April 03, 2017 Monday
Bedtime Story
The Three Greek Problems of Geometry
If you get the impression that I am constantly hesitating from
getting into the meat of the proof of the incompleteness theorems, you are not
wrong.
It is so because I may myself not be at ease in going about it.
Perhaps it is so that I have a feeling that we are still not
adequately equipped to understand the import of these landmark theorems.
Just going back a bit, the axiomatic method of Greeks simply
accepts certain basic axioms and postulates as “truths” without any proof.
I have put the “truths” in quote because here, in this case, I do
not have a precise definition of truth.
Mathematicians, at least of today, you must understand are not
seriously concerned with truth.
Yet, for centuries, the truthfulness of these axioms were never
debated; After all, who would debate a postulate such as this:
“It is possible to draw a straight line from any point to any
other point”.
These presumably self-evident truths, the axioms form the
foundations of mathematics.
The derived theorems from them form the superstructure of
mathematics.
This axiomatic method turned out to be very powerful and extremely
useful for centuries and hence it greatly impressed some of the most powerful
minds of all times.
It was, to some extent, even tried in other fields of sciences
such as in physics.
Newton’s laws of motion have a truly Euclidian approach.
Yet, for centuries, it was only geometry that was firmly and
solidly based on axioms.
It remained so for centuries till the timeline hit the nineteenth
century.
Of all the centuries that human apes have lived through, it is the
nineteenth century that is truly transformative (and in some sense disastrous as
the greed of industrial revolution was fully unleashed upon the fragile planet).
Mathematics made terrific advancements in this revolutionary
century (1800 to 1900), almost all of it occurring in Europe, including parts
of European Russian Empire.
Some of the problems that were put forth by Greeks that laid
unsolved forcenturies were finally put to rest.
The elementary problems that the Greeks had put forth were as follows;
Using an unmarked straightedge and compass, is it possible to carry out these 3
geometrical maneuvers:
1. Trisect an angle
2. Duplicate a cube and
3. Square a circle.
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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