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Holden

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Re: Arithmetic is Drudgery
« Reply #15 on: February 09, 2017, 11:08:25 pm »
I think this experiment of yours in really interesting. I mean studying the same math book which you studied many decades back .
People are not only killing themselves but also others because of math.

http://www.hindustantimes.com/bhopal/bhopal-man-murdered-parents-for-forcing-him-to-study-maths-say-police/story-2gzSmNqMqijM3u5ewV2vUM.html
 
Also,I think what you have said numerous times,we need to study math not as a mathematician(what is a mathematician!) but as,say, a Machinist.

« Last Edit: February 09, 2017, 11:12:15 pm by Holden »
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Holden

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Re: Arithmetic is Drudgery
« Reply #16 on: February 12, 2017, 11:03:28 pm »
Mathematics is said to have saved Bertrand  Russell from suicide.
It has certainly given you a sense of direction too.
Do you think I can study just the topics (in maths) which interest me immensely?
I mean a bit like literature- I have not read all the so-called canonical works but only the ones which “spoke” to me ,so to say.
Like Schopenhauer says that a work of art is like a prince,you should let it speak first to you,do you think it applies to maths as well?

I am studying 3-variable simplex method at the moment.I will tell you all about it should I comprehend it myself.

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Re: Arithmetic is Drudgery
« Reply #17 on: February 13, 2017, 04:21:18 pm »
I have a book on the way, inspired by your interest, on Linear Programming.   I will also be looking into this with you.

Chapters 3, 4, 5 appear to be promising, and there is an updated appendix(C) for implementing linear programming with a spreadsheet (like LibreOffice Calc, the author using Microsoft Excel)

You can probably learn just enough about Vectors and Matrices and solving systems of equations to understand the Simplex Method.

Which books are you using for guidance?
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

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Holden

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Re: Arithmetic is Drudgery
« Reply #18 on: February 13, 2017, 09:53:43 pm »
Im using Operations Reseearch by Hamdy Taha .You see I never studied physical sciences in college,mostly what one could call social sciences.Like I told the other day I came across a problem,which I could solve easily by generating random numbers in excel.However,I think I can convert it into a linear programming problem.Last night I was learning the Gauss Jordan Elimination method.Mathematics as a way of
running away from the world is something to think about.
There are couple of other things I may be looking at in math.

There is something very elegant about maths.For one,it certainly took my mind off misery last night.I have framed my problem by myself into equations and will now apply the algorithm to them.Maybe that is it Mr.H,maybe I should hop from one math concept which interests me to another.
And keeping away from thinking about this horror movie we are trapped in.Math as a way of exorcizing or at least keeping away for a while,the ghosts and demons.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
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Re: Arithmetic is Drudgery
« Reply #19 on: February 16, 2017, 02:22:33 pm »
I am reading through Daniel Solow's Linear Programming: An Introduction to Finite Improvement Algorithms, so I am going to play around with LibreOffice Calc Solver (which is Open Source, free as in "free beer").   There is a short tutorial here for LibreOffice Calc and Micosoft Excel, including the version for Mac.    You told me you have a MAC at home.  You can install LibreOffice on a MAC.  Then you would use LibreOffice Calc which has a Solver like Excel.  I'm not sure if you are working at home or at the office.


Thanks for letting me know what you are tinkering with.  Although I am quick to dive directly into the calculations and computational aspects, as I suspected, the real thinking must occur in setting up the "objective function" and the constraints.  I will have to work through some "baby problems" before getting a feel for that.  I suppose it takes years of experience to develop a knack for formulating linear programming problems.

I am finding the subject interesting.  It is a nice diversion from re-learning from the ground up about reading and "doing" proofs.  The same author of the LP book, Solow, also has a book on proofs.  The 6th edition is on Library Genesis (solution manual at higheredbcs.wiley.com).  This 6th edition includes 3 sections from an earlier work, "The Keys to Advanced Mathematics" ... I found an old copy of the original "The Keys" for five dollars.

Solow starts from a geometric motivation, where you change the inequalities to equalities, and graph the lines to get an idea of feasible and infeasible solutions.  It leads into the algebraic solutions ... I am going to read through the entire book as it looks worth getting a feel for this, but I couldn't resist going directly to Appendix C to see how it is implemented in a spreadsheet.   Reverse learning? 

Meanwhile, Fräulein Kitty is getting worse ... she just lays around and has difficulty walking.   She started growling a great deal this morning.   I had put out the dough to take her to a vet last time, and he really didn't give too much information.  He suspected she sprained her leg ($100 for that).   I was against bringing her back ... It's not that I'm cold hearted, but if she is aging, there is nothing to do but comfort her and wait it out.   

The Mother had a neighbor bring her back to the vet, as I was against the idea.   It is what it is.  What's he going to do, prescribe pain medication?

UPDATE:  You guessed it right, she needed pain medication: she's got arthritis.
« Last Edit: February 17, 2017, 09:18:56 am by Ιδιοτεσ-5150 »
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Holden

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Invitation to embrace Dionysian Maths
« Reply #20 on: February 17, 2017, 12:36:02 pm »
First off ,sorry for not writing sooner-internet problem.
For me,  pure maths is problematic. It's just that it is often taught with no connection to the real world, and that makes it FAR more difficult for me to learn. The endless march of "Axiom, definition, lemma, lemma, proof, corollary" makes me lose interest rapidly. I am a visual thinker, and if I can't visualize something in my head, I struggle more than I otherwise would.
The dominant impulse among maths professors and so-called experts is to try to reduce trial and error. These people overestimate the prowess  of scientific knowledge. Intolerant of the messiness of trial-and-error volatility, they avoid small errors and the essential feedback those errors provide to the students. The end result is to create something that is steadier and more predictable, but fundamentally boredom inducing.  The stimulation of randomness is denied. But the effort to avoid small mistakes and minor pains makes larger ones more severe. Ironically, the imposition of false stability with the intention of avoiding mistakes makes math yawn inducing.

University atmosphere is marked by a preoccupation with pure math  , the systematic smoothing of the world’s jaggedness, and the stifling of trial and error. Math professors have made a religion of rationalism, optimization and efficiency.I  embrace the “Dionysian” maths: the dark, visceral, wild, untamed, hard to understand.This is about looking at reality as the mess it is .

