Should I Become a Mathematician?

[domhal]

Interesting reading. I too had been wandering recently about retention of details from courses past - in particular how it seemed like I had forgotten too much!

As an example I was looking over some analysis recently (I took analysis most recently in the Spring of this year) and had to show that the set of all limit points of E is closed. Feels like it took far too long to do!

Nice suggestion Mathwonk, I'll see how that goes.

 

[balletomane]

Thanks so much, Mathwonk.

 

[mathwonk]

one thing i do myself, is i always try to reproduce material without, looking at any books or notes. that's how i see how much i have understood. this year i am teaching grad algebra and i try every day to write down the lecture without consulting any book or even my own notes from the previous course. i do not always succeed and when i do not, i make a point of noting what it was i overlooked.

 

[courtrigrad]

so basically if one can write down what he remembers perfectly, he is an expert in that area?

 

[CPL.Luke]

^ pretty much, if you are able to reproduce all material on command perfectly without any reference, then you are obviously very knowledgeable on the subject. And if you are very knowledgeable about a particular subject, then doesn't that make you an expert?

 

[mathwonk]

i am not writing out the material entirely from memory, but reworking it, as I only remember the details vaguely.

As I rework it and rethink it I begin to understand it better, and add more and more new insights every year, than I had before. it should keep getting easier as time goes by. It should boil down to a few basic precepts, and not remain a long list of facts to remember.

One should also try to solve problems, using the ideas, and to think of ones own proofs for the theorems, simplifying and improving the ones one may have seen before.

 

[JasonRox]

One should also try to ,solve problems, using the ideas, and to think of ones own proofs for the theorems, simplifying and improving the ones one may have seen before.

Yeah, always a good idea if possible. Some proofs are just like the way it is.

I tend to attack proofs from time to time, which results with my own proof that I enjoy better.

 

[mathwonk]

here is a tiny example: i always liked group actions in terms of orbits and stabilizers, as very visual.

But i disliked cycle notation for permutations, as overly compoutational.

now i realize a cycle is just an ordered orbit, and i like them much better, and can also see why certain little conjugation formulas hold.

e.g. if s is a permutation then a standard useful algebraic fact is that

if (123...k) is a k cycle, then s(123...k)s^-1 = (s(1)s(2)...s(k)).

in orbit terms this just says if a certain element takes a to b, then if i conjugate it with an element s, the result takes s(a) to a to b to s(b), i.e. takes s(a) to s(b).

this can be visualized. i.e. if i rotate one vertex to another then fix that one, then rotate back, it is the same as fixing the original vertex. so the conjugate by s, of a rotation fixing vertex a, fixes vertex s(a).well anyway, i guess you need to find your own way here.

 

[mathwonk]

heres a deeper one: the hard part of the proof of the inverse function theorem is that a smooth map taking 0 to 0, and having derivative equal to the identity at 0, maps some open nbhd of 0 onto an open nbhd of, 0.

i finally realized that the linear approximation definition of derivative, guarantees that the original map is homotopic to its derivative on some nbhd of 0, hence wraps the unit sphere the same number of times around 0, qed.

the proof still takes work, but this is it in a nutshell, conceptually.

 

[mathwonk]

the point is that i am always turning every concept over and over in my mind, trying to make it my own. I want to live there, to see the objects, and not have to depend on some memorized argument to understand them.

i hope this comes across in all my explanation here. i am never parroting some learned formulas, unless that all i have to offer. then i say so. for this reason for a long time some people failed to understand that my explanations of strange objects, were even about the same objects they had memorized versions of.

to a mathematician the objects are real, not dependent on some book learned representation of them.

 

[AlbertEinstein]

I haven't gone through all those 18 pages, yet I have a question: does speed of problem solving a necessary quality of a mathematician?

 

[mathwonk]

no. depth matters, and creativity matters. not speed. unless you compete with faster people for the same results. if so, do not tell them how far you have gotten unil you are finished.

of course you have to be fast enough to finish before you die. that's about it.

 

[JasonRox]

of course you have to be fast enough to finish before you die. that's about it.

I'm certain that if I'm given enough time I can prove Riemann's Hypothesis. I'm just not fast enough!

