{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

Late last night, riding the wave of a full day working through the last section of the 1960 School Mathematics Study Group module (unit 23), Introduction to Matrix Algebra (see page 219) [Frank Allen, et al], I was experiencing a kind of elation, what psychiatrists might robotically refer to as "a manic state."

I was very enthusiastic about the representation of quaternions as 2X2 matrices.  This morning I woke up refreshed and curious, since, unlike an alcohol binge, a manic episode did not leave me with a hangover.

Following my curiosity, I put my pencil and paper work down for a moment, then uncharacteristically fired up the computer before noon to search the Interwebs to try to make sense of the apparent isomorphism.

Elsewhere, when I saw quaternions, they were in the form q = {R, i. j, k}, a real number added to 3 complex numbers, where i, j, and k were different "directions" or rotations ... that is, i, j, and k were like the imaginary i in the Complex Plane, but quite different just the same.   That is, i, j, and k are not "imaginary numbers in the complex plane," but something different, "quaternion imaginary".   Thus, the quaternion in this form has four dimensions, one real and the other three?.     (see chapter 1 of Naive Lie Theory [or not ... no need to follow me into this ... you can view a half hour visualization-explanation at your leisure ... meanwhile, I formally apologize for the chaotic nature of this post ...] )

While looking through some materials from Library Genesis which would have been inaccessible otherwise (books far too expensive to even consider), I saw that I was trying to make sense of the matrix representation, and wanted to relate the 1960 SMSG text notation to what is used in the q = {1, i, k, j} form.  The Appendix of the book I am just about through presents quaternions in the guise of matrices exactly as described in this rare clue in a post here:  Isomorphism of quaternions with a matrix ring over real numbers.   In the appendix (research exercises) of the Frank Allen 1960 experimental high school text, instead of the labels I2, I, K, J, it uses I, U, V, W. 

Also, where quaternion q = a + bi + cj + dk
= matrix( [ [a + bi, c + di], [-c + di, a - bi] ]), where z = a + bi, w = c + di, and the other two are the negative conjugate of w and the conjugate of z,
the matrix in the Allen text is matrix( [ [x + yi, u + vi], [-u + vi, x - yi] ]).   
That is, z = x + yi, w = u + vi.

So, I have figured out that in the classical form, this maps to quaternion q = x + yi + uj + vk.

The notation begins to make sense to me, and I see the isomorphism, but there is still such great potential for confusion that one really has to have their head together to approach this material; that is, one mustn't panic.  One must find a way to stay calm and spell things out to oneself, at ones own pace.   Certain others may cause great confusion and havoc.  How to reduce the havoc and confusion in oneself?  That's the tricky part.  You can see just from my brief description how "i" is used in two different ways [not the same i], and "I" is used in different ways with the different systems.   Add on top of this that no one but the one seeking to understand cares about such things, and you have a perfect recipe for mental isolation bordering always on the brink of collapse and disintegration.

While I do not believe in any "supernaturals," what the natives of North America referred to as "the inivisibles," I do trust the authors of the SMSG movement, so I will try to temporarily forget the form in which quaternions were first developed (by William Rowan Hamilton around 1835: q = a + bi + cj + dk), and focus primarily on considering quaternions as an algebra of matrices.  There must be a method to their madness, a reason why they believe quaternions are more easily presented in the guise of 2X2 matrices with complex numbers as entries.


Now, the Lovecraft Connection:  It was in these moments that I reflected upon something I read about the life of HP Lovecraft, how he experienced some kind of deep psychological despair at the time he might have been thinking of pursuing higher education, and I suspect that the root of this despair had something to do with similar experiences of what might be described as "a feeling of inevitable defeat."  (still in the process of finding a better way of describing this particular species of depression).

It is a feeling that one may only be able to fathom such an extremely small degree of mathematics, and even then, only with particular notation and form one is comfortable with.

This led me to search for information about HP Lovecraft's struggles with higher mathematics:

Whipple Van Buren Phillips' death in 1904 greatly affected Lovecraft's life. Mismanagement of his grandfather's estate left his family in such a poor financial situation they were forced to move into much smaller accommodations at 598 (now a duplex at 598-600) Angell Street. Lovecraft was so deeply affected by the loss of his home and birthplace he contemplated suicide for a time. In 1908, prior to his high school graduation, he suffered a nervous breakdown and consequently never received his high school diploma. S. T. Joshi suggests in his biography of Lovecraft that a primary cause for this breakdown was his difficulty in higher mathematics, a subject he needed to master to become a professional astronomer. This failure to complete his education (he wished to study at Brown University) was a source of disappointment and shame even late into his life.

