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

Maybe you are a bitter young man who does not like people.  This will not hamper your pursuit of mathematical contemplation.  In fact, you can enter a realm that is millions of miles away from popular culture and its concerns.

Reading this over, I wanted to clarify that I do not consider you bitter.  I only presume you might be somewhat bitter in regards to your work associates.

See private message.

I have to say that I am coming from an entirely different orbit than this individual with the superhuman memory. 

In fact, just now, I am working through a problem where I have to write out an proof involving perpendicular and parallel components of vectors, and, even though the exercise is from my old high school textbook, it is one of the problems toward the end of a set, which tend to be more difficult and time consuming.   

Even with a solution manual, it takes a great deal of tedious writing, keeping track of subtleties of the notation.  I guess our engagement with mathematical computation is a very personal and subjective process.  I can certainly not sit back and trust that my brain will spit out solutions in my mind's eye.

Hence, I make use of paper and pencil and would never encourage anyone to make calculations in their heads.

Of course, if one is writing a proof in order to "show" why and how a statement is true or false, then doing things in one's head is not even an option.  In fact, the calculations most likely will not involve any numbers at all.   One will be working with algebraic symbols.   Everything must be written down in a clear and exact manner.

I guess we each have our own personal understanding of what mathematics actually is.  To some, it is simply the realm of numbers and computations.  To others it may have more to do with geometric and analytical representations.

In the real world, i suppose numerical approximations are the most useful; but these do not come into play when proving why a statement is true.

Finding your comfort zone can be a long process, but it is well worth sacrificing the vanity of the ego in order to discover just what area of mathematics you are comfortable exploring and investigating.

Maybe there is no authority other than your own anxiety-level that can tell you where your comfort zone is.

Personally, I find that I actually enjoy studying things again and again over the years when I see myself as kind of slow.  If I were to delude myself and consider myself some kind of math whiz, then I would feel horrible, like a fraud.   As long as I detach from the ego and accept the slow pace, and my own slow thinking processes, then a great deal of misery is overcome.

So, you have the right idea, I think. 

Those positive thinkers who want everyone to develop confidence fail to recognize just how liberating it can be to be resigned (to think as slowly as it takes).

Taking the pessimistic approach, one will not feel ashamed of going over foundations and fundamentals.  In fact, one might spend years just working at this level and have no delusions about ever getting into much more complex areas.

There is a point when one might even feel relieved to be a mere mortal when it comes to this.  The pessimistic student will aim low - and will not be disappointed.   The optimistic and ambitious student with a great deal of confidence and a huge ego might be the one who hangs himself from a tree or throws herself off a tall building during a fit of panic at being some kind of phony.

It's best that we are honest with ourselves.  In fact, acknowledging that we are slow thinkers when it comes to arithmetic, algebra, computing, proving theorems, etc, then we will not be too self-deprecating when we witness ourselves staring blankly into space ... waiting for an idea to materialize.

I think maths can be very interesting.

"Warum willst du dich von uns Allen
Und unsrer Meinung entfern?”  –
Ich schreibe nicht euch zu gefallen,
Ihr sollt was lernen. ~ Goethe


"Why wilt thou withdraw from us all
And from our way of thinking? " –
I do not write for your pleasure,
You shall learn something.

One thing I have learned is that I am definitely not a genius.  I think it's good to have a more realistic awareness of our level of competence.  I refuse to live a lie, and so I force myself to face difficulties.


I feel as though I am the outcome of a failed experiment.  I really believe my educators had the best intentions, and that, through self-study throughout my life, I probably know a great deal more than I give myself credit for; but, then again, not much comes easy to me.   I really have to think and concentrate in order to understand what I'm looking at.   Ego smashing myth destroyer, I. 

It is difficult to access how much I actually know, or what the "I" is that is supposed to be the repository of the so-called knowledge.

Why am I still obsessed with the fundamentals?  It is obvious that I will never be employed in any capacity which would require even the most rudimentary knowledge of mathematics. 

As far as levels of difficulty is concerned, this seems to depend on the presentation.

It is easy enough to see that x times 1/x is 1.  Is it necessary to be able to prove this in a formal manner?   

And what was the intention of my educators in teaching in a formal and rigorous manner?   Most educators are concerned with the education of the youth.  I feel that, after a certain age, we are considered a lost cause even if we once showed potential.

