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I am a professional mathematician, and I regularly meet other mathematicians.
I have come to wonder if there is something like a slight neurosis, specific to this activity.

To be more precise, let me grossly describe the mathematician's activity:
The mathematician tries to solve a problem. This problem is usually sufficiently hard to not be solvable without an intense concentration. There are two types of concentration here:

  • A local type: the intensity in the moment (i.e., all the thoughts are focused on this single problem).
  • A global type: the commitment in time (i.e., work on this problem for months, even years).

Usually, the local concentration can be sufficiently intense so that the mathematician loses the feeling of hunger, thirst or sleep. In this state, there is usually no room for a healthy relationship with others.
It is almost as if the mathematician has a state near that of autism (see Asperger syndrome).

The global concentration permits one to develop in the time a more and more intense concentration on the problem. The mathematician is committed "body and soul" into solving the problem and could neglect all other aspects of his life like social relationships, married life, children...
It is almost as if the mathematician has become more and more autistic.

So ok, autism is a severe neurosis and it's usually irreversible. It's the reason why I speak about a slight neurosis specific to mathematicians, because it's usually reversible...

In light of what I wrote:

Is the mathematician's activity psychologically healthy ?

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    $\begingroup$ This is an awesome question, genius. You could substitute mathematician's with other very complex high cognitive load activities. When I walk across campus from the engineering to the liberal arts section, I become more aware of how different the mental expression is in the people studying the different fields. Of course one can argue their autism is why they are in the complex field in the first place but I like the part of the question that asks if it creates more of it. $\endgroup$ – Greg McNulty Jul 3 '13 at 0:54
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    $\begingroup$ I don't know why you think of mental disorders and autism. Your description reminds me most of meditation and the lives of Indian sannyasin or Chinese hermits. $\endgroup$ – user3116 Sep 14 '13 at 20:35
  • $\begingroup$ @what: I like your comment. What do you think of the following: In some sense, be extremely focused on spirituality is like be a Sannyasin, and be extremely focused on something else is like autism. $\endgroup$ – Sebastien Palcoux Jul 17 '14 at 13:31
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    $\begingroup$ @SébastienPalcoux In clinical psychology there is one criterion that almost all behaviors have to fulfill to be considered pathological: they have to prevent normal everyday functioning. Many writers close themselves up for a few weeks to finish their novels. They don't sleep, wash or eat. But then they come out, take a shower, play with their kids, socialize etc. It would only be considered pathological, if you don't have any friends at all, no family, never wash or cut your hair, etc. Being extremely focussed on what you do is called "flow" and it causes a state of bliss. It is very healthy. $\endgroup$ – user3116 Jul 17 '14 at 14:13
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    $\begingroup$ @what: see the following documentary Paul Erdos - N is a number (Mathematics Asperger) The man made of Maths. Extract of the abstract: << If he were alive today he would be diagnosed with aspergers, but he had fully productive aspergers. >> $\endgroup$ – Sebastien Palcoux Jul 28 '14 at 10:22
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Well that looks like the behavior of any person with a strong passion and focus for his work. There are plenty of these around!

I guess it would be more common in any field of work were people already have dedicated a significant part of their life to it, and where it is almost a prerequisite. Being a mathematician selects and cultivates people able to display consistently that kind of behavior. Otherwise you just don't succesfully study high level maths, or fail at it. That is why you may observe many occurences of this phenomenon in your environment, when you work with other mathematicians.

Nothing to worry much about i think, there are plenty of people like you around, and this is not directly related to mathematics.

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  • $\begingroup$ Thank you for your answer. I'm agree with you on the point that mathematics attract people predisposed to this behavior. $\endgroup$ – Sebastien Palcoux Jun 30 '13 at 17:58
  • $\begingroup$ But my point is also that mathematics promote and develop particularly this behavior, without limit, because contrarily to others domains, the mathematicians access easily to infinite areas ! $\endgroup$ – Sebastien Palcoux Jun 30 '13 at 18:09
  • $\begingroup$ If this behavior is characteristic of a slight neurosis, it may not be a good idea to develop it into a serious one, isn't it ? $\endgroup$ – Sebastien Palcoux Jun 30 '13 at 18:16
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There's a very small percent of people who enjoy the adrenaline of mental exhaustion. While that signals most people to stop, there are people who will continue exhausting themselves. This isn't physiologically healthy. You need to recognize when you're worn out and rest. Don't get hyper-focused on your problem.

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  • $\begingroup$ Your answer is about what I call the concentration of "local type", there is also an equivalent for the "global type". Anyway, the mathematician's activity seems more perverse that we believe : in general, without an hyper-commitment of our mental and our life, it seems not possible to be creative and productive. $\endgroup$ – Sebastien Palcoux Jun 30 '13 at 21:02
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This kind of behaviour also occurs in fields that use/rely on mathematics, such as my 2 of my main fields - physics and programming. Both employ a significant amount of maths concepts to understood and applied, often overlapping in places.

As an example, I have been known to work for over 12 hours in a single block to solve a problem.

