1학기, 10주차.

두 번째 모듈이 시작되었다. 첫 모듈을 끝내고 나니 그저 퍼져 있고 싶어진다. 이게 모듈제의 단점이라면 단점, 아니 그냥 단점이다. ㅋㅋ

– 어제 자동차를 샀다. 오피스메이트가 다니는 교회 담임목사가 이 지역을 꽉 잡고 있는 딜러와 오랜 친구라고 해서 그 쪽을 통했다. 혼다 어코드 2008년형. 7,200불 줬다. 2010년대에 나온 차를 사고 싶었는데 조금 아쉽다. 차알못에 초보운전이라 이제 갈 길이 멀다. 벌써 겨울이 걱정된다.

1학기, 10주차.

– 한편, Winter is here. 일주일 새 바람이 매서워졌다. 아직 11월도 되지 않았는데 이러면 1-2월을 어떻게 날지 걱정이다. 주말에 히터를 청소할 계획이다.

1학기, 10주차.

– 슬슬 체력이 달리는 느낌. 운동을 해야 하는데… 후. ESL을 다시 한 번 욕해 본다. 2학기에는 꼭꼭 운동 다녀야지. 지금은 간신히 팔굽혀펴기 좀 하는 수준… 유산소가 필요합니다 ㅜㅜ

– 전반적으로 첫 모듈 교수들이 티칭에 더 능했다. 과목별로 강의가 3번씩 있었는데… 다들 수업이 지루하다. (절대로 “만만하다”는 뜻이 아님) 살아남아라, 용사여!

 

1학기, 9주차.

첫 모듈이 끝났다.

– 성적이 나쁘지 않게 나왔다. 아직 공식적인 건 아니지만 미시와 거시는 모두 1등일 것이다. 경제수학은 기대 이상이었다. 기말고사 평균이 상상초월로 낮아서 발생한 일인 듯. 기말 문제가 길고 어려워 시간 관리가 쉽지 않았기 때문인데, 난 푼 문제만큼은 제대로 풀었고 그 결과 예상보다 좋은 성적을 받았다. 어드미션 프로필 / 코스웍 퍼포먼스 / 리서치 퍼포먼스가 전부 별개라고 하지만, 어쨌든 기분 좋으니 된 것 아닌가? (근데 시험 못 봤으면 또 저거로 정신승리했겠지. 인간이란…)

– 동기들에게 모듈 리뷰 워크샵을 진행하자고 제안했다. 모두 참석하겠다는 의사를 밝혔다. 동기가 8명으로 적고, 출신국가가 5개국이라 특정 국가 출신들끼리 파벌을 형성할 일도 없으니 남은 것은 함께 살아남는 것이다. Spillover effect!

– 오늘 미주리 경제학과에서 박사과정 3년차 계신 분이 놀러 오셔서 하루 종일 시간 가는 줄 모르고 떠들었다.  실제로 뵌 건 처음이었지만 정말 즐거웠다. 미국 와서 한국어를 가장 많이 사용한 날.나도 차를 사면 시카고나 미시간, 오하이오에 있는 지인들을 방문할 수 있을 텐데… 지금으로서는 답이 안 나온다.

– 수업이 조금 더 rigorous했으면 좋겠다. 퀄 지나면 어차피 내가 보는 분야만 볼 것이다. 그러니 코스웍 때 좀 맛을 많이 보고 싶은데. 사람마다 의견이야 다르겠지만 나는 코스웍 수업은 다양한 내용을 깊이, 그러니까 빡세게 가고 퀄 시험 자체는 쉽게 가는 것이 가장 좋지 않을까 생각한다. 경영대 소속이라서 좋은 점도 많지만, 아무래도 경제학 전공이 아닌 학생들도 함께 끌고 가다 보니 미시-계량 시퀀스 밀도가 조금 떨어지는 것 같아서 아쉽다.

