September
14, 2017 Thursday
Bedtime
Story
End of "The Notion of Proof"
What
Tarski is implying is that a sentence will be recognized as a theorem of it
merely by inspecting its shape and comparing its shape with the sentences that
came before it in the list.
One
will not even have to deploy the complicated step of deducing and reasoning.
Such
a system will eliminate the need of intuition to a very large extent.
Such
a system has surely not been adequately devised for languages but a far as
mathematics is concerned a lot of it has been formalized.
Even
the most complicated theorems have formal proofs.
With
this we have finished the second part of Tarski’s paper – The Notion of Proof.
Now
we move on the third and the final part – The Relationship of Truth and Proof.
Mathematical
logic with its formalized notion of proof completely revolutionized the notion
of proof leaving no room for intuition for which there never existed any clear
defining idea.
Yet
with this success came an unforeseen trouble; the very simplicity of the method
carried within it germs of problem that would gradually grow to become a
monster.
Once
there was this well established notion of proof, it then had to be linked to
the notion of truth.
After
all, the whole purpose of going into so much trouble of curbing the vagueness
of intuition in the process was to get to the truth.
The
entire new process of formalization was only valuable if all the sentences that
were derived from it proved to be true and secondly if it was assured that all
true sentences would be derivable using this system.
So
Tarski raises two questions in this respect.
(a)
Is the formal proof really an adequate procedure for acquiring truth?
(b)
Does the set of all formally provable sentences coincide with the set of all
true sentences?
Let
us try to study these two questions through a very simple example.
Consider
the number theory which is nothing but arithmetic that deals with natural
numbers.
Numbers
are such entities which are very intuitive and with which we all are familiar
with, even the uneducated and uninitiated.
We
will take it for granted that number theory has been formalized with highly
restricted vocabulary.
I
hope you recall we had discussed this concept of limited vocabulary on August
26, 2017 bedtime story titled “Developing Semantically Restricted Language.”
We
shall deal in greater detail with the vocabulary of such a formalized number
theory in the nights to come.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
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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