Wow ... I remember my nervous breakdown in the last year of high school.  I ended up graduating, but it was just the beginning of something horrid ... and, even all these years later, even after having completed a difficult major that entailed much higher mathematics than I faced in high school, I am still determined to master the material that I suspect played a part in my mental breakdown. 

I have been careful in my approach, and I refuse to allow myself to be overwhelmed or intimidated by "post graduate material" that is incomprehensible to me.  I am determined to take things in stride and grapple at a level at which I can actually comprehend and digest what I am studying.

With this post, I wanted to bring attention to this "adolescent mental collapse associated with difficulties with mathematics," as I think it would interest Holden that I actually have this in common with Lovecraft, although I have been fortunate enough to keep at it with persistence throughout my life, minus the 13 year drinking binge after graduating from Rutgers University in 2002.   

So, there is something you (Holden) and Lovecraft and myself have in common, although we each deal with the psychological challenges in our own peculiar ways.  Lovecraft incorporated weird mathematics into his stories, such as Dream in the Witch House, whereas I am obsessed with mastering the curriculum that was supposedly presented to me during a period of my life when I was suppressing a mental collapse.  I think that this might have something to do with why I lack confidence in "writing proofs."

While I am no avid reader of HP Lovecraft, from the handful of stories I have read, I do, at times, feel like a protagonist in such a story, not so much having to do with super-intelligent squid-like extraterrestrials who were here before the earliest [indigenous] humans, but more having to do with the subtle art of not allowing oneself to be driven insane by one's own brain.

While there is our evident connection in our mutual high regard for the clues left to us in the writings of Arthur Schopenhauer, I believe that our destinies might also have some kind of intersection having to do with our sincere and raw CURIOSITY about the discipline of mathematics.   In sharing the details of my obsession with studying math(s), it is my hope that you will be better able to incorporate [and HONOR] your own interest in mathematics in the midst of your torturous commitments to your employer which leave you feeling drained, defeated, and "disintegrated."

In my most honest moments, I think that the breadth and scope of mathematics overwhelms me, and if I continue to peck away at it, I do so with my tail between my legs.
_______________________________________________________________________________

an aside:  an uncanny coincidence?  While trying to come up with a title for this thread, when I finally settled on "Primary Cause for Lovecraft's Breakdown Was His Difficulty in Higher Mathematics," I noticed that this uses exactly the maximum limit of permitted length!  Do you believe in ghosts?   Maybe Schopenhauer did entertain the idea, as may be discerned from reading parts of  On the Will in Nature.

Bertrand Russell reports that Schopenhauer told people that certain of the paragraphs were written by the "Holy Ghost."  I can attest to this, as I have vivid recollections of Schopenhauer coming right out and stating this - although I always suspected it was written "tongue in cheek."

Also:  "Moreover, Schopenhauer had experienced animal magnetism and ghosts; he also had a sense of being mildly clairvoyant."   from The World as Weird (Schopenhauer Workshop)

The reason for mentioning "ghosts" is that I can't help but feel the guiding hand of this weird mixture of mathematicians and "educators" who were involved in the SMSG [academic think tank] project of the 1960's financed by the United States National Science Foundation.

I doubt the fact I got a C+ in advanced ordinary differential equations because I got sick the last 3 weeks will matter if I publish 6 significant papers on additive number theory over the next 2 years,do you guys?

This individual is suffering from egomania.

No, the C+ in  Advanced Ordinary Differential Equations will not matter, but not as a condition that Cotton Seed publishes 6 SIGNIFICANT papers on addititive number theory over the next 2 years.

You see, the 6 papers, no matter how significant, do not matter either.  They would not matter to, say, Farrokh Bulsara, for whom nothing really mattered.

This poor Cotton Seed is a victim of his own ego.

I think that we will be spared such torments.  We can look forward to a life of study outside the elite halls of academia.

I will be content if my notes get uploaded to the internet before I get run down by an automobile, and perhaps the notes might be of interest to some solitary students of the future who need to fill their days waiting for the welfare check to arrive, as an alternative to blowing thier minds with drugs and booze.

