Our Philosophical Descent into Madness

General Category => Why Mathematics? => Topic started by: Dog on October 21, 2015, 12:54:35 pm

Title: How to Attain a Studious Life
Post by: Dog on October 21, 2015, 12:54:35 pm
I have found that rushing through material too fast or trying to cover too much leads to frustration and kills motivation.  When I find myself skipping the exercises, that is a sure sign that I am not genuinely interested in even trying to understand what I'm looking at.

So, I decided to slow down and do some of the exercises in pencil.  This has actually helped keep me calm. 

Oh, the reason for this thread:  I may have come up with an imaginative way to make autodidactic study a more enjoyable experience.   If one can "pretend" one is in some kind of futuristic prison complex, imagine how delighted one would be to be granted the privilege of holding onto a little notebook computer in one's cell (with mathematical software like Sage and SymPy/NumPy/SciPy [Python] and loaded with mathematics and programming texts), with blank sketchbooks and pencils and erasers ...

One would be quite content in that cell.   :D

It's a matter of using one's imagination so that what you are studying becomes like the Holy Grail.

Title: To be a contently miserable weirdo
Post by: Dog on October 22, 2015, 08:23:23 pm
Maybe I have been approaching this the wrong way.  How about a different approach?  How about I get used to the fact that I am not always going to be enthusiastic about studying things that can often be difficult for me to understand.  How about I allow myself to be somewhat depressive and uninspired?

Is there a way to enjoy one's misery in a way so as to not really care too much one way or another?  I mean, if one expects to be inspired and enthusiastic and passionate about something, and then it turns out to be very dry, boring, and uninspiring, then one might be quick to give up ... totally.

Suppose one were to face the unpleasant truth that one is interested in something that is essentially boring and dispassionate.    One could be studying congruencies, and be like, "the inverse of 9 modulo 10 is 9 since 81 (mod 10) is 1" ... OK, I see ... it is what it is ... Do I care?   Well, it's a mathematical reality revealing some kind of inner nature or rules of modular arithmetic ... When a textbook includes too many exercises asking me to prove this or that, I find myself wondering why in the world I would need to do that.  I mean, proofs are a social activity, and I appear to be on my own with this.

In the book, Hitch-Hiker's Guide to the Galaxy, there was a depressed robot.  Tonight I find myself leaning in that direction.  With so much hype in the world over such meaningless things as World Series and Presidential Elections, I can at least have a private inner realm where there is no hype, where there is anti-hype.

I am not a professional justifying his existence or looking for validation of my craft.  I shamelessly depend on government relief.  I'm like a "social misfit" for whatever reasons ... too much intellect, not enough teeth, who knows?   It is what it is.

One of my favorite fiction writers when I was a teenager was Kurt Vonnegut.  He became a very bitter old man.  Good for him.  I commend his honesty and his bitterness ... at least it is genuine.

So, yes, I am interested in math and pessimistic philosophy. 

This may sound like a contradiction, but the way I am going about "enjoying the process of studying" is to become relatively depressed and at my own natural pace experience little moments (breakthroughs) where I can say, "well ... that's KIND OF INTERESTING ..."  but without passion, without hyped up enthusiasm.

If there is any invisible sentient intelligence observing us from another dimension, they will know me as the one "who does not blow smoke up the a-s-s of the <<<Creator>>>" ...

I am just a chimpanzee that can read, write, and do  little math.  So much for the miracle of birth.
Title: Re: How to Attain a Studious Life
Post by: Dog on October 23, 2015, 10:55:53 pm
I supposedly experienced a nervous breakdown when I was 18.  I wonder if it had something to do with my being torn between the realities of our world (Native American reservations and the nightmare world faced in inner cities) and higher mathematics.

I rebelled against "computers" when I was a teenager.

OK.  Fast-forward to 1998.  I left my job as a maintenance worker to formally study mathematics and computer science (at age 30).  While I did well academically, I never followed through and got a job in the industry.  Maybe I experienced yet another nervous breakdown at age 36 or so ... I seem to reach a point where I become frustrated and angry and depressed all at once, a prime candidate for the epidemic of alcoholism, right?

Now, for whatever reason I am meddling with math textbooks again, trying to focus on areas of math having to do with algorithms and data structures ... programming. 

I swear I still experience moments when I see how this material would make a youth just want to die.

I have a low frustration tolerance.  I understand why one would prefer to mop floors and clean toilets rather than study math or programming ... and yet there is another part of me that enjoys dabbling as long as it is at my own pace.

I am a very frustrated man.  I do not lash out at others over my frustrations but tend towards self-destruction.  I think this is why it is best I do not drink alcohol.  There is a great deal of repressed anger in me.  The authorities must know this about me.  I have to keep my cool.
Title: Re: How to Attain a Studious Life
Post by: Dog on October 24, 2015, 11:15:36 pm
I will keep updating this thread.  Keep in mind that, while I may maintain a degree of enthusiasm throughout the day while studying, and while I have a great appreciation for having access to computers and even the Internet, I do reach a point where I lose interest.  This just means I have had enough for one day.  The thing is, that is usually when I post here.

In other words, when I am engrossed in study with a high level of interest, I most likely won't be typing about those feelings. 

It's like when someone once read my "poems" - which I don't seem to write any more (it was just a phase, I guess) - the person said my poems seemed very angry and negative.  That's when I would write them. 

So by the time I check in here to write on our little wall, I may have reached my limit as far as "enthusiasm" goes.

I just like to be honest.  What a revolutionary concept:  honesty.

The more honest I am, the more I have to admit how uncertain I am. 
Title: How to Attain a Studious Life: Slow Motion
Post by: Dog on November 11, 2015, 09:56:31 pm
If a professional psychiatrist were to see me in action while I was configuring, building, and installing the compilers last night, he/she might point out in a snide manner that I was "in a manic state".   And yet, I was very focused, and I was driven to get the environment right so that I could do what I am doing this evening ...

The demons that drive me in a state of hypomania are as valuable as the angels that calm me down to a slow-motion pace.

What makes actual compilers and debuggers so valuable while reading through a text such as Josuttis's The C++ Standard Library is that I can just place any given code into some text file, include the necessary header files, compile it, and then trace through the code with a debugger.

While reflecting upon the dynamics of a compiler can be overwhelming, once the environment is set up, the debugger itself can become our Teacher.

Suppose I am looking at an explanation of scope and lambda functions.  Reading it on a page is fine, but for whatever reason, tracing through the code in a debugger and inspecting the values of the expressions when inside and outside functions gives one an intuitive feel for the scope.

The GNU GCC debugger, gdb, has many useful commands that aid our understanding, such as displaying values and even inquiring "whatis x" - where it returns the type.

Perhaps the greatest advantage to autodidactic education is that a solitary student does not have to feel ashamed upon being delighted to learn something basic and fundamental, whereas in a classroom or formal setting, there may be discomfort and much posturing by others, which totally robs one of the pleasure of discovery.  Instead, one most likely becomes discouraged, feeling inadequate.

How does one embrace this beginner's mind and resist the irrational tendency to look down upon the little details as "simple", and romanticizing the complex?

You know, I don't have to be forever reading Schopenhauer in order to be living in a manner that Schopenhauer would smile upon ... Every time we hide in a cave-like dwelling to enjoy our mental faculties, we are implementing his philosophy. 

And yet this is not necessarily anti-social behavior.  Studying a book is not anti-social.  Other human beings organized knowledge in a presentable format to help seekers looking for a better understanding.  We may go on about how we despise the human condition, but there are moments when I am humbled by my dependence on other human beings.  The fact that we use these symbols to communicate ...

Perhaps I'm a bit mystified and hypnotized and even somewhat brainwashed.

I mean, I think I secretly worship computers and the scientists behind the scenes, and this most certainly conflicts with my youthful rebellion and glorification of living more harmoniously with the Natural World.  The cold rain does not love me. 

OK ... I will stop this flow of automatic writing and return to my quasi-religion - "programming and mathematics".   :-\

Afterthought:  When I use the term "worship" - adoration - I do not mean to imply awe, wonder, or extreme enthusiasm.  There is no mirth or fanatical praise.  The emotional connection has no passion whatsoever.  It's more like a curiosity or a seduction. 

There is surely an element of wizardry involved.  I remember my curiosity with mythological wizards when I was a child.  I used to have a poster of a wizard in his room filled with open books.   I tend to be especially attracted to books that may contain secret knowledge.

