Life as Perpetual Misery Machine
General Category => Why Mathematics? => Topic started by: gorticide on April 08, 2017, 12:51:35 pm

Something occurred to me in a flash of the reoccurring frustrations I experience during my stubborn lifelong engagement with mathematics. Has anyone ever applied "defamiliarization" to the process of studying mathematics?
I suspect not.
Also, this might be what Holden and I have been experiencing throughout our lives when it comes to mathematics.
It doesn't matter if I am looking at a Physics problem, a problem from a Differential Equations text, a Linear Algebra text, a Multivariable Calculus text, or even just some high school "precalculus text".
Sure, the computational aspects are mechanical and even boring; but when I am in a state of  the only word that seems to make sense here  DEFAMILIARITY, then it is as though I had never thought about the problem before.
When I had started attending a local community college at age 27, I had taken a test so I was allowed to start with "Calculus". Even though I had been out of high school for almost 10 years, I was determined to dive right in, and I did well. I most enjoyed all the algebra and trigonometry one must use when one encounters calculus problems. I would learn on the fly, greatly appreciating the algebraic and trigonometric pointers the instructor gave us along the way.
What I learned enabled me to pull off A's in Physics I and II ... and I even went so far as to get an A in "Multivariable Calculus" after transferring to the State University. Still, something was missing. I was never very confident in what I was doing.
I never want to experience such pressure to race through subjects like that again. In fact, you might say that my entire existence represents an authentic refusal to jump through such hoops. If I were to pick up a precalculus text, it will still take me a year to go through it if I am honest; and I have to tell you this not only frustrates me, because I want to be getting back into the physics and more advanced mathematics, but it also makes me suspect that this world is totally full of shiit, and that a great deal of formal education involves selfdelusion. Tests, exams, grades, diplomas all reinforce the delusions of mastery. Then there are poor honest devils like Holden and I who are just too honest to play along, to "fake it until you make it".
It makes me very suspicious and paranoid.
It makes me want to scream.
I look at the physics books and realize that it will take me several years ... and a good year, most likely, before I even motivate myself to start. I've had a few false starts already.
One almost envies those who just say they're not "good at math" or those who just resign themselves to alcoholic oblivion.
I want to allow the "inner Artaud" to speak, if I may be so spontaneous: I despise all those who are responsible for projecting a social hierarchy on levels of mathematics.
I want so badly for a professor of some advanced mathematics course to openly admit that most of his students are selfdeluded egotists who would be just as challenged going over some of the more difficult exercises found in less advanced mathematics courses.
Holden, I have an idea for our own private school of mathematical philosophy. It will be disguised as a joke, as it must be. We can refer to ourselves as Mathematical Defamiliarists.
What do you think my "problem" is, Holden? I ask you, with all honesty and sincerity.
I seem to be able to hold my own with whatever I decide to study, but by experimenting with "going back to more fundamental and elementary topics", I see that I have a low frustration tolerance. I get frustrated when computations are too [easy] mechanical, but I become equally frustrated when I draw a blank when asked to write a formal proof.
In another post you mention Heidegger. I have absolutely no patience for any obfuscators.
When people write mathematics textbooks, they are most likely using other mathematics texts as guides.
No one seems to be willing to expose what we really are when presenting material.
Let me be perfectly clear about Defamiliarizing mathematics.
All one needs to do in order to experience this is to pick up a math text meant for advanced high school students, and work through the exercises. Pay special attention to the more difficult problems ...
Then imagine the act of coitus and how many will pride themselves on replicating their DNA. It is a cruel joke.
We are slime programmed to ejaculate, and an understanding of mathematical ideas does nothing to ensure one's survival as an organism.
So I am applying a term preserved for the artists, defamiliarization.
When applied to one's personal efforts to educate oneself, this will either lead deeper into insanity (a religion of one) or will have a liberating effect, serving as an antidote against the Delusion of Acquired Knowledge.
Rather than saying you are "slow with mathematics", you can declare yourself to be a "Mathematical Defamiliarist"! :D
You understand my sick sense of humor, right?
I am kind of mocking all the complicated jargon.
Do you remember Cioran's comment about the kind of aphorisms he wanted to produce?
He wanted something that could be explained to a drunkard or whispered into the ear of a dying man.
We all die like animals, hopefully not run over by a motor vehicle, but this is quite probable.
Why does this animal study mathematics? What is "advanced mathematics"?
If "advanced mathematics" is taught in a high school, does this mean it ought to be easier than "elementary mathematics" taught in a college? WTF!
Aha ... do you see what I am getting at here?
Madness!!!!
This is exactly the kind of egodeflating situation I wanted to force myself into.
You know, when I would run into some frustrating confusion working on heavy duty integrals, I could still comfort myself saying to myself, "Well, this is integral calculus!"
I guess I wanted to work with this old "Modern Introductory Analysis" text from my suicidal high school days so that I could face the anxiety of running into confusion even with something more basic, simply because it is presented in such a formal manner.
Holden! Do you realize what a break through this kind of approach could be?
Rather than studying mathematics to "become smarter", one might study mathematics to "feel more stupid", even if one is becoming smarter the stupider one feels and stupider the smarter one feels.
Am I making any sense? It's as though I enjoy the process of discovering just how thin of a grasp I have. There is this great relief to face that I don't really understand all that much. There is something refreshing about this realization.
Maybe I study math because I like to feel stupid. What I mean is, part of me longs to be in touch with reality, and not to be deluded. Maybe feeling stupid is a sign of a special sort of intelligence. The converse might also be true: those who feel very smart may just be deluded.
Up is down and down is up. The Inversion Principle?
This could be called a kind of "gospel of defamiliarization." :P
AFTERTHOUGHT: Once again, I catch myself using this word, "stupid", when a more accurate term would be "disoriented".
Once I get passed the ego, I experience a subtle thrill in the realization that I am learning something new. It is particularly thrilling due to the fact that this is a textbook I don't remember learning anything from when it was our book senior year in high school. As I have confided several times already, I experienced an existential meltdown that year, what must have been an actual nervous breakdown.
So, all these years later, my humility is rewarded. It took great humility to pause from working through "multivariable calculus", to forget about physics for awhile, and to commit myself to this strange mathematical engagement with the old "New Math" presented in Dolciani's "Modern" Introductory Analysis.
I don't know how to explain this thrill, but I do know that I would keep all this to myself if it were not for the kind encouragement Holden has given me.
I remember a couple years ago, when I could barely sober up enough to walk into the public library without causing a disturbance, I was communicating with Holden.
He would repeatedly urge me to get back into mathematics.
So, you see, I have to remove myself from the mentality of "high school vs community college vs state university vs ivy league".
I am in competition with no one. The beast within me is the honest one, and it is he that has the beginner's mind, it is he who is learning. The false ego who would prefer to run before we can walk, well, he must be overthrown if the beast in me is to study in peace.
Yes, I know there are more important things I am obligated to be concerned about, but, for the moment, I am caught up in my own little drama, content in my little cell in the penal colony of existence.
What do you think about this idea of defamiliarization?

