Should I Become a Mathematician?

[Mandelbroth]

I'm having trouble understanding how to apply specific topics to specific events. For example, I enjoy solving systems of linear equations, matrix operations, and the like; however, I have no idea how this knowledge can translate to a research topic, job, etc.. Basically, I understand the application portion when I'm looking at textbook examples, but cannot seem to come up with my own applications.

Bottom line is, I really enjoy linear algebra and numerical analysis, but have little idea how to use these outside of the popular applications (cryptography, computational fluid dynamics, etc.).

Any thoughts are greatly appreciated.

Does mathematics need to have applications?

If I may give my opinion amongst the more experienced-backed opinions of the others who are probably better to answer this, you're fine. There is a difference in severity of the problem (see the following examples), but if I'm understanding you correctly, you should be alright.

There are two extremes for this kind of situation. If we have a problem like...

Solve the following system of equations: \begin{matrix}x+y=2 \\ x-y=4\end{matrix}​

...and you have trouble applying methods of linear algebra (or elementary algebra, for that matter) to that, you're probably in trouble.

However, I gather that you might be somewhere near the other extreme. If you look inside a physics book containing advanced topics such as relativistic necromancy (note: not an actual physics topic) and don't automatically think "I can apply eigendecomposition to this matrix and create a whole new subfield of relativistic necromancy!", you're probably okay.

 

[dkotschessaa]

Mathwonk and others,

This is a bit personal and I was going to journal it for myself, but I'm putting it out there despite the exposure.

I realized this morning (while meditating actually) that I still have a lot more anxiety about mathematics than I realized. I am in my senior year now and considering graduate school (at least a masters).

I realize that there are people who suffer from "math anxiety" and typically because of this they do not like mathematics and avoid it. But what about someone who does like mathematics? I realize that even though I've done well in most of my previous classes (though not extraordinary) I still worry about failing the next one. "I did ok in the last class, but this next one is more difficult!"

I know we're not psychologists here but I was wondering what your read on this is and how you may have experienced it. Some beliefs that are hanging me up are "I'm too old (37) to be doing this." "I am not naturally talented." and so forth. These are deep seated and I know consciously they are not a fact. I want to relax into it and enjoy it more.-Dave K

 

[Crake]

Mathwonk and others,

This is a bit personal and I was going to journal it for myself, but I'm putting it out there despite the exposure.

I realized this morning (while meditating actually) that I still have a lot more anxiety about mathematics than I realized. I am in my senior year now and considering graduate school (at least a masters).

I realize that there are people who suffer from "math anxiety" and typically because of this they do not like mathematics and avoid it. But what about someone who does like mathematics? I realize that even though I've done well in most of my previous classes (though not extraordinary) I still worry about failing the next one. "I did ok in the last class, but this next one is more difficult!"

I know we're not psychologists here but I was wondering what your read on this is and how you may have experienced it. Some beliefs that are hanging me up are "I'm too old (37) to be doing this." "I am not naturally talented." and so forth. These are deep seated and I know consciously they are not a fact. I want to relax into it and enjoy it more.


-Dave K

Hey, sorry that I'm not going to address your issue. (I don't have anything to offer you, honestly. I believe it's best to wait for the "pros").

I'd like to know more about your meditation habits. I'm thinking of starting to meditate, but I'm not sure if it's going to change anything tbh. Would you say meditation helped you? How so?

 

[mathwonk]

well i still have math anxiety, e.g. before posting on mathoverflow. once e.g. i asked a question about what some fancy theorem in algebraic geometry means. the first comment was from someone who was astonished that I didn't already know, because i am supposed to be an algebraic geometer. If i couldn't handle looking dumb like that, I would never get my questions answered.

The point is we are all ignorant but we are in there striving because we are interested in learning. I have occasionally also explained a few things to some really smart people who just didn't happen to know that one thing.

We are often afraid we will look dumb by asking a question, but actually one of the best ways to learn from someone is to let them look smart, by explaining what they know to us. people love to answer questions when they feel smart by answering them. they appreciate our giving them the chance to enlighten us, provided we allow them to enjoy the spotlight.