Hmmmm ... Just writing in a spontaneous manner here has made my interest in programming and its roots in the algorithmic methods of mathematical thinking clearer to me.  It is not so much religious worship as it is a love of knowledge, a lust for the acquisition of understanding something obscure and mysterious. 

I don't have to come to any profound conclusions about my reawakened interest.  To say I worship knowledge is a bit too much.  It is far too easy not to be impressed with anything and to curse the cosmic accident that brought forth life on this planet; and yet, as long as I do exist, and since I cannot reverse the event, as the manifestation of the organic consciousness of The Thing in Itself, I am drawn to some ideas and repelled by others.  It's how IT is wired.

When a "students and masters of a discipline or craft cherish the obscure nature of its core knowledge, it becomes like wizardry, but the elder wizards appear to be attempting to organize the unmanageable amounts of knowledge into usable and expandable libraries - to eliminate the necessity of "programmers" to have to start from scratch.   

Polish scientist, Alfred Korzybski, called this tendency "time-binding".  We are symbol using animals who time bind.

Unlike Korzorbski, I do not praise scientists or think scientists ought to rule society (although it might be better than a police-military rule or a corporate businessmen rule). 

I do not glorify humanity.

Writing always leads me to contradictory conclusions.

I appreciate technical tutorials and references, and much of my own notes are becoming more and more technical, more and more details about technical procedures or just keeping track of myself as if I were some kind of laboratory experiment ...

Here, though ... here I let my fingers just type as though I were kind of thinking while I am typing, rather than thinking before I type.  Long pauses between paragraphs ...

Maybe trying to learn to decipher more and more code is, for me, a way to occupy my mind lest I decipher the universe the way Schopenhauer and others did.
Title: Unbroken Spirit & Exploratory Studying
Post by: Dog on November 15, 2015, 11:02:58 am
All the effort I put into setting up a lab environment has not been a total waste of energy.  While I was able to set up the GiNaC library in Linux on my main notebook computer, and I can use the gdb-dashboard for stepping through the code, analyzing the data types and all that, I also have a smaller machine with an old VGA monitor and standard keyboard hooked into it that, for some buggy reason that I will not allow to drive me crazy, is just not compiling code with GiNaC libraries.

What I have decided to do is run Windows on that machine, using the free Visual Studio Community 2015 debugger, which I humbly submit is very cool, to step through code that it can compile.  The GiNaC libraries are a side interest, and I have a great deal to explore in the Standard Template Library as it is.

So, this is ultimately what I enjoy doing the most, learning through analysis with debuggers.  I refer to this as "exploratory studying".  I put much "work=energy" into setting up the compilers so as to have my own little real-time lab environment, and, "God willin' and the crick don't rise," I will continue to cherish it whenever possible (until the wheels fall off  :P).

I will take deep breaths realizing that every little breakthrough in understanding I experience during my "lab sessions" will certainly enhance my capacity to comprehend what I read in the hard copy texts that I plan to carry around in the spirit of the Unofficial School of the Weirdo-Rejectionist Scholar-Warrior.

As I foresee not having the leisure to take detailed notes from the Internet, the investment in actual texts has become necessary, and, wouldn't you know, it is a great feeling to have the actual texts in my hands.  I did not make compulsive purchases, but acquired the handful of texts that are "holy" to me at this point in my life.   Maybe some days I may even snatch Schopenhauer (or Cioran) or Husserl (or Merleau-Ponty) or even Lovecraft (or Ligotti) from the shelf in order to sneak in a couple pages throughout the day of mandatory psychiatric (behavioral) "treatment".

What I am exploring now is the Standard Template Library in C++.  Armed with compilers and debuggers, I compile the example code from a text, and run it in a debugger.  With a beginner's mind I explore the nature of the data structures and algorithms. 

For instance, with gdb-dashboard (in Linux) I can type "dashboard expressions watch t" and "whatis t"

type =  std::tuple<int, float>

With gdb, you set a breakpoint at main() with: b main, then run
Stepping through the code with s, and just calmly analyzing what you can.

In Visual Studio, there are also many ways to explore the code in debugger mode.  It is one of the things, like the Power Shell, where I throw all my anti-Microsoft prejudices aside, and just appreciate what the Visual debugger has to offer in Slow Motion Mode.

Can't we all just get along?  :-\

So, side by side (http://whybother.freeboards.org/math-diary/cc-compilers-and-debugging/msg1302/#msg1302), GNU GCC with the gdb-dashboard on one machine, and as a supplement, Visual Studio in debugger mode ... stopping the code and stepping through it as slowly as I need to.  This is Exploratory Studying.  What I see while "in" these debuggers, like I said before, will enhance my comprehension while going through texts.  The words may become more "alive".

This kind of studying has the****utic value in for the following reason:  a certain passion transforms the discipline from a boring technical interest to a rewarding and satisfying activity.

Why was I so particular in choosing who to invest energy in "following"?   It's not a game.  What we choose to focus our attention on is what becomes our life-world.

When we are restrained or constrained by forces more powerful than us, where the representatives of our society are in our face with intrusive and often denigrating questions, this is the illusion of CONTROL I think the character from Ishmeal was referring to.  We can choose to reflect upon the tenuous nature of our social status, or what Schopenahauer called the image we make in other people's heads.  It is secondary.   What goes on in our own heads?

I can be torn from my little programming laboratory, but the code is in my head.

I can't help but keep hearing those words of Virginia Woolf over and over again.

[Knowledge] "is open to everybody. I refuse to allow you to turn me off the grass. Lock up your libraries if you like; but there is no gate, no lock, no bolt that you can set upon the freedom of my mind."

dinosaur (https://www.youtube.com/watch?v=1KKdl8dJ_4E)
Title: Continuously Sparking Imagination
Post by: Dog on November 22, 2015, 11:38:11 am
I want to have fun learning.  Does this lead to contradictions when I hit a wall?

I must keep repeating to myself, "My intellect has exhausted itself in order to demonstrate its own limitations."

Do I consider it fun to exhaust my intellect in order to demonstrate its own limitations?

I may bash academics and professionals, but most if not all my sources for learning are from the academic world or from industry.  I can't be a purist.  It is difficult enough to remain interested and not be overcome by a sense of futility without burdening myself with guilt and shame.

Am I still having fun learning?

I think so ... as long as I can handle the ego-squashing activity of deferring to the knowledge and experience of superior intellects. 

An analogy is called for.  There are cultures on this planet who teach their young how to make fire without flints or matches, or to build shelters from natural surroundings - to find food and water.  If I were dropped into such a society, I would have a great deal to learn from a 12 year old.  It would be humbling, but do I want to learn or not?

Likewise, technology is always changing.  Even C++ has transformed in the past 20 years since I was first exposed to it.  I was always terrified of Assembly Language, and I am curious about the low-level differences between 32-bit and 64-bit architectures.   I am just curious.  This is a spark I can refer to as "interest".

I am curious, therefore I am interested.  This has become very personal for me.

It seems as though I may be attempting to validate the 8 years of serious study.  After that I experienced about 13 years of psychopathic depression ... and I lost interest in what I had studied because I felt all the work was not worth it.

Why have I become interested again?  What has sparked my curiosity? 

Maybe it has to do with the fact that in 1993 or so I was drawn to material that only a post-graduate would be interested in, or there would be warnings about prerequisites.

Now, I at least can research material without the voice in my head telling me, "You are a janitor.  Maintenance workers cut grass, clean toilets, chop up fallen trees with chainsaws.  You are not qualified to study what you think you are drawn to."

In a sense, I have paid some dues.  I have been knocked around by the school of hard knocks.  I am just another strange character in this science-fiction reality ...

There are many different kinds of people in this world.  Whatever I have been drawn to over the past 6 months has a lot to do with having a computer before me.  How do I interact with the machine?   I make it dual boot so as to have access to Linux while not totally alienating myself from the Microsoft world.  I leave records here on this message board of the kinds of things I have been tinkering with.  Those are the things that have my interest ... not online poker.

In order to retain motivation and interest, it may help to see how the dots are connected in a causal chain of psychological events.  One thing leads to another. 

OK, so there are things I wish I would have known by now and never really got around to learning.  Like I said, this is very personal for me.  I can't allow the fact that I appear to have a kind of "unemployable personality" to prevent me from pausing to study the kinds of things I wish I would have understood better by the time I reached this age. 

