April 19, 2017 Wednesday
Bedtime Story
David Hilbert Disliked Infinity and Kept It Away
Yet, even in this exhaustive and highly disciplined Hilbert’s
program, the fear of infinity (thanks to Georg Cantor) always lingered.
Hence Hilbert made it very clear that in all the axiomatic systems
that would be tested out using this formal calculus, no reference will be made
to infinites, either to infinite numbers, or to sets or to formulas.
Infinities were outlawed, banished and made an outcast.
Currently I am reading a book “Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics” whose author is
John Derbyshire who studied mathematics at the University College London before
immigrating to America and becoming a computer programmer (at the Wall Street)
and later on a full-time writer.
In the book he categorically states that the term infinity is
meaningless in the sense that it should NOT be taken as a number.
Let me quote him and give you his take on infinity.
“The notions of the infinite and the infinitesimal created serious
problems in math during the early nineteenth century, though, an eventually
they were swept away altogether in a great reform.
Modern standard analysis does not admit these concepts.
They linger on in the vocabulary of mathematics…the usage,
however, is only a convenient and imaginative shorthand for more rigorous
concepts.
When
I say that the harmonic series adds up to infinity, what I really mean is given
any number S, no matter how large, the sum of the harmonic series eventually
exceeds S. See―No “infinity.”
The
whole of the analysis was rewritten in this kind of language in the middle third
of the nineteenth century.
Any
statement that can’t be so rewritten is not allowed in modern mathematics.
Some
people ask me – What is infinity divided by infinity?
I
can only reply, “The words you just uttered do not make sense.
That
was not a mathematical sentence.
You
spoke of ‘infinity’ as if it were a number.
It
is not.
You
may as well ask, ‘What is truth divided by beauty?’
I
have no clue.
I
only know how to divide numbers.
‘Infinity’,
‘truth’, ‘beauty’―those are not numbers.”
I
end the quotation from the book here.
So Hilbert’s fear of infinity or rather infinities is justifiable.
So the procedures that were applicable to his proposed new formal system
would be “finitistic” in his words and the proof of consistency shown or
derived would be “absolute”.
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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