### X+Y and the Special Triangle

Asa Butterfield as Nathan Ellis in X+Y.

How can mathematicians grapple with abstruse concepts that are, for the majority of people, beyond comprehension? What mental processes enable a small proportion of people to produce mathematical work of remarkable creativity? In particular, is there a connection between mathematical creativity and autism? We revisit a book and a film that address these questions.

The Mathematical Mind

In The Mind of the Mathematician, Ioan James, a mathematician and Michael Fitzgerald, a psychiatrist, examine the behaviour and personality traits that tend to fit the profile of a mathematician (Fitzgerald & James, 2007). They discuss the life and work of some twenty great mathematicians. Some were what may be called neurotypical; others had behaviour patterns that would be associated with conditions such as autism, often in a relatively mild form. Specifically, Asperger Syndrome has been linked to exceptional creativity.

Autism often involves areas of strength, such as a remarkable attention to detail and the ability to concentrate on a topic for long periods. People on the autistic spectrum frequently have enhanced capacity for deep concentration. Relatively unconcerned with social norms, they may focus undistracted on a specific problem. They may also tend to be perfectionists and be attracted by the logical consistency of mathematics.

It is very difficult to identify precisely the source of mathematical creativity. The proof of a mathematical theorem follows a path of rigorous reasoning based on clearly-defined logical rules. But the discovery of new mathematical results involves imagination, experimentation and intuition. Intuition is a kind of unconscious perception: we often have insights that appear as if from from nowhere, although they are invariably triggered by previous (conscious) mental effort, often involving prolonged and intensive concentration.

Fitzgerald and James profile twenty well-known mathematicians of the past. However, their choice of cases is far from random: they selected cases “whose personalities we find particularly interesting.” Nineteen of those chosen already had biographies in a collection, Remarkable Mathematicians, by of one of the authors (James, 2002). Only three of the twenty were female. There is undoubtedly strong selection bias affecting their conclusions.

In many of the cases, concluding comments are along the lines “ [Name] … displayed some traits of Asperger syndrome.” For example, in the article on Hamilton, Fitzgerald and James conclude that “his truly exceptional mind, eccentricity, and creative nature all point to him being an Asperger genius.” Likewise, “Hardy certainly had Asperger traits” and, for Fisher “these are all characteristics of Asperger syndrome”. While there are undoubted links between Asperger Syndrome and mathematical genius, it would be unsafe to draw any quantitative deductions from this book.

X+Y and the Special Triangle

A number of films have central characters having exceptional mathematical abilities. Many of these are based upon real individuals, although they vary in the level of realism with which their characters are portrayed. One such film, X+Y, is a story about a boy, Nathan, who is passionate about mathematics and who also has an autism spectrum condition. The film was reviewed by clinical psychiatrist Simon Baron-Cohen (2015).

Baron-Cohen identifies three factors — Autism, Sex and Mathematical talent — as forming a “Special Triangle” (see Figure). He notes that these three factors occur together more often than chance would suggest. X+Y is an excellent drama, based loosely on a real person, Daniel Lightwing, who was diagnosed by Baron-Cohen at his clinic in Cambridge to have Asperger Syndrome.

X+Y was based on an earlier film, Beautiful Young Minds, about six young students competing to be selected for the British team at the International Mathematical Olympiad. All the students were male, and half of them were diagnosed to be on the autism spectrum. While this proves nothing, Baron-Cohen argues that the Special Triangle — sex, autism and maths — is attracting attention from researchers.

Nathan, the subject of X+Y, struggles to understand other people’s behaviour, he winces when anyone touches him, he cannot engage in light-hearted banter and is incapable of figuring out what other people are thinking or how they might be feeling. He says little and, when he does speak, his words can be undiplomatic and lacking in empathy. Social conversation is so unlike mathematical arguments, which proceed, by well-defined logical steps, to a clear conclusion. Such social interaction poses great challenges for people with Asperger Syndrome. Indeed, for a real person with this form of autism, mathematics can be a gateway to social acceptance and inclusion.

Baron-Cohen presents arguments to support the hypothesis of the Special Triangle. First, autism is diagnosed far more frequently in males than in females; the ratio may be as large as four to one. He states that the ratio of males to females at the highest levels (above the 99th percentile) of the SAT-Math tests is 2:1. Of course, there are many, many reasons why this is so. Third, mathematicians have higher levels of diagnosed autism than the general population.

The film X+Y portrays the positive aspects of autism rather than concentrating on the negative aspects such as social communication difficulties. It gives us an insight, raising our awareness and increasing our understanding of the difficulties facing people with autism spectrum conditions. The film is a superb drama, handling delicate issues in a sympathetic and sensitive manner, and is both enlightening and emotionally engaging.

Sources

${\bullet}$ Baron-Cohen, Simon, 2015: Autism, sex and maths: the special triangle. The Lancet, Psychiatry. PDF.    Online.

${\bullet}$ Michael Fitzgerald and Ioan James, 2007: The Mind of the Mathematician. Johns Hopkins University Press. 1st Edn., 196 pp. ISBN: 978-0-8018-8587-7.

${\bullet}$ Ioan James, 2002: Remarkable Mathematicians. Cambridge Univ. Press, 433pp. ISBN: 978-0-5215-2094-2.

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