April 18, 2017 Tuesday
Bedtime Story
Differentiating Mathematics from Meta-mathematics
The important thing that needs to be emphasized about the
meta-mathematical statements is that even though they apparently contain the
mathematical signs and formulas, they do not contain the mathematics of it.
This is dicey and needs to be cleared.
Let us take a simple English statement.
Bombay is the densest populated city of the planet.
Here the word Bombay represents Bombay symbolically.
It does not mean that in this sentence I am literally placing the
entire city of Bombay in this sentence (leave aside the point that it is
impossible).
But now look at this sentence when I wish to speak about the word
Bombay and not the city that it represents.
Bombay is a short and sweet English name for the city.
Though this seems to be all right, it is wrong.
The correct way would be to put the word Bombay within a single
quotation mark to make it clear that it is about the word Bombay we are referring
to and not the city.
‘Bombay’ is a short and sweet English name for the city.
All this discussion should clear the air around these two words of
mathematics and meta-mathematics.
All the work done by the mathematicians including their theorems, proofs
and conjectures is mathematics.
All the description and discussion about what they do, in short
all my bedtime stories on mathematics is meta-mathematics.
The distinction between these two is of high importance for
mathematicians.
Differentiating these two apart was critical for mathematicians to
show the logical structure of mathematical reasoning.
It allowed the creation of a formalized calculus where signs and
symbols were bereft of any meaning.
This allowed getting rid of all the concealed assumptions and
hidden meanings.
Moreover, exact definitions and rules of operation and logical deduction
were made explicitly clear and strict.
This whole idea was Hilbert’s baby and it was on this distinction
of pure mathematical logic from the descriptive mathematics that he was hoping
to construct the system of absolute proof.
The system of formal calculus would analyze signs, symbols,
combine them to form formulas and from these formulas derive further formulas.
Using explicitly stated rules of operation exhaustive and careful
study of expressions derived from a particular axiomatic system would be
studied and to make sure that contradictory result were not arrived at.
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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