They have turned the wolf into a dog and man himself into the man’s best domesticated animal.” – Friedrich Nietzsche

While cursing and bad language can be a sign of dog-like status and total ignorance –the “canaille” which etymologically relates such  people to dogs; ironically the highest status, that of free-man, is usually indicated by voluntarily adopting the mores of the lowest class. Consider that the English “manners” isn’t something that applies to the aristocracy; it is a middle class thing and the entire manners of the English are meant for the domestication of those who need to be domesticated.

P.S.I am sorry about the cat.I hope she feels better soon.
This world is insane.

« Last Edit: February 17, 2017, 12:54:35 pm by Holden »
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Re: Arithmetic is Drudgery
« Reply #21 on: February 17, 2017, 06:41:53 pm »
They have turned the wolf into a dog and man himself into the man’s best domesticated animal.” – Friedrich Nietzsche

There does seem to be a tendency to equate book-learning with becoming domesticated.

This is why I embrace my entire existence as a contradiction to the images of studious individuals becoming domesticated and servile scientists for the industrialists' war machine economy.

Holden, I will overdose on mathematics.  I will become stupefied by rationalism.  My efforts will not be rewarded in any way, other than making me far too preoccupied with studying to be of any use to society.

The really weird aspect of all this is that immersing myself in this study of the purely rational abstractions of mathematics may very well drive me insane ... so Dionysus rears its wild and untamed head after all.

I really do think these pursuits will eventually drive me cuckoo.

Quote from: Holden
For me,  pure maths is problematic. It's just that it is often taught with no connection to the real world, and that makes it FAR more difficult for me to learn. The endless march of "Axiom, definition, lemma, lemma, proof, corollary" makes me lose interest rapidly.

Yes, some of it is dry and boring.   It is an invisible machine .... it is machinery itself ...

My brain is mostly irrational, as it is made of the same stuff as the sun and the earth, the will, the product of copulation. 

When I sit (or stand or pace around) forcing myself to continue this mad mathematics obsession day after day, I imagine myself a somewhat cerebral and melancholy protagonist who is undergoing invisible transformations, where I suspect that I'm bound to be learning something ... more than I would if I had continued to seek oblivion daily.

It is a relief to be able to state that all this studying is not making me any more intelligent.

Evidently, becoming more intelligent is not my goal.

After all, I'm just enduring time, enduring Being-Time.

I am one of those scribblers in the madhouse Schopenhauer would joke about. 

Yes, my thinking is becoming ever more slow.  My greatest blessing is that I am not required to perform, that I am not being manipulated into jumping through hoops or seeking approval.  I am free to think as slowly as necessary.  This is the key for me: slow thinking.  Although, if I sit there drawing a blank for a few minutes (a few seconds sometimes), I go straight for the solution manual!   :-[

There is something humorous about the whole thing.  I mean, mathematics and science is pushed on the youth, where it is supposed to be a means to landing a career with some giant corporation.  I don't think the social engineers ever meant for many of us to become useless Hikikomori spending eternity trying to understand the exercises in the textbooks.

Maybe they should affix warning labels on the math books.

It is said that one does not read a math textbook like a novel, that you could spend an hour (at least) on one page.  Evidently, food and shelter and eyesight and sanity are taken as a given.

As you say, the world is a messy place.

How inconvenient that we have to eat and defecate and plug devices into electric sockets.

It makes one feel so pathetic.  I try to take solace just knowing I am not watching much television.   In a real Catcher in the Rye sense, I study mathematics to block out celebrity culture, to detach from the narratives being pushed in the media.

I search for guidance in a shelf filled with math books.   My favorite Oracle is a solution manual.

... a permanent student ... I am never satisfied with my level of understanding ... always going back over fundamental concepts.

I look at it this way, at least I'm not trapped in a room with strangers smoking c-r-a-c-k  c-o-c-a-i-n-e.

At least I am not compelled to stand on the corner begging for a couple dollars so I can get a drink.

This math madness is not such a bad way of life, really.  I am living an unwritten philosophical novel.  Of course I am not as funny as Ignatius Reilly, and I am not naive enough to believe in "the people".   You just have to watch the ZooTube videos of people rushing the stores during the holiday sales or drive on the highways to get a sense of what "the people" are all about.

I am living a peculiar life.  I often wonder how many like me exist ... We are Legion.

Revelation:  Enjoying mathematics is not a requirement for studying it.  It is only necessary to become obsessed with a desire to understand something you are curious about.

Interest does not imply delight.

Obsession does not imply love or passion.

And, most importantly, hard work does not imply inborn talent.

All this studying is not making me feel any smarter.  In fact, it has the opposite effect.  Still, at this point I have developed some kind of psychological need to continue, come what may.

I just hope I can remain interested.  That's all that is required to study mathematics.

How does one stay interested?

There are times I feel I am in a self-imposed prison, and I can actually feel my body cry out, "Is this all there is?"

And yet I have learned enough from Schopenhauer, and even from Dostoyevsky's Crime and Punishment, to know that, yes, this is as good as it gets, to be able to be still engaging with the books, training the mind to focus, forcing the irrational, impulsive, restless brain to act as though there is some kind of purpose to what it is looking at.

One of the delusions of putting things down in writing is that there is some "orchestrator", the rational mind, the frontal lobes, the neo-cortex, the intellect, who can direct the will to do its bidding.

Isn't it the other way around?   Doesn't the cerebral cortex serve the will?

This may be where things get rough.  The blind will does not, is not capable of, attaching any importance to the abstractions we deal with in mathematics.   You know what the will wants.  It wants to replicate.  It wants ... copulation.  It wants food ... It is trapped in this condition.

So students around the world are "bribed", coerced, enticed to study such abstractions by convincing the will that such activity will secure future food, shelter, transportation, etc.

To study as an end in itself, as a worthwhile activity to stimulate the brain, well, there is no guarentee one will be able to sustain a lasting interest. 

I think of those you mention who long for a fistful of flour.  Then, there are others who travel far and wide, planning one "vacation" after another.  Around and around they go ...

