Other Should I Become a Mathematician?

[teleport]

OMG toss a coin n times and count the number of combinations such that successive heads never appear. What do you get? Fibonaci! Yep, that's sharp. Does anyone know any other kool/crazy/magical thing like this?

 

[Werg22]

I think people confuse the two a lot. Especially when you first meet a person.

The one's I met generally have allot of pretence in their voice and seem almost vexed that you address them.

 

[teleport]

Well the majority of the ones I know in my class are just genuinely funny. My calc prof has a wild sense of humour, but terrifies everyone by asking surprise concept questions in class. I wonder if this is somehow related to the way pure math is. Don't you have the feeling sometimes that pure math is just a very hard infantile game? Combinatorial math reminds me a lot of this for example. What can I say, its just fun.

But math usually takes a lot more effort than many other disciplines so this can take its toll on the person as well. No math people like that around here that I've seen. There are a lot of stupid stereotypes out there. I guess they feel insecure that good looking people can be good in math too. But then again, most just feel insecure in front of good looking people, like if they own the beans...

 

[PowerIso]

The one's I met generally have allot of pretence in their voice and seem almost vexed that you address them.

Well, if that's the case, I suggest you meet new nones :).

Anyway, when do people generally start studying for the Math GRE?

 

[mathwonk]

if you go to a top school, the gre may be less important. but probably nowadays it is wise to take it as a way of comparison with others.

when i was a student, and i realize this is somewhat out of date, as far as i knew no one studied at all for the gre. it was assumed that if you had learned the material in your courses that was sufficient.

i hope i do not come across merely as a wiseguy, but there is more than a little truth to the idea that if you just learn the subject, you will do well on the test over that subject.

hearing myself say these things, i am looking over my shoulder for the guys coming to burn me at the stake, or at least with a straightjacket. so few people believe today that merely learning the material will suffice. everyone has bought into the BS that you need a leg up, and advantage, some special test Kaplan prep, or edge.

this is the kind of thinking that used to be restricted to confidence men and card cheaters.

tests have one purpose, to determine whether you have learned the stuff. not whether you paid some cynical, ignorant hustler $500 to prepare you.

this is so far from being the norm today that i expect some counter messages to follow this immediately, from young people who think they know better than i do, how to succeed. to them, bless you, maybe you do, but please spend a little time thinking about what i said.

after all i have achieved most of what many of you are hoping to obtain.
could it be that my old fashioned advice is not insane after all? try it, it can't hurt you.moral: if you learn the subject, someone will notice and appreciate it, and you WILL get a position.

 

[tronter]

and the means of learning the subject material do not matter (i.e self study)?

 

[J77]

Mathwonk, you won't be burned at the stake for saying the truth.

However, there is the problem that to get into PhD positions in the US, you need the grades from the tests -- at least, that's how I perceive it (am I right?).

As far as I'm concerned, this is completely wrong. Like you say, you don't necessarily have to learn the subject to pass the tests; you can educate yourself to pass a test rather than learn the subject.

Thankfully, it doesn't work like this over here in Europe -- at least, not yet.

I would rather meet the candidate in person and my judgement as to whether or not I'd take them as a student would be based somewhat on their grades (NOT totally) but mainly on their enthusiasm for the research topic and general personality; it's much easier to work with a personality than someone who talks like they're reciting a textbook!

 

[J77]

and the means of learning the subject material do not matter (i.e self study)?

You need to have some grades on paper -- so purely self-study wouldn't work -- but, conversely, you don't need straight A's, a 1st, or 4.0s, afaic (this is the top grade, right?)

 

[pivoxa15]

Mathwonk, can you recall any academics who did poorly in their undergrad studies? If so how poorly? And how did they make it?

 

[mathwonk]

since you ask, i myself had a 1.2 gpa for the first year or so (out of4.0), got kicked out of school, worked in a factory for a year, got back in, did ok and got in grad school.

i never took gre, but as a junior took honors advanced calc, and got B+/A-.
Then as a I senior took one grad course in real analysis, and got an A.

