Should I Become a Mathematician?

[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."

 

[ekrim]

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 theior own theories, whileothers have no theories at all."

I read that and was surprised by such honesty in a preface. A bit encouraging and discouraging at the same time. Unusual that libraries are removing textbooks on the subject in some sort of Salem PDE Hunt

 

[mathwonk]

well it means one should study carefully the basic examples: heat, wave, laplace.

i have spent 30 years looking mostly at the heat equation myself, and some at laplace.

another simple equation many people spend their whole lives looking at is the dbar equation, the one that vanishes on holomorphic functions. obviously if all holomorphic functions satisfy one pde, it is hard to have a general theory that covers all pde.

but i suspect there is some theory, such as a classification into elliptic, and so on..., types,,,,and the theory of the "symbol".
there is the theory of the index of an elliptic operator.

see hirzebruch, 3rd edition, page 187, for some remarks on the atiyah singer theorem on these objects. or the seminar by palais et al.

 

[pivoxa15]

pivoxa, when the problem is a proof, you know if you have it or not, so no answers are needed. when it is a calculation, to be siure you have it right, you need two ways to do it so you can compare answers.

Do you think its a big step for someone to go from needing the answers from the back of the book to confidently doing the exercises without answers?

Because when someone can do that then the possibilties seem endless since most books don't have answers supplied.

 

[mathwonk]

well i do not recall ever needing answers from the back of the book in college, or even knowing they had them there, but i often want to look at the answers in the back of our calculus book now when teaching it.

so i think there is a difference in the kind of problems that were given in my college courses, mostly proofs, compared to the tedious problems in the courses i teach now, almost all computations.

so i do think getting away from needing answers is a big step, but it is partly a matter of getting away from trivial computational oriented courses, and partly self discipline of not letting yourself look back there.

the same thing applies to all work, on paper and on the board. some of my students put their work up and then turn to me to ask, is that correct? but a good student looks at his own work to see if it looks correct.

you just have to start building the habit of checking your own work, and stop asking mama, or papa to reassure you.

 

[Cincinnatus]

I'm the grader for a lower level calculus classes at my school. I recall one time when the answer in the back of the book was wrong. Nearly the entire class got this question wrong. A large number of them even did all the correct work, then crossed out the right answer and replaced it with the wrong one from the back of the book!

 

[quasar987]

ow!

 

[Diffy]

Hi,

I am 4 years out of college, with a BS in mathematics. Ever since graduating I have dreamt and thought (almost daily) about mathematics. I love it. At work when someone mentions a number, I think about whether or not it is a perfect number, a square, a cube, what the prime factorization is. I constantly re-read My Abstract Algebra College book and try problems in my spare time. I have a strong desire to learn more about math. I have purchased books since graduating, and have tried to get through them on my own. Unfortunately, I often find that on my own, learning is very time consuming. I believe with help and guidance though, I am fully capable to take my understanding to the next level. I may not be the best test taker, but I CAN work extremely hard. I have a strong desire to learn more, but there are several barriers that stand in my way.

I have a full time job. It is not conceivable that I will go back to school full time. My lifestyle demands the income of a full time job. I am young, but I do have others who depend on me. I have a family. Time with them is not negotiable, they need me, and that’s that.

Part time may be possible. Two hours at night for classes, I can manage a few times a week. I can study and do course work at home. (Who needs sleep? That’s what coffee is for, right?)

At my current job, I am an analyst. I get to study trends, and deal with large quantities of numbers. I don’t get to apply much pure math, but certainly High School Algebra, and sometimes Calculus. Perhaps I am mathematician already… But I don’t think so, not when there is so much more I can learn in Graduate level classes.

I am not sure if I am looking for advice, or encouragement or what. But please comment.

 

[mathwonk]

talk to hurkyl.

 

[mathwonk]

i want to recommend a precalculus book by one of the greatest masters of all time, euler:

https://www.amazon.com/dp/0387968245/?tag=pfamazon01-20and i also recommend as the greatest geometry book of all time, the one by euclid.

https://www.amazon.com/dp/1888009195/?tag=pfamazon01-20and a free online copy:
http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

hartshorne also has a nice companion volume, stemming from his course at berkeley,

called geometry, euclid and beyond.
https://www.amazon.com/dp/0387986502/?tag=pfamazon01-20i just realized my previous book recommendations do not follow closely enough the famous dictum (by abel?) to read the masters, not the pupils. inded i have not done this enough myself in my career.