I'm for staying put and cracking open a textbook ... and diligently working through the exercises, even as the Beast Within resists.  You see, this Beast, the Lizard Brain, the Old Brain, the sub-cortex, all it wants is DEEP PLEASURE, the kind of pleasure when is likely to experience while intoxicated, stupefied ... that sweet euporic "high" ... ahhhh, that's what the blind will seeks:  relief from the burden of its own existence.

This is an ancient struggle between Appollo and the Dionysus.   Of course, Dionysus is the god of intoxication, and, well, I suppose I have been a proponent of giving in to such impulses. 

Part of me would like to permanently sleep, to never be roused.

So, which wolf do we feed?   I dare not feed the wolf with the insatiable appetite for inebriation and euphoria.

For me, the calm (often boring) routine/ritual of testing my "knowledge" and "math skills" is less an academic discipline and more an artform, as in the art of doing time, and, in a real sense, detaching from the blind will [desire] for [deep] pleasure (release from the burden of existence).

Maybe you also want to engage in mathematics this way, not so much as a means to mastering a craft or "tranferable skill", but more as an end in itself, as a practice which will relinquish the absolute authority of the subcortical Lizard Brain over the neocortex frontal lobes.   

To train my animal being to find some contentment in thinking, cerebrating, manipulating abstractions - all in the neocortex, maybe it won't be as dominated by the insatiable blind will.

For me, then, maybe the study of mathemtics simply represents world of abstraction a million miles away from the bullshiit being broadcasted far and wide as the "historical narrative".   

I don't want to constantly be thinking about world events or even reality, for that matter.

I guess I am okay with being lost in my own little world, a kind of personal and hopefully harmless kind of madness.  Maybe I just won't take anything too seriously, especially myself.

I mean, so some exercises are easy, and these make me feel like I am doing "baby math".  Other exercises require more thinking, and these expose me to the lazy nature of my brain.

Fortunately I am not subjected to some military boss who is cracking the whip or threatening to send me to the front lines.  When there is any kind of pressure or stress involved in performing any kind of computations, calculations, or explaining concepts, I would most likely prefer to clean toilets or drink booze in the woods.

I only like math(s) when I can goof off with it.  You see,  the people who make their living primarily off their skill at stressing their employees (or students or clients) cannot access me, for the moment, anyway.

« Last Edit: February 18, 2017, 10:26:23 am by Ιδιοτεσ-5150 »
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Gorticide @ Nothing that is so, is so DOT edu

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Re: Arithmetic is Drudgery
« Reply #22 on: February 18, 2017, 07:16:06 pm »
Quote from: Holden
I think this experiment of yours in really interesting. I mean studying the same math book which you studied many decades back .

People are not only killing themselves but also others because of math.

???

Yes, I had to invest in a rare copy of the Solution Key.  For many of the exercises, I swear, I just wouldn't have known where to begin - and this is after graduating from the State university (with honors, 2002).   No wonder I was so perplexed in 1984.  I was in rebellion against a mandatory Computer Science class ... I was more interested in Philosophy, Mythology, the Occult, and even "Eastern Religion".  It just was a very bad math year for me.   Since I have no recollection of doing any of the exercises, I am going through every single of the over 4400 problems.   I am suddenly inspired to do the text justice, giving it my attention this time (33 years later).  I owe this to myself, and it really is an almost mystical encounter "beyond the bounds of time".  I suppose I am putting some other things on hold.  One thing I do have a vague recollection of is the formal notation.  Set notation is used everywhere.  I always liked to see that kind of notation even years later.  Now I know where I first encountered it.

This is a very humbling experiment.  I have explained my reasons, and it is having a kind of calming effect on me.  I mean, I was not in the right frame of mind back then ... many emotional problems ... my parents had divorced when I was 12, and I was drinking steadily throughout my high school years.  I all caught up to me in my last year, which is the year we must have been using Modern Introductory Analysis (Docliani). 

I remember when I was preparing to attend the community college.  I passed the test so that I could just start with Calculus in 1994.   I figured all I learned in high school was behind me.

It is fascinating to me that on so many of the exercises I have no choice but to sneak a peak at the solution key.  It's just the way the text approaches the material - extremely formal.   Like you said, axioms, theorems, definitions, using results from previous exercises, etc.

I never imagined that this is what I would be doing at age 50.   I just want to approach all this in my current state of mind, where I have no emotional drama in my day to day life.  No heartache, no fears about the future (the future is already here, and according to the standards of that high school, I have already "failed"; that is, I am a "big time loser")   :D

How liberating!

Also, not being a professional of any kind, I have so much more freedom as far as being honest with myself.   As long as I find the material challenging, I will continue.   I am not beyond this level.  I am fortunate not to be paralyzed by an ego that would be wounded returning to a high school textbook.

This has nothing to do with ego-building.  My foundations are shaky.  I want to rebuild the basement!   

Peace to you, my rare and unique friend!
« Last Edit: February 18, 2017, 07:26:17 pm by Ιδιοτεσ-5150 »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Re: Arithmetic is Drudgery
« Reply #23 on: February 18, 2017, 07:44:21 pm »
Quote from: Holden
University atmosphere is marked by a preoccupation with pure math  , the systematic smoothing of the world’s jaggedness, and the stifling of trial and error. Math professors have made a religion of rationalism, optimization and efficiency.I  embrace the “Dionysian” maths: the dark, visceral, wild, untamed, hard to understand.This is about looking at reality as the mess it is .

I am considering approaching each day with the attitude of one with amnesia.   ;D

Regardless of how each of us approaches our obsession with maths (there, I've conceded to use this term), it is safe to say we will be engaged entirely in our own orbits.  What I mean to say is that I think we must refuse to hold ourselves to the same standards that aspiring "professionals" subject themselves to.  That's the least we can do for ourselves.   Even if we think about mathematics every hour of every day of our lives, we owe it to ourselves to proceed in a casual manner with a minimal amount of stress or pressure.  This could be one of the greatest benefits of declaring ourselves 100% non-professional, non-academic amateur hobbyists.

To be blunt, we may shamelessly incorporate the solution manuals in a nontraditional manner ... reverse learning.

Have you come across in your readings of Schopenhauer his repeatedly mentioning the great differences between individuals in the way one individual's mind is wired compared to another?

