June
22, 2017 Thursday
Bedtime
Story
End of Arithmetization of Meta-Mathematics
As
I was saying last night, the strings of symbols within the Principia are
essentially meaningless.
In
fact they were devised to be so.
Yet
at times it makes sense to give them a “meaning”.
This
is so because meaningless string such as ‘Sub (243,000,000, 17, 243,000,000)’ acts
as a formal emblem of some very complicated mathematical calculation.
For
instance, just consider this meaningless string:
‘ss0
+ ss0’
Though
by itself totally meaningless, it is a formula of the Principia that mirrors
“two plus two”.
In
effect, that meaningless string of symbols indirectly represents four.
With
this much, I have now completed two of the three parts of Gödel proofs.
The
first part was Gödel numbering.
The
second part that we have just finished was the Arithmetization of
Meta-Mathematics.
Now
we will go to the final part of his proof, the very heart of his argument.
This
is the longest of the three parts and I dare say the part that will test the
limits of your logical reasoning and memory.
To
understand this argument part, you will need to constantly recall whatever we
have been discussing for the past few weeks or even months.
Remember
what I had written about mathematical reading?
First
let me enlist very broadly the five main points of his argument.
Then
we shall tackle each of them individually and in greater depth and yet not
getting too mathematical.
(1)
Gödel at first constructed a formula G of Principia that represented the meta-mathematical
statement:
“The
formula G is not demonstrable using the rules of Principia”.
(Any
formula that can be derived using the rules of Principia by default becomes the
theorem of Principia.)
Apparently
this formula is claiming of itself that it is not demonstrable or derivable.
To
a certain extent, Gödel had constructed the formula G on the lines of
Richardian number.
If
need be, you can go back (May 24, 2017 to May 25, 2017) and look how the
property ‘n is Richardian’ was created.
Gödel’s
argument is very much analogous.
We
have not completed this first point and I shall carry it forward in the nights
to come.
After
all, there is only so much our evolved brains (or rather regressed brains) can
handle.
It
may surprise you, but the fact is that human brain size peaked with the
Neanderthals some 28,000 years ago at an average of 1500 cm3.
Since
then it has been shrinking with the current average cranial capacity of about
1440 cm3 in males and 1240 cm3 in females.
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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