Why do you suppose so many people come on here and answer questions for free for years and years?

And I think meditation can be helpful in achieving balance and calm.

 

[Cod]

However, I gather that you might be somewhere near the other extreme. If you look inside a physics book containing advanced topics such as relativistic necromancy (note: not an actual physics topic) and don't automatically think "I can apply eigendecomposition to this matrix and create a whole new subfield of relativistic necromancy!", you're probably okay.

This part. I can work through problems (regular and applied) if the text "gives me the information". I just can't take linear algebra and apply it to something on my own, like the example you provided.

What are things I can do to help myself? Or do I just keep chugging at different topics I like and let it "come to me" eventually?

 

[Mandelbroth]

This part. I can work through problems (regular and applied) if the text "gives me the information". I just can't take linear algebra and apply it to something on my own, like the example you provided.

What are things I can do to help myself? Or do I just keep chugging at different topics I like and let it "come to me" eventually?

Most of the time, you just have to think about it long enough. A good example comes from my economics class.

The other day, we discussed elasticity of demand and the formula for revenue. I noticed that, if the elasticity was equal to 1, the revenue did not stay the same (by the formula we were given), dispite what we were told. I thought about it a little, and then I noticed that, if we took the limit of part of the equation for elasticity, we got a formula $\varepsilon_D=-\frac{P}{Q(P)}Q'(P)$, which rather obviously implied the statement about if the elasticity was 1.

It just takes some extra pondering, I think.

 

[Mpopyft]

Hey guys; can anyone recommend some tough textbooks for math and science high school and calculus level. Not the 100$ new ones but some old ones such as some listed in this thread already?

 

[mathwonk]

i like this old edition of thomas calculus, 3d edition, 1965. for under $5.

http://www.abebooks.com/servlet/Boo...=an=george+thomas&bsi=120&kn=calculus

 

[theoristo]

Hi,I would like to ask if anyone had seen this book
Gems of Geometry John Barnes https://www.amazon.com/dp/3642309631/?tag=pfamazon01-20 which seems to be a geometry fun textbook or is it?Geometry is a beautiful subject and my friend claim this book make anyone fall in love with it.

 

[Cod]

Most of the time, you just have to think about it long enough. A good example comes from my economics class.

The other day, we discussed elasticity of demand and the formula for revenue. I noticed that, if the elasticity was equal to 1, the revenue did not stay the same (by the formula we were given), dispite what we were told. I thought about it a little, and then I noticed that, if we took the limit of part of the equation for elasticity, we got a formula $\varepsilon_D=-\frac{P}{Q(P)}Q'(P)$, which rather obviously implied the statement about if the elasticity was 1.

It just takes some extra pondering, I think.

Nice work. From now on, when I go through specific subjects, I'll try to apply it to something on my own once I have a solid grasp of the information. Thanks for the advice.

 

[Mandelbroth]

I'll never understand how a person like Ed Witten majored in history.

"Let $n$, the number of presidents, be an integer..."

can learning topology help me to design electric and electronic circuits better? I have not finished analysis... But if topology helps me in some way to design efficient systems, then I could self study both analysis and topology in these four years of my electronics engineering...

I can't see how it wouldn't.

Nice work. From now on, when I go through specific subjects, I'll try to apply it to something on my own once I have a solid grasp of the information. Thanks for the advice.

You're welcome.

 

[modnarandom]

I think someone might have mentioned it earlier, but what did people who did Part III in Cambridge think about it? Why did you go there? Who would benefit from it?

 

[Prince1]

I am a high school student and I want to get the most rigorous math education available in algebra and geometry. I was thinking the SMSG books from yale univ, but that may be outdated (they use stuff like "truth sets"). How about this plan:
Starting of with basic math by lang
Algebra by gelfand
Lang's geometry/kiselev geometry
gelfand trigonometry
Gelfand and sullivan's precalculus/"graphs and functions"
Is this enough to give me the strongest, most rigorous background in algebra and geometry? Or should I consider the yale univ SMSG books as well? Thanks.