Whereas we may struggle with constructing proofs, there are those individuals who design software which automate the building of proofs.  Hell, they automate the step-by step solutions to finding antiderivatives, if they exist.  I don't want to be a hater.   I, for one, promise myself to move at my own pace and to commend myself for my humble efforts, conceding that I am certainly not an egghead or wizard.

I remember that there are those walking barefoot in the mud, desperate for a fistful of flour.

There is no shame in spoon-feeding our brains only what it can digest.

This brings me some peace of mind, especially when I behold the enormity of the realm of mathematics and science.   I study as a curious human organism who has nothing better to do.  Sometimes I catch myself almost enthusiastic.  At other moments I question my sanity.

Even if I were working as a janitor, I believe I would be pecking away at mathematics in the evenings and on meal breaks ... You see, even though we may look forward to the inevitable  moment when we draw our last breath, there is a chance we may be here for awhile. 

We might as well take a second and third look at some concepts, at least the very fundamental concepts.

I guess it is important to me that I acknowledge that I am obsessed.  This is an obsession.  As I said before, possessing great talent is not a necessary requirement when one has such an obsession.   :)
« Last Edit: February 19, 2017, 09:28:32 am by Ιδιοτεσ-5150 »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Holden

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Re: Arithmetic is Drudgery
« Reply #24 on: February 19, 2017, 10:10:03 am »

What is Dinoysian Mathematics?
Imagine you are in India. You are walking through a crowded, bustling, noisy street market, full of activity. Say,you're  in the city of Calcutta, but it could be any one of dozens of cities here. You walk up to one of the stalls, selling coconuts. It is manned by a largely uneducated twelve-year old boy from a poor background.
"How much is one coconut?" you ask.
"Thirty-five," he replies with a smile.
You say, "I'd like ten. How much is that?"
The boy pauses for a moment before replying. Thinking out loud, he says: "Three will be 105; with three more, that will be 210. (Pause) I need four more. That is . . . (pause) 315 . . . I think it is 350."
How well did our young coconut seller do?
If you think about it for a moment, it's clear that the boy isn't doing it the quickest way, which is to use the rule that to multiply by 10 you simply add a zero - so 35 becomes 350. The reason he doesn't do it that way is that he doesn't know the rule. He's never learned it. Despite spending a couple of years in school, he has almost no mathematical knowledge at all in the traditional sense. What arithmetical skills he has are self taught at his market stall. Here is how he solves the problem.
Since he often finds himself selling coconuts in groups of two or three, he needs to be able to compute the cost of two or three coconuts; that is, he needs to know the values 2 x 35 = 70 and 3 x 35 = 105. Faced with your highly unusual request for ten coconuts, the young boy proceeds like this. First, he splits the 10 into groups he can handle, namely 3 + 3 + 3 + 1. Arithmetically, he is now faced with the determining the sum 105 + 105 + 105 + 35. He does this is stages. With a little effort, he first calculates 105 + 105 = 210. Then he computes 210 + 105 = 315. Finally, he works out 315 + 35 = 350. Altogether quite an impressive performance for a twelve-year old of supposedly uneducated background. If you asked him to take a pencil-and-paper test that comprised exactly the same arithmetic problems that had been presented to him during the purchase, he will do much better when presented with market- stall word problems requiring the same arithmetic and much worse when virtually the same problem is presented to him in the form straight abstract sum .If you  were to say: "I'm going to take four coconuts. How much is that?" The boy would reply: "There will be one hundred five, plus thirty, that's one thirty- five . . . one coconut is thirty-five . . . that is . . . one forty."
Let's take a look at this solution. Just as he had in the exchange I described earlier, the boy began by breaking the problem up into simpler ones; in this case, three coconuts plus one coconut. This enabled him to start out with the fact he knew, namely that three coconuts cost Rs.105. Then, to add on the cost of the fourth coconut, he first rounded the cost of a coconut to Rs.30 and added that amount to give Rs.135. He then (apparently, though he did not verbalize this step precisely) noted that the "correction factor" for the rounding was Rs.5, and added in that correction factor to give the (correct) answer Rs.140.
If you then asked him on the formal arithmetic test, to multiply 35 and4; here is what he would say; "Four times five is twenty, carry the two; two plus three is five, times four is twenty." He then writes down "200" as his answer.
Despite the fact that, numerically, it was the same problem he had answered correctly at his market stall, he got it wrong. If you follow what he said, it's clear what he was doing and why he went wrong. In trying to carry out the standard right-to-left school method for multiplication, he added the carry from the units-column multiplication (5 x 4) before performing the tens-column multiplication, rather than afterwards, which is the correct way. He did, however, keep track of the positions the various digits should occupy, writing the (correct) 0 from the first multiplication after the (incorrect) 20 from the second, to give 200. The children are absolute number wizards when they are at their market stalls, but virtual dunces when presented with the same arithmetic problems presented in a typical school format.These poor children demonstrate that they could handle arithmetic in the appropriate context, when the numbers meant something to them, it seems clear that meaning plays a major role in our ability to do arithmetic. When the children carry out computations at their stalls, both the numbers and the operations they performed on them had meaning, and the operations made sense. Indeed, the children were quite literally surrounded by physical meanings of the arithmetical procedures , in contrast the essence of school mathematics, which the  children are not able to do, is that it is entirely symbolic - i.e., it operates on symbols that are devoid of meaning.
The problem is that humans operate on meanings. In fact, the human brain evolved as a meaning-seeking device. We see, and seek, meaning anywhere and everywhere. We can't avoid it. I don't think the human brain can perform genuinely meaningless operations at all.
If I'm right, then that means that a crucial component of mathematics education is making sure that the student is able to construct (or otherwise acquire) appropriate meanings for the various abstract concepts and methods he is faced with.

Did Gauss himself not say that number theory is the queen of mathematics.Could we not plug in numbers to make math problems more meaningful?
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Re: Arithmetic is Drudgery
« Reply #25 on: February 19, 2017, 03:27:23 pm »
I have found myself doing exactly that at times, plugging in numbers.  For instance, if I am asked to prove by counterexample that an assertion is false, my first impulse is to plug numbers in.

For all x, u, y, and v in {real numbers}: if x < u and y < u, then x*y < u*v.

The trick not to try to find a case where x*y > u*v, but simply a case where x*y is not less than u*v; in other words, showing a case where x*y = u*v is sufficient in order to contradict the conclusion, x*y < u*v.

