Rote Learning Vis-à-vis Physical Comprehension

#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.

#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.

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.....

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.

#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.

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.

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

Re: # 41 sorry physical / visual problems creeping up
The fist line from King Lear is wrongly written , so all lines, it has good rhyme
"oldest have born most
We that are young
shall never see so much
Nor live so long" Albany ( King Lear)"
what a poetic prose style writing I read when,young and then always remembered

Masadi mian,

An Indian poll recently conducted polls in Pakistan and India. One poll was about the founding fathers:

Jinnah's approval ratings in Pakistan 97% in India 43 %

Gandhi's approval ratings in Pakistan 29% in India 85%

So your wishes are not going to come true.

Your failure to argue with facts and to repeat the illogic of the Mullah freaks who called Jinnah kafir-e-Azam in the 1940s shows your desperation.

So keep repeating your lies. Pakistanis are done believing liars like you who are their biggest enemies. And I speak for the common man... not you- a third rate reject.

Masadi the genius writes:

"Yes, I am for real and I am the worst nightmare of a-holes of the West"

Manto writes "Masadi mian,

An Indian poll recently conducted polls in Pakistan and India. One poll was about the founding fathers:

Jinnah's approval ratings in Pakistan 97% in India 43 %"

One things third rate lawyers should never do is throw out polls to social scientists as "proof". What kind of nonsense is this "approval rating"- approving of what? Of course they are going to approve just like the poll that Zia ul Haq conducted asking Pakistani's if they "wanted Islam"- what were they going to say? That the Church of MAJ has taboo attached with rejection is why you get these high poll numbers, dig a little deeper with REAL questions and you'll get to know just how deep the resentment is with lackeys of the West...

Masadi mian,

What about 43% in India? Is there a Church of MAJ there as well?

We went through the discussion in detail and you failed to answer my points. So there is no point really in going in circles. You are utterly incapable of civilised academic dialogue based on facts.

Gill Sahib,

Nice to read this article. You have highlighted the superiority of practical comprehension over rote memorization. Obviously I have to agree with it. But in addition, I recognize another form of understanding which, I think, is superior to even practical comprehensions. I can share my following article which can explain my view in detail. This is one of my early writings so it can lack maturity of expression:

Practical and Intellectual Understanding

http://khuram.wordpress.com/2006/08/23/practical-and-intell ectual-understanding/

Practical Understanding is very good. But "Intellectual Understanding" is even more good. With "practical understanding", you just "practically implement" already existing theories. But with "intellectual understanding", you invent and make new theories! I have explained this point in my above-referred article.

regards!