Author Topic: Individualized Regrouping  (Read 3788 times)

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Nation of One

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Individualized Regrouping
« on: September 09, 2020, 07:06:21 am »
For my own nervous system, whatever it is we mean when we utter or write the symbol "I" in English, "Ich" in German, the so-called "Subject," is deluded enough this morning to make an attempt to proceed with plans to organize some notes (of a purely mathematical nature) to be scanned up into The Wayback Machine.

It is actually a clever postponement of staring directly into the Abyss.

Before proceeding with this projection of a project, I will gather some related posts for my own amusement (to order my consciousness about, to direct my animal being, to give an illusion of order to my existence):

1.  An Invitation to Explore So-called "Pure Mathematics"  (from 4 April 2017)

Quote from: Hentrich
I was doing some research on Mary Dolciani and what was considered "New Math" in the 1960's and 1970's.   Evidently, she died in 1985, the year I was a senior in high school where Modern Introductory Analysis was the text we used. 

In 1987, the book called "Introductory Analysis" was published.  She is listed as a co-author even though she had already deceased.  Most of the material overlaps, but there is a some attention paid to Calculus.


... I suspect that what was considered "New Math" was how the material is presented, using set theory notation, rigorous proofs, definitions, etc

Maybe when they "reformed" the curriculum, they included some applied mathematics.

All I know is that I am looking upon my current study as the manifestation of a desire to become more familiar with pure mathematics.  For whatever reasons, I feel much more comfortable thinking of it as "pure mathematics" than as "pre-calculus".   I don't need to bone up on pre-calculus as I have a firm grasp on the algebra, geometry, and trigonometry used in "the Calculus". 

My entire reason for going through "Modern Introductory Analysis" [Mary P. Dolciani, Edwin F. Beckenbach, Alfred J. Donnelly, Ray C. Jurgensen, William Wooton] as well as "Introductory Analysis" [Mary P. Dolciani, David Myers, Robert Sorgenfrey, John Graham] is to, as I already stated, become more comfortable using formal notation, and to finally, over 30 years later, learn how to construct such proofs.  This is where I am still a little baffled.

Why do you think it makes such a difference to me what I call it (referring to this as pure mathematics as opposed to "precalculus")?   I think it is because the material is presented in such a different way than in texts that deal more with computational, mechanical drills.  It has to do with the presentation being formal as opposed to just mechanical computational drills.

The exercises actually force me to think, and I realize this is the kind of mathematics that I find so challenging, as opposed to the calculation-obsessed drills of differentiation and integration, which I have certainly had my fill of.

One might wonder why I can't just study everything at once.  Why do I feel I have to stop dead in my tracks, putting Multivariable Calculus, Vector Calculus, and even Physics on the shelf?

This is just the way I am wired.  As this exploration will take 10 to 15 years, I figure it is best to rebuild my foundations, retrain my brain to be more comfortable with the more "pure" approach, before continuing with the "applied mathematics" one finds in Physics, Differential Equations, Computational Physics, etc ...

I really sympathize with the youth who aspire to become "mathematicians".  Do they yet realize how life can get in the way and totally mock their plans?

My main question is:  why do you think it matters to me what I call it?   I suspect that I would not be able to justify to myself studying "precalculus" as it might indicate something remedial - and it is not remedial, it is challenging to me to approach the material in such a formal manner.

Maybe clarifying that I am trying to better understand "pure mathematics" is a way to make it clear once and for all that I have no intentions of putting anything I learn to practical use.  Besides that, since the Physics and other "applied mathematics" texts are so intimidatingly fat, I figure I have the rest of my life to get into them, and I don't want to drown in that swamp of books until I face my uncertainties concerning pure mathematics.

As long as I am here on this earth for no particular reason, I think I owe it to myself to spend as much time as possible concentrating on the aspects of mathematical training that are mysteriously absent in university education. 

Even in courses such as Mathematical Reasoning, where the focus is on writing proofs, the foundations of the applied mathematics courses are never really explored.

