June
21, 2017 Wednesday
Bedtime
Story
Distinguishing
In
formal logic there exists a difference between the terms ‘the numeral for x’
and ‘the number x’
The
word numeral denotes symbol that represents a specific number.
So
numeral is a representation that can be written down, erased or copied.
A
number on the other hand is a concept that numeral represents an abstraction
that literally cannot be written down or erased.
Let
me clarify this even further with an example.
When
you make a statement about a month and say that the month of June has thirty
days.
The
symbol thirty is a numeral but the property of thirty days in a month is a
number.
This
point will bear out if I will point out the fact that you can substitute the
wordy thirty in the sentence with the Arabic number 30 or the Roman number XXX.
So
we can replace one numeral with another as they are all signs and symbols for
one number concept.
So
we can replace one numeral with another but cannot replace it with number or
concept.
So
similarly in a formula that contains a numerical variable, we can only replace
it with a numerical expression (such as ‘0 x 0’ or ‘ss0 x ss0’), and not with a
number.
This
is why, when we were speaking meta-mathematically about the sub function, it
was stated “All this new number asks of you is to take the formula of the old Gödel
number x and wherever in it you come across the presence of the numerical
variable ‘y’, substitute it by the numeral for x”, rather than number for x.
This
is just to bring into record the linguistic precision that is so highly
required in formal logic.
Now
once we have agreed to the sub function being recursive primitive, then as per
the Correspondence Lemma, there has to be a formal expression inside the
Principia that exactly mirrors it.
Let
us label this as ‘Sub (x, 17, x)’, the S here being capital letter, thereby
differentiating this typographical concept from its informal arithmetical
counterpart.
It
should be borne in mind that while the term
‘sub
(243,000,000, 17, 243,000,000)’ would denote an actual number, the term
‘Sub
(243,000,000, 17, 243,000,000)’ would represent a string inside the Principia.
Just
in case if you are wondering where this huge number 243,000,000 turned out
from, you need to go back some nights back.
We
had deconstructed this Gödel number to discover that it is represents the
formula ‘0=0’.
Although
strictly speaking, the term
‘Sub
(243,000,000, 17, 243,000,000)’ is a string and hence is meaningless, it makes
sense to consider it as having meaning.
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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