Author Topic: Defamiliarization in Mathematics ?  (Read 715 times)

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_id_Crisis_

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They should have called it Novel Math
« Reply #15 on: November 11, 2017, 02:03:14 pm »
Last night I was reading The new math : a political history, circa 2015, by Christopher Phillips.

The more I think about it, the more I think that movement ought to have been called "Novel Math" rather than "New Math".

novel - original or striking especially in conception or style

It's hard to believe that the Federal Government of the United States actually funded this program, School Mathematics Study Group, where mathematicians were placed in charge of writing a series of experimental textbooks, mostly focused on secondary schooling.

I won't go into details as I have written about my obsession with texts written by Mary Dolciani or Frank B. Allen or Bechenbach, Donnelly, Wooton, Sorgenfrey, Graham, Myers, or even Swanson, Sharron, Kane, Brown, Jurgensen ... You know the texts I mean, the ones that presented the material in a very novel and formal manner grounded in set theoretic notation.

This strange event is like science fiction to me.  That movement came and went rapidly.  It was strong for maybe 10 years and was rejected, replaced by a "Back to Basics" revolt.

But, for that brief period, mathematicians wrote some textbooks for high school mathematics, presenting in a manner that might prepare the minds for pure mathematics.

The revised editions of the Dolciani texts which appeared in the 1980's and even the ones from the 1990's which contained computer programming exercises using BASIC or PASCAL are really awesome, I think.

What I keep pointing out is that even if the "New Math" movement is seen as a failed experiment, I, for one, appreciate many of the textbooks from that period and I have hunted them down, along with any solution manuals I could find.

I even located a couple obscure texts by Frank B. Allen, one which experimented with presenting Linear Algebra to high school students (from 1961) which made it all the way to Germany.  I found a rare inexpensive copy at abebooks.com along with teacher's commentary.  I don't know why I am so fascinated and obsessed with such texts.   Maybe it has to do with the fact that there was a long decade between when I graduated high school (1985) and when I attended university, finally studying Linear Algebra, Multivariable Calculus, and Mathematical Reasoning (writing proofs) in the year 2000. 

The material is presented in a novel way, and those texts bridge a certain gulf between computational mathematics (algorithms) and the more abstract mathematics involved more with reasoning and less with "plug and chug" rote learning.  I feel these texts offer an extremely rare presentation of "school mathematics" from the perspective, not of educators, but of "pure mathematicians".   It had never been done before, and I doubt it will ever be presented this way again, not at that level, anyway.

The book I linked to, The new math : a political history,  is interesting.  It explains the political climate in which the project was funded.  What a unique event in the history of textbook publishing!   

Maybe my life's work might be intimately tied up with this phenomenon. 

They thought they were writing those texts for high school students and high school teachers (back in the 1960's).  In this science fiction saga, it turns out that a 50 year old Steppenwolf ends up being the unexpected receiver of the mind treasure, the gong-ter.

It is for this reason that, whenever I hear this phenomenon referred to as "The New Math" (used in a derogatory manner), I will refer to it as "A Novel series of mathematics textbooks created by the School Mathematics Study Group back in the mid-twentieth century".

 The "New Math" may have been mocked and rejected by popular culture, but I am one who appreciates those efforts, and I will treasure going through their texts, filling in the gaps and bridging the gulf between the technical computational math hacker and the elite "pure mathematician" ... The buzz word "new math" is misleading.   It's now so old school, and I might go as far as to use the term, "uncanny," to describe the feeling I get when I approach a textbook older than I am that presents "modern mathematics".

What was called "modern algebra" is now called "abstract algebra".

What was "modern analysis" is now called "real analysis" even though NJ Wildberger has some nasty things to say about so-called real numbers, like the fact that there is no way to represent them as a decimal number?  or is his gripe that there are an infinite number of real numbers between two points on the real number line?
« Last Edit: November 11, 2017, 03:49:55 pm by Non Serviam »
He [Arthur Schopenhauer] has been the most radical of all troublemakers. He was defiant. ~ (Marcuse?)