I can't allow myself to be overcome by the sense that "I am too old to start over," for it is not even possible or desirable to start over.   I think I can blend my study of C++ STL (generic programming with templates) with some kind of disciplined study of 64-bit Assembly ... and continue to explore abstract algebra, group theory, and even the Scheinerman book, "C++ for mathematicians".

We go through these lives, and each of us has our own particular life-world.  The television portrays a global narrative, a national narrative, etc ... but is that how we really experience reality, as citizens and consumers and clients and inmates and employees?

Isn't there a personal inner life that trumps the meta-narratives?

One thing I do have in my favor is that I certainly can't help but value access to information.  I can honestly say that I treasure having access to texts and other mediums.
Title: Mind Shift: Caring Enough To Think Slowly
Post by: Dog on February 20, 2016, 11:47:04 am
Since our local search engine is, let's face it, close to useless, I could not easily find the thread about slow thinking and repetitive learning styles, but this is as good a place as any to document an example from lived experience.

Suppose I am "working" through the exercises in a text I consider to be "elementary" just as a review, and I come to a problem that involves some thinking about trigonometric identities.  Now, let's assume I have a stack of textbooks I want to "get to," and one on top, say a Linear Algebra text, that I am most committed to, almost feeling as though I am just goofing off by working through Joyner's Differential Calculus and Sage.   I come to a problem where I have to prove that the limit (as h approaches pi/2) of cos(theta - h)/cos(2*theta - h) is -tan(theta).

Sage computes result as sin(theta)/sin(2*theta)

There is the temptation to skip this pesky little problem since I want to "get to" something else.

Aha, and this is an example of what I am referring to when I plead the case for SLOW THINKING in the spirit of bringing some "fun" into the learning process. 

This would involve what I am calling a mind shift.

This is the benefit of learning outside the confines of an academic or corporate setting: to have the inner (intellectual) freedom to respect one's own lack of clarity enough to CARE about the what and the why and the wherefore. 

[enter demon, Subscript One]: Now, it's getting late in the game, Herr Hallar ... do you have time to understand things in such a thorough manner?

[enter second demon, Subscript Two]: I'm afraid I'm going to have to INSIST we move slowly through this material, sir!  Do not piisss on trigonometric relations while calculating limits in an effort to go racing up the metaphorical mountain so as to get down to the serious business of trying to understand what differential equations mean!  Are you afraid you shall be suddenly ripped from your chamber and sent into the abyss before reaching your intended destination?  :o

[Subscript One]: There's no time to goof off and actually understand!!!! 

[Subscript Two]: No time?

[Steppenwolf]: I must face that which perplexes me with courage and show my disdain for those who act is if everything is trivial.  I have to agree with Subscript Two.  We can afford to go on a tangent if it promises a mind shifting experience.

master stroke (https://www.youtube.com/watch?v=N73NdiH0cBk)

[enter third demon, Subscript Three]:  Hint:  sin(theta) == cos(2*theta), as in sin(pi/6) == cos(pi/3) = 1/2 

second hint:  tan(-theta) = -tan(theta)

tan(-pi/6) = -tan(pi/6)
cos(pi/6 - pi/2)/cos(pi/3 - pi/2) == cos(-pi/3)/cos(-pi/6) = -tan(pi/6)

A few more tid-bits:

cos(-2*theta) = cos(2*theta) = sin(theta)

cos(-theta) - cos(theta) = sin(2*theta)

tan(-theta) = -tan(theta)
Where I get jammed up is that sin(theta)/sin(2(theta) = tan(theta), not -tan(theta)

Note that cos(theta - pi/2) = cos(2*theta) = sin(theta)

Also, cos(2*theta - pi/2) = cos(-theta) = cos(theta)

so, as limit of h approaches pi/2,
cos(theta - h)/cos(2*theta - h) approaches cos(2*theta)/cos(theta) = sin(theta)/sin(2*theta) = tan(theta), not -tan(theta) !!!!

- and Sage agrees!   Those who publish textbooks sometimes make mistakes.

Another CAS computes result of limit as 1/(2*cos(theta)) which is the same.

Check:  1/(2*cos(pi/6)) = 1/(2 * (sqrt(3)/2)) = 1/sqrt(3) = sqrt(3)/3

And after spending the entire morning on "one easy little exercise," while my understanding isn't exactly intuitive, one thing is clearer to me, and that is the definite relationships existing between the trigonometric identities.

Do you think that the authors of such texts have that kind of time to consciously lead the attentive reader into specific conceptual arenas?  They must exist in eternity.  I mean, there has got to be some kind of transcendental realm where "time" no longer presses down on us.  That's the "place" where we can actually enjoy the thinking process and momentarily transcend drudgery.

Maybe ...  :-\
Title: Re: How to Attain a Studious Life
Post by: Holden on February 20, 2016, 03:11:29 pm
They must exist in eternity.  I mean, there has got to be some kind of transcendental realm where "time" no longer presses down on us.  That's the "place" where we can actually enjoy the thinking process and momentarily transcend drudgery.

Our minds are so alike that sometimes I find it extremely strange! Just today I have been reading about just such a "place".
Dog, dog, dog, dog, dog, dog, dog, dog, dog, dog, dog, dog, dog, dog, dog, dog, dog.

Wait, was that a word? Didnít it mean something a minute ago?

This fascinating psychological phenomenon ó when a word loses meaning after being repeated over and over without interruption ó is called semantic satiation.Where it really gets interesting?

This leads to other curious phenomena, one in which you want to experience a type of semantic satiation, like during meditation when you sit down to meditate and employ a mantra word such as a mathematical formula to chant repeatedly.

I wonder if whats being done with "OM" in the video could also be done with mathematics.
From the "Dreams in the Witch-House":
Old Waldron, who had curtailed his activities before, would have made him take a rest - an impossible thing now that he was so close to great results in his equations. He was certainly near the boundary between the known universe and the fourth dimension, and who could say how much farther he might go?

But even as these thoughts came to him he wondered at the source of his strange confidence. Did all of this perilous sense of imminence come from the formulae on the sheets he covered day by day?

Can semantic satiation help me to cross the boundary between the known universe and the fourth dimension?
Title: Re: How to Attain a Studious Life
Post by: Dog on February 21, 2016, 01:54:09 pm
I am not so sure I want to find the fourth dimension, but I suspect that we live in a 3-dimensional world simply because we are organisms equipped with apparatus for mental representation limited to 3 dimensions.   

I respect the limitations of the world-representing apparatus "I" (the thing in itself) am equipped with.

I do not think it is possible to OM myself into perceiving higher-order dimensions.  We handle 2-dimensions OK on paper, but even 3-dimensions gets a little tricky when trying to represent it on flat 2-dimensional space.  Something Cioran wrote that is very depressing but may be true in the end.  He wrote that there can be no "Instant India".   Ciroan wrote some depressing things, but I found this especially depressing.  It made me think that the only solution was a barbaric drowning in hard alcohol.  He wrote an attack against the "Instant India" that has been fashionable in the west for a couple of centuries now.  I mentioned it somewhere where Cioran seems to be promoting living as a beggar, that this would be authentically getting the gist of ancient teachings, quite the opposite of holding a position on the philosophy (of Asian Studies, perhaps) in some University, driving to a meditation temple in one's Volvo or Volkswagen.   

As for the trigonometric identities, I find them fascinating.

I love how circularly connected (pun intended) certain branches of mathematics are.  I mean, I want to understand how to solve differential equations, so this leads me back to Integral Calculus where there is use of trigonometric identities ...

These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

I am tempted to take some time to transcribe such a list from wikipedia (https://en.wikipedia.org/wiki/List_of_trigonometric_identities) into a special high-quality art notebook and call it Techgnosis.

So, as you can see, I have my hands full just with 2-D and 3-D.  That's where my attention is focused.   :)

If I compare this intellectual journey to climbing a mountain, where the mathematics of classical and quantum physics is the mountain itself, then I see myself, personally, as spending many days at a time in a camp site off the beaten trail going on deep tangents.   (LIST OF INTEGRALS OF TRIGONOMETRIC FUNCTIONS (https://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions)

One thing leads to another.  I first loved Calculus because it forced me to use what I really enjoyed, Algebra.   Now that I want to learn about differential equations, I find I keep going further back to fundamental ideas ... Integrals .... and, as I said, this leads to an appreciation for trigonometry (https://en.wikipedia.org/wiki/Inverse_trigonometric_functions)

Do you remember when Henry Fool tells Simon, "I'm gonna blow a hole this wide in this world's idea of itself!" ?

