May
26, 2017 Friday
Bedtime
Story
Even the Reasoning/Logic Used by Gödel is Mentally Punishing
Just
to be clear, the idea of mapping per se is not totally a novel idea.
Co-ordinate
geometry for instance translates or maps geometry into algebra or even in
construction of actual maps wherein the shapes on the surface of a sphere like
our planet is projected into flat rectangular papers.
Yet
the way Jules Richard deployed the idea of mapping was unique.
This
sparked the idea inside Gödel’s mind and using it he showed that
meta-mathematical statements about a formal logical system of arithmetic can be
represented by arithmetic formulas within the system.
This
statement written above is a very difficult one for any non-mathematician or
even a mathematician not trained in formal logic.
But
it is foundationally critical for the proof of Gödel’s theorems as we will see
when we go deep into the theorems themselves.
He
used the mapping in a totally revolutionary way, devising and proving that
within his formal logical system neither his arithmetic formulas were map-able
to certain meta-mathematical statements nor these same formulas could be corresponded
to the negation of such meta-mathematical statements.
This
again is a profound statement and will not be comprehendible at this stage.
But
since these arithmetical formulas ought to be true and codify the truth of the
system, the system was incomplete.
Gödel
then went further.
He
found a way to construct a number theory statement that corresponded to the meta-mathematical
statement:
“The
formal logical system is consistent’.
He
went on to show that that particular number theory statement was not
demonstrable or impossible to drive within that formal system using the
established rules of inference.
It
is only possible to derive that number theory statement if additional rules of
inference were invoked which makes the validity and the consistency of the
formal system questionable.
If
you now think that we are done with Gödel’s incompleteness theorems, you are
badly mistaken.
All
this was the background knowledge to prepare and educate you (and me more than
you) before going into the meat of his theorems.
The
problem of taking up a bedtime story as complicated as this is twofold:
One
is my own lack of education in serious and higher mathematics and formal logic.
Before
I present anything to you, I need to at first understand it myself and be clear
about it.
That
takes time for an average ape like me.
And
by the way, I always find it more enthralling to write about something that is
new to me and that which is more painful to write (definitely a serious streak
of mental masochism).
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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