I applied to columbia, brandeis, and a few others, but grades were not that great.
I got into Brandeis, and when i arrived, I was as good as the others, only less well prepared. I lost focus during the vietnam war and left again with only a masters.

'then 4 years later, i began studying again, then just drove over to seattle and took the phd quals at uw.

i passed them and they offered me a slot in the class, but i got a better offer from utah.
so i went to utah, and that's where i finished.

 

[tronter]

well you need a degree in some subject not necessarily math right? Its not necessary to have a math degree to get into grad school for math?

 

[mathwonk]

right. you need to know something and have some ability and impress someone.

i also meant that to do well on tests, the right oreoaration is not test preparation, but real rpeparation.

in high school i never prepared specifically for sat tests (US college scholastic aptitude tests). i took them only once, without ever having seen one before, and got something over 1530/1600.

 

[tronter]

Yeah, for example, if one self studies Analysis by an expert like Dieudonne/Simmons, he would probably be more prepared than one who is taught Analysis from a more contemporary text.

Or if one self studies Algebra by Hungerford/Lang, vs. someone who is taught algebra using Beachy/Blair etc..

I think self study forces you to develop your own perspectives of math rather than following a professor's.

 

[morphism]

Yeah, for example, if one self studies Analysis by an expert like Dieudonne/Simmons, he would probably be more prepared than one who is taught Analysis from a more contemporary text.

Or if one self studies Algebra by Hungerford/Lang, vs. someone who is taught algebra using Beachy/Blair etc..

I think self study forces you to develop your own perspectives of math rather than following a professor's.

Then again if you're taught by someone who knows what they're talking about, they could tell you something you would not find in any textbook, or summarize an entire chapter in one single, brief but illuminating comment!

Of course if you don't study things on your own, they will never sink in.

 

[pivoxa15]

I think that if you can work through a maths book and do everything single excercise then its as good as getting it taught by someone.

Its when you can't do some excercises then you will need someone to teach you so that you can ask them questions. Most people fall into this category so they need to be taught.

 

[mathwonk]

i agree: in order, the best is probably to study from a top book like dieudonne, next best is to be taught by someone good/ but not great who understands it (like me), third is to read a mediocre book like all the ones they use in college nowadays, 4th is to take it in high school from someone who thinks all he needs to teach calc is to have taken it in college.

 

[pivoxa15]

Mathwonk, do you know anyone who are betting at abstract concepts rather than doing concrete examples? i.e most of the time if you tell them to think abstractly they get it right but tell them to think of concrete examples they fail or not as good as when thinking abstractly? Most people would be better at thinking concretely wouldn't you say.

 

[Ki Man]

How much maths would one need to do very basic physics research? Calc II?

 

[mathwonk]

well yes i know people who think very differently. i myself like specific examples, as did perhaps david mumford, whereas people usually say grothendieck thought very abstractly. but mumford told me i believe, that grothendieck also started from concrete examples but very quickly generalized them.

i find it easier to solve problems by thinking of simple cases and then generalizing, rather than thinking generally from the start. but these differences do exist in different people. it is probably no more healthy to force people to think in one way or another, than to try to change a good shooters natural technique.

 

[pivoxa15]

well yes i know people who think very differently. i myself like specific examples, aS did perhaps david mumford, whereas people usually say grothendieck thought very abstractly. but mumford told me i believe, that grothendieck also started from concrete examples but very quickly generalized them.

i find it easier to solve problems by thinking of simple cases and then generalizing, rather than thinking generally from the start. but these differences do exist in different people. it is probably no more healthy to force people to think in one way or another, than to try to change a good shooters natural technique.

I was just about to ask about Grothendieck. If even he starts off with concrete examples then it would be fair to say that no one would start off abstractly?