Well, I would like to embrace a feeling of indifference to the large egos of physicists working in N-dimensional space by realizing my cup is over-flowing in R^3.  I am serious about this.  My plate is full.  I am engaged in a spiritual battle with a mindset that spans many cultures, not merely the culture of Taker Prison, and that mind-set is based on "hierarchies". 

I hate to sound mystical, but the entire apparatus of the mathematics for physics would crumble without being grounded in the fundamentals.

For this reason, I will honor any text I approach by always pausing when necessary to review necessary grounds. 

Of course, at such a rate, one might not ever reach the top of the mountain.  I am not trying to reach that ... I am content just camping out somewhere on this mountain.   ;)

With this being said, please don't think I don't appreciate the passage you quoted.  Is that from Lovecraft?  I vaguely remember having read that.  Passages like that makes me suspect Lovecraft did a great deal of dabbling informally in higher mathematics.  He did not enter the university, but that did not banish him from a love of learning.

Did all of this perilous sense of imminence come from the formulae on the sheets he covered day by day?

Indeed ...  :o

It is kind of eerie that the words "qualitas occulta" - which we both admit to being somewhat obsessed with - were the very words that brought someone into our semi-private discussions.  That this has further motivated me to divert my attention from Number Theory and Abstract Algebra into more traditional branches of mathematics which are used in Physics, has forced me to adapt (psychologically) to the chaos of what we call "the mind".

I have no choice but to approach these subjects with humility.  I mean, fortunately I already have an interest in Linear Algebra and Multivariable Calculus ... so, preparing for this "disciplined self-study" of Differential Equations is pretty much where I peak out.   You see, I will be satisfied with this ... if I want to revisit physics with a "revived, revamped, and expanded" mathematical maturity, well ... if that's where it leads ... so be it.

I appreciated that one video about chaos theory, and how some are applying it to psychology.  I just want to spontaneously flow in the moment and not "minimize" what I am thinking about today.  I don't want to "glorify" theoretical physicists.   When push comes to shove, they must exert just as much mental concentration to visualize physical phenomena mathematically as anyone else.  I don't believe in "quantum gods" ...  ;)

Theoretical Physicists, Mathematicians, Engineers, and Scientists of all stripes have to eat food, sleep, poop, piisss, and the rest.  On an empty stomach, the brain does not function.  I mention this because I am constantly reminded of this when I have to put down the pencil and eat bread or drink carrot juice ... just because ... well, we are, as Kurt Vonnegut said, chimpanzees who can read and do a little math.  To be an organism is a humiliating experience.  hmmm ... I need water ...

You do understand that my goal is not to be a "great mathematician" who makes some kind of breakthrough discovery, but to humbly cherish gaining understanding of very basic and fundamental concepts.

It think it is vital to find enchantment in thinking about whatever we are able to grasp at the moment ... in a grain of sand, so to speak.  I do not want to try to appear to know more than I do.  You understand that it's all I can do to gain a tenuous grasp of the material.

Holden, I am the eternal student ... I am not attempting to master this craft, but merely able to demystify it a little for my own satisfaction.

If ever I sound like a "hater of professional scientists and engineers," it may simply be an underlying resentment against what I perceive to be a classist hierarchy of servile asslickers posturing for social status and ego-inflating positions ...

Studying this stuff is an ego-destroying activity for me, and I study it in defiance of the meritocracy.   I want to reach just beyond my intellectual capacity, and then come back around and focus on the level that I can comfortably comprehend.

It helps if I see myself as a chimpanzee, for then I will be patient with my frustrations when I face problems that incomprehensible. 

I do not ever want to be one of those who says, "Oh, that, I studied that, I know all about that.  It's boring."

Rather, I am more like, "Oh, that, yes, well I studied it formally but, ever since, I have felt as though I barely skimmed the surface ... and now I am left in a state of perpetual curiosity ..."

In the past I have used language such as "servile scientists" ... Both my grandfathers were scientists, a chemist and a mechanical engineer ... one helped build the atomic bomb.  My father studied refrigeration and continues to install walk-in freezers for semi-wealthy businessmen who own chains of liquor stores and what not.  Me?  Well, I am the skinny version of Ignatius Reilly.  I consider myself a misfit, an outsider ... I felt no shame cleaning toilets or collecting garbage cans.  Like Kafka, I may have been paying a kind of karmic penance ... you know, for grandpa's designing some little pressure relief valve component which may or may have helped with the construction of an atom bomb.

I am no space monkey.  I look at those who want to travel to Mars the way all the other animals on this planet would look at them ... as a pathetic joke to be mocked.

So, I am quite the frustrated chimpanzee, huh?

I can honestly say that I love mathematics, and I am fairly certain that I love computers ... This will not change the fact that I am a moody ape who seems most content when hiding away studying ... but I have contempt for anyone who would want me to compete or jump through hoops or act enthusiastic about "the space program" or the "arms race."

Just because I love mathematics and programming does not mean I want to get aboard the Starship Enterprise.

Hooooo Haaaaa

Title: Re: How to Attain a Studious Life
Post by: Holden on February 22, 2016, 12:16:23 pm
Both my grandfathers were scientists, a chemist and a mechanical engineer ... one helped build the atomic bomb.

That's interesting.You certainly come from a long line of German mathematicians.I cannot help thinking that its only with you that your lineage has reached its culmination & with it,its end. It made me think of The Kreutzer Sonata by Leo Tolstoy:

But," said I, with astonishment, "how would the human race continue?"

"But what is the use of its continuing?" he rejoined, vehemently.

"What! What is the use? But then we should not exist."

"And why is it necessary that we should exist?"

"Why, to live, to be sure."

"And why live? The Schopenhauers, the Hartmanns, and all the Buddhists,
say that the greatest happiness is Nirvana, Non-Life; and they are right
in this sense,--that human happiness is coincident with the annihilation
of 'Self.' Only they do not express themselves well. They say that
Humanity should annihilate itself to avoid its sufferings, that its
object should be to destroy itself. Now the object of Humanity cannot
be to avoid sufferings by annihilation, since suffering is the result
of activity. The object of activity cannot consist in suppressing its
consequences. The object of Man, as of Humanity, is happiness, and, to
attain it, Humanity has a law which it must carry out. This law consists
in the union of beings. This union is thwarted by the passions. And
that is why, if the passions disappear, the union will be accomplished.
Humanity then will have carried out the law, and will have no further
reason to exist."

"And before Humanity carries out the law?"

"In the meantime it will have the sign of the unfulfilled law, and
the existence of physical love. As long as this love shall exist, and
because of it, generations will be born, one of which will finally
fulfil the law. When at last the law shall be fulfilled, the Human Race
will be annihilated. At least it is impossible for us to conceive of
Life in the perfect union of people."


"Strange theory!" cried I.

"Strange in what? According to all the doctrines of the Church, the
world will have an end. Science teaches the same fatal conclusions.
Why, then, is it strange that the same thing should result from moral
Doctrine? 'Let those who can, contain,' said Christ.

Do you think its possible to have no interest in algebra,calculus,linear programming-no interest in any particular topic/branch of mathematics & yet be obsessed with mathematics,with mathematics- in- itself?
Title: 999: How to Attain a Studious Life
Post by: Dog on February 22, 2016, 03:05:53 pm
I am repeatedly astounded that we have been fortunate enough to have this medium for our correspondence.

As I am inclined to babble on and on, I will try to answer your question directly.

Yes, I do think it is possible to  be obsessed with mathematics while not committing oneself to any particular discipline (with no interest in algebra,calculus, or programming). 

Perhaps it all depends on how one is wired. 

Myself, I am comforted by some guidance and structure when approaching "mathematics" since it is so monstrously vast, where so many regions are incomprehensible to me.

You might be drawn to Chaos theory and fractals ... and, as you mentioned, Number theory ... It does seem a little grotesque to compartmentalize into so many branches, I know.

Some people are wired to zero in on what they feel is "the ultimate" area of study, almost the scientific version of Hegel's "Absolute".   Many theoretical physicist revere Quantum Mechanics as some kind of holy grail, presenting it to the lay public as a sort of streamlined modern day mysticism for an elite inner circle of "quantum gods".   ::)

Myself, when I venture into reading about Quantum Physics, I look at a few pages and back away from the text shaking my head - only because it appears to be utterly incomprehensible to me.

