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I am a Master's student in math who is doing pretty well so far. I aced my first two classes (real analysis) and my summer session in probability theory is going well, too. Even before my Master's study, I already thought of a PhD. Nonetheless, there are concerns:

  1. My good grades were the result of grit, not brain, meaning that I had to spend time to understand what the book is telling me. Furthermore, a lot of times during exam and homework, I had to try and fail a few times before arriving at the solution.

  2. I did not study math during undergrad. Before my Master's study, I gulped down three semesters of calculus, plus LA and DE in one year. So compared to other students, I am already behind. In the next two months, I will learn complex numbers on my own and review my linear algebra.

  3. I am a professional in a field that has nothing to do with math or research. After the holiday, I will speak to my advisor about taking a thesis class. It's not an insurmountable obstacle, but, in an unrelated field, I do have less time to concentrate on math.

At the end, how big is the jump from Master's to Phd? Especially after what I said in (1), I do worry that it is beyond my ability. I do not hold anyone responsible for my decision. So, please kindly offer your best assessment of my situation.

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    I don't think a bunch of strangers on the Internet can possibly answer this question for you. Do you know a professor in real life who will be honest with you? – Austin Mohr Jul 1 '13 at 2:17
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    My good grades were the result of grit, not brainDo not listen to the Impostor Syndrome! Everybody has to try and fail a few times to truly learn anything. – JeffE Jul 1 '13 at 5:28
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    Coursework is one thing; doing a PhD is about doing research. You can be great at coursework and not be a good researcher, and vice versa. Don't let struggles with coursework make you dispirited about doing research. – aeismail Jul 1 '13 at 6:32
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    Difficult enough that you won't really understand until you do it yourself :) – cartonn Jul 1 '13 at 21:47
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    I wouldn't worry about abilities. Maybe it is not the right question: the right question what do you want to do (after)? Do you need a PhD or helps to you? Be careful: a PhD doesn't guaranteeing you a job or higher payment. – Greg Aug 1 '14 at 7:27
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I'll answer your title question and sub-questions in three words (then add to that): Difficulty is relative.

Noam Elkies finished his PhD in math at the age of 20. The average graduate student would probably take anywhere from 4 - 5 years. That doesn't mean Noam was a god among graduate students when he was a graduate student, and that's the reason he was able to finish his thesis faster than Joe Schmoe (though he is quite exceptional) and Joe Schmoe should just give up. Far from it. The fact that you're doing well in your master's courses is a good indicator you're well qualified for the coursework part. Working on a PhD thesis means producing original research. Producing original research means dealing with not so cut-and-neat problems like ones you'd encounter in your homework or textbooks. It means working with problems that you're not sure are close-ended, open-ended, or neither. In other words, different skill sets are needed for research in addition to the problem-solving heuristics which you've already acquired from your coursework.

Since you've never wet your foot it seems to me you have no indication of your ability to do research . So what do you do? Just go for it! You only live once as far as we know, and it seems to me that your real problem is your self-imposed doubt. Get that out of your head. Research is about staring and thinking about a problem long enough then coming up with an answer which turns out to be completely wrong and in that, the process repeats until you get your "Eureka!" moment.

You should also stop worrying about others being ahead of you. There'll always be someone faster and better than you (unless your name is John Von Neumann :-) especially in academia. A good algebraic geometer from UGA by the name of Roy Smith didn't get his start in advanced mathematics until his late 20's, after working in industry as a meat lugger. He received his PhD at 35. Over 30 years later, he's still going strong. What you should learn from him is if passion calls for it, then there should be no stopping you. If you fail at a career, then that's that. It's not the end of the world. But if you never bother to even try, then you'll never know and you'll always have a "What if?" moment.

So just to sum all of this up: Apply to a PhD program and see what happens.

https://mathoverflow.net/questions/7120/too-old-for-advanced-mathematics/45644#45644 (if you're interested in Roy's post)

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There are a few different things going on in this question, I'll try and address them separately.

First of all, heed JeffE's words about the impostor syndrome. The idea that some people are effortly geniuses (which is inevitably followed by "but not me") is untrue and distinctly unhelpful.

Moreover trying and failing a few (or many!) times before coming to a solution is the very definition of doing research! The important question you should ask yourself is if you enjoy the challenge of struggling against a problem; if so, then research would be a great fit.

As for your background, you seem to have a plan to address the gaps in your background, and if you're doing well in your master's programme, I wouldn't worry too much about being behind the other students. Another challenge may be if you have a full-time job; doing a Ph.D. with a fulltime job is by no means impossible, but requires a some great time management skills. Anyway there's plenty of resources on this forum to help with this issue.

Finally, as for the question in the title, I would say it depends the most on what you plan to do with your Ph.D. after you graduate. I'm a recent graduate (in math) from a well-known North American school, and the quality of the Ph.D's varied greatly. I think it really depends on what you want to do with the degree. If getting a Ph.D. is the farthest one wants to go in academia, the standards for graduating are pretty low, and I think can be achieved not too strenuously. If one wants to be competitive in the academic research job market, the bar is quite a bit higher.

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As Austin Mohr, pointed out, it is unlikely a stranger on the net fully answer this for you, but what you are experiencing is similar to what I experienced, so I can tell you how it was from my perspective.

Given what JeffE commented, studying through true 'grit', does involve 'brains' as you are developing and applying problem solving skills and stategies - which has clearly been effective, as you mention, you have good grades. So, it seems you have developed effective study techniques.

Point 2 is pretty much not an issue, based on the skills and strategies you developed in point 1.

If you do an unrelated thesis, as you mentioned in point 3 - look at this way, you'll be developing the research and synthesis skills needed for a PhD.

What I found about the transition between the Masters and PhD, was that it wasn't so much of a leap, but a case of using the skills developed a lot more thoroughly.

I hope this helps.

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In your shoes, if I were applying for a PhD program, I might be thinking in terms of an "ABD" (All but dissertation.)

From the sound of it, "grit not brain" will get you through your PhD courses. You likely will pass the "comprehensive" examination. And then the fear is that you will "freeze up" when it comes time to write the dissertation because you are behindhand in "natural" (as opposed to synthetic), talent compared to others.

You will have to balance these real concerns against the benefits of "taking courses" and getting as far as you can, before possibly running up against a brick wall with your thesis.

This was basically my story (some decades ago). I was "counseled out" of a PhD program by a dean who observed that I had the preparation and the brains for the program, but lacked the "spark," "thirst," or drive that would see others through, but maybe not me.

What's worse, as a history undergraduate with mostly As, I had trouble finding a senior thesis topic, and spent the first part of my senior year wondering if I would be able to graduate, and thinking about writing a "trivial" paper that would earn a C. In the middle of the first semester, I stumbled on a viable topic that earned me departmental honors.

I can see one of three outcomes for you. 1) A "light bulb" will go off in your head at some point before you approach the end of the PhD program. If this happens, you're fine. 2) You will somehow squeak through, not quite knowing how you did it, and come out of the process a bit shell-shocked. 3) Neither of the above will happen, as you feared, in which case you might "bail out" as an "ABD."

I can't predict the outcome for you, but am sharing my experience. Just understand what the risks are, take into account your circumstances, and (hopefully) make the best decision for you.

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