Is this healthy? If it impinges on necessities such as sleep, eating, social life - then yes, it can potentially be somewhat harmful; however, people with a passion for what they do, often achieve great results that no only benefit themselves, but potentially others too.

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    $\begingroup$ Is there a term for "being intensely into something to miss meals, do not leave house and intensely believe that the work is important, meaningful, etc?" Is it (hypo)mania? $\endgroup$ – Alex Stone Jul 1 '13 at 0:04
  • $\begingroup$ @AlexStone en.wikipedia.org/wiki/Hypomania - yes, it does seem like a good fit. $\endgroup$ – user3554 Jul 1 '13 at 0:10
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    $\begingroup$ It is not hypomania. It is more like OCD. $\endgroup$ – guest43434 Jul 1 '13 at 1:20
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If you define mental disorder as any behavior not applying to (more or less arbitrary) social norms, then yes, the activity you describe would probably be considered mental disorder. However, the same would apply for example to:

  • homosexualism
  • most hobbies
  • asceticism and religious devotion
  • playing and listening to music

The last may seem odd, but Plato have written in 'The Republic' that musicians are not needed in his ideal state, because there's no use of them.

As for now, being common or being social norm is not the crucial part of determining if something is mental disorder or not. The most important parts are that some behavior is causing objective suffering and it has the nature of disability.

Is your mathematical drive causing you objective suffering? Well, sometimes you are frustrated for sure, but objectively it makes you happy and gives you aim in life. Has it disabiliating nature? No way, mathematics are so important in contemporary society!

Some symptoms are similar, but it's just like suspecting ADHD because you don't sleep a half of the day.

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Simon Baron-Cohen developed a relevant tool for this question: the autism-spectrum quotient (AQ).
A test is available here.

Here are the abstracts of two papers of Baron-Cohen et al.:

"The Autism-Spectrum Quotient (AQ): Evidence from Asperger Syndrome/High-Functioning Autism, Males and Females, Scientists and Mathematicians":

Abstract: Currently there are no brief, self-administered instruments for measuring the degree to which an adult with normal intelligence has the traits associated with the autistic spectrum. In this paper, we report on a new instrument to assess this: the Autism-Spectrum Quotient (AQ). Individuals score in the range 0–50. Four groups of subjects were assessed: Group 1: 58 adults with Asperger syndrome (AS) or high-functioning autism (HFA); Group 2: 174 randomly selected controls. Group 3: 840 students in Cambridge University; and Group 4: 16 winners of the UK Mathematics Olympiad. The adults with AS/HFA had a mean AQ score of 35.8 (SD = 6.5), significantly higher than Group 2 controls (M = 16.4, SD = 6.3). 80% of the adults with AS/HFA scored 32+, versus 2% of controls. Among the controls, men scored slightly but significantly higher than women. No women scored extremely highly (AQ score 34+) whereas 4% of men did so. Twice as many men (40%) as women (21%) scored at intermediate levels (AQ score 20+). Among the AS/HFA group, male and female scores did not differ significantly. The students in Cambridge University did not differ from the randomly selected control group, but scientists (including mathematicians) scored significantly higher than both humanities and social sciences students, confirming an earlier study that autistic conditions are associated with scientific skills. Within the sciences, mathematicians scored highest. This was replicated in Group 4, the Mathematics Olympiad winners scoring significantly higher than the male Cambridge humanities students. 6% of the student sample scored 32+ on the AQ. On interview, 11 out of 11 of these met three or more DSM-IV criteria for AS/HFA, and all were studying sciences/mathematics, and 7 of the 11 met threshold on these criteria. Test—retest and interrater reliability of the AQ was good. The AQ is thus a valuable instrument for rapidly quantifying where any given individual is situated on the continuum from autism to normality. Its potential for screening for autism spectrum conditions in adults of normal intelligence remains to be fully explored.

and "Mathematical Talent is Linked to Autism":

Abstract: A total of 378 mathematics undergraduates (selected for being strong at “systemizing”) and 414 students in other (control) disciplines at Cambridge University were surveyed with two questions: (1) Do you have a diagnosed autism spectrum condition? (2) How many relatives in your immediate family have a diagnosed autism spectrum condition? Results showed seven cases of autism in the math group (or 1.85%) vs one case of autism in the control group (or 0.24%), a ninefold difference that is significant. Controlling for sex and general population sampling, this represents a three- to sevenfold increase for autism spectrum conditions among the mathematicians. There were 7 of 1,405 (or 0.5%) cases of autism in the immediate families of the math group vs 2 of 1,669 (or 0.1%) cases in the immediate families of the control group, which again is a significant difference. These results confirm a link between autism and systemizing, and they suggest this link is genetic given the association between autism and first-degree relatives of mathematicians.

Questions: Is the AQ of a mathematician high (in average) because of his/her activity? Or was it high before? Or, was it slightly high before and then increased by the activity?