You see, even if I do experience a major breakthrough in how to bridge the gap between how we generally "use" mathematics and the structure of  the "pure mathematics" that supports those computations, my status in this world is "lunatic" - just another mad scribbler ... a total outsider. 

What I am getting at is that, even were I to experience a gradual yet profound shift where I suddenly developed a level of mathematical maturity to actually be able to consider myself an amateur mathematician, I stand no chance of ever being trusted to instruct or teach, not even adults, as I have, in no uncertain terms, been diagnosed as a potential psychotic.

So, tell me Holden, does one spend a lifetime trying to develop mathematical maturity just so as to be "permitted" to stand in front of a classroom and crack jokes?

Of course not.  Whatever the intentions were of my early educators, they have created a monster in that this world of ours is not concerned with whether or not we have developed mathematical maturity.   We need to be housed indoors.  We need to eat food.  We are a burden to ourselves. 

How can I justify my existence?   Evidently, there is a conflict between my being-for-others and my being-in-itself.   They only wanted me to understand mathematics so that I might - what - become an employee of some corporation that designs technology for collecting data to sell to advertisers?   

How did Adorno put it? 

I quoted him in Dead End, in the chapter, The Impossibility of Being Hentrich:

Horkheimer and Adorno articulate the situation I experience in everyday life in the following:  “Anyone who does not conform is condemned to an economic impotence which is prolonged in the intellectual powerlessness of the eccentric loner. Disconnected from the mainstream, he is easily convicted of inadequacy.”

From Enlightenment and Deception: “The most intimate reactions of human beings becomes so entirely reified, even to themselves, that the idea of anything peculiar to them survives only in extreme abstraction: personality means hardly more than dazzling white teeth and freedom from body odor and emotions. That is the triumph of advertizing in the culture industry: the compulsive imitation by consumers of cultural commodities which, at the same time, they recognize as false.”

Another quote:

... this society ... reproduces to a certain degree only the lives of its faithful members.

The standard of life enjoyed corresponds very closely to the degree to which classes and individuals are essentially bound up with the system. The manager can be relied upon, as can the lesser employee Dagwood - as he is in the comic pages or in real life.  Anyone who goes cold and hungry, even if his prospects were once good, is branded. He is an outsider; and, apart from certain capital crimes, the most mortal of sins is to be an outsider.

In films he sometimes, and as an exception, becomes an original, the object of maliciously indulgent humor; but usually he is the villain, and is identified as such at first appearance, long before the action really gets going: hence avoiding any suspicion that society would turn on those of good will. Higher up the scale, in fact, a kind of welfare state is coming into being today. In order to keep their own positions, men in top posts maintain the economy in which a highly-developed technology has in principle made the masses redundant as producers. The workers, the real bread-winners, are fed (if we are to believe the ideology) by the managers of the economy, the fed. Hence the individual's position becomes precarious.

Under liberalism the poor were thought to be lazy; now they are automatically objects of suspicion. Anybody who is not provided for outside should be in a concentration camp, or at any rate in the hell of the most degrading work and the slums.

I have long since stopped caring that I am an outsider, and I certainly can not base my current obsession with developing mathematical maturity in the ridiculous hopes of becoming useful to society as I grow older.  I am fully prepared to be seen as a joke.  I tell you this because I sense that you are also prepared for such an existence.

I would like to continue to communicate with you for a long time.  We know we have in common our gratitude to Arthur Schopenhauer for spending his life communicating to us, his target audience - the few.   Now, this paradox where we both find mathematics at once difficult and intriguing, this may also prove to be an area where we might both greatly benefit in discussing.

I have let you know what my current experiment in self-education entails.  I have been able to collect what I have come to see as unique books on the subject.  I am committed to investing most of my energies to these texts over the next few years.  Maybe their target audience was the "top 10%" of the excelling students, but I was certainly not in the best frame of mind as an adolescent for learning that formal side of mathematics, what I now know to be "pure mathematics".  The one magical aspect of being a survivor of a failed experiment of such grand proportions (I did not succeed in committing suicide at age 19) is that I have some hindsight.   It doesn't matter that the targeted audience of the "Dolciani series" were college-prep high school students.   The material is no longer presented in this manner to high school students, and not even to students at community colleges.  In fact, one would not see material presented in such a formal and rigorous manner until graduate school, and by then they expect you are "mature" - so there is no way in hell they are actually going to help bridge these huge foundational gaps!