 

[dustbin]

I've had instructors who say "truth sets" so I don't think that aspect is necessarily outdated... I'm sure they would still make fine books (I've only read a little), but that's if you can stand the typewriter typesetting.

Here are some other "lists" for you:

https://www.physicsforums.com/showthread.php?t=307797&highlight=pure+math+high school
http://www.artofproblemsolving.com/Store/curriculum.php? (They have classes, too)

 

[Prince1]

Thanks a lot. The writing style isn't an issue. I have gone through AoPS, but it isn't too rigorous. So should I go with SMSG or my other list (lang, kiselev, gelfand etc)? Or a combination of both?

 

[dustbin]

I'd personally do the Lang/Kiselev list. Be sure to check out some of the other "theory" books on the first link a gave you. Particularly the ones on inequalities.

 

[mathwonk]

In my opinion, the best geometry book is euclid, and the best guide to it is hartshorne: geometry: euclid and beyond,.

 

[Tobias Funke]

the best geometry book is euclid, and the best guide to it is hartshorne: geometry: euclid and beyond,.

I agree. Also, the Dover edition has its own commentary with plenty of good stuff to go along with Hartshorne, which is a great book but not absolutely necessary (but if you don't have the Dover edition of Elements with the commentary, it might be necessary!). Whatever coordinate geometry you need, which obviously isn't in Euclid, is probably in Gelfand, although I haven't seen his books for a while.

 

[QuantumP7]

It's not going to impress Harvard

I'm curious. What WOULD impress Harvard or MIT or the other top math programs?

 

[Calabi_Yau]

This is an interesting thread. I'm a freshamn in college, studying Physics but right now I'm seriously pondering about switching to a maths degree. I have always been good at math, and every math teacher I had, told me I was talented at it. However, I got into physics mainly because I read 3 years ago Kaku's Parallel Worlds, and having watched many science tv programmes about the marvels and excentricities of the cutting edge theories in theoretical physics I decided that it was that I wanted to do.

Recently, I have read the book "The Man Who Loved Only Numbers" which portraits the life of the great mathematician Paul Erdös, and my attentions shifted to math again. Basically, when I read about maths I want to become a mathematician and when Iread the lectures of Feynman I want to become a phycist again. So I guess I'll be working on something related with mathematical physics.

The problem is that I don't know whether I should better major in physics and minor in math, or do the opposite instead, since in my country it's impossible to double major at once. Porbably I'm majoring in Physics, with a minor in maths concerning some topics about abstarct algebra, differential geometry and galois theory. But I really don't know. That's my story so far lol, I'd like to read about those who are passing through the same, or already have. It seems I will only get an answer through personal experience.

 

[Mandelbroth]

This is an interesting thread. I'm a freshamn in college, studying Physics but right now I'm seriously pondering about switching to a maths degree. I have always been good at math, and every math teacher I had, told me I was talented at it. However, I got into physics mainly because I read 3 years ago Kaku's Parallel Worlds, and having watched many science tv programmes about the marvels and excentricities of the cutting edge theories in theoretical physics I decided that it was that I wanted to do.

Recently, I have read the book "The Man Who Loved Only Numbers" which portraits the life of the great mathematician Paul Erdös, and my attentions shifted to math again. Basically, when I read about maths I want to become a mathematician and when Iread the lectures of Feynman I want to become a phycist again. So I guess I'll be working on something related with mathematical physics.

The problem is that I don't know whether I should better major in physics and minor in math, or do the opposite instead, since in my country it's impossible to double major at once. Porbably I'm majoring in Physics, with a minor in maths concerning some topics about abstarct algebra, differential geometry and galois theory. But I really don't know. That's my story so far lol, I'd like to read about those who are passing through the same, or already have. It seems I will only get an answer through personal experience.

