May 31, 2017 Wednesday

Bedtime Story 

Continuation of Mathematical Reading 

Reading mathematics is an arduous task.

Each line requires effort, concentration and high degree of cognitive activity; trying to read mathematics at the rate of your average book will eventually lead to frustration, despair and eventually loss of interest.

In fact simply reading mathematics may not be sufficient.

You may need to write it down in your own language, even not in mathematical language but at least in your words the concept.

Bedtime storytelling, in that manner, has proven to me quite useful.

Even better is what mon ami does; to narrate the idea to someone else verbally.

Mon ami does it naturally; but for an average ape like me that would need preparation of at least few hours, or may be one entire evening.

The other thing about mathematics and mathematical writings are the absolute preciseness of the words that is alien to most readers of fiction and poems.

Again I quote Edward Rothstein:

“The meaning is rarely completely transparent, because every symbol or word already represents an extraordinary condensation of concept and reference.”

One of the notorious phrases that is commonly encountered in mathematics is “It follows from that”.

This simple four-word phrase is perhaps the most deceptive statement ever created.

What this phrase actually means to say is that the next logical conclusion that the author derives from it can be explained using a series of steps but since it would occupy a lot of space and since it isn’t too farfetched if you cared to work it out, the author is giving it a skip.

It is up to the reader to try to pause, give it a little thought and then to work it out or simply assume that the author has done his homework and carry on reading.

The last thing about reading mathematics is being aware of your own knowledge and preparedness on the subject that is being written about.

A writer of mathematics is most likely a mathematician himself who tries to write for the sake of average public.

The problem that the reader is confronted with then is that the expectation of the writer of that word “average” isn’t all that average.

The intended audience of the writer is expected to either possess and if lacking then attain a certain level of mathematical maturity.

So at times in order to read a mathematical text, one might have to work up and brush a bit to become the intended audience of the author.

No wonder which so much hard work expected, both mathematics and mathematical writings could never gain popularity.

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

May 30, 2017 Tuesday

Bedtime Story 

Understanding Why Reading Mathematics is Difficult

Mathematics and formal logic is vast and huge, thanks to the thousands of great minds who have worked on it for 5000 years and maybe more.

Even though an extremely tiny fraction of few human apes actually contribute anything new to mathematics, given so many years and our ability to document and store information, great accretion and advancements have been made - just like evolution and origin of life - in it.

In fact so much of knowledge has now been accumulated that a modern average child and for that matter, even an adult finds it both overwhelming and scary.

Even the books that are supposed to be easy reading on mathematics aren’t that plain sailing.

So how should one read a book on mathematics (mark you, I am insisting on ‘book on mathematics’ rather than a ‘mathematical book’)?

When you read a story or a book on mathematics, try to get a sense where exactly you are located.

Is it algebra (the fashion of this century), numbers, geometry, analysis or logic?

In what epoch of history of mathematics is the story you are reading dealing with.

Getting a historical perspective of any idea or event gives a distinctive satisfaction - much like a piece of a jigsaw puzzle falling in the right place - besides adding an element of human interest to a subject so abstract.

Remember, the mathematics that is taught in high schools to most average apes or at least the mathematics that most average human apes seem to know is extremely primitive; even more antediluvian than what was known in the medieval European ages of 12^th century.

So it is quite normal to know very little mathematics.

Hence reading mathematics becomes a slow process as we are all guilty of serious ignorance in this arena.

Besides, mathematics demands thinking and asking questions.

It will even mean going back to the chapters that you read and rereading them again with the new perspective that the later chapters gave you.

It will also need searching some interesting points on the internet and YouTube as they are littered with fantastic videos.

Let me quote Edward Rothstein once again:

“Reading mathematics is not at all a linear experience; …Understanding the text requires cross references, scanning, pausing and revisiting.

A three-line proof of a subtle theorem is the distillation of years of activity.

Reading mathematics involves a return to the thinking that went into the writing.”

In popular literature, too much is written and too many words are used to depict scenes or emotions.