In this world which rushes the youth from grade school to high school full steam ahead toward Calculus and Physics, there is not much "basic training" for those who need to be spoon fed "pure mathematics".

The reason I chose to major in Computer Science was because I was getting a grant from the government, and it was taken for granted that I would find some kind of job/career afterwards.

So I focused on applied mathematics.

Now I have the benefit of living without any hope of ever being employed as anything other than a janitor, so I am free of the delusions that I study so as to find a place in the so-called "modern day workforce".   I simply don't have the necessary servile temperament to be some kind of "professional scientist".

My failure to find any kind of vocation has liberated me to do nothing with my life.

That is why I have this opportunity to return to the text that was presented to us (at age 17) as "AP Calculus", and to approach it with my own personal initiation into "Pure Mathematics".

There seems to be a psychological aspect to the mind set of the student.  If one approaches the text as an invitation to Pure Mathematics, then there is the potential for a quasi-religious encounter; but if one sees it as "precalculus" or a stepping stone to a career based in applied mathematics and science, well, it somehow gets contaminated by the student himself/herself with questions such as "what use is any of this?"

It doesn't have to have any use.

Student:  "When am I going to USE any of this?"

Teacher: "When you find yourself on welfare living off foodstamps, if you choose to continue to explore these areas of knowledge, you can lock yourself in a fortress of solitude and resist the conspiracy against you, the conspiracy that wants to see you crawling on the floor looking for a speck of crack co-caine."

"Should you find yourself in a similar predicament as H.P. Lovecraft and countless others like him, and if you have no particular talent for writing stories, you can spend many years apparently doing nothing whatsoever while developing what we call mathematical maturity."

Student:  "But, if I may be so bold to ask, oh wise teacher, what is the use of mathematical maturity?  Will it put food on the table?  Will it pay the landlord?"

Teacher: "Well, probably not; but you will develop an inner life of the mind in the process, and this might afford you enough detachment from practical concerns that you will be satisfied living on bean soup and sleeping on the floor."

"How's that for motivational encouragement?"

2. Solution: Lowering expectations of How "Mathematically Mature" I will become

Quote from: Hentrich
OK, you know my story.  I graduated from the university at the ripe old age of 35 in 2002.

Lowering my expectations of how "Mathematically Mature" I will become might actually be a manifestation of maturing.  Schopenhauer counsels us not to try to be something we are not as it will only cause us grief.

I do not expect to become a mathematician, but I am prepared to spend my life "tinkering with mathematics".  I'm just a tinkerer ... my plate is full.

If I were ever condemned to be held prisoner for a long period of time, I would become severely depressed unless I could continue studying in and just beyond my comfort zone.


If the jail cell were to be transformed into the little room I dwell in today, I would have been filled with thankfulness.   I refuse to ruin the delight of learning by constantly biitching about how "I will never be a mathematician" or "I must have smoked too much crack and drank too much booze" or "maybe I am a little brain damaged". 


I still have to think very hard to figure out how to solve problems (especially involving writing a formal proof) that I am finding in these special books, "Modern Introductory Analysis" as well as "Introductory Analysis".

I wonder why this material is not covered in this manner in community college or university.   Maybe I am a rebel monk who has stumbled quite accidentally upon forbidden knowledge by stubbornly looking precisely where one would never look in a thousand years.  First of all, who is going to look in a "high school" text?   Secondly, who would think that the material would be presented in a superior manner way back in the 1970's and 1980's than it is now?

OK, so I am a little ridiculous and a bit comical, but I am going to "make believe" I am a toothless monk of an ancient cult, older than the Abrahamic triad, older than the Devil even.   To paraphrase Lovecraft, I daydream of being influenced by a qualitas occulta far older than even the aborigines of North America ... non-human ...

And I am guided by these specters ... guided to these old vintage math books which take on a glow ... saying, "When the student is ready, the Teacher appears."

This student is old.  This student is ready.  How I shun the elite academic mathematicians and just want to be left alone to tinker ... to allow myself to say, "I don't get it."

And to allow myself to be "retarded" ... and to begin to grasp something, and to make genuine breakthroughs that would seem pathetic to the snobs of academia.   

