# What is Mathematical Maturity, really?

87 views

### Seraph-sama

Aug 14, 1999, 3:00:00â€¯AM8/14/99
to
I have been told before that although I can do higher-level mathematics like
abstract algebra at my age (16), I will not be able to study it the *right way*
unless I'm extremely special. It was nearly impossible, I was told, to have
mathematical maturity at my age, which has made me stop and think about it. I
never ran across any definition of mathematical maturity, but I always assumed
I had it. I assumed it meant the understanding and appreciation of the nature
of proof and mathematical concepts, and an understanding of the whole of
mathematics in general, and what it actually is. The only thing it could
possibly be is to be able to visualize and understand a definition or theorem
right after it has been defined/proved. For example, last night I read the
proof of the existence of Sylow p-subgroups, and although the proof was very
easy to follow, it is hard to "visualize" the existence of such subgroups,
although I could probably do it if I put in a lot of effort. Is this
visualization of all theorems the essence of mathematical maturity? If not,
what is considered to be mathematical maturity, really? Can math professors and
twenty-something-year-old grad students visualize theorems like Sylow's
theorem, or in general almost any new theorem to which they are introduced and
had proven to them, with ease? That I might be fatally inferior has shook me up
for the past couple of days, and I appreciate a helping hand from

P.S. I am not trolling.

---

Seraph-sama
16/m, so don't call me "sera" or nothin'

### Michael Cohen

Aug 14, 1999, 3:00:00â€¯AM8/14/99
to
Like all kinds of maturity, people mathematically mature at different ages, and
some never do. Don't worry about mathematical maturity; you reached it early.
(By the way, the next stage will be to start formulating your own conjectures and
trying to prove them.)

Seraph-sama (serap...@aol.com) wrote:
: I have been told before that although I can do higher-level mathematics like

--
Michael P. Cohen home phone 202-232-4651
1615 Q Street NW #T-1 office phone 202-219-1917
Washington, DC 20009-6310 office fax 202-219-2061
mco...@cpcug.org

### Herman Rubin

Aug 14, 1999, 3:00:00â€¯AM8/14/99
to
In article <19990814163145...@ng-da1.aol.com>,
Seraph-sama <serap...@aol.com> wrote:

It is my quite considered opinion, based on my own observations
as a student and teacher and researcher, that everything you
have been told is completely wrong. The time to learn to
understand mathematical reasoning, even for the average person
capable of doing it, is even before the teens; few have the
opportunity (I did not).

Geometric visualization is NOT that important, or even that much
used in situations where one would think it is at first glance.
There is an intuition about proofs, and what will work, which has
nothing to do with "visualization". It is relatively easy for
some of is to think abstractly, and this is really what it takes.

Considering the success rate, those who can do research are
extremely special. The ones who can overcome the hobbles placed
by those who have students memorize, compute, and visualize are
somewhat special. When it ocmes to understanding mathematics,
use whatever works, and visualization is highly overrated.

As for age, we will get more good mathematicians when we teach
them abstract mathematics before teens; I would like to see
people like you learn abstract algebra much earlier.

This does not mean that one should never use visualization, but
to attempt to rely on it is stultifying. There are few uses of
it in abstract algebra and set theory, and not that much in
probability theory and combinatorics. Even analysis often needs
to ignore visualization.

> I have been told before that although I can do higher-level mathematics like
>abstract algebra at my age (16), I will not be able to study it the *right way*
>unless I'm extremely special. It was nearly impossible, I was told, to have
>mathematical maturity at my age, which has made me stop and think about it. I
>never ran across any definition of mathematical maturity, but I always assumed
>I had it. I assumed it meant the understanding and appreciation of the nature
>of proof and mathematical concepts, and an understanding of the whole of
>mathematics in general, and what it actually is. The only thing it could
>possibly be is to be able to visualize and understand a definition or theorem
>right after it has been defined/proved. For example, last night I read the
>proof of the existence of Sylow p-subgroups, and although the proof was very
>easy to follow, it is hard to "visualize" the existence of such subgroups,
>although I could probably do it if I put in a lot of effort. Is this
>visualization of all theorems the essence of mathematical maturity? If not,
>what is considered to be mathematical maturity, really? Can math professors and
>twenty-something-year-old grad students visualize theorems like Sylow's
>theorem, or in general almost any new theorem to which they are introduced and
>had proven to them, with ease? That I might be fatally inferior has shook me up
>for the past couple of days, and I appreciate a helping hand from

>P.S. I am not trolling.

