March 31, 2017 Friday

Bedtime Story 

Stating the Two Incompleteness Theorems

The paper had to be translated into English which itself proved to be a hugely cumbersome task.

There were two reasons for it being so.

One for the difficulty and the novelty of the subject per se and secondly and most importantly, for the personality of the author.

One of the translators was Jean van Heijenoort, a Frenchman of Dutch ancestry.  

He described Gödel as the most sedulously punctilious individual he had ever known.

Between them they “exchanged a total of seventy letters and met twice in Gödel’s office in order to resolve questions concerning subtleties in the meanings and usage of German and English words.”

Not only did Gödel took his mathematical work very seriously, but even the translation to English had to be meticulous.

Now let me state his theorems.

Gödel’s incompleteness theorems as is suggested by the plurality of the noun, are not one but two.

Let me first state the two theorems in English (remember, the paper was originally written in German).

First Incompleteness Theorem:

“Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.”

It must be brought to the notice that this theorem in the original paper was published as “Theorem VI”.

The formal system that Gödel took into consideration was Principia Mathematics of Whitehead and Russell.

Second Incompleteness Theorem:

“Assume F is a consistent formalized system which contains elementary arithmetic. Then F ⊬ Cons (F).”

That strange symbol ⊬ between F and Cons (F) stands for “does not prove”.

The term Cons (F) stands for consistency of F.

I will explain in detail what this statement means in detail.

This second theorem in the original 1931 paper was written as “Theorem XI”.

Now having stated these theorems, let me go step by step what these theorem imply.

First of all, these theorems restrict themselves to formal systems of arithmetic.

Now what do we mean by a formal system of arithmetic?

We shall take up this formal system 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

March 30, 2017 Thursday

Bedtime Story 

November 17, 1930 The Incompleteness Theorems Come into Print

Mathematics is totally deductive.

But deductive from where?

It got to start from some place.

This starting point in mathematics are the most primitive assumptions called the axioms.

Its axiomatic origin was giving rise to lots of flaws and inconsistencies in mathematics that was giving men like David Hilbert nightmares.

Anything could be flawed, but not mathematics.

That was the idea then as still many of us are led to believe so even today.

David Hilbert was an optimist who thought that with rigor and determination, these little flaws would be ironed out and mathematics could be built up on solid axiomatic foundations.

Russell had tried to smoothen things out and yet while working on his Principia he had discovered a paradox in set theory that we had discussed in one of our bedtime stories.

Little did Hilbert know that his dreams were about to be shattered for good by a little know man then.

On September 7, 1930 Kurt Gödel had unleashed the first hint of his incompleteness theorems in a round table gathering of the Conference of Epistemology.

All of my bedtime stories that I have narrated, starting from Cantor and continuing up to Hilbert, are linked and form the basis for the thesis that Kurt Gödel chose to work upon in 1929.

If you recall, it was the 1928 Hilbert-Ackermann book “Principles of Mathematical Logic”, an introduction to the first-order logic where the famous question was posed:

“Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?”

Gödel took this question very seriously and decided to write a thesis on it under the guidance of Hans Hahn.

In all likelihood, all his contributions to mathematics has not given him as much recognition as the fact that he had Kurt Gödel as his student. 

Hans Han also contributed indirectly to the founding of the Vienna Circle.

It did not take long for Gödel to make the breakthrough.

Within the year of 1929, Gödel defended his doctoral dissertation that he called the completeness theorem.

The year following the dissertation, on November 17, 1930 the Vienna Academy of Science published the incompleteness theorems under the title:

“On Formally Undecidable Propositions of Principia Mathematica and Related Systems.”

The language was German and the journal was Monatshefte für Mathematik. 

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

March 29, 2017 Wednesday

Bedtime Story 

This is all that I have to say about Mathematical Notations

This Bourbaki group introduced the notation 𝝓 for empty set and terms used in set theory such as injective and bijective.

More importantly, the Bourbaki group laid a great emphasis on the use of notations and minimizing imprecise language wherever possible.

It was finally this group that made mathematics possess that scary look that sets fear in the hearts of the bravest of men.

This is clearly visible if you make a study of papers on pure mathematics that was published before the 1940s and after.

Post Bourbaki, pure mathematics papers had a dramatic reduction in texts and a manifold increase (to borrow a term from mathematics) in the notations.

With this huge increase in notation, came the great division.

Now clearly, there were two kinds of people in the world; People who understand and are at ease with the mathematical notations and the rest who aren’t.

Clearly I like most average apes fall in the second category.

This is all that I have to say about mathematical notations.

Now it’s time I take you to Kurt Gödel and his incompleteness theorem.

This is a difficult subject to write upon so please bear with me if you are left dissatisfied.

Of course, I will time and again need to digress since without laying the foundations and establishing the background, the understanding of the incompleteness theorems will leave you discontented.

Moreover, it does not make for a good story telling if one does not have the liberty to back track and shift the background sceneries.

It was in 1931, just when the Nazis were appearing on the horizon, that a young man of 25 at the University of Vienna published a paper in a German scientific magazine.

