June 16, 2017 Friday

Bedtime Story 


String of Symbols in a Formal Calculus Should Not Be Assigned "Meaning"

Last night we were juxtaposing equivalent arithmetical statements that lie outside the Principia Mathematica to those that lie within it.

It is critical to get this concept straight as it will form a vital part of the argument in the actual proof.

The example that we had discussed last night was more abstract and may have been difficult to register.

So today we shall take a much simpler example that would register better with our non-mathematical, irrational brains.

Let us consider the following mathematical statement:

“Two plus two is not equal to five”.

This is an English statement that happens to be true.

Another way of writing it more mathematically would be:

2 + 2 ≠ 5 

This is mathematical but not formal.

Why this is not formal?

This is because the notations used here have meanings.

To make it formal, symbols need to be used that strictly speaking, have no meanings.

That would look like this:

‘~ (ss0 + ss0 = sssss0)’

This is a meaningless string of symbols that happen to be true.

The function of addition is primitive recursive.

That allows us to invoke Correspondence Lemma which in turn mandates that this formula ought to be a theorem of Principia.

Strictly speaking, the theorems of Principia should not be assigned the label true or false.

In this case the arithmetical statement that this theorem expresses happens to be true.

This was a simple example that highlighted an arithmetical statement being presented in both its informal and formal way.

The second point that needs to be stressed is that the theorems of the Principia must not be tagged with the notion of true and false or meaningful and meaningless.

It is only its meta-mathematical counterpart outside the Principia that can be assigned these values.   

Let us go one step further and consider one more example, this time a tad more difficult.

Let us examine prime numbers or the concept of a natural number being a prime.

We shall perhaps examine it 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:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg