July 07, 2019 Sunday
Bedtime Story
The Basis of Arithmetic is ZFC which are Non-Logical Axioms
Euclidean geometry is still as “true” as
ever in spite of its Non-Euclidean counterpart omnipresently breathing down its
neck.
Non-logical axioms are formulas or
assumptions that are limited to specific theories of mathematics.
Non-logical axioms are starting assumptions
of a particular structure such as group theory or Euclidean geometry.
In this sense non-logical axioms are not
tautologies meaning that they are only true in one specific interpretation and
only one.
Thus non-logical axioms are best considered
as assumptions or postulates and not tautologies since their “self-evidence-ness”
disappears when applied or viewed from different systems or mathematical structures.
So all modern mathematical theories that we
use all the time are built upon non-logical axioms which in mathematical
discourse are simply referred to as axioms with “non-logical” term usually left
out but assumed to be present by the very few who study mathematics deeply.
It is understood that they are not true or
self-evident in any absolute sense but are mere assumptions to built up a logico-deductive
system.
Unlike geometry or rather Euclidean
geometry whose axioms are often taught and clearly specified to children there
are several branches of mathematics such as arithmetic (very widely used),
real-analysis and complex-analysis (less used directly by most apes) whose
axioms are never specified or made explicit.
Did you ever recall yourself being taught
any axiom of arithmetic?
Were you ever taught the theoretical basis
of natural numbers?
Were you ever taught or proven why 0 = 0?
Are any of our school children familiar with
the name of Giuseppe Peano or for that matter even the school teachers or the
private tutors who charge them a bomb for their training?
Are you aware of the existence of the
set-theoretic definition of natural numbers?
In his Principia Mathematics Bertrand
Russell took several hundred pages to prove the validity of the proposition 1 +
1 = 2 starting from primitive symbols, strings of symbols, primitive ideas,
primitive propositions and constructions.
Yes, 1 + 1 = 2 is a proposition that
requires proof starting from propositional logic and logical axioms!
Actually all of arithmetic, real and
complex analysis are based on the axioms of Zermelo-Fraenkel set theory with
choice or ZFC (developed by Ernst Zermelo and Abraham Fraenkel in 1922 and 1925
based on their interest in set theory and axiomatizing it) which came centuries
later after the natural number system had been widely in use and practice.
Besides Zermelo and Fraenkel others too
have developed their own axioms of set theory on which lie the foundations of
several areas of mathematics.
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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