April
29, 2017 Saturday
Bedtime
Story
How Number Theory Was Formalized in Principia Mathematica
Formalization
of number theory as carried out in Principia was worked out in four steps:
[1]
Establishing a whole new vocabulary.
This
was done by conceiving a long and comprehensive list or catalog of whole new
signs and symbols to be used in this formal system.
[2]
Preparing a whole new grammar or the Formation Rules.
The
formation rules would determine which strings of symbols are acceptable to
combine and are syntactically valid.
Syntax
as you might know from any language that you may have learnt later in your life
(say Russian in my case), is concerned with the rules for construction or
transforming of symbols and words of a language.
This
is all the transformation rules is concerned about.
It
has no bearing on the meaning or the semantics of the language.
Syntax
in linguistics governs the order in which verb, subject and object should be
arranged to construct a well-formed sentence.
Syntax
also describes the way punctuation needs to be used and hence syntax is a
subset of grammar.
Grammar
encompasses all the components of language such as morphology, semantics, phonology
and of course, syntax.
Example
being ‘I liked the apple’
So
even though you can say ‘The apple I like’ and the meaning will be conveyed, it
is not following the rules of syntax and hence is not a well-formed sentence.
[3]
Transformation Rules or the Rules of Inference.
I
had written to you about the three famous rules of inference, namely modus
ponens, modus tollens and contraposition.
These
transformation rules describe the structure of formulas from which other
formulas are derivable.
[4]
Selection of Axioms, or Primitive Formulas
These
form the foundation of the whole formal system and the starting point of
further operations.
Any
formula or formulas that are derived from these primitive formulas using the
transformation rules would go on to become the theorems of the system.
In
such a system, a proof of a theorem is the demonstration of the sequence of
formulas that started out from the axioms and led to the generation of a
certain derived theorem using the transformation rules.
The
next to come following these four elementary stuff is the propositional
calculus.
We
shall take up the story of propositional calculus in the nights to come.
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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