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To Padmasrinivasan -- Letter from an Asylum

Discussion in 'Snippets of Life (Non-Fiction)' started by ojaantrik, Dec 26, 2009.

  1. ojaantrik

    ojaantrik IL Hall of Fame

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    Dear Padma:

    I enjoyed each little morsel of your post 'Square vs. Round'. Especially the way you begin, connecting geometry to arithmetic. Of course, you don't use the word 'arithmetic'. You say 'mathematics'. Since geometry is a branch of mathematics itself, I have to admit that I was puzzled for a while. But then I tried and interpreted your statement my own way.

    I didn't feel too confident about it though, because whenever I brush against mathematics, I feel scared. I don't have happy memories of mathematics from junior school. Normally I would fail each and every exam and was no more than a laughing stock for classmates. The teachers usually made me stand on the bench or outside the classroom as an object of tourist attraction as it were for the rest of the school. They are scary memories from youth and it is best that I speak about them in detail some other day. Suffice it to say for now that elders at home as well as outside home never ceased to puzzle over what they called God's greatest miracle: The fact that I managed to clear school and enter college! They were almost sure that I had resorted to unfair means, an ancient counterpart of modern day rigging at the alter of our Great Indian Democracy.

    Anyway, going back to your little post on the endlessly many ways a square or a round manifests itself, I thought I should add a few words about the relationship between geometry and arithmetic and the meaning of endlessness, picked up from the annals of mathematics. I am doing this knowing fully well that it will run tangentially to what you were discussing. In fact, I won't blame any ILite if this post reminds her/him of the delicious Carroll rhyme:

    He thought he saw a banker's clerk
    Descending from the bus
    He looked again and found it was
    A hippopotamus!

    Well, going back to arithmetic and geometry, I couldn't help recalling what are called numbers in arithmetic. They are things like 1, 2, 3, ... and so on. Right? No, not right. I have left out some of the commonest numbers, such as 1/2, 3/49 and so on. Hmm...

    So, what are numbers? They are the collection of integers, such as 1, 2, 3, ... , all the fractions and all the negatives of integers and fractions. Right? No, not right.

    Well, why not? To understand the problem, we need to define what we mean by fractions first. What are fractions then? Quite obviously, they are numbers that result from dividing an integer by another. There can be no argument about this, right? Yes, this time it's right.

    The mathematicians though are a crazy lot. They are hard to satisfy. The moment they discover that you think you have solved a problem (the problem right now being what constitutes the set of all numbers), and gone to sleep (perchance to dream too), they wake you up in the middle of the night and drown you in useless nonsense. They tell you things which you believe no one ever could possibly bother about.

    Well there was this guy called Georg Cantor who lived in the second half of the nineteenth century and early part of the twentieth. And he had this irritating habit of raising questions that disturbed one's peace of mind. He wouldn't let you sit down and enjoy the infinite expanse of the sky and marvel at God's creativity. Instead, he would poke you in the ribs and keep on asking what infinity meant, arithmetically speaking. I think he never stopped asking such questions to himself either and lost a good deal of his sanity in the process. What else do you expect? Right? Yes, quite right.

    Well the way he landed on this infinity question was chilling to say the least. If we collected all the numbers we defined a little while ago, integers and fractions I mean, you will certainly have a collection of infinitely many numbers. For example, 1, 2, 3, ... itself is an endless, and hence infinite, sequence of numbers. And recall that we had added to it the fractions and the negatives. Surely, that is infinitely large. Right? Yes, right indeed.

    Well this crazy chap had the temerity to pronounce that even if this is infinitely large, it was possible to create yet another infinity that was larger than our first infinity! Hmm...

    "Come on man! Leave us alone," said every sensible person around. But Cantor wouldn't listen. He kept on insisting that there were more numbers than the ones we noted above. And the famous example was the number 'square root of 2'. Well, there is no point trying to prove a funny result here, even though the proof is quite simple. And that funny result is this: You cannot express 'square root of 2' as a fraction. And quite clearly, it was not an integer either. But can you deny that such a number exists?

