September 22, 2017 Friday
Bedtime Story
Deciding the Truth of a Mathematical Statement - From Leibniz to Ackermann and Hilbert
As I was saying, Leibniz dreamt of a
machine that would, by manipulating symbols, be able to determine truth value
of mathematical statements.
To bring to fruition this goal, he realized
long back then, the essential prerequisite was to devise a formal language.
Even though Leibniz is not known for this,
but he was one of very important logicians in the history of mathematical
logic.
He never actually published anything on
logic specifically under that rubric, but there is lot of work on this subject
done by him amongst his unpublished drafts.
Also the fact that he did so much pioneering
work in other areas of mathematics, calculus and its notations in particular,
that they overshadowed his work and ideas on formal logic.
(Not to forget the famous Leibniz-Newton
Calculus Controversy)
The idea that Leibniz had raised informally
was given its proper form by David Hilbert and Wilhelm Ackermann in 1928 that
now goes by the famous name of Entscheidungsproblem or the decision problem.
It is said of Leibniz that he had ideas in
his brain concerning formal logic that would come to their full fruition in the
early twentieth century, some two hundred years after his demise.
He saw in the uniqueness of prime
factorization a way to exploit the prime numbers for some universal
characterization, a proposal with striking similarity to the concept of Gödel
numbering.
Leibniz was certainly far ahead of his
times.
But even when the decision problem was
posed by Hilbert and Ackermann, it had almost nothing to do with computation
but more to do with foundations of mathematics.
You know how obsessed Hilbert was in laying
a rock solid foundation to mathematics.
The way Hilbert and Ackermann posed their
question was targeted towards their concern of being able to know the
truthfulness of a statement of first-order logic.
It was the very essence of Hilbert’s
Program to get things in mathematics so firmly founded that there would remain
no doubts.
He dreamt of a mathematics where it be
would be possible to have an algorithm (series of logical steps) that could
decide the truth and falsity of any mathematical statement from the axioms
using the rules of logic.
This dream of Hilbert is now famously known
as the problem of decidability or in German Entscheidungsproblem.
One of the drawbacks of this last point was
that way back then the definition of algorithm was not completely clear.
In this question too Gödel had made his
mark by contributing his completeness theorem of first-order logic.
Gödel’s completeness theorem forms the
first of the two cornerstones of first-order logic and is a fundamental theorem
of mathematical logic.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
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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