Proof of Existance of Hell on Earth
Mike Powell
Just found this group....
I was interested in learning a bit about what goes on here,
and am in search of a little wisdom... I thought the FAQ
would be a good place to start...
The FAQ opens with:
"Start with a mod p representation of the
Galois group of Q which is known to be modular. You want to prove that
all its lifts with a certain property are modular. This means that the
canonical map from Mazur's universal deformation ring to its "maximal
Hecke algebra" quotient is an isomorphism. To prove a map like this is
an isomorphism, you can give some sufficient conditions based on
commutative algebra. Most notably, you have to bound the order of a
cohomology group which looks like a Selmer group for Sym^2 of the
representation attached to a modular form. The techniques for doing
this come from Flach; you also have to use Euler systems a la
Kolyvagin, except in some new geometric guise."
[END QUOTE]
My response to this being "WHAT IS THIS LANGUAGE!?!?"
Dragging myself through the beginning of the FAQ line by line, word
by painful word is very near to proof that there is hell on earth.
I communicate with words and pictures... and yet the words that appear
in the FAQ are foreign to me... apparently, they are the words of folks
who speak of all things mathematical... and I have not a clue as to
what they are saying.
And I have a deep jelosy of those who do...
I consider myself to be a very technically competent person... I've
had my hands in electronics for ages, been doing 3-D computer animation
for many years (self-taught), a licensed pilot, electrical systems
maintenance tech for military aircraft, computer programmer etc. etc. etc.
I'm now a technician at Hewlett-Packard.
And yet, every so often I come up against some numbers... arranged
and separated by mysterious symbols... I feel that they must mean something,
that they aren't simply random scribbles. And usually when I read any
accompanying text, it all looks like gobbldegook to me. My head spins
out of control. Blinded by the sudden apperance of clouds I begin to
fall into a slow death spiral, ending in a sudden "whomp!" as I slam
the book closed.
I do not learn well by memorization... I find that my retention comes
only from a solid understanding of something. I don't do well with
street names.
I have a true desire to understand all forms of math, but I go into this
mental death spiral so easily when confronted with the language of
math, that I don't get far. I must be missing some basic element in
my approach to math... You know what I hate most? I hate when I'm reading
a text explainng some mathematical acrobatics that goes like: " Under
these circumstances, the probability of a sputtered atom reaching the
substrate without colliding with an argon atom is
[much mathematical gobbledegook deleted]
which is at least a reasonable chance."
Ug... "Is *WHAT* a reasonable chance? How the heck can I know when you
switch languages on me like that!" It's painful and frustrating for me,
yet I am deeply drawn to some hint of magic I perceive in math...
I've been through some low-level college math courses... I guess I
learned something, but never any breakthrough... and it has
only made my frustration worse. And being generally smart hasn't
helped me a bit... a people assume that being 'smart' means that I
am probably a math wiz, so they bring problems to me, and I end up
refering them to someone else, as often as not.
I'm in a real jam here...
Some of you will no doubt have no idea what I'm whining about... but on the
chance that someone will, someone who has been where I am now, and
pulled themselves out of this death spiral, I'd like to ask your help.
Where might I have gone wrong? Is there a path to success that I should
explore? What did you do to succeed?
-Mike-
(dizzy as hell, but not beaten)
john baez
The FAQ opens with:
>"Start with a mod p representation of the
>Galois group of Q which is known to be modular. You want to prove that
>all its lifts with a certain property are modular. This means that the
>canonical map from Mazur's universal deformation ring to its "maximal
>Hecke algebra" quotient is an isomorphism. To prove a map like this is
>an isomorphism, you can give some sufficient conditions based on
>commutative algebra. Most notably, you have to bound the order of a
>cohomology group which looks like a Selmer group for Sym^2 of the
>representation attached to a modular form. The techniques for doing
>this come from Flach; you also have to use Euler systems a la
>Kolyvagin, except in some new geometric guise."
>My response to this being "WHAT IS THIS LANGUAGE!?!?"