When I was studying math quite a bit in 2014 I guess I was using the defamiliarisation technique in math unconsciously as I was reading HPL simultaneously and makes use of this technique a lotmost horror writers do.These experiments of yours in mathematics are very intriguing to say the least.I have very little experience of these matters.What is evident is that they are working for you.I guess so long as they continue to work for you,and I hope they do continue to work for you for a long time to come,I think you should stick to them.
I would think about this matter some more.

Sometimes the technique is not at all intentional, and I suppose it occurs on a subconscious level. It can be very subtle, like simply a different kind of notation.
Using notation for ordered pairs, (a,b), in order to represent vectors, and performing various operations, there is this sense of learning something new, but then when I replace the notation with something more familiar to me, which I am unable to type here, not just because I neither know how to use TeX and this site does not support TeX, but even when i write certain letters in [ ] square brackets, they get interpreted by the editor as some kind of "code".
I am so very much a pencil and paper man, a man who even prefers cursive writing to typing on a keyboard  especially when it comes to anything having to do with mathematics beyond arithmetic. There's just so many subtle abbreviations and notation that make it elegant.
I see the rest of the world using the fancy formatting on websites that support TeX, and even the TINspire calculator's on board computer algebra system uses very "pretty printing", with sigma notation and the integral symbol and all that.
Holden, I am far behind the eight ball on the Learning Curve of Sisyphus, but I choose to focus on the old school fundamentals, which is why I am content working with old books with yellowing pages and fresh cheap notebooks.
I was hoping we might borrow this term, defamiliarisation, from the world of literature as a way to nurture a less hostile perspective on our own feelings of sudden bewilderment when studying mathematics, like when something you might have thought you were familiar with suddenly becomes alien and strange, when it seems altogether new. Rather than becoming too discouraged, we can use the bewilderment as an opportunity to think with a Beginner's Mind.