I am easily overwhelmed, you see, and, much like you, I don't want to feel as though my "appreciation of the realm of mathematics" is inadequate just because I have not mastered a particular branch.

However, speaking for just myself, I find that when I finally do find a source, such as a textbook, that is just a little beyond my comprehensibility, if I find myself respecting the effort the authors of the text have invested in trying to guide the reader/student to understand the subject (or branch or field), then, should I become engrossed in an exciting adventure, I tend to be less "nasty" and "mean-spirited" toward the "specialists" ... and attempt to put into words what I do and don't comprehend.

For example, since I have become curious about differential equations, I resisted the temptation to go diving into some incomprehensible physics or engineering text. I was very fortunate to find a book for five dollars on Amazon that really appears to be directed at my level of comprehension.

See Differential Equations & Linear Algebra (http://www.amazon.com/gp/product/0536357560?psc=1&redirect=true&ref_=oh_aui_detailpage_o03_s00)

There are "exploratory laboratory" exercises in a supplemental Interactive Differential Equations (http://www.aw-bc.com/ide/) ... I am curious about the authors' encouragement to learn how to "write in mathematics" where they promote "suggested journal entries" to practice writing about mathematics, which for me, has always been frustrating.

I suppose I am wired in such a way, where my mind can easily be overwhelmed by the chaos of it's own thinking, that I cherish some kind of "grounding" ... and, yes, guidance from experienced "mathematicians".

Remember how over the summer I was just exploring Number Theory, and I stumbled upon (purely by chance) William Stein's Elementary Number Theory: Primes, Congruences, and Secrets (http://www.maths.tcd.ie/pub/Maths/Courseware/NumberTheory/Stein.pdf)?  I was fascinated with the Chinese Remainder Theorem.  That text had a huge effect on my life, encouraging me to begin a special series of "Math Diaries" (which I now call Computational Sketchbooks).  It is also where I first learned about Sage - Stein is the founder, and, hence became intrigued with Python on whose libraries of modules much of Sage is built.

Quote from: Holden
    Both my grandfathers were scientists, a chemist and a mechanical engineer ... one helped build the atomic bomb.

That's interesting.You certainly come from a long line of German mathematicians.I cannot help thinking that its only with you that your lineage has reached its culmination & with it,its end.

It's end?  Yes, this appears to be the case unless the space monkeys lock me in a planet-of-the-apes cage (***blankets and bananas***) with a beautifully exotic specimen for the purpose of squirting in the last few drops of my evil seeds.    :P 

As you phrase it, I see "its end" more as a Schopenhauerian/Nirvanic culmination ...

As for a long line of mathematicians, no, no, and again, no.   It was just the fathers of both my parents.  My parents have absolutely no interest in mathematics.  As for great grandparents, I have heard stories about a Swedish great grandfather who was a heavy drinker and a hater of religion.  I feel a great connection with him.   Another great grandfather (of German lineage) committed suicide like Schopenhauer's dad, and I believe he was also a businessman like Heinrich Schopenhauer, but not quite as wealthy.   I think he owned a factory that crafted coffins.  Can that be true, or is that some kind of joke?   Well, I'm sure it wasn't a factory.  It wasn't a coffin boutique!  Maybe they were built on demand ... People are always dying.  :D

He committed suicide Mr. H?  And crafted coffins?   ??? ;D

How ironic!

In the tradition of Kurt Vonnegut Jr., I might as well spill the beans, since you alone display an interest in such hilarious details.

It was just Charlie Weber, the engineer who worked for RCA designing "pressure relief valves for water heaters" (and forced by the US government to help out on some minor detail with the Manhattan Project) ... and Carl Hentrich, the chemist who worked for DuPont (and tried to convince my father --- Mr. Bill, who constructs walk-in freezers for countless merchants from India who have set up shop [mostly liquor stores]  ;) on the East coast ---- to take a job at DuPont as a janitor).   When my father rejected this game plan, my grandfather suggested he study "Refrigeration" since it could not be "outsourced" ... it's hands on and expensive to ship the parts, unlike "information technology" which is zipped and zapped around the planet at the speed of 3◊10^8 meters per second.

Too much information?

I am not ashamed of my grandfathers.  In fact, they appear to be the source of my mathematical inclinations.  I'm just more wired like my Swedish great grandfather as far as my insubmission (http://whybother.freeboards.org/general-discussion/trouble-with-being-cioran/msg1895/#msg1895) goes, and a little like my father when it comes to disassociating myself with any class of people who seem pretentious or pompous, such as ... well, servile corporate scientists.   :-\

No offence to my grandfathers, I hope.


I suspect that I inherited my hand-writing directly from my great grandmother on the H side.

My "sober" hand-writing, that is. 
Title: Re: How to Attain a Studious Life (Why Mathematics?)
Post by: Dog on March 14, 2016, 01:18:59 pm
After reading this, I changed this forum's title from "Math Diary" to "Why Mathematics?" ...   

Quote from: Warren Henning
Recently Iíve been working my way through several lower-division undergraduate math textbooks. Why? Why spend your spare time learning math, which is challenging and sometimes dry?

Because math is too beautiful, too powerful, and too important to be reduced to mere mechanical calculation the way most of us experience it in school. That is not what math is really about. What matters in math, and what gives it its beauty, is reasoning and connections between ideas.

Reading books at my own pace lets me try a subject out without fully committing to it and making it a necessity that I find work based off of it. It confers the full lightness of being a beginner, exploring in a free, untutored fashion. A degree program may be a good choice for some people. If thatís you, fantastic. This article is for people who canít or wonít commit to a conventional academic program.

The full article is: A Software Engineerís Adventures In Learning Mathematics (https://medium.com/@warrenhenning/a-software-engineers-adventures-in-learning-mathematics-62140c59e5c#.yxtrgo76h)

I know we don't care for the title, "Software Engineer".  It sounds pretentious.  I prefer the term "code monkey."    8)

Regardless of what Warren classifies himself as professionally, as a fellow student of Mathematics, the Queen of Science (with Number Theory as the Queen of Mathematics), I like his suggestions.  I can tell that he must have a similar passion [LIFELONG OBSESSION ?].  I want to lift some of his tips here in our thread about How to Attain a Studious Life, giving him the credit, of course.  My own objective is to strip the requirement of usefulness from the subject, as if something needs to be useful or career-oriented in order to be worthy of our devotion.

We already know that most worthwhile pursuits are lonely paths ... but it's good to chronicle some advice by fellow math junkie, even if he pays homage to military training and anti-slack concepts such as "discipline".  Myself, I would like to transform the "difficult nature" of studying mathematics into "slacking off" ...  I will see if we can come to some consensus on just what it is that makes an activity mentally painful in one psychological setting, and mentally stimulating in another psychological setting.

Quote from: Warren Henning
Cultivating Disciplined Habits While Walking A Lonely Road

Thereís something going on when attempting any challenging course of study. Weíre striving to build within ourselves the discipline to achieve a worthwhile, ambitious long-term goal in small, manageable increments. We are training ourselves to become disciplined, effective learners. We are sculpting our minds and emotions to become used to working hard in our spare time.

This is challenging in an unusual way: for the most part, we are alone.

Making time to study with work and family life can be very challenging. We live in a noisy, busy world, and math demands the deepest concentration, preferably in large, uninterrupted blocks of time. Just finding a quiet place free of distractions can seem impossible.

If possible, do not procreate ... and try to avoid living in apartments where you might be bombarded by intruders.  It might help if you drop out of the work-force altogether like a Japanese Hikikomori.  Yes, resign from the species, and find time to think and reflect.  The Establishment would like to have a monopoly on mathematics education.  They just can't stand not to be making money off some poor sucker who just wants to understand mathematics better, and is not particularly fond of how corporate drones and a-s-s lickers manage to destroy "the love of learning" with their pressure and stress.  If you inherit a little 4 cylinder vehicle and need to insure it just to gather groceries, stocking the shelves at the grocery store may be necessary to keep the car legal.  For now, let's bypass the complications of keeping your organism fed and dry.   :-[

I like this section on Facing YOUR Demons ... [the "Why Bother?" demons?]:

Quote from: Warren Henning
Facing Your Demons

Iíve had to contend with many self-defeating mental blocks I wasnít initially even aware were impacting me. Iíve thought at various times that I didnít even have the right to try teaching myself because math is somehow the exclusive domain of established experts. If you arenít already a master by now, donít even bother, part of me thought. That ďdonít even botherĒ sentiment paralyzed me for years.

