July 06, 2019 Saturday
Bedtime Story
What Drove Wittgenstein to Restlessness
So if a statement is a tautology it is
true, if a contradiction false and if none of the above then it can be placed
into the category of neither.
In his words in Tractatus
Logico-Philosophicus (first published in German in 1921), “A tautology’s truth
is certain, a proposition’s possible, a contradiction’s impossible.”
But in order to achieve this he needed to
put this in a form a primitive proposition (the most basic logical axiom) which
had the very essence of this statement and then that primitive proposition
would be the basis of all logic that would ensue.
In his words:
“The big question now is, how must a system
of signs be constituted in order to make every tautology recognizable as such
in ONE AND THE SAME Way?
This is the fundamental problem of logic!”
Just see from the passage below how unhappy
and deeply unsatisfied he was with the then current understanding of science
and mathematics (remember this was all after Newton, Einstein’s Annus mirabilis
papers, Boltzmann, Gottlob Frege but just about nine years before Gödel would
unleash his incompleteness theorems):
“The whole modern conception of the world
is founded on the illusion that the so-called laws of nature are the
explanations of the natural phenomena.
Thus people stop today at the laws of
nature, treating them as something inviolable, just as God and Fate were
treated in the past ages.
And in fact both were right and both wrong;
though the view of the ancients is clearer insofar as they have an acknowledged
terminus, while the modern system tries to make it look as if everything were
explained.”
In short, to Wittgenstein the physical laws
of nature discovered even by the greatest of the physicist such as Maxwell,
Faraday and others were not resting on solid foundations of mathematical logic.
So obsessed was Wittgenstein with this
problem and so important it was for him that he felt if he failed to address it
and come to its solution he should end his life.
A negation of a tautology or a logical
axiom is unsatisfiable and thus tautologies are the perfect logical axioms.
If I recall correctly we did study
tautology to a far greater extent than tonight during my bedtime stories on
mathematical logic.
Now we shall move on to the non-logical axioms
which seem to a paradoxical terminology since after all axioms are supposedly
self-evident truths that do not even need to be proved.
It would also seem –at least apparently -
to make the entire edifice that is built upon them to be illogical and thereby
fallacious (probably to the delight of True-Believers who would wish nothing
more than that).
As we shall see things really aren’t so
depressing and mathematics has learnt to cope well with such discrepancies
without bringing down the entire establishment.
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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