Other Should I Become a Mathematician?

[Darkiekurdo]

Does not being able to solve problems that involve proving something mean I am not going to succeed in mathematics/physics?

 

[quasar987]

Does not being able to do smashes the first time you play tennis mean you're never going to be able to play?

 

[Darkiekurdo]

No, but isn't proving a creative thing?

 

[quasar987]

IMO, the creativity part comes in when inventing new theories and figuring out results not yet known. In 99% of cases, proving propositions is usually simply a matter of applying definitions and manipulating logic to show your proposition is indeed consistent with the definition and theorems you previously proved.

"If only I had the theorems. Then I should find the proofs easily enough." --Riemann

 

[ekrim]

No, but isn't proving a creative thing?

Art is a creative thing, but you need to learn the fundamentals of brush strokes, color mixing, etc. before painting masterpieces. In the same way you need to development the proof fundamentals as tools through which you can express creativity

that being said, I picked up Rudin from the library today and have been trying to get some fundamentals by working his proofs. i haven't decided if its more arduous or humbling yet

 

[mathwonk]

try some of my books.

 

[ekrim]

try some of my books.

I actually looked up artin, but couldn't find him in the library, so i reached for the familiar name. any others you might recommend? the rudin isn't impossible though, i think id have the same trouble with anything theory based

 

[threetheoreom]

Is it strange that after I read a chapter of Spivak's book I have no idea how to do the problems? And after I read the chapter again I still can't do the problems.

Which of Spivak's books? Some or many of us have not seen any of his books (although others of us certainly have - not me, though); we may like to know which is the one you are finding trouble understanding, in case we might like to obtain a copy to use for study. Is it his Calculus book which you find difficult?

All enthusiasts of Spivak's books, please give your discussions about this.

Also, something interesting - do a web search and you can find a wikipedia article on Michael Spivak.

Yes he is using Spivak's Calculus book.

Maybe a bit strange, but please consider that spivak himself said some of the problems are so difficult ( marked by the asteriks ) that even the brightest students will have to be really interested to continue trying to solve them. One problem in the second chapter he says " if you have figured or looked up the answer" That said it is worth working through these problems because they not only enhance your conception of the topic being discussed but also teach you problem solving.


I recommend that you work out even the examples that he shows ever so clearly, because that clarity can fool you into think you understand such and such.

P.s i use the same book.

 

[morphism]

I like Herstein's view on the matter: "Many [problems] are introduced not so much to be solved as to be tackled. The value of a problem is not so much in coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver." (Taken from the preface to the first edition of Topics in Algebra.)

I think the same applies to Spivak's problems (many of which come from Courant, by the way); in fact, some are notoriously difficult. So don't feel disheartened if it takes you a lot of time and effort to do a small portion of them.

 

[threetheoreom]

I like Herstein's view on the matter: "Many [problems] are introduced not so much to be solved as to be tackled. The value of a problem is not so much in coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver." (Taken from the preface to the first edition of Topics in Algebra.)

Well said

(many of which come from Courant, by the way)

really i am not aware, also i think the larger amount of problems in the later editions transcended from the first in the 1960's. I am not familiar with the problems of Courant's book nor the publication date but i think Spivak's preceeds R. Courant book, again i am not sure.

 

[quasar987]

Courant is an old folk... When Spivak received his doctorate, Courant was 76.

http://en.wikipedia.org/wiki/Michael_Spivak
http://www.genealogy.ams.org/html/id.phtml?id=15162
http://en.wikipedia.org/wiki/Richard_Courant

 

[Darkiekurdo]

Thanks guys. I felt stupid because I couldn't do the problems after reading the chapter several times.

 

[threetheoreom]

Courant is an old folk... When Spivak received his doctorate, Courant was 76.

Okay thanks

 

[mathwonk]

From studying both books, (and having been a Harvard student myself), I believe Spivak may have learned from Courant, probably as a student in honors calc at Harvard, around 1960. Courant's book of course dates from the 1930's.And my suggestion to try reading some of "my books" for proofs, was a suggestion to try some of my free books from my webpage, which is visible in my public profile.

 

[teleport]

Does anyone know of "Undergraduate Algebra" by Lang, and "Advanced Calculus" by Taylor? These will be my 'main subject' books for the upcoming year. I think I will particularly like the first one. I read yesterday as much as I could (I was not able to avoid it!), and the explanations and proofs seem very clear. The solution of many exercises seem to use techniques used in the proofs of the theorems, a thing that I just love. It actualy makes me feel I learned a cute trick.