A rather spontaneous revelation occurred to me while lying on the floor on my belly involved in my daily ritual of enduring existence, that is, working through math(s) exercises.
I found a particular set of problems extremely boring, to the point where I felt like a child in grade school going through the motions with very little brain power exerted.
I am in the process of putting together this solution manual, so I am not skipping any problems.
Well, I found that there was an alternative approach, and using this method made the boredominducing problems a little more interesting. Is this not in some sense a defamiliarization process, where something familiar to the point of boredom can be made more interesting using a more exotic or more elegant method of calculation?
The problems were trivial, where the student is asked to derive a scalar equation after having found the slope from the given points P and S.
Now, most students, including myself, would use the familiar pointslope form of an equation:
y  y1 = m(x  x1), where (x1, y1) are the coordinates of either P or S.
One then manipulates this algebraically into the scalar form a*x + b*y = c (with no fractions as coefficients).
I was so bored that I wondered why I was spending my life this way. My brain had turned off, and I was just going through the motions.
And yet, the purpose of this particular drill was not just to derive the scalar equation by any means necessary, but, I suppose, to use the algebra of vectors and a theorem such as X dot n = P dot n, where X = (x,y), n = (a,b) is the vector normal (perpendicular) to the line, and "dot" means taking the "inner product" (dot product).
What made these alltooboring and routine exercises so much more interesting was what I want to refer to as "mathematical defamiliarization".
Rather than using the familiar and boring method, using the unfamiliar method seems more elegant and sophisticated. It had more MEANING.
Given points P and S, you can find the slope m, true; but you can also find the normal vector n.
n will be perpendicular to (PS).
Once you have n = (a,b), then you derive the scalar equation with: (x,y) dot (a,b) = (x1, y1) dot (a,b)
a*x + b*y = a*x1 + b*y1
This actually felt more elegant than simply using the pointslope form: y  y1 = m(x  x1).
I think that this quite accidentally serves as another example of the benefits of defamiliarization in mathematics. It can make things less boring, less routine, and more interesting, which makes it more satisfying.

Don't be mislead by the fact that some of the exercises can be boring and routine.
I have found that the last handful of exercises in almost every section requires referencing the solution key (Dolciani's Modern Introductory Analysis  high school math text 1980's [from 1960's]).
So ... Maybe it will be good for me to kneel in the dirt and dig some holes and drop some plants in the dirt. :\
You know, returning to that foundational material is not some kind of punishment to myself, nor is it an exercise in egodestroying humility. I'm sure there is some kind of method to my madness.
I think I will keep some of my day to day frustrations and breakthroughs to myself as some of the insights can just as easily be scribbled in a private notebook. The thing is, I am becoming disillusioned in such a way that can be described as liberating. I do not have superhuman powers of computation. Even problems from advanced level high school texts require me to think as hard as I can.
My "above average" intelligence is not all its **** up to be, but I will stop complaining about the limits of my capacity, since I probably have more math skills than many. You know the way life is, on the interior of our minds, that is. My own image of myself can sometimes be unforgiving. As I said once before, I berate myself.
I'm going to try to give myself a break and just stay out of my own way. There is a part of our own brains that must represent society's value judgments, whatever Sickmind Fraud referred to as the SuperEgo. I have to overthrow that part of the brain so I can just putz around with whatever level of mathematics I am focusing on at the moment. If not now, when?
If I don't review this material now, I never will. I mean I would just keep pushing myself into more and more complicated areas. No, I owe this to myself. I want to rebuild the foundations from my current perspective, after having seen behind the curtain so to speak.
Oh, by the way. I did read the Erich Zann story last night. I remember reading that awhile ago. It was a little better the second reading. Lovecraft, like Schopenhauer, mentions the phenomenon of sleeping and dreaming ... even going so far as to suggest that we are more what we are while asleep than when awake.

Brainstroming
Desribe what you have been referring to as "the Dolciani series" in one word and then list synonyms.
novel
neoteric, different, odd, peculiar, special, strange, uncommon, unique, unusual, innovative, atypical, unfamiliar, exotic, unknown
These words are screaming "WEIRD":
So, let's list some synonyms for weird:
awful?, creepy?, curious, eccentric, eerie, freaky, horrific, magical, mysterious, peculiar, strange
So, the overlap of "novel" and "weird" is peculiar and strange.
And THAT's why the material seems to be "collegelevel" not high school level.
It is the weirdness, the novelty of the presentation, that I want to revisit. Back then I did not appreciate the novelty since I knew nothing else. Only now do I recognize how uncommon and unique the presentation was; and I think a devoted study of this series will fill in gaps that I did not even know existed ...
Of course, I will never be a mathematician, for, to be a mathematician is to imply that this is one's profession. Nowadays, academics such as Alain Badiou would like to similarly professionalize philosophy.
I guess we will have to be satisfied with not being anything in particular.
With Holden I can say that I will never be a mathematician or a philosopher  and yet!
I will never be a professional mathematician or an academic philosopher.
My philosophy is a way of life and the study of mathematics is my hobby, my obsession. Talent is not a requirement.

