Anna Sfard has contributed seminally to the role of communication in the learning and teaching of mathematics, especially through her book Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing.
Here we discuss the curious dilemma posed by the pivotal role of communication in the historical and individual development of mathematical thought, and recent findings in neuroscience that show mathematical experts do not interpret mathematical statements in the brain’s language centers.
The dilemma is essentