The Should I Become a Mathematician? Thread

homeomorphic

I am a master at making dumb mistakes. That's part of why I did so much better when I got past high school math and lower division math. In the long run, it doesn't matter that much, as long as the mistakes are inadvertent ones. In "real world" situations (including research), you can check your work 20 times if you want to get it right.

homeomorphic

Anyone can afford to be a mathematician to some extent. In America, all you have to do is do really well in high school and you can get a scholarship. Then, in grad school, you usually get paid. Even if you don't go to college, you can still teach yourself quite a bit on your own.

mathwonk

you might try becoming a mathematician who spends more time with her family. you could start a trend.

dkotschessaa

I recently started getting invited to gatherings with our math department, and it was funny to start finding out how many of the professors were married to each other. I had no idea, because most of the women kept their last names. So, I guess that's one way!

-Dave K

mathwonk

there are at least 5 couples in our department such that both spouses are either professors or instructors.

Mépris

Good to know. That was my intended course of action. (go outside the "syllabus" if I feel like it but then when there's exams, I focus on those)

A lot of what motivated my initial question was that I had some ~12 exams within the span of 3-4 weeks and they were all exams that are much in the vein of the usual standardised testing...

Does anyone here have any experience with the Jerry Shurman (at Reed College) notes on single variable calculus? I'm currently checking out Apostol and Spivak using the free previews available on Google Books and Amazon, before choosing which of the two to buy. Shurman says that he learned from them, Courant and Rudin.

Mathwonk, I read on another post that you used Sternberg and Loomis after Spivak back in the day. What do you think about this course compared to the modern alternatives - Apostol's second volume, I guess? Would one be correct in assuming that the current MATH 55 course at Harvard assumes (equivalent?) knowledge of both that book and Spivak?

Mariogs379

Thought I'd update. This 6 week real analysis class covers the first 6 chapters of Rudin. I'm finding the homework hard but we have a midterm on Monday; he showed us the one from last year and it looks *relatively* easy (definitely compared to the HW). Anyway, thinking I'm gonna take RA II, and some classes in the fall, decide about applying to grad school the following year.

In short, material's harder than I appreciated but also much more interesting. I think I'll enjoy it even more once I get more comfortable with some of the concepts (I feel like I spend a lot of time trying to understand Rudin's language/terminology/general technical writing even when he's conveying a *relatively* basic idea. A good example is his def. of convergence; easy now, but was a bit confusing at first. Tho I think once I'm able to get the ideas more easily, it'll be even more rewarding.

Thoughts?

eliya

Wowza. Six chapters of Rudin in six weeks? How many times do you meet every week?

I'm not sure what you meant by ``thoughts?", I'll take it that you ask how to understand the material quickly. I don't think there's a tired and true method to expedite one's understanding other than practice in time. I'll also add that if you manage to understand the ideas in Rudin in 6 weeks, then you're doing fine. Also, this stuff takes a lot of time to understand. With that being said, try the following:

Write definitions, proofs, concepts, whatever you see fit really, in your own words. By explaining the ideas to yourself, you'll start figuring out how you understand things, and how to approach them. So next time you read a definitions or a proof, you'll be faster.

Get a few more books from your library. Sometimes Rudin is terse, and sometimes those proofs are hard. Other authors expand on the material more than Rudin. It'll be worth it to look some stuff up in those books. I recommend Charles Chapman Pugh's Real Mathematical Analysis. It has the same breadth and depth as Ruding, although sometimes the author does things with less generality.

Read about some of this stuff on Wikipedia. I tried to avoid Wikipedia for a long time, because I was afraid that I'll read an entry that was edited by some crank. All entries I've encountered were nicely written, explained the ideas in depth, and have a nice way of tying things together (how one theorem relates to another, why it's important, generalizations, etc.)

Good luck!