They did not write for our pleasure, but so that we might learn something that unified all mathematics, not just to enable us to compute and calculate.   I want to honor them by devoting my adult mind to the material. 



"Warum willst du dich von uns Allen
Und unsrer Meinung entfern?”  –
Ich schreibe nicht euch zu gefallen,
Ihr sollt was lernen. ~ Goethe


"Why wilt thou withdraw from us all
And from our way of thinking? " –
I do not write for your pleasure,
You shall learn something.


May our correspondence have strong legs to stand on as we each get through this life, for we do not seem to be chasing the elusive "pleasures" of entertainment and amusement; but, rather, we want to learn something!

Maybe we can learn to laugh at the brute fact that human society is not at all concerned with our true education.  The only metric human society seems to be able to measure with is money-value. 

All my writing is a kind of scream.  The passion and frustration is not directed to anyone in particular.  As you know, I am only an appendage of the blind will which is our inner essence.  Whatever knowledge I gain does not even belong to "me" since my consciousness will vanish without a trace.

So why do I collect these old math books and proceed to worship with them as though they contained mysterious powers?  I devote my attention to them and find that I actually have to think; and I so very much want to nurture whatever it is these authors are trying to pass on to us.   What is it?  Could it be PURE MATHEMATICS?

And, if this is the case, who gives a flying fuuck if 95% of the jobs in industrial civilization do not require even basic algebra!   I don't care about what employers want.    Should children and teenagers be motivated to understand pure mathematics so that they can get a job?    It won't work because most employers aren't concerned with what you actually understand.  Most employers don't understand it themselves.   Nobody cares! 

They just want to rate you and measure your "level of productivity" or competence.  In most jobs, I'm afraid the main trait they want is obedience. 

Well, I do care about learning pure mathematics.  I care about it deeply.   

Society praises honesty and then lets it starve. You end up like a dog that's been beat too much.  You learn to keep your thoughts to yourself since your intellectual integrity is very likely to get on other people's nerves.  An honest thinker becomes unfit for polite society, and - hence, unemployable.   Were they teaching  me math in a formal and rigorous manner so that I would be "employable" in the future workforce?  It's been a total disaster.  My love for knowledge might even be considered a mental disorder.

They worship technology and praise the managers and engineers who use mathematics as a tool, but there are few who are concerned with helping adults approach this subject in a meaningful manner.

Maybe the target audience better suited for the Dolciani series would be passionate adult seekers such as ourselves.   For teens, it might be pearls before the swine.

I know I never gave those books the attention they deserved.  Now I will.  This is a very personal matter for me.  It is not something many people are willing to discuss. 

Ich sollt was lernen!

Sometimes I become angry because I feel I am not learning anything new, that I am just wasting my time, that I could be studying something more challenging.  And yet, if I take a deep breath, and just allow myself to think ... then I'm OK.  I already know many things like second nature, and it is sometimes very annoying to force myself to explain the processes taking place in a detailed formal manner.   

This is the difficult part for me.  I am forcing myself to work on problem sets that are only difficult due to the novelty of having to use the formal language. 

I think I am driving myself insane, fighting my own inner being.  It's hard to explain.  Part of me is demanding to be allowed to work on more advanced mathematics, while the Driver (consciousness) really feels that this experiment will help me develop better thinking habits. 

The main quality I require is patience.  I guess I just want to witness myself in the act of thinking.  I get frustrated very easily ... no wonder I used to like to get drunk or smoke herb. 

Why am I putting myself through this?  I am some kind of Steppenwolf with a dual nature.  There is a method to my madness.  I am frustrated with the brain between my ears.  It is stubborn, as though it has its own will.  I want to force it to concentrate, to work through one textbook after another, one problem set after another ... but it will protest, "this is stupid."

You should hear my brain complain!

Very interesting experiment.  I wonder how it will turn out.

AHA!  So that's it.  I am just an observer.  I am not in control of what this brain does or how it thinks.  I make plans and it delights in sabotaging those plans.   

Designer babies?  I can't figure out where the transhumanists are coming from.  It's a creepy world, that's for sure. 

I guess we are facing reality squarely.   I think the H+ folks are similar to the gorts who want to colonize Mars.  I don't even want to contemplate the contents of their minds or why they think there is any meaning whatsoever in such idiotic schemes.