So we can plug in negative and positive ones.

let x = -1 and u = 1
let y = -1 and v = 1

then x < u, y < v; and we have x*y = (-1)*(-1) = 1 and u*v = (1)*(1) = 1;
hence xy = uv, and we have proven by counterexample that the assertion is false.

These are the easiest kind of proofs where we just have to show a counterexample, and it invariably involves plugging in numbers.

I try not to type up too much about specific exercises since I don't want to bore you with tedium. 

What the boy in the market place is doing is actually the definition of multiplication, where multiplication is a series of additions. 

I would be stumped on how to go about teaching arithmetic to children.   There were supposedly educational experiments in Africa, where "it takes a village to raise a child".   What they have been doing there for a long time is having children teach children.   Their philosophy is that a child a little beyond the level being taught has a clearer idea of the nature of the conceptual mental blocks that can cause difficulties, and they are far more patient in helping a child to gain insight.

My nephew was home-schooled, and he would experience a mental block when looking at the formal definition of absolute value.  You see, when he saw "-x", he read "negative x", but the definition means the opposite sign of x.   So, here, "-x" is a positive number.

You know, abs(x) = x if x > 0 and -x if x < 0.

say x = -5

then abs(-5) = 5; and I could see how frustrated my nephew would become because, in his mind, x is 5 and -x meant -5.  In the definition, -x means -(-5) = 5.

He would become quite angry. 

When I was stuck in a "welfare compound" in 2003, after graduating from the university, unable to find any work, and after receiving some kind of "note" from a psychiatrist notifying the officials that I was no longer required to "seek employment", that I had been diagnosed as bipolar with rapid cycling between mania and chronic depression, even as I was hitting the bottle (vodka) regularly, on days when I was sober, I would tutor an entire family of Puerto Rican brothers, from grade 3 to senior in high school.  They were all genuine biological brothers.  I am not using the word the way "Jesus" would refer to everyone as "Brother".   

I was able to help the senior in high school the most, with the algebra and trigonometry, but algebra stretches throughout the grades, so the 7th and 8th grader also got much from my tutoring.  We were all in the same welfare motel ... for over a year.  One of the brothers called me el maestro.   The two youngest ones didn't get much out of it.   The youngest one was too full of pep and vinegar ... paradoxically, he was like "the boss" ... a real character.

It is fascinating for me to consider youth in India who might not make use of multiplication by 10 (and just adding the 0 to the number) since I, and all the world, including the Arabian Algebraists, know that the concept of 0 and the number system based on 10 comes directly from India.

I guess people forget that mathematical symbols and the sophistication of manipulations belongs to the realm of culture and is not something people are born with.  I suppose any kind of education, even one filled with years of being shuffled through corridors like sheep, is not something to take for granted.   

And yet, when we are really honest, we must admit that "counting" and the sense of "sets" have to be inherent in the human brain.   We have to suspect that chimpanzees know the difference between 2 bundles of bananas and 4 bundles of bananas.

How many leaves does the squirrel need to build a sufficient nest?   Do they have a language?  Can one female squirrel signal to her daughter that she needs "at least 3 more loads of leaves"?

The weed man is very good with arithmetic.  He knows multiplication by ounces, grams, and pounds.  He does not care at all about how one proves that 1 < 0 is a false assertion.  He knows 100 dollars is not less than 0 dollars. 

When you refer to something as Dionysian Mathematics, I take this to mean "free play" as opposed to rigor, learning by discovery as opposed to proving theorems and corollaries. 


Maybe the Apollonian way is the way of rigor, formalism, proofs, and Dionysian is trial and error, free-play, exploration and discovery.

I think my experiment with going back 33 years to engage with a text that made me feel totally inadequate has to do with seeing if, after much more experience with applied mathematics and computations, using algorithms, vectors, trigonometric functions, etc, I might be less intimidated, or at the very least, more careful and attentive to the rigor and formalism.

The thing is, I am old enough to be patient and not too defensive about protecting a fragile ego.   I can admit to myself when formalism and rigor are psychologically painful.  I can laugh it off and realize that deep down inside I am a brute who is impatient with such rigor.

Perhaps I long to develop a bit of sophistication ... and I can tell the youth that it took me a lifetime to develop just a small degree of mathematical maturity.   Meanwhile there will be those math wizards who are very comfortable with rigor and formalism, and they might be like that at an early age.  I still think more has to do with culture and environment.  If they have been groomed for this ...

I was reading online a math text where the author suggested one implore "God" to help one understand.  I didn't want to look any further.  How could I?   How can one sit there writing a mathematics text and suggest to the reader to pray to God to "help us understand the math"?

At that point I become more than a little disgusted.

Not understanding something is nobody's fault. 

Does one care to understand?  Does one care to learn a rule that will make their daily calculations easier?

What is the motivation to learn anything, for that matter?

I am certain that the Division of Vocational Rehabilitation did not invest in my further education so that I could take up mathematics and computer programming as hobbies that would motivate me to resist the strong compulsion to drink myself into oblivion.  They intended for me to become trained and ... EMPLOYED.    The State wants us EMPLOYED.

And yet there are many who study things like foreign languages simply to improve their minds ... culture?

Maybe my interest in mathematics is a longing for culture.   I want to receive a small kernel of the culture of humanity, the culture of mathematics ... as a human being on this planet, as an aging human being on this planet.   I do not want to be a barbarian.   Although, I recall, as a teenager, I often envied aliterate peoples who made their living fishing or hunting/gathering.   There is great irony in imagining Albert Einstein foraging for edible roots in the ground and constructing a shelter out of leaves and twigs.   These are the kinds of things that had me in such a confused state as a teenager.  I began to suspect that non-alphabetic "uncivilized" peoples knew a great deal more about actual living than the most well-trained scientists of the Industrial World.

And so I study ... the language of mathematics.   Free-play or rigor, by discovery or by proof ...