I'm just getting started with Galois theory. I consider it a subfield (no pun intended) of abstract algebra. That being said, I haven't really seen any good real world applications for either abstract algebra or Galois theory. Differential geometry, on the other hand, is my candidate for a foundation for modern physics. Manifolds are an important part of contemporary studies of physics, so you will definitely want to take that. All three of them are beautiful subjects with many aesthetically pleasing results, though, so if you really like mathematics I would definitely advise taking all three.

I used to think I wanted to be a doctor of medicine. Then, I figured out that the real world is kind of boring to study. Math is where it's at. If you are really considering going into mathematics, I think you should go the distance.

 

[R136a1]

I'm just getting started with Galois theory. I consider it a subfield (no pun intended) of abstract algebra. That being said, I haven't really seen any good real world applications for either abstract algebra or Galois theory.

One thing you can study is coding theory. You'll see how things like finite fields and ideals are applicable to generate good codes.
For (finite) groups, they are very applicable in chemistry. Just google it and you'll find a lot of hits.

 

[George Jones]

I'm just getting started with Galois theory. I consider it a subfield (no pun intended) of abstract algebra. That being said, I haven't really seen any good real world applications for either abstract algebra or Galois theory. Differential geometry, on the other hand, is my candidate for a foundation for modern physics. Manifolds are an important part of contemporary studies of physics, so you will definitely want to take that. All three of them are beautiful subjects with many aesthetically pleasing results, though, so if you really like mathematics I would definitely advise taking all three.

The combination of abstract algebra and differential geometry is extremely important in theoretic physics. Continuous symmetries (both spacetime and "internal" symmetric in quantum field theory) are modeled by representations of Lie groups, which are groups that are both groups and differentiable manifolds, with the group operations being differentiable.

In fact, right now, I am reviewing the relationship between the spacetime Poincare group, its Lie algebra, and relativistic wave equations.

 

[IGU]

One consideration for you might be that you can't do physics without math, but you can do math without physics.

 

[Calabi_Yau]

One consideration for you might be that you can't do physics without math, but you can do math without physics.

That is correct, but I think those who start in physics can change to maths easier than those who start in maths can change to physics. That is, in my opinion, because during a physics course you acquire the basics and the the skills necessary to do maths (although with less rigour). But if you finish maths and want to pursue physics, you'll have a greater deal to catch up, you may be an ace in mathematics but know nothing about the underlying principles of mechanics or electromagnetism for example.

 

[deluks917]

I'm curious. What WOULD impress Harvard or MIT or the other top math programs?

For Grad school doing well on the Putnam is considered very impressive by about half the professors at top schools. The other half think it looks good but is somewhat overrated.

However impressing half the professors at these schools is pretty likely to do a lot for your admissions chances. However doing well on the Putnam is exceedingly difficult.

 

[Seydlitz]

This is perhaps already asked before, so excuse me if I have not conducted a search beforehand in the thread, but my question is this:

How can one prepare for international sort of competition like Putnam, and IMO? In this case, I don't even dare to think to solve the majority of the problem, I just want to know what topics or what one should learn in order to be able to solve at least one or two questions in the competition, considering that their level are significantly higher in comparison to ordinary math problems given in textbook and day to day activity?

To deliver the point further, I don't even understand what is being asked by the problems (I just skimmed through one Putnam past paper.) I've never learned formal math so to say beyond application of calculus in high school and A-Level, but when I read through physics olympiad question I know at least what the question means even though I don't know the answer to it.

Can these advance problem-solving skills be learned? Again, I don't even think of participating in those competitions, but I'm hoping to learn some of the skills that could be eventually useful in my university study.

 

[QuantumP7]

I asked a Putnam Fellow this question. He said that the best way to do really well on the Putnam is to practice. Go over the old questions, and practice a lot! I'm going to do this all of 2014, and take the Putnam in December 2014. I'll let everyone know how it turns out.

 

[Cod]

Not sure if its been posted, but here is a link a lot of math and computer science book reviews (more in-depth than an "everyday book" review) done by multiple university professors from around the globe: http://www.cs.umd.edu/~gasarch/bookrev/bookrev.html

The focus is more on CS, but there are a good bit of math books.

