July
14, 2017 Friday
Bedtime
Story
The Dilemma That Mathematicians Are Left Facing With Post-Gödel
Of
course, so far no one has proven that these hypothesis and conjectures (Goldbach,
Twin Prime and Riemann) are unprovable.
I
am almost done with Gödel’s incompleteness theorems and would like to move on
to other men on whom Gödel’s work had profound and revolutionary affect.
One
of the ideas that Gödel’s work gave rise to, which you can very obviously see,
is the issue of undecidability.
By
undecidability here I am specifically referring to the dilemma that
mathematicians are faced post-Gödel.
How
are they to know before they jump on to spend their entire lives finding a
proof of a theorem whether the proof actually exists?
Before
Gödel, it was taken as a gospel fact that any mathematical truth is provable.
The
other problem that Gödel’s work might be touching upon is the issue of
complexity of some proofs.
Take
for example the proof of Fermat’s Last Theorem published by Andrew Wiles in
1995 in the form of two papers in the journal Annals of Mathematics.
The
two papers have a forbidding length of 129 pages and took seven years of Andrew
Wiles life.
Just
look at the dichotomy that exists between the childishly simply stated Fermat’s
last theorem and the titles of the two papers that Andrew Wiles published.
That
will give you a rough idea how complex the proof must have been.
Fermat’s
Last Theorem, formulated in 1637, very simply states that no three distinct
positive integers a, b and c can satisfy the equation
an
+ bn = cn
where
n is an integer greater than 2 (n ≥ 3)
You
may not be able to prove this one, but at the very least it has such a benign
simplicity attached to it that even a child will get it (unlike the formulation
of Riemann hypothesis that I shared with you last night).
Now
compare this simple theorem with the two papers that Wiles published and which
form the proof to this problem.
“Modular
elliptic curves and Fermat’s Last Theorem”. Wiles, Andrew (1995). Annals of
Mathematics.
“Ring
theoretic properties of certain Hecke algebras”. Taylor R, Wiles A (1995).
Annals of Mathematics.
Wiles
had written the second paper with Taylor.
Now
it is very evident from the title of the papers that the proof required for
this theorem was seriously complicated that incorporated the work of many other
specialists.
So
much so, that at the time Andrew Wiles and Taylor published their papers, it
was said that there were only a handful of mathematicians capable of
understanding the proof.
So
it is possible, highly probable in fact, that there may exist such proofs so
complex that will be beyond the capacity of even the most intelligent and some
that will be so long that even a whole life time will not be enough to
construct it.
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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