 

[courtrigrad]

Can Spivak's book (Calculus) be used as an intro to analysis text (i.e. before a real analysis course?) How would you change or edit the following curriculum:

Calculus 2
Calculus 3
Linear Algebra And Differential Equations
Computer Science
Intro to Analysis (Spivak Calculus)
Real Analysis
etc..Also what are your opinions about Real Mathematical Analysis by Charles Chapman Pugh?

Thanks

 

[Ki Man]

Any good math books for High School students to read?

I'm sure those calculus boks you recomended are great but I'm pretty sure they will be a tad too advanced

 

[mathwonk]

spivak wrote his book for college students who were very bright but had no calculus, so it could precede calc 1 and 2, but maybe a course in calc from say thomas would be wise.

a good book that anyone can read in high school, is "calculus made easy", by sylvanus p. thompson, about 100 years old.

it was a book studied in high school by one of my friends when we took the spivak course as freshmen in college.

this thread is so long no one can be expected read it all but i think i have already given recommendations for high school, junior high, etc. let me try to find them.

 

[mathwonk]

yes, they are in post #8, page one of this thread.

 

[mathwonk]

a high school student can and should try to read anything he/she likes. actually high school students may be brighter than college students.

and often more motivated. so plunge right in.

i read principles of mathematics in high school, and it had lots of great book references at the ends of the chapters. then i went to the college library and looked up those books

i still remember sitting in the stacks puzzling over the proof that there exist an infinite number of prime integers.

let p1,...,pn be any finite set of primes. and then consider the integer

N = 1+p1p2...pn, 1 plus the product of all the primes pi.

we claim no pi divides N, because if say p1 did divide N, then since

1 = N - p1p2...pn, p1 would also divide 1, which is false.so none of the primes pi divide N. But N is larger than 1, so we claim some prime must divide N. I.e. among all divisors of N greater than one, there is a smallest one say q.

then q cannot have any factors larger than 1, or they would be smaller factors of N.

so q is a prime factor of N, but q cannot equal any of the pi. so the finite list p1,...pn, is not the full list of all primes.

hence there is an infinite number of primes.

 

[Ki Man]

actually high school studnets ARE BRIGHTER THAN COLLEGE STUDENTS. (i think i was brightest when i waS ABoUT 15, at least based on my IQ scores.)

thats what i was thinking too =P

I need stuff that would be available at a local library... would those be on their shelves

 

[mathwonk]

public libraries do not have good math books in my experience, but maybe that's because i live in the south. you can look. but most people can gain access to a university library somehow. and many books are available free, like my graduate algebra text, which is accesible to anyoe with enough patience and who knows something about matrices. so maybe the place to start is learning matrices. there should be books on that available most places. again my book on my website is free, but very concise. there are many free books on linear algebra on the web. ill send you some adresses if you cannot find them.

 

[Ki Man]

I live in the west, near disneyland. there's lots of universities and bookstores around here. I probably live within a half hour drive of at least 10 state colleges and universities and caltech is around 40 minutes away or less

i guess for now i could ask my math teachers to let me borrow books but maybe next year i'll start looking into more sources of books

E: what's 'your site'

 

[mathwonk]

heres my website:

http://www.math.uga.edu/~roy/

 

[mathwonk]

have you noticed? even though there are almost 14,000 hits to this thread it still is not a sticky?

 

[JasonRox]

It might even surpass the hits for Physicists!

 

[Ki Man]

maybe they kind of assume mathmaticians are a level of sub-physicists

physics is nothing without math, but math is still just math =]

My math teacher let me borrow a book called freakonomics bye Steven D Levitt and Stephen J Duber

 

[johnnyp]

I love this thread - very insightful. And I definitely like it unstuck; stickies are labeled in such a way that it's bound to get unnoticed some time or the other--just keep it the way it is!

Mathwonk -- I have a question, if you don't mind. I'm currently taking real analysis. I'm understanding everything so far but the homeworks have always been [for the most part] difficult, and that's not what my conception of math was -- I was never stuck in Calculus and Differential Equations, unlike now. Is this a danger sign? Should I not pursue math as a major? It's just that in analysis, I seem to need significantly more time to solve problems (mostly proofs) than I would have in Calc and ODE's.

