June
25, 2017 Sunday
Bedtime
Story
Gödel was Mindful of the Subversive Implications of his Paper
The
last two lines of Gödel’s paper reflect modesty, or perhaps a fear that lurks
in the mind of any scientist who is set to publish a path-breaking paper.
Very
much like Charles Darwin, Gödel was mindful of the subversive implications of
his paper and was apprehensive of ripples of shock waves it was bound to send
into the universe of mathematics.
That
is the reason why he was gentle in his conclusion that his arguments need to be
further proved for other formal systems.
In
other words, he was soothing some nerves beforehand in declaring that the proof
of the generalization (applicability to other formal systems) of his radical
idea would follow in his ensuing papers.
Little
did he realize that the arguments that he developed in his paper were so
convincing that there was little need of any further elaboration.
Hence
it is crucial to stress that Gödel’s results were not an outcome of some flaws
or insufficiency of the system of Principia Mathematica.
Gödel’s
implication would hold firm and true to any system that is structured upon the
arithmetic of integers, something that involves as basic as addition and
multiplication.
Of
course, only truly brilliant minds were capable of grasping the ramification of
Gödel’s proofs.
One
such mind was the great John von Neumann.
When
Neumann saw this paper, with exasperation he exclaimed, “It’s all over”.
Now
my dear fellow cousin apes, it is time to go back and quickly look at the five
points under which Gödel’s reasoning is being scrutinized.
We
shall tear apart and probe each of these five arguments in depth.
This
will probably be the most punishing chapter of Gödel bedtime series, but
perhaps also the most rewarding if you are able to comprehend the logic.
So
buckle up and get set go.
(1)
If you go back and look up (unless you have an incredible memory like mon ami),
you will find that the formula ‘Dem (x, z)’ has been defined.
Within
the Principia, it mirrors the meta-mathematical statement:
‘The
sequence of the formulas with the Gödel number x is a proof for the formula
with the Gödel number z.’
Now
let us add to this formula the existential quantifier of the predicate logic
“there is”.
Then
we get:
‘(∃x)
Dem (x, z)’
What
does this formula say?
It
simply says, ‘There exists a sequence of formulas (with Gödel number x) that
embodies the affirmation of the formula with Gödel number z.’
Or
in more simple terms, ‘The formula with Gödel number z is demonstrable.’
As
usual, there is no need to hurry and we will carry it on in the nights to come.
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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