Q:

How to tell if I'm good enough for graduate school?

I'm concerned about my level of preparation for graduate school. I have decent grades (3.64 GPA, 4.0 in math), great recommendations lined up, and some OK research experience (no publications), but I really don't feel like I'm ready for the next step. 
I'm interested in applied math, and was a physics/computer science double major for most of undergrad, switching into math just last semester. I'm taking Analysis 1 now as a first semester senior, and it is the only pure math class I've ever taken (though I've taken a bunch of applied courses and semi-pure courses, like advanced udergrad ODEs). I'm doing very well in the course, and I'm really enjoying it, but I feel that, because I'm so late in joining mathematics, I'll be really behind everyone when I get to graduate school, if I do get in somewhere decent. My scores on practice tests for the math GRE are in the mid 600's, and that makes me feel like a phony as well (although I haven't really studied that hard yet, maybe they'll go up? Also, its a good thing many applied programs don't care about the subject test...). Sure, I have some skills from my background that other students don't have, such as non-trivial software development and an ability to solve difficult physics problems, but when I look at the profiles of the 'good' people on this website, or talk to my friends that have been doing math olympiads since they were 3, I feel like there is just no way I can be good enough for a career in mathematics given my 'late' start.
Is it possible to catch up with others at this stage? Will I be doomed to a life of constantly being behind those who are smarter than me? Has anyone ever built a strong career from my kind of situation before?
I just wish I had some more time...

A:

At my university we use baby Rudin, Dummit and Foote, and Munkres in the first year analysis, algebra, and topology courses (respectively), which is the average schedule for an incoming grad student.  You can also test out of these and move to the 2nd year if you happen to be good in one of those fields (or two, if you're really exceptional).  My opinion is that most math majors could handle this schedule given an appropriate amount of studying.  Worst case you can opt out at a master's if you decide you don't want to do research.
As far as being stupider than other people goes: of course you are.  Unless you're the smartest person on earth, there are smarter people, and if you spend your time sitting around thinking about who's smarter than who, they're all you're going to pay attention to. If you're getting a math degree, you're probably pretty smart, and that's good enough.
I embraced my own stupidity long ago, and in doing so, I gained the ability to ask questions without losing face.  Instead of worrying about other students knowing I'm not a genius, I can talk my problems out with them, gain their insight, and learn from my own mistakes.  For whatever reason, people in the mathematical field have this tendency to try to convince you that you're too dumb to do anything. Don't listen to them.  If I miss a proof, I'll get mad, reread the chapter, work book problems, harass the professor, and stop sleeping until I understand everything. Be stubborn, not smart. You've got to have grit.

A:

I will take your question in reverse. 
We all wish we had enough time. I certainly wish that I could live to see the Starship Enterprise, and I would be happy to mop the floors there if I could be so fortunate.
Yes (emphatically) you will be doomed to a life of constantly being behind those who are smarter than you. You really need to accept this because it is reality. You may be suffering from a very common thought paradigm. Many people, especially when they are younger, base their own self worth on their ranking among their peers. The truth of the matter is that no matter how good, talented, or gifted you are, there is probably (degree of probability very near unity) someone that is much better than you. 
Stop competing with your peers. Take neither pleasure nor pain in how your performance compares to your known peers. Shut the thought of comparing your own performance with that of your known peers out of your mind forever, and take neither joy nor sadness in these sorts of results in the future. You need a real peer to compare your self worth and performance against. I can tell you from very deep experience that this peer is you. You can spend the rest of your life trying to outdo yourself, and you will actually succeed sometimes. After that success you will still have your biggest challenge in the mirror. If you want true joy out of success in life, then push your own envelope of ability, and when you "outdo" yourself, take pleasure there. You really are and will always be your own greatest competitor, take my sage advice here. 
I have a secret for you. I will share it here on stackexchange, but as it turns out, most people will not actually take advantage. By definition, most people are mediocre. This applies to the smartest people ever born. Einstein was probably to the right of 4 standard deviations in antimediocrity, who was known as a genius (that sucked at math). He actually was bright, but his true gift was that he shot for the stars. You can take a lesson from what I say, and take comfort in the fact that most people, even the brightest, are mediocre in life. All of the natural born talent in the world rarely results in an Einstein. The bigger factor usually breaks down to a willingness to do the hard work. You can top the brightest of minds just by shooting for the stars in your own effort. Attempt to outdo yourself and only yourself daily. 
GRE tests are crap. Shut that whole deal out of your mind. Most schools do not take them seriously, they just require them.
When you graduate, you will be ready for graduate school. By virtue of your diploma you have actually prepared for graduate school. Shut this malarkey about other people being better, or the things that you do not know out of your mind. 
That being said, you should do a few things to prepare, and you should start doing them tonight and for the rest of your life. 
Do not be a mediocre slackard like the average person. When given a choice, take the harder class and not floating in the swimming pool 101. 
When you are taking a math class (or any class) do not wait for a teacher to tell you what to learn and when. Come up with your own questions, and enjoy the struggle of discovery. Grab knowledge by the cohojones!
Do not be afraid of failure. Do be afraid of not doing your best.
You definitely need to do well in your classes when you are taking classes. After those classes, if you really want to get strong in math then get yourself into the habit of doing about 6 hours per day of just math, and plan to continue this for several years (until you decide to retire). Review your studies, explore new ideas, read, write, solve problems, and attempt the proofs. Euler did it until the day he died. We call him a genius, and I am absolutely convinced that Euler was one of the greatest mathematical minds in all of history, but I suspect that much of his success came from the fact that he spent all of his days doing the math with a spirited unquenchable passion for all things math that continued even through blindness and ended on a gentle summer day when he probably died in the midst of a great problem in applied mathematics. 
Physics and computer science are actually ideal, probably optimal skills for a degree at any level in applied mathematics. Capitalize on this. Programming skills are especially important, and you should not let them go away. You will definitely need them, so do not discount them. Computer science is really a domain of mathematicians that became its own offshoot. We developed the first computers, the first languages, and most of the greats of computer science were actually mathematicians. Just being able to program will give you a serious edge, so do not discount this.
Go to school anywhere they will take you my friend and don't look back. 

