I feel a kinship with those who have a passion for math and I want to help you with your decision. Your passion for mathematics is a vital factor in achieving a Ph.D. in that subject. Of course, a high IQ, including a capacity for abstraction, is also a necessary criterion because you will be in a field that attracts the very brightest people from around the world.
One of the best ways to assess your mathematical potential is to review your academic performance in mathematics as well as your performance on high school math olympiads and on the undergraduate Putnam Contests. Of course, these provide only a rough approximation of your mathematical ability, because some people are late bloomers and their academic performance is not evident in the early stages of their career. Other extremely intelligent people do not perform well in competitions because their problem-solving abilities are constrained in environments with time limits. They perform best when they can ruminate over problems extending over long periods of time. So, you will need to factor these caveats into your decision.
Prior to conducting research for a Ph.D., your study of mathematics involved learning and applying mathematics that had been previously created. You studied theorems and their proofs and then used the theorems to solve other problems. However, when you embark on a Ph.D., you are attempting to expand the frontiers of mathematics, by solving previously unsolved problems or creating new theorems. This is the point where imagination and creativity enter the picture and you have to get a solid grasp of everything that is known about a specific problem and combine it with a more global awareness of the mathematical tools you can apply to its solution.
A mathematician once asserted that achieving a Ph.D. is mathematics is harder than a Ph.D. in chemistry, because a chemist predicts that A combines with B to produce C. If his prediction is incorrect and the result is D, he publishes a thesis titled, “A combines with B to produce D.” If a mathematician makes an assertion that theorem A and theorem B imply a new theorem C and it turns out to be untrue, the mathematician comes up empty and has to begin anew.
So, if you embark on a Ph.D. in mathematics, you must have a strong tolerance for failure and an unrelenting resolve to succeed. This may occur amidst concerns about finances, connections with your thesis supervisor and the lurking fear that you may come up empty. Such trials and tribulations associated with repeated failures, false starts and perceived triumphs were beautifully, and humorously described by Paul Halmos in his book titled, “I Want to Be a Mathematician.” In that classic, he describes the rollercoaster ride of thinking that you have solved a problem and then discovering an error in the proof. And then after numerous false starts, you discover what you believe to be a rigorous proof of your theorem. Then you go before a thesis committee of experts in your field who have perused your thesis with a fine-tooth comb, and you hold your breath that you and your thesis supervisor have not overlooked a small glitch that will collapse the entire edifice.
As long as you are psychologically prepared for this challenge, and are confident in your mathematical ability, it’s a journey that will increase your capacity to deal with adversity. Equally important is the confidence you will gain in your ability to understand any concept as long as you invest the time to master it. Most importantly, if you secure an academic appointment you will be able to pursue your life’s passion. All the things worth having in life come from extreme effort and intensity, and if this is your dream and you believe you have the capability to achieve it, I wish you luck and celebrate your willingness to embark on a significant challenge.