Herr Kaspar,

I dont know,no one really knows, if he'd be even alive after the next five minutes. But if I am cursed to live long enough then yes, I think I can hack programming/coding.For the moment it is elementary math only for me.
In horror movies and novels they have pentagrams and latin and what not. Its interesting.
Please note that I admire your dedication to math and that I have taken a leaf out of your book.
And much to my surprise I am beginning to grasp some mathematics.

You know what Kant and Schopenhauer called math and geometry? Empty forms.What are they filled with ? WILL. Its better to get lost in these empty forms.

But I make no promises. I do not even know for sure for how long I might continue to draw breath.I dont  say  this  to  get  any  sympathy,   it  just  happens to the brute truth.

I have a nervous temperament like that of Lovecraft's and tend to get overwhelmed quite frequently.
I wonder why evolution throws up an extreme introvert like me.oh that's right, the process happens to be blind.
My father is math phobic(though an engineer) and he told me that I might have inherited his incompetence in math and so he made me to choose non-math subject in high school( though I had scored a distinction in grade 10).

That path took me to a time where I found myself doing dishes for 16 hours a day in a hotel. I don't blame him. He possesses as little free will as I do. I used to think of myself as Gorky(who also worked in small hotels for a time).

Well, I am not my father. For one thing, unlike him, I was not foolish enough to drag another sentient being into this meat grinder.Its a matter of time really, my colleagues would love to see me die and I don't much care about life either, but if I continued to live then ,well, who knows, one day I might share some code of my own with you here on this message board. Stranger things have happened.

Unlike Lovecraft,who was married, if only for a time, I have no romantic delusions. I am not waiting for any special lady. I just don't know.I have heard that in France they teach kids philosophy in high school.They only taught me misery in mine.

I hate people. And math helps to keep away from them.Only the bastards really messed up my mind in high school. I guess the bottom line is, I need some time. Over the last one year I am made some gains in mathematics. I don't know. Hobbes was right . Everyone wants to kill everyone else.

Keep well.

(Beware of typos):

Is that code Visual Basic from a Microsoft Excel sheet?

It would be interesting to translate that to C++ or even in a python file to call from command line.

You might like to learn C++ and Python so that you could choose which things you like and customize commands for your personal use.   I am a living testament to the fact that developing your mathematics skills and devoting a large portion of your energy in engaging with the mathematics through programming may not be very lucrative, but it is ultimately liberating when you have developed some basic skills and can create the software YOU need to assist you with whatever level of mathematics you choose to focus on.   Mathematics opens a world of novel-programs you might create on the fly over the years. 

I had started with writing C++ code for Linear Algebra work (2015), which led to me revisiting a Fraction class (type) I had created in 1999 during a C++ class at a community college (while working full time with State Park Service).   To make a long story short, after creating programs for solving systems of linear equations stored in matrices, I was able to use my Fraction class in a RREF program so that the output in matrix form had each element in rational form.  Holden, to me, this was a quasi-mystical experience, and it all came out of the wetware of my nervous-system engaging with the mathematics and somehow translating this into code the machine understands.   As for communicating with those in your monkey sphere about such interests?  It's not even necessary.  Many people have absolutely no i nterest in such things.


I am now studying ways to generalize programs such as the Row Reduced Echelon Form programs, where it would take parameter <typename T> rather than <double> or <Fraction>, which is a Pair<Integer, Integer>.  It is Generic Programming (algorithm-oriented) as opposed to Object-Oriented.  It's kind of exciting, but, as usual, one must be very willing to be confused and a little overwhelmed at all times as this is what it feels like to be learning.

As a young man, it was too much for me.   Too many intrusions and masters and chiefs and bosses.  You have to be left alone in order to feel honestly stupid or stumped, and each day you are a new creature who is not the same as the one who woke up yesterday.    First and foremost you will want to get a grip on the anxiety first and understand that you are under no obligation to study maths, programming, or even philosophy or literature, for that matter.

You are under no obligation to breathe.

You can only do so much in a day.  if you have committed to a text that is taking longer than you thought, this is a very good thing.   That means you are engaging with it, but you are kept jumping through hoops so much that by the time you unwind yourself, there is little mental energy left for your innermost interets which have been drowned out by your ties to the social fabric.

It is, as henry Fool said, how great men topple.

