Rote Learning Vis-à-vis Physical Comprehension

#31 Posted by AlephNull

Could you please recommend a basic reading on non-standard analysis. I have seen it being applied to a problem where essentially you have to join the two closest points opposite an infinitely small hole (non-convexity)in a particular way. The text was very intimidating for me.

#25 Posted by freethinker:

Dear Dr. Gill,

This is my last post on the particular example. You write:

"In this particular case, one doesn't need to use any limiting value of x because delta y = 0 for all values of x. y = constant is a straight line parallel to the x-axis; its gradient (dy/dx) is zero for all values of x."

I am sorry but you do not seem to get my point. IF the gradient is defined as dy/dx, then it is NOT AT ALL CLEAR that the gradient is zero for ANY x (forget all x). This is because delta x needs to converge to zero for dy/dx to have meaning. WITHOUT the concept of limit, dividing by zero would leave things UNDEFINED (see Aleph Null's #31, and an earlier post by khurram). Convergence to zero is very different from zero (see iron mask's beautiful example in #15).

Regards.

Gill writes "There is no short cut"

Of course there is a short cut for everything in the Prozac nation aka USA. Take a pill and it is supposed to fix everything, send a couple hundred dollars and out pops a PhD. And then we have "scholars" like you who spit out in tape recorder fashion what other books have already said or other people have said in "high school of the rote tradition" manner without any thought whatsoever. Then you read my report on the state of education in Pakistan and the "rote methodology", censor that article and write your own article titled "Rote Learning..." come on man, get real, perverts don't come in any purer form than yours...ahh but there was MAJ, he doomed us all to misery, made us politically weak, and then many more than us in India, you cannot approach that level of perversion...sorry my bad...

The status of infinitesimals – quantities that are smaller than any positive real number but not zero - was a thorn in the side of mathematicians for more than a century. Their use seemed problematic even though they were being employed to derive results that were seen to be obviously correct. Newton was so worried about the shaky foundations of the infinitesimal calculus that he had invented, that he used classical geometry to prove, in the Principia, results that he had obtained via calculus. An 18th century churchman, Bishop George Berkeley, launch a broadside against mathematicians, asking why, if they could accept and do business with such dubious notions, they could not accept the truths of revealed religion. It was left for Cauchy in the 19th century to put infinitesimal calculus on a rigorous foundation using the idea of limits.

There is however a 20th century approach to this problem – non-standard analysis, pioneered by the late Abraham Robinson - which shows that arithmetic with infinitesimals can be put on a logically sound footing. It is based on the observation that, as a consequence of the incompleteness of arithmetic, we can adjoin ‘nonstandard’ natural numbers – numbers all of which are larger than any standard natural number – to the natural number line without any inconsistency. Infinitesimals can then be defined as reciprocals of nonstandard natural numbers and we can proceed from there. Nonstandard analysis can be used to provide slick derivations of results whose traditional proofs are unwieldy.

IB "I happened to check your blog - and I fail to get the proof! what excatly is your point?"

You wouldn't know, after all is said and done he asks if Laila was male or female...perverts with a pea brain that believe in a rote monger like Gill wouldn't know...

In #28 read " majority collectively towards a minority because of the greed of one man"

as " majority collectively punishing a minority because of the greed of one man

Manto writes "Is this guy Masadi for real?"

Yes, I am for real and I am the worst nightmare of a-holes of the West that want to perpetuate the same old MAJ BS in this country to take it further down into the gutter than even what it is in right now. What I do know for sure is that the fate of the Indian Muslim was a direct consequence of the majority collectively towards a minority because of the greed of one man, the MAJ BS, a reaction to obtaining a non-viable country that was to become a whore of the West, thus causing debilitating suffering to the Muslims of this region both inside and outside India.

Today, I had to get some papers Xeroxed, so I went to a corner shop. This guy walks in and asks the shopkeeper, "When is Shab e Miraaj?" The shop keeper says, "People of an enslaved nation do not partake in these things, I am worried about load shedding and you are asking me this question. People who are enslaved cannot have a religious identity." Then he turned to me and said, "kyun saab jee, theek baat hey?"