 

[mathwonk]

i have just ordered copies of archimedes' and euler's works. maybe next time i lecture on archimedes' method i will actually know what i am talking about.

 

[timur]

How can a foreigner become a math graduate student in US? May be it is too practical question but can you please give a suggestion or a link?

 

[mathwonk]

most of our best grad students are foreigners. come on down! apply to any school by looking at their websites. try uga.math.edu

 

[J77]

and i also recommend as the greatest geometry book of all time, the one by euclid.

https://www.amazon.com/dp/1888009195/?tag=pfamazon01-20

Thanks mathwonk.

I haven't seen this complete edition -- and at only $25 it's extremely good value, even with the European delivery charge!

 

[teleport]

Mathwonk, what would u say of the decision of a math prof to, in preparation for a test for his students, suggest doing every single exercise in the book in addition to the ones given by him in class, for him to end up giving a problem in the test worth 1/3 of the mark, which covered nothing of his suggested exercises for prepartion, that had only been briefly mentioned in class, and was anything but trivial (in fact the single hardest question of the test).

In fact, I felt proud that I did almost all of his suggested problems for prepartion, given this is my first year doing pure math, but felt demoralized after all the work to see that all that was almost not covered at all in the test. So I had to use pure ingenuity, no knowledge at all, to figure out the problem. But this wasn't enough. This is not my main complain, however. I tend to see it as, if you have done the work, you should somehow be rewarded by being albe to use the knowledge/techniques gained from it.

Weird, somehow I feel sympathy with what he did. He didn't care much of what I know. I would say he cares more with me thinking and being creative. This is definitely a wake up call for me. I will be looking for some revenge on the next test. I can now better predict the next one.

 

[mathwonk]

well when i was a student i had similar stories but i do not have any now that i am a prof. usually now it is that the student does not realize that the test was essentially identical to, or a direct outgrowth of, the exercises that were assigned.

i would say however that you probably learned more studying for this test than anyone in history. so it can't be all bad.

 

[mathwonk]

by the way i also just bought myself a copy of the works of archimedes, so i can judge for myself how similar his ideas were to what we call calculus today.

 

[teleport]

well when i was astudent i had similar stories but i do not have any now that i am a prof. usually now it is that the student does not realize that the test was essentially identical to, or a direct outgrowth of, the exercises that were assigned.

i would say however that you probably learned more studying for this test than anyone in history. so it can't be aLL BAD.

In this case I can't be more explicit of how disconected that question was from my preparation; because of this, no, it wasn't a direct outgrowth of the exercises assigned. If that were the case, I would have painfuly accepted it without bringing it up to debate. In fact, there was not a single exercise assigned with the "theme" of the problem.

Well I don't know if I prepared more than anyone else in history for the test since I managed to do the problems in two days over the weekend. But of course, how can I prove this when the test didn't give me the chance to. A test that tests? The topic of the problem was briefly mentioned in class, not a single exercise assigned given by the prof related to it, nor even mentioned in the book. Of course, given the absence of any attenuation by the prof on the significance of this topic, I naturally ignored it. Oh well, I just don't want to think more of this.

 

[mathwonk]

you nonetheless remind of my junior course in probability where the prof had assigned a very difficult problem for homework which no one got, so he devoted a whole class to its solution. unfortunately i was absent from that one class. then on the test the sob gave that as a problem worth 25% or more of the whole test. i got a B , and in spite of my best efforts made a B in the course, when i wanted so badly an A.

but so what? I still became the world famous intellect I am today.

 

[teleport]

Ha, that shall raise my mood. Thanks.

This situation reminds me of a talk between two criminals (wait). One criminal is confessing to the other that he is going to have a child. He also mentions that his dad didn't treat him well in his childhood. So the other criminal asks: so that means you are going to give your son everything you father didn't give you? The other criminal looks back surprised and says: no! That means it is my time to have fun!

This might actually apply to my prof and ur probability one.

 

[teleport]

So yea, those profs might have decided that if they were going to spend time grading, they might as well have some fun along the way, even though they probably went through the same things as students. BTW, my prof has made it official, he expects us to think even in tests! It was bizarre: he said in class he was "tricky" while giggling suspiciously. Oh boy, this is gona be some ride.