April 01, 2017 Saturday
Bedtime Story
Understanding Formal System (aka Logical Calculus)
What is a formal system in mathematics?
A formal system of arithmetic like a game of chess is a deductive
apparatus with a set of axioms (like the pieces on the chess board) with very
rigid rules of symbolic manipulation in inference (exactly like the rules
permitted for each piece on a chess board).
A formal system is highly mathematical but it need not be always
so.
Way back between 1664 and 1665, the Dutch philosopher Baruch
Spinoza wrote a book that is considered a masterpiece in Western Literature.
This outstanding work titled “Ethics, Demonstrated in Geometrical
Order” is built exactly on the principles of Euclid’s Elements and can be
considered a formal system in philosophy.
In Ethics, Spinoza starts with basic definitions and axioms, and
from there goes on to derive propositions and corollaries.
It is not very relevant how truthful his conclusions were; the
fact remains that it can be truly considered a non-mathematical formal system.
I adore Spinoza for many reason, but one important reason was that
though he earned a living by grinding away lenses and making optical
instruments, in private he was a scholar of very high caliber.
He corresponded with top philosophers and scientists of his times
including Leibniz.
Yet he sought no fame, turning down rewards, honors and teaching
positions throughout his life.
Let me return to the topic of formal system.
Any formal system will have the following four components:
[1] A finite set of symbols
[2] A grammar that allows construction of well-defined formulas
from the symbols
[3] A set of axioms (the axioms themselves are well-defined
formulas)
[4] A set of inference rules (also called transformation rules)
One of the most common or basic inference rule is the modus ponens or modus ponendo ponens.
It is Latin which when translated to English means – the way that
affirms by affirming.
What it says is this: A implies B and A is asserted to be true, so
therefore B must be true.
In the notation of present mathematical logic, this statement can
be abbreviated to:
A
→ B, A ⊢
Modus ponens is one of the most common arguments forms in logic
that even we as average apes use intuitively without ever having been taught.
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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