I can't remember the last time I went fishing.  At least I grew some tomatoes this past summer.   :-\

Maybe my ancestors have only recently crawled out of the caves of Europe.  So be it.  Now I live in a world in between ... not knowing enough about hunting, gathering, or even fishing ... and not efficient enough to earn a living as some kind of scientist.  My only option was to stand in line for one of the redundant jobs required "semi-skilled" labor.   This world is Hell, but I hide away and live in a world of exercises and solutions ... textbooks ... searching the internet for information and guidance ... When the last fish is dead, all money will be worthless.  I depend on exploited labor to knock the sacred cow on the head so I can fry its liver in a wad of butter.   Are we no longer barbarians simply because of mechanized agriculture?  Maybe we are more barbaric than ever.   As John Trudell used to say, "Civilization is not civilized."

The Greeks and even the Romans may have viewed the Teutonic Tribes of Northern Europe as barbarians, but look how they took to philosophy a couple thousand years later!   :-\

Alcohol ... I do not possess the gene to break down the alcohol.   I am susceptible to being destroyed by it ... rapidly.

Anyway ... Your example just got me thinking about how culture is passed through the generations, and that it is not something passed on through the DNA.   A child might be born in the land where so much sophisticated thinking developed, but the demands of his daily existence may force him to get by with rudimentary education.

If one's parents require him or her to go into the market to sell product, then there will be trouble for him or her if they are caught hiding in the hills with math books or any kind of literature at all, I suppose.

I knew a cop who would beat the crap out of his kid when the kid got overly enthusiastic about Calculus.   He didn't want his kid "thinking he was smart".  The cop did not respect such education and most likely found it effeminate, or at least not very "macho" or "practical".

Some other parent might shelter a child from the world specifically to allow them to study in a fortress of solitude.   

Maybe it is a matter of luck and fate how much "culture" we are exposed to.

It is said that Industrial Civilization is culture-destroying, and yet we witness how much culture is acquired through conquerors via militaristic endeavors.   

As far as money goes, I pay close attention to my arithmetic and never try anything "fancy".   :D

The inner barbarian doesn't fuuck around when it comes to dollars and cents.

That's when decimal place errors can really be devastating. 

So I certainly applaud careful arithmetic.  That is preferable to Gaussian formulas.

Let me ask you.  Do you prefer pencil or paper when working with math?   Not just arithmetic, but any kind of math.   Do you write down the linear equations for a linear programming problem in pencil or pen?

I always use pencil for anything other than arithmetic, and I frequently make use of erasers.

I am error-prone. 

Now, the type of math I am studying at the moment requires a great deal of thought before the pencil hits the paper.  It's not the kind where I can compute or calculate then check for errors.

Actually, as I have said before, this kind of math that requires proofs is not my cup of tea at all; but I am forcing myself to RELEARN HOW TO THINK, and how to develop patience for SLOW THINKING as opposed to quick calculations which crunch numbers through a machine (algorithm).

Again, I appreciate these conversations.

Maybe your Dionysian Mathematics is a call for a less formal, less rigorous, almost blasphemous free-style mathematics.   I don't blame you for rejecting the edifice of pure formal mathematics.  I do enjoy learning as much of their language as possible.   My goal is not to become a mathematician, but merely to be able to read enough mathematical notation so as to be able to study a few of the most fundamental subjects in applied mathematics ... In other words, I humbly submit that I intend to use mathematics, not to "do" mathematics.  Although, I would like to be able to read and write a few basic proofs.

I have a similar attitude toward computer programming.  I only ask of myself to be able to write command line programs that perform some kind of numerical computation. 

In the meantime, may we not believe the Great Lie that this is "civilization" ...

« Last Edit: February 19, 2017, 08:16:15 pm by Ιδιοτεσ-5150 »
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Re: Arithmetic is Drudgery
« Reply #26 on: February 19, 2017, 08:28:59 pm »
Quote from: Holden
the other day I came across a problem,which I could solve easily by generating random numbers in excel.However,I think I can convert it into a linear programming problem.Last night I was learning the Gauss Jordan Elimination method. 

Mathematics as a way of running away from the world is something to think about.

There are couple of other things I may be looking at in math.

I know I tend to sometimes write very lengthy posts, so I am breaking this response up into parts.

Gaussian Elimination helps to put a matrix in row echelon form, while Gauss-Jordan Elimination puts a matrix in reduced row echelon form.  This is exactly what I was getting into last year at this time.  Remember "rref".  I left the c++ code here at the end of the post: rref.txt
y enjoy doing all the calculations, but there tend to be many complicated fractions with endless opportunities for human error.

I know you don't want to get into programming.   Maybe it is best you study Gauss-Jordan elimination with pencil and paper ... Using matrices ... A computer solves these much faster.  I used to really get into doing these by hand.  Using a program that takes the augmented matrix in as input is kind of fun.  After doing rref (that's what I call Gauss Jordan Elimination) by hand, you can appreciate how fast a computer is if you have the code working well.  In fact, even if doing Gauss-Jordan Elimination by hand has a calming effect on you, if you are in the mood to carefully work through addition and multiplication of fractions, you still will appreciate a computer program that takes the augmented matrix as input ( [A | b] ) and returns the vector x (of solutions x1, x2, ... xn) at least as a way of checking your tedious arithmetic calculations.

Eh.   It's fun though.  I have to admit.  Solving systems of equations with Gauss Jordan Elimination is one of the first things I started to get back into last year when I became interested in mathematics again.   I remember wanting to know more about Linear Algebra.

It's a cool thing to be looking at Holden.  I even remember when employed with the park service sitting at my kitchen table on a day off ... It was 1993 or so.  I wanted to remember how to solve systems of equations.  I remember mentioning to my live-in concubines step-father, who was an engineer for AT&T.   I had ordered the just released programmable scientific graphing calculator, TI-85 ... the one just prior to the ground-breaking TI-92.   Anyway, my obsession with number crunching has outlived any promptings to reproduce.

Mathematics as a way of running away from the world is something to think about.

Some days it works out.  While there is no way to run away from the world-as-will, since it is all up in our veins and sinews, I am sometimes able to block out the world of other people.  In other words, a means to entering a state of mind where public opinion has minimal effect on you.  You become too engrossed in inner developments and transformations to take the world of social status too seriously at all.
« Last Edit: February 20, 2017, 05:59:26 pm by Ιδιοτεσ-5150 »
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Absolute Value Proof
« Reply #27 on: February 20, 2017, 10:12:22 am »
Quote
My nephew was home-schooled, and he would experience a mental block when looking at the formal definition of absolute value.  You see, when he saw "-x", he read "negative x", but the definition means the opposite sign of x.   So, here, "-x" is a positive number.