The paper was titled: “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”.

Now you can see why I had to tell you so much about the history of mathematical notations and the daunting work of Alfred North Whitehead and Bertrand Russell.

It was and still is a preeminent masterpiece of mathematical logic and foundations of mathematics.

Masterpiece it may have been, but Gödel seemed to have found something not quite right about it.

That itself speaks volumes about the beautiful mind; the audacity to challenge the prevailing authority, even someone as towering as Bertrand Russell.  

As you all know, mathematics, more particularly geometry, is not an experimental science which are often described as inductive.

In other inductive sciences, there are theories that rest on the outcomes of experiments.

Mathematics is totally deductive.

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

March 28, 2017 Tuesday

Bedtime Story 

The Bourbaki Group and Their Éléments de mathématique

Nicolas Bourbaki was a group of French mathematicians that was formed in Paris.

Hence it should actually be called the Bourbaki group or officially “Association of Collaborators of Nicolas Bourbaki”.

It was founded by the Paris born (1906) French mathematician André Weil who eventually ended up becoming a faculty member of the Institute for Advanced Study, Princeton.

(The same institute that gave shelter to helpless but brilliant German minds such as Kurt Gödel and Albert Einstein).

Between 1930 and 1932, he spent two years at the Aligarh Muslim University, situated in the gigantic state of Uttar Pradesh of India.

These two years it seems had a deep impact on his psyche as since then onwards, in spite being an agnostic, he became greatly attracted to the Hindu philosophical thought.

It was André Weil who organized the first meet of this singular group on December 10, 1934 in the basement of a grill room in Paris.

The group included even his sister Simone Weil, though she was more of a philosopher and a mystic rather than a mathematician or logician.

The aim to collect this group was to produce and publish books on mathematics that would be used for teaching.

One of this group’s masterpiece was a 12-volume Elements of Mathematic.

In French, it goes by the name Éléments de mathématique.

If you happen to notice, the word mathematics has been intentionally kept to singular mathematic to convey the message that the set of twelve volumes is in fact a single entity as a whole.

The twelve books in the series are as follows:

1. Set theory

2. Algebra

3. Topology

4. Functions of a Real Variable

5. Topological vector spaces

6. Integration

7. Commutative algebra

8. Differential manifolds

9. Lie groups and algebras

10. Spectral theory

11. History of Mathematics

12. Algebraic Topology

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

March 27, 2017 Monday

Bedtime Story 

From Principia Mathematica to Nicolas Bourbaki

It seems till date only a handful of men have read Principia Mathematica (they obviously got to be mathematicians since most average mortals don’t dare even go near that book).

Of those who have read it, I wonder how many were able to fathom its reasoning that were laid as foundation of mathematics from pure logic.

Not only this book makes for an impossible reading, even printing and publishing of this treatise turned out to be a monumental task.

Most of us barely use mathematical notations while typing on the word document on our computers.

Those that do on rare occasions, like I had to do for my bedtime stories, will realize what a task it is to get them printed out in the manner one would like to; more so when one has to write complex equations.

The old word document does not even have the software for most of mathematical notations; only the new one does.

Even with that, it is a painstaking job to get the equations in the desirable manner.

Just imagine what it must have been like for Whitehead and Russell to get their Principia Mathematica published way back in 1910 loaded with the weirdest symbols and notations possible.

Russell had to get special fonts made for the mammoth text.

Now, of course, thanks to Microsoft word, fonts are just few clicks away.

Not then in 1910.

Back then, fonts had to be made out from the real lead metal; in fact it is said that Russell was seen carting wheelbarrows full of lead fonts over to the Cambridge Printing Press.

Printing and publishing of Principia Mathematica was not intellectually taxing but also seriously physically demanding.

And understanding more so.

For the entire exercise of Principia Mathematica was to show that nearly whole of mathematics could be derived from logic using only the barest of the starting or primitive axioms.

For Russell, logic was even more primitive than mathematics and logic, to his consideration, was ‘pure thought’.

If we go back to the history of mathematical notation (and not logic), the next big development to happen was Nicolas Bourbaki in France somewhere in 1935.

You may wish to ask “Who is Nicolas Bourbaki?”

Well, there is no one by the name of Nicolas Bourbaki, at least in the context to my writing.

It is a pseudonym for a group of young French mathematicians from various French universities who had formed a collective in 1934.

We shall take up this curious Bourbaki 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

March 26, 2017 Sunday

Bedtime Story

How Peano and Principia Mathematica (1910) Revived Begriffsschrift (1879)

As I had stated, the outstanding Begriffsschrift got scant attention.

It was the student of Gottlob Frege Rudolf Carnap, along with Ludwig Wittgenstein and of course the monumental Principia Mathematica that revived and gave this treatise a new breath of life and deserving recognition.

After Gottlob Frege, Peano further worked on these notations.

In fact, all of the notations that I had used in my previous bedtime stories were not that of Frege but more of Peano’s.

