August 09, 2018 Thursday
Bedtime Story
The Three Problems of Mathematics that Express its Limitations
Why is there no general formula for the
roots of fifth or higher degree polynomial equations in terms of the
coefficients of the polynomial using only the five algebraic operations of
addition, subtraction, multiplication, division and radicals (square roots,
cube roots and so on)?
Please read back this question over and
over again and then try to get a glimpse of the minds that are not only capable
of raising such a kind of question but spending their lifetime trying to crack
the solution.
One has to be passionately obsessive (I thought
of using the word insane though to me it is always the average ape that appears
to be the one who is most irrational and lacking reason) about abstract algebra
to be capable even to be showing the barest of interest in raising such an purposeless
question.
An average ape like me would not even know
from where to even begin attacking such a problem were it posed to me as a
challenge with an award of say 100,000 dollars which I would love to acquire.
While Ruffini and Abel together provided a
definite proof for this fact, they did not answer the why of the question.
Furthermore both of them were very well
aware, as perhaps mentioned earlier, that some quintic equations do have a
solution such as (x-1)5 = 0
So it is clear that not all polynomials of degree
higher than 4 do not have roots that can’t be expressed in radicals.
So what is it, (if anything at all), that
determines whether a given quintic or a higher polynomial equation could be
determined to be solvable?
Framed this way, the problem is reminiscent
of the famous undecidable problem or the decision problem which perhaps we may
take up later some night.
In fact, to the English mathematician and
mathematical physicist, John D. Barrow the works of Ruffini, Abel and Galois
appear very similar to the limitations of ‘compass and straight-edge
constructions’ of the past and in the future, to Gödel’s incompleteness
theorems.
All these three problems, namely ‘compass
and straight-edge constructions’, Gödel’s incompleteness theorems and
unsolvability of quintics and higher polynomials are a common expression of
some kind of fundamental limitations that is inherent in mathematics which is considered
to be the most sound of all sciences.
Let me quote an extract from one of the
several books written and published by Barrow (he is yet another prolific English
writer of popular science and mathematics books with at least 22 of them under
his cap) ‘Impossibility: Limits of Science and Science of Limits’.
“Abel, aided by the work of Galois, finally
established an impossibility theorem.
Later, he discussed the general question of
solubility in mathematics and, sounding a little like Hilbert many years later,
he realized that any attempt at complete understanding of mathematical problem
must have two means of attack; one to find explicit solutions; the other, to
discover whether solution is possible or not.”
We shall continue with the abstract from
the book ‘Impossibility: Limits of Science and Science of Limits’ in the nights
to come.
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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