"Learning math is never a waste of time." ~ Ivan Savov

"Programming is understanding."  ~ Kristen Nygaard

{breathing piece of defecating meat destined to die}

raul

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Re: Defamiliarization in Mathematics ?
« Reply #16 on: November 11, 2017, 03:43:49 pm »
Hentrich,
I am not in a position to say anything about mathematics. The only thing I can say here is that almost thirty years ago the Ministry of Education changed the entire school program. Although this program had many flaws, at least one could learn but now the students have so many subjects they can hardly learn well. It is all confusing. As I understand there is a conspiracy against education here, there, everywhere. We have the Facultad Politécnica whose budget has been reduced. After all why would they need mathematicians if it is better to spend that money on blond secretaries for the  politicians. This world is a joke.

I think with that those vitamins you take, you are going to endure many nights of hard study.

Stay safe.

_id_Crisis_

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Re: Defamiliarization in Mathematics ?
« Reply #17 on: November 11, 2017, 06:47:08 pm »
Something very weird was going down back then with these math textbooks.  Since when have mathematicians ever been as involved in writing secondary school textbooks?  And it was being funded by the federal government.  It's like a lost treasure trove!

You know the rare text that I was very curious to investigate, Introduction to Matrix Algebra (Frank Allen et al, circa 1961)?

LOOK WHAT I FOUND in pdf format  ---> Introduction to Matrix Algebra, Student Text (unit number 23).  I am just curious to see how they approached it.  Nowadays I suspect most high school math texts are filled with photographs - a lot of gorty fluff.

That is the digitized version of this book selling on ebay which is in Germany somewhere.   That book got around.


It's aimed at high school students - part of the SMSG experiment.   Maybe it will contain some insights into basis, dimension, linear independence, etc.


I think that these might spark your interests and reawaken a deep curiosity in the Weirdness of Mathematics.   I haven't really looked at any of them yet, so I am not certain.   I intend to focus on the Dolciani series since I am most interested in the exercises; but at least there are plenty of pdf files of those old experimental texts, in case we are curios and have the inclination to explore.

Here is a link to
A Guide to the School Mathematics Study Group Records, 1958-1977


I will try to leave hyperlinks in this thread when I find direct links to the pdf files.

Unit 17 (Intermediate Mathematics) corresponds to what would be called "Algebra 2" in the secondary schools.   

Some other research into SMSG lead to these:  School Mathematics Study Group

Holden, please follow that link, then right click on the texts to choose download.

You might be interested in 09 to 20, then 37 and beyond ... It could be like when you found those old math books in Sanskrit ...

You never know, this could turn out to be some Weird Lovecraftian Tale and we are being personally invited to explore it. 

Then again, why bother?

What were they trying to accomplish with these texts?

My plate is already full ... but maybe I will get around to looking through some of these.  Who know?


PS:  Raul, there is a good PDF reader called Evince.  There is a version you can download that will run on Windows XP.

« Last Edit: January 30, 2019, 04:40:44 pm by Kaspar the Jaded »
He [Arthur Schopenhauer] has been the most radical of all troublemakers. He was defiant. ~ (Marcuse?)

"Learning math is never a waste of time." ~ Ivan Savov

"Programming is understanding."  ~ Kristen Nygaard

{breathing piece of defecating meat destined to die}

_id_Crisis_

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A Template for Adult Mathematics Education?
« Reply #18 on: November 12, 2017, 10:29:40 am »
Quote from: Raul
As I understand there is a conspiracy against education here, there, everywhere. We have the Facultad Politécnica whose budget has been reduced. After all why would they need mathematicians if it is better to spend that money on blond secretaries for the  politicians. This world is a joke.

I think with that those vitamins you take, you are going to endure many nights of hard study.