I become fixated on certain books and they become the Holy Grail to me ... in my own little world where I am this child growing to be an old man rather swiftly and almost happily.


3. Review of SMSG inspired textbooks (part 5: ANALYSIS ++)

_______________________________________________________________
Like the H-Diaries thread, I will be pecking away at this "uploading of the math diaries" throughout the seasons just to be able to rest in some peace when these bones return to the Devil.  The thread will contain links to the notebooks uploaded to archive.org [Gorticide].  I am a living man, meaning a bundle of nerves, so I cannot imagine how I will manage to follow through with the task.   I suppose I wish to give the Old Gods a chuckle.    :P

Most of the notes are written in cursive, so these are for the most curious students only.
« Last Edit: September 09, 2020, 07:56:30 am by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

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Nation of One

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Re: Individualized Regrouping
« Reply #1 on: September 09, 2020, 09:01:00 am »
This must be where the existential exhaustion kicks in.   There are but a limited amount of breaths we inhale and exhale.  There is only so much energy.  There is the dusting and housework.  There is the continual feeding of the organism ... the keeping oneself alive, dry, hydrated, etc.

Where does logic and set notation get squeezed in?  In between power naps?  In between fits of despair?
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: Individualized Regrouping
« Reply #2 on: September 09, 2020, 04:18:06 pm »
Mathematics ,for me,is almost as puzzling as my existence is and the two are not necessarily separate.Indeed,for me, a lot of the mystery that surrounds existence is because of mathematics.For me, a sad,painful existence is not a hindrance, rather a motivation for studying mathematics.

And even if I failed to learn mathematics, I could always, learn to revel in the suffering caused by my failure to learn it. One more failure, in a life made up of failures-that is okay.
« Last Edit: September 09, 2020, 04:23:10 pm by Holden »
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

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Re: Individualized Regrouping
« Reply #3 on: September 12, 2020, 09:04:00 am »
It takes a great deal of radically subjective defiance simply to muster the courage to stand up to Nature Itself and cry foul.

We are thrown into a life where we are introduced to intellectual realms which are trampled under foot by the Head Gorts in Charge, along with their army of careerists looking for job security on the front lines of the local Slave Patrols in the global industrialized medical/entertainment Farm Complex.

I pity the students who develop a passion for mathematics, philosophy, or ancient spiritual practices, and are then exposed to the epidemics of madness which plague our species at this point in our "devolution."
« Last Edit: September 16, 2020, 06:53:11 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: Individualized Regrouping
« Reply #4 on: September 12, 2020, 07:02:33 pm »
Such students ,the kind your mention, must tell themselves ,frequently, that the path they have chosen for themselves might well take them to the edge of a forest where they would be required to starve themselves to death.

Cioran mentions that he read somewhere in the Upanishads that a truly honest man can continue to live on earth for just a couple of weeks before he must embrace death.

Such math students must realise that the only way to study math is the way Van Gogh painted-without any hope of ever making a farthing while one is still around.

In India such students would be run out of town. No one knows that I study mathematics in my room, if they ever came to know, then all I would be offered would be endless mockery and taunting. Only people in their late teens are supposed to study such things in India.

If the Buddha were here today, he would be have been given a good scolding for wasting his time thinking about useless things,he would have been put to work,maybe offered an axe to cut down the tree he liked to sit under and sell off the wood in order to contribute to the GDP.

Its true. One who wishes to live in such a society, must first of all realise that,for all intents and purposes, he is dead already. Only then can he study.
He must be like the violet in the middle of the forest, no human eye would ever see.
How can one man withstand the combined force  of  the whole of one's society.A society which produces for its own sake-both goods and people.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

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Re: Individualized Regrouping
« Reply #5 on: September 12, 2020, 10:48:10 pm »
Quote from: Holden
If the Buddha were here today, he would be have been given a good scolding for wasting his time thinking about useless things,he would have been put to work,maybe offered an axe to cut down the tree he liked to sit under and sell off the wood in order to contribute to the GDP.