You know, abs(x) = x if x > 0 and -x if x < 0.

say x = -5

then abs(-5) = 5; and I could see how frustrated my nephew would become because, in his mind, x is 5 and -x meant -5.  In the definition, -x means -(-5) = 5.

He would become quite angry.

Lo and behold, after my first cup of very strong coffee, while smoking my first cigarette of the day in my rocking chair outside, I come across an exercise that I decided not to start last night after midnight when my brain had had enough math for the day.

It was the exact concept that had my nephew confused so many years ago.

Now, since our communication is for the most part private, I can muster up the courage to be honest.   I drew a blank.

When x < 0, I too have a mental block when I see abs(x) = -x.

This means abs(x) = abs(-x) = -x, which is somewhat confusing mainly because -x is a positive number!

Quote from: Holden
Did Gauss himself not say that number theory is the queen of mathematics.  Could we not plug in numbers to make math problems more meaningful?

And this is all I could do to make sense of it.

I imagine x being -5.

So -x = 5

In other words, x < -x  [-5 < 5]

For whatever reasons, this is difficult for my brain to process.  It was not just my nephew.

When doing a proof, we can't plug in numbers.

For a real number x, the absolute value of x is defined to be: abs(x) = x if x >= 0, -x if x < 0

I can see exactly where our confusion might arise.  The absolute value of x is -x when x < 0.   With numbers, we are used to abs(-5) = 5, and this is correct.   Here, x = -5, so the absolute value of -5 is -(-5) = 5.

The thing is, when x < 0, -x is a positive value.

That is, when x = -5, the absolute value is -x = 5.

As long as the discussion just involves plugging in numbers, the definition of absolute value makes sense.

Ironically, the confusion arises when it comes to the proof, which involves only symbols.  I say that the confusion is ironic because the purpose of a mathematical proof is to make things more clear; and yet we find "plugging in numbers" more clear.

OK, remember, I do not intend to bore you.  The purpose of this post is to point out the uncanny nature of day to day existence.  I just mentioned this very thing yesterday, and this morning I face it head on first thing in the morning. 

The exercise is to prove the theorem:  If a > 0 and x <= -a or x >= a, then abs(x) >= a.

As is usually the case for me first thing in the morning, my mind draws a blank.  There is even a part of me deep down inside that romanticizes the lifestyle of reporting to the liquor store at 9AM for a bottle of Fireball. 

I refuse to deceive you or pretend that I can produce the proof from scratch in my head.

What is the solution?   What was the solution expected of my 17 year old self back in 1984?

I consult the Oracle.  What is given?

HYPOTHESIS:  a > 0 AND x <= -a OR x >= a.

All the hypothesis is is the restatement of the p part of the conditional "p implies q", which is in the form "if p, then q".

We have to show both cases, (1) where -x >= a; (2) where x >= a.

(1) if -x >= a, then -x > 0 [since our hypothesis tells us a > 0, so, by the Transitive Property of Order, -x >= 0]

All this is saying is that, in case 1,  -x is a positive number.  [THIS IS THE ROOT OF THE CONFUSION]

abs(x) = abs(-x) = -x  (by the definition of absolute value: abs(x) = x if x >= 0, -x if x < 0

Therefore, abs(x) >= a.

(2) If x >= a, then x >= 0

abs(x) = x

abs(x) >= a

Therefore abs(x) >= a for all x.
_______________________________________________________
Granted, this is not even a very difficult proof.  Still, it was necessary for me to find the solution, which certainly was not residing in between my ears!

Where is the mental block when it comes to writing down proofs of this kind?

I have to be able to break it down into some kind of algorithm or pattern that I can apply off the top of my head, that is, while sitting in a rocking chair smoking a cigarette (away from the computer).

This is pretty basic stuff, so, if I am drawing a blank now, this is not something I can just ignore.  I have to grab the bull by the horns so as to get to the source of my mental block.
 
Let's follow the thought process to see if we kind derive some kind of algorithmic process.

We start with the theorm:  If a > 0 and x <= -a or x >= a, then abs(x) >= a

We first state the hypothesis:  a > 0 and x <= -a or x >= a

Taking the first part of "x <= -a or x >= a", we zoom in on "x <= -a":

x <= -a is equivalent to the statement "-x >= a"

Why/How?   Because when we divide or multiple by a negative, the inequality sign changes direction.

So this is case (1).  From "-x >= a" [AND from the hypothesis a > 0], we note that -x >= a > 0, so -x > 0]

So -x is a positive number.  Plugging in numbers, this corresponds to -(-5) = 5. (when x = -5)

In symbols, abs(x) = abs(-x) = -x.

This is the root of the confusion because we see -x as -5, when it is really -(-5) = 5.

For case 2 we just take the second part of "x <= -a or x >= a" which we find is more straightforward and intuitively appealing.

Quote from: Holden
There must be a way to make math more likeable.

I find it very depressing to think that there might be no way out.

Regardless of how mathematicians write and read proofs, for us, it may be helpful to plug in numbers to make sense of what the symbols represent.   While I may seem to be really picking this apart, I am sure Schopenhauer would get a real belly laugh, perhaps even slapping his knee, since situations like this might be exactly what he was reffering to when he wrote: 

Quote from: Schopenhauer
Incidentally, it may here be remarked that many minds find complete satisfaction only in what is known through perception. What they look for is reason or ground and consequent of being in space presented in perception. A Euclidean proof, or an arithmetical solution of spatial problems, makes no appeal to them. Other minds, on the contrary, want the abstract concepts of use solely for application and communication. They have patience and memory for abstract principles, formulas, demonstrations by long chains of reasoning, and calculations whose symbols represent the most complicated abstractions. The latter seek preciseness, the former intuitiveness. The difference is characteristic.

You see, Holden, where you seek intiutiveness, the mathematicians are seeking preciseness.

Schopenhauer is articulating the very thing that threw me for such a loop when I was 17.  I thought I was attending this academy so as to be given INTUITIVE UNDERSTANDING, but they were dealing with PRECISENESS ... the cart was before the horse.  How was I to be precise when I lacked intuitive understanding.  Now that I have spent my life acquiring some intuitive understanding, only now can I go back and learn "preciseness".

