February 15, 2017 Wednesday
Bedtime Story
Computability Theory Has Its Origins in the Study of Computable Functions
At the very heart of computability theory or recursion theory lies
almost pure mathematics and logic.
The primary questions that the recursion theory addresses are as
follows:
Question 1:
“What does it mean for a function on the natural numbers to be
computable?”
Question 2:
“How can noncomputable functions be classified into a hierarchy
based on their level of noncomputability?”
(For example, real numbers
are not countable and hence
they are noncomputable).
It is interesting to know that such a theory that went on to
become a branch of computer science was never pre planned to become so.
These strange questions that this theory seeks to ask largely came
out of the mathematical inconsistencies that so much bothered men like Hilbert.
In fact, one of its pioneers was that German high school
mathematics teacher and the co-author of Hilbert’s book Wilhelm Ackermann.
If you recall, his book “Principles of Mathematical Logic” for the
first time had precisely posed the problem of completeness and decidability.
No one in those days could have imagined that these questions
would have something to do with computability, or at least the computability as
understood by today’s computer scientists.
Ackermann along with Gabriel Sudan (Romanian) in the early 1920s
began to study whether functions are computable.
First of all, let us be clear what the word function means in
mathematics.
Function is one of the most basic concepts of mathematics, as
basic as “numbers” or “set”.
A function in mathematics relates an input value to a
corresponding output value.
The input value is in the language of mathematics known as the
argument.
The corresponding output value technically is known as the
function value.
One can even use the word mapping though it is less used.
It is formally written as follows:
f:X
Y
The values that will be assigned to X are called arguments for the
function f.
For each argument x there will be a corresponding and unique
function value y.
It is written as y(x) and we can say that the function associates
y to x or maps x to y.
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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