I am glad we have this thread separate from the one where I list all the various textbooks in the "mysterious" Dolciani series for I have tracked down and received a most unique edition of an Algerba textbook which, by section 27, covers precisely what I was finding so "unfamiliar" in the Introductory Analysis text I mentioned.
I am not suggesting you search for this book as I have noticed it can be quite expensive. I happened to get lucky so I pounced on it.
For future reference: Student edition Algebra 1 c 1985 or 1988 or 2006 by Dolciani, Swanson, and Graham (https://www.gettextbooks.com/isbn/9780395430569/)
The teacher's edition (https://www.amazon.com/gp/product/0395343747/ref=oh_aui_detailpage_o04_s00?ie=UTF8&psc=1) (which is more suited for selfstudy, I think) at Amazon is the correct ISBN, and the authors name + page numbers are listed wrong. What I did was to contact seller and verify that this was in fact by Dolciani/Swanson/Graham c 1985 and not "Totten, Douglas Smith" (misprint).
Those specific authors (Dolciani/Swanson/Graham) are important, for there are other editions (Dolciani/Brown/Cole) which make no mention whatsoever of the particular topic of interest, that is, proving theorems using the field axioms of real numbers.
I am so relieved to that the book exists. You see, Holden, it is not any of the algebraic methods that I feel are weird. It is this way of writing formal proofs with the axoims and properties of the real numbers. None of this is covered in any courses I took in college, and I suspect that, for math majors, they might expect one to already be familiar with this. So, I was able to track this rare source down that presents the material for beginners. It's a rare thing. From what I have researched, this may be the only such book to attempt to teach it at the beginner's level.
My idea is to scan certain pages and email them to you when I settle down. This way you can decide for yourself whether such material qualifies as "the Weird".
I am rushing out to visit my father in the hospital. Internal complications. I am not sure about details.
Before going, I wanted to post here in case I am involved in some vehicular misadventure. At least I will die knowing I passed on this clue to you as far as where to look for just what I have been trying to become more familiar with. A teacher's edition has solutions. That is the copy I found for less than what the student's edition goes for. Maybe I have invisible allies?
Over the summer, I would like to introduce you to the material to see if you have ever seen such formal proofs at the level of basic field axioms. For some reason, it all seems so strange to have to prove that (a) = a, and to use the fact that 0 = (a + a) in order to prove it.
Got to go. Stay safe!

Before I head out into the stormy weather so my mother can "go to church" ::) I want to type up just a small example of the very basic kind of proof I am trying to familiarize myself with. It may seem alltootrivial, but it is the simplicity of it that gives me some difficulty.
Observe the following and just consider what it might be about such a procedure that I find so "defamiliarized". (not a word, but what else can I call it?)
If I do become more familiar with this kind of familiarity, I hope I never forget this feeling of the uninitiated acolyte.
It seems too obvious to require proof, but I find the necessary mental gymnastics is taking some getting used to. I confess that my mathematical activities have been heavy on the computational and problem solving end, but with very little formal discipline when it comes to proving even such simple statements as the following. It is not my intention to jest. I am serious. I want my brain to patiently learn these "tricks". I need references for the details, as in when to use the word axiom as opposed to when to use the word property or even principle.
I'll get there if I can find the humility to face my ignorance without egoinflicted shame.
_________________________________________________________
Prove: For all real numbers a and b, if a = b, then a = b.
PROOF
STATEMENTS REASONS
______________________________________________________________
1. a and b are real numbers 1. Hypothesis
2. a + (a) = 0 2. Axiom of additive inverses
b + (b) = 0
3. a + (a) = b + (b) 3. Transitive axiom of equality
4. a = b 4. Hypothesis
5. a + (a) = a + (b) 5. Substitution principle
6. a is a real number 6. Axiom of additive inverses
7. a + [a +(a)] = a + [a +(b)] 7. Additive property of equality
8. a + [a + (a)] = (a + a) + (b) 8. Associative axiom of addition
9. a + 0 = 0 + (b) 9. Axiom of additive inverses
10. Therefore, a = b 10. Identity axiom for addition
Since, for most of my fairly long life, my interaction with mathematical concepts has been strongly based in "computational, mechanical problemsolving" algorithms and techniques, having to write the formal reason for each minute operation does not come secondnature to me at all.
To me, I am studying formal pure mathematics, here, in other words, Weird Mathematics.