The size of the calculus book scares me. Would u recommend sticking to it or to look for another book? Note though, the first statement does not completely imply the question. Thanks

 

[ekrim]

From studying both books, (and having been a Harvard student myself), I believe Spivak may have learned from Courant, probably as a student in honors calc at Harvard, around 1960. Courant's book of course dates from the 1930's.


And my suggestion to try reading some of "my books" for proofs, was a suggestion to try some of my free books from my webpage, which is visible in my public profile.

ahh, ill definitely try that, thanks dr. mathwonk

 

[teleport]

Mathwonk, I will definitely use your elementary algebra notes as reference during the next semester. Thank you. Are your other algebra notes designed for an introduction into abstract algebra? This is what I'll have next year.

 

[mathwonk]

my math 843-844-845 notes are a detailed introduction for students who have studied matrices and determinants.

 

[threetheoreom]

I am using the 4000 notes, particularly the notes on polynomials.

 

[mathwonk]

the 4000 notes are for our intro to algebra course, but we precede that course by an intro to proofs course (3200) where i tend to pre teach much of the same stuff, at least the elementary number theory part.

they also have usually had alinear algebra course from adams and shifrin beforehand (math 3000). then there is 4010 course introducing groups.
afterwards students should be ready for my 843 notes, but my 843 notes are pretty self contained on groups, so could be an introduction to them.

i do use matrices though, and only teach them later, in the 845 notes.

how are the 4000 notes going? those were actual class notes as handed out, and not rewritten, so may lack some organization or editing.

heres the catalog descrioption:
MATH 4000/6000. Modern Algebra and Geometry I. 3 hours.
Oasis Title: MOD ALG & GEOM I.
Undergraduate prerequisite: (MATH 3000 or MATH 3500) and (MATH 3200 or MATH 3610).
Abstract algebra, emphasizing geometric motivation and applications. Beginning with a careful study of integers, modular arithmetic, and the Euclidean algorithm, the course moves on to fields, isometries of the complex plane, polynomials, splitting fields, rings, homomorphisms, field extensions, and compass and straightedge constructions.
Offered fall, spring, and summer semesters every year.
MATH 4010/6010. Modern Algebra and Geometry II. 3 hours.
Oasis Title: MOD ALG & GEOM II.
Undergraduate prerequisite: MATH 4000/6000.
More advanced abstract algebraic structures and concepts, such as groups, symmetry, group actions, counting principles, symmetry groups of the regular polyhedra, Burnside's Theorem, isometries of R^3, Galois Theory, and affine and projective geometry.
Offered spring semester every year.

 

[teleport]

Abstract algebra, emphasizing geometric motivation and applications. Beginning with a careful study of integers, modular arithmetic, and the Euclidean algorithm, the course moves on to fields, isometries of the complex plane, polynomials, splitting fields, rings, homomorphisms, field extensions, and compass and straightedge constructions.

Yeah, we will have almost all of this plus groups. Unfortunately, we will not have constructions. Damn, when will someone show me with detail why the circle can't be squared! My problem is that I'm not seeing any, not even convergence of classical geometry with other modern subject. I've looked into my future courses, and classical geometry seems to be ignored. Is this becoming common in many universities, or is it just mine?

BTW, we will also cover Galois theory! (I think at a fundamental level, but don't know)

 

[mathwonk]

since circles and lines have equations of degres 2 or 1, it turns out that the coordinates of points obtained by intersecting them, satisfy equations of degree 2 or 1 over the field generated by the coordinates of the points determining the lines and circles themselves.

thus constructible points have coordinates which lie in an extension of Q which is composed of successive quadratic extensions. by multiplicativity of degree of field extensions, this means they lie in field extensions of degree 2^n for some n.

hence points whose coordinates satisy irreducible cubics/Q for instance cannot be constructed. this is why an angle of 20degrees (i believe) cannot be constructed, so one cannot trisect a 60degree angle.

similarly a point whose coordinates do not satisfy any rational equations at all, such as pi, cannot be constructed. this is why a circle cannot be squared. the detailed proof is in jacobson's algebra book, complete with a proof that pi is transcendental.

 

[teleport]

Wow what a mouthfull! Well, if part of the general purpose of providing the explanation is to motivate us in the studies of abstract algebra to get answers to such problems, then u got me there. I would absolutely love to understand all of this which was mentioned.