Especially if your first course in upper level math is with Analysis from Rudin. Rudin isn't a bad book, and in fact I like it quite a bit, however, it's a little hard for beginners

In fact, I think that practice and time will help you understand things more quickly

mathwonk

in my opinion loomis and sternberg is a show offy book (my book is harder than yours) and the two volumes of apostol or the two volumes of spivak, or of courant, are much better.

grendle7

Just to let you know, it's MUCH harder to get into Berea, than Harvard. G'luck! Out of the "foreign students" pool of accepted students, only 30 aspiring applicants can be chosen, out of thousands. I'd still apply, if I were you. Just cross your fingers for good outcomes, from crazy probabilities. They usually prefer to accept "brilliant" foreign applicants who are living under crisis conditions, really deserve going to college, and/or wont ever have a chance at it; like that talented math-wiz living in Homs, Syria right now.

Either way, it's a great liberal arts school. In my opinion, you could get a great mathematics education there because it seems that their mathematics students graduate with a broad knowledge in mathematics, ranging from pure mathematics, applied mathematics, and statistics/probability; which is ideal, I think. Check out their mathematics courses! The only problem is, though, that they don't offer much variety in mathematics courses :b

And, have you considered, the best one of them all for math (in general), the University of Waterloo? It's in a town close to Toronto, Canada. I'd go there, if I didn't mind getting into debt; "Lulz."

By the way, unless you want to be chocking in debt after you graduate, then go to Colorado College! I'm infatuated with their block plan and great academic programs; and the MAGNIFICENT LOCATION; but it's totally not worth graduating with $130,000+ in debt.

Lol

Mépris

Coincidence I came back to see this post. I read it before it was edited.

I think my grades may actually be just good enough to get me into Waterloo but it's really not worth the money...that I don't have. I don't know much about Colorado; it looked nice and has financial aid on offer, but it's very limited, as with most liberal arts colleges. I probably won't apply there. There's also the issue of limited coursework but few math/physics majors mean that one can try get some "independent study" thing going on. It doesn't mean grad-level courses, though.

Yeah, I read that about Berea. It's definitely going to be competitive but I believe it's free to apply, so I might as well give it a shot. There's also a list of those "free to apply to" colleges, somewhere on CollegeConfidential. It's easy to find - in case you can't find it, lemme know and I'll try dig it up.

Another thing about liberal arts colleges is that bar a few (Amherst and Williams, being one of those), there just isn't much money to give to international students, which makes the competition even fiercer. It makes more sense to apply to larger colleges. Casting too wide a net is also not a very good idea. Too many essays, too much money on application fees, etc but some people can manage that just fine. ;)

This looks interesting:
http://en.wikibooks.org/wiki/Ring_Th...rties_of_rings

It's the post below, on another thread, that made me ask the question. I had also, per chance, stumbled upon the book, which is available for free on Sternberg's website.

In spite of its "show offy" nature, is the book any good? As for Spivak, are you referring to "Calculus on Manifolds" or is there another text which comes after "Calculus"?

mathwonk

It depends on your definition of "good". I have already stated that i think it is not as good as the other three I named.

Of course Loomis - Sternberg is very authoritative and correct and deep and well written. But the show offy aspect refers to very little attempt to make it accessible to anything like an average student, or to cover what is really needed by that student.

Differential calculus is done in a Banach space, possibly infinite dimensional, essentially the last case anyone will ever need. Most people will benefit far more from a careful treatment of calculus in 2 and 3 dimensions instead.

E.g. after giving all the definitions of differentiation in infinite dimensions, most applications are to finite dimensions. Even the brief discussion of calculus of variations is apparently influenced by Courant who devotes a chapter to it.

The treatment of the inverse function theorem again in Banach space is overkill, and gives little intuition that is actually needed in everyday practice. The implicit function function should be understood first for single valued functions of two variables.

Loomis is an abstract harmonic analyst. His own personal preference is to render everything as elegant as possible, not as useful or understandable.

But make up your own mind. These books are available in many libraries. Just because my course of lectures from Loomis left me feeling very disappointed, with little intuition, and almost deceived as to what is important in calculus, does not mean it may not help you.

If you read Loomis and Sternberg at least you will learn that a derivative is a linear map. That's a lot right there. Indeed that's about all i got from loomis, but it has been very helpful. But I recommend Fleming, Calculus of several variables more highly. Loomis used that book officially in his course, before writing his own.