As for joining me in this mathematical sojourn (I had to look up the meaning of this word, sojourn), rest assured that it is very far removed from such ambitious projects as transhumanism or human colonization of outer space.

Of course, there are those, including myself at times, who might think I really am crazy or stupid for trying to force myself to find pleasure in studying things I find particularly difficult.

I am glad you are open to discussing the subtle difficulties in studying mathematics, and that, like me, you can imagine dropping out of the workforce to just spend your life trying to understand the underlying structure of mathematics.   They should be careful who they teach certain things to, for we can really go off the deep end studying and thereby become quite useless to the captains of industry.

I am going to give an example of just how the novelty of the formal and rigorous presentation appears so strange and, well, difficult.

You understand that I had been getting back into Linear Algebra and Calculus, working with Gaussian elimination, studying Infinite Series, exploring the techniques of integration.  Well, these methods are relatively straightforward even though they can appear intimidating.  There is the use of abbreviations which expand into more fundamental operations.  That's neither here nor there.  It's just that when  I studied calculus in 1994 at a local community college, my interest in mathematics was restored, rejuvenated.

Anyway, there is something about constructing proofs for some of the basic operations with the real number system that are requiring me to "start over" and move in baby steps.   It is so formal that I can see why, as a teenager, I might have been overcome with grief over how something so simple, like addition, had suddenly become so very complicated.   The formality and rigor was lost on me.  It annoyed me that something I had become so familiar with had suddenly been transformed into what I found to be unnecessarily complicated.

For instance, in proving the conditional statement, "If 7 = 5 + 2, then 7 + (-2) = 5."

Clearly, we can see that the hypothesis, "7 = 5 + 2,"  and the conclusion, "7 + (-2) = 5," are both true, so the conditional statement as a whole is true.  And yet, have you ever been exposed to a formal proof of such a simple statement?

I am only half joking when I say that the level of detail reminds me of god damn assembler language low-level instructions.   

I have to find a way to become less annoyed by this requirement for rigorous proof, for this has formed some kind of mental block against the rigor of pure maths.  I may have to "train my brain" to be able to make this great leap into pure mathematics.  It's a paradox that I have to "go back in time" in order to move forward.

It transforms the familiar process of addition into a weird series of statements that take slow motion photographs of the minute details of the thought processes involved; hence, the formality.

An example - a proof of the above conditional statement: If 7 = 5 + 2, then 7 + (-2) = 5

(1):     7 = 5 + 2     [Hypothesis]
(2):    7 + (-2) = (5 + 2) + (-2)    [Substitution principle]
(3):    (5 + 2) + (-2) = 5 + [2 + (-2)]       [Associative axiom of addition]
(4):    7 + (-2)  = 5 + [2 + (-2)]       [Transitive property of equality]
(5):    2 + (-2) = 0       [Axiom of additive inverses]
(6):    Therefore, 5 + [2 + (-2)] = 5 + 0      [Substitution principle]
(7):    7 + (-2) = 5 + 0        [Transitive property of equality]
(8'):    5 + 0 = 5       [Axiom of zero (the additive identity)]
(9):    Therefore, 7 + (-2) = 5       [Transitive property of equality]


This is called a direct proof.  To make the proof shorter and easier to follow, steps involving substitution and the properties of equality are not stated.  Steps 2 to 9 could be replaced by the following chain of equations;

7 + (-2) = (5 + 2) + (-2)     [Substitution principle]
            = 5 + [2 + (-2)]     [Associative axiom of addition]
            = 5 + 0                [Axiom of additive inverses]
            = 5                      [Axiom of zero]
Therefore, 7 + (-2) = 5    [Transitive property of equality]

This proof is based on the field axioms.

Is it any wonder they no longer teach mathematics this way in the high schools?  In fact,  this kind of teaching was only implemented for a couple decades.   I was at the tail end of it ... I never wanted mathematics to be like that, but this is evidently the language of actual mathematicians as opposed to "math educators".   There was none of that in the community college when I took calculus and physics, and only something similar in a course called Mathematical Reasoning at the state university.

And yet, now, at age 50,  I am really curious to see if such formality and rigor might fill in many gaps and serve as a bridge to engaging with mathematics on a deeper, structural level.

  The preceding proof suggests the following theorem.

