May
02, 2017 Tuesday
Bedtime
Story
Kurt Gödel's Criticism of the Notations of Principia Mathematica
Before
I list out the four axioms, just have a look at how some of the primitive ideas
are defined.
Take
for instance negation.
Negation
is defined as follows: “If p is any proposition, the proposition ‘not-p’, or ‘p
is false’, will be represented by ‘~p’”.
Though
most of the notations used in the Principia were invented by Whitehead, some of
them were taken or modified from Giuseppe Peano’s Formulario mathematico (1895)
and some from Gottlob Frege’s Begriffsschrift (1879).
Principia
may have impressed many but there was one man who was intensely critical of the
notations of this seminal tome.
That
was Kurt Gödel and I quote these 3 lines:
“It
is to be regretted that this first comprehensive and thorough-going
presentation of a mathematical logic and the derivation of mathematics from it
[is] so greatly lacking in formal precision in the foundations (contained in #1
to #21 of Principia [i.e. section #1-#5 (propositional logic), #8-14 (predicate
logic with identity/equality), #21 (introduction to relations theory)] that it
represents in this respect a considerable step backwards as compared with
Frege.
What
is missing, above all, is a precise statement of the syntax of the formalism.
Syntactical
considerations are omitted even in cases where they are necessary for the
cogency of the proofs.”
I
end the quote.
This
criticism is also a reflection on the personality of Gödel.
This
was not a timid man daunted and in awe of famous and established.
Anyways,
let us go back to the axioms of which there are four.
I
shall discuss two of them tonight and the other two subsequently.
[1]
(p ∨
This
in simple language means, if either p or p, then p.
Its
equivalent in ordinary language would be something like this:
If
(either Navin is an average ape or Navin is an average ape) then Navin is an
average ape
[2]
p ⊃
This
in simple language means, if p, then either p or q.
Its
equivalent in ordinary language would be something like this:
If
astrology is popular, then (either astrology is popular or alternative medicine
is a thriving industry)
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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