– 두 번째 모듈은 첫 모듈에 비해 훨씬 부담이 적을 것 같다. 미시2는 일반균형과 후생경제학, 거시2는 Stochastic Dynamic Programming, RBC, Asset Pricing, Unemployment를 다룬다. 대충 뭔지는 알고 있는 내용들이다. 확률통계(Probability and Statistics) 역시 지금까지 공부했던 수리통계학 내용을 벗어나지 않는다. 수업을 Hogg and Craig로 하니까 대충 말 다 한 것이지. 푸아송이나 지수분포는 솔직히 볼 일이 없어서 좀 잊어버렸지만;; 다시 보면 기억나겠지. 아무래도 들고 온 Casella and Berger를 좀 읽으면서 내공을 쌓는 시간을 보내야 할 것 같다. 이 수업 들으면서 그냉 널널하게 시간 보내면 나중에 분명히 후회할 날이 올 것이다.

– 미시이론을 어느 정도로 깊게 공부해야 할까? 첫 모듈 미시이론은 그리 깊게 들어가지 않았다. 8주 동안 선호체계부터 불확실성 하의 선택이론까지 나갔으니 표준진도에 맞춘 것이지만 깊이는 별문제다. 그냥 MWG+Rubinstein 수준? 석사 때도 했던 내용이라 익숙해서 그럴지도 모르겠지만 설렁설렁 했다는 인상을 지우기 어렵다. Jehle and Reny나 Kreps – Old or New – 를 리뷰 때 적당히 skimming할 예정. 거시가 확실히 가장 demanding하고 수업을 잘 따라가야 한다. 다행히 거시 교수들이 가장 티칭에 열정적이고 탁월한 것 같다.

 

어쨌든 박사과정 생활 아직까지는 즐겁게 하고 있다. 다들 대학원생에게 life가 없다고 불평하는데, 물론 나도 동조하지만, 나는 이런 생활 양식을 사랑한다. 내가 멍청해서 지식을 더 스펀지처럼 빨아들이지 못한다는 사실을 제외하면 만족스럽다고 할 만하다. 학부 때도 이렇게 살았어야 하는데… 는 지나고 하는 소리가 맞다. 학부 시절 보냈던 그 숱한 방황의 세월이 있어 지금 이렇게 안정적으로 공부에만 집중할 수 있다고 생각한다. 매일매일 “방어의 사자” 발터 모델Walter Model 원수의 어록을 떠올린다.

Every minute that we lose will cost us great losses later that we will not be able to afford. We must push forward now, otherwise we risk everything. Hurry yourself with the technical aspects, a lot of time has already been lost.

 

Go ahead!

1학기, 8주차.

모듈 1 종료 직전. 학기 절반이 지나갔다.

#. 거시와 경제수학 기말고사 모두 결과를 기다려봐야 한다. 두 과목 모두 작년보다 현저히 어렵게 출제되었다. 수요일로 예정된 미시 시험은 무엇이 나올지 종잡을 수가 없다.

#. 동기들에게 리뷰 워크샵을 만들자고 제안할 생각이다. 방법은 차차 고민해보아야 할 듯.

#. 한편 이런저런 생각이 많아진다. 이론을 엄밀하고 깊이 있게 배운다는 것은 무슨 뜻일까? 내 직관은 그저그런 수준에 머물러 있는 것 같다. 첫 술에 배부르겠냐만은, 글쎄… 지금 충분히 엄밀하고 깊이 있게 배우고 있는 건가? 간단한 질문인데, “나는 이런 문제를 낼 수 있는가?” 대답은 No에 가깝고, 그 간극을 좁히는 방법을 모르겠다. 한편 지금 느껴지는 장벽은 누가 가르쳐서 넘어가는 게 아니라 내가 열심히 머리를 굴려야 하는 영역이라는 생각도 든다.

 

…아무튼 1년차의 지상목표는 퀄 통과니까 시키는 대로 공부하자.

The Productivity Slowdown and the Declining Labor Share: A Neoclassical Exploration (NBER w23853)

1년차 때 논문 열심히 읽으면 망한다(..)는 얘기를 많이 들었다. 그래서 요즘은 감 잃지 않게 NBER 메일링 받으면서 출퇴근길에 초록만 훑어본다. 그런데 이번 주에 나온 페이퍼가 눈길을 끌었다.