I said "bilkul theek" and I am happy that the people of this nation are finally waking up. Then they guy who asked the "Miraaj" question asked the shopkeeper, “Then why did Quaid e Azam make Pakistan", and the shop keeper says, "Quaid e Azam ney jhak maree thee". I laughed as I left the shop on whose wall was the "jhak man's" photograph and said, " Soon the people of this nation will rise up against the BS of both the Quaid and the Mullah variety, enough is enough, the time of our liberation is nigh, one that involves social justice for all". No lackeys of the West like the MAJ will be tolerated, new currency will be printed. Enough of the same old BS. The High Priest of the Church of MAJ and all its franchises and whore houses will be shut down. Then will the humanity in Pakistan rejoice......Inshallah!

Kaptain: #26

I don't really know much about your situation, e.g., your background, etc. I may say one thing though. If you're not sufficiently knowledgable in mathematics, it's not the end of the world. There are other intellectually creative fields also. For instance, there is poetry. But if you must come back to mathematics, I suggest you should read some books on the history of development of mathematics. Two of such books which I read and found inspiring are as follows:

1. Mathematical Thught From Ancient to Modern Times (in 3 volumes), by Morris Kline.
2. Men of Mathematics, by by E.T. Bell.

These books provide a background on the kinds of problems which inspired great mathematicians. If after reading these books, you feel that you should know more about mathematics itself, you should take suitable courses at some college to learn. There is no short cut. If you start with applied mathematics instead of pure mathematics, you might be able to sustain your interest better. Mathematics is dry but for those who are really into it, it is simply beautiful.

Wishing you well,

Mohammad Akram Gill

Couldn't develop the passion for maths, it was made to look dreadful.

Any ideas, Mr. Gill, how can I make a comeback to Maths? Its been years I've been through it. But its the logic of nature.

GT: #23
In this particular case, one doesn't need to use any limiting value of x because delta y = 0 for all values of x. y = constant is a straight line parallel to the x-axis; its gradient (dy/dx) is zero for all values of x.
Mohammad Akram Gill

Which board, GT? I will look it up right now. Thanks.

#22 Posted by freethinker:

Gill, no I agree that your point is broader and I appreciate your point. As far as my point 2 is concerned, of course it can be proved given the definition of a limit. My earlier point was that we could at times intuitively understand dy/dx without going into the fundamentals.

Cheers.

GT: #12
I used dy/dx just as an example to make a point. I purposely stayed away from the beautiful (Calculus is indeed a very beautiful branch of mathematics) intricacies of calculus because the topic of my article was much broader. However, in your example #2, we do not necessarily need to believe intuitively that dy/dx = 0, it can be actually proved that dy/dx = 0 when y is a constant.
I was hoping that some young readers might share their experiences as students regarding rote learning. In my time in Pakistan, quite a bit of teaching was also by rote.
Wishing you well,
Mohammad Akram Gill

#19 Posted by KaalChakra:

Kaal:

Did you get to see the "quote"? I posted it sometime back in one of the boards.

Iron (with due apologies to Gill for the deviation):

Two things:

1. Asadi is a thinking man, a bit short-tempered. It would be worthwhile if you could be nice to him. There is a lot beyongd math.

2. Since you are interested. Do you know that all integrals under the chi-square distribution are approximations done with polynomials! Check it out. As an exercise try doing your own approximation using say 'mathematica'. Your approximations would be much sharper than chi-square tables published in the 60s (at the boundaries). Keep the degree at 1.

Still, GT, don't you love iron's 'visualizing' approach? If he tried hard enough, he might even be able to 'visualize' -3/2! :)

For the rest of us, mathematics indeed is the language of gods, not to be confused totally for anything mundanely 'real'. In that sense, jayp"s (#1) approach has an appeal.