April 30, 2017 Sunday

Bedtime Story 

Basics of Propositional Calculus or Zeroth-order Logic

Propositional calculus is also known as the sentential calculus and also as the zeroth-order logic.

Notice that the word ‘calculus’ and ‘logic’ are very often used interchangeably.

Propositional calculus is developed with the help of two kinds of signs, one being variable and the other being constant.

The variable signs include letters like p, q, r… but they can have sentences replace them.

These variable signs are also known as “sentential variables”.

The constant signs are either connectives or signs of punctuation.

They are otherwise known as “sentential connectives”.

Examples of sentential connectives are as follows.

‘ ∨’ is a symbol for ‘or’

‘ ⊃’ stands for ‘if…then…’

‘.’ Stands for ‘and’

Punctuation marks can be assigned to parenthesis brackets namely   ‘ ( ‘ or ‘ ) ‘.

Following this, the Formation Rules determine how these elementary signs can combine and how sentences or formulas can be formed.

Let me give few examples what kind of formation rules were set up for this formal system.

It isn’t that difficult as you would expect from mathematics.

If a symbol ‘S’ is a formula, then it negation ~(S) is also a formula by default, assuming the tilde sign stands for negation.

This tilde sign or grapheme (smallest unit of a writing system of any given language) in present modern day English stands for “approximately”, “about” or “around”.

Further, if S1 and S2 are formulas, then (S1) ∨ (S2), (S1) ∧ (S2), or (S1) ⊃ (S2) are also formulas.

Same thing holds true with sentential variables.

So if ‘p’ is a formula, then its negation ~(p) is also a formula.

Similarly, these formulas (p) ⊃ (q), ((q) ∨ (r)) ⊃ (p) are also allowed.

Now let me show you what is not allowed.

‘(p)(~(q))’ cannot be a formula.

This is so because though both ‘(p)’ and ‘~(q)’ are formulas by themselves, in the above denied formula there is no sentential connective between them.

Similarly a formula like ‘((p) ⊃ (q)) ∨‘ is not allowed.

It is simple and I want to think for yourself and tell me why.

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

April 29, 2017 Saturday

Bedtime Story 

How Number Theory Was Formalized in Principia Mathematica 

Formalization of number theory as carried out in Principia was worked out in four steps:

[1] Establishing a whole new vocabulary.

This was done by conceiving a long and comprehensive list or catalog of whole new signs and symbols to be used in this formal system.

[2] Preparing a whole new grammar or the Formation Rules.

The formation rules would determine which strings of symbols are acceptable to combine and are syntactically valid.

Syntax as you might know from any language that you may have learnt later in your life (say Russian in my case), is concerned with the rules for construction or transforming of symbols and words of a language.

This is all the transformation rules is concerned about.

It has no bearing on the meaning or the semantics of the language.

Syntax in linguistics governs the order in which verb, subject and object should be arranged to construct a well-formed sentence.

Syntax also describes the way punctuation needs to be used and hence syntax is a subset of grammar.

Grammar encompasses all the components of language such as morphology, semantics, phonology and of course, syntax.

Example being ‘I liked the apple’

So even though you can say ‘The apple I like’ and the meaning will be conveyed, it is not following the rules of syntax and hence is not a well-formed sentence.  

[3] Transformation Rules or the Rules of Inference.

I had written to you about the three famous rules of inference, namely modus ponens, modus tollens and contraposition.

These transformation rules describe the structure of formulas from which other formulas are derivable.

[4] Selection of Axioms, or Primitive Formulas

These form the foundation of the whole formal system and the starting point of further operations.

Any formula or formulas that are derived from these primitive formulas using the transformation rules would go on to become the theorems of the system.   

In such a system, a proof of a theorem is the demonstration of the sequence of formulas that started out from the axioms and led to the generation of a certain derived theorem using the transformation rules.

The next to come following these four elementary stuff is the propositional calculus.

We shall take up the story of propositional calculus 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

April 28, 2017 Friday

Bedtime Story 

Principia Mathematica and its Connection with Kurt Godel's Incompleteness Theorems  

Many mathematicians were still not convinced of the notion of Frege-Russell thesis that mathematics could be reduced to pure logic.

Perhaps that is the reason why logic eludes most textbooks of mathematics.

Yet Principia was definitely a major advancement as far as the idea of proof of consistency in mathematics was concerned.

Principia provided three things for the other mathematicians to work upon:

[1] It provided a completely novel and very comprehensive system of notation with the aid of which all the statements of number theory could be formally coded in a standardized manner.

[2] The rules of inference were made very explicit.

In short, Principia was the first ever formal system, almost a kind of experimental device or an instrument that could be used to investigate the entire system of number theory as a pure un-interpreted calculus.