Its true. One who wishes to live in such a society, must first of all realise that,for all intents and purposes, he is dead already. Only then can he study.

This leads us back Full Circle to the Tyranny of Public Opinion.

« Last Edit: September 13, 2020, 09:06:03 am by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

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Re: Individualized Regrouping
« Reply #6 on: September 20, 2020, 11:23:05 pm »
It is true that one must live as though one were already dead, that is, with an eye on one's death and what one might wish to leave to the coming generations.  We will pass quickly.  There's just enough time for a quick joke, so squeezing in any encyclopedic solution manuals to obscure series of texts is most likely aiming too high, but one can certainly give it Hell.  A lifetime of work, a secret magnum opus ... may be done in vain, or just might have a small cult of receivers. 

Either way, there is a spooky sensation in concentrating on one's effect postmortem, as opposed to day-to-day local street politics, gossip, etc.
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

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Re: Individualized Regrouping
« Reply #7 on: September 24, 2020, 09:49:34 am »
With self-development, there is no One Right PathTake your choice and run with it.  Don't let anyone make your decisions for you. 

The math I study may serve no purpose other than as a personal practice for me to force myself to think, and as a way to defiantly focus on my own path ... and, of course, most importantly, to help me keep my eye OFF the ball.  That is, with this obsession with understanding proofs and theorems, I might discover a secret passageway out of the prison of personality (individual versus society) altogether.

It is not such a bad thing for a wolf to fail dog-obedience training class.   
« Last Edit: September 24, 2020, 11:39:12 am by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: Individualized Regrouping
« Reply #8 on: September 24, 2020, 05:46:20 pm »
I think I am being able to solve many maths problems that I might have struggled with an year back.

But I do employ a great deal of brute force to get to the solution and my attempts are hardly ever elegant.Not to mention the fact that I would take probably 50 minutes to solve a problem where a competent mathematician might take 4 or 5.

I do it because as long as I do it I do not need to get involved with the other people most of whom are horrible.

Maybe someday my solutions would get more elegant ,or maybe not.
But I do like what I am doing with maths a great deal.

A long time back I watched a documentary about a mountaineer who was trying to climb the Everest or K2 or some such mighty mountain.Well, he slipped and lost his legs.

After a few years of physical therapy and with artificial limbs he decides to come back to the same mountain.

He is slow and lumbering,but,well,at  least he is there.

Sometimes I go round and round in circles trying to solve a problem and after an hour or so I wring the bastard's neck.

When I look up at the solution in the book I manage to follow most of the arguments and the steps but on my own,while often I manage to reach the solution,it takes me a lot of time to do that.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

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Re: Individualized Regrouping
« Reply #9 on: September 24, 2020, 06:08:22 pm »
I find myself mesmerized by having once familiar notation becom familiar once again.   To witness where I was truly first exposed to the notation puts things into perspective for me.   I was not ready as a young man.   Now, I suppose I am more settled down, and seeing as there is nothing to be had in this world, I can give the material some attention.   There is a definite delight I experience when scribbling with pen or pencil something like f = { (x, y) | y = x + 3, x in Reals}.

or even the little circle for composition of functions:  f o g = f[ g(x) ]

[ f o ( g o h) ](x) = f[ (g o h)(x) ] = f( g[h(x)] ) ...

I prefer writing with pen/pencil as opposed to typing, so, when writing mathematically-oriented programs, I really have to be in a particular mood.  It does not happen on command.

I must read my old code as though it were written by someone else altogether.  I read my mathematics notebooks the same way.  Concentration is always required, and I find it difficult to trust my own thinking.   I really am still quite uncomfortable with writing any proofs whatsoever.   I am not ashamed of this, just being honest.    The closest thing to an "organic proof" I can get is writing a computer program which applies the theorems.
« Last Edit: September 24, 2020, 07:46:38 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

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Re: Individualized Regrouping
« Reply #10 on: September 26, 2020, 10:33:01 am »
As for my discomfort anxiety while attempting to write a proof, there is this sense of being ill-equipped as a wordsmith to come up with the proper way of stating the situation.
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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To the Man with Van Gogh Eyes
« Reply #11 on: October 07, 2020, 04:01:11 pm »
Gauss was a  contemporary of Schopenhauer and so I am quite sure he would have know him.The two people he did mention in the preface of his chief work,namely, Plato and Kant, considered mathematics to be of great value.
In fact, mathematics is at the very heart of not only Plato's but also Kant's thought. I have been studying mathematics a great deal, in fact, all the time when I am not working as a slave and yet, most people, at least most who are around me, seem to be better at it than I am.

