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Modern Mathematics as Religion

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Nation of One:
Here is an example of being able to disagree with someone's premise while remaining extremely interested in that person's ideas.  I am intrigued by many of the things he writes.  I was searching for something he had written about how hardly any academic or professional mathematician is interested in revisiting high school mathematics since they simply don't have the time (preoccupied with professional obligations).

While tracking down the reference, I discovered this:  Set Theory: Should You Believe?  (N J Wildberger)


--- Quote from: Wildberger ---Modern mathematics as religion


Modern mathematics doesn't make complete sense. The unfortunate consequences include difficulty in deciding what to teach and how to teach it, many papers that are logically flawed, the challenge of recruiting young people to the subject, and an unfortunate teetering on the brink of irrelevance.

If mathematics made complete sense it would be a lot easier to teach, and a lot easier to learn. Using flawed and ambiguous concepts, hiding confusions and circular reasoning, pulling theorems out of thin air to be justified `later' (i.e. never) and relying on appeals to authority don't help young people, they make things more difficult for them.

If mathematics made complete sense there would be higher standards of rigour, with fewer but better books and papers published. That might make it easier for ordinary researchers to be confident of a small but meaningful contribution. If mathematics made complete sense then the physicists wouldn't have to thrash around quite so wildly for the right mathematical theories for quantum field theory and string theory. Mathematics that makes complete sense tends to parallel the real world and be highly relevant to it, while mathematics that doesn't make complete sense rarely ever hits the nail right on the head, although it can still be very useful.
 
So where exactly are the logical problems? The troubles stem from the consistent refusal by the Academy to get serious about the foundational aspects of the subject, and are augmented by the twentieth centuries' whole hearted and largely uncritical embrace of Set Theory.

Most of the problems with the foundational aspects arise from mathematicians' erroneous belief that they properly understand the content of public school and high school mathematics, and that further clarification and codification is largely unnecessary. Most (but not all) of the difficulties of Set Theory arise from the insistence that there exist `infinite sets', and that it is the job of mathematics to study them and use them.

In perpetuating these notions, modern mathematics takes on many of the aspects of a religion. It has its essential creed---namely Set Theory, and its unquestioned assumptions, namely that mathematics is based on `Axioms', in particular the Zermelo-Fraenkel `Axioms of Set Theory'. It has its anointed priesthood, the logicians, who specialize in studying the foundations of mathematics, a supposedly deep and difficult subject that requires years of devotion to master. Other mathematicians learn to invoke the official mantras when questioned by outsiders, but have only a hazy view about how the elementary aspects of the subject hang together logically.

Training of the young is like that in secret societies---immersion in the cult involves intensive undergraduate memorization of the standard thoughts before they are properly understood, so that comprehension often follows belief instead of the other (more healthy) way around. A long and often painful graduate school apprenticeship keeps the cadet busy jumping through the many required hoops, discourages critical thought about the foundations of the subject, but then gradually yields to the gentle acceptance and support of the brotherhood. The ever-present demons of inadequacy, failure and banishment are however never far from view, ensuring that most stay on the well-trodden path.

The large international conferences let the fellowship gather together and congratulate themselves on the uniformity and sanity of their world view, though to the rare outsider that sneaks into such events the proceedings no doubt seem characterized by jargon, mutual incomprehensibility and irrelevance to the outside world. The official doctrine is that all views and opinions are valued if they contain truth, and that ultimately only elegance and utility decide what gets studied. The reality is less ennobling---the usual hierarchical structures reward allegiance, conformity and technical mastery of the doctrines, elevate the interests of the powerful, and discourage dissent.

There is no evil intent or ugly conspiracy here---the practice is held in place by a mixture of well-meaning effort, inertia and self-interest. We humans have a fondness for believing what those around us do, and a willingness to mold our intellectual constructs to support those hypotheses which justify our habits and make us feel good.
--- End quote ---

The reference I was searching is at the end of this next quote.  I will pull it out and leave these excerpts for posterity and as a reminder to myself of what motivates me to study what I do.


--- Quote from: Wildberger ---The problem with foundations

The reason that mathematics doesn't make complete sense is quite easy to explain when we look at it from the educational side. Mathematicians, like everyone else, begin learning mathematics before kindergarten, with counting and basic shapes. Throughout the public and high school years (K-12) they are exposed to a mishmash of subjects and approaches, which in the better schools or with the better teachers involves learning about numbers, fractions, arithmetic, points, lines, triangles, circles, decimals, percentages, congruences, sets, functions, algebra, polynomials, parabolas, ellipses, hyperbolas, trigonometry, rates of change, probabilities, logarithms, exponentials, quadrilaterals, areas, volumes, vectors and perhaps some calculus. The treatment is non-rigorous, inconsistent and even sloppy. The aim is to get the average student through the material with a few procedures under their belts, not to provide a proper logical framework for those who might have an interest in a scientific or mathematical career.

