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As a biomedical sciences student who also minors in Mathematics, I am often amazed by some of my classmates who really appear to be born for Mathematics (with regards to both their abilities and passions). Is there any research from a psychology/neuroscience perspective that looks at how the brains of these exceptional individuals (or indeed someone who likes Mathematics) function differently to typical individuals?

If indeed there is not systemic evidence showing their brains are different, is there a systematic way we can describe how they view the world differently from people not trained in Mathematics?

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Psychology

The first major source I would go to is the Study of Mathematically Precocious Youth (SMPY). It is a major longitudinal study following several cohorts of mathematically talented individuals. There are many publications regarding their interests, talents, life paths, achievements, et cetera. The study is headed by Camilla Benbow and David Lubinski.

If you could only read one article describing their psychology, I would refer to Lubinski's review article of 100 years of findings.

Lubinski (2016). From Terman to Today: A Century of Findings on Intellectual Precocity

However, there are many interesting publications from the study, and if you're interested I recommend further looking at their publications.

Neuroscience

Historically, neuroscience as a field has suffered a lot from small sample size studies (Button, Ioannidis, et al (2013)), and only in recent years do we get really large scale studies. I personally don't know of any findings that compare mathematicians with non-mathematicians that have sufficient power.

The closest thing I know of to what you're asking, is Richard Haier's book "The Neuroscience of Intelligence" which summarizes what we know and don't know about "intelligent" brains. He emphasizes that no single study is definitive, and that no story about the brain is simple. I recognize that just "intelligence" is not precisely what you asked, but I still think you would find it relevant to what you're curious about.

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I'm not aware of any comparative study between mathematicians and normal human but there is some research on professional mathematicians:

Stanislas Dehaene has done some research on how the brain of professional mathematicians work, specifically a study suggestion that mathematical reasonings does not rely heavily on language:

https://www.pnas.org/content/113/18/4909.short

They have also compared humans and monkeys doing math:

https://www.sciencedirect.com/science/article/pii/S0960982215007290

Finally, there is a work by Semir Zeki on how the mathematician's brain find beauty in math:

https://www.frontiersin.org/articles/10.3389/fnhum.2014.00068/full

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