The world is a wicked joke, yes.   While education is praised, formal education is most often mocked.  In general, money talks, education walks.

Of course, in reality, humans have always learned in the normal course of their lives.

Modern conditions may require formal education in formal courses, but who can be trusted to be teaching what is relevant to us?  After all, I have witnessed the disappointment most graduates must experience.  Even after receiving A's or B's in difficult and challenging courses, when all was said and done, I was only more aware of how very little I knew and how little one can really learn in such environments. 

El mundo humano esta rebosante de enfermedades mentales.

One hears of "wars of the womb".  People still gather to dance and sing with children screaming with excitement. 


Back in the mid-twentieth century when money from the federal government (in the USA) was invested in the creation of the "modern mathematics" textbooks, they had the youth in mind, primarily "high school students", or what is known as "secondary schooling" as opposed to primary schooling (from elementary school up to "junior high school" (7th and 8th grades),   

In other words, it was a systematic approach, an attempt to cultivate intellectual discipline.   Who knows what the intentions were of the politicians.  I suspect it was motivated by the desire to advance military technology in some kind of imagined international competition between nation-states.  Whatever the motivations behind the money which put that program into action, the results were chaotic and unprecedented.   It really caused distress and confusion for so many.

My concern is not with institutions of compulsory education.

Also, as far as adults are concerned, the purpose of lifelong learning is most often highly job related.  This approach presents too narrow and limited an understanding of the nature, aims, and purpose of "lifelong education."

While lifelong learning may be mostly associated with technological development, I imagine that, when it comes to formal and rigorous study of mathematics, the concept of lifelong learning must be extended to encompass the needs of those who have fallen (or slithered) out of the work force.  To be blunt, the concept of lifelong learning must be extended to encourage the study of formal mathematics as a way to develop certain mental habits and engage in something more akin to art and music.

Too often our governments seem to only be concerned with training future scientists who will work in the military-industrial complex, but most are destined to participate as prisoners or guards in the prison complex, or as patients in the psychiatric wards or drug/alcohol rehabilitation centers.   The world is a horrible joke.

Lately I have been keeping my posts here short, but I could ramble on and on about my ideas concerning the development of more meaningful ways to enjoy long stretches of leisure time.   Many people are mentally wounded by too much time on their hands, and they want to keep busy with some kind of steady employment.   

I suppose there are others who only wish they had more leisure to engage in a personalized curriculum of lifelong learning.

In other words, while the motivation for having mathematicians write a series of mathematics texts was based on the political and military ambitions of the collective so-called leadership, my interest in their underappreciated and most often mocked achievement is to spend my time and energy revisiting mathematics in a formal manner - but not necessarily in a formal setting - for the personal rewards and satisfactions that this confers.

This encourages one to be in one's own orbit.

If certain societies wished to broaden the scope of institutional learning, they might wish to create more opportunities for adults to pursue further education.

My intentions are not to persuade politicians or their masters, the rich industrialists, to invest in creating such opportunities, but to point out that the individual need not wait for such opportunities to arise.  In fact, quite subversively, I wish to remove the careerist-oriented outlook from the equation altogether.

I want to study mathematics as an end in itself and not to become a better tool for the Overlords.

Many adults view learning as an opportunity to improve their professional and economic positions.  However, current socio-economic conditions lead to many adults feeling like losers amidst the present-day developments.  They are forced to participate in adult education courses and have not chosen to do so.  They do it because they must. 

I suppose that I ought to clarify that I only speak for myself and that I choose the lifelong study of mathematics as some kind of reincarnated Pythagorean, and not as a gort trying to find gainful employment in the military-industrial entertainment-prison complex.

I am suggesting the motivation of embracing lifelong learning of mathematics solely for the benefit of one's mental health and the cultivation of one's personal mental life.

It's a crazy idea that is not likely to find much support in mass society where people need shelter and access to food.

And so I simply represent one man stubbornly devoted to keeping his own head together in the midst of a world drowning in chaos and confusion.