I hope you give yourself permission to pursue your own interests, even if you donít have the traditional background of experts in your field of interest.

Fortunately, as an old gortbuster warrior, I do not defer to the authority of the gort establishment.  So, it looks like I am good to go on the pursuit of my own interests.   :D

Quote from: Warren
Average People Have Things To Add, Not Just Alpha Geeks

If youíre like me, the academic-industrial complex of research labs and paywalled research journals is a bit intimidating. There seems to be no room in such a place for mere mortals. And yet there are so many problems to solve in our world, and MIT-educated geniuses so few in number, that they will undoubtedly overlook things us average folk can be in a better position to actually do something about than they. Researchers often donít focus on delivering complete solutions to problems their work may be useful in addressing; by necessity, they focus on the fundamental issues, and once those are figured out, they often move on to the next research problem. Valuable insights that could be useful to many languish in obscurity behind horrid paywalls. Letís not make the established, traditional experts out to be more than they are, either; a lot of them are really only capable in their specific niche and lack a lot of valuable complementary skills you already possess.

It seems a great tragedy that many of our best and brightest in industry wind up working on better ways to sell ads online or allocate capital in financial markets. The amazing things they could do with their highly advanced skills go unrealized. Because of this, I think the wider group of intellectually curious amateurs have an opportunity to truly capitalize on the advances made in the sciences.
[italics added by H]

[This next segment is simply beautiful ... This Warren character is a true math preacher!  Great stuff!]   :'( [tears of enlightenment]

Advocation Of Autodidacticism In Math, Science And Engineering

To be honest, I feel nervous even publishing this. I left it sitting as a draft on Google Drive for well over a month after spending hours writing and editing it. What will the turbogeniuses who already know everything I aspire to learn and are also probably better software engineers to boot say? Many of them are several years younger than I am, already more accomplished in every way. I just have to learn to live with that.

The prize I have my eye on far overshadows any anxieties or reservations I have. A world of eternal truths and cold, subtle beauty awaits. We will fill whole notebooks with thoughts, scribbles, dead ends and epiphanies. We will construct a body of knowledge which we can truly claim for ourselves and which we can never be stripped of. Let us grant ourselves permission to be beginners, to be uncertain, to have more questions than answers, to be ignorant but motivated to learn. Let us dare to begin.

Iíve found searching Amazon and the wider Internet for books to be quite rewarding. You can be surprised by what you find.

Let us dare to begin!

More importantly, may we continually cultivate the Beginner's Mind by embracing our "not-knowing" state ...

Title: What virtues does studying mathematics inculcate?
Post by: Dog on March 22, 2016, 10:52:44 am
I was searching for articles related to "mathematics and humility" as I am hoping some kind of psychological transformations might make the endless lifelong encounter with mathematics less frustrating for me.

It is said that when mathematics is practiced with great devotion and humility, it involves the transformation of both mind and heart which leads to wisdom. 

What virtues does studying mathematics inculcate? (https://www.quora.com/What-virtues-does-studying-mathematics-inculcate-besides-logical-thinking)

Patience. Math is hard. It takes time. Try to make progress today, but also know it will be there for you tomorrow.

Humility. Related to the above. Math will surpass your abilities on some days. That's fine. It does that to everyone sometimes.

This next one I just love:

Intellectual honesty. You won't get anywhere until you admit to yourself what you don't know.

In other words, sometimes our confusion may not stem from ideas in Calculus or Differential Equations, but in the Algebra and Trigonometry we are using to solve problems in "other subjects" that make heavy use of algebra, geometry, trigonometry, and even general number theory.  It is great to have to continually return to fundamental ideas.

Zenlike mental calm. It is impossible to do math if you are feeling overly emotional. To study math is thus in some sense to rehearse inner stillness.

An aesthetic appreciation for rationality.  You know that in some sense it will always be beyond you, but the slow, steady accretive work of understanding one piece at a time has its deep, life-long satisfactions.

Willingness to tolerate bad explanations.  Changing gears, a lot of pedagogy in math is terrible, perhaps because people generally don't understand math all that well, or just don't know how to talk about it clearly. By wading through bad explanations, you can learn how to mentally translate something complex into something that makes sense to you. This is a useful skill in other domains too.

An appreciation for complexity and for the limits of our ability to understand things.  I guess this is the same as humility above, but doing math makes you realize that most people, most of the time, probably don't know what they are talking about. Math, like programming, chess, or most complex pursuits, is an antidote to human BS because BS doesn't get you anywhere in trying to figure out something mathematical.

Title: Re: How to Attain a Studious Life
Post by: Dog on March 26, 2016, 10:37:46 pm
To motivate myself to forge ahead and continue studying ... I reflect upon those who are physically trapped in jail cells, frequently in pairs so as to NEVER have any true solitude and always be tormented by the presence of another human being in close quarters.

I reflect upon how precious is the access to paper, pencils, computer algebra systems, textbooks, coffee, tobacco, peanut butter, eggs, etc ... a blanket and pillow ...

Sometimes it helps to imagine oneself in some open air prison where you will spend your existence in your room, walking around outdoors (dodging automobiles) ... and you are studying to understand ... and you cherish being one who knows that there really is no one to know, nowhere to go, and no one to know. 

Study as though you are doing something forbidden rather than required, like a chattel slave sneaking a book of higher mathematics back into his quarters, then refusing to report to the master in the morning ... come what may.
Title: What virtues does studying mathematics inculcate?
Post by: Dog on April 02, 2016, 11:15:05 pm
Patience, humility, and intellectual honesty.

What are the antitheses of these virtues?

Frustration, arrogance, and self-deception.

I tend to focus on more difficult material in the morning, but when my brain feels too stretched, later in the evening, I have been going over some more fundamental and elementary material.  During such "casual" study sessions, I am humbled by the fact that, regardless of the "level" of the exercises, I find that I still have to CONCENTRATE and THINK.

This requires patience and humility, true; but the most amazing virtue these sessions inculcate is intellectual honesty.  I have to accept that some problems are not quite as straight forward as they first appear.

Take equations involving absolute values.  These require a little more thought.

|x - 3| + |x - 4| = 1

It seems too "easy" to pay attention to ... what a pompous and arrogant professor might refer to as "baby math," but one should never be ashamed to stop and think.  That's why I enjoy going over more basic reviews later at night, giving my brain a little rest, and at the same time, developing a kind of humility in that I do not feel "above" having to think about more elementary material.

I remember hearing a story about how Einstein had difficulty counting his change.   He was prone to make arithmetic errors.   

Just think of how vulnerable all human beings are to miscalculating, where the error is arithmetical or algebraic. 

I really want to develop this kind of humility and patience and intellectual honesty.

Studying mathematics as an antidote to human bullshiit.

It is an antidote to human BULLSHIIT because BULLSHIIT doesn't get you anywhere when trying to figure out something mathematical.

Title: Euclid: How to Attain a Studious Life
Post by: Holden on April 03, 2016, 10:26:22 am
Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.

O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
Title: Re: How to Attain a Studious Life :: With Defiance
Post by: Dog on April 11, 2016, 09:15:11 pm
I think that there is an element of defiance in my renewed obsession with specific core subjects of mathematics and physics.  These subjects are associated with careers in science and engineering.  Maybe, when I finally reached the goal of graduating from the university back in 2002, already fairly "old" (35), when it began to dawn on me that I was not suited for corporate employment, I believed that all that education had been in vain.  So, I sunk into a depression and rebelled in self-destructive ways.

Now I approach those same subjects with an entirely different attitude.  I am still rebelling, but now in a fairly healthy manner.  I now study defiantly.  I do not use the word defiantly lightly.   Now, even if I feel, for whatever reason (for all I know it is my own fault just for being a cantankerous pessimist and anitcapitalist) that there is no place for me as a professional scientist, I want to study to develop as a thinking human being, a problem solving higher order primate.

I think Schopenhauer would encourage me to indulge in these studies.  For one, he had great respect for physics.  Another aspect has to do with experimentation.  Perhaps due to the pressure to prepare students for technical work in industry and business, the way these subjects are taught can be haphazard, where students do a great deal of cramming, graduating filled with doubt and very little confidence.

Maybe I wish I had this attitude back in 2003 when I first went on welfare for emergency assistance, but, I went on a downward spiral.  It is what it is.  Now, even though I have all the textbooks I will want to study, I found myself repeated the same scenario, trying to study 6 subjects at once.  Now, here is where the defiance comes in.  There is no need to proceed in this manner. 

