July
13, 2017 Thursday
Bedtime
Story
Implications of
So
let me wind up Gödel’s theorems by summarizing the implications they have for
mathematics.
For
one, these theorems placed a kind of absolute limitation on the axiomatic system
of mathematics, or simply mathematics.
There
might be conjectures out there that may be true but it would be impossible to
prove them using the standard axioms.
Goldbach’s
conjecture is the most popularly cited example though the theorem says there
ought to be many more such conjectures out there.
Before
Gödel came along, it was a given that if there is any true statement in
mathematics, it is bound to have a proof.
It
is just a matter of time before one can derive it.
Fermat’s
Last Theorem took 358 years from the time Pierre de Fermat first conjectured it
in 1637 in a margin of a copy of Arithmetica to 1994 when Andrew Wiles released
the first successful proof.
But
now with Gödel, the truth was getting delinked from proof in mathematics.
The
naive belief that mathematics is consistent and won’t give rise to
contradictions was dismantled (though so far no serious inconsistencies have
been encountered except for those involving Cantor’s infinities).
I
told you earlier that there was a belief earlier that Gödel’s theorems were
very theoretical and were not really applicable to real arithmetic.
After
all, no one is sure of Goldbach’s conjecture is unprovable.
Similarly,
no one knows for sure if the twin prime conjecture and Riemann hypothesis are
unprovable.
For
now, I will simply state the above mentioned conjecture and hypothesis but will
not go into explaining them.
By
the way, in spite of mathematics being considered such a precise science, no
one knows for sure the exact difference in mathematics between a hypothesis and
conjecture.
The
twin prime conjecture states that there are infinitely many primes p such that
p + 2 is also a prime.
By
the way, twin primes are two prime numbers that have a prime gap of two such as
41 and 43.
Now
let me state the Riemann hypothesis and see what you can make of it.
Riemann
hypothesis states that the Riemann zeta function has its zeros only at the
negative even numbers and complex numbers with real part ½.
Now
mon ami, just to understand what this conjecture/hypothesis says requires a
deep knowledge of modern mathematics including serious analysis/calculus.
So
all these three conjectures (Goldbach, Twin Prime and Riemann) could be
unprovable within the current mathematics and yet be true.
This
is what Gödel’s proof implies.
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.
The real part of every non-trivial zero of the Riemann zeta function is 1/2
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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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