If you want a very high powered book that also does things in banach space, but manages to be very useful, in my opinion, there is dieudonne's foundations of modern analysis. he also perversely adheres to a credo of making life harder for the reader by banishing all illustrations from his book. but it is good book with a lot of useful high level information. not easy to find elsewhere. he explcitly states however that one should not approach his book until after mastering a more traditional course, (e.g. courant).

Another book Loomis used that I do not recommend either is the super show offy book by Steenrod Spencer and Nickerson. As one reviewer put it roughly, this book is more about the ride than the destination. However I do have all these books on my shelf, I just don't look at them all very much nor with the same pleasure.

Your last quote from me above is a historical account of life at Harvard in the 1960's, not a personal recommendation, indeed to some extent the opposite.

Spivak's second recommended book is indeed calculus on manifolds, an excellent place to learn the most basic several variable calculus topics.

now that i reflect, i am not familiar so much with sternberg's (second) half of the LS book. i only heard him lecture once and was quite impressed with his down to earth and insightful approach. maybe that half of the book would suit me more.

but i'm not much into physics.

In my opinion you are spending more than enough time here asking for advice, i.e. "dancing around the fire", and need to get to work in the library reading some of these books.

mathwonk

well you provoked me to go look at LS and i did in fact like Sternberg's chapter 12 on integration.

This whole discussion is beginning to remind me of a friend telling me that his brother warned him off of reading a famous algebra book, so i myself also avoided it for years.

Finally I was required to read some of it and found it wonderfully clear. When I went back and asked my friend's brother he said he never said it was bad, just "tedious". by which he seems to have meant overly detailed, just what I appreciated about it.

so please take what we have said with a grain of salt and try to get a good look at these books yourself.

Even Loomis' half of the book helped me in the section on "inifinitesimals" and his slick proof of the chain rule.

But the abstract implicit function theorem in terms of projections from a product of banach spaces, there left me wondering what Mumford even meant when he said the theorem simply says you can solve for some of the variables in terms of the others.

mathwonk

oh, also the intro to LS says plainly that apostol, spivak, and courant are suitable prerequisites for their book. if that includes both volumes of those books, i would agree.

RJinkies

Hi sahmgeek......


Quote: 'I am seeking advice on a receiving a math degree (e.g. Master's + Secondary Ed cert) however I have very little formal math training beyond high school......

.....Given that I would need to start from scratch, I wondered if taking the basics at a community college (Cal 123, Linear Algebra Abstract Algebra, Finite Math, and ODE) and, of course, doing very well....

....stay home with my 2 very young children and would most likely need to go back to school part-time. I might have some time during the day to work on math, but most of my free time would occur in the evening, 8pm and later. I am concerned that this isn't enough free time to really study this subject.....

.....I am not very concerned about my intellectual capabilities, but with my time constraints perhaps this is an unrealistic goal given the rigorous nature of math. I do, however, like the idea of studying math for it's own sake, even if the end result is purely for personal gain......

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Well, I would think that there shouldnt be any problem for self-study, if you feel that you can put in the time to read the textbooks and do all of the problems.

With the rule of thumb for an Ivy League school, it's about 48 hours of class work a week with a full schedule. And you do that for 12-15 weeks to complete one semester or half of the textbook. One can figure out how to pace yourself pretty okay on your own. As long as you know what the good textbooks and supplementary textbooks [1-3 texts - old and new] should be, and the books fit best with your learning style.

It sounds like you could actually learn the subject better and on your own terms, setting your own schedule as long as you're motivated to get the most out of the textbook by reading all the pages and doing all the problems 95% of the time.

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As for goals, that can change semester by semester as you master one more notch in the textbook ladder, and your interests may change, and perhaps your direction... If you want to take some courses later, by all means, but I'd probably do most of the work on my own, but it all depends how much time to spend with the family, and how quickly you wanna zoom up the ladder of checking off the courses that you got a lettergrade in.