    May be you would challenge Cantor to show the existence of the number. He would have his own way of convincing you, but there is an easier proof. A geometrical proof, a proof that connects geometry to arithmetic! Well here is the proof. Consider a right angled isosceles triangle. The two sides between which the right angle is nested are chosen to have a length of 1 inch each. OK? What is the length of the hypotenuse of this triangle? According to the Pythagoras theorem, it should be 'square root of (1 plus 1) = square root of 2' !!! In other words, there definitely exists a straight line whose length is a 'square root of 2' inches. Hence, square root of 2 is a pretty respectable number. Right? Hmm...

    So geometry comes to the aid of arithmetic. Geometry tells you that there are things in heaven and earth Horatio that arithmetic cannot quite capture.

    Well, this last bit was a lie of course. Arithmetic has a way of writing these numbers, or at least their decimal expansions. But I will not torture you with it.

    The great thing that people realized was that the Egyptians were aware of such numbers. They were aware because they used geometry to build pyramids. And once in a while they came across these funny lengths. Of course, they knew about particular examples only. It was Pythagoras who proved the general theorem.

    And what did Cantor do? He showed there were endlessly many numbers like the 'square root of 2'. So endless they were that when you collected them, they gave rise to an infinity that was larger than the infinity of the integers and fractions. He called these irrational numbers as opposed to the integers and fractions, which were rational. Irrational numbers indeed! Well what else do you expect from a madcap?

    And do you know that one of these irrational numbers is what is usually denoted by the Greek alphabet 'pi' (pronounce pie). It is pretty complicated to prove that 'pi' is irrational, but that's beside the point. (In fact, the easiest proof I am familiar with is 3 pages long!)

    What's important is that it has a bearing on your post. For the area of a circle (your round I mean) is exactly ('pi') times the square of r, where r is the radius of the circle!

    And that completes the circle of reasoning for me. The links between geometry and arithmetic, squares and circles, little infinity and LARGE INFINITY !

    Sorry, need to take your leave now. The doctor's ordered a straitjacket for me. And a padded cell of course.

    He knows that I flunked my exams at school and never recovered from the shame of it all!



    oj :crazy


    Photograph of Georg Cantor​

    Georg_Cantor.jpg
     
    Last edited: Dec 26, 2009
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  2. knot2share

    knot2share Gold IL'ite

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    Nomashkar OJ ji............

    I am nearly tempted to say
    "Come on ! Leave me alone,".........heheh
    I have read this post of yours a few times trying to make an attempt to understand the hidden meaning...but I failed to do so until I reached the formula for the area of the circle in my nth attempt. Now I get the relationship between the "square" and "round/circle" and that too the arithmetic and geometry relation. I am tired now..........Aur waise bhi itnee raat hue aap yeh sab sochte kaise hain?

     
    Last edited: Dec 26, 2009
  3. iyerviji

    iyerviji Finest Post Winner

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    Dear OJ Sir

    You say you were not good at mathematics but here you have taught us so many things. Mathematics was my favourite subject in school.

    One thread you had written Dear Mindi, now Padma, next is JPatma I think
    Enjoyed reading your post. In every thread of yours there is something to learn.

    Regards
    viji
     
  4. Kamalji

    Kamalji IL Hall of Fame

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    Dear OJ,

    You say yu were not good at Maths in school, and i say i was pretty good in maths.

    But today u have taught me, tat u are the master of mathematics and writing.Good i am drinking tea now and reading this, after whiskey, i shudder i would have fainted by now.HAHA

    Well i learnt enough maths to be able to earn some money, beyond tthat i dont wish to learn.That example of isoscleles right angled traingle is a stroke of genius.Yes i still remember the formula, the square of teh hypotenuse,is equal the sqyares of the other two sides, so that is a classic emaple.


    As viji rightly pointed out, the next one could be jaya, or it could be me int he firing line,but i am ready with a bullet proof Vest, better qualtiy than the ones the trhee cops wore on 26/11.:biglaugh

    simply marvellous.