So would my question be equivalent to asking whether anyone can run before they can walk? Offcourse some can run very soon after they can walk but all start off walking first.

 

[mathwonk]

well it just isn't safe to try to rule out anyone's doing things differently. mind you i agree with you, but there are different minded folk out there.

 

[J77]

Then again if you're taught by someone who knows what they're talking about, they could tell you something you would not find in any textbook, or summarize an entire chapter in one single, brief but illuminating comment!

Of course if you don't study things on your own, they will never sink in.

Yep -- that's the exact balance.

From primary school, you're always told to do your maths homework because that's how the ideas/methods sink in.

However, what you get from any textbook is the basics, and the basics will only take you so far. We're back to this same old discussion of, "if I read everything will I be an expert?". The answer is obviously no, simply because the stuff that experts are working on hasn't been put down into textbooks yet.

 

[J77]

I think that if you can work through a maths book and do everything single excercise then its as good as getting it taught by someone.

Yes -- but being taught by someone isn't about just learning what's in the book. If not, anyone with a bit of confidence could stand up there and lecture a chapter every week -- with only a basic grasp of the work behind the exercises.

A good counter example is the seminar way of teaching, where you go to seminars every week to discuss a topic but you're not tested on it. The tests come from basic textbook exercises that you should do between seminars.

 

[J77]

How much maths would one need to do very basic physics research? Calc II?

To do research you can learn the methods when, and if, you need them.

It's a lot more relaxed when you're not restricted to a timetable, leading up to tests.

But, of course, to get into that research position you'll need to have/show a certain aptitude in all aspects of calculus ;-)

 

[J77]

I was just about to ask about Grothendieck. If even he starts off with concrete examples then it would be fair to say that no one would start off abstractly?

There are plenty of books out there which start abstractly and end abstractly ;)

 

[pivoxa15]

There are plenty of books out there which start abstractly and end abstractly ;)

Okay but wouldn't you say the authors who wrote them actually thought about concrete examples first. Same as the reader as he/she would along the way think up of concrete examples. I guess there is also how you define what is concrete and abstract.

Some may think basic set theory is abstract. Others may not.

 

[mathwonk]

i repeat my warning about generalizing the way others think, and i do so from experience. i have been talking to certain people, and i would seize on a specific concrete example, only to have these people say how unfamiliar that was to them, and they would begin to hit their stride when we took a totally abstract view of the topic. these were often strong algebraists, perhaps with no need to visualize the matter, e.g. christian peskine, with whom i had such a conversation.

 

[mathwonk]

here is a description of peskine's 1995 book, an algebraic introduction to complex projective geometry:

"This is an excellent textbook for those who seek an efficient and rapid introduction to the geometric applications of commutative algebra."

1. Rings, homomorphisms, ideals, 2. Modules, 3. Noetherian rings and modules, 4. Artinian rings and modules, 5. Finitely generated modules over Noetherian rings, 6. A first contact with homological algebra, 7. Fractions, 8. Integral extensions of rings, 9. Algebraic extensions of rings, 10. Noether's normalization lemma, 11. Affine schemes, 12. Morphisms of affine schemes, 13. Zariski's main theorem, 14. Integrally closed Noetherian rings, 15. Weil divisors, 16. Cartier divisors,

just look at those topics!
observe that affine schemes appear in chapter 11, instead of chapter one, as they do in my notes. notice also that if you search in his book for affine schemes, (on amazon), there does not appear a single actual concrete example in any of the pages 145-150, where he is discussing them, although some abstract discussions there are entitled "example". Note Zariski's main theorem does appear, which I seem to recall was his thesis topic.

 

[ekrim]

So far in PDE's I'm finding it nearly impossible to learn generally theory first (if at all). But the methods seem very haphazard, so mimicking examples is about the best I can do.

 

[mathwonk]

as the great v. arnol'd says at the beginning of his lectures on pde, "in contrast to ode, there is no unified theory of pde's. some equations have their own theories, while others have no theories at all."