May
13, 2017 Saturday
Bedtime
Story
Goldbach's Conjecture as an Example of Gödel's Incompleteness Theorem
There
exists even today and preserved very safely both physically and digitally some
167 letters of correspondence between Euler and Goldbach.
If
possible, I will try to send a copy of this historic manuscript to you or post
it right down here after tonight’s bedtime story.
In
a letter dated June 30, 1742 Euler wrote:
“That…every
even integer is a sum of two primes, I regard as a completely certain theorem,
although I cannot prove it.”
Since
Euler, no one has ever proved it even though this theorem has been shown to
hold good up through 4 X 1018.
Expressing
an even number in this Goldbach’s fashion, that is, as a sum of two primes is
called a Goldbach partition.
Let
me show you few examples.
6
= 3 + 3
8
= 3 + 5
10
= 7 + 3 = 5 + 5
20
= 7 + 13
22
= 11 + 11
30
= 13 + 17 = 23 + 7
100
= 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53
That
will be enough for now as I would like to divert too much into Goldbach’s
conjecture.
The
whole point of narrating the story of Goldbach and his conjecture was to show
that this conjecture which apparently seems to be a true statement regarding
numbers or number theory, has still found no proof.
This
is exactly what Gödel’s second conclusion was meaning to say.
Here
we have in Goldbach’s conjecture a beautiful mathematical statement that
appears to be very true but apparently (at least so far) it appears to be
non-derivable from the axioms of number theory.
Now
let us make a postulation.
What
if we modify the axioms of number theory or make some additions so that
conjectures like Goldbach’s become derivable?
Will
that not solve the problem or take care of the difficulty that we encountered?
Gödel’s
replay is an emphatic no.
The
conclusions of his theorems make it explicitly clear that even if a formal
system like the Principia Mathematica were to be augmented with new rules and
new axioms, we would still end up with additional arithmetical truths that
would be unprovable and hence underivable from that expanded axiomatic formal
system.
We
will study in detail how Gödel came to these conclusions.
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.
Letter dated June 7, 1742 written by Goldbach to Euler. It is in the margin that Goldbach wrote:
"Every integer greater than 2 can be written as the sum of three primes"
"Every integer greater than 2 can be written as the sum of three primes"
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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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