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Aug 14, 1999, 3:00:00â€¯AM8/14/99

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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.

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'

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.)

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

Aug 14, 1999, 3:00:00â€¯AM8/14/99

to

In article <19990814163145...@ng-da1.aol.com>,

Seraph-sama <serap...@aol.com> wrote:

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

>mathematicians and "real" grad students.

>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

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

> "real" grad students.

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

grad school. Once you get in, you don't need to worry much about grades,

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.

Chris Hillman (who has dreadful qualifications to be a Graduate Advisor)

Home Page: http://www.math.washington.edu/~hillman/personal.html

Aug 14, 1999, 3:00:00â€¯AM8/14/99

to

In article <19990814163145...@ng-da1.aol.com>,

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

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

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

|age, which has made me stop and think about it.

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

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

|age, which has made me stop and think about it.

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

an interesting plan with regard to abstract algebra. I had already

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

that instead of taking the undergraduate algebra course the following

year, I should take the graduate algebra course. This all went well.

It also was helpful in graduate school (to answer a previous posting

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

like about mathematics.

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"

|grad students.

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

steadily after their degrees, who I would have assumed would readily

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

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.

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.

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

as a place to ask.

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

the other grad students.

You asked in another thread about math problems. I think that a

good way to get math maturity is to do the exercises in the math

books that you are reading. Math Olympiad problems tend to be

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

reading, what you already know, and ask for good book recommendations.

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

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,

close your eyes in holy dread,

for he on honeydew hath fed,

and drunk the milk of Paradice!"

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.

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

mathematics into graduate school.

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.

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.

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.

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.

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.

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.

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.

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.

>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!

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.)

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

Aug 17, 1999, 3:00:00â€¯AM8/17/99

to

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).

the ability to read and understand _badly_ written math books (rather

than well-written ones as chris hillman suggested).

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 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

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.

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.

>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

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."

#

# Ritter von Haske

# "Adventures of a U-boat Commander".

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.

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

remarks, go ahead and put this through your photocopy machine

too.

Keith Ramsay

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