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

### Chris Hillman

Aug 14, 1999, 3:00:00â€¯AM8/14/99
to

Seraph-sama (serap...@aol.com) wrote:

> last night I read the proof of the existence of Sylow p-subgroups, and
> although the proof was very easy to follow, it is hard to "visualize"
> the existence of such subgroups, although I could probably do it if I
> put in a lot of effort.

Some books discuss the constructive proof of Wielandt.

> Is this visualization of all theorems the essence of mathematical
> maturity?

Not everyone thinks visually, but of course you need some way of
understanding the basic facts of life in any theory you hope to master.

> If not,what is considered to be mathematical maturity, really?

Having mastered enough of the most central concepts of modern mathematics
to be able to read well-written textbooks on a new subject with a high
level of understanding (not neccessarily at first reading, and assuming
one is doing problems as one reads).

> Can math professors and twenty-something-year-old grad students
> visualize theorems like Sylow's theorem, or in general almost any new
> theorem to which they are introduced and had proven to them, with
> ease?

In my experience of observing the most talented among my peers, some
people do have uncanny instant insight in some form suited to the
structure present (e.g. a "number sense" in number theory, or accurate
visualization in differential geometry). A very few people have the gift
of not needing to understand things concretely (e.g. in terms of
understanding very well a handful of simple but nontrivial examples of the
theory at hand), but are able to work comfortably with abstractions
without need of examples.

> That I might be fatally inferior has shook me up for the past couple
> of days, and I appreciate a helping hand from mathematicians and

According to the above, most mathematicians would probably count as only
modestly talented (visualize ;-/ me raising my own hand here). I think
most are simply generic smart people with the same ability to think of
abstract structure in some concrete way as any other generic smart person
would naturally have (but this latent talent would only be developed by
formal mathematical training, except in the rare case of born
mathematicians gifted with extraordinary talent far beyond what is
possessed by the generic smart person). Oh, but I should add that
according to this guess, the average working mathematician might be gifted
more with generic high intelligence than with purely mathematical or
musical talent, but they do tend to be far, far, far more hardworking than
the average generic smart person! Some people would draw a distinction
between people who do very well on the Putnam exam and people who don't,
but have sufficient creativity, work ethic, and raw intelligence to
partially make up for the lack.

Perhaps someone should say that history shows that only mathematical
geniuses make really important contribution to mathematics. But I'd add
that even "minor" mathematical achievements usually require a fascinating
and completely fulfilling intellectual journey, fully worth the devotion
of many years of searching and thinking. I have never discovered anything
as important as Poincare, but I'm not sure he had all that much more fun
discovering things than most of us have had on our best days :-) Kind of
like a Zen version of not expecting to compete in a swimming event in the
Olympics, but training hard for the fulfillment of becoming the best
swimmer you can be, consistent with taking the time to splash around and
generally having fun in the water.

And if you decide at some point along the way that you'd rather spend more
time splashing than swimming, another benefit of graduate school is that
math grad students are remarkably inventive of ways of intriguing ways of
wasting time :-/

IOW, the sooner you can decide to just have fun learning math (and the
sooner you recognize that this involves a goodly amount of pain or effort
for most students), the more likely you are to have an enjoyable time in
and you'll have plenty of opportunity to learn about things like system
administration on the side, if you like working with computers. (It's a
good idea to major in an "applicable area" or, if you must major in number
theory (say), to pick up some "employable skills" on the side, and I think
most departments now recognize these practical facts of life.)

I'd add that in my experience there is a strong positive correlation
between enjoying grad school (particularly thesis research) and somehow
finding a way to pursue a mathematical career; e.g. you are likely to
spend a lot of time teaching lower level math courses, and if you are good
at that, you are already good at one of the most important things working
mathematicians do, and if you are good at research and writing up your
discoveries, you are good at two of the other most important things that
working mathematicians do, in which case, despite the generally very
competitive job market, you probably have a fair shot of finding a tenure
track job in a math department.

### Steve Leibel

Aug 14, 1999, 3:00:00â€¯AM8/14/99
to
In article <19990814163145...@ng-da1.aol.com>,
serap...@aol.com (Seraph-sama) wrote:

> I have been told before that although I can do higher-level mathematics like
> abstract algebra at my age (16), I will not be able to study it the
*right way*
> unless I'm extremely special.

Well Galois had more mathematical maturity at 19 than the next two hundred
years of algebraists, but he didn't have enough social maturity to keep
from getting shot to death in a duel. So I'd advise you to stay away from
politics and women, like Galois should have.