I have going to stick to some kind of plan stretching from January 2016 into the summer throughout the autumn, and well into 2017, maybe even stretching into 2018.

Maybe my notes on this process will be more meaningful to some youth of the future than some kind of existential novel or collection of diatribes against being born.

The acquisition of the hard copy textbooks motivates me to remain stable and grounded.  To be blunt, the little library encourages me to "nest", to settle down for a good decade.  This will also serve my mother well, since she requires a companion at this time in her life. 

I lost the notes I kept from the university, so I am making these notes even better, hoping that I will have yet another opportunity to review this stuff, this time using notes which are more explicit than just haphazardly "searching the Internet".

In the past I have associated defiance with drunkenness, drums, and loud guitars ... Dionysus.

Now I suspect that studying math and science, when there is no motivation to become a working "scientist" may be a kind of stubborn defiance.  I will be developing skills that are useless for making money, acquiring security, automobile, et cetera.  I know that would sound like a false statement to the youth who associate scientific education with careers as nuclear physicists for the military.

In a world where professional athletes and TV celebrities are worshiped, studying for the purpose of filling one's head rather than one's bank account is a defiant way of life, all things considered.

In this way I am intend to flip the script.   Rather than be made to feel like an intellectual deadbeat non-breeding pariah of society, in my own mind, at least, I can live a somewhat heroic life ... where the hero happens to be an antihero in terms of not representing the wealth-warped values of his contemporaries.

There is one slight aspect of the autodidactic method:  when there is a mistake in the solution manual, a glaring fundamental inconsistency, this can cause unnecessary frustration.

It can be something very fundamental, such as a 2x3 matrix times a 3x1 vector yielding a result with dimensions 3x1 when you know the result should be 2x1.

The only good thing I have to say about such blunders is that encountering them can be a test of one's confidence.  What is one to do?  Complete the problem anyway, or move onto the next one?
Title: Re: How to Attain a Studious Life
Post by: Dog on April 14, 2016, 11:40:38 am
I cannot express strongly enough how worthwhile an endeavor it is to give Sheldon Axler's Algebra and Trigonometry (http://libgen.io/search.php?req=Sheldon+Axler+algebra+and+trigonometry&open=0&res=25&view=simple&phrase=1&column=def) its due.  While I initially presumed this would be a simple review of material, I have so far been pleasantly surprised by the density and diversity of the exercise sets.  If one is self-motivated, one will appreciate how many of the exercises really force one to think like an ancient scholar, like Euclid would.

This has altered my approach, and I will proceed with mindfulness ... seeing myself not so much as a former student living an uneventful life, but as one who aspires simply to be a learned citizen of the world. 

I know that you [Holden] are more attracted to Number Theory and the more esoteric branches of mathematics, but I have never seen an approach to Algebra and Trigonometry like Sheldon Axler's.  The exercises are a treasure trove, where one can momentarily put aside the realization that it would be better never to have been born, and get lost in analytic geometry, embracing this material as one of the ancient scholars would have, or even as an escaped slave who has come to lead a scholarly life outside of academia.  The Internet has made such scenarios possible.  (The intersection of the modern and ancient worlds?)  This technology is revolutionizing the way we educate ourselves. 

What is also cool about this particular textbook is that many of the solutions are worked out in detail at the end of each section, eliminating the need to track down a solution manual.

This book is all about the exercise sets, which would compel you to sharpen your pencil and stretch your mind.  I thought this book would be used just for reference, but it is turning out to demand my full attention.

For me, this is where patience and humility and intellectual honesty come into play. 

My goal is to stretch my mind and force myself to think carefully as opposed to racing through to get it over with.  Most of what I know of algebra, trigonometry, and analytic geometry I associate with calculus.  I have not allowed myself to approach these subjects as ends in themselves since 1993 when I was preparing to go to community college to take calculus in 1994, ten years after graduating high school.  I want to revisit the fundamentals to get a better appreciation for how much algebra (analytic geometry) and trigonometry is the foundation of the more advanced subjects.  Many things that I associate with calculus are really algebraic or trigonometric.  I can never consider my study of the fundamentals as complete.

Now I suddenly found myself grappling with problems in Differential Equations and Multivariable Calculus, thinking I would just pick up where I left off in 2002.

While I could work through the exercises, I was not calm, and even kind of frantic.  I want to redevelop strategies for problem solving.

I think that Sheldon Axler's textbook will help me rebuild my preliminary foundations.

Also, there is color pdf for Dale Hoffman's Contemporary Calculus (http://scidiv.bellevuecollege.edu/dh/Calculus_all/Calculus_all.html), freely available on line.  That series of texts are sold in separate volumes, actually less expensive freshly printed at lulu DOT com than used from Amazon.  For instance, CC IV is $18 to print at lulu, but shockingly like $40 used through Amazon.  No, it's not reachable through Library Genesis (yet).  The author (Hoffman) encourages self-motivated students to make use of these online resources rather than investing in the texts. 

But, the font is small so you might be better off using the online pdf in color (also since you can't be lugging books around while traveling around for your employer).
Title: Re: How to Attain a Studious Life
Post by: Holden on April 14, 2016, 12:31:40 pm
Thanks.Much appreciated!
Title: Re: How to Attain a Studious Life
Post by: Dog on April 14, 2016, 01:49:27 pm
Although it is not necessary, since we each march to our own drum, it would be interesting to see where our dialogue goes were we to suddenly find ourselves on the same page.

Some of the exercises in Axler's textbook, as I have said, are quite novel.

novel - new and not resembling something formerly known or used
         - original or striking especially in conception or style

Granted, there are a handful of exercises that can be "easy," but many others that one might expect to be easy end up requiring some thought, nothing too severe, but just enough make the text worthwhile.

I have a shelf filled with texts, and I figured that going through this one early on can only enhance what I get from the more advanced subjects.

From the investigations I have been making on the Internet, it appears that this itching to gain a better understanding of subjects one figured one was "finished with" years ago is not so rare at all.  I had read several comments by people doing something very similar to what I am engaged in now. 

No matter what level I am working at, whether advanced or elementary, I still find myself making errors, but fortunately they are careless errors, not really many logic errors.  I mean, I will copy the wrong equation initially and solve with the wrong initial equation or other values initially copied down differently than was in the text.  Then I am off and running, calculating correctly, but basically solving a different problem, so, of course, when I go to verify my answer with the solution manual, there is that WTF moment.

[Yes, I invested in a magnifying glass with LED light]

So, the main errors occur from "dumb errors" which are not as bad as totally not understanding what one is doing.
Title: Re: How to Attain a Studious Life
Post by: Dog on April 20, 2016, 03:31:33 pm
Going through the Hoffman text (http://scidiv.bellevuecollege.edu/dh/Calculus_all/Calculus_all.html), chapter 5, even though I am anxious to get into chapter 6 since I want to understand direction fields better, I am finding the core of Hoffman's approach is in the exercises, which is great.  When I need clarification, I have been looking for clarification in the Internet.  It has been money well spent paying for the internet connection and an almost required expense for any student (of whatever age) who wants to study in a non-traditional manner.

I am sure there are countless others studying in this manner.  Not everyone who enjoys learning is going to be looking for credentials or grovelling around tables on "Career Day" looking to be hired as an intern.  Some of us have given up on doing what we love for a living.  That doesn't mean we have to stop learning.

I'm going to ride this until the wheels fall off, this life of the lifelong student of mathematics.

I may have mentioned this before, but I feel as though I am a self-ordained monk of his own order.

I may start to point out those who offer helpful explanations without any monetary compensation for the time they take.  The least I can do is point out a few.

For instance, while researching how to go about deriving a formula for finding the volume of an ellipse rotated about an axis, I found a comment left by a Mr Harish Chandra Rajpoot (http://math.stackexchange.com/users/210295/harish-chandra-rajpoot).  There is a reason why it is the last comment on the page as he answered the question so succinctly.

Proving the Volume of an Ellipsoid (http://math.stackexchange.com/questions/946198/proving-the-volume-of-an-ellipsoid)

Do we need to justify our interest in mathematics?  The restrictions are imaginary social constructs.  The Kingdom of Mathematics is within us.   :D

Anyway, although I don't have to point it out, I will anyway:  For the time being I am spared from many of the horrors of existence.  I understand how fortunate I am to be able to do what I'm doing.  The world is filled with nightmarish suffering, especially when it comes to losing one's health or one's wits.