This is called "number theory," man. You are apparently reading the
section on Fermat's last theorem, and maybe you see why most people in
math aren't working on Fermat's last theorem. I agree that this
shouldn't be near the beginning of the FAQ!! There is a lot that is
more readable. However, something like this should be in the FAQ,
preferably with bright red warnings before it, because there are lots of
people interested in how Wiles (may have) proved Fermat's last theorem.
>Dragging myself through the beginning of the FAQ line by line, word
>by painful word is very near to proof that there is hell on earth.
Huh, it's odd that you even bothered; when I come to mathematics that I
do not understand at all I don't bother reading it. If the terms are
not defined it is almost pointless. (With practice one can sometimes
figure out what the definitions must be, but I couldn't tell a Selmer
group from a.... some other group I know nothing about.)
>I communicate with words and pictures... and yet the words that appear
>in the FAQ are foreign to me... apparently, they are the words of folks
>who speak of all things mathematical... and I have not a clue as to
>what they are saying.
>And I have a deep jealosy of those who do...
Aha, NOW the truth comes out.
>I consider myself to be a very technically competent person... I've
>had my hands in electronics for ages, been doing 3-D computer animation
>for many years (self-taught), a licensed pilot, electrical systems
>maintenance tech for military aircraft, computer programmer etc. etc. etc.
>I'm now a technician at Hewlett-Packard.
Sounds good. I bet, however, that you did not, one day, simply hop in a
plane for the first time, rev it up, and fly off....
>And yet, every so often I come up against some numbers... arranged
>and separated by mysterious symbols... I feel that they must mean something,
>that they aren't simply random scribbles. And usually when I read any
>accompanying text, it all looks like gobbldegook to me. My head spins
>out of control. Blinded by the sudden apperance of clouds I begin to
>fall into a slow death spiral, ending in a sudden "whomp!" as I slam
>the book closed.
Go a bit more slowly!!! It's not like a TV show where you can turn it
on halfway through and expect to follow it. Math texts are very often
read at the rate of a page or two a day. (Actually I read the whole
book in one day, then read the whole book again the next day, and so on
until it slowly makes sense. But that's a matter of taste.)
>I do not learn well by memorization... I find that my retention comes
>only from a solid understanding of something. I don't do well with
>street names.
Then math is the thing for you. Memorization is very little of what
math is about, less so than any other subject. Most subjects have some
CONTENT! That is, there are such-and-such kings of England and you
simply have to learn who they were to be up on British history, and
there are such-and-such continents and such-and-such bacteria.... math
is much more the kind of thing where you can figure it out yourself
if you happen to forget some --- not quite, but much closer.
>Where might I have gone wrong? Is there a path to success that I should
>explore? What did you do to succeed?
You need to relax and not expect it to be quick, and realize that the
process of learning math is a process of gradually struggling with
understanding something until it clicks. Start with something easy that
you already think you know and try to understand it better. I don't
know how much math you know. The quadratic formula? Can you derive it
or have you just memorized it? If you've only memorized it, why not
derive it yourself. Or some trig identity or some such thing. That's
one approach. Another is to do somes stuff like this: say you have a
pyramid of balls that has a triangle with, say n ball on each side at
the bottom:
x x x
x x
x
(here n = 3), then n-1 on the second level:
x x
x
and so on, and finally 1 on top:
x
Figure out a formula for how many balls there are, as a function of n.
If that's too hard, first figure out a formula for how many there are in
a triangle that has n balls on each side.
Or something like that.
Alex Lopez-Ortiz
writes:
>
>
> Where might I have gone wrong? Is there a path to success that I should
> explore? What did you do to succeed?
As the keeper of the FAQ, only I can answer your question my son:
Try buddhism zen.
And then when you have grasped the meaning of all things impossible,
you would be ready to take the step that only a selected few are
allowed to take:
A college degree in mathematics!
Then an only then will you be able to crack the non-sensical statement
that opens my FAQ. You will learn then that this statement, intended to
drive the uninitiated away from Mathematics (as egipcian curses
did in other times), is actually a joke that greets the weary mathematician
who happens to rest his mind and thought in the lines of my FAQ.