The notations that were used by Gottlob Frege had a strange two-dimensional layout that not only made reading it cumbersome but printing it out economically impossible.

Peano went through the notations of Frege and transformed them to a more linear, more manageable and far more aesthetically refined.

As I had narrated once, so obsessed was Peano was with mathematical notations that he devised a whole new language Interlingua that was essentially Latin but bereft and denuded of all grammar.

Which means to say not only was Peano unhappy with the state of mathematics but also with daily use of language.

He also summarized all of the known mathematics till then in his Formulario Mathematico.

The notation that he used in this Formulario Mathematico caught on really well (unlike his Interlingua), and was used by Bertrand Russell and Whitehead in the Principia Mathematica.

This use of notations was not casual; Russell had met Peano in the famous twin conferences of philosophy and mathematics in 1900 that were held in Paris.

(Yes Sir, the same one where David Hilbert set the agenda for the mathematicians of the time and those to come by announcing his list of 23 unsolved problems.)

As I had stated before, at the onset of the conference Peano offered Russell a copy of his Formulario Mathematico.

Russell was deeply and profoundly impressed by Peano’s notation and it was this encounter that probably implanted the seed in the mind of Russell that would flower on to become Principia Mathematica.

When it comes to mathematical notation, there is no text that can match the sheer intensity, depth and extent that was deployed in Principia Mathematica.

In fact, some claim that it is the most notation-intensive work ever generated by a non-machine (considering the fact that computers use codes all the time).

You really need to see just one page of this monumental work; it is simply incomprehensible for the mind of an average ape.

Sometimes one wonders whether Whitehead and Russell ever meant it to be read by the human apes.

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.

                   

  A Typical Page from Principia Mathematica

             List of "Definitions" (about 500) are listed at the end of Volume 1 with most fanciful notations possible. No wonder very few human apes have ever went on to go through this landmark work.   

             

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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

March 25, 2017 Saturday

Bedtime Story 

Material Conditional and Negation

Material conditional is a logical connective or a binary operator that is symbolized either by arrow → or by the symbol ⊃ .

So if I write:

P → Q or if I write P ⊃ Q, then it implies

If P then Q.

The P is called the antecedent of the conditional.

Q is called the consequent of the conditional.

In simple language, it is equal to the word “implies”.   

This material conditional is critical in logic as it helps in establishing the truthfulness of a statement.

In an ordinary language, it would look like this:

If there is an infection, there will be fever.

If there is fire, there will be smoke.

The next thing that Gottlob Frege developed in his Begriffsschrift was the idea of negation.

Negation is also otherwise known as logical complement.

Negation is an operation that converts a proposition say p into “not p” that is written as ¬p.

It is also another way of saying that a proposition is true when p is false.

Thus it can be applied as an operation of truth value.

It is also a single-argument or a unary logical connective.

Take for example this statement.

If a person is not dead, then we cannot call it a murder.

The negativity of a fact has a logical implication.

I shall not go into the depth of this remarkable book on mathematical logic but it is clear that Gottlob was fully aware how monumental his work was.

He was not shy in stating it and quiet proudly he complements himself in his own preface.

In the preface of Begriffsschrift, he writes:

“If the task of philosophy is to break the domination of words over the human mind […], then my concept notation, being developed for these purposes, can be a useful instrument for philosophers […]

I believe the cause of logic has been advanced already by the invention of this concept notation.”

Yet, this monumental work of piercing intellect went largely ignored.

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

March 24, 2017 Friday

Bedtime Story 

The Language of Mathematical Logic

One of the important notation among the quantifier is the turned A or ∀.

This turned A in a sans-serif fond today is the universal quantifier that signifies “given any” or “for all”.

The other famous quantifier that he came up with was the existential quantifier.

It is denoted by the turned E or ∃ . 

This existential quantifier implies “there exists” or “there is at least one” or “for some”.

All this may seems very alien as mathematical logic is the least taught subject in high school curriculum or that was the way it used to be when I was in the school.

Then Gottlob Frege developed the idea of predicate.

Predicate as I had discussed in one my bedtime stories is a function that can be true or false depending on its value.

One can consider predicate to be a property of certain variable say x.

This is written mathematically as P(x).

Here the, x becomes a predicate variable.

So now we have three new terminologies, namely quantifiers, predicate and predicate variable.

Quantifiers are used together with predicate variables.

Let me give you an example here.

Suppose you want to say that there exists a natural number such that when it is multiplied by itself, it results in 36.

In mathematical logic, it is written as:

∃n∈P(n,n,25)

The symbol ∈ means element of or in.

This symbol establishes the relationship.

So if I wish to say that “x is an element of A”, I would write:

x ∈ A

The inverse of it would be is that “A contains x”.

That is stated as:

A ∋ x

There is also a negation of the set membership.

So to say that x is not an element of A, we write:

x ∉ A

Further, he introduced what is known as material conditional.

We shall take up the material condition in the nights to come.

Isn’t it fun mon ami?

It is just like learning a whole new language.

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