 

[superduck]

about studying mathematics: questions

Hello,
I am a Japanese student of university. I am a philosophy of science major. But, to tell the truth, I really want to be atheoretical physicist. Unfortunately I have big lack of mathematics and everything academic skills because of I have got a mental illness sinse I was a high school student. But, I'll never give up my deam to be a theoretical physicist. Then, I am studying mathe matics by myself ( I am in correspondence course). I have to start from high school level math. You recommended several books. It is very helpul. But, I want to ask you about geometory textbook. Are thete any good books? At the moment, I am thinking to use "Foundation Mathematics" by K Stroud. Do you know this book? If you know this, is this book useful for studying high school math? I have another question. Which is better way to study mathematics, to use a thick multiple textbook like which is carried algebra and trigonomketory and geometory etc,or separated books which is carried one topic specially?
I'll be grad if you answer me.

 

[matqkks]

Use Engineering Mathematics Through Applications by Singh because it has complete solutions online to all the questions.
For Linear Algebra use 'Linear Algebra Step by Step' by Singh. Again it has complete solutions to all the problems in the book so ideal for distance learning.

 

[TheKracken]

Hello, currently I am at a community college and after tons of reading and thinking I have decided I want to be a math major.
Anyways, I also want to join the military for one term (usually 4 years), this is a something I want to complete for many reasons including the honor, the family tradition and just in general feeling responsible for serving my country.

Would it be best to join now that I have 15 college credits and would go up a rank or would it be better to join after college when I would be an officer. My goal would be to go back to academics and possibly get a Phd in pure mathematics, but I feel like a 4 year term in the service would cause me to forget most of the material.

Does anyone have anything to say about this topic? I have also considered going the NSA route to serve my country, but it just isn't the same.

Thank you everyone.

 

[homeomorphic]

Would it be best to join now that I have 15 college credits and would go up a rank or would it be better to join after college when I would be an officer. My goal would be to go back to academics and possibly get a Phd in pure mathematics, but I feel like a 4 year term in the service would cause me to forget most of the material.

I'm not sure, but throwing 4 years of military service would be making an already extremely difficult path even more difficult. You need recommendation letters to get into grad school. That could be tricky if the last time you took a math class was 4 years ago.

I'm not sure you forget all the material, though, if you know what you are doing when you learned it. I can't comment much on math, since I never stopped doing it, but I'm working on programming a game right now, and I basically can still program, even though I didn't really do any programming for the last 8 or 9 years. So far, I've barely had to look anything up. That's from taking 3 programming classes. Sure, I'm a bit rusty on some stuff that I haven't had to use yet, but I'm sure it will come right back. Plus, programming is not one of the subjects I did the best job of learning--most of the stuff I've forgotten could probably be attributed to lack of understanding of the motivation (i.e. what's the point of object-oriented programming, and how does it help you in concrete situations?). With the basic stuff like iteration and functions, it's easier to remember because you see why it's useful and as soon as you think about writing a program that does this or that, the need for them is obvious--that, and because it's simpler, and you use it over and over again if you take the next couple CS classes. I actually think taking a break from programming after the first two classes and then having to remember it later when I took data structures is one reason why I still remember a lot of it now. It almost seems like I know it better than when I was taking that data structures class, having to remember back to the previous class a couple years earlier. When you have to work to remember, that's one of the things that implants things in long term memory more firmly.

You just have to have a good strategy for making it stick. Learn how long term memory works. If you really know how to learn, the knowledge lasts a lot longer. So, that could be a possible solution, if you can figure out that puzzle of how to make the best use your own mind.

 

[drewb]

Wow! This thread is really comprehensive... and humbling. I have a long way to go if I want to become a mathematician.

I'm just finishing up my BS in Astrophysics. I'm thinking about making a thread asking for advice on what to do next. :P

Thanks for all this!

 

[mathwonk]

bless you. and good luck!