April 15, 2017 Saturday
Bedtime Story
David Hilbert Designs the Concept of "Absolute Proof"
Let me once again state the Russell’s Paradox or Antinomy.
Let A be a set of all sets that are not members of themselves.
In case of such a set, if A is not a member of itself, then by
definition it must contain itself.
On the other hand, if it contains itself, then it contradicts its
own definition.
In short, the set A is such that it is only possible if it is
impossible by its very definition.
The contradiction is fatal and so you see that something as
elementary as a definition of set can create so much disharmony when apparently
all the reasoning to begin with was so clear.
So you see how elusory the problem of proving consistency of a
mathematical system is.
Using of non-infinite models is not satisfactory as it fails to
reveal the totality of the system which though in near-sight seems fairly
robust, but on far-sight is fraught with hidden inconsistencies waiting to
surface when the right mind brings them forth.
Therefore something else, something totally novel had to be
invented that would give absolute proof of consistency of a system; something
that need not depend on the presumed consistency of another system.
So the idea of coming up with what is known as “absolute proof”
came up or was in the air so to speak.
David Hilbert proposed a way of such a kind of absolute proof.
It was to construct a totally new formalized deductive system with
completely new symbols.
These symbols would be like meaningless signs that would be
manipulated with highly precise stated rules.
If I bring the analogy of the chess board here, that would make
for a great illustration.
It is like replacing the knight (which represents a mounted
cavalry) with a block of wood and assigning a whole new way it would move.
It is as if you are taking away the quale of “knightness” from
this knight chessman and making it a totally nameless abstract object that can
only be manipulated in a very specific manner.
These empty symbols and signs would be governed and manipulated by
explicitly assigned specific rules.
Anything that will be derived from these signs and symbols by the
strict assigned rules will hence come out as strings of meaningless symbols or
marks.
One of the crucial purposes of this formalization business is to
get rid of any subjective reasoning that would otherwise creep in the usual
mathematics.
Taking away the meaning brings into play pure logic.
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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