April 17, 2017 Monday
Bedtime Story
Understanding Hilbert's Meta-mathematics
Formalizing mathematics is almost like going to its most basic
roots and starting afresh.
It is as if one is rewinding the centuries and millennia of
evolution of mathematical notations, discarding all of it and starting anew
with fresh symbols where only the few initial axioms and rules of manipulation
reign supreme.
This kind of putative meaning is what David Hilbert called
“meta-mathematics” or about the mathematics.
Hilbert made an attempt to demarcate mathematics from
meta-mathematics.
Mathematics according to him should merely be a system of
meaningless signs.
Meta-mathematics on the other hand states meaningful statements
about mathematics.
Essentially Hilbert wanted to acknowledge that mathematics can be
looked at in two ways; one is from within the system and one from without the
system or from outside the system.
A simple example will clearly illustrate this point.
The proposition that “5 + 5 = 10” is mathematics.
The proposition “‘5 + 5 = 10’ is valid” is meta-mathematics.
Another statement that belongs to meta-mathematics is as follows:
“If the inequality signs ‘>’ or ‘<’ are to be used in a
mathematical formulas, then on either side of these signs there has got be
mathematical expressions.”
To further illustrate the idea of meta-mathematics, consider these
following three equations.
y = y
1 = 1
1
1
Now all these three formulas belong to mathematics.
Yet, the statement:
‘y’ is a variable
Belongs to meta-mathematics.
Similarly the statement given below belongs to meta-mathematics.
The formula ‘1 = 1’ can be derived from the formula ‘y = y’, as
‘y’ is a variable and ‘y’ can be replaced with 1 on both the sides to get the
result.
Another statement of meta-mathematics would be:
‘1
1’ is not a theorem of a
formal system Y.
First of all, this equation cannot be derived from the first
equation.
Secondly, this statement is telling us something about the
mathematics involved here.
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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