Alex "running-on-an-overdose-of-cheese-and-crackers" Lopez-Ortiz
>
> -Mike-
> (dizzy as hell, but not beaten)
--
Alex Lopez-Ortiz alop...@neumann.UWaterloo.ca
Department of Computer Science University of Waterloo
Waterloo, Ontario Canada
Keith Ramsay
|Blinded by the sudden apperance of clouds I begin to
|fall into a slow death spiral, ending in a sudden "whomp!" as I slam
|the book closed.
ba...@guitar.ucr.edu (john baez) writes:
|Go a bit more slowly!!! It's not like a TV show where you can turn it
|on halfway through and expect to follow it. Math texts are very often
|read at the rate of a page or two a day.
Right. Sometimes less. Sometimes it makes sense to spend a day or more
on one point.
|(Actually I read the whole
|book in one day, then read the whole book again the next day, and so on
|until it slowly makes sense. But that's a matter of taste.)
(It's because John Baez is a physicist at heart, isn't it.) :-)
Pacing is important. Perhaps the lack of an obvious rule governing the
time required to do or learn some piece of mathematics explains why
people find the process confusing.
On the one hand, people sometimes suppose one "should" be able to learn
or do some mathematics at a given pace. This may be, as Mike Powell
indicated for example, because one is intelligent and normally good
at learning new things. People then are startled (and suffer from a
"death spiral" :-) ) when they hit an unexpected obstacle, and perhaps
conclude that there is some mysterious "knack" to mathematics, which
they simply don't have.
On the other hand, I see people apparently expecting a potentially easier
mathematical task to require X amount of time, where X may be a long time.
They've envisioned doing things the hard way, really gritting their teeth
and straining. I've noticed that sometimes a certain piece of mathematics
develops a false aura of difficulty in my mind too, which is dispelled when
I stop worrying about the difficulty, and start simply working on it at
whatever pace it demands. I remember how as graduate students we'd sometimes
"psych each other out" by commiserating or referring with trepidation or awe
at some piece of mathematics which we didn't understand yet. It would sometimes
occur to me later that my idea of how difficult it would be to learn it was
based on hearsay.
I'd be interested in reading what people might have to say about these
psychological issues as they affect mathematical research work.
I have sometimes felt the effects of time-pressure anxiety. Sometimes I feel
as though I'm spreading myself out too thin, starting assorted projects which
I then have to stop without producing any concrete results, because I don't
think I'll have the time to finish them. Sometimes I try to concentrate on my
"main line" of research, but find it difficult to maintain such a singleness
of direction when going through a relatively boring or slow phase of the work
(writing things up at the end, especially) in the face of distractions.
(Perhaps I'll write up the story of my "paper from Hell" after I'm finished
with it.)
What have people found useful for coping with these difficulties?
Keith Ramsay
john baez
>ba...@guitar.ucr.edu (john baez) writes:
>|(Actually I read the whole
>|book in one day, then read the whole book again the next day, and so on
>|until it slowly makes sense. But that's a matter of taste.)
>(It's because John Baez is a physicist at heart, isn't it.) :-)
Heh, how come they only say that here on sci.math? Over on
sci.physics they try to insult me by saying stuff like "only a
mathematician would think that." Guess I must be doing *something*
right. I think, actually, that the difference is more one of intuitive
versus rational temperament. Being intuitive, I find it almost
impossible to follow a sequence of rational steps until I have a misty
notion of where they are leading; once I get that intuition, the steps
make sense and are not that hard. Other people find it impossible to
tolerate reading things until all the definitions and details are firmly
in place. One has to find ones own best strategies on this sort of
thing, I guess, and then not be unduly swayed by what *other* people
think is the "right" way to learn things or work problems. (On the
other hand, it's good to spend a certain amount of energy now and then
stretching ones cognitive styles by going against ones own grain... as a
matter of exercise. Only by knowing ones own grain can one do this
effectively.)
