August 13, 2019 Tuesday
Bedtime Story
Fitch's Paradox of Knowability
Post Gödel and Tarski saw an American
Logician and Sterling Professor at Yale University by the name of Frederic
Fitch come out with what is now known as Fitch’s paradox of knowability which
in some sense poses a challenge to the knowability thesis.
Fitch’s paradox states that the existence
of an unknown truth is unknowable.
The very bizarre corollary that follows
from this statement is that if all truths are knowable then all truths are in
fact known.
The proof of this theorem (Fitch’s paradox)
was published by the author in a 1963 paper titled ‘A Logical Analysis of Some
Value Concepts” wherein this particular theorem was merely a minor part of the
whole paper.
The theorem can be proven using formal
logic in the Gödel manner but can also be logically expressed in common English
language.
So let me put the proof for you.
Let us suppose that p is a sentence that is
an unknown truth.
This implies that the sentence p is true
but it is not known that is true.
In that case “the sentence p is an unknown
truth” is true.
If all truths are knowable (as is asserted
by the knowability thesis) it should be possible to know that “p is an unknown
truth”.
But this is not possible because the moment
we know “p is an unknown truth” we also know that p is true which makes the
sentence “the sentence p is an unknown truth” a falsity.
Therefore, the statement “p is an unknown
truth” cannot both be known and true at the same time.
Therefore, if all the truths are knowable,
the set of “all truths” (we once again return to Cantor’s set theory) must not
include any of the form “something is an unknown truth”.
Thus we have proved that if the knowability
thesis is true then there must be no unknown truths.
Therefore all truths must be known.
Quod era demonstrandum or “what was to be
shown” has been demonstrated.
Men like Bois-Reymond were neither
mathematicians nor logicians but somehow they had an innate belief that many of
the biological problems have no solutions or are to complex and therefore
unexplainable.
I think by the amount of progress biology
and computer science has made, particularly the field of powerful transistors,
artificial intelligence, artificial neural network and deep learning,
Bois-Reymond might regret having made that list of seven “world riddles”.
There just might be a little hope that perhaps
the Latin phrase “Ignoramus et ignorabimus” might not come in future as a major
threat.
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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