> For example, last night I read the

> proof of the existence of Sylow p-subgroups, and although the proof was very
> easy to follow, it is hard to "visualize" the existence of such subgroups,
> although I could probably do it if I put in a lot of effort.

If you can "easily" follow the Sylow theorems you've got plenty of
mathematical maturity in my opinion.

> That I might be fatally inferior has shook me up
> for the past couple of days, and I appreciate a helping hand from
> mathematicians and "real" grad students.
>

Well the purpose of grad school is exactly to make most of the people
there feel fatally inferior. That has nothing to do with mathematical
maturity, and everything to do with separating out the amateurs from the
pros.

My advice to you would be to ignore these naysayers and study what you like.

Steve L

### Keith Ramsay

Aug 15, 1999, 3:00:00â€¯AM8/15/99
to
In article <19990814163145...@ng-da1.aol.com>,

serap...@aol.com (Seraph-sama) writes:
| I have been told before that although I can do higher-level
|mathematics like abstract algebra at my age (16), I will not be able
|to study it the *right way* unless I'm extremely special. It was

|nearly impossible, I was told, to have mathematical maturity at my

Mathematical maturity is a catchall phrase describing those aspects
of mathematical ability which do not have to do with knowing specific
mathematical facts and techniques of computation. By ability I don't
mean innate ability so much as learned ability, although certainly
people differ in how readily they acquire mathematical maturity.

When I was your age, what I lacked mostly was *not* mathematical
maturity, but a good plan of study. There is a sort of plan available
in schools, but it generally is a one-size-fits-all plan. There may be
certain alternative plans, but in lots of places they don't seem very
well prepared for really fitting them to the individual student. You
should try to figure out what sort of study is good for you.

I also worried about my abilities and what to do with them, but I have
been finding (over a long period of time) that worries about ability
before getting started _doing_ aren't very useful. Once you are started
doing mathematics, the question really is, "How is it going?" And there
are many different kinds of things which might be relevant to that. It
is partly a matter of how mathematics fits into your life overall.

So how is it going? :-)

For a long time, boredom was a big problem. Have you been feeling
quite bored at times? Boredom seems to be pretty common with teenagers
though some of us were more bored than others. It's much better for
you if you can do activities which leave you feeling engaged and
active. [This is not just about mathematics, either. Mathematics was
part of it, but I was naive in thinking that the key to avoiding
boredom was to do advanced mathematics. That can help, but it's just
one piece of the puzzle. The people whose only strategy for avoiding
boredom is to do more work tend to get into trouble later on in life.
I mean, you can try to be Erdos, but even if you were to have the
ability to be Erdos, there are various ways in which such a lifestyle
can fail, and leave you depressed or worse. Suppose you put all your
eggs in one basket, and then somebody comes along and smashes it?]

One issue for me was how to advance so as not to get stuck being bored
in math classes. For a long time I was ahead of the class I was taking,
in the sense that the material presented was almost all things I knew
already. It was worse also because the system was set up so that we did
a lot of review anyway (and I'm not sure what they ought to have done
differently, since many of the other students appeared really to have
forgotten what they learned, and needed refreshing). One year, in class
I learned a definition of "ray" I hadn't known, and that was it.

I was too shy as a teenager to approach the adults in charge of my
education with this sort of issue, although I wish I had. There were
two subjects which I did manage to get a jump on: calculus and abstract
algebra. I learned enough calculus on my own to be placed into a class
in the middle of the term, and when I got to college, they gave us a
math placement test, which placed me out of calculus. So, one question
you might ask is, do you have your "AP calculus" solidly under your
belt? (I haven't been watching your posting enough to have a good idea
of what you seem to know and not know.) This is a fairly well-defined
body of material; there are standard tests which will probably tried
on you; hence you can basically find out what you need to know to place
yourself out of it (assuming this is something you want to do).

When I got to college, I had a class with Professor Sally, who proposed
learned some abstract algebra on my own, and now we did something a
bit more organized. We had a textbook, Herstein's _Topics in Algebra_.
"Everyone should have the opportunity to do all the exercises in
Herstein." :-) He had me study my way through it doing the exercises.
Once he was satisfied I was picking it up adequately, he concluded
year, I should take the graduate algebra course. This all went well.
of yours), as I was able to pass the qualifying exam my first semester
without having to take any "supplemental" oral exams, which meant not
having to take any courses for a grade.

You might be amused to know that on the first day in that graduate
algebra course, we were asked to write our names and whether we had
been exposed to three theorems: the Jordan-Holder theorem, Sylow's
theorem, and something else. Not whether we knew them, but whether
we had seen them. A lot of the students hadn't seen Sylow's theorem.