I don't want to come across as a spoiled brat who only cares about his own personal agenda.

I think I may post less about the details of my little world.
Title: Re: How to Attain a Studious Life
Post by: Dog on April 22, 2016, 04:03:25 pm
How Philosophical Pessimism and Depressive Realism might be conducive to attaining a studious life.

Question:  Do you think being slightly depressive puts one in a state of mind conducive to the kind of continuous daily effort involved with the commitment to studying specific subjects?

Put in a more crude manner, what else am I going to do?  I might as well be boning up on my problem solving skills.

When I become unenthusiastic, and the world seems to be passing me by, I quietly and slowly continue to take notes and prepare to tackle the next set of exercises.  This somehow makes it clear to me that living a boring uneventful life is precisely what is necessary.  It is a blessing I would not wish for, but a blessing just the same.
Title: Re: How to Attain a Studious Life
Post by: Dog on July 27, 2016, 08:45:49 pm
"If war is too important to be left to generals, then, for analogous reasons, mathematics education may be too important to be left to mathematicians."

 ~ Jerry King (author of The Art of Mathematics, c.1992)
Title: Re: How to Attain a Studious Life
Post by: Dog on August 12, 2016, 07:55:42 pm
In all honesty, I get much more from working through "novel" exercises in interesting textbooks than from taking notes from the examples.   

One great benefit of the Internet is that, rather than just skipping an exercise that is too confusing, unfamiliar, or otherwise taxing, access to the internet must give one more courage in tackling more of the exercises.

It is fun to learn when there is no pressure to "perform", when the attention you give something is motivated by genuine interest and curiosity, not by the demands or requirements of society (family).

I think I enjoy working through exercises in textbooks because it allows me to carry on a continuous written dialogue with myself ...
Title: Re: How to Attain a Studious Life
Post by: Dog on April 17, 2020, 02:56:36 pm
I have come to the conclusion that I harbor secret perceptions of Schopenhauer as classist in so far as his condemnation of "the drudgery of arithmetic" as being beneath the dignity of a human being.  Schopenhauer himself displays a definite "artistic" distaste for computation (reckoning), stating that machines could be made [programmed] to carry out such tasks far more efficiently without the wear and tear on the sensitive brain that might be prone to becoming machine-like itself if it trained itself to operate as such.   He was an honest man, after all.

And yet there are those, with  far less wit or brilliance as Schopenhauer, who find great mental stimulation in studying the ways humankind has computed, calculated, reckoned with advanced numerical computations. 

I have to make a truce in my heart with Schopenhauer, accepting the subtle snobbery shown by certain classes, such as an intellectual class, that has contempt for those who value reckoning algorithms and the wondrous realm of computing and calculating.   The thing is, there is an evident aristocracy beyond the economic/educational/cultural, and that is Nature's Aristocracy, and in the Natural Order, I must error on the side of caution and take Schopenhauer at his word, that spending many years with the tedium of computing might not be so good for those who prefer to sleep at night.

Sorry, Herr Schopenhauer, I will end up in the madhouse yet!
I am sure to be mumbling SHH-ope-IN --HHH-AA-Uer ... with my dying breath, wondering what on earth am I to do now but smirk ... What kind of crazy dream is this?  For Madman Only?

For Hesse's Harry Hallar, it was some great Whoever, and for me, most likely I will have to face an image of Schopenhauer, my guiding spirit through-out.  I have to face that I must have burn myself out many times over, and the only reason I enjoy studying programming and mathematics is out of the Mental Stimulation of Pure Cognition.  There is something precious about committing oneself to a fat and challenging text, getting through the slumps, capturing a string of days of deep concentration ...

I am a low down and dirty "elegant programmer" who continues to pay homage to obscure mathematics texts, showing the work like a science-fictionalized monk.  I appreciate your presence in this Cave of Thought.
Title: The Setbacks of Learning (Mathematics as a Hobby)
Post by: Dog on April 02, 2021, 08:58:16 am
As has been the case in the past, I find myself having to look for inspiration from what I have written in the past.  It is as though I am reading someone else.

Now I am trying to see if I might gently guide myself back to more fulfilling rituals.

It is difficult to commit to exhaustive studies and rigorous proofs when it all seems to have been done in vain.   So, rather than seeing myself as guiding Holden, I am going to allow myself to see if a previous version of my "self" might guide this current bundle of nerves back into math mode ...

From Mathematical learning (and math as a hobby) (https://3quarksdaily.com/3quarksdaily/2011/08/mathematical-learning-and-math-as-a-hobby.html)

Quote from: Rishidev Chaudhuri
It is an oddly well-kept secret that mathematical learning is a very active process, and almost always involves a struggle with ideas. To a large extent, this is due to the nature of mathematical intuition: grasping a mathematical idea involves seeing it from multiple angles, understanding why it's true in a broader context and understanding its connections with neighboring ideas. And so, when you sit down to read through a proof or the description of an idea, you rarely do just that. Instead, digestion more often involves settling down with a pen and a piece of paper and interrogating the concept in front of you: ďWhat is this statement saying? Can I translate it into something else? Can I find a simpler case that will help me gain insight into this general context? What about this makes it true? What would be the consequences if this statement were false? What contradictions would I encounter if I tried to disprove it? How does this concept reflect those that have gone before? How do the various assumptions used to prove this statement factor in? Are all of them necessary? Are there other ways to frame this fact that seem fundamentally different?Ē And so on. And this interrogation often involves taking your pencil and paper on long digressions, slow rambling explorations of ideas that help clarify the one you're trying to understand.

Similarly, proving a mathematical statement or solving a problem is an unfolding of false sallies and blind alleys, of ideas that seem to work but fail in very particular ways, of realizing that you don't understand a problem or a concept as well as you thought. And again, these are not wasted. In almost every case, if someone were to just give you a proof or a solution and you didn't either try to come up with it first or actively interrogate it once you had it (which is almost the same thing), you'd learn that the statement was true, but learn very little about why it was true or what it meant for that statement to be true. And much of the learning in a math class happens not in the lectures but afterwards, in the time spent on problem sets (and, if you had a choice between attending the lectures and doing the problem sets, you should always pick the latter).

Unfortunately, most people make it through a high school mathematical education without being taught this. This has unfortunate consequences and makes mathematical learning exceedingly vulnerable to expectation and self-belief, so that it is often seen as something you either can or can't do, and many people see the struggle as a sign of a lack of ability rather than as an intrinsic part of the learning. There are certainly children who, for whatever accident of genetics, upbringing or attentional prowess start out by being quicker at math. But this seems swamped by differences in temperament and confidence, or by the effect that initial quickness has on confidence. How you engage with the setbacks of learning seems more important than how quick you are.

This was strongly brought home to me when running math classes. There would inevitably be two groups of people who could take the same amount of time to solve or almost solve a problem but be quite differently convinced about their mathematical ability (which, over a semester, ends up being self-fulfilling). Some students, ten minutes into wrestling with a problem, would find progress difficult and take this as a sign that they were learning what didn't work, were spending time understanding the problem, were edging towards a solution, were exercising their reasoning ability and so on. Others would start off anxious and ten minutes in, at about the same stage of reasoning, would come to me convinced that they were never going to figure it out, and that they were dumb or not good at math. And yet the two groups didn't seem to have wildly differing levels of intuition and for the second group reassurance that they were participating in the right process or helping them follow the path they were on, even if it was headed in the initially wrong direction, would often lead them to the same solution. Strangely, while some of the job of a math teacher seems to be to help with mathematical intuition, a large part of the job seems to be palliative, compensating for something that they should have been told or learned but hadn't: be patient with yourself.

One of the inevitable tragedies of specialization is that most people don't take classes in most areas after college or high school. For some this is compensated by an amateur interest in history, say, or philosophy. But for the variety of reasons I mentioned, the reasons that make students think that mathematics proficiency is an extreme example of a natural talent and that it is hopeless to do math without this essential ability, few people seem to maintain an amateur interest in mathematics or study mathematics recreationally.

If it isn't clear already, I think this is a huge pity, especially because it is often motivated by a false assessment of one's mathematical ability. And it is also a pity because most people stop doing math just at the point when the fun stuff starts, just when they've worked through most of the tedious arithmetic and are finally ready to embark on sweeping journeys of abstraction. It's like taking dance classes but never going dancing.