A:

It feels good to be sunny and encouraging when giving advice.  But if one gives sunny, encouraging advice regardless of the situation then it seems to be of little use other than as a pep talk.  If one gives sunny, encouraging advice when the situation is actually rather bleak, then this could actually be irresponsible.
There is also the question of who is giving you the advice.  Anonymous answers to math questions are worth almost as much as any other kind of answer: you can assess for yourself whether the answer is correct.  Anonymous career advice seems to me to be worth little.  In this case, I am concerned that the OP is asking advice about a PhD program and a career in mathematics, and it seems to me that many of the answerers have not completed a PhD program and/or do not currently have a career in mathematics.
I got a PhD in mathematics at Harvard University in 2003, and I am currently a tenured, associate professor in the mathematics department at the University of Georgia.  I have already directed the PhD thesis of one student, and I have two students currently doing thesis work under me.  I have also been on the thesis committees of about a dozen more students over the last few years.  I tell you all this so you can weigh my credentials against the following pronouncement:

The academic job market in mathematics is currently extremely tight.

People usually like to say that the market is cyclical, meaning that if it is bad now one can wait a few years for it to get better.  I think it is true that there is a lot of variation in the market: when I was on the market for tenure track jobs in the 2005-2006 academic year, it sure seemed like there were more jobs available then than there had been only a few years before.  (In fact, in 2003 I went on the postdoctoral job market.  From all the jobs I originally applied to I got exactly one offer, and that came late enough for me to worry that I was not going to get anything whatsoever.  I reiterate that I was graduating from Harvard University, and my thesis adviser was famous and influential even by the standards of the faculty there.)  But proponents of the "cyclical" theory will usually not go so far as to suggest the period: in other words, so far as anyone knows, the job market will be as bad as it is now or worse for five years or more.  Most people seem to agree that a necessary condition for serious improvement is for the local, national and international economies to substantially improve: exactly when that will happen is anyone's guess.
When I started graduate school in mathematics I had a similar attitude to the OP: I wasn't looking farther ahead than getting a PhD in mathematics.  That seemed to me to be an extremely daunting goal: it seemed somehow presumptuous and silly to plan farther ahead than that.  But when I knew I was about to graduate and feared that I would have no academic job whatsoever I cringed at my previous naivete.  I remember well that my sister called me and tried to cheer me up: "Isn't it good enough for now just to have a PhD in mathematics?"  That was a good question.  I reflected upon it and the answer was a resounding no.  A PhD is a preprofessional degree par excellence: it prepares you for an academic research career.  If you can't continue on in that career, then having the PhD seems worthless.  Well, it might be worthwhile if you had a backup plan (or even a different plan A) as to what to do with your PhD in mathematics.  I most certainly did not.
Thus one thing I would tell the OP is that he is not asking exactly the right question.  As Betty Mock indicates, sure, there are places where a minimally competent person (and even the little information the OP provides convinces me that he is more than minimally competent; a score in the 600's on the GRE math is distinctly above average, and many of the weaker programs do not require the exam at all) can be assured of getting a PhD.  The question should be

Why do I want a PhD in mathematics?  What will I do with it afterwards?

If you have a specific professional career in mind for which a PhD is either required or very helpful: great.  Please keep that goal in mind at every step of the process, especially the application process.  If you are doing it because you want an academic career in mathematics...please think carefully about what I've said and how tight things are right now.  I know a lot of extremely bright people who played the academic game for as long as they could and eventually had to cut their losses.
If you run the numbers in American academia, you see that math departments everywhere are fighting tooth and nail to replace each departing faculty member's line.  On the other hand graduate enrollments are not significantly shrinking and many departments without PhD programs are trying to start up small PhD programs.  The arithmetic is therefore discouraging.
Although getting a PhD from a top program is definitely not a guarantee of future employment and getting a PhD from a lesser program is not a death sentence, statistical studies show that a much higher percentage of graduates from the top ten programs get desirable academic jobs than graduates from the next forty or so programs, and much beyond the top 50 programs it is possible but quite unusual for graduates to pursue academic careers (by which I mean involving some mixture of research and teaching).  So here is some less sunny, but I hope more responsible advice to the OP:

Apply to a wide range of schools.  Apply to some top ten programs -- why not?  Apply to some which are just outside of the top ten but still very good.  And so forth: I would say not to apply to schools which are not in the top hundred unless there are very exceptional circumstances.  Then based on where you get in, you can decide whether or not you are "good enough" for graduate school.
My graduate program is about the 50th best.  We get some very good students, students who could go on to serious academic careers if the market would bear it.  But we are having trouble placing our excellent graduates in academic jobs, and every year some people leave academia who didn't want to.  We have a few graduates, including recent ones, who are doing extremely well.  So getting a PhD from the 50th best program and then trying to make an academic career is a real struggle, but it can be done.  If I were you, I would think carefully about whether or not to enroll in a program like UGA's and try to figure out how to "beat the odds" from the very beginning.  If you can't get into a program as good as UGA's, then in my opinion it would be a better career move to spend a year improving your dossier and trying again to get into a more reputable place.

