
pyramid:
1.In Physics the Michelson-Morley experiment showing that
the speed of light is invariant with respect to the speed
of source and observer and thus falsifying the mechanistic,
Dogma.
2.Russell's Paradoxes calling in question the foundations
and sense of naive PL and founded in it naive Set Theory.
Question arose, if apparently local breaches in otherwise
solid pyramid may be patched locally, or should lead to
global reconsideration of the whole structure.
In Physics Einstein has chosen the global approach, revised
entire Physics and laid the cornerstone of the Second
Enlightenment with its new Rationality.
Logicians did not notice that Russell's Paradoxes wrecked
only the naive PL and try in vain, till our days, to patch
it rather than to scrap the wreck and conceive in its place
a new rational Logic.
2.1.Russell's Set-Theoretical Paradox
It seems to make sense to inquire, for any set, whether it
is member of itself or not. The set of horses, e. g., is
certainly not a horse. The set of all sets, on the contrary,
being a set, is clearly member of itself.
Therefore it seems reasonable to ask the same question with
regard to "S" the 'set of all sets that are not members of
themselves'. The answer is alarming:
S IS MEMBER OF S IFF S IS NOT MEMBER OF S.
(IFF="if end only if").
A contradiction derived from apparently plausible assumptions.
2.2.Russell's Logical Paradox
Russell's Set-Theoretical Paradox shows that an apparently
legitimate set may lead via sound inference to a logical
contradiction thus proving that something is rotten at the
very base of the naive Set Theory.
Actually, the situation was much more serious. The Paradox
turned out to be essentially logical and in no way depending
on some perhaps out-of-the-way peculiarities of sets:
We may inquire whether a property applies to itself or not.
Property of being red is not red while property of being
abstract is abstract. Calling "impredicable" the property of
not-applying-to-itself we conclude that
IMPREDICABLE IS IMPREDICABLE IFF IT IS NOT IMPREDICABLE.
3.Aftermath
It has been said that Russell's "Logical Paradox" shattered
Logic, just like the MM experiment shuttered Physics.
However, Logic should not be confused with PL. The Paradox
indeed (and not surprisingly) wrecked the noumenal naive PL
with its feeble foundation in misinterpreted linguistic
expressions. One would expect PL, like mechanistic Physics,
to be dropped and replaced by some redefined and refounded
rational Logic. Yet, such is the sticking power of Dogma,
such is the dread of quitting Naive View and of following the
"Sapere Aude" (Dare to reason) of Horace and Kant, that all
Logicians (some considered as leading thinkers of 20th
Century) tried vainly to save and rebuild the collapsed
structure upon its rotten base rather than construct a new
edifice upon new rational foundations.
As result, the established Logic became and stays till our
days an extraordinary proliferation of competitive remedial
systems all noumenal, ill-founded and meaningless.
This critic which may appear severe is justified in more
detail in CA2.INTRODUCTION TO PREDICATE LOGIC.
Somebody's got to say at last that the Emperor is naked.