The word un-interpreted is very significant.

Principia created the un-interpreted calculus that had this system of meaningless novel symbols whose strings combined and transformed in explicitly and precisely stated rules of operation.

An uninterpreted function in mathematical logic is one that has no other property than its name and n-ary form.

This arity of a function is the number of arguments that the function takes.

No wonder it is an impossible book to read and comprehend for most average and even intelligent apes.

Yet Principia Mathematica was the prototype, the first ever system of formal calculus ever devised, particularly when it came to number theory.

As I had said earlier, George Boole had something similar in mind that he could never realize for himself.

Yet, he had laid enough foundations for other mathematicians to build up upon it.

If you recall the title of Kurt Gödel’s landmark paper, its ends with the phrase “Principia Mathematica und verwandter Systeme” which translates to “Principia Mathematica and related systems”.

So Kurt Gödel in his paper was referring primarily to this prototypical formal calculus system devised by Whitehead and Russell but secondarily also to any such systems that may or would came out later.

Now I shall attempt to try out a more difficult feat; to take a portion of Principia and show how the formalization of a deductive system was achieved.

And still further, how the absolute consistency of such a system can be established.

Only with this knowledge in hand (or rather in our brain), can we even begin to understand Gödel’s genius.

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

April 27, 2017 Thursday

Bedtime Story 

George Boole Coverts Aristotelian Syllogism Into Notations 

Here is an example how Boole converted Aristotelian syllogism into symbols and notations, thereby revolutionizing logic.

Aristotelian syllogism applies deductive reasoning to arrive at a conclusion based on two or more propositions.

So here we have two propositions followed by an Aristotelian conclusion.

Only some men find mathematics interesting.

No woman finds mathematics interesting.

Conclusion: Nobody interested in mathematics can be a woman (Mind you, this is not an absolute truth but a conclusion based on the top two propositions).

In Boolean symbolism, this can be written as:

 M  ⊂  ma   (men belong to group of math lovers)

 W  ⊂  -(ma)   (women belong to group of non-math lovers)

 ∴ ma ⊂ -(W)    (therefore math lovers cannot be the women group)

The dash represents the negated relationship.

There is another way of this Boolean symbolism wherein the two characteristics can be combined.

Let me show how that goes.

M-(ma)  = 0 (the group that does not love mathematics and has men does not exist). Here an assumption is being made that all men love mathematics.

Wma = 0 (the group that has women and loves mathematics does not exist)

∴ -(M)ma = 0 ( ∴ the group that has no men and math lovers cannot exist) 

The above two examples should give you quite a fair idea what Boole was trying.

What George Boole started and never completed fully was brought to its grand finale in 1910 by Whitehead and Russell through their chef d’oeuvre Principia Mathematica.

This first “complete” work of true and pure mathematical logic attempted and nearly-successfully demonstrated, (and thereby making it a very tedious book), that all number-theoretical notions can be expressed and defined in terms of pure logic.

It made a bold claim that all the axioms of number theory can be derived from very few propositions that can be given the label of logical truths.

Hence Principia Mathematica was almost Hilbert’s dream come true; it had reduced the problem of consistency of mathematics, particularly the consistency of number theory to the consistency of formal logic.

So now all that remained for the absolute proof of the consistency of the axioms of number theory was to show that the axioms of logic were consistent.

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

April 26, 2017 Wednesday

Bedtime Story 

In the Most Endearing Words of Mon Ami, "Formal Logic Subsumes All Math" 

In the story of mathematics, written by far better and greater writers and mathematicians then me, logic always seems to get left out.

Great emphasis is laid to geometry, algebra and number theory and even applied physics, but somehow logic tends to get no space, not even a mention.

For Kurt Gödel, mathematical logic was “a science prior to all others, which contains the ideas and principles underlying all sciences.”

According to my best friend and the most voracious reader and most knowledgeable man of mathematics that I know of personally, “Formal Logic subsumes all math”.

The honor for codifying logic for the first time is attributed to Aristotle who was the tutor to Alexander the Great till the mighty emperor reached 16.

Aristotle taught the future mighty emperor somewhere around the period of 340 BC at the “Shrine of the Nymphs” in Mieza, a village in ancient Macedonia.

His work Prior Analytics on deductive reasoning is not only the first of its type, but stood to be the only one till the dawn of that incredible nineteenth century.

In his Prior Analytics, Aristotle presents and identifies valid and invalid form of arguments called syllogism.

Syllogism as taught by Aristotle is an argument that consists of at least three statements.

There should be at least two premises and in the end, a conclusion.