In the first year of university the student encounters calculus more seriously and some linear algebra, perhaps with some discrete mathematics thrown in. Sometime in their second or third year, a dramatic change happens in the training of aspiring pure mathematicians. They start being introduced to the idea of rigorous thinking and proofs, and gradually become aware that they are not at the peak of intellectual achievement, but just at the foothills of a very onerous climb. Group theory, differential equations, fields, rings, topological spaces, measure theory, operators, complex analysis, special functions, manifolds, Hilbert spaces, posets and lattices---it all piles up quickly. They learn to think about mathematics less as a jumble of facts to be memorized and algorithms to be mastered, but as a coherent logical structure. Assignment problems increasingly require serious thinking, and soon all but the very best are brain-tired and confused.

Do you suppose the curriculum at this point has time or inclination to return to the material they learnt in public school and high school, and finally organize it properly? When we start to get really picky about logical correctness, doesn't it make sense to go back and ensure that all those subjects that up to now have only been taught in a loose and cavalier fashion get a proper rigorous treatment? Isn't this the appropriate time to finally learn what a number in fact is, why exactly the laws of arithmetic hold, what the correct definitions of a line and a circle are, what we mean by a vector, a function, an area and all the rest? You might think so, but there are two very good reasons why this is nowhere done.

The first reason is that even the professors mostly don't know! They too have gone through a similar indoctrination, and never had to prove that multiplication is associative, for example, or learnt what is the right order of topics in trigonometry. Of course they know how to solve all the problems in elementary school texts, but this is quite different from being able to correct all the logical defects contained there, and give a complete and proper exposition of the material.

The modern mathematician walks around with her head full of the tight logical relationships of the specialized theories she researches, with only a rudimentary understanding of the logical foundations underpinning the entire subject. But the worst part is, she is largely unaware of this inadequacy in her training. She and her colleagues really do believe they profoundly understand elementary mathematics. But a few well-chosen questions reveal that this is not so. Ask them just what a fraction is, or how to properly define an angle, or whether a polynomial is really a function or not, and see what kind of non-uniform rambling emerges! The more elementary the question, the more likely the answer involves a lot of philosophizing and bluster. The issue of the correct approach to the definition of a fraction is a particularly crucial one to public school education.

Mathematicians like to reassure themselves that foundational questions are resolved by some mumbo-jumbo about `Axioms' (more on that later) but in reality successful mathematics requires familiarity with a large collection of `elementary' concepts and underlying linguistic and notational conventions. These are often unwritten, but are part of the training of young people in the subject. For example, an entire essay could be written on the use, implicit and explicit, of ordering and brackets in mathematical statements and equations. Codifying this kind of implicit syntax is a job professional mathematicians are not particularly interested in.
The second reason is that any attempt to lay out elementary mathematics properly would be resisted by both students and educators as not going forward, but backwards. Who wants to spend time worrying about the correct approach to polynomials when Measure theory and the Residue calculus beckon instead? The consequence is that a large amount of elementary mathematics is never properly taught anywhere.

But there are two foundational topics that are introduced in the early undergraduate years: infinite set theory and real numbers. Historically these are very controversial topics, fraught with logical difficulties which embroiled mathematicians for decades. The presentation these days is matter of fact---`an infinite set is a collection of mathematical objects which isn't finite' and `a real number is an equivalence class of Cauchy sequences of rational numbers'.

Or some such nonsense. Set theory as presented to young people simply doesn't make sense, and the resultant approach to real numbers is in fact a joke! You heard it correctly---and I will try to explain shortly. The point here is that these logically dubious topics are slipped into the curriculum in an off-hand way when students are already overworked and awed by all the other material before them. There is not the time to ruminate and discuss the uncertainties of generations gone by. With a slick enough presentation, the whole thing goes down just like any other of the subjects they are struggling to learn. From then on till their retirement years, mathematicians have a busy schedule ahead of them, ensuring that few get around to critically examining the subject matter of their student days.
--- End quote ---

There it is:  From then on till their retirement years, mathematicians have a busy schedule ahead of them, ensuring that few get around to critically examining the subject matter of their student days.

Yes, they are most likely surfing the learning curve of Sisyphus, learning to use computer algebra systems or use LaTeX to get the ever-so-beautiful mathematical notation to magically appear on their websites.

I have even heard such mathematics refereed to as "baby math," which immediately turned me off to the [academic] author of the text.

While I admit that the way the subjects are rigorously presented by mathematician/educators from the 1960's/80's such as Mary Dolciani - and even more so by Frank Allen, may have been a bit too much for high school students, at this point in my life, I really feel what could be described as a religious feeling toward the way the material is presented - based on Set Theory and "the set of real numbers" ...