I discovered that in the following extract of the book on Grigori Perelman by Masha Gessen "Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century":

More than forty years after Hans Asperger, a British psychologist named Simon Baron-Cohen came to study autism and Asperger’s syndrome and figured out several things that seem to me to be very useful in understanding Grigory Perelman. First, Baron-Cohen suggested that the autistic brain was lopsided in a particular way. Where a neuronormal brain has the ability to both systemize and empathize, the autistic brain might be excellent at the former but is always lousy at the latter - causing Baron-Cohen to dub the autistic brain “the extreme male brain.” Baron-Cohen defined systemizing as “the drive to analyze and/or build a system (of any kind) based on identifying input-operation-output rules” and theorized that great systemizers might be at increased risk for autism. When he tested this theory on a population of Cambridge University undergraduates, it turned out that the mathematicians among them were three to seven times more likely than other students to have a diagnosis of an autistic condition. Baron-Cohen also developed the AQ, or the autism-spectrum quotient, test, which he administered to adults with Asperger’s or high-functioning autism as well as to randomly selected controls and Cambridge students and winners of the British Mathematical Olympiad. The correlation between math and autism and/or Asperger’s was proved again: mathematicians scored higher than other scientists, who scored higher than students in the humanities, who scored roughly the same as the random controls. I took the AQ test too when Baron-Cohen e-mailed it to me, and scored as high as Baron-Cohen would probably expect a former math-school student to score, which is very high. Grigory Perelman, as far as I know, never took the AQ test and certainly cannot be diagnosed by someone who has not talked to him, though after I spent an hour on the phone describing Perelman to Baron-Cohen, the famous psychologist volunteered to fly to St. Petersburg to evaluate the famous mathematician - who sounded so very much like many of his clients - thus joining the long list of people who had volunteered help that Perelman did not welcome.
Had Baron-Cohen chosen Russian rather than British mathematicians as his subjects, the results would probably have been either the same or even more clearly pronounced. After all, Russian mathematical prodigies are often grouped with others of their kind in environments that are especially tolerant of their particular brand of weirdness. The tradition of forgiving mathematicians their autistic rudenesses dates back as far as anyone can remember. Many memoirs of Kolmogorov cite his peculiar manner of walking away in midconversation, demonstrating both his utter disregard for social convention and his pragmatic approach to socializing, which is typical of Aspergians: once he had received the information he sought, he had no further use for communication. In one instance, Kolmogorov, then a dean at Moscow University, was accosted in a hallway by a man who said repeatedly, “Hello, I am Professor Such-and-Such.” Kolmogorov did not answer. Finally, the professor said, “You do not recognize me, do you?” Responded Kolmogorov: “I do, and I realize that you are Professor Such-and-Such.” In the Aspergian world, conversations are exchanges of information, not exchanges of pleasantries. Most of Kolmogorov’s students cited another of their teacher’s typically Aspergian traits: what they called his “temper” and what were actually frightening episodes of apparently uncontrollable rage. That Kolmogorov’s marked social problems did not impair his career is a measure of the degree to which a sort of Aspergian culture was built into the larger Russian culture of mathematics.

This extract should be balanced with the following extract coming from the Book Review for Notices of the AMS by Donal O’Shea:

Gessen argues that the people who surrounded Perelman sheltered him from ordinary reality, allowing him to mistakenly believe that the world is as he thinks it should be. This elaborate narrative is totally conjectural—Gessen has no evidence about what Perelman believes. Undaunted, she goes on to diagnose Perelman with a full-blown case of Asperger’s syndrome. I simply don’t know enough to evaluate these claims and am entirely unconvinced. Everyone agrees that Perelman lives simply, so why not make the simpler assumption that he wants privacy and does not want to be encumbered with fame or money? Perelman’s recent refusal of the million-dollar Clay Millennium award suggests this, particularly since the Clay Institute made it clear that Perelman would not have to participate in any public ceremony.
Even putting aside the evidentiary questions, I found the second half of the book offensive. I felt uncomfortable reading about a living individual who wishes to remain out of public sight. Publicly diagnosing someone with a serious psychological disorder without consultation seems ethically questionable, not to mention presumptuous. Doing any sort of mathematics requires precision, careful attention to meaning, and concentration. Gessen’s account of British psychiatrist Simon Baron-Cohen’s autism-spectrum quotient test, and the purported strong correlation between high-functioning autism and mathematical ability in a test population, runs dangerously close to medicalizing precisely these traits. Gessen’s presumption does not end with psychiatric expertise. She opines freely on Perelman’s work, characterizing it as solving the “very, very complicated olympiad problem” into which she has Hamilton casting Thurston’s geometrization conjecture. She cavalierly ranks top mathematicians in descending order from those who open new fields by posing questions no one has thought to ask (such as Poincaré and Thurston) to those who devise ways to answer those questions (such as Hamilton) to the bottom of the top, those poor souls (such as Perelman) who take the last steps in completing proofs. Mathematicians will easily discern the depth of Gessen’s mathematical ignorance, but others will not, and it is depressing to see Perelman’s inspiring achievement and powerful new ideas reduced to psychobabble: “Speaking of the imaginary four-dimensional space, he referred to things that could and could not occur ‘in nature’. In essence, he [Perelman] was able to do in mathematics what he had tried to do in life: grasp at once all the possibilities of nature and annihilate everything that fell outside that realm - castrati voices, cars, anti-Semitism, and any other uncomfortable singularity.”

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