There are other issues than advancing through the educational system,
of course. Why do you do mathematics? If you are reading mathematics,
simply because you enjoy it, you shouldn't let worries about whether
you're doing it "the right way" deter you, any more than one would let
concerns over whether one is listening to music "the right way" keep
one from listening to music. It is possible to appreciate either one
more or less deeply, but no reason to deprive yourself.

|I never ran across
|any definition of mathematical maturity, but I always assumed I had
|it. I assumed it meant the understanding and appreciation of the
|nature of proof and mathematical concepts, and an understanding of
|the whole of mathematics in general, and what it actually is.

That's a decent definition. When people mention "mathematical
maturity" as a prerequisite to reading a book, they often have in
mind that although the book does not *require* prior knowledge of
material a student doesn't know, it still might be difficult for them
to read, if they are not accustomed to the process of studying
mathematics. The author wants to brace the would-be reader for this
possibility. If you don't find a book too hard for your purposes, then
usually this isn't a problem.

One cornerstone to "Mathematical maturity" is a kind of acquired knack
for reading and doing proofs. I hardly would consider it unlikely that
a 16 year old would have this. I agree with Rubin that it would make
sense to start earlier with it than we do. When we had a Euclidean
geometry class with proofs, for students roughly your age (something
which students have told me is not so common anymore), we at least had
a visible entryway for getting into the swing of proving things, and
not just a few exceptional ones of us. Some people who had always found
arithmetic rather dull will report later in life that they found that
one course a refreshing change. It's something a lot more people could
probably appreciate than now do. Finally, they got a taste of what we

On the other hand, some mathematics students don't get this knack
of proofs until a number of years later, or barely at all.

|The only thing it could possibly be is to be able to visualize and
|understand a definition or theorem right after it has been

|defined/proved. For example, last night I read the proof of the

|existence of Sylow p-subgroups, and although the proof was very easy
|to follow, it is hard to "visualize" the existence of such subgroups,

|although I could probably do it if I put in a lot of effort. Is this

|visualization of all theorems the essence of mathematical maturity?

I think more to the point is how well you followed the proof. It's
possible (and a lot of people have this problem at least sometimes)
to read along in a book, or attend a lecture, and say to oneself, "Ok,
sure, uh huh, no problem, this makes sense," only to find later that
there was actually a lot that one didn't follow or didn't follow
correctly. If you want to tell whether you've understood the Sylow
theorems, there are various things to do. Try writing out a proof
yourself so you can see how much you remember of it. Textbooks have
exercises about the Sylow theorems, and you can try picking some. Try
one which looks less than obvious, and work it out. Try to extend the
theorems a little bit. Are there cases where a group of order n is
guaranteed to have a subgroup of order k, where k isn't a prime power
divisor of n? (Well, of course. But try investigating it.)

*Some* mathematicians barely use visualization at all, although I
think ordinarily we use some. It's often not what you would think of
as a direct representation of a mathematical object, but just kind of
a mental doodle helping to hold things together, much like people will
think verbal representations of ideas to themselves as they think.

|If not, what is considered to be mathematical maturity, really? Can

|math professors and twenty-something-year-old grad students visualize
|theorems like Sylow's theorem, or in general almost any new theorem

|to which they are introduced and had proven to them, with ease? That

|I might be fatally inferior has shook me up for the past couple of
|days, and I appreciate a helping hand from mathematicians and "real"

I would recommend not worrying about whether you're inferior,
especially at such a point in your life, let alone if you're already
reading group theory. But even if at this point you didn't seem
especially talented, now is not the time to think your options are
closing in on you.

There are a number of mathematicians who didn't have any particular
interest in mathematics until they were several years older than you
are (and a few who got into it later in life). Some of us just
gradually developed "maturity" over a period of years, some very
early, some later on. There are some who have a kind of epiphany
partway through their undergraduate work, and "maturity" arrived
relatively quickly. Paul Halmos writes:

The day when the light dawned-- I remember the circumstances and
the scene-- Ambrose and I were talking a seminar room on the second
floor of the mathematics building, and something he said was the
last candle that this blind camel needed. I suddenly understood
epsilons and limits, it was all clear, it was all beautiful, it was
all exciting. [...] It all clicked and fell into place. I still had
everything in the world to learn, but nothing was going to stop me
from learning it. I just knew I could. I had become a mathematician.

I wouldn't suppose you to be inferior from any difficulties in
visualizing you might have.

