September 26, 2017 Tuesday
Bedtime Story
A Set of Postulates for the Foundation of Logic 1932
As mon ami says, in the name of atheism I
was carrying out my own jihad on disbelief which in a way is no different from
any other religion.
So let us leave aside the religious beliefs
of Church and return back to our primary theme of the story, which is the
Entscheidungsproblem and the contribution of Alonzo Church to it.
Church too began to ponder over the problem
that Hilbert and Ackermann had posed: Is there a decision procedure to
determine if any particular statement of first-order logic is true or false?
Church too was interested in the
foundations of mathematics.
In 1932 Church published a paper in the journal
Annals of Mathematics titled “A set of postulates for the foundation of logic”.
He republished this paper in 1933 with
further refinements.
It was in these papers that Church first
introduced lambda-calculus or λ-calculus.
You might think from the name that this
lambda calculus must be that formidable calculus that so scared you in high
school or maybe even worse than calculus since it has that dreadful lambda
added along to calculus.
You will be surprised to know that the
calculus of Church has nothing to do with mathematical analysis that involves
differentiation and integration.
What is common to both the calculi though
is the notion of functions albeit considered in totally different ways.
I discussed a lot about functions in my
previous bedtime stories.
The function that Church was dealing with
is specifically known as computable function.
Computable function has a direct bearing on
the problem that was posed by Hilbert and Ackermann.
And on the other side, computable functions
have a deep-seated link with algorithm that in the modern world forms the basis
of computer programming.
Computable functions are formalized analog
of the intuitive notion of algorithm that we all very familiar with even if we
have never been programmers in any serious sense.
To put it most simply, a function is
computable if there is an algorithm for it.
Or in other words, if there is a certain
input, using certain finite number of steps, its output can be worked out for
that function.
Let me try to explain lambda calculus in as
simple way as it is possible.
I am no computer scientist without any
formal training either in mathematics, or in formal logic or in computer
science.
Whatever little I do know on these subjects
is largely inspired by mon ami’s fascination for these subjects and my
fascination for his fascination.
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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