I will not delve too much into Aristotelian logic but suffice to say that it remained the corner stone of logic till the entry of George Boole and his seminal works “The Mathematical Analysis of Logic” (1847) and “The Laws of Thought” published in 1854.

George unfailingly and unwaveringly accepted Aristotelian logic, but he wanted “to go under, over, and beyond” it in three very diverging ways:

1. To provide and arm Aristotelian logic with mathematical foundations

2. To extend the class of problems it could treat

3. To expand its range of applications 

For this, Boole had to develop a special type of algebra for logic with specific new notations never ever tried before.

Boole in short found a way to convert Aristotelian syllogism or arguments into symbols and notations and thereby inventing symbolic logic.

This can be illustrated with a simple example.

We will take up the example 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

April 25, 2017 Tuesday

Bedtime Story 

The Third Rule of Inference: Contraposition

The third rule of inference is called contraposition.

In general English, the word ‘contrapositive’ refers to either converse or inverse.

In logic, contraposition applies to conditional statements.

For instance,

If the sky is dark and cloudy, then it is bound to rain.

Conditional statements have the “if…then…” relationship.

In other words, it has an antecedent and consequent.

In our example, dark and cloudy sky is the antecedent condition and the rain is the consequence.

The contrapositive of a conditional statement has its antecedent and consequent inverted and flipped.

So the contrapositive of the statement that we took as an example would be:

If it is raining, then the sky ought to be dark and cloudy.

The third rule of inference i.e. contraposition states that a conditional statement is logically equivalent to its contrapositive.

So if we again refer back to our example, what the contraposition rule says is this:

The statement ‘If the sky is dark and cloudy, then it is bound to rain’ is logically equivalent to ‘If it is raining, then the sky ought to be dark and cloudy’.

All these examples should suffice to give a general idea of the rules of inferences that are used in logic.

You will notice that all of these were never taught yet they were all implied throughout our education in nearly all the subjects, except may be in the Bible class that I had the misfortune to painfully endure.

Let us now go back to the Euclid’s theorem on the infinity of primes.

In the proof, if you seek to go back few nights ago, was a statement:   

“Then q can either be a prime or not be a prime.”

The rule of inference that was used to get this statement is known as the “Rule for Substitution for Sentential Variables”.

According to this rule, a statement can be derived from another that contains a variable by substituting any statement for each occurrence of a distinct variable.

In our case, the variable was p and the distinct variable that we replaced it with was q.

I know it is banal and may be overdone, but advances in formal logic brought out into open the rules that were being applied almost subconsciously for ages.

It can even be said that mathematicians and logicians of ages have been using reason for thousands of years without explicitly being aware of the underlying principles.

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.

                           

  

                

             

Advertisements

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

April 24, 2017 Monday

Bedtime Story 

Greek Stoicism and Logic

To some extent, more or less, correctly or erroneously, wantonly or unknowingly, most Greek philosophers took interest in logic.

Diodorus Cronus, a Greek philosopher who lived somewhere around 300 BC, was a Stoic philosopher who is said to have first developed propositional logic.

The tenants of Stoicism are fundamentally materialistic; in the sense that they resort least to magical beliefs, spirituality and all other kinds of false comforting and irrational yearnings.

Once you get rid of all this superfluous dust and opacity that mask the reality, one is naturally left with just logic and observations (Experiments came a lot later).

Though the word stoic as used currently refers to someone who represses feelings and is indifferent to pain, or rather someone who is nonchalant to pain, grief, or joy or pleasure, the stoic philosophy as a whole is quite different from this simple understatement.

It is something else about stoicism that appeals to me.

Stoicism, in my view, is a way of life which involves practice of logic, attaining knowledge through the use of reason (and through reading of the works of others), self-dialogue, contemplation of death, training to remain in the present moment and daily reflections on everyday problems and investigating its solutions.

Stoics place highest premium on the following four virtues:

1. Wisdom (not in the sense of multiple degrees or letters)

2. Courage (not in the sense of battling lions or warriors)

3. Justice (not in the sense of court and legal jurisprudence) and

4. Temperance (moderation or voluntary self-restraint that in turn amounts to strong discipline of character)

Besides their philosophy on living, the stoics take great interest in understanding nature.

It is obvious that the Greek stoics’ understanding of nature was primitive and had to be so because they simply did not possess the technology to investigate nature.

They did not even possess the number 0, at least not in the sense the way we use it!

Yet I find that at the very least, they paid a very minimal emphasis on unnatural, magical or superstitious.

That is a huge progress considering the beliefs an average modern human ape holds in spite of accretion of a huge wealth of knowledge concerning the origins of life, the universe and complexity from simplicity in general. 

I shall now halt my discourse on stoicism with this much and continue my story on the rules on inference.

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.

                           

  

                

             

Advertisements

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