I also hope that you realize that lack of outstanding mathematical
ability isn't "fatal" at all. I've known some very good mathematicians,
certainly more productive than I have been, people who got degrees
from first rate universities and continued to publish good work
have gotten careers as academics, who did not, because conditions were
sufficiently poor. Some who could have stayed left because they didn't
enjoy the conditions they were working in. One guy I know decided he
didn't like doing work which was so little appreciated by others. He
could solve problems, but nobody seemed to care.

So even if you turn out to be quite good, there is no guarantee that
you will be paid to do mathematical research. On the other hand, even
if you were not very good at it at all, there isn't anything worse
that is going to happen to you than having to find some other line of
work. Just think about the happiest people you know of-- many of them
are non-mathematicians, aren't they? :-) Mathematics can be a very
happy way of life, but you can be happy without it too.

Actually, to be honest, if your talents are not as great as you had
thought, something worse can happen to you than merely not being a
mathematician, but only if you react badly.

I heard of a fellow who was diagnosed with manic depression when
(during a manic episode) he proved a result in category theory, but
fell into a deep depression when he discovered it was a good result,
but was already known. This is not an issue of absolute ability or
lack of ability, but of what significance you assign to it. Taniyama
is supposed to have become despondent when he found he was unable to
do something like nonabelian class field theory (which has still only
partially been done by anybody). Everybody has to deal with the
existence of things that they can't do. Enjoy doing mathematics, and
don't worry about what you won't be able to do.

Keith Ramsay

### Pertti Lounesto

Aug 15, 1999, 3:00:00â€¯AM8/15/99
to
Seraph-sama wrote:

> I was told, to have mathematical maturity.

Mathematical maturity, or rather lack of it, is a label attached
on graduate students, who are interested in other topics in
mathematics than the topic of the dominating research group
of a mathematics department.

### Stephen Montgomery-Smith

Aug 15, 1999, 3:00:00â€¯AM8/15/99
to
When I was 16 years old, I also was very interested in mathematics,
and I faced a lot of resistance from those around me, especially
teachers.

The best way to aquire this "math maturity" is to learn as much
math as possible. Instead of struggling to get into math classes,
I would do precisely what you are doing, and read books, and learn
by yourself. I would get good book recommendations from
sci.math. If you get stuck on something, well you have sci.math

I don't think that anyone truly "visualises" Sylow subgroups. I
mean, some people explain it by drawing diagrams, but the diagrams
don't really mean anything. I think that if you understand the
proof, then you already have "math maturity" - indeed if you signed
up for a graduate class, you would find that you had more of it than

good way to get math maturity is to do the exercises in the math
a bit contrived. I mean, they are fun to do, but you don't get
to learn "real math" that way. If you find the exercises too
easy (which I suspect that you do), you should seek out a harder
text book.

I would put a post on sci.math, explaining which books you are

Let me give you one book you might enjoy - Graph Theory by Bela Bollobas.
It is a branch of math that requires little background knowledge. His
exercises range from easy to very very hard. It is quite easy to find
unsolved problems in graph theory, which you have a chance as anyone of
solving. (And they will be much easier than the twin primes conjecture.)

Finally, if you really find olympiad problems easy, well you must be
extremely special. I myself find them very hard. The ability to do these
problems does not improve with "math maturity". While being good at such
problems is not a guarantee of math success, it certainly is a good start.

Seraph-sama wrote:
>
> I have been told before that although I can do higher-level mathematics like
> abstract algebra at my age (16), I will not be able to study it the *right way*

> unless I'm extremely special. It was nearly impossible, I was told, to have
> mathematical maturity at my age, which has made me stop and think about it. I

> never ran across any definition of mathematical maturity, but I always assumed
> I had it. I assumed it meant the understanding and appreciation of the nature
> of proof and mathematical concepts, and an understanding of the whole of

> mathematics in general, and what it actually is. The only thing it could

> possibly be is to be able to visualize and understand a definition or theorem
> right after it has been defined/proved. For example, last night I read the
> proof of the existence of Sylow p-subgroups, and although the proof was very
> easy to follow, it is hard to "visualize" the existence of such subgroups,
> although I could probably do it if I put in a lot of effort. Is this

> visualization of all theorems the essence of mathematical maturity? If not,

> what is considered to be mathematical maturity, really? Can math professors and
> twenty-something-year-old grad students visualize theorems like Sylow's
> theorem, or in general almost any new theorem to which they are introduced and
> had proven to them, with ease? That I might be fatally inferior has shook me up
> for the past couple of days, and I appreciate a helping hand from
> mathematicians and "real" grad students.
>