 

[ultranet]

this is well guided posts. I am goin to check out some books...
thanks

 

[JasonRox]

I love this thread - very insightful. And I definitely like it unstuck; stickies are labeled in such a way that it's bound to get unnoticed some time or the other--just keep it the way it is!

Mathwonk -- I have a question, if you don't mind. I'm currently taking real analysis. I'm understanding everything so far but the homeworks have always been [for the most part] difficult, and that's not what my conception of math was -- I was never stuck in Calculus and Differential Equations, unlike now. Is this a danger sign? Should I not pursue math as a major? It's just that in analysis, I seem to need significantly more time to solve problems (mostly proofs) than I would have in Calc and ODE's.

If this is your first time doing proofs, I wouldn't worry about it. That seems to be normal for first timers.

Just be sure to strictly justify each step during a proof. Read lots of proofs too. And justify each proof you read. Don't just read along.

I come by proofs I don't like myself sometimes. Feeling as though there could be another way, then I try it out myself. Sometimes I get a new and sometimes I don't. If I don't, I then learn to just enjoy that proof a little more.

I wouldn't worry about it for now. Just keep practicing. If you're determined, good things are bound to happen.

 

[mathwonk]

good point johnnyp. this way (unstuck) when it dies it will fade away gracefully.
no, analysis is just harder than those other subjects, we all think so.
and it does not prevent one from being a mathematician to find analysis hard.
there are three kinds of thinking in math, algebra, analysis, and geometry.
i.e. finitistic, infinite (limiting), and visual.

few people are good at all of them. i am very visual-geometric. i majored in algebraic geometry because it was halfway in between algebra (hard for me) and geometry (easier for me.

no slight intended, but topology to me seemed "too easy". i found the challenge of seeing the geometry behind the algebra stimulating.

analysis on the other hand was painfully hard. I did ok in complex analysis of several variables while i tried that topic, but my head hurt when I was thinking about it.

It felt pleasant the whole time I pondered geometry or topology. i wanted to enjoy myself, not suffer. You cannot get a PhD taking several years, if you are suffering the whole time. It is hard enough in the best of circumstances.

that said, one should not avoid the subject one finds hard, as it too will be useful learn as much of it as you can, and try to change your attitude to it. work with someone who likes it and try to see why they think it is beautiful.

 

[mathwonk]

i like arnol'd's definition of math: "that branch of physics where experiments are cheap."

 

[courtrigrad]

mathwonk, does one pick up proof writing techniques when they learn real analysis? I know some institutions offer classes that teach students how to write proofs. Would it be better to learn the technique by yourself?

 

[mathwonk]

learn it as soon as possible, from any source that helps. learn it in as many ways as one can. better not to wait until reals as then it is very hard and coupled with very hard topics too.

i started learning it in high school, from the book principles of mathematics, by allendoerfer and oakley. i also took euclidean geometry, whose absence is one of the main reasons proofs are no longer understood by today's students.

i.e. removing geometry proofs and inserting AP calculus from high school I think is a prime culprit for our current demise as a math nation.

 

[courtrigrad]

ok, yeah I will use the textbook by Solow then. https://www.amazon.com/How-Read-Pro...bbs_sr_1/ 102-6215276-8882554?ie=UTF8&s=booksAlso what programming languages do you think one should learn? Should he learn Java? Because I think knowing a programming language will be extremely helpful (or am I wrong)?

 

[Ki Man]

http://www.math.uga.edu/~roy/

nice picture :]
were you talking about the links to all of the algebra notes and everything?

 

[mathwonk]

i forget what i recommended. the linear algebra notes link is a very condensed review of linear algebra for a strong student who either wants to work out all the theory for himself, and is already good at proofs, or has learned it before and wants to review for a PhD prelim.

the 843-4-5 notes are for a detailed first year grad algebra course, for any grad student or upper level good undergrad student, or bright motivated high school student who knows whaT a matrix is. actually even that is reviewed in the 845 notes.

the rrt notes are for advanced students who know some complex analysis.

the research papers are for people interested in prym varieties and other abelian varieties, and the riemann singularity theorem.