For decades I was always at the mercy of at least three or four overlords who might choose to torment me just because they could.   Literature was non-existent.  It is quite a Twighlight Zone scenario.   Fortunately I worked for the parks, so I was often left alone to complete my tasks.  You, on the other hand are exposed to many business-oriented folks who rely on commercial software to manage companies.  It would be good for you to be able to approach problems removed from business and commerce. 

I may be a freak in the family since I suspect both my grandfathers' interest in mathematics or science was tied to their career-oriented mentality.  They were bred to seek social status.  Their status as "scientists" would spare them from facing the front lines during the "Great Wars with the Fatherland (ouch! what a mess, to be German American during the "world wars" jee-wizz. )

Neither of my grandfathers relied on their interest in mathematics as a reason to exist.  I have always suspected that success as a professional scientist/engineer must have more to do with primate politics, telling the right jokes, going on "business trips," etc.    Myself, however, I enjoy healthy bowel movements mainly to continue this narrative I am living out, which ends with me as a bag of bones.   I want to study more math.   I want to learn about Concepts in C++ with templates (generic programming).  There are some mathematical entities I wish to see if I can create TYPES of, like a mixed number that has (N*sqrt(R))/D, where I have to deal with all the nasty arithmetic when multiplying or adding the real numbers and the integers.   I have run into some interesting limitations.  Over time, I may explore building numerical beasts of my own insane imagination.   So, you see, Holden, it is really easy to be as useless to industry as a dope-fiend when you become a math-junkie who is actually quite passionate about the whole enterpise of ENGAGING WITH MATHEMATICS AND COMPUTING.   My brain kept itself sane in a jail cell by computing prime factors with pencil, paper, long division.  I refered to it as "a game I made up."   It kept me sane.  Rough cats respected it.  When they saw me computing, they left me alone, seeing as I might have discovered a way to keep this tormented sentience occupied, almost relishing in having a grip on these numbers which become the center of my mental universe.   So, I am not my grandfathers.  i am not a traditional "scientist."   Nor am I a "math-hater" (both my parents).  Just another freak in the Family of Man.   We don't fit.   No one can stop you from peering into the terra incognita.  Who am I to dictate what you will discover by cutting your own path?

It is said that "living well is the best revenge."   Some of the most content moments in memory are those when I was left alone long enough to spark that small kindle of flame ... that INTEREST.

Maybe the most interesting way to introduce yourself to languages which would allow you to experiment with your own non-commercial personal command line "commands/programs," would be to translate the Visual Basic to a routine in C++ or Python.   Maybe you might find one prebuilt, and all I would have to do is talk you through compiling it or running it.    Whatever it is, nothing is ever a rush.

I've gotten great delight from buiding programs that complete the square and factor trinomials, operations which I wish to be able to understand well enough as to write the code to automate it.  When you begin to check your work with it, or have the code output information while your building it to help you see what effects what (hacking away), you might become curious or interested.   

There is much to distract you, and many who would devour all your energies, leaving you fit for nothing but passive entertainment.    You will want a long stretch of days when you are not rising filled with anxiety to report to a "supervisor," "superior," ... uh ... Sorry, Holden.  Somehow you have to be patient enough for your body and mind to be relaxed enough for "tinkering."

Such a fortress of solitude is difficult to muster from this world.   You are doing all you can do.   Somehow I have managed to do something most men my age would never dream of doing:  revisiting HIGH SCHOOL texts, applying programming skills picked up through my personal interests.  Note that "programming" is not something one learns while studying Computer Science, but an endeavor you can learn on your own through experimenting and exploring.   Holden, trust me, much of the high school material is more challenging than the college level courses, since all the foundations are covered, not taken for granted.   It is great that you begin with foundations since you will forever return to them.  When you create programs assisting you with some drills at this level, it is exciting in a mad-scientist way.   You can factor a number into primes.    It's not co-caine, but something better.  Mental stimulation that you might enjoy simply by making the effort to understand.  As a whole, it is a gigantic alien artifact, but we might come to understand the parts we have time to devote some attention to during our short span of time as living individuals.

There is nothing stopping any of us from being a private scholar.  Even some prisoners are able to maintain a mental life with minimal resources.

I understand how discouraging it can be.   We do live in a nightmare world.  How to give yourself enough sanctuary from the madness in order to enjoy your own brain, or discover how your brain learns.    For me, it requires much writing.   I have to take notes, working through books slowly with pencil, notebooks for notes, other notebooks for "finished work," a great deal of scrap paper.    One of those gadgets you suggested for working things out might be appropriate.