The Productivity Slowdown and the Declining Labor Share: A Neoclassical Exploration

We explore the possibility that a global productivity slowdown is responsible for the widespread decline in the labor share of national income. In a neoclassical growth model with endogenous human capital accumulation a la Ben Porath (1967) and capital-skill complementarity a la Grossman et al.

주제도 주제고, 저자진도 휘황찬란하다. Gene Grossman (Princeton), Elhanan Helpman (Harvard), Ezra Oberfield (Princeton), Thomas Sampton (LSE). 차마 초록만 읽고 넘어갈 수 없어서 서론과 결론 정도 더 읽어 보았다(???). 논문의 핵심은 생산성 증가율이 하락하면 요소소득이 노동에서 자본으로 옮겨가는 경향이 있다는 것이다. 저자들의 칼리브레이션에 따르면 대략 생산성증가율 1%p 하락할 때마다 자본소득분배율이 2~6%p 올라간다(당연히 노동소득분배율 하락). 이 결과는 현재 관찰되는 미국 노동소득분배율 하락을 절반에서 전부 설명한다.

The Productivity Slowdown and the Declining Labor Share: A Neoclassical Exploration (NBER w23853)
γL, γK는 각각 노동/자본생산성증대 기술진보율, a는 생산함수의 모수.

 

논리는 대충 이렇다. 경제성장률이 하락하면 실질이자율이 하락하고, 사람들이 교육을 더 받기로 선택한다. 이는 노동-자본 대체탄력성이 비탄력적일 때 (자본-인적자본 보완관계에 의하여) 기업의 자본상대수요를 증가시킨다. 다시 말해 주어진 labor expense 하에서 자본소득분배율이 상승한다는 것. 다만 아직 인과를 말할 단계는 아니라고 거듭 강조한다.

솔직히 무엇이 새로운 것인지는 잘 모르겠다. 대가들이 흔히 하듯 간단한 모형으로 디테일한 논의를 전부 엮어낸 느낌인데(전형적인 “참 쉽죠?”) 관련 참고문헌을 다 읽을 수도 없고. 저자들이 제시하는 셀링 포인트 중 하나는 대체탄력성 실증분석 결과가 분분한 이유를 설명한다는 것이다. 피케티 21세기 자본 논쟁에서도 문제시되었던 바로 그 대체탄력성 맞다. 저자들에 따르면 몇몇 실증연구에서 대체탄력성이 1보다 큰 이유는 인적자본 선택을 감안하지 않아서 그렇다는 것. 국가간 데이터를 이용한 연구가 특히 이 문제에 취약하다. 이 모형은 (쉽게 짐작할 수 있듯이) 인적자본선택을 내생화해서 이 문제를 해결하는 한편 대체탄력성이 1보다 작아야 한다는 이론적 결론을 재확인한다.

아무튼 대충 그렇다. 정리를 다 하진 못하겠다.

초록을 옮기면 다음과 같다.

본 연구진은 세계적 생산성 둔화 현상이 노동소득분배율 하락을 야기할 가능성을 탐색한다. Ben Porath (1967) 식 내생적 인적자본축적, Grossman et al. (2017) 식 자본-숙련 보완관계를 추가한 신고전파 성장모형에서, 균제상태(steady-state) 노동소득분배율은 자본생산성 증대(capital-augmenting)·노동생산성 증대(labor-augmenting) 기술진보율과 정의 상관관계를 갖는다. 戰後 미국 데이터를 이용해 칼리브레이션한 결과 1인당 소득증가율 1% 하락은 최근 미국 노동소득분배율 하락을 절반에서 전부 설명한다.