I guess that what I am trying to say is that the nerdiest of the squares from the mid-twentieth century may have unintentionally passed down a template that can be used by those who were not their intended audience at all.  To be more specific, adults might discover that mathematics is not what they or their teachers have been lead to think it is.  What most people think of as mathematics is only one particular system.

Myself, I was always drawn to mind-altering drugs.  When you think about it, retraining the mind to think about mathematics in a novel way does alter the mind.

Maybe I am looking for a way to tap into the mind-boggling hipness of the absolute nerd.
« Last Edit: November 12, 2017, 11:53:44 am by Non Serviam »
He [Arthur Schopenhauer] has been the most radical of all troublemakers. He was defiant. ~ (Marcuse?)

"Learning math is never a waste of time." ~ Ivan Savov

"Programming is understanding."  ~ Kristen Nygaard

{breathing piece of defecating meat destined to die}

raul

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Re: Defamiliarization in Mathematics ?
« Reply #19 on: November 12, 2017, 04:58:14 pm »
Hentrich,
"El mundo humano esta rebosante de enfermedades mentales."
Sin ninguna duda, Profesor, ya que nuestro mundo es un gran manicomio y nosotros somos los pacientes.
Y los psicópatas son los médicos.

"One hears of "wars of the womb".  People still gather to dance and sing with children screaming with excitement.  "
As usual, among millions of tragedies every second, in the countryside a mother was denounced to the police for offering her two daughters for G.(guaranies) 50,000, roughly USD 10.00. The older girl had a baby suppossedly as a result of a **** by her own uncle. According to the police, the minors were given USD 10,00 each session and with that money the mother drank alcoholic beverages.

"My concern is not with institutions of compulsory education."
Almost everything is compulsory in this life. Compulsory birth, compulsory death, compulsory work or jobs,etc.etc.

"Many people are mentally wounded by too much time on their hands, and they want to keep busy with some kind of steady employment."
Our mental and physical wounds are/were a result of birth. 

"I suppose that I ought to clarify that I only speak for myself and that I choose the lifelong study of mathematics as some kind of reincarnated Pythagorean, and not as a gort trying to find gainful employment in the military-industrial entertainment-prison complex."
Pythagoras would have welcomed you and Holden in their sacred schools. These ancient Greeks and Romans studied the way you want to pursue your studies. Clearly much has been lost with these ancient learning traditions. The modern may know more but they do not have that mystique. All lost for "gainful employment in the military-industrial entertainment-prison complex."

Nerd or losers. We are attacked with these labels. We all are losers. We lose our innocence, we lose our youth, we lose our little pleasures, we lose our health, we lose our illusions. Who is not a loser?

Stay well and continue with the vitamins.
 





_id_Crisis_

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Re: Defamiliarization in Mathematics ?
« Reply #20 on: November 12, 2017, 10:36:01 pm »
Quote from: Raul
Almost everything is compulsory in this life. Compulsory birth, compulsory death, compulsory work or jobs,etc.etc.

This is true.  Maybe this is why I am able to embrace my decision to revisit mathematics with such devotion, since it is not compulsory nor demanded of me.   If this were forced on me, I most likely would rebel. 
He [Arthur Schopenhauer] has been the most radical of all troublemakers. He was defiant. ~ (Marcuse?)

"Learning math is never a waste of time." ~ Ivan Savov

"Programming is understanding."  ~ Kristen Nygaard

{breathing piece of defecating meat destined to die}

_id_Crisis_

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Re: Defamiliarization in Mathematics ?
« Reply #21 on: November 15, 2017, 11:04:01 pm »
Thoughts inspired by my reading from The New Math: A Political History

Note about "the New Math" phenomenon of the 1960's which the Dolciani series was a reflection of:

When I promote these texts, I am not saying that the experiment was NOT a failure, but only pointing out that those texts are still valuable to anyone who desires a conceptual and structural understanding of the nature of mathematics.  After the decline of the movement, it was asserted that, for the general population, training in mechanical skills and memorization techniques was what was desired.  Hence, the movement known as "Back to Basics."