You will eventually want to become familar with some kind of computer algebra system such as Sage or SymPy.   Please do not be intimidated by any of it, Holden!   Explore it.   You are free to become a silent, secret scientist on the down-low.  Keep it to yourself, and it becomes a bit sacred.   

This craft, programming, becomes extremely valuable when you understand that you can build simple mathematical tools for yourself.  They do not have to meet anyone's standards but your own.   Professionals might not like this attitude, but I find it empowering.   Like I say, it's the greatest revenge.  You can make your own mathematical tools.  Give yourself several years and even a lifetime ... it's in our world all around us.    Knowledge may be like religion.  it becomes hypostatized.  Your own understanding is your own personal religion, Holden.   Stick with the books you have been working with.  Organize your work ... I mean, allow yourself to THINK on paper.   Allow yourself to show yourself your work.    Your mental life is within you, and you might be most drawn to books that contain a mixture of method and theory.   

You are deeply philosophical, so putting any kind of time-frame restrictions (the way institutions of learning do) would cause unnecessary anxiety.   Life is always getting in your way and draining you.   Give society a little less and keep some mental energy in reserve for your own private agenda.   Conserve your energy at work-place by disallowing co-workers from vampiristically sucking out all your energy.  Then, if you get through the day with a reserve of mental energy, you might eek out a couple hours invigorated, or, maybe enough resentment to call in sick ... just don't call in dead.   

I felt obliged to reply.   It's a long path.   I helped a neighbor with installing their break pads this morning, and now I plan to run some code through a debugger to understand a part that is very cryptic to me.

The main thing is to allow yourself to be somewhat confused.  Much has to do with heat and energy levels.   You will understand more when you are still.   I wish life were not what it is.  It is a miracle we are able to remember anything at all.   

I would never want to write a technical manual, but I suppose that might be my destinty one day.  Maybe I might be the Schopenhauer of technical manuals, or the Schopenhauer of "comments in source code" (he's so descriptive!  that algorithm was pure poetry, elegant artwork)   

Acting goofy is a coping mechanism.  In reality, I am probably a lot brighter than I give myself credit for.  But that is neither here nor there, as life is still what it is.   If any of my wits give me a slight edge or advantage, I will say thank you and cherish my little gifts like a secret treasure.

No matter how much I may flatter myself for my lifelong interest in mathematics and philosophy, and no matter how well I might measure up agasinst my "peers" or in a random concensus of the general population, I fully grasp that I am a pitiable human creature whose sustenance is dependent upon often grotesquely complicated and ugly machinery.  That is, in the midst of the ego-flatterring consensus that studying mathematics is something commendable, I get it that, as an animal creature, I am pathetic, just as pathetic as the average gort or mall-rat.   I suck at the teat of the Mother Ship and am just as much a space pod as the most ignorant flag waving sportsfan who drives a big pick-up truck.

I am cancer with a conscience, and I can do a little math and write a little code ... am I interested in this as a monkey or am I interested in this as a man?   Who is the monkey and who is the man?   I think I am interested in mathematics and programming as a MONKEY.  What I mean is I want to monkey around with math and computing in my own way.   I am not concerned so much with contributing as a man among men in a group of men, although, maybe if I ever wind down to where I might get along with others ... that might happen --- but it looks like I am interested in this stuff primarily as a monkey-creature with nothing better to do with his time on this earth, which appears to have no purpose whatsoever - the entire infinity, eons upon eons of striving to remain afloat.

I might be the guy in the room who can figure out the math problem, and I also might be the monkey who gets so frustrated with coming up with the wrong answer that I go bat-shiit-crazy.    "Technical types" run amuck.   

I chnage from momnet to moment.  People are generally liars (demons) who lie to confuse you.  They use langauge militaristically as a weapon to control and manipulate (and most often to deceive).   We may be drawn to scholarly type efforts as they at least sometimes appear to be authentic and not merely psoturing for social status.   So much is politics since we are a bunch of god damn apes.   I am surprised I was permitted to survive.  I mean, I have been outspoken.  I guess i am fortunate nobody listens to me, and people are too overwhelmed (psychologically) with the burden of their own existence, to have energy to worry about anything I say or do (as long as I don't disturb "the peace" - whatever that is).