We explore the possibility that a global productivity slowdown is responsible for the widespread decline in the labor share of national income. In a neoclassical growth model with endogenous human capital accumulation a la Ben Porath (1967) and capital-skill complementarity a la Grossman et al. (2017), the steady-state labor share is positively correlated with the rates of capital-augmenting and labor-augmenting technological progress. We calibrate the key parameters describing the balanced growth path to U.S. data for the early postwar period and find that a one percentage point slowdown in the growth rate of per capita income can account for between one half and all of the observed decline in the U.S. labor share.

 

핵심 문단은 대충 이 정도.

When we solve for the balanced growth path, we find simple analytical formulas for the long-run factor shares. If we further assume—in keeping with the empirical evidence—that the elasticity of intertemporal substitution is less than one, then the labor share in national income is an increasing function of the rates of capital-augmenting and labor-augmenting technological progress. Therefore, a productivity slowdown of any sort results in a decline in the steady-state labor share. The mechanism operates through optimal schooling choices. When growth slows, the real interest rate falls, which leads individuals to target a higher level of education for a given level of the capital stock. Inasmuch as skills are capital using, this reduces the effective capital to labor ratio in the typical firm, which in turn redistributes income from labor to capital, given an elasticity of substitution less than one.

How important is this redistributive channel quantitatively? To answer this question, we take parameters to match the average birth rate, the average death rate, the rate of labor productivity growth, the internal rate of return on schooling, and the factor shares of the pre-slowdown era in the United States, as well as a conservative estimate of the elasticity of intertemporal substitution. One key parameter remains, which can be expressed either in terms of the composition of technical progress in the pre-slowdown steady state or as a measure of the capital-skill complementarity in the aggregate production function. We are cautious about this parameter, because Diamond et al. (1978) tell us that it cannot be identified from time series data on inputs and outputs, while our formula tells us that it plays a central role in our quantitative analysis. We consider a range of alternatives, including some derived from estimation of the cross-industry and cross-regional relationships implied by our model. In all of the alternatives we consider, a one percentage point slowdown in secular growth implies a substantial redistribution of income shares from labor to capital, representing between one half and all of the observed shift in factor shares in the recent U.S. experience.

In this setting, if human capital is more complementary with physical capital than with raw labor and if the elasticity of substitution between physical capital and labor is less than one (holding constant the level of schooling), then the rate of labor productivity growth and the share of labor in national income will be positively correlated across steady states. Accordingly, a slowdown in productivity growth—such as has apparently occurred in the recent period—can lead to a shift in the functional distribution of income away from labor and toward capital. The mechanism requires a fall in the real interest rate, which has also been part of the recent experience. When the interest rate falls relative to the growth rate of wages, individuals target a higher level of human capital for any given size of the capital stock and state of technology. When human capital is more complementary to physical capital than to raw labor, the elevated human capital target implies a greater relative demand for capital. With an elasticity of substitution between capital and labor less than one, the shift in relative factor demands generates a rise in the capital share at the expense of labor. Moreover, if the productivity slowdown is associated with a deceleration of declining investment-good prices or with a fall in the rate of disembodied capital-augmenting technical progress, then the model predicts a slowdown in the annual expansion of educational attainment, which also matches the data in recent economic history.

Finally, we have focused in this paper on exploring a potential explanation for recent trends in the labor share. But it is possible that our story holds broader sway in economic history. Figure 7 shows the evolution of the labor share in the United States and the United Kingdom since the beginning of the twentieth century and the evolution of labor productivity in each country over the same period. Evidently, these two variables have been temporally correlated throughout modern history. For example, the period from 1900 until approximately 1930 was a period of slow productivity growth in the United States and United Kingdom. It was also a period of an historically low labor share. When productivity growth subsequently accelerated, the labor share rose in tandem. While we are cautious about drawing firm conclusions from such casual observations, it is possible that productivity growth and the functional distribution of income have been linked for quite some time.

 

The Genius Fallacy (Jean)

천재에 관한 환상을 버리도록 권고하는 글 하나 더… (원문 링크) 참고로 글쓴이는 MIT 졸업한 양반이다. ㅋㅋㅋㅋㅋ 이런 글이 원래 다 이 모양일 수밖에 없다는 걸 잘 알고 있다. 상징자본을 갖고 있는 사람이 발언권을 갖는 법.