The geometry proofs are not coming naturally to me.
The psychology is interesting. I mean, I never felt ashamed studying integral calculus or differential equations no matter how many times I referred to the solution manuals.
And yet, with the geometry, well, I don't know. It surely has deflated my ego quite a bit. I know that this was my intention all along, but I am not sure I am all that delighted about how accurate my assessment had been (as far as my need of a total overhaul and reeducation) in a rigorous manner.
It has been humbling, but I am not reacting in an emotionally immature manner. In fact, now I am sure that this has been the right way to proceed for me, that is, to rebuild my skills using the texts I hunted down before becoming too engrossed with physics and advanced mathematics.
I just have to not be too hard on myself.
When I find myself wondering what kind of life I am living, I resist the tendency to become depressed and force myself to look upon these days and years as an unusual opportunity. I do not trust the suggestions of most people on the Internet as far as using these online courses. I have my own plan and my own agenda IF THEY DON'T MIND.
There was another author besides Docliani who may have been even more radical in his axiomatic approach, a certain Frank Allen ... his books are even harder to track down than the Dolciani series (and forget about the solution keys!) ... I found an inexpensive copy of a geometry text by him from 1973 (An Introduction to Proofs) as well as an old copy of a 1970 Algebra with Trigonometry: A Logical Approach.
You see, I am not just interested in "high school mathematics," but a specific way of presenting the material with setbuilder notation and the inclusion of axioms. Even though I find it difficult, I figure that with continual exposure to it, I will eventually merge with the Weirdness of it all.
from a comment at Amazon left in 2007:
I have an MS in Math from Ohio State, and my wife and I home school our three children. We've been home schooling now for several years and it is approaching time for us to figure out our algebra/geometry/trig (or the equivalent) program. As a former graduate student in math, I know about this thing out there lurking under the surface of college math. It is the proof, of course. Somehow what a "proof" is, what "math" is, and what it is all good for has all gotten extraordinarly lost in a way that goes far beyond even the scope of secondary school education.
This basic problem can be heard reverberating in ancient videos of Feynman lecturing to the public on the role of mathematics in physics (and how rigor is not particularly useful). It can be seen in the mathematics curricula of undergraduate programs all over the nation that pander to other departments' needs, cutting out most of the actual math content and reducing the math major to a generalist in the mathematical sciences rather than a specialist in mathematics. At any rate, it is much, much bigger than even math ed or math ed reform and will stop any meaningful progress in math ed reform, for that matter, since it is a basic disagreement on the necessity and/or intellectual value of rigor (and, in many cases, what "rigor" even is for that matter).
At any rate, it's too bad these books are out of print  victims of a war far greater in magnitude than even the math wars. The New Math of the 60s was as close as it gets to mathematics being handed down to society by its mathematicians, and we threw it all away. Frank Allen's books are not just books written to pay lip service to the movement, but truly written in the spirit of the times by a real advocate of the New Math. In any case, these books are probably the very best algebra books I have ever seen as of this writing. If you put them together with a good geometry program that at the very least proves the Pythagorean Theorem, you will have youself one first class high school education.
Unfortunately, Frank Allen will never receive the vindication he deserved. But, perhaps he imagined that there might be people like me that would happen upon his work and find it immeasurably valuable in an antiintellectual world so dominated by politics that only the most vulgar displays of superficial mechanical proficiency are ever even noticed while everyone frantically attempts to "Beat the Joneses" with whatever latest gimmick they can get their hands on.
Another comment from 2012 about the Geometry text:
This book is a timepiece that harkens back to the day when it was decided by the mathematics establishment that Mathematics education needed an overhaul in terms of adding additional rigor to various courses. Although it can be argued if this movement was a success, this book is still a very nice rigorous treatment of geometry that lays the foundations for further study in mathematics. The author, Frank Allen, was a champion of the new math movement and was a very influential force. If you have had high school geometry and want to see a watertight logical representation of the material this book is a good choice.
_______________________________________________________________
You know, Holden, in a way, my relationship with mathematics is kind of nightmarish ... if only for the fact that I seem to be doing all this studying in vain ... And, throughout my life, all these "returns to math" may be the result of being shocked by how challenging geometry was for me at age 15/16. I mean, that was when I was experiencing a kind of identity crisis which has lasted throughout my life. I still don't really know where I stand with myself.
The only thing that prevents me from tormenting myself with selfmockery is that I have developed a sense of humor about the whole situation. It is this sense of humor and being resigned to be a deadbeat loser that differentiates me from the "official" (younger) traditional math students who may suffer great anxiety over this to the point of wanting to commit suicide.
Since I am already ok with my low social status, I have no fear of failure. I'm already a total failure by society's standards. This is very liberating for me.
I can finally allow myself to look into the kind of math that left me stumped ...
My humble view of my own mental capacity can sometimes almost drain the nightmare of its ability to torment me.
I can laugh at those who compete for status since my goals are kind of pure and uncontaminated.
_____________________________________________________________
I know my struggles are pathetic in comparison with those with real gutwrenching issues to deal with. This is why I have not been writing too much lately about the psychological challenges I am facing while forcing myself to remain devoted to this path I have chosen (as an alternative to alcoholic oblivion).
I will not be detered even if I come to see myself as a madman/freak.
*** On a brighter note, the first Zucchinis have appeared ... the work I put into the garden may bring some wholesome delight into this life (tomatoes, squash, cucumbers, peppers, and zucchini). It is the highlight of this life of quiet desparation ...
My father is eating again.
I hope you and Raul (and Mr Maughan) are doing alright.
My nephew and his wife are displaced in South America ... so I am kind of concerned about his wellbeing. There is nothing I can do.
I'm just thankful I have a cot and a pillow to lay my head on ... and even a garden.
I am one of the lucky ones, I suppose.
I will not allow the world of competition to rob me of this awareness of my many blessings.