While I enjoy calculating, computing, and especially algebraic manipulation, the reason I am force-feeding myself the old "modern mathematics" is because I am still very impressed with the SMSG's emphasis on the creative, flexible, structural thinker. 

The sensibility represented by the "Back to Basics" movement epitomized the backlash against the new math in the 1970's.  Proponents of Back to Basics claimed that overemphasis on mathematical structure had impeded the ability of students to calculate, and they encouraged schools to return to the "basics" of drilled exercises and rote memorization of arithmetic facts.  (Phillips, 2015)

In other words, the masses did not appreciate the mathematicians efforts to improve their minds, but only wanted their children to develop competence in computational skills.  And yet, when it comes to the approach to mathematics I want to take now, where I am already comfortable with my computational skills, I long to develop the kind of understanding Dolciani, Allen, and the minds behind SMSG's "modern mathematics" were concerned with.  They were not interested in helping students calculate quickly or accurately. 

It's not so much that the masses want their education "dumbed down," but that they can't stand to exert energy (and money) into anything that can not be put to practical use in daily life.  Maybe this situation is related to what Vonnegut was pointing out in Harrison Bergeron from the Welcome to the Monkey House collection. 

I don't think anyone should be concerned with what works for mass society, especially when it comes to your own personal lifelong education.

These reform movements in systematic education of the youth have political motivations. 

Higher test scores and the ability to calculate quickly does not imply mathematical understanding nor mathematical maturity. 

I really think that the teachers and parents who resisted (where Dolciani and other mathematicians who were writing text books were coming from) may have had firmly ingrained ideas about what mathematics is (the computation of numbers; that is, arithmetic) and may have been put off my the mention of "abstract algebra" such as sets, fields, or rings.

I have a different reaction to being exposed to those more abstract concepts.  Suddenly I get a glimpse of an underlying structure.  Of course, when this happens one tends to slow down, to think more slowly - because you are more into it, not just mechanically calculating or memorizing arithmetic "facts".

I understand that there are far more pressing concerns for most people and that I am in my own little world, but Phillips's book was released in 2015, and I have been very obsessed with the novelty of the books I intend to study exhaustively over the next couple of years.  When I say I am revisiting high school mathematics, I ought to make it clear that it is not just any old high school math that I am interested in.

I am specifically interested in how mathematicians would present it to the youth if they were in a position to do so, and, for a brief time in the mid-twentieth century in the United States, they were.   I caught the tail end of this when I attended a private high school in the 1960's, but by the time I got to the geometry, I began experiencing emotional disturbances which must have had an impact on my ability to concentrate.

So, maybe these texts weren't so hot for kids and teens.  I don't know what to think about all that.  What I do know is that, now, as an "old man," I am ready for the "new math," or, as I prefer to call it, the "novel approach to developing my conceptual and structural understanding of the nature of mathematics."

I am someone with computational skills who realizes I have never been and will never be a "mathematician."  This does not mean I am going to stop trying to develop a deeper appreciation for what mathematics really is and to approach it in a novel way with a Beginner's Mind. 

I like to compute and calculate, but I sense there is a deeper aspect to mathematics which Mary Dolciani and others were trying to inculcate on a mass level which ended up being pearls before the swine.

Man, I know this stuff is not important compared to the everyday problems people face such as paying for visits to the doctor and making sure they have a roof over their heads with some heat coming from the pipes or vents.  I understand that access to nutritional food takes precedence over understanding set notation or the logic of algebra.