So, you see, I may have a religion after all.  I just happen to be a self-ordained unorthodox priest of the craft/cult.  :-\

raul:
Hentrich,
Clearly you can be the High Priest of the Mathematics Cult. I read that the Pythagoreans sacrificed an ox after discovering the 47th Proposition of Euclid. They also believed in the the transmigration of the soul and which included the transmigration of human souls into the bodies of animals.Pythagoras strictly forbid the consumption of meat.

The Pythagoreans believed that the human soul was trapped in a continuous cycle of death and reincarnation and the way to be free from this cycle was to reach a high understanding of the universe through introspective thought and philosophical study. They also believed that the number one was very important.

Of course we are living different times. Worshipping numbers can lead you to wear a straight jacket!

Stay well

Nation of One:

--- Quote from: Raul ---Clearly you can be the High Priest of the Mathematics Cult.
--- End quote ---

No, definitely not a high priest, unless, of course, I fall off the wagon.   :D

The thought of simply being some kind of weirdo who is able to keep himself occupied with studying mathematics from morning until late at night, day after day, season after season, makes me feel like I have discovered a Fortress of the Mind, a Mental Sanctuary, in which, for the relatively small price of used books, pencils, notebooks, and computer, I maintain a rich inner life a million miles away from all the hype in TV land.


--- Quote from: Raul ---I am not sure but the U.S. author Ray Bradbury  said that we need a little fantasy to endure reality. We need a sense of humor in life. Studying mathematics is an excellent way to endure life.
--- End quote ---

This Mathematics Cult can be a joke disguised as a religion or a religion disguised as a joke.  The high priests are most surely the academics as well as the renegade physicists who can only find work as computer programmers.  Me, I am certainly no high priest, but just a devoted but wayward acolyte.

Either way, the important thing is to be able to approach the subject with zeal outside the confines of compulsory education systems and under no pressure whatsoever to justify the time spent.  Now, if I were married, I suppose these interests would be discouraged unless I could apply them to earning money.  There is something terribly subversive about the idea of devoting a great deal of mental energy into an endeavor that does not have gainful employment as a goal.

This is what makes it a kind of fantasy, for I am in my own little world.  I have great disdain for NASA (space monkeys) and the servile scientists who devote their lives to engineering aircraft and bomber jets for the Department of Defense.  Fortunately for me, I am not the kind of person NASA or the Department of Defense would be interested in. 

So, if I am in this "religious order" - the Mathematics Cult - at all, it is as a self-ordained monk, layman, hobbyist, and math junkie.

In this way, those who have been thrown overboard by Industrial Society might eek out a humble existence as a kind of math monk who may be spiritually and intellectual nourished by lifelong learning without requiring the approval of credential-granting institutions, without begging to be exploited by an employer who might prevent one from approaching areas of interest in a deep and meaningful manner.

Personally, I like this idea of combining fantasy and humor with subjects as traditionally serious as mathematics and religion.

Even the series of notebooks I have filled over the last couple years each have the half-joking/half-serious subtitle, "A Mathematical Diary by a disciple of The Weird."

The subtitle was inspired by something Holden had written about Cthulu, Lovecraft, and mathematics ... about how "weird" mathematics can appear, where you intuitively suspect that the meanings behind some of the notation hold some mysterious truths.

Although he frequently employed mathematical concepts, Lovecraft did not consider himself an adept mathematician.  In a letter to Maurice W. Moe in 1915 Lovecraft remarked:   

"Mathematics I detest, and only a supreme effort of the will gained for me the highest marks in Algebra and Geometry at school.  In everything I am behind the times."

Although Lovecraft professed to dislike mathematics, he was very interested in physics and picked up mathematical notions through these interests. (See Lovecraft, Selected Letters 1911-1924)
 

--- Quote ---Some have complained that Lovecraft should have spent more time writing fiction and less time corresponding. They argue that Lovecraft’s doing so would have produced more horror fiction for the world to enjoy. However, many have discovered that Lovecraft’s letters are just as enjoyable, if not more enjoyable, than his fiction. In his letters, Lovecraft doesn’t have the constraints placed upon him that he does in writing fiction. He is free to describe his philosophy, his interests, and his dreams, the descriptions of which are sometimes superior to his fiction.

One thing seems quite clear: Lovecraft’s fiction may never be considered literature by academia—but his correspondence makes it very clear that he was “a man of letters.”
--- End quote ---

Library Genesis

raul:
Hentrich,
Thank you for your response. In this world the study of mathematics,if it is not for financial purposes, is a subject for nerds. You are a dedicated student. Here from time to time I hear that Paraguayan math students cannot keep up with the level of more demanding universities in the U.S. and Europe. But the thing is that they do not care if these students are able to reason logically and attain knowledge. They are only cannon fodder to the industrial society. But even if these men and women ,with excellent grades, have no room in national universities and schools.

