November 22, 2016 Tuesday
Bedtime Story
Galileo's Paradox was published in his last book: The Discources and Mathematical Demonstrations to Two New Sciences (1638)
Galileo's Paradox was published in his last book: The Discources and Mathematical Demonstrations to Two New Sciences (1638)
We all have a vague intuitive sense of the words finite and
infinite.
Yet we are all unaware the emphasis on rigor and precision of
definitions that pure mathematicians demand of their subject.
Finite and infinite have troubled nearly all the mathematicians of
the past including the pre-Socratic Zeno of Elea (somewhere around 450 B.C.)
and Galileo.
A famous mathematical paradox on infinity has been named after
Galileo that describes the essence of trouble one get into while dealing with infinity.
Galileo had published this paradox in the form of a dialogue
between 3 fictional men who go by the name of Simplicio, Salviati and Sagredo.
The book “The Discourses and Mathematical Demonstrations Relating
to Two New Sciences” was Galileo’s final treatise published in 1638 that was
gleefully banned by the Roman Inquisition.
The Christianity of Medieval ages was like the Islam of today, far
different and ominous from what it portrays itself at present.
In the dialogue, Simplicio first states that some numbers are
squares (such as 4 and 9 are of 2 and 3) whereas other are not (such as 5 or
13).
Salviati then states that it must follow that the sum of all the
squares and non-squares ought to me more than just the squares.
Salviati further states that it is also true that very square has
to have a root and hence every square will have a corresponding root.
Salviati then poses a question.
What if I were to ask how many roots there are?
The roots must be same as there are squares.
Also the number of squares must be same as there are number since
every number can be squared.
But this goes against our first statement which said that there
are more numbers than there are squares as not all numbers are squares.
Sagredo (playing the role of an idiot like me) then asks what can
be inferred from this paradox.
Salviati (playing the role of wise man) says we can conclude the
following:
First, that the numbers are infinite, their squares are infinite
and so are their roots.
Secondly, the terms such as “equal”, “greater” and “lesser” are
not applicable to infinite quantities but only to finite.
Cantor disagreed with Galileo and like him, nearly 400 years
later, he was brutally disparaged, maligned, vilified and castigated so much so
that he was thrown into bouts of depression, madness and eventually
suicide.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night and my fellow cousin ape.
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, may I
suggest this large collection of Kids Songs:
The book has 4 parts in the form of 4 days and in each day a different aspect of physics is discussed
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