경제학과 교수가 했다는 말에 주목할 만하다.

“You have to be a super star to succeed in a department like ours, (..) I want undergraduate researchers who are stars.” 그리고 그는 글쓴이의 이메일에 답장을 보내지 않았다. 흔히 일어나는 일이다… 글쓴이는 컴퓨터과학과 교수에게서 더 나은 조언을 들을 수 있었다고 한다. ㅠㅠ 안 그래도 모교 교수님(UChicago 박사)이 하시던 말씀이 있다. “탑스쿨 교수들은 코호트에서 스타를 찾는 것이 목표다. 내가 대학원생이던 당시 1년 위에 Ivan Werning (MIT, 2014 JBC Medalist)이 있었다. 교수들은 그 이야기를 하면서 우리 코호트엔 스타가 없고, 망했다는 말을 서슴지 않고 했다.” 이 이야기를 5번 넘게 들은 것 같다. 이 글쓴이도 같은 에피소드를 얘기하고 있다. 평가란 본디 냉정한 것이나, 교육학적으로 저걸 바람직하다고 할 수 있을까.

…아무튼 나 같은 범재는 존버정신으로 가야지 뾰족한 방법이 없다.


The Genius Fallacy

During our department’s PhD Open House this past spring, a student asked what I thought made a PhD student successful. I realized that my answer now is different than it would have been a few years ago.

My friend Seth tells me I need to build more suspense in my writing*, so let me first tell my life story.

The whole time I was growing up, I was slightly disappointed that I wasn’t some kind of prodigy. It seemed that my parents were telling me every day about so-and-so’s toddler son who was playing Beethoven concertos from memory, so-and-so’s daughter who, as an infant, had already completed a course on special relativity. In order to give me the same opportunities to demonstrate my genius, my parents spent all their money on piano lessons, gymnastic classes, writing camps, art camps, tennis camps, and extracurricular math classes. Unfortunately, nobody ever said, “This is the best kid I have ever seen. I must take her away from her family to train her for greatness.”

“Child prodigies have hard lives,” my father would tell me, probably trying hiding his disappointment. “It can be difficult for them to make friends because others can’t relate to how gifted they are.”

“Just work hard, be a nice person, and try to be happy,” my mother would tell me. “You didn’t know how to cry when you were born. I’m glad you’re able to talk in full sentences.”

Despite the comforting words from my parents, there was always a part of me that held out hope of discovering a secret prodigious talent. But the angst of not being a prodigy was small compared to the existential angst of being newly alive and so I mostly tried to work hard, to be a nice person, and to be happy. This got me all the way to college, where I thought I could leave all this prodigy nonsense behind me.

In college, I discovered that the pressure to be immediately and wildly gifted came in another form. In my first two years of school, I attended many talks and panels by professors telling us what we should do with our lives. I attended a research panel in the economics department, where one of the professors kept repeating the word “star.”

“You have to be a super star to succeed in a department like ours,” he said about what it meant to be on the tenure track in the economics department. “I want undergraduate researchers who are stars.”

I didn’t know what a star was and I didn’t presume to be one, but I liked the professor’s research, so I emailed him my resume and said I would like to work with him.

He never wrote back.

I resigned myself to not being a star. I took hard classes with people who had medaled in math, informatics, and science olympiads, wondering how it would feel to do the problem sets if I had such a gifted, well-trained mind. I also became concerned about my future. What was my place in a world that worshipped instatalent?

It all began to change when I began to talk more with the professors in the Computer Science department. Despite my lack of apparent star quality, my professors seemed to like answering the questions I asked them. They pitched me projects I could do, and before I knew it I was applying to PhD programs and preparing to spend the next few years doing academic research. As I was graduating, I spoke with my one professor to get advice about my future in research.

“Research isn’t just about smarts,” my professor told me. At the time, I thought this was a white lie that professors told to their students who weren’t prodigies.

