The top 3 answers to the MathOverflow question Which mathematicians have influenced you the most? are Alexander Grothendieck, Mikhail Gromov, and Bill Thurston. Each of these have expressed serious concerns about the community.
Grothendieck was actually effectively excommunicated by the mathematical community and then was pathologized as having gone crazy. See pages 37-40 of David Ruelle’s book A Mathematician’s Brain.
Gromov expresses strong sympathy for Grigory Perelman having left the mathematical community starting on page 110 of Perfect Rigor. (You can search for “Gromov” in the pdf to see all of his remarks on the subject.)
Thurston made very apt criticisms of the mathematical community in his essay On Proof and Progress In Mathematics. See especially the beginning of Section 3: “How is mathematical understanding communicated?” Terry Tao endorses Thurston’s essay in his obituary of Thurston. But the community has essentially ignored Thurston’s remarks: one almost never hears people talk about the points that Thurston raises.
I don’t know about Grothendieck, but the two other sources appear to have softer criticism of the mathematical community than “actually functioning as a cult”.
The links you give are extremely interesting, but, unless I am missing something, it seems that they fall short of justifying your earlier statement that math academia functions as a cult. I wonder if you would be willing to elaborate further on that?
The most scary thing to me is that the most mathematically talented students are often turned off by what they see in math classes, even at the undergraduate and graduate levels. Math serves as a backbone for the sciences, so this may badly undercutting scientific innovation at a societal level.
I honestly think that it would be an improvement on the status quo to stop teaching math classes entirely. Thurston characterized his early math education as follows:
I hated much of what was taught as mathematics in my early schooling, and I often received poor grades. I now view many of these early lessons as anti-math: they actively tried to discourage independent thought. One was supposed to follow an established pattern with mechanical precision, put answers inside boxes, and “show your work,” that is, reject mental insights and alternative approaches.
I think that this characterizes math classes even at the graduate level, only at a higher level of abstraction. The classes essentially never offer students exposure to free-form mathematical exploration, which is what it takes to make major scientific discoveries with significant quantitative components.
I distinctly remember having points taken off of a physics midterm because I didn’t show my work. I think I dropped the exam in the waste basket on the way out of the auditorium.
I’ve always assumed that the problem is three-fold; generating a formal proof is NP-hard, getting the right answer via shortcuts can include cheating, and the faculty’s time is limited. Professors/graders do not have the capacity to rigorously demonstrate to themselves that the steps a student has written down actually pinpoint the unique answer. Without access to the student’s mind graders are unable to determine if students cheat or not; being able to memorize and/or reproduce the exact steps of a calculation significantly decrease the likelihood of cheating. Even if graders could do one or both of the previous for a single student, they are not 30x or 100x as smart as their students, making it impractical to repeat the process for every student.
That said, I had some very good mathematics teachers in higher level courses who could force students to think, and one in particular who could encourage/demand novelty from students simply by asking them to solve problems that they hadn’t yet learned to solve. I didn’t realize the power of the latter approach until later (and at the time everyone complained about exams with a median score well under 50%), but his classes were always my favorite.
Thank you for all these interesting references. I enjoyed reading all of them, and rereading in Thurston’s case.
Do people pathologize Grothendieck as having gone crazy? I mostly think people think of him as being a little bit strange. The story I heard was that because of philosophical disagreements with military funding and personal conflicts with other mathematicians he left the community and was more or less refusing to speak to anyone about mathematics, and people were sad about this and wished he would come back.
Do people pathologize Grothendieck as having gone crazy?
His contribution of math is too great for people to have explicitly adopted a stance that was too unfavorable to him, and many mathematicians did in fact miss him a lot. But as Perelman said:
Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.” He has also said that “It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated.
If pressed, many mathematicians downplay the role of those who behaved unethically toward him and the failure of the community to give him a job in favor of a narrative “poor guy, it’s so sad that he developed mental health problems.”
I don’t really understand what you mean about math academia. Those references would be appreciated.