> P.S. I am not trolling.
>

> ---
>
> Seraph-sama
> 16/m, so don't call me "sera" or nothin'

--

Stephen Montgomery-Smith ste...@math.missouri.edu
307 Math Science Building ste...@showme.missouri.edu
Department of Mathematics ste...@missouri.edu
University of Missouri-Columbia
Columbia, MO 65211
USA

Phone (573) 882 4540
Fax (573) 882 1869

http://math.missouri.edu/~stephen

### Main Night

Aug 16, 1999, 3:00:00â€¯AM8/16/99
to
>There are few uses of
>it [visualization] in abstract algebra and set theory, and not that much in
>probability theory and combinatorics.

in set theory I made up my own visualizations. I visualized the U as like a
cup that you fill with two different things, so that you get the total union of
ingrediants. I visualized the (upside down U) as a scooper that scoops up a
bunch of different things, and anything that only gets scooped up once (ie is
only in one of the two sets) falls back out.

In combinatorics i wasn't as creative as in set theory, but i visualized the
formulae themselves (i have [very] slightly photographic memory).

I havent taken abstract algebra or probability, but from the "everyday"
probability i've learned, I know i hate it so far.

Sam Main Night Alexander
"Beware, beware,
His flashing eyes, his floating hair,
Weave a circle 'round him thrice,
for he on honeydew hath fed,
and drunk the milk of Paradice!"

### Dave L. Renfro

Aug 16, 1999, 3:00:00â€¯AM8/16/99
to
I've read a lot (over a thousand) of the old
posts to the discussion groups posted at
<http://forum.swarthmore.edu/discussions>
during the past month or so, and I must say
that some of the best I've come across are the
replies during the past week to Seraph-sama's
various questions on being a mathematician.

I'm calling these posts to the attention of
some of my best math students from the past
three years (during which time I was a teacher
at a selective state-run boarding math-science
type high school)--students now majoring in
mathematics and who most likely will continue

Just thought you'd like to know your efforts are
appreciated ...

And, to Seraph-sama, "mathematical maturity" is a relative
term, but unless one is using the term with extremely
high standards in mind, I'm sure you have it. Here's
a test for it that was used (mostly in jest) back during
my undergraduate days: As soon as all the "it is clear that ...",
"a routine verification shows ...", etc. type statments
in Serge Lang's ALGEBRA (his graduate level text) really
are (to you) as Lang claims, then you've reached mathematical
maturity. Lang's book was used for a two semester
graduate algebra sequence at Univ. of North Carolina
and, as I recall, one of the goals for this course was to
raise their beginning graduate students' maturity to the
point that they could successfully read Lang on their
own around midway into the second semester.

### Axel Harvey

Aug 16, 1999, 3:00:00â€¯AM8/16/99
to
On Sat, 14 Aug 1999, Steve Leibel wrote:

> Well Galois had more mathematical maturity at 19 than the next two hundred
> years of algebraists, but he didn't have enough social maturity to keep
> from getting shot to death in a duel. So I'd advise you to stay away from

> politics and women, like [ sic ] Galois should have.

This is what consumer advocates would call a tied sale: the young man
wants advice on mathematics and we take the opportunity to sell some
old-time morality along with it. If Galois had a social conscience and
was successful with the ladies, then it seems he had more general
maturity than most men then or now.

> My advice to you would be to ignore these naysayers and study what you like.

Yep.

### David Petry

Aug 16, 1999, 3:00:00â€¯AM8/16/99
to
In general, the essence of "maturity" is this:

RESPECT THE RULES OF THE GAME.

Keep in mind that in any social context, there
are typically many games being played. If
someone calls you immature, he is likely
really saying "quit playing your game, and
start playing my game".

And it should go without saying that you are
free to choose what game you play, regardless
of what other people may tell you.

### Herman Rubin

Aug 16, 1999, 3:00:00â€¯AM8/16/99
to
In article <19990815204827...@ng-ft1.aol.com>,

Main Night <main...@aol.com> wrote:
>>There are few uses of
>>it [visualization] in abstract algebra and set theory, and not that much in
>>probability theory and combinatorics.

>in set theory I made up my own visualizations. I visualized the U as like a
>cup that you fill with two different things, so that you get the total union of
>ingrediants. I visualized the (upside down U) as a scooper that scoops up a
>bunch of different things, and anything that only gets scooped up once (ie is
>only in one of the two sets) falls back out.