 

[ekrim]

Hey Mathwonk, I'd greatly appreciate if you could tell me what's the standard/classic text for PDE's. Unfortunately my professor teaches by copying directly from the book (McOwen) to the board. It's a little weak on theory and leaves me unsatisfied.

 

[mathwonk]

i am ignorant in pde, but i myself like vladimir arnol'd's books, and i personally have his text on pde.

i gather there is no systematic theory of pde's as there is for ode's, so one studies the classically important special cases, like: heat equation, wave equation, and laplace equation.

i myself have studied the (several variables complex) heat equation quite a bit, and of course the laplace equation is important in all complex analysis, since both real and imaginary parts of holomorphic functions satisfy it. harmonic functions are also important in geometry.

but i know nothing about the wave equation.

but i recommend arnol'd for auxiliary reading in any course.

 

[ekrim]

i am ignorant in pde, but i myself like vladimir arnol'd's books, and i personally have his text on pde.

i gather there is no systematic theory of pde's as there is for ode's, so one studies the classically important special cases, like: heat equation, wave equation, and laplace equation.

i myself have studied the ehat equation quite a bit, and of course the laplace equation is important in all complex anakysis, since both real and imaginary parts of holomorphic functions satisfy it. harmonic functions are also important in geometry.

but i know nothing about the wave equation.

but i recommend arnol'd for auxiliary reading in any course.

I checked out Arnol'd today and although the language is rough for me I think it will be a good supplement. The preface was interesting; I didn't realize PDE's were such a ruthless and improper branch of math. Thanks for the advice

 

[PowerIso]

Mathwonk, would you happen to know the type of math needed before a person studied chaos dynamics?

 

[kaisxuans]

huh?

Mathwonk, would you happen to know the type of math needed before a person studied chaos dynamics?

There is such thing as maths needed in chaos dynamics? Tell me about it

 

[Werg22]

Biology is this way >>>.

 

[mathwonk]

i searched on google and found some notes with this intro:

"These are class notes written by Evans M. Harrell II of Georgia Tech. They are suitable for an introductory course on dynamical systems and chaos, taken by mathematicians, engineers, and physicists. Students are expected to have completed two years of calculus and basic courses on ordinary differential equations, linear algebra, and analysis."

 

[PowerIso]

There is such thing as maths needed in chaos dynamics? Tell me about it

Well, I am going to take a course called Nonlinear Dynamics and Chaos I don't know much about it but the course info reads as followed: Dynamical systems associated with one-dimensional maps of the interval and the circle; elementary bifurcation theory; modeling of real phenomena.

 

[PowerIso]

i searched on google and found some notes with this intro:

"These are class notes written by Evans M. Harrell II of Georgia Tech. They are suitable for an introductory course on dynamical systems and chaos, taken by mathematicians, engineers, and physicists. Students are expected to have completed two years of calculus and basic courses on ordinary differential equations, linear algebra, and analysis."

Ah thank you :D

 

[ekrim]

Poweriso, at Umiami, applied math majors can take this two course sequence:

"MTH 515: Ordinary Differential Equations, 3 credits.
Linear systems, equilibria and periodic solutions, stability analysis, bifurcation, phase plane analysis, boundary value problems, applications to engineering and physics.
Prerequisites: MTH 311 and either MTH 211 or 310.

MTH 516: Dynamics and Bifurcations, 3 credits.
Bifurcation of equilibria and periodic solutions, global theory of planar systems, planar maps, nonlinear vibrations, forced oscillations, chaotic solutions, Hamiltonian systems, applications to engineering and physics.
Prerequisites: MTH 515 or permission of the instructor."

Hope this helps.

 

[J77]

Mathwonk, would you happen to know the type of math needed before a person studied chaos dynamics?

You could browse here: http://www.scholarpedia.org/

 

[teleport]

me,me, I want to be a mathematician! Ha, sorry for this, I'm just happy today. That girl likes me! You know who you are:!)!

 

[mathwonk]

hey hey hey ! congratulations! celebrate, take her to a nice restaurant.

 

[teleport]

I sure will! But I know one thing I should not talk about there: MATH!

 

[mathwonk]

you are wiser than your years.

 

[Kummer]

Why is this "I want to be an mathematician" thread much larger than the "I want to be an enginner" thread?

 

[teleport]

Well, did you also ask this in the eng thread? You should, this kind of questions sometimes get subjective answers. :) Also, check, you might be quoting something else. But no need rectifying yourself. I do know that this thread originated earlier (not sure of the time difference). Other than that, it might be showing some (maybe to you) interesting stat.

