July
08, 2017 Saturday
Bedtime
Story
This
astounding proof and bewildering conclusion of Gödel has a tendency to get
misconstrued.
The
proof of incompleteness theorems does not proclaim that there is a lack of
meta-mathematical proof of the consistency of the Principia.
What
is claims is that such a meta-mathematical proof even though may exist will not
be mirrored within the Principia.
In
case if this leaves you with a bitter taste of dissatisfaction, consider a
geometry analogy.
You
remember we had discussed the Greek problem of trisecting an angle with a ruler
and compass?
If
not, please do go back and search it out and reread it.
Just
because it has been proven (by Pierre Wantzel) that an angle cannot be
trisected by a compass and straightedge does not mean that an angle cannot be
trisected at all.
In
fact, all one needs extra to trisect an angle, besides compass and
straightedge, is a fixed distance marked on the straight edge.
Very
similar is the case with Gödel’s proof too.
It
must be pointed out that meta-mathematical proofs or rather perfectly
reasonable meta-mathematical arguments have already been accomplished that
establish the consistency of formal systems.
One
of those was devised by yet another compatriot and contemporary of Gödel,
Gerhard Gentzen who too was a mathematician and logician.
The
life of Gerhard Gentzen was short (spanning a mere 36 years from 1909 to 1945)
and marked by bizarre vicissitudes of life as he was trapped in the Nazi politics
and propaganda of Aryan supremacy.
Yet
in those short 36 years he made major contributions to both the foundations of
mathematics and proof theory.
Gentzen
was a student at the famous University of Göttingen during the reign of David
Hilbert in 1930s.
Göttingen
reigned supreme and was the center of mathematics thanks to Hilbert’s gift for
detecting out talents and surrounding himself with them.
Gentzen
was a student of the Swiss mathematician and logician Paul Bernays who was an
assistant and close collaborator of David Hilbert.
His
second Habilitation was on the subject that we have been discussing all along –
the axiomatic of the propositional calculus of Principia Mathematica.
In
1922, Göttingen appointed him as an extraordinary professor.
In
1933, the Nazis came and started to strip naked the University that had
harbored and was still possessed with exemplary great minds.
One
of its victims was Bernays who was dismissed from his position for being
“non-Aryan”.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic engineer
and a former professor at BARC, does and teaches real mathematics and physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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