February 25, 2017 Saturday
Bedtime Story
Why the Need To Formalize Mathematics?
One of the reasons of formalizing mathematics by the intellectual
giants of the human apes is to understand what lies beneath the entire edifice
of mathematics.
You see mon ami, before the Greeks, in Mesopotamia and in other
civilizations, mathematics served as a real useful tool in the real world of
daily affairs.
You needed mathematics for business, for division of land in
agriculture, in the markets where trading was done and by the royalties for
bargaining at the times of war and matrimonial alliances.
Euclid for the first time (at least what we are aware of), decided
to detach mathematics from all the worldly affairs by formalizing it using a
set of basic axioms.
His basic definitions about points and lines were not really
concerned with the world we live in.
This was one method of formalizing mathematics.
There was yet another type that is well less known as it is well
less spoken about.
That was the formalism of logic by Aristotle.
As I told you in my previous bedtime stories, both the Greek
school of logic and the Hindu Nyaya School of logic developed pretty much
nearly at the same time.
What was common to both was that even though both the schools
tried to bring rigor into logic, they both failed to connect mathematics with
logic or bring mathematics into logic.
It was only in the 1700s and 1800s when men in Europe and England
such as Gottlob Frege, George Boole, George Peacock and Augustus De Morgan
began to seriously apply mathematics and mathematical rigor to logic.
Bertrand Russell was like David Hilbert, another one those
brilliant optimists who thought that by tightening up all the basic and
relevant definitions using logic, he could rigorously derive every known aspect
of mathematics.
He believed that even elusive topics such as infinites and liar’s paradox
could be tackled if sufficient logical rigor was applied at the very
fundamental level.
Yet he had doubts.
For while examining the liar’s paradox he came across yet another
inherent inconsistency; the paradox of self-reference.
Is it possible for a set of all sets that do not contain
themselves to in fact contain itself?
To solve this great conundrum, Russell introduced to mathematical
logic the idea of types of sets.
This was not enough though.
He along with Whitehead decided to completely formalize
mathematics by writing it down with a completely new set of symbols.
It was well known even much earlier that all the languages that
had been invented by the human apes were to imperfect for mathematics.
Even as early as in the late 1600s, Gottfried Leibniz had brought
in the idea of introducing a completely new notation for mathematics.
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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