How I have come to be obsessed with studying these books offers a clue as to why I frequently refer to my day to day reality as existential science fiction.  All the politics and actual government funding that went into that SMSG phenomenon which inspired Dolciani and others to create the series of texts, of which I chose about six or seven to focus on [Algebra I, Geometry (Jurgensen), Algebra 2 and Trigonometry, Introductory Analysis, Modern Introductory Analysis, Analytic Geometry, Matrix Algebra] - it was a freak event in history, and it did not take long for the texts to be rejected.   So, when I feel this strong compulsion to pay attention to the material and how it is presented, it's as though I am in contact with GHOSTS; but not the European conception of "phantoms" or "spectres", but in the way Robert Pirsig talks about "thoughts as ghosts" in Zen and the Art of Motorcycle Maintenance.


Maybe what I was trying to express with "the Defamiliarization of Mathematics" was to force oneself to approach it in a novel way, in a way where what you thought you were familiar with appears to have deeper layers.

I think this is why it is beneficial to consider different number systems.  For the novelty.

2 + 2 = 4 in base ten.

In base 3 (with digits 0, 1, and 2), 2 + 2 = 11

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Numerological Note:  This was post number 4444 at whybother.freeboards.org
« Last Edit: November 16, 2017, 09:43:22 am by Non Serviam »
He [Arthur Schopenhauer] has been the most radical of all troublemakers. He was defiant. ~ (Marcuse?)

"Learning math is never a waste of time." ~ Ivan Savov

"Programming is understanding."  ~ Kristen Nygaard

{breathing piece of defecating meat destined to die}

_id_Crisis_

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Re: Defamiliarization in Mathematics ?
« Reply #22 on: January 30, 2019, 01:05:07 am »
A reminder of the SMSG (early 1960's experimental high school math texts) : some links

I am just finishing up my notes from going through all the exercises in Introduction to Matrix Algebra, circa 1960.  I am presently going through the APPENDIX, which contains "Research Exercises."   The first exercise covers the topic of quaternions in the guise of matrices.   It is so interesting that I feel like I have ingested some mild drug (no, it's not coffee!  It's well passed midnight).

I made it through the text in a few weeks, filling nearly two entire sketch pads.  Fortunately, awhile back I had tracked down the Teacher's Commentary (called "Unit 24") --- found it somewhere in Germany of all places off ebay for about only $12.   SMSG was based in the United States, so I was excited to find the movement had found its way all the way across the pond to that mysterious place I am forbidden to love.

Whoever sold it to me must have known what a blessing it was to me.  The solutions in the Teacher's Commentary were essential to fully appreciating the material.  (I had to highlight this in red to clarify that, for many of the exercises, I most likely would not have even known where to begin, as far as what kind of solution was expected, if I had not happened to be destined to be the receiver of such an odd gem/relic.)  I really feel blessed to have gone through hardbacks, and to have acquired them at such generously low prices.  It appears nearly impossible to even track down a digitized scanned version of the Teacher's Commentary for the Matrix Algebra module, and there is no site of another hard copy, at least not one that anyone is willing to part with.  It is for this very reason that I have basically recorded in my own writing in large sketch pads nearly the entire contents, altering some of the notation to suit my tastes, and adding explanations when I was able to do so.  So, in effect, I pulled a modern day monk project off in a few weeks, and I think my hand-written copy may take on a certain glow, as long as I don't make too many smudge marks.  It's all in pencil!   

I just can't bring myself to work with a pen doing math(s) of any kind.  I constantly need an eraser.  So be it.  That is my preference.  I know this will make the writing disappear much sooner, but, hell, we all fade away fairly quickly relative to "geological time," in a flash actually.

If you (Holden) ever go through the digital version (called Unit 23), do not hesitate to request detailed solutions to problems.  I would scan pages from my own notes as well as original Teacher's Commentary (unit 24), and upload to Dropbox (which is full), deleting several files from the voice diary of a drunken madman.

Note that the Analytic Geometry UNIT 64 module DOES have the Teacher's Commentary UNIT 65 module digitized.  It covers conic sections and even 3-space.