So what do you have here? I do not see the concept illustrated
well; truth tables, and even Venn diagrams, do a much better job,
and truth tables generalize to more complicated cases.

>In combinatorics i wasn't as creative as in set theory, but i visualized the
>formulae themselves (i have [very] slightly photographic memory).

Memorizing formulas does not help in understanding.

>I havent taken abstract algebra or probability, but from the "everyday"
>probability i've learned, I know i hate it so far.

Probability is usually poorly taught; "equally likely" tends to
confuse students. Combinatorics is a TOOL in probability, which
needs more flexibility.

Abstract algebra is basic, and is relatively easy if you look
for the abstract ideas as abstractions, not as generalizations.
Nothing taught before, except the ideas of variables, proofs,
and possibly some of the number systems, helps to any great
extent, and can hinder.

### Robin Chapman

Aug 17, 1999, 3:00:00â€¯AM8/17/99
to
In article <mm9pgi...@forum.swarthmore.edu>,

And I suppose the next stage of MM is to be confident
when you are right and Lang is wrong :-)

--
Robin Chapman
http://www.maths.ex.ac.uk/~rjc/rjc.html
"They did not have proper palms at home in Exeter."
Peter Carey, _Oscar and Lucinda_

Sent via Deja.com http://www.deja.com/
Share what you know. Learn what you don't.

### Pertti Lounesto

Aug 17, 1999, 3:00:00â€¯AM8/17/99
to
Robin Chapman wrote:

We can refine the scale of MM and say that the next stage
is to find out the books where Lang has copied his text.

We can continue the scale of MM and say that the next
stage is when you develop Lang's presentation further.

### Main Night

Aug 17, 1999, 3:00:00â€¯AM8/17/99
to
>We can continue the scale of MM and say that the next
>stage is when you develop Lang's presentation further.

This theory, though interesting, is flawed, for if we adopt it, it forever
places an upper bound on the MM of Lang himself!

### Torkel Franzen

Aug 17, 1999, 3:00:00â€¯AM8/17/99
to
serap...@aol.com (Seraph-sama) writes:

> I never ran across any definition of mathematical maturity, but I
> always assumed I had it.

Speaking as a teacher, I would say that the mathematically mature
student is one who, when faced with the problem "Prove that
every fnorgle is a glunk" starts by looking at the definitions of
"fnorgle" and "glunk".

(The remaining students just look blank or flap their arms at
random.)

### Lee Rudolph

Aug 17, 1999, 3:00:00â€¯AM8/17/99
to
Pertti Lounesto <Pertti....@hut.fi> writes:

>Robin Chapman wrote:
>
>> And I suppose the next stage of MM is to be confident
>> when you are right and Lang is wrong :-)
>
>We can refine the scale of MM and say that the next stage
>is to find out the books where Lang has copied his text.
>

>We can continue the scale of MM and say that the next
>stage is when you develop Lang's presentation further.

Who was it who, in a review of which book by Lang, wrote
approximately "A new book by Lang makes ever other book
on the subject look like a pedagogical breakthrough"?

Lee Rudolph, who lived in the same apartment building as
Lang in the summer of 1968, and who was told--but did not
see for himself--that Lang had three or four typewriters
scattered about on different desks, and moved from one to
the next while working concurrently on an equal number
of pairwise-distinct books

### jdo...@my-deja.com

Aug 17, 1999, 3:00:00â€¯AM8/17/99
to
i think it ought to be clear by now that "mathematical maturity" means
than well-written ones as chris hillman suggested).

### Kyle R. Hofmann

Aug 17, 1999, 3:00:00â€¯AM8/17/99
to
On Tue, 17 Aug 1999 13:05:30 GMT, jdo...@my-deja.com <jdo...@my-deja.com> wrote:
> i think it ought to be clear by now that "mathematical maturity" means
> the ability to read and understand _badly_ written math books (rather
> than well-written ones as chris hillman suggested).

I think it also implies dissatisfaction with badly written math books and
an appreciation for the well written ones.

--
Kyle R. Hofmann <rhof...@crl.com> | "...during the years between 960 and
1000 there was great activity in the production of homilies ... [ The
Blickling Homilies ] voice the almost universal belief that the world would
end in the year 1000." -- The Concise Cambridge History of English Literature

### Pertti Lounesto

Aug 18, 1999, 3:00:00â€¯AM8/18/99
to
"Kyle R. Hofmann" wrote:

> jdo...@my-deja.com <jdo...@my-deja.com> wrote:
> > i think it ought to be clear by now that "mathematical maturity" means
> > the ability to read and understand _badly_ written math books (rather
> > than well-written ones as chris hillman suggested).
>
> I think it also implies dissatisfaction with badly written math books and
> an appreciation for the well written ones.

