Grothendieck’s mindset embodied what polymath Alan Turing described as mathematical reasoning itself, “the combination of two facilities, which we may call ‘intuition and ingenuity.’” Grothendieck’s approach was to “dissolve” problems by finding the right level of generality at which to frame them. Mathematician Barry Mazur, now at Harvard, recalls conversations with Grothendieck as having been “largely, perhaps only, about viewpoint, never about specifics. It was principally ‘the right vantage,’ a way of seeing mathematics, that he sought, and perhaps only on a lesser level its byproducts.”
Grothendieck’s unique vantage point and thought style contributed to his genius. But they were also his undoing. The prospect of mathematical madness has been debated ever since Pythagoras, often described as the first pure mathematician, went on to lead a strange cult. Isaac Newton, Kurt Goedel, Ludwig Boltzmann, Florence Nightingale, and John Nash all attained mathematical prominence before succumbing to some type of psychopathology, including depression, delusions, and religious mysticism of the sort engendered by psychosis.
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Thinking styles lie on a continuum. On one end is mechanistic, rule-based thinking, which is epitomized in minds that gravitate to math, science, engineering, and tech-heavy skill-sets. Mechanistic cognition is bottom-up, concerned with the laws of nature and with objects as they exist in the world, and stands in contrast to mentalistic thinking. Mentalistic cognition exists to decode and engage with the minds of others, both interpersonally and in terms of larger social forces. It is more holistic (top-down) and humanistic, concerned, broadly speaking, with people, not with things. This mindset makes loose, sometimes self-referential inferences about reality. If “hypermentalistic,” too much meaning will be attributed to events: All coincidences are meaningful and all events are interconnected.
Every mind lies somewhere on this diametric cognitive spectrum. And as with many spectra, at each extreme the signature thought style is dialed up too high to be fully functional. Autism, in this conceptualization, is an extreme form of mechanistic thinking. It stands in contrast to psychotic disorders, characterized by false beliefs in the sentience of inanimate objects and delusions about the self and others. Reading minds is the lingua franca of mentalistic cognition, and symptoms of psychosis are essentially mind-reading on steroids.
Extreme cognitive styles map onto genius in that autism is in some cases associated with high intelligence. General intelligence, after all, includes the ability to quickly master rule-based, highly abstract thinking. And psychotic spectrum disorders, including bipolar disorder, schizotypy, and schizophrenia, are disproportionately diagnosed in highly creative individuals (they’ve been most often measured in artists, musicians, and writers) or in their first-degree relatives. Grothendieck’s broad-spectrum thought style represents both off-the-charts intelligence and unparalleled creativity. It is within this rarified space that genius may reside. And the overshoot toward either pole—or to both—may, by the same token, engender mental illness.
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The genius-madness debate has gone off course in asking whether creative individuals are at greater risk for developing mental illness than are their noncreative peers. Some are, some are not. The matter is confounded by the degree of giftedness in play. While creative types are more mentally stable than are noncreatives, the correlation reverses in the presence of exceptional creativity. Dean Keith Simonton, a professor of psychology at the University of California at Davis, finds that extraordinarily creative individuals are more likely to exhibit psychopathology than are noncreative people. He dubs this the “Mad Genius Paradox.”
An inability to filter out seemingly irrelevant information is a hallmark of both creative ideation and disordered thought. The state, known as reduced latent inhibition, allows more information to reach awareness, which can in turn foster associations between unrelated concepts. The barrage accounts for both the nonsensical ideas seen in psychosis and for novel thinking.
Over the centuries mathematical and artistic minds (and those with both gifts, such as the writer David Foster Wallace) have opined that their accomplishments flowed from the same liminal zone that harbored their greatest challenges. “The ideas I had about supernatural beings came to me the same way that my mathematical ideas did,” John Nash stated when asked why he’d once believed in space aliens.
And yet divergent thinking, while necessary for creative leaps, is hardly sufficient. In that direction lies a mind’s unravelling. Cognitive control and high intelligence must also be present, both to manage the informational cascade and to make novel use of it. “There are abnormalities of the brain that, when they co-exist with certain cognitive strengths, allow visionary thought to occur,” says psychologist Shelley Carson, who lectures on creativity at Harvard and is the author of Your Creative Brain.
“High productivity is associated with both intelligence and with high creativity, whether of a schizotypal or an autistic nature,” states Rex Jung, a neuropsychologist who studies creativity and intelligence at the University of New Mexico in Albuquerque. “These unusual characteristics are all distributed around the edges of a normal bell curve, making the possibility of [these minds] producing something new much more likely.”
The exceptional intelligence required for genius-level contributions to mathematics may not just optimize divergent thinking. It may also delay or prevent mental illness in those who are susceptible, at least for a significant period of time. Among men, the typical age of onset of schizophrenia or other psychotic spectrum disorders is in the late teens or early twenties. Yet Grothendieck, Newton, and Nash did not demonstrate thinking that could be characterized as delusional or psychotic until later in life: Nash at 30, Newton and Grothendieck well into midlife. From a neuronal perspective, the normal process of demyelination that begins in the mid-forties leads to a weakening of executive networks that are neuroprotective, explains Jung. Myelin function impacts processing speed, which is a key individual difference in intelligence, “it makes sense that someone who is highly intelligent and has a propensity to mental illness might begin to experience symptoms in this age range.”
Carson believes that these men were initially protected from illness not only by their brilliance but also by their drive to create.
This idea is echoed in the words of Pierre Cartier, among Grothendieck’s most accomplished contemporaries, who wrote that while he wished to avoid diagnosing his peer, he nonetheless considered Grothendieck’s output a buffer in his precarious mental state. “His capacity for scientific creation was the best antidote to depression, and the immersion in a living scientific milieu (Bourbaki and IHES) helped this to take place by giving it a collective dimension.”
It has been said that the ultimate mathematician is one who can see analogies between analogies. Such was the case for Grothendieck. He used metaphors, often of buildings, such as la belle demeure parfaite (the perfect mansion), to describe the solutions he sought. This is captured even in the language used by others to describe Grothendieck’s work: “Mathematicians like to walk along narrow little paths in unknown landscapes, looking for beautiful scenery or just for precious stones, but [Grothendieck] started by building a highway,” wrote Valentin Poenaru, a mathematician who knew him well. “Where some might build an acrobatic bridge between two distant mountain tops, he would just fill up the space between.”
Might the twin poles of ultramechanistic and ultracreative thinking be the variables needed for mathematical genius in particular? As Carson sees it, “People who have cognitive disinhibition added to high systematizing ability ‘see’ systems that others cannot see— visions interlocking and put together.”
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Cognitive disinhibition is integral to the creative process: It underlies loose, associative thinking as well as the inability to filter out seemingly extraneous information. This type of thinking occurs when the executive control network, responsible for higher-order cognitive tasks, ramps down in favor of the default mode network. This brain network is active in the absence of explicit attentional goals; it comes online when daydreaming or parsing the minds of others—which accounts for its dominance in mentalistic thinking. Imaging studies confirm that both highly creative individuals and those at risk for psychotic spectrum disorders exhibit unusual patterns of connectivity between the two networks.
The default network and the central executive network normally operate in dynamic tension, a hallmark of mental health as well as of high IQ. Anthony Jack, a neuroscientist who studies these regions in his Brain, Mind and Consciousness Lab at Case Western Reserve University, explains that “creativity is the only desirable area in which the seesaw is disrupted. During ‘aha’ moments you have engagement of both. Genius comes from blending the two sides. The problem is managing that blend while maintaining stability.”
