April 13, 2017 Thursday
Bedtime Story
Consistency of Euclidean Geometry Was An Assumption
It was near universal consensus by the mathematicians of all ages that
the Euclidean geometry was consistent in spite of the fact that it was never
proven to be so.
Simply because Euclidean geometry vibes along perfectly fine with
our daily experience, its truthfulness and hence its consistency was assumed.
This kind of inductive reasoning and proof was never meant for
mathematics.
Mathematics is based on absolute deductive reasoning; anything and
everything either needs to be proved or disproved.
As long as neither of these two is achieved, then no matter how
sacred the axioms are, they can at best be conjectures; meaning it can be
plausibly or possibly true.
David Hilbert tried another way out of this dilemma.
He resorted to coordinate geometry, converting all the axioms of
Euclid into algebra or algebraic truths.
So the point was converted to a pair of numbers, a straight line
was defined using a first degree equation with two unknowns, a circle by a
relation between numbers using a quadratic equation and so on it went.
Hence the entire arena of geometry was converted to algebra.
If the axioms satisfied the algebraic model, then apparently its
geometry ought to be consistent.
This was an impressive feat and yet again this method had the same
previous flaw.
It too relied on the consistency of another system which was
algebra.
So proving the consistency of an axiomatic system remained
elusive.
One of the key reasons as you will notice is that even though the
consistency is assured for a limited amount (even though vast by our human
standards) of the known theorems, there is no guarantee that the axioms will
hold good for an infinite number elements.
In short, how can we rest assured that just because so far it has
worked well, in future too the axioms will hold on water tight not leading to
mutual contradictions?
As you are well aware, nearly all the postulates of mathematics
are NOT limited to finite number of test cases.
Even such a simple proposition in elementary arithmetic that
states:
“Every integer has an immediate successor that is different from
any integer that came prior to it.”
So if we were to check the veracity of this axiom (yes, it is an
axiom) using a finite set from one to say one thousand, then we are assured
that the axiom holds good and we can be pleased.
This is an inductive method of reasoning that works well for
physics.
Alas, in mathematics we are dealing with infinite integers.
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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