April 28, 2017 Friday
Bedtime Story
Principia Mathematica and its Connection with Kurt Godel's Incompleteness Theorems
Many mathematicians were still not convinced of the notion of
Frege-Russell thesis that mathematics could be reduced to pure logic.
Perhaps that is the reason why logic eludes most textbooks of
mathematics.
Yet Principia was definitely a major advancement as far as the
idea of proof of consistency in mathematics was concerned.
Principia provided three things for the other mathematicians to
work upon:
[1] It provided a completely novel and very comprehensive system
of notation with the aid of which all the statements of number theory could be
formally coded in a standardized manner.
[2] The rules of inference were made very explicit.
In short, Principia was the first ever formal system, almost a
kind of experimental device or an instrument that could be used to investigate
the entire system of number theory as a pure un-interpreted calculus.
The word un-interpreted is very significant.
Principia created the un-interpreted calculus that had this system
of meaningless novel symbols whose strings combined and transformed in
explicitly and precisely stated rules of operation.
An uninterpreted function in mathematical logic is one that has no
other property than its name and n-ary form.
This arity of a function is the number of arguments that the
function takes.
No wonder it is an impossible book to read and comprehend for most
average and even intelligent apes.
Yet Principia Mathematica was the prototype, the first ever system
of formal calculus ever devised, particularly when it came to number theory.
As I had said earlier, George Boole had something similar in mind that
he could never realize for himself.
Yet, he had laid enough foundations for other mathematicians to
build up upon it.
If you recall the title of Kurt Gödel’s landmark paper, its ends
with the phrase “Principia Mathematica und verwandter Systeme” which translates
to “Principia Mathematica and related systems”.
So Kurt Gödel in his paper was referring primarily to this
prototypical formal calculus system devised by Whitehead and Russell but
secondarily also to any such systems that may or would came out later.
Now
I shall attempt to try out a more difficult feat; to take a portion of
Principia and show how the formalization of a deductive system was achieved.
And
still further, how the absolute consistency of such a system can be
established.
Only
with this knowledge in hand (or rather in our brain), can we even begin to
understand Gödel’s genius.
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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