Herr Raskolnikov,
I hope your father and mother are well there. So your nephew is still in Ecuador? By the time he comes back I am sure he will have no problems with Spanish. And gardening is a beautiful job.
You say: “I have no fear of failure. I´m already a total failure by society´s standards. This is very liberating for me.” I wonder if those people who attack you do not realize that they do not have the social status of, let´s say, the Queen of England. I am sure that lady has never used a broom in her life. Lucky her and the Prince Consort. Do you think people are going to call her “a loser”. I think not.
You see here there is a big problem with students who want to enter the university. Most fail in Castilian or Spanish and mathematics. For many years I have heard that Paraguayan students do not meet the requirements of the university. But they say that because they are only interested in getting technical jobs. Parents,specially, are afraid that their children won´t get into the job market. Stay well.

Hola Raul,
I think mi sobrino had to leave Ecuador in a hurry and most likely will end up in Columbia ... He and his wife trust in "the Universe," so I worry, since I have no such faith that this universe cares one iota for any of us in particular.
It is tough here as well for the youth who aspire to enter the university. Many join the military as a way to pay. Myself, I say I am liberated from all that since, for me personally, studying has nothing to do with employment. Actually, it never has. For me and countless others, there has never been any chance to get "into the job market". If we want a job, cleaning public toilets is it for us.
No matter how much we study, we will not be granted social status.
Do you believe in conspiracies? I sure do. There have been open conspiracies to replace entire populations with more servile workers who will accept harsher conditions out of desparation and hunger for prosperity (social status). Maybe I have been targeted (blacklisted) for suicide, and my continued interest in geniune education has foiled the plot. Unfortunately for those who see my banishment as a punishment, I embrace it as an unusual opportunity to lead a scholarly lifestyle.
The youth ought to study out of interest in the subject, even if it is a technical subject. This way, no one has the power to deny them entrance into the temple of their own minds.
I know I must sound impractical considering I depend on government relief payments, but education and employment may have nothing in common. Those who seek training for some specific job are like insects in a giant hive. It is called "specialization".
I'm afraid some of us will never be turned into insects, and so we exist as domesticated animals on the prison farm plantation. We have learned to use our uselessness to our advantage.
We live as a kind of parasite, but a humble parasite with a conscience, getting by with as little as possible.
I suppose I would never be trusted to instruct the youth since my entire "philosophy" or "psychology" seems to be rooted in a firm acceptance of the limitations of our mental capacity. Whereas the established status quo (the marriage of academia with corporate competetive values) encourages displaying false confidence, what I call credentialism. They worship social status and seem to take morbid pleasure in displaying their status symbols. Ostentatious consumption.
Not me. I do not measure myself by the commodities I acquire. Books, computers, calculators, notebooks ... none of these are displayed as wealth, but simply enjoyed by me as personal treasures that motivate me to stay out of trouble so I might continue to have access to them.
Looking for employment is like asking to be abused and tormented.
Imagine how many sadistic and cruel personalities are placed in positions of authority.
As for the royalty of England, there is a similar phenomenon here with "celebrities".
I don't spend too much energy hating those in such postions since I do not envy them in any way.
At the end of the day, ones world is most influenced by the head on the shoulders perceiving that world; so, literally, one type of person might experience more peace of mind in a prison cell than a totally vain nincompoop would experience in a palace. (paraphrasing Schopenhauer  he expressed it much better, I'm sure).
I really believe that there are very few "professionals" with "degrees from universities" and comfortable salaries who would ever be inspired to revisit "high school mathematics" presented in a formal and rigorous manner. Some might view my efforts as commendable while many more would find it worthy of mockery.
You see, their status symbols, access to gadgets, and obsession with being kept entertained has destroyed their capacity to put any mental effort into anything that does not "pay off in dollars or sex or status".
I say I am liberated because I have contempt and disdain for the society which has rejected me.
I have no desire to instruct their children, nor do I ever want to put myself in a position where I am at the mercy of cruel, insensitive blockheads.
No ... I am quite content out to pasture. Let me be considered useless and lazy, for I do not share the same values as the "productive and ambitious gortinsects" who aspire to land some kind of "career opportunity" with a corporation.
Like you, Raul, I prefer to consider myself "no good".
I don't like bullshiit, and I am too old to pretend I ever want to play those games ...
If everyone thought like me, civilization would most likely collapse.
Oh well. It is not my responsibility to ensure the trains run on time.
I just don't care.
I study because I feel somewhat stupid, not because I feel particularly smart.
Too many enjoy making others feel stupid, and so I suspect that there is a conspiracy against true education as the great majority are either victims of the conspiracy or aspire to be coconspirators.
It's best to trust your paranoia and suspect a great deal of lying taking place. Mindfuuckery.
Many people are very full of shiit. What is social status anyway?
I have gotten respect from people in the most unlikely places, psychiatric wards and county jails ... Some people would see me as a "natural philosopher" as opposed to an "academic philosopher".
Perhaps I am also a "natural mathematician/logician" as oppossed to a trained academic professional mathematician.
I admit up front that my ignorance is astounding, and I have no false expectations of ever becoming well trained in the craft.
Now, at my age (we are about the same age, you and I Raul), I like to think of my current studies as a spiritual discipline aimed at cultivating humility and deeper intellectual satisfaction. I do not need society's approval nor its status symbols in order to practice this "studious way of life".
I think you are correct, Raul, in your assessment that I may be unknowingly, a distant disciple of Pythagoras.
Also, I like to show my disdain for all the phonies by making it perfectly clear that I am inwardly focused on my intellectual development, and do not study in order to win their favor or to become one of their useful slaves.
From Dead End:
November 2006
Intrusive thoughts … unfulfilled desires … society – a network of lies and deception. We live in a society which encourages playing a role, being phony. People seek the rewards of a good reputation. Individuals low on selfdeception are at such a disadvantage in social life that this increases anxiety levels, even leading to a psychopathological personality. An extraordinary amount of energy is devoted to “impression management,” the effort to establish credibility, to acquire the finances needed to purchase the paraphernalia (designer clothes, luxury automobiles, etc) to impress with. I have long since stopped caring about such superficial status symbols.
The “old” natives of the “New” World knew how to handle the phenomenon of resentment caused by the aristocratic snobbery brought over by the lords and masters from across the pond. They recognized that such snobbery was used by a small handful of Europeans in order to shame all the other Europeans (which the natives thought were actually slaves, since they all seemed to have a “boss”). They (the natives hostile to such gortdom) just got rid of anything anybody wanted. They didn’t own property and they dressed in rags. They laid low and let the aristocrats, egalitarians, sycophants, and governmentpaid assassins all look on them as worthless.