It is quite the opposite in mathematics; mathematics says a lot with a little.

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

May 29, 2017 Monday

Bedtime Story 

Edward Rothstein is a Ray of Hope for an Average Ape

The essay “How to Read Mathematics” is a sagacious one for any bibliophilic even if he lacks mathematical insight.

In the very beginning of the essay is a quote by Edward Rothstein who is an author of book titled, ‘Emblems of Mind: The Inner Life of Music and Mathematics’ (Times Books, 1995) that goes like this:

Mathematics is “a language that can neither be read nor understood without initiation”. (In general, that statement can be universalized for almost anything).

Rothstein, by the way, has nothing to do with mathematics; he holds M.A. in English literature from Columbia University.

Yet he is writing on the beauty of mathematics (and music).

This just shows that it is not impossible to appreciate mathematics even if one lacks a total inclination for it.

Most of us will never do any serious mathematics; and most possibly are not capable of any serious mathematical work.

In fact, most average apes, including advocates, judges and doctors even question the teaching of higher abstract mathematics in schools (They are quite satisfied with the pedagogy of its basic applications).

Luckily for us that is not a barrier to know mathematics, to enjoy what mathematicians do and yes, even try it out a little bit for yourself.

I have discovered a great website that keeps feeding me mathematical problems from time to time.

https://brilliant.org/

The problems range from simple to very difficult.   

The questions offered to you are such that they stay at the level that you are comfortable with unless you feel like challenging yourself and intend to be a masochist and enjoy torturing yourself.

Then you can go for higher levels and feel free to terrorize yourself.

Believe me, most great achievers were certainly masochist; no one achieved anything sitting comfortably in their bedroom watching a nice movie.

I have come to enjoy a little bit of mathematics through this website and this is all that I “do” in mathematics.

Doing apart, I realized that even reading mathematics is quite an ordeal; it is very different from all other forms of readings that most of us adults do.

(Reading law and court verdicts are also similarly challenging as one has to seek out the logical connectives, reasoning and nitty-gritty points that have to be sought-for in the vast details of data out there).

So let me tell you a few things what I learnt from reading mathematics and a bit of amateur writing on it.

You will have to wait for it yet another night mon ami.

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

May 28, 2017 Sunday

Bedtime Story 

Levels of Abstraction

Lingering a little bit longer on abstraction, I would like to emphasize that abstraction is a mental process that is very commonly used by us apes.

It is a technique to generalize or simplify something specific by leaving out the concrete, specific details.

The abstraction can have different levels.

Douglas Hofstadter in his most unusual and difficult masterpiece ‘Gödel, Escher, Bach: An Eternal Golden Braid’ (1979) illustrates six levels of abstraction in something as banal as a newspaper.

Level 1: A publication

Level 2: A Newspaper

Level 3: The San Francisco Chronicle

Level 4: The May 18 edition of The San Francisco Chronicle

Level 5: My copy of the May 18 edition of The San Francisco Chronicle

Level 6: My copy of the May 18 edition of The San Francisco Chronicle as it was when I first picked it up (as contrasted with my copy a few days later: in my fireplace, burning)

The higher up you go, the more the abstraction.

So abstraction is inherent in most of our thought process (the book attempts to explain how cognition emerges from single neurons using concepts from computer science and mathematics such as self-reference and recursion); the only difference is that mathematics requires abstraction of a degree so intense that is lacking in average apes.

Having said all that, I can’t help but expect you to be overwhelmed with a deluge of nihilism leading to complete renunciation of mathematics.

That is but natural; Most of us will make for pathetic mathematicians.

Yet, there is hope – just a tiny bit – but it is there.

No one can stops us from enjoying and reading on mathematics, which is meta-mathematics.

Lack of ability to do any meaningful and original mathematics does not imply that one cannot savor the beauty of mathematics and knowing about those rare few apes who CAN do original mathematics.

But even that takes immense effort.      

    

Before I go into the meat of Gödel’s brilliant theorems, I wish to share with you something what mon ami shared with me.

