Rote Learning Vis-à-vis Physical Comprehension

Re: # 15
Study , real analysis goldburg than name calling Masadi.

Re: # 28 Masadi is real. People can read his book if you want to become free from western domination mentally.
Mr. Masadi you thoughts are right. I asked you to write about G man and you and I agree. ( As you know I have disagreed with you some times).
Even with all slander and criticism by Nanto man
I just felt about G man as in KingLear, Kinglear and her ungrateful
" Oldest have bosrn most
We that are young
Shall never see so much
Nor live so long" Albany ( King Lear- Bill Shakesare)
To my surprise you can find same echo in what Einsten said about G man after he had departed.

Re: # 32
Whatever I wrote supporting you is rejected so I do not want to waste my time. In usa heard is fast PhD programmes just like fast food at 3rd rate and 4rate univs. It is absurd and laughable when they talk of producing 1000 Phds per year who are mostly worthless like fast food burgers which is imposed dirty and unhealthy food imposed on brown and dark people by american food factories. They should ban them. They produce 4000 calories worth food per person in usa. 2400 calaries needed consume all those calories produced by food factories resulting in fat and obase population. As you said slavary pervades even in food. Stop food facories, already young women and men of pakistan are carrying loads from these food factores by becoming obase. But local brown elites are so happy digging their graves with western foods to please food factory owners.
Good mr. Mr. Masadi.

Re: # 38 I am appaled by sloppy definition of derivative.
It is with horroe I read when dy= o. It goes against basic understanding. The derivative is simply rate of change of one function with respect to variable. Dy/dx is rate (ratio) of change of for small change of x. This small change is very small and you can make as small as possible but never equal to zero. arithmatically you can think and visualise increment in x can be 0.00000000001 but can not be zero as we can not divide by zero.
It my feeling as a amature tutor that different people learn different way. Many times its waste of time by tutor to go in classical definitions. As the students uses these tools and finds solving simple problems the ideas crystalize in young mind slowly and there is learn and digesting time and one can not rush through. Many time as students advance things not clear to him start clearing. This is learning process and is development. Initially crude or sloppy ideas is not problem and like rough diamond as students starts learning ang and thinking it slowly starts sparkling. The duty og good teacher to help and guide him and provide mental tools the student will cut himself like diamond cutter and will shine and delight to teacher. Just like in mine there are gems, sparkling stones and diamonds teacher should always keep in mine most is coal and should be happy as coal is powerful when you burn, he will become engineer and serve society.
Some people are not intelligent enough to grasp finer points but many things can be taught mechanically and in few terms they get essence as mathematics it self is tool to analyse and study things. But teacher with maths has more responsibility as some times even good student can have problem as language of mathematics some time itself is obstuse and difficult. A teacher should try to give practical and physical examples to make it simple and obvious. There is misunderstanding in areas of mathemtics studies masters are very difficult to understand, that is smoking something. People who have profound depth and wisdom in those areas ( there are teachers who have lots of information and no deep knowledge start show of talking obscure).have great ability to simply the discussion and remove fluff and go to essence. I always try to give examples while teaching and make tution seeker give his moneys worth.
For people who want to study cheaply if possible buy old times published russian subsidised , you will find them very cheap in old karachi books in used section.
They are translated and they are written in very sinpme language and most are complete text books.3 volumes on real analysis and one book on calculus of variations of variation with hard cover only 125 rs. The book sellers are very happy to get read of as now no body uses text book and class notes which are so sloppy and full of mistakes corrected appendix will be larger than books. I understand this is fast food time but people who wrote such garbage should be whipped.
I have not written here any thing supporting MASADI. So thete is chance it may get posted. Whenever I write anything supporting him it is rejected.
Good day and good luck.

Re: # 37 Mr. GT ...I will suggest all of you interested in what you are talking to understand to study basic course in Real Analysis. The book I can recommend is Real Analysis by mr. Goldberg. Writing here little blindly is fad but I know there much more to than that.
Have good day.

#36 Posted by AlephNull

Aleph Null, thanks a ton.

Iron,

I briefly skimmed through the paper you reference. I am a bit confused and so should read it more carefully when I have the time. What confuses me is the following:

Let n stand for null.

Axiom 9 (I believe) states that n=-n. But then n=0. In which case we are back to square one and there is nothing different.

So suppose there is something else. Now take axiom 6 (I believe) which states that for all a and b: a-b=a+(-b).

But then Axioms 6 and 9 imply that n=0 and again we are back to square one.

If you have read the paper do let me know what mistake I am making.

GT #34,

Goldblatt’s ‘Lectures on the Hyperreals’ in the Springer GTM series constructs basis analysis from the ground up without using epsilon-delta. There is also a very slim book by Edward Nelson, ‘Radically Elementary Probability Theory’, that develops probability theory using a minimal amount of nonstandard analysis. These are the only two books I have any acquaintance with – there may be many other good ones.

re: 34 and re:31
there is something called transreal numbers. You can check out a guy called James Anderson ( http://www.bookofparagon.com) who did some work in this area (some publications (http://www.bookofparagon.com/Mathematics/PerspexMachineVIII.pdf).
If you go to the website, he seems to have used this idea ofor developing what he calls a Perspex Machine and has developed some sort of a proof for this,.

There is a lot on wiki as well (but I would go a bit easy and sound a note of caution with the the wiki criticisms).
Check it out - who knows might just kindle some thing anew here.....

#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