Then she told me something that turned my worldview upside down. “My biggest concern for you, Jean, is that you need to start finishing projects,” she told me. “You need to focus.”

It was then that I began to realize that maybe the myth of the instagenius was but a myth. I had gone from interest to interest, from project to project, waiting to find It, that easy fit, that continuous honeymoon. With some projects I had It for a while, long enough to demonstrate to myself and others that I could finish. Then I moved on, waiting to fall in love with a problem, waiting for a problem to choose me. What I had failed to see was that this relationship with a problem didn’t just happen: I had to do my share of the work.

Still, I clung to the dream of the easy problem. At Google, employees get to have a 20% project: a side project they spend the equivalent of one day a week working on that may or may not make its way into production eventually. In graduate school, my 20% project was looking for an easier project–a project with which I had more chemistry, a project with fewer days lost to dead ends and angst. One of my hobbies involved interviewing for internships in completely different research areas. Another one of my hobbies was fantasizing about becoming a classics PhD student, despite knowing no ancient languages. (I once took an upper-level literature seminar on Aristotle with the leading world scholar on Homeric poetry and I thought he had a pretty good life.)

But because I like to finish what I started, the PhD became a process of learning to persevere. Instead of indulging the temptation to switch projects, advisors, or even schools, I kept going. I endured something like five rounds of rejections on the first paper towards my PhD thesis, and multiple years of people telling me that maybe I should find another topic, because I didn’t seem in love. Eventually, I learned that every problem that looks like it might be easy has hard parts, every problem that looks like it might be fun has boring parts, and all problems worth solving are full of dead ends. I finally learned, in the words of my friend Seth, that “the grass is brown everywhere.”

And this shattering of my belief in instagenius has shaped my conception of what makes a student a star. There was a time when I, like many people, thought that the superstars were the ones who sounded the most impressive when they spoke, or who had the most raw brainpower. If you asked me what I thought made a good researcher, I may have said some other traits like creativity and good taste in problems. And while all these certainly help with being a good researcher, there are plenty of people with these traits who do not end up being successful.

What I have learned is that discipline and the ability to persevere are equally, if not more, important to success than being able to look like a smart person in meetings. All of the superstars I’ve known have worked harder–and often faced more obstacles, in part due to the high volume of work–than other people, despite how much it might look like they are flying from one brilliant result to another from the outside. Because of this, I now want students who accept that life is hard and that they are going to fail. I want students who accept that sometimes work is going to feel like it’s going to nowhere, to the point that they wish they were catastrophically failing instead because then at least something would be happening. While confidence might signal resilience and a formidable intellect might decrease the number of obstacles, the main differentiator between a star and simply a smart person is the ability to keep showing up when things do not go well.

It has become especially important for me to fight the idolization of the lone genius because it is not just distracting, but also harmful. Currently, people who “look smart” (which often translates into looking white, male, and/or socioeconomically privileged) have a significant advantage for two main reasons. The first reason has to do with self-perception. Committing to hard work and overcoming obstacles is easier if you think it will pay off. If someone already does not feel like they belong, it is easier for them to stop trying and self-select out of a pursuit when they hit a snag. The second reason has to do with perception by others. Research suggests that in fields that value innate talent, women and other minorities are often stereotyped to have less of it, leading to unfair treatment.

And so I’ve written this post not just to reveal my longstanding delusions of grandeur, but also to start a discussion how the myth of instagenius holds us back, as individual researchers and as a community. Would love to hear your thoughts about how we can move past the genius fallacy.

Related writing:
The Structured Procrastination Trap
The Angst Overhead
Five Things More Important About a Research Project than Being in Love
On Quora: How common is it for PhD students to do work in projects that they are not passionate in?

* Seth also tells me the main idea of this blog post is the same as Angela Duckworth’s book Grit. I guess I should tell you that you could read that instead of this. On the subject of the lack of originality of my ideas, you should also read what Cal Newport has to say about the “passion trap.”