>On the one hand, people sometimes suppose one "should" be able to learn
>or do some mathematics at a given pace. This may be, as Mike Powell
>indicated for example, because one is intelligent and normally good
>at learning new things. People then are startled (and suffer from a
>"death spiral" :-) ) when they hit an unexpected obstacle, and perhaps
>conclude that there is some mysterious "knack" to mathematics, which
>they simply don't have.
That's definitely very common. When I tell people "what I do" at
cocktail parties, they typically see fit to relive the story of their
"death spiral." E.g., "I thought I was good in mathematics until
trigonometry..." or geometry, or calculus, or differential equations, or
whatever. I have this image of a bunch of airplanes under enemy fire,
one after another getting hit, crashing and burning, with only a few
making it.... Sad! Of course most people won't become mathematicians,
but I think that there's some way people "go down in flames" that is
very traumatic to them.
>On the other hand, I see people apparently expecting a potentially easier
>mathematical task to require X amount of time, where X may be a long time.
>They've envisioned doing things the hard way, really gritting their teeth
>and straining. I've noticed that sometimes a certain piece of mathematics
>develops a false aura of difficulty in my mind too, which is dispelled when
>I stop worrying about the difficulty, and start simply working on it at
>whatever pace it demands. I remember how as graduate students we'd sometimes
>"psych each other out" by commiserating or referring with trepidation or awe
>at some piece of mathematics which we didn't understand yet. It would
>sometimes occur to me later that my idea of how difficult it would be
>to learn it was based on hearsay.
Heh, this is true too! I remember being psyched out about Calculus as a
high school student, and even resolving at one point to become a
mathematician but never use Calculus! I think this was because my uncle
had tried to explain derivatives to me once, and the process of taking
the derivative of x^2 seemed very mysterious. But he also slipped me
Silvanus Thompson's "Calculus Made Easy" and in a while I learned the
stuff, so by the time I took a class on it I knew what was up.
I also remember as a junior high school student thinking that the
computer science concept of a "loop" was absolutely inpenetrable. I
think that in this case it was simply badly explained! I hate to
imagine that I was *that* dumb!
I think in college and grad school a lot of intimidation goes on, as you
say. Before entering MIT for grad school, I remember a Harvard
"freshman" grad student explaining how they were studying Hartshorne's
Algebraic Geometry and that a "scheme" was something utterly
incomprehensible. Schemes have that kind of reputation, no? --- and so
does cohomology theory for people at a certain level. (Remember on
sci.math how someone's professor told them that NOBODY understood
cohomology theory? How terrible.) Something that still holds me in a
certain amount of terror is spectral sequences. My friend Minhyong Kim
says the right approach is to completely not bother with understanding
them and just learn how to *use* them to compute things, and then learn
why they work.
For the most part, though, I have not been too intimidated about
mathematics, perhaps because in my student days I constantly kept a vast
pile of math books around, on all possible subjects, and read bits and
pieces of them in a completely random way, until things started making a
vague global kind of sense, so that when someone talked about this or
that it would usually set *some* sort of bells ringing... be it only the
tintinnabulation of my overheated pate. In other words, I may not have
really known much about K-theory, or Sturm-Liouville equations, or
whatever, but as long as I had *some* sort of dim image of them I would
be able to keep sharpening that image whenever I heard or read more
about them, whereas when I know *nothing* about a word I tend to turn
off when I see it. Which after all is what most people do when they see
an equation, however simple.
>I'd be interested in reading what people might have to say about these
>psychological issues as they affect mathematical research work.
>I have sometimes felt the effects of time-pressure anxiety. Sometimes I feel
>as though I'm spreading myself out too thin, starting assorted projects which
>I then have to stop without producing any concrete results, because I don't
>think I'll have the time to finish them. Sometimes I try to concentrate on my
>"main line" of research, but find it difficult to maintain such a singleness
>of direction when going through a relatively boring or slow phase of the work
>(writing things up at the end, especially) in the face of distractions.