And now for my second cigarette and a hard boiled egg.
« Last Edit: February 20, 2017, 10:14:25 am by Ιδιοτεσ-5150 »
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Dolciani: Modern Introductory Analysis c.1964, 1980
« Reply #28 on: March 24, 2017, 04:45:23 pm »
Reminder:  When using the local search engine here, it makes all the difference in the world if you perform the search from within the Forum Section where the suspected post is hiding in.

For example, searching "Dolciani" gives no results from the index page of this site, but when I go into the Why Mathematics? forum and perform the same search, bingo!

Now, I want to point out that this text is no ordinary "high school" math book.

Introducing this material to me when I was 17 was a sad waste of energy.  Only now am I able to appreicate how special that text was.  I mean, they do not present the material this way in other books ... not even at the universities.   In fact, I always wondered why I was drawn to set notation, but I only came across it in very specific areas, such as Set Theory.

See comment:  here:

Quote
Who won the Nobel prize for literature in 1967?, April 12, 2014

Actually, no one knows or cares, and the winners books are likely selling for a buck on Amazon used. This book is truly great literature that has changed the course of many lives. A like new copy sells for many times the original cost. Simply the GREATEST high school mathematics text ever written. Dolciani should have won!


The Dolciani textbook, Modern Introductory Analysis has the following features that are worthy of note:

Mathematical soundness and pedagogical aptness characterize the treatment of all topics.  Students are led to discover mathematical concepts and relationships by using intuition, inductive reasoning, and analogy.  The role of proof is stressed thoughout the book.


There would be no way to explore this old textbook with any kind of "ereader device".  Remember I found an old copy for $7 with free shipping from abebooks?   Well, there are, of course, 4400 exercises, and I will be working through this until I am done.  I figure I will be involved with this book until around July, so I would most likely have a homemade solution manual for you by the end of the summer ...  You have pleny of time to consider this.  I won't be offended if you say this is not your cup of tea.

I may be able to digitize the scratch pads where I am doing the work and mail a copy to India.  I'm not making any promises.   If you are interested in finding a used copy, that is, if you ever find yourself curious to inspect this different kind of math book, then I would make copies of the solutions with scanner.   Where there is a will, there is a way.   If not, then I will still, most likley, make some kind of digital copy of the notes and the worked out solutions FOR POSTERITY, but I won't mail them to India unless you have acquired this text and find yourself stumped.   It would be nothing to be ashamed of.   The formal notation is something kind of alien at first glance.

How different my values are from those "ubercool" machines that make money you mention in the Wombs for Rent thread!  I love to behold the discolored and aged pages of that old book.   The book and I have aged.  While my memory is not what it was when I was 17, my life experience gives me patience where the 17 year old would become overwhelmed.

By the way, I looked into your options for acquiring that text, and what sells for $10 here sells in India for over 13,000 rupees (around $200) ! YIKES!!! 

Notice the same books:   INDIA       USA

It would cost less for me to send it to you myself, much less, in fact.

Unless you go through abebooks.  I looked into it.  $14 for the book and about $9 for shipping to India.   If you are ever interested sometime in the future:  Abebooks ships to India

Anyway, what makes the approach of Mary Dolciani (and her "crew") so unique is that she was an actual mathematician.  Most textbooks these days are produced by eductors as opposed to mathematicians.   

Set notation is used throughout, for example, a line is defined as { (x,y): (x,y) + t*v_ }

Of course, v_ is written differently, with the "ray-arrow" drawn above the v. 

In python code, since I cannot put such a symbol over a variable, when I want to clarify (to myself) that it is a vector, I just add the underscore, where force_ = vector(0,1,0) for x=0, y=1, z=0.

The notation used in the Dolciani text actually slows me down in a good way, and I can feel my brain actually intuitively clicking ideas together, where it silently "gets it", that this notation makes you see a line in two dimensional space as a set of points (x,y) such that P + t*v_, so L = {P + t*v_} is a vector equation that is the same as (x,y) = (a,b) + t(c,d) = (a + c*t, b + d*t).

Oh well, The Biological Mother is pestering me to cook the Basmati rice ... I better get back to work.   :P
« Last Edit: March 24, 2017, 05:46:07 pm by Raskolnikov »
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The Aesthetics of Drawing Curly Braces { } By Hand
« Reply #29 on: March 25, 2017, 12:46:30 pm »
I think one of the main reasons I am committed to continue my engagement with old used math textbooks, inexpensive scratchpads in which to work through exercises with care, solution manuals, magnifying glass, pencils, pencil sharpener, erasers, etc, has a lot to do with aesthetics.

It is truly ironic that I am comforted by the aesthetic aspects of writing in cursive, writing by hand the symbols, such as the curly braces, { }.   You see, while many may be getting up to speed with LaTeX, obsessed with transcribing mathese onto web pages and other electronic documents, presently I have no such concerns.

I marvel at keeping a steady hand, writing larger as the pencil becomes duller ... and, there are moments when I am less focused on the computational calculating aspects of the mathematics, and find myself getting some kind of elevated enjoyment in writing these symbols by hand.

I notice that, even when I am using the ereader, a great deal of the literature I am into is from over 100 years ago ... von Hartman (Philosophy of the Unconscious), Paul Carus (History of the Devil), Baudelaire (Flowers of Evil), to name a few.

On the queue:  Lovecraft, Dostoyevsky, and even Christopher Marlowe ...

I think of Raul as well as my childhood friend's German mother who is 90% blind and bed ridden.  When I close my eyes, I see the Will (the Unconscious?) more clearly.  It is hunger.  It is anxiety and fear. 

Without math and literature I believe I would go mad.

In my dreams I see all the suffering bodies ... a horror to behold.

When I use the phrase, "the negative path", I mean to suggest that we do not seek happiness so much as we want to minimize our suffering.

It is best we daily wrap our minds around the idea of death less we become erroneously attached to our "plans" and "projects".  In an instant, our eyes can be closed and our so-called reality exposed as Maya.
« Last Edit: March 25, 2017, 03:33:11 pm by Raskolnikov »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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