August 10, 2018 Friday
Bedtime Story
John D. Barrow on Impossibility in Mathematics
Tonight we shall continue with the abstract
from the book ‘Impossibility: Limits of Science and Science of Limits’ written
by English mathematician John D. Barrow.
“Only in this way could a problem be
closed, because:
‘To arrive infallibly at something in this
matter, we must therefore follow another road.
We can give the problem such a form that it
shall always be possible to solve it, as we can always do with any problem.
Instead of asking for a relation of which
it is not known whether it exists or not, we must ask whether such a relation
is indeed possible…
When a problem is posed in this way, the
very statement contains the germ of the solution and indicates what road must
be taken; and I believe there will be few instances where we shall fail to
arrive at propositions of more or less importance, even when the complication
of the calculation precludes a complete answer to the problem.’
Interestingly, Abel at first thought he had
found a solution for the degree five problem.
But before his paper could be published he
found a mistake and, as a result, started to see the problem in a completely
different light.
This vital change of perspective led
ultimately to his proof of the impossibility of the very result that he thought
he had once established.
It appears that Abel’s work did not give
rise to deep philosophical and theological speculations about why it was that
solubility stopped at degree 4.
Clearly, it could have done.
Equations of higher degree certainly have
solutions.
We can solve some of them by inspired
guesswork, approximations, and so forth (as could have mathematicians in Abel’s
days) but Abel’s proof seemed to have opened up a gap between what human
reasoning could achieve and what was true in the transcendental world of mathematical
truths, or in the mind of God.
Many of the philosophical issue raised by
Gödel’s theorem could have been stimulated by these discoveries that there are
limits to our ability to solve algebraic equations and to the scope of
ruler-and-compass construction, but they were not.
There are many analogies between the two
lines of enquiries.
Both Abel and Gödel attacked problems that
everyone expected could be solved.
Both displayed remarkable flexibility of
mind in establishing an impossibility theorem: Abel did a last minute
about-turn after thinking he had got a ‘possibility’ theorem, and Gödel had
actually been proving the completeness of smaller logical systems than
arithmetic (this was his doctoral thesis work) just months before announcing
his impossibility theorem for arithmetic.”
This abstract highlights the point that in
the history of mathematics both before and after Abel and Galois, there were
instances of certain types of impossibilities detected by the most brilliant minds.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
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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