Hola Profesor,
¿Qué tal por ahí? Aquí hace frío en Asunción.
Yes, you are right in saying that the universe does not care for any of us. I hope that your nephew arrives there safe and sound as you say in English. Yes, here many join the military or the police and government just to have a fixed salary. Sometimes it makes me angry that the old generation, people in the seventies, say that it is not worthy of a person to work for the state or government. They do not realize or do not want to realize that times are very different. Forty or fifty years go there wasn´t this population growth that we are having now. There were few trained people at that time. But the main issue is that they view work or jobs as something that one needs in order to contribute to society. Besides I think there is this need of having a boss or bosses all the time.
I agree with you that studying has nothing to do with employment. But the companies now want young people with interact in the social networks such as Facebook, Instagram, Linkedin, etc. They also want young people who are debtclean, as we say here. If you buy a TV set and suddenly you can´t pay anymore, well your name is registered in a private company called Informconf (Información confidencial). Imagine, a private company dealing with confidential information of thousands of people here and unless you pay all, you are unable to apply for a regular job because the companies check all the information. Very unfair.
Yes, I wouldn´t mind cleaning public toilets either. It is not a problem. Many people considered ridiculous that I wiped my late mother´s behind and also taking her to the wash in the bathroom.
I believe there are powerful societies conspiring all the time. Specially, when General Dwight D. Eisenhower spoke of the industrialmilitary complex and the assassinations of Kennedy, his brother, Robert and many other strange situations in this world. Conspiracies are part of this show called history.
What you say is a big truth, they want servile workers who will take any jobs to attain a high social status. I also think specialization will be a big problem. I think that those who are viewing us as losers will have a hard time in a not distant future. Technology is making most traditional careers and professions obsolete. Besides those who are forty now are doing the impossible to stay healthy and handsome and have the latest gadgets. There was this strange term called “metrosexual”. I don´t know the new term but it is not far from the older one.
Yes, we are humble parasites with conscience. We are not “useful” or “productive”.
“Looking for employment is like asking to be abused and tormented.” This sentence is totally accurate.
Celebrities. I don´t watch much TV but I hear some reporters on radio stations talking about some Kardashian women, they are celebrities there. They envy their wealth, status, all the media behind them, flattering them or praising them. It is the show business.
“If everybody thought like me, civilization would most likely collapse”. I think that when the collapse comes you will be able to cope with it. Stay safe.

It's cold there? Really?
Anyways, I thought you might find it kind of funny that I am getting sick of all this math.
It must be human nature. I will persevere. I am committed to it now. It's an obsession, a psychological addiction, and a great distraction to keep me out of trouble.
What else would I do, write a book? I get so tired of reading my own words.
Oh well, if I feel like I know so little, imagine how less I would know if I didn't spend so much time studying. Still, I think of all those trapped in prison cells and hospitals and standing armies.
I am as content as possible in such a world as this. You know, I have to keep things in perspective.
Are your eyes causing you much distress these days?

