September
06, 2017 Wednesday
Bedtime
Story
The Four Founding Principles of the Axiomatic System
From
this understanding arose two principles which would be applied in laying of the
foundations of the future mathematics.
The
first principle was to chose a minimal number of primitive sentences that would
be called axioms and that would be assumed true once again as they seemed to be
self evident.
You
will argue that nothing changed from before.
Yes,
I do agree that you have a valid point there but the key difference here was
that these primitive sentences would be kept to a bare minimum whose numbers
would remain constant once accepted.
The
foundation would have weakness but at least an attempt was being made to keep
the shortcomings as low as possible.
What
was really different this time was the second principle.
The
second principle made it absolutely clear that no more sentences would be
considered true until and unless they were derived from these axioms or from
other sentences that had earlier been derived using these axioms.
All
these provable sentences as you are very well aware now go by the name of
theorems.
Two
more principles can be added to make the idea of axiomatic system complete.
These
two principles closely resemble the two mentioned earlier.
Even
before the axioms or the primitive sentences in mathematics, it was decided to
have a list of some primitive terms.
These
terms have no definition in the sense that they seem to be predefined and as
Tarski puts it “directly understandable”.
Their
meaning once again seems to be self-evident.
The
second principle (which actually becomes the fourth principle) was to not
accept any new terms unless they were definable in those already established
primitive terms or any term that were defined using those established primitive
terms.
These
four principles are the fundamental basis of the axiomatic system and any
theory (or theorem) developed on this basis is termed axiomatic theory.
This
would enable anyone going over such a theory to be aware that at the very root
of it lie some simple and accepted assumptions.
Even
though the mathematics that had developed in the great ancient civilizations
such as Mesopotamia, Babylon, Egypt, India and China were very sophisticated,
they were neither explicitly nor inherently based upon the axiomatic method.
The
almost single-handed acclaim for the development of a full-fledged axiomatic
method is given to Euclid of Alexandria.
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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