수학은 천재나 하는 것인가? (Terrence Tao)

천재로 유명한(..) 테렌스 타오 교수가 블로그에 쓴 글. 내가 수학전공자는 아니지만 범재로서 학업의 자세를 생각하는 데 도움이 된다.

원문 링크


Better beware of notions like genius and inspiration; they are a sort of magic wand and should be used sparingly by anybody who wants to see things clearly. (José Ortega y Gasset, “Notes on the novel”)

Does one have to be a genius to do mathematics?

The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities.

The popular image of the lone (and possibly slightly mad) genius – who ignores the literature and other conventional wisdom and manages by some inexplicable inspiration (enhanced, perhaps, with a liberal dash of suffering) to come up with a breathtakingly original solution to a problem that confounded all the experts – is a charming and romantic image, but also a wildly inaccurate one, at least in the world of modern mathematics. We do have spectacular, deep and remarkable results and insights in this subject, of course, but they are the hard-won and cumulative achievement of years, decades, or even centuries of steady work and progress of many good and great mathematicians; the advance from one stage of understanding to the next can be highly non-trivial, and sometimes rather unexpected, but still builds upon the foundation of earlier work rather than starting totally anew. (This is for instance the case with Wiles‘ work on Fermat’s last theorem, or Perelman‘s work on the Poincaré conjecture.)

Actually, I find the reality of mathematical research today – in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck – to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of “geniuses”. This “cult of genius” in fact causes a number of problems, since nobody is able to produce these (very rare) inspirations on anything approaching a regular basis, and with reliably consistent correctness. (If someone affects to do so, I advise you to be very sceptical of their claims.) The pressure to try to behave in this impossible manner can cause some to become overly obsessed with “big problems” or “big theories”, others to lose any healthy scepticism in their own work or in their tools, and yet others still to become too discouraged to continue working in mathematics. Also, attributing success to innate talent (which is beyond one’s control) rather than effort, planning, and education (which are within one’s control) can lead to some other problems as well.

Of course, even if one dismisses the notion of genius, it is still the case that at any given point in time, some mathematicians are faster, more experienced, more knowledgeable, more efficient, more careful, or more creative than others. This does not imply, though, that only the “best” mathematicians should do mathematics; this is the common error of mistaking absolute advantage for comparative advantage. The number of interesting mathematical research areas and problems to work on is vast – far more than can be covered in detail just by the “best” mathematicians, and sometimes the set of tools or ideas that you have will find something that other good mathematicians have overlooked, especially given that even the greatest mathematicians still have weaknesses in some aspects of mathematical research. As long as you have education, interest, and a reasonable amount of talent, there will be some part of mathematics where you can make a solid and useful contribution. It might not be the most glamorous part of mathematics, but actually this tends to be a healthy thing; in many cases the mundane nuts-and-bolts of a subject turn out to actually be more important than any fancy applications. Also, it is necessary to “cut one’s teeth” on the non-glamorous parts of a field before one really has any chance at all to tackle the famous problems in the area; take a look at the early publications of any of today’s great mathematicians to see what I mean by this.

In some cases, an abundance of raw talent may end up (somewhat perversely) to actually be harmful for one’s long-term mathematical development; if solutions to problems come too easily, for instance, one may not put as much energy into working hard, asking dumb questions, or increasing one’s range, and thus may eventually cause one’s skills to stagnate. Also, if one is accustomed to easy success, one may not develop the patience necessary to deal with truly difficult problems (see also this talk by Peter Norvig for an analogous phenomenon in software engineering). Talent is important, of course; but how one develops and nurtures it is even more so.

It’s also good to remember that professional mathematics is not a sport (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest “score”, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications. For these tasks, mathematics needs all the good people it can get.

Further reading:

How to be a genius“, David Dobbs, New Scientist, 15 September 2006. [Thanks to Samir Chomsky for this link.]
The mundanity of excellence“, Daniel Chambliss, Sociological Theory, Vol. 7, No. 1, (Spring, 1989), 70-86. [Thanks to John Baez for this link.]