In this industrial society beauty is much more appreciated. The beautiful gets more money than those who devote their time to study math and science.  That is the reason there is much investiment in these Miss World contests where  blonde top models are the some kind of royal figures, at least for some time. But then after they turn 25, they are considered old. They were squeezed to their last drop. The ugly, the deformed and the old are forbidden. Even here we prefer blondes. Probably we have something of a Nazi Aryan with blue eyes inside us.

H.P. Lovecraft was a man ahead of his time but probably born in the wrong century. I read his biography a little. His books sell millions now but in his time he suffered much because he was not "financially empowered".

Stay well.

Nation of One:

--- Quote from: Raul ---In this world the study of mathematics,if it is not for financial purposes, is a subject for nerds.
--- End quote ---

Yes, there is that stigma here as well, but at my age I no longer care about such things. 

I find life to be very depressing, and sometimes mathematics just doesn't cut it for me, or I find myself feeling inadequate in that it takes a great deal of effort to remain "dedicated".  It is a real mental and spiritual battle to study regardless of what we may appear to be to others.

I had realized a very long time ago that there is not much worth doing in this world.  There's just so much hype. 

I'll be the first to admit that many would see my lifestyle as pitiful, but when i wake up in the morning and am able to revive my interests in a text I am determined to work through, this saves me from getting sucked into the ZooTubes.

The main thing for me is to find some kind of inner sanctuary which is not at all dependent on how others would judge me.

As I have said before, chasing oblivion lead me nowhere.

And yet, I am surely behind the eight ball.  What I mean by this is that I want nothing to do with competing with others who have far more education, skill, and talent than i ever will.  This is why I consider my engagement with mathematics more of a "religion" than a hobby or a vocation.

My interest in programming is always revived by my engagement with mathematics, and I don't need to master anything particularly advanced in order to be mentally stimulated.

Of course, a bad toothache can just about sabotage everything, so our state of mind at any given time is very tenuous - hence, the underlying horror of being an organism made of flesh, blood, bones, nerve endings, excrement, etc.

I really have a great deal of sympathy and empathy for the youth who may feel overwhelmed by what is before them, and I have no regrets about having not been chosen by "a nubile nyphet" to take my seed and transform it into yet another frustrated and nervous Hentrich who must find his way in a hostile world.

My paying deference to mathematics is not so much because I love it so much, but that it is one of the only things I have enough interest in to remain committed to some kind of plan, where I move in a slow and steady direction.  I am sure none of this will lead anywhere.  In fact, those who blow their minds with drugs may be better off.

The main purpose for my studies is that it keeps my mind engaged and keeps me out of trouble so I can be around to help my mother as she ages.   I consider access to math texts and computers as a reward and motivation to stay out of trouble.

We are in the giant prison farm, as you say. 

So, I don't mind if people were to see me as some kind of "nerd" - even though I do not care for such labels.  At this point I am very attached to these books - psychologically attached ... to the extent that I would suffer terribly were I suddenly torn from my current endeavors.

It's not even something I care to discuss with others.  I don't mind if people do not understand me. 

And yet, because I do not discuss much with anyone (especially since I do not venture out anymore except to gather groceries), I very much appreciate my correspondence with Holden and yourself.

You see, the subject of mathematics is so very vast.  It is easy to become overwhelmed and discouraged.  Someone at my age who is studying the way I do might even be suspected of escaping from reality.  And yet, this is my reality.  I have witnessed how we human beings can be locked in cages.  I have seen that our own bodies and brains are used against us to torment us.

And so, my approach to studying has a kind of Twilight Zone, Science Fiction, Horror, Existential Twist to it.  I am in this strange reality along with you and Holden and countless others, and I do what I do until I am unable to do it. 

I don't believe in marching in the streets.  I don't vote.  I do not cheer on any sports teams, and I've only purchased on music CD over the last 4 years, and that was Roger Waters's recent one.   Hell, I don't listen to much music anymore.

When I drank alcohol, I always wanted to listen to music - and even sing, even though I can't carry a tune.

So, you see, I am not I.  I am not me.

Whoever I am now, he is obsessed with studying mathematics, and not just any mathematics.  I want to study the math that I supposedly already know.  You see, I want to find some delight in having to think about how to go about solving problems - so I only like to go over those problems which are a bit challenging, but not impossible.

As I have said, i am in my own little world, and maybe someone of a different temperament might even want to end their life or go out there and raise some hell; but I have had enough trouble.  So many vampires, sociopaths, and chaos out there!

I am living my own science fiction story where I am the strange protagonist.

Aren't we each the main character in a bad existentialist film?

As Holden says, we might even already be dead, and what we experience as our lives is really playing out in minds in a few microseconds like in the film Jacob's Ladder.

Take care.

- Mike, the Math Junkie

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