April 14, 2017 Friday
Bedtime Story
Why Mathematicians Are Wary of Making Even "Obvious" Assumptions
Even when it comes to something as elementary as the set of all
natural numbers, it is simply impossible to inspect all of its endless elements
of this infinite set.
Hence the inductive method of proof will simply fail to be
applicable as a proof.
Many of you will feel exasperated and say isn’t it obviously
“clear” that an integer has to come after an integer and it always has to be a
new one.
History of mathematics has shown that whenever we apes have gone
careless and made such seemingly ben.
This inspite of the fact that everything was based on an apparent
sound and clear thinking.
Especially when it concerns infinites.
I hope you recall my stories on Georg Cantor on how this great
mathematician proved the existence of multiple infinities.
This was achieved through his ingenious one-to-one correspondence
technique.
What was even “worse” for the establishment was his continuum
hypothesis.
In continuum hypothesis, he not only talks about the two different
infinities, that is, the infinities of the natural numbers and that of the real
numbers but goes a step further.
He proposed (though did not prove) that there exists no set whose
cardinality or power lies between that of natural numbers and the real numbers.
The cardinality of natural numbers is designated by aleph-naught ℵ
You can write the continuum hypothesis algebraically as follows:
ℵ0 < ⎮S ⎮< 𝖈
ℵ0 < ⎮S ⎮< 𝖈
Continuum hypothesis says that there is no set that can satisfy
the above equation.
So if contradictions can come up in mathematics dealing with set
theory which in turn deals with numbers, is it not possible that similar
contradictions can come up with numbers which too are infinite?
After Cantor, Bertrand Russell too found that there exists
contradiction in elementary logic.
I surely must have narrated this Russell’s paradox or antinomy in
one of my bedtime stories.
It is childishly simple when stated.
We will take it up in the nights to come.
Hope you will recall it by then.
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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