April 20, 2017 Thursday
Bedtime Story
What Mathematical Proofs Lack
The fact that I have placed the word absolute in the last night
bedtime story within quotation marks should be self revelatory of what followed.
Reading further will shed more light.
So for instance if the number theory was being studied with this
finitistic method of logical calculus, that it needed to shown that such
contradictory results such as “1 = 1” and its negation “~ (1 = 1)” could not
both be derived with the stated explicit rules of deduction from the axioms or
the initial formulas.
In short, Hilbert’s idea of absolute proof of consistency would be
to show by finitistic methods the impossibility of deriving contradictory
theorems or formulas in a given formal mathematical system.
So far so good.
Now we can bring into picture Principia Mathematica of Andrew
Whitehead and Bertrand Russell first published in 1910.
We all agree that mathematical proofs are full proof and very rigorous.
Yet they suffer from one deficiency.
There is a lack of explicit or clear-cut rules of inference.
This might not be a statement to which you will agree so readily.
So let me try convincing you with an example.
Consider Euclid’s Theorem that Euclid published as Proposition 20
in the Book IX of his master piece “Elements”.
I know most of us apes are not aware of this theorem unlike the
famous Pythagorean Theorem which is very well hammered into our brains in our
schools.
I do not why it so, even though Pythagoras of Samos did his work
around 540 BC whereas Euclid of Alexandria wrote his Elements somewhere around
300 BC, some 150 years or so after Pythagoras.
If Pythagorean Theorem is a fundamental relationship in Euclidean
Geometry, then Euclid’s Theorem is a fundamental statement in number theory.
Euclid’s Theorem is quite a simple statement: That there are
infinitely many prime numbers.
I shall keep tonight’s bedtime story short and then perhaps
tomorrow, if I continue to exist and be cognitively active, will publish the proof
of Euclid’s theorem as stated in the Elements.
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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