But whether a book is written badly or well depends on the level
of abstraction and background information of the reader. Thus,
we conclude that "mathematical maturity" is not only a function of
ones own previous cognitive structures, but also of the environment:
are the existing structures sufficient to assimilate new information
offered without substantial restructuring.

### Douglas J. Zare

Aug 19, 1999, 3:00:00â€¯AM8/19/99
to
Axel Harvey <a...@cam.org> wrote:
>On Sat, 14 Aug 1999, Steve Leibel wrote:
>
>> Well Galois had more mathematical maturity at 19 than the next two hundred
>> years of algebraists, but he didn't have enough social maturity to keep
>> from getting shot to death in a duel. So I'd advise you to stay away from
>> politics and women, like [ sic ] Galois should have.
>
>This is what consumer advocates would call a tied sale: the young man
>wants advice on mathematics and we take the opportunity to sell some
>old-time morality along with it. If Galois had a social conscience and
>was successful with the ladies, then it seems he had more general
>maturity than most men then or now.

Don't be too hard on Galois. He was just proving that not all real-life
problems can be solved by radicals.

>> My advice to you would be to ignore these naysayers and study what you like.
>
>Yep.

One should worry about whether one is learning to read between the lines
and whether one is prepared to appreciate mathematics on a deep level
rather than just solve the exercises or learn the material in a course.
The Sylow theorems themselves are not so important that every
mathematician should have learned them, but rather they and their
consequences provide wonderful examples of what one can hope for when
thinking albegraically. (It is a fun exercise to try to see how to
organize an undergraduate mathematics class for hypothetical
mathematicians who learned everything except the intrinsic material.
Sometimes I meet these people, since different countries emphasize
different subjects.)

On the other hand, age alone does not prevent one from having mathematical
maturity, and advice to the contrary frequently stems from the hope that
strengths must be balanced by weaknesses. Indeed, if it were not so
damaging, it would be amusing how offended and upset many people are when
they encounter the precocious.

Douglas Zare
za...@math.columbia.edu

### John R Ramsden

Aug 21, 1999, 3:00:00â€¯AM8/21/99
to
On 17 Aug 1999 06:48:54 -0400, lrud...@panix.com (Lee Rudolph) wrote:

>Pertti Lounesto <Pertti....@hut.fi> writes:
>
>>Robin Chapman wrote:
>>
>>> And I suppose the next stage of MM is to be confident
>>> when you are right and Lang is wrong :-)
>>
>>We can refine the scale of MM and say that the next stage
>>is to find out the books where Lang has copied his text.
>>
>>We can continue the scale of MM and say that the next
>>stage is when you develop Lang's presentation further.
>
>Who was it who, in a review of which book by Lang, wrote
>approximately "A new book by Lang makes ever other book
>on the subject look like a pedagogical breakthrough"?

Possibly Andre Weil, who I gather didn't see eye to eye with Lang.
Apparently Lang wrote (and circulated?) an unpublished manuscript
detailing various less than complementary things about Weil, but
never having read it I can't be more specific.

>Lee Rudolph, who lived in the same apartment building as
>Lang in the summer of 1968, and who was told--but did not
>see for himself--that Lang had three or four typewriters
>scattered about on different desks, and moved from one to
>the next while working concurrently on an equal number
>of pairwise-distinct books

---

John R Ramsden # "No one who has not shared a submarine
# with a camel can claim to have plumbed
(j...@redmink.demon.co.uk) # the depths of human misery."
#
# "Adventures of a U-boat Commander".

### Keith Ramsay

Aug 21, 1999, 3:00:00â€¯AM8/21/99
to
In article <37be3b8d...@news.demon.co.uk>, j...@redmink.demon.co.uk (John R

Ramsden) writes:
|Apparently Lang wrote (and circulated?) an unpublished manuscript
|detailing various less than complementary things about Weil, but
|never having read it I can't be more specific.

Perhaps you are thinking of Lang's file where he critiques the calling
of the modularity conjecture "the Weil conjecture"? People now usually
call it Taniyama-Shimura, or occasionally Taniyama-Shimura-Weil.

Someone I knew gave a lecture with Lang in the audience in which he
accidentally referred to a curve satisfying the conjecture as a "Weil
curve", and Lang reproached him for that, giving him a copy of the
file.

Toward the end of the file, there is a letter from Weil in which he