For all real numbers b and c, (b + c) + (-c) = b

b and c are real numbers       [Hypothesis]
b + c is a real number           [Closure axiom of addition]
-c is a real number                [Axiom of additive inverses]
(b + c) + (-c) = b + [c + (-c)]       [Associative axiom of addition]
                     = b + 0             [Axiom of additive inverses]
                    = b                    [Axiom of zero]
Therefore, (b + c) + (-c) = b     [Transitive property of equality]

Then we can use this theorem to prove other theorems.   This theorem we just proved is called the Cancellation Property of Addition, so, once proved, we can just list [Cancellation property of addition] whenever we use the fact that "if a + c = b + c, then a = b".

This all begs the question, "Why bother?" - and I confess to not being able to give you, or anyone else for that matter, a good reason why I even bother.   George Carlin quit school in 8th grade.  He wanted to "burn down the math building".  Maybe I am a miserablist, and since I find this stuff rather depressing, it keeps me grounded.  In other words, I am only happy when I am depressed.  I have to wonder if I even "like" mathematics.  Sometimes I suspect that I do not like it at all, not at this level of rigor and formality.  Hence, the inevitable identity crisis which is sure to ensue. 

Ironically, I WANT to like it!  Does this make sense?  A great deal of the material I skip because it is not necessary for me to just work through exercises for the hell of it.  I only work through the exercises that are challenging to me, such as the ones involving proofs or the extras at the end.

Well, Holden, whether I like it or not, this    It's something I want to force myself to face ... a spiritual exercise which demands great humility and patience.  The parts that are not challenging make me doubt this decision, thinking I might have wasted money on the books, but when I come upon any exercises requiring these direct proofs, I am reassured that this was a good move.  It's not the kind of thing you can just track down on the internet.

I still experience a conscious resistance to such formality and rigor, but now I have become stubborn in the opposite direction.  Now I want to force whatever part of me that is repulsed by the level of detail to actually slow down and "get into it".   It takes a certain amount of faith or "hope" that in the process of finally taking such matters seriously, inner transformations will occur in my head which might crystallize into the attainment of that mythological awareness they call "mathematical maturity".

The only way I am able to take my present obsessions seriously is to be so detached from mainstream society, and to kind of "pretend" I am undergoing a mysterious initiation into a secret order of "monks" who are called to see behind the curtain, to firmly grasp that the mathematical operations we take for granted are based upon foundational axioms dealing with that set of real numbers represented by the number line.  These same axioms can be used to determine if sets other than numbers are groups or fields.  That's why it is presented as such, as the underlying unifying structure is transferable to other areas of mathematics.

I think I have repressed my early exposure to this kind of rigor since it proves to be so useless in day to day practical existence - in "getting things done".

Now I am hoping I can not only accept the rather uselessness of this formal rigor, but even try to embrace the uselessness, or at least try to get a little psyched about the weirdness of it.

There is definitely a sense of "defamiliarization" in that we are not accustomed to stating such minute details about operations we just take for granted.

Oh well, Holden.

I think that what we may have in common when it comes to mathematics is that we are honest about our contradictory and conflicting feelings toward it as a whole.

It hurts my brain to revisit this material, which is, I suspect, all the more reason I am so obsessed with focusing on this for a few years before devoting my mind to "advanced calculus and computational physics".  For all I know, my study of this foundational material will be it for me before I die in some "vehicular misadventure".

So, I can't become too concerned about the time spent going through the more challenging exercises in this special series of old neoteric high school textbooks. 

Like I said, the only way I have been able to remain committed to this endeavor is to proceed as though I were some kind of holy and deranged madman.

Don't think that I take our communications for granted.  If it were not for you, I really would have no one to complain to about my struggles.  Not only does no one care, but hardly anyone, including current students of so-called "engineering" mathematics, would encourage me to pursue these nagging details the way you do.  I think that you have an intuitive appreciation for how significant it is for me to come to terms with these frustrations with mathematics at this point in my life.  I feel that it makes no sense to proceed until I can unify mathematics in my own mind, in my own little world.

Note about when to expect to feel despair:   

After a long day of studying late into the night, you may feel that you are not getting anywhere.  Get a good night's sleep.   As Raul suggests, take a siesta.

In the morning you may find that your understanding and rudimentary skills are improving, and that you can take some delight in being able to approach a problem in a calm manner free of that paralyzing despair from the previous night.