No matter, I would continue to study, there are many things that I do understand now that I did not ,say, two and a half years back.It is not explicitly mentioned in any of Van Gogh's biographies I have read, but I really think one of the strongest reasons why he liked painting so much was because he did not have to be put up with other people while he was painting.

Mathematics is a lot like that.When I am trying to understand a problem,often, I manage to understand just some steps and not the others but the fact that while I am doing so I don't have to think of the people I work with and others ,is a real blessing.I don't aspire to be a Gauss. But I would very like to be able to understand at least basic mathematics before my time is up and I am not sure how much more time I might still have with me.

To tell the truth, sometimes studying mathematics feels like a one sided love affair. Thankfully, I am no stranger to being spurned.

« Last Edit: October 07, 2020, 04:03:47 pm by Holden »
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

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Re: Individualized Regrouping
« Reply #12 on: October 07, 2020, 04:54:13 pm »
Men have called me mad but the question is not settled, whether madness is or is not the loftiest intelligence - whether much that is glorious - whether all that is profound - does not spring from the disease of thought - from modes of mind exalted at the expense of the general intellect.

— Edgar Allen Poe

In order to even approach mathematics without "performance anxiety," one would have to surmount the tyranny of public opinion, which flaunts the lifestyles of the rich and famous, and makes minced meat out of malnurished students of mathematics.  It's an evil world, indeed.

I have invested in blank notebooks with no lines, some 2B pencils, erasers ... and the books I had wished to focus on which I had tracked down several years back, have been back-tracking to show pre-requisite material, and must thank Jupiter just for being able to keep the kindle going to burn the fire of enthusiasm.

That is the power, right there.  So, it is more akin to keeping a fire going to keep from freezing to death, as opposed to a one way unreciprocated passion for intimacy.  Going around the ideas and the notation long enough for one to become ever more familiar with them may present a calm mood where the exploratory parts of the brain are stimulated.  This must require a powerful sense of inwardness, since it all exists in the mind, as an abstraction.  I have dained to preserving my work simply as an edifice of how I would go about organizing and solving ... as a resource for the rare archeologist of such things.  I imagine our species will survive in one form or another, even if in some military base underground.   Maybe one of their offspring will be Fated to stumble upon them on the archives.

It is a labor of love.  I am making use of this energy and passion to preserve some kind of capsule documenting a transformational period in education, a presentation that is not likely to occur again.

Some time this winter the digital versions will be forthcoming, but I want also to upload the related notebooks containing the exercises, which form the real crux and value of the project as a whole.  I am pecking away at both ends of this "work".

It will take a long time to upload all the notebooks, but I will work backwards from where I am at now ... that is, I am still working through the last two texts, which are, the most challenging of all.  Using two different versions/editions of the text have to do with transformations from tables to calculators/computers, one from 1964, the other from 1988.  I find it is important to include both.  Calculating without instruments is a curious phenomena which demands things such as patience.  To the outside world, I might be made out to be an eccentric loser, but I know in my heart that I must be a rare specimen to take on such a task, and to have made so much headway on it already!

It is as Schopenhauer said of his work, that it felt as though the Holy Ghost were doing the work.

I think I might be able to share my "style" (of approaching mathematics exercises) with some others in the future, after I croak.  I think the academics call it an immortality project.  I am also paying honor to those mathematicians who attempted to present mathematics with formality and rigor to high school students, even if we were not ready for such depth.
« Last Edit: October 08, 2020, 02:54:30 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~