I think you will find the approach they took refreshing in comparison to our 21st century track.
« Last Edit: January 30, 2019, 04:30:53 pm by Kaspar the Jaded »
He [Arthur Schopenhauer] has been the most radical of all troublemakers. He was defiant. ~ (Marcuse?)

"Learning math is never a waste of time." ~ Ivan Savov

"Programming is understanding."  ~ Kristen Nygaard

{breathing piece of defecating meat destined to die}

_id_Crisis_

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Revelation:  Those proof-based texts they were experimenting with in the 1960's/1980's [Frank Allen et al including Mary Dolciani and Edwin Beckenbach] were far too majestically advanced for high school students, hence the nervous breakdowns.  They could prove useful to those who are drawn to the Quest for Teaching Themselves Things Their Teachers May Not Have Understood.

Where these texts/courses have proved invaluable to me as an aged mathematically inclined computer programmer /SLASH/ computationally inclined math hobbyist [dead-Beat pHilosopher], is that I can combine the old school hand-written techniques with the bleeding-edge Open Source Computer Algebra Systems as well as personal collection of math-oriented C++ programs [all inspired by re-working through the Encyclopedic Thought Provoking Exercises, including proofs of theorems translated into computer programs as a tired old toothless  & bald-headed Twig Man, rather than as a suicidal teenager].  Eureka.  It's all upside down and backwards as per usual.   Maybe my working diligently through these texts over these years, leaving a trail of organized notes and programs is actually a work of significance for older students rather than "traditional high school jailbirds".

I mean, the books can be approached as a Religious Vocation [rather than as Robo-Scientist-Business-WizKid Training] which may lead to nervous breakdowns in more sensitive students not well-groomed for such Mental Pressures during the topsy-turvy chaos of adolescence [contemplating the philosophy of suicide may take precedence over even the most captivating cerebral abstractions, cognitions, and cogitations].

Now, as one who most likely "ought to be dead or not born," it is a real blessing, this reawakening interest and devotion to an Edifice of Thought.

Also, thoughts of my nephew and how a study of the Cartwright biography of Schopenhauer might help him deal with the estrangement from his own mother, my sister.   Whereas the power of Schopenhauer's intellect may have been what blew my nephews mind as a teenager, I think that the living pain of Schopenhauer's lived-reality, all that intense and painful agony/hatred/anger---and-universal-compassion-for-all-that-is, I think that as one ages, these issues become something that has shaped character (damage is done?).  My mother shows me affection, so in this way Schopenhauer's emotional life is more closely akin to my nephew's than my own.

« Last Edit: May 28, 2019, 11:48:19 pm by H »
He [Arthur Schopenhauer] has been the most radical of all troublemakers. He was defiant. ~ (Marcuse?)

"Learning math is never a waste of time." ~ Ivan Savov

"Programming is understanding."  ~ Kristen Nygaard

{breathing piece of defecating meat destined to die}

Holden

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Re: Defamiliarization in Mathematics ?
« Reply #24 on: May 29, 2019, 04:26:58 am »
I am starting to see how programming  is ,in its essence, mathematics. However,  in order to  do justice to the subject I must, for the time being ,focus  exclusively  on mathematics.  I  am developing, slowly but  surely,better understanding of  mathematics.
I mentioned  modal logic  as  it paved  way to the "rebirth"  of  metaphysics after  the analytic  philosophers  had thought  they  had  killed  it off  for good.

There  is  a disanalogy between  the statement that  gold   is  Au  and the statement  that pain is a brain state,say, brain state  H.
In the case of gold ,there could  be a gold mimic  that is not Au.In such a case,we could say that this  gold mimic is  not gold. It is a contingent fact, if it is a  fact  at all,that  everything  that looks and  feels  like gold is gold.

With pain on the other hand,the situation is different, because nothing could be a pain  mimic.If something feels like pain ,then it is pain..


I  really hope your nephew reads  Schopenhauer's biography.
I am just a sad little green  tortoise  who crawls and crawls..