>(Perhaps I'll write up the story of my "paper from Hell" after I'm finished
>with it.)
Oh boy, this is a whole separate interesting issue in itself, but I am
momentarily talked out.....
Michael Weiss
Part of the problem, no doubt, is that many texts are atrociously written.
(I have a friend who claims that analysts in general have a better writing
style than algebraists. I stick this in here to provoke flames.) V. I.
Arnold, one of the best mathematical stylists going, puts it this way:
It is almost impossible for me to read contemporary mathematicians who,
instead of saying ``Petya washed his hands,'' write simply: ``There is
a $t_1<0$ such that the image of $t_1$ under the natural mapping $t_1
\mapsto {\rm Petya}(t_1)$ belongs to the set of dirty hands, and a
$t_2$, $t_1<t_2 \leq 0$, such that the image of $t_2$ under the
above-mentioned mapping belongs to the complement of the set defined in
the preceding sentence.''
Cohomology, for example, becomes quite clear if motivated via the Cauchy
integral theorem, or electrical networks (Bamberg and Sternberg, vol.II),
or even (much to my surprise) carrying in elementary arithmetic (James
Dolan's recent posts).
It would be a Good Thing if someone made up a list of attempts by authors
to furnish intuitive underpinnings for mathematical concepts--- arranged by
concept. (E.g., "differential form: see Misner, Thorne, and Wheeler,
"Gravitation", and Spivak's first tome on differential geometry.)
Mark Hopkins
>Huh, it's odd that you even bothered; when I come to mathematics that I
>do not understand at all I don't bother reading it. If the terms are
>not defined it is almost pointless. (With practice one can sometimes
>figure out what the definitions must be, but I couldn't tell a Selmer
>group from a.... some other group I know nothing about.)
a symptom of a lack of grounding in the subject. The logical recourse
is to backtrack and solidify the foundation first by becoming deeply
familiar with each of the founding pillars, and preferrably working out
the problems and examples. After all the pillars have been erected,
the edifice can be constructed and the notation will make perfect intuitive
sense.
In a rush, it shouldn't take more than a few days. The slowdown that most
people observe and experience arise from trying to skip stages, work 1, 2
or 3 levels above the optimum, and consequenctly running aground whilst
trying to bash meaningless and ungrounded abstractions into one's own head.
Basically I come up with the following time estimates:
Time to assimilate information Number levels above current
0 0
5 days 1
3 weeks 2
6 months 3
Thus, a direct transition 0 -> 3 would be 6 months, a two-layered
transition, 0 -> 1 -> 3 would be 5 days + 3 weeks or about a month, and
a three-layered transition 0 -> 1 -> 2 -> 3 would be 2 weeks and a day.
So in essence, the slower you go, the quicker you'll get done.
It's reminiscent of light polarization...
###
Michael Weiss
>Cohomology, for example, becomes quite clear if motivated via the Cauchy
>integral theorem, or electrical networks (Bamberg and Sternberg, vol.II),
>or even (much to my surprise) carrying in elementary arithmetic (James
>Dolan's recent posts).
Matthew P Wiener writes:
Homology as presented by topologists always seemed intuitive enough to
me geometrically, so by the time I got to cohomology I was used to it.
This leaves a few questions unanswered. How was homology presented--- in
particular, did the text or instructor explain how homology groups "count
holes" before, say, plunging into the Eilenberg-MacLane axioms for a
homology theory? If not, did you figure this out on your own? Or was it
at once obvious?
When you encountered cohomology for the first time, how was *that*
presented? Did you ask yourself, "Why are we defining cochains, cocyles,
and so on this way?" Did you have enough background (in say, complex
variables or differential geometry) so you could immediately come up with
geometric examples of bilinear pairings, a la de Rham cohomology or Hodge
theory?
For that matter, do you enjoy being thrown a bunch of definitions without
any motivating examples, seeing some neat theorems, and puzzling out the
intuitive picture on your own? I'm quite serious--- I used to enjoy that a
great deal. Now I've gotten lazy and I wait for the topic to turn up on
sci.math.