That is - ‘How to Read Mathematics’.

We shall discuss this essay before indulging ourselves further with Gödel.

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

May 27, 2017 Saturday

Bedtime Story 

The Qualities of a Mathematical Mind

As I was saying, the problem of taking up a bedtime story as complicated as Gödel’s incompleteness theorems is my own lack of any decent mathematical training.

But the second reason is even more important.

Gödel’s paper is seriously grueling, back-breaking and mentally exhausting requiring mastery of 46 preliminary definitions and many basic propositions before the main results are reached.

Serious mathematics requires high mental abstractions and formal logic perhaps even more so.

I am not educated enough or perhaps to put it more correctly, not dedicated enough to go into that kind of depth and understanding of abstract mathematics involved here.

In mathematics, abstraction involves extracting and isolating the underlying essence of a concept, removing its connection from any real world physical constructs and then generalizing it to other related or even non-related phenomena. 

The hard fact and blunt truth is that highly abstract concepts are cognitively very challenging to grasp.

Conceptual assimilation of such high level of abstractions requires a certain level of mathematical maturity.

Mathematical maturity is a mixture of mathematical experience - which can be acquired through conventional education - and insight that is generally genetic (either you have it or you don’t).

Mathematical maturity demands following traits from the risen ape:

[1] Fearlessness in the face of symbols; ability to read and understand mathematical notations

[2] Capacity to generalize from a specific example to a broad concept

[3] Capacity to handle increasingly abstract ideas

[4] Capacity to shift to learning by rote to learning through understanding

[5] Ability to link a geometrical representation with an analytical representation

[6] Ability to convert verbal problem into a mathematical one

[7] Ability to recognize mathematical patterns

[8] Ability to move back and forth between geometry and analysis

[9] Ability to draw a line between what you know and what you do not

[10] Ability to teach yourself

[11] And of course, ability to remain focused

Yes, no one said mathematics is easy.

Bertrand Russell in 1931 wrote that the words of ordinary language are not sufficiently abstract.

What he was meaning to say is no matter what the level of abstraction found in ordinary language may be, it is nothing in comparison to the depth encountered in mathematics and mathematical logic.

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

May 26, 2017 Friday

Bedtime Story 

Even the Reasoning/Logic Used by Gödel is Mentally Punishing

  

Just to be clear, the idea of mapping per se is not totally a novel idea.

Co-ordinate geometry for instance translates or maps geometry into algebra or even in construction of actual maps wherein the shapes on the surface of a sphere like our planet is projected into flat rectangular papers.

Yet the way Jules Richard deployed the idea of mapping was unique.

This sparked the idea inside Gödel’s mind and using it he showed that meta-mathematical statements about a formal logical system of arithmetic can be represented by arithmetic formulas within the system.

This statement written above is a very difficult one for any non-mathematician or even a mathematician not trained in formal logic.

But it is foundationally critical for the proof of Gödel’s theorems as we will see when we go deep into the theorems themselves.

He used the mapping in a totally revolutionary way, devising and proving that within his formal logical system neither his arithmetic formulas were map-able to certain meta-mathematical statements nor these same formulas could be corresponded to the negation of such meta-mathematical statements. 

This again is a profound statement and will not be comprehendible at this stage.

But since these arithmetical formulas ought to be true and codify the truth of the system, the system was incomplete.

Gödel then went further.

He found a way to construct a number theory statement that corresponded to the meta-mathematical statement:

“The formal logical system is consistent’.

He went on to show that that particular number theory statement was not demonstrable or impossible to drive within that formal system using the established rules of inference.

It is only possible to derive that number theory statement if additional rules of inference were invoked which makes the validity and the consistency of the formal system questionable.

If you now think that we are done with Gödel’s incompleteness theorems, you are badly mistaken.

All this was the background knowledge to prepare and educate you (and me more than you) before going into the meat of his theorems.

The problem of taking up a bedtime story as complicated as this is twofold:

One is my own lack of education in serious and higher mathematics and formal logic.

