September
19, 2017 Tuesday
Bedtime
Story
Conclusion of Tarski's 1969 Paper
So
when restricting the language did not suffice, we went to the third stage and resorted
to dividing even this restricted formal language into object language and
meta-language.
In
this new setting with certain very specific restrictions we did manage to tame
the ghost of Liar Paradox.
We
did ensure that the paradox was not generated but along the way we also managed
to generate a result, a conclusion, that has far reaching (and perhaps
upsetting) implications.
The
significance of the result is profound no matter even if it is negative and
perhaps even pessimistic to our human likings.
Unequivocally
the result states that as far as mathematics is concerned, there is no branch
of it where the notion of provability is a perfect substitute to the notion of
truth.
The
belief that formal proofs can serve as foundation for establishing truth of all
mathematical statements was truly shaken up.
It
was a serious blow to the rock-solid axiomatic system that Hilbert had wanted
to establish.
Yet
all is not lost.
There
are some tangible gains to be made out for the wreckage of the demolished
formal axiomatic system.
We
now have a precise and adequate definition for the notion of truth in a formal
restricted language.
Though
the restrictions are present in the object language, the notion of truth can be
freely used in meta-language.
On
the other hand, though the notion of proof does not have hundred percent
correspondence with the notion of truth, formal proof still remains the gold
standard and the only method of proving the true mathematical statements.
The
only difference is that post Gödel and Tarski, we are now aware that there
could exist true mathematical statements which may not be provable.
If
and when we do encounter such sentences or conjectures, in those cases we may
have to refresh the existing axiomatic systems with further axioms or new rules
of inference.
When
we do so, we will have to use the principles of notion of truth as the acting
guide.
This
is crucial as we would not like to add on a new axiom or a new rule of proof
which is not true or which will end up generating false sentences.
Of
course, by adding such a true axiom we will also generate more true sentences
which will not be provable and thus will force us add further true axioms and
this has the potential to carry on back and forth endlessly.
Thus
at the end all we can say, to sort of reconcile with harsh unpleasant truth
that we ended up with, is that the notions of truth and proof though not
exactly mutually agreeable are not at war either with each other.
They
live in peaceful coexistence.
This
is how Tarski ends this beautiful paper which mind you has Gödel’s work written
all over it, especially in the last part when one is trying to bring in
together the notions of truth and proof.
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.
Advertisements
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:
No comments:
Post a Comment