Quite a while back, people could teach high school math where i was with at least a minor in math and maybe a major in education or something else... [like a major in physics and a minor in math and some education courses]

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But it sounds like you wanna do a BA/BSc in Math and well just know what the basics are, and then add the stuff that interests you to your liking. As you finish off one textbook and then go to the next tier, you get to choose your own path pretty much.

a. getting your Calculus I II III ..... and IV [aka Vector Calculus]
b. Taking you Analysis courses and thinking of them as one stream of at least four semesters to like Real Analysis - say from RG Binmore/Apostol/Rudin/Hartle/Strogatz/Royden....
c. Linear Algebra - and up
d. Differential Equations - and up into PDE and Non Linear Dynamics/Chaos
e. Complex Analysis [Applied if you're for Physics, Pure for just math, or maybe both]

You can always figure out if you wanna go into [most people might only do 5 courses worth [20%] of these....

f. Geometry - like Coxeter's book
g. Number Theory
h. Mathematical Logic and Set Theory
i. Abstract Algebra [helpful with Analysis to get into Topology]
j. Topology - Munkes and Guilleman as the main two books
k. Probability
l. Differential Geometry and Tensor Calculus - like Synge's book [what you'd want after Vector and for say Wheeler's Gravitation]
m. Mathematical Physics stuff [like if you took Symon and then Goldstein in physics] and then wanted to go into the mathematical side of LaGrange and Hamilton
n. Fluid Mechanics [if you're more physicist/engineering curious]
o. Continuum Mechanics [if you're more physicist/engineering curious]
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I tend to think of Grad School as basically what textbooks did you find cool in Third and Fourth Year, and sometimes the rest of those books are your grad school classes [like Royden] or the supplementary reading in those texts...

Me i would choose a Mathematical Physics like route where you can get the best of the Applied and the best of the Pure, i dont think people think of things as Pure Math anymore like Hardy....


I think of it as, spend the 400 hours on each textbook, master things on your own with completing the reading of one chapter as your self-mastery, and then doing all the problems in that book, as the proof of your self-mastery.

That way you dont get hung up on midterms and finals, you see the ladder of math or science as a bite filled chunk as a single chapter, accomplished usually in a week with maybe 20 hours of effort, getting to that goal of the last chapter and 400 hours clocked on your mental library card for your own textbook, using your own dining room table as your own little uni.

RJinkies

Hi PrinceRhaegar


Quote: my second semester of college as a mechanical engineering major, but I'm thinking about switching to math. The reasons are simple; recently I've found that I'm better at math than any other subject (especially physics....

.....I just think math is cooler than any other subject I've seen so far. The reason I'm really hesitant to do so is because firstly, I have no idea what I'd do with my degree after I graduate, and secondly, and this may seem a bit shallow, I know that I'll likely be making more money as an engineer....

.....In a perfect world I'd major in math and get a job as an engineer (or at least in an engineering company). This is because I love math and I feel like I'd get a TON of satisfaction out of doing useful stuff for the world while also doing what I love.....


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Honestly, it sounds like the ideal path is to do both, and just take that extra one or two years for your B.Sc and do a double major

There are people out there that sound a lot like you and they do things like get a Mechanical Engineering degree and double it with a Physics Degree...

if you really wish to slow it down, and you got zero problems with the textbooks, you can almost accomplish it all, and think of it as engineering as a hobby and math as a hobby, and then think of the engineering stuff as your income...

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Some bizarre and brilliant souls in 5-6 years end up with a satisfying thing of doing four Bachelor degrees. [maybe 6-7 for ordinary mortals with the same goal]

a. Mechanical/Aerodynamic Engineering
b. Physics Degree
c. Math Degree
d. Electronics Engineering Degree

since there is considerable overlap and his future goals worked out that he used most all of it in his career.... though he wasnt as deep as some that just took one and only one path...

But you can be 65%-80% fluent in two courses with a Double Degree.


so there is a LOT you can accomplish with an extra 1.5 years of your life, that these sorts of things are possible.

The Hardest thing is knowing how to self-study and how much effort to put into things, and not fearing failing or exams anymore.... the second dilemma is what really makes you happy, and a career may or may not conflict, if you just put some extra time into things.

but sure if you go up a ladder in academia you do tend to end up stuck there, where people who get a physics degree who almost wanted to go to grad school in pure math, and they find they *had* to pick one or the other, but if it's a hobby, or circumstances are right, you can sometimes slide into both worlds.... all depends how happy you are, and you like the results..

mathwonk

or perhaps, just start with one small significant goal, like learning calculus, and do it well. then go further.