Herr Raskolnikov,
Yes, we are in winter but winter here is not like winter in the US. Now it is cloudy. Well, we are in July. Time passes very quickly.
Mathematics is a healthy obsession or addiction. You already pointed out how wealth, social status, fame, sex are dangerous obsessions. Let´s not forget those who are addicted to the power of government. For them power at all costs sanctifies/justifies the means.
Mathematics, as you say, it keeps you out of trouble and it has helped you to think more clearly and logically. Also reading those difficult authors such as Schopenhauer, Cioran, Mainlander, Caraco, Nietzsche and other authors helped you develop and improve reasoning skills and critique. Of course one can have a different worldview.
I think you are already writing a book. This blog is a book where you are sharing thoughts, views, and opinions.
Yes, as I have myopia I have problems in taking the bus specially at night. I see with my left eye with glasses 50 percent and the right eye 30 percent. Although I saw little before; now I have a distorted or blurred vision in the right eye. The treatment with Avastin helped clean the fluid in the back of the eye. I have to go for a eye checkup from time to time. Stay safe.

Herr Raskolnikov,
I read your email. I hope your nephew is allright. I also hope he contacts you wherever he is. Stay well.

Last night I was reading The new math : a political history (http://libgen.io/search.php?req=The+New+Math%3A+A+Political+History&open=0&res=25&view=simple&phrase=1&column=def), 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 gongter (http://www.ligmincha.org/en/boenbuddhism/resources/glossaryofbonterms.html).
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 midtwentieth 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 socalled 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?

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.

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) (http://indexof.co.uk/Mathematics/Introduction%20to%20Matrix%20Algebra.pdf). 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 (https://www.ebay.com/itm/IntroductiontoMatrixAlgebraSchoolMathematicsStudyGroupStudentText/323635593077?hash=item4b5a2f3f75:g:RHEAAOSwXI1cL~Qa:rk:2:pf:0) 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, 19581977 (http://www.lib.utexas.edu/taro/utcah/00284/cah00284.html)
I will try to leave hyperlinks in this thread when I find direct links to the pdf files.
Unit 17 (Intermediate Mathematics (http://files.eric.ed.gov/fulltext/ED135625.pdf)) corresponds to what would be called "Algebra 2" in the secondary schools.
Some other research into SMSG lead to these: School Mathematics Study Group (https://drive.google.com/drive/folders/0B4wpC387vwzeTzVzdlM1dVBFTkk)
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 (https://wiki.gnome.org/Apps/Evince/Downloads).

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 midtwentieth 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 nationstates. 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 militaryindustrial 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 socalled 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 careeristoriented 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 socioeconomic conditions lead to many adults feeling like losers amidst the presentday 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 militaryindustrial entertainmentprison 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 midtwentieth 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 mindaltering 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 mindboggling hipness of the absolute nerd.

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 militaryindustrial entertainmentprison 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 militaryindustrial entertainmentprison 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.

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.

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 forcefeeding 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 (http://libgen.io/search.php?req=The+New+Math%3A+A+Political+History&open=0&res=25&view=simple&phrase=1&column=def))
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 midtwentieth 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 (http://extraconversion.com/basenumberconversiontable/base3todecimals.html).

Numerological Note: This was post number 4444 at whybother.freeboards.org

A reminder of the SMSG (early 1960's experimental high school math texts) : some links (http://stern.buffalostate.edu/newmath/SMSG/SMSGTEXTS.html)
I am just finishing up my notes from going through all the exercises in Introduction to Matrix Algebra (https://files.eric.ed.gov/fulltext/ED135631.pdf), 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 handwritten 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 (https://files.eric.ed.gov/fulltext/ED135631.pdf)), 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 (http://stern.buffalostate.edu/newmath/SMSG/64analygeomStud.pdf) module DOES have the Teacher's Commentary UNIT 65 (http://stern.buffalostate.edu/newmath/SMSG/65analygeomteaccomm.pdf) module digitized. It covers conic sections and even 3space.
I think you will find the approach they took refreshing in comparison to our 21st century track.

Revelation: Those proofbased 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 [deadBeat pHilosopher], is that I can combine the old school handwritten techniques with the bleedingedge Open Source Computer Algebra Systems as well as personal collection of mathoriented C++ programs [all inspired by reworking through the Encyclopedic Thought Provoking Exercises, including proofs of theorems translated into computer programs as a tired old toothless & baldheaded 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 RoboScientistBusinessWizKid Training] which may lead to nervous breakdowns in more sensitive students not wellgroomed for such Mental Pressures during the topsyturvy 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 livedreality, all that intense and painful agony/hatred/angeranduniversalcompassionforallthatis, 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.

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.