April 10, 2017 Monday
Bedtime Story
The Formality of Mathematics is Crucial
The formality of mathematics lies in the fact that the validity of
mathematical conclusions are grounded in the structure of statements rather on
any subject matter.
So a mathematical statement should never be meant to be taken as a
verdict for share market, or fruits or the physical universe.
It must be hammered into our minds once and for all that a
mathematician is not concerned either with the truth of the assumptions or the
truth of the arrived conclusions.
All that a true mathematician needs to care is whether the
purported conclusion arrived from the stated assumptions is a necessary logical
consequence of the stated assumptions.
This is the crux of mathematics.
Let us consider the example of the great mathematician David
Hilbert himself.
In 1899, David Hilbert in his book “The Foundations of Geometry”
had proposed 20 axioms.
Yet, even before these 20 axioms, Hilbert listed three primitive
terms and three primitive relations.
The three primitive terms were point, line and plane.
The three primitive relations were:
[1] Betweenness (note that such a word does not exist in English
grammar)
[2] Lies on
[3] Congruence
Now just because these terms are in English and we seem to be
familiar and understand them, we tend to get very comfortable with them.
This is exactly what Hilbert opposed.
Hilbert made it clear that when it comes down to mathematics, all
the familiarity with these terms must be ignored and all the “meanings” to be
shed off.
The only meaning that should be associated with these terms is the
one assigned to them by the definitions of the axiom system.
So the point or the line as defined by the strict rules of
mathematics is not what we understand or intuit by those very same words.
Mixing up the “point” of Hilbert’s definition with the
understanding of point that we take for granted will give rise to paradoxes.
This is exactly what Bertrand Russell meant when he said:
“Pure mathematics is the subject in which we do not know what we
are talking about, or what we are saying is true”.
Many of us will find this very surprising as mathematics is often
associated with ultimate truth or reality.
Such was the belief even at the beginning of the nineteenth
century held by many mathematicians.
And that is what restricted their thinking.
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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