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I am interested in math and have been thinking about going for grad school and becoming a research professor since graduating from college. However, a few things that happened recently really make me hesitate a career in academia. Let me quickly describe two recent incidents that changed my mind

I recently met a math phd from an Ivy league school. We together read the proof of Radon Nikodym derivatives on a Tuesday night and a week later, when I called him, what surprised me was that he remembered every single detail of the proof, even the page number, the notations and symbols used by that book. The proof we read relied on a few other theorem named in numbers like thm 12 or thm 14. What amazed was that he even remembered the numbers of the thms and could perfectly match the thm number with the statement and he has never read that particular version of the proof. Well, maybe many of the people here can do this easily, there is no way I could remember things that well. Furthermore, even at his Phd program, few people land assistant professor positions even after years of postdoc.

I am taking a summer math class heavily relying on matrices. The professor teaching the class does all of his writing for the class on his tablets and we see it through the projector. One program with his software is that it is hard to scroll back in pages; so once he is done with one page, he does not like scrolling back to the previous ones. Many times, I noticed that he could remember every single number in a 3*3 matrix calculated a couple of pages ago perfectly without making any mental effort to memorize those numbers. That is something I could never do and I was quite impressed with his memory. So I went back and checked his CV. It turned out that he was very successful when he was my age. In fact, he already got his phd in math when he was 25 years old from MIT. I am 25 years old now and I am not close to getting into a decent math phd program.

Many great scientists including mathematicians are genius people. John von Neumann has graphic memory, Einstein has IQ of 190 and the list goes on and go.

So practically speaking, do people with average intelligence really stand a chance to be successful in academia? Is it worthwhile for someone like me to spend years in life to pursue a phd in math in the hope of making a research faculty?

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closed as primarily opinion-based by Will Jagy, Cameron Williams, user223391, M. Vinay, Gerry Myerson Jun 12 '16 at 4:33

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ I think when you live and breathe mathematics, recalling details becomes exceptionally easy. Once you have a large library of different proofs, techniques, and research papers in your brain, you remember how a particular piece of work was different from everything else more easily. Chess masters can remember a board layout perfectly, but only if the board was an actual game and not just randomly scrambled. $\endgroup$ – JasonM Jun 12 '16 at 1:07
  • $\begingroup$ The question, as it stands, is less about math and more about cognitive psychology. Downvote. $\endgroup$ – MathematicsStudent1122 Jun 12 '16 at 1:52
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    $\begingroup$ @MathematicsStudent1122 I disagree. It's rather a question suited for Academia SE but I can see why OP would post here, since departments in academia have completely different expectations and profiles of how and who they hire. $\endgroup$ – galois Jun 12 '16 at 2:07
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    $\begingroup$ I know that intelligence tests have their flaws, but I'd be shocked if very many tenured mathematics professors have IQ below 120, which is significantly above the average of 100. $\endgroup$ – Forever Mozart Jun 12 '16 at 2:32
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    $\begingroup$ @Cat: I disagree. Pretty much every postdoc, professor, etc. is committed to $N$-hour work weeks for unnervingly large values of $N$, and yet there's considerable variation in their success. The best advice I've gotten about math is that you should go into mathematics iff you find yourself thinking about it and doing it at every opportunity: at breakfast, late at night, in the bathroom, etc. There's no shortage of hard work at the higher levels of math. $\endgroup$ – anomaly Jun 12 '16 at 3:38
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The answer to the question in the title is no, but eidetic memory is not the same as intelligence. That having been said, the ability in question is really about remembering the important parts of a proof and how they fit together; memorizing lists of arbitrary numbers and facility with arithmetic computation are irrelevant. It's similar to the way chessmasters think about a game: It's not a series of arbitrary board positions, but rather a collection of related objects that link together in a way that makes sense. If you asked me to, say, present the proof of the Whitney embedding theorem from scratch, I could come up with a basic outline of the proof from memory (construct a mapping onto some large $\mathbb{R}^N$, use Sard's theorem to reduce to $\mathbb{R}^{2n+1}$, then use the Whitney trick to reduce to $\mathbb{R}^{2n}$), and I could name a couple of books where you're likely to find a full proof. If you ask me about my thesis or what I'm currently working on, I could tell you all the gory details, probably including theorem numbers in papers. If you ask me about a minor but well-known result in an area outside my particular subfield, I could probably at least steer you toward a likely reference. Photographic memory for irrelevant details is certainly not required, or even particularly beneficial, to a mathematician.

The answer to your last question is probably no, regardless of your intelligence, because math academia is insanely competitive. About fifty years ago, getting an academic position involved your advisor calling up his contacts and asking one of them to give you a job; now, there are a couple hundred applications per position. There's a steady stream of PhDs in math looking for jobs in academia and math research, but it's not like we're making more universities, and there's obviously very little turnover at them. Success depends not just on your ability, but on factors outside your control: whether you hit the big break in your research in time, whether your advisor is useful, whether your advisor gets along well with you, whether your particular area of research in popular and sexy when you're looking for postdocs, etc. It's awful, but there's literally no other oppportunity to work on nontrivial pure math. If you're not obsessively interested in mathematics and desperately want to spend your life doing math research, it's not worth your time and effort and sanity. Being extraordinarily good at mathematics is necessary for getting a tenure-track academic position in math, but (unless you want to take "extraordinarily" to the ridiculous extreme) it's not sufficient.

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  • $\begingroup$ The mention of chunking information "similar to the way chessmasters think about a game" is, in fact, grounded in research; some of the early studies were done by de Groot (wiki). $\endgroup$ – Benjamin Dickman Jun 12 '16 at 3:25
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    $\begingroup$ Ah, "chunking" was the word I was looking for. Thanks! $\endgroup$ – anomaly Jun 12 '16 at 3:38
  • $\begingroup$ You're welcome! I used a similar bit of reasoning/terminology in an answer about memorizing the details of proofs for a question on MathEducators; cf. MESE 2226. $\endgroup$ – Benjamin Dickman Jun 12 '16 at 3:47
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This answer consists of some impressions I formed of your question.

  • Feats of memory are party tricks. While it is essential for success in any cognitive pursuit to have a strong memory of your interests, that comes with the intense focus that is even more essential for success. Behind these effortless demonstrations are years of training that you have not even begun.

  • IQ is irrelevant and speculation about the superhuman capacities of our favorite idols is actually damaging. I know more Ivy League mathematicians than you do and none of them ever talked or, as far as I can tell, thought about this measure (well, maybe one did. I won't say whom I think it is). I don't know anyone's IQ.

  • The intersection of math PhDs and people of average intelligence is empty. That said, the average is defined by IQ, and since that is irrelevant...if you want to prove yourself of above average intelligence, you have only to obtain a degree in math.

  • Don't go into academia. It's depressing and there are many equally rewarding careers that enhance rather than degrade your self-worth. The only people who should ignore this advice are the ones who have already demonstrated to themselves that it's false. Your question makes me think that you're not one of them. You like math? Then learn some math. Learn to program! You can apply either to the other one and there are endlessly many companies that give you the opportunity to combine them.

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    $\begingroup$ +1 "Behind these effortless demonstrations are years of training that you have not even begun." It took me so long to realize this; it's impossible to overstate how much more experience a professor has than an undergraduate, even a very good one. $\endgroup$ – Eric Stucky Jun 12 '16 at 3:48

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