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.”