Before I present anything to you, I need to at first understand it myself and be clear about it.

That takes time for an average ape like me.

And by the way, I always find it more enthralling to write about something that is new to me and that which is more painful to write (definitely a serious streak of mental masochism).

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

May 25, 2017 Thursday

Bedtime Story 

The Fatal Paradox: n is Richardian if, and only if, n is non Richardian

Let us go once back and understand the Richard’s Paradox in its other variant.

Let us suppose that n is Richardian.

But this is only possible if n is assigned a definition that does not define it.

This means that n is not Richardian contradicting our assumption.

Now let us suppose that n is not Richardian.

In that case it will have its defining property ‘n is Richardian’ attached to it.

But then by definition makes it a Richardian which goes against our assumption.

In short, n is Richardian if, and only if, n is non Richardian.

This makes the statement ‘n is Richardian’ both true and false.   

Now you may be wondering how on earth does this paradox of Richardian numbers fit into our story of incompleteness theorems.

That is an excellent question indeed.

Jules Richard in framing his paradox had invented something that was never done before: he had invented the idea of mapping meta-mathematical statements about the formalized mathematical logical system to the arithmetical formulas within the system.

If you go back and consider his paradox, this idea of mapping was not the primary intent; the mapping was being done primarily to generate a paradox or an antinomy.

Let us look at the example that we took previously to get this idea straight.

The following statement, ‘The real number the integer part of which is 1 and the nth decimal place of which is 0 if n is even and 1 if n is odd’ unambiguously defines the real number 17.10101010… which represents the fraction 1693/99.

You would surely recall this statement in one of my bedtime stories not so long ago.

The statement ‘The real number the integer part of which is 1 and the nth decimal place of which is 0 if n is even and 1 if n is odd’ is a meta-mathematical statement.

It lies outside the arena of arithmetic.

The numbers themselves namely the rational number 17.10101010… or 1693/99 are themselves part of arithmetic which were linked to the above mentioned meta-mathematical statement.

I urge you to keep this in your memory as these ideas will gain ascendancy and become prominent when we will go into the heart of Gödel’s theorems and logic.

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

May 24, 2017 Wednesday

Bedtime Story 

Completing Richardian numbers

Now comes the second part of the Richard's  paradox that brings in the concept of mapping.

It now asks us to map or link or associate each definition of integers that we have so arranged (length-wise and lexicographically) with each natural number starting from one. 

So the first shortest definition in alphabetical order will correspond to number 1, the next definition in the arrangement to number 2 and so on.

You will naturally understand that the definition that has been assigned to each natural number may not, or rather most likely will not, fit the natural number.

For instance, number 8 might end up getting tagged with the definition ‘not divisible by any integer except for 1 and itself’.

This paired definition to number 8 certainly does not describe the property of 8.

And in most instances it is bound to be so.

Yet, in some rare instances the definition that has been assigned to a natural number might actually end up in being the property of that number.

For instance, if the natural number 17 gets tagged with the same definition ‘2-digit integer that is not divisible by any integer except for 1 and itself’, then in this case the definition happens to describe this number unwittingly but very suitably.

We shall denote the former example when the integer is not paired with the definition that describes it property as Richardian. (This is the whole crux of the paradox)

Hence, a natural number will be Richardian if the definition to which it has been mapped with lacks its property.

So we have to a state where defining ‘x is Richardian’ is just another way of saying ‘x lacks the property that the defining expression of a serially ordered set mapped to it suggests’.

As you will understand, most, or nearly all the natural numbers will be Richardian as it is more likely them to be paired with the definition that does not define them.

At least apparently, the property of being a Richardian is itself a numerical property of integers.

This implies that it should also be among the serially ordered definitions that were created above.

Therefore, the property of ‘x is Richardian’ must be assigned to some natural number.

Let that number be n.

So now if we ask the question, is n Richardian?

You can obviously foresee a fatal paradox popping its dirty head out now.

If you don’t, please go over the whole schemata once again.

We shall go over once again 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.

                           

  

                

             

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