    Regards

    kamal
     
  5. Padmini

    Padmini IL Hall of Fame

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    Dear Oja da,
    Really I am spell bounded and still cannot get out of the voracious knowledge you have depicted in your post!!! You have gone to the depth of integers and geometry.When you mention Egyptians
    used geometry to build Pyramids, these thoughts run in my mind which i think I can share( You may think me as a cellmate to you)
    Life itself as we know it is inextricably interwoven with geometric forms, from the angles of atomic bonds in the molecules of the amino acids, to the helical spirals of DNA, to the spherical prototype of the cell, to the first few cells of an organism which assume vesical, tetrahedral, and star (double) tetrahedral forms prior to the diversification of tissues for different physiological functions. Our human bodies on this planet all developed with a common geometric progression from one to two to four to eight primal cells and beyond.
    The square root of 2 embodies a profound principle of the whole being more than the sum of its parts. I am yet to get a feed back from you for my posts ( I won't get because i am treating the matter in a very light manner:) without attributing any philosophical attitude!)Reading your posts make me think " miles to go before I sleep"
    with love
    pad
     
  6. Cheeniya

    Cheeniya Super Moderator Staff Member IL Hall of Fame

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    My dear OJ
    What an insightful piece of writing! I am sure Padma will feel greatly rewarded by your own take on Mathematics, Arithmetic and Geometry. I have been a member of the Masonic Fraternity for over 3 decades and Freemasons talk of the seven liberal arts and sciences namely Grammar, Rhetoric, Logic, Arithmetic, Geometry, Music and Astronomy. Of these, Arithmetic and Geometry are the very backbone of Masonry. A typical Masonic verse talks of the seven liberal arts and sciences thus:

    “The grammar rules instruct the tongue and pen,
    Rhetoric teaches eloquence to men;
    By logic we are taught to reason well,
    Music has claims beyond our power to tell;
    The use of numbers, numberless we find;
    Geometry gives measure to mankind.
    The heavenly system elevates the mind.
    All those, and many secrets more,
    The Masons taught in days of yore."

    The basic mathematical functions teach masons:
    “Add to your knowledge
    Never subtract from the character of your neighbor
    Multiple your benevolence to your fellow creatures and
    Divide your means with those in need.”

    Incidentally, a favourite limerick of mine teaches the value of  as
    'Tis a favourite project of mine,
    A new value of pi to assign.
    I would fix it at 3,
    For it's simpler, you see,
    Than 3 point 1 4 1 5 9”

    I guess this is a subject one can ramble on and on!
    Sri
     
  7. Padmasrinivas

    Padmasrinivas Silver IL'ite

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    Dear OJ-da,

    A simple and innocuous post on choosing between Square or Round containers has spawned such an illuminating post from you!

    Oh, oh! Geometry and Arithmetic (and Algebra and Trigonometry = Mathematics?) Never realised that I'd be caught out by the 'Mathematical genius' here:hide:! (Not kidding, OJ-da, I say this in all sincerity:thumbsup)

    Well, integers or fractions, rational or irrational, real or natural, finite, infinite, Cantor's transinfinite numbers, Pythagoras' Constant 1.41421356 or Pi....

    :spin That's me, head going round in circles .... you have brought in so many eminent ILites like K2s, Viji, KMji, Padmini, Cheeniya sir to elaborate...

    My words seem totally inadequate, OJ-da!

    Thanks so much,

    Regards,
    Padma








     
  8. ojaantrik

    ojaantrik IL Hall of Fame

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    @ k2s

    Thanks for reading n-times! Actually, the post had little to do with Padma's post, but I know that I wouldn't have written it had I not read her piece. Quite obviously, I couldn't go too deep into the matter. Nor was it necessary. But sometimes I get these attacks of lunacy -- they are increasing with age -- and I go on a fantasy ride! To tell you frankly, I don't know if I wrote sense or nonsense!! :bonk

    oj
     
  9. ojaantrik

    ojaantrik IL Hall of Fame

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    @ viji

    Thank you so much. It's nice to write posts in the form of letters. I don't know to whom my next one will be directed. By the way, I still remember one of your earlier posts. I told people how much I liked it. It explained why you write and it was a post that I loved for its flavour of pure innocence. I have not been able to read you lately. But sooner or later I shall catch up. Unfortunately, I go through these bouts of depression quite often. And on those days I become quite dysfunctional. On those days, I can think of nothing but my incompetence.

    oj
     
  10. ojaantrik

    ojaantrik IL Hall of Fame

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    @ kamal

    I think I have now found a form, or a structure, or may be a model for my posts. Letters. Cheeniya has his unique structure. Perhaps I could give my posts a character by sticking to the letter form. I think I can write better when I address a person. Of course, these are open letters. Nonetheless, they have this personal communication form that suits me best. At least, that's the way I think it is for the timebeing. Let's see!!

    The math thing was just for fun. The real idea was trying to communicate to a person.

    oj
     

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