The top 3 answers to the MathOverflow question Which mathematicians have influenced you the most? are Alexander Grothendieck, Mikhail Gromov, and Bill Thurston. Each of these have expressed serious concerns about the community.
Grothendieck was actually effectively excommunicated by the mathematical community and then was pathologized as having gone crazy. See pages 37-40 of David Ruelle’s book A Mathematician’s Brain.
Gromov expresses strong sympathy for Grigory Perelman having left the mathematical community starting on page 110 of Perfect Rigor. (You can search for “Gromov” in the pdf to see all of his remarks on the subject.)
Thurston made very apt criticisms of the mathematical community in his essay On Proof and Progress In Mathematics. See especially the beginning of Section 3: “How is mathematical understanding communicated?” Terry Tao endorses Thurston’s essay in his obituary of Thurston. But the community has essentially ignored Thurston’s remarks: one almost never hears people talk about the points that Thurston raises.
I don’t know about Grothendieck, but the two other sources appear to have softer criticism of the mathematical community than “actually functioning as a cult”.
The links you give are extremely interesting, but, unless I am missing something, it seems that they fall short of justifying your earlier statement that math academia functions as a cult. I wonder if you would be willing to elaborate further on that?
I’ll be writing more about this later.
The most scary thing to me is that the most mathematically talented students are often turned off by what they see in math classes, even at the undergraduate and graduate levels. Math serves as a backbone for the sciences, so this may badly undercutting scientific innovation at a societal level.
I honestly think that it would be an improvement on the status quo to stop teaching math classes entirely. Thurston characterized his early math education as follows:
I hated much of what was taught as mathematics in my early schooling, and I often received poor grades. I now view many of these early lessons as anti-math: they actively tried to discourage independent thought. One was supposed to follow an established pattern with mechanical precision, put answers inside boxes, and “show your work,” that is, reject mental insights and alternative approaches.
I think that this characterizes math classes even at the graduate level, only at a higher level of abstraction. The classes essentially never offer students exposure to free-form mathematical exploration, which is what it takes to make major scientific discoveries with significant quantitative components.
I distinctly remember having points taken off of a physics midterm because I didn’t show my work. I think I dropped the exam in the waste basket on the way out of the auditorium.
I’ve always assumed that the problem is three-fold; generating a formal proof is NP-hard, getting the right answer via shortcuts can include cheating, and the faculty’s time is limited. Professors/graders do not have the capacity to rigorously demonstrate to themselves that the steps a student has written down actually pinpoint the unique answer. Without access to the student’s mind graders are unable to determine if students cheat or not; being able to memorize and/or reproduce the exact steps of a calculation significantly decrease the likelihood of cheating. Even if graders could do one or both of the previous for a single student, they are not 30x or 100x as smart as their students, making it impractical to repeat the process for every student.
That said, I had some very good mathematics teachers in higher level courses who could force students to think, and one in particular who could encourage/demand novelty from students simply by asking them to solve problems that they hadn’t yet learned to solve. I didn’t realize the power of the latter approach until later (and at the time everyone complained about exams with a median score well under 50%), but his classes were always my favorite.
Thank you for all these interesting references. I enjoyed reading all of them, and rereading in Thurston’s case.
Do people pathologize Grothendieck as having gone crazy? I mostly think people think of him as being a little bit strange. The story I heard was that because of philosophical disagreements with military funding and personal conflicts with other mathematicians he left the community and was more or less refusing to speak to anyone about mathematics, and people were sad about this and wished he would come back.
His contribution of math is too great for people to have explicitly adopted a stance that was too unfavorable to him, and many mathematicians did in fact miss him a lot. But as Perelman said:
Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.” He has also said that “It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated.
If pressed, many mathematicians downplay the role of those who behaved unethically toward him and the failure of the community to give him a job in favor of a narrative “poor guy, it’s so sad that he developed mental health problems.”
What failure? He stepped down from the Steklov Institute and has refused every job offer and prize given to him.
From the details I’m aware of “gone crazy” is not a bad description of what happened.