 

[teleport]

O LORD, school begins tomorrow. With it, four pure math classes +... May the hand of Gauss and Cauchy move my pencil in a constructive way.

 

[PowerIso]

Why is this "I want to be an mathematician" thread much larger than the "I want to be an enginner" thread?

It's simple, math people like to talk to other good looking people.

 

[teleport]

Ha, yeah. BTW you forgot to mention how we love when engineering girls come to ask for our help (I guess I should also mention the mechanic for the other sex to be fair, but assume it implied)

 

[ekrim]

Ha, yeah. BTW you forgot to mention how we love when engineering girls come to ask for our help (I guess I should also mention the mechanic for the other sex to be fair, but assume it implied)

That's why I'm in engineering; for the chicks man.

 

[teleport]

That's why I'm in engineering; for the chicks man.

Ha, peace and good luck.

 

[qspeechc]

I am a first year student, and I would like to major in pure maths. But here's the problem: the 1st year maths course is boring, in my opinion. It is 70% calculus, and I find calculus a dry subject. The other topics covered, such as vector, binomial theorem and such, those were more interesting. If I don't find the first year work interesting, does that mean maths is not for me? I feel like I am in a real crisis, because I thought I loved maths, but this first year maths course has really bored me. I feel it is too much routine. As a result I virtually never work. I always study for test for a few hours the night before, and I do ok, I am averaging 74%. I have even considered switching to mechanical engineering. Is maths still for me?

 

[Kummer]

I am a first year student, and I would like to major in pure maths. But here's the problem: the 1st year maths course is boring, in my opinion. It is 70% calculus, and I find calculus a dry subject. The other topics covered, such as vector, binomial theorem and such, those were more interesting. If I don't find the first year work interesting, does that mean maths is not for me? I feel like I am in a real crisis, because I thought I loved maths, but this first year maths course has really bored me. I feel it is too much routine. As a result I virtually never work. I always study for test for a few hours the night before, and I do ok, I am averaging 74%. I have even considered switching to mechanical engineering. Is maths still for me?

So then just take final exam to skip all those courses. That is what I did, it saved me a lot of time.

 

[PowerIso]

I am a first year student, and I would like to major in pure maths. But here's the problem: the 1st year maths course is boring, in my opinion. It is 70% calculus, and I find calculus a dry subject. The other topics covered, such as vector, binomial theorem and such, those were more interesting. If I don't find the first year work interesting, does that mean maths is not for me? I feel like I am in a real crisis, because I thought I loved maths, but this first year maths course has really bored me. I feel it is too much routine. As a result I virtually never work. I always study for test for a few hours the night before, and I do ok, I am averaging 74%. I have even considered switching to mechanical engineering. Is maths still for me?

Averaging a 74% and wanting to major in my math is difficult. You have to do the dirty work before you can get to the real fun stuff. I can only speak from my personal experience, but I find even if a person doesn't like calculus, if they want to major in math, they give it a good go to get the grade and be able to go to graduate school. Just motivate yourself and keep pushing forward. However, I have to ask, why did you want to major in pure math?

 

[mathwonk]

unfortunately many first year calc courses are not taught from the viewpoint of future math majors. that's why at uga, and chicago, we have a special course for them taught from spivak, and taught by outstanding profs.

not all courses do a good job of rpesenting what the subject is about. math is really not dry. take a look at what is mathematics by courant and robbins, or some of the many books recommended earlier in this thread.

this thread is now so lengthy that many questions asked here are already answered in earl.ier parts of this thread.

have you read the general guidelines and advices which began this thread on becoming
mathematician? i recommend it. in fact peruse the whole thread.

surprizingly, since there are now 900 posts, i easily reviewed the entire thread recently in a short amount of time. of course i did not reread every word of my own advice.

 

[qspeechc]

However, I have to ask, why did you want to major in pure math?

Because before I hit University, I used to love doing maths problems. Not necessarily ones covered in the high school syllabus. The joy of solving a difficult problem that you've been at for a long time! The excitement of arriving at a simple answer to what looked like a comlpex question. The joy of finding connections and the way the mathematics works! I used to love maths, but this first year maths course has really bored me, and now I think majoring in pure maths is not for me.

unfortunately many first year calc courses are not taught from the viewpoint of future math majors

I agree, that is the way with most of my first year courses.

have you read the general guidelines and advices which began this thread on becoming
mathematician? i recommend it. in fact peruse the whole thread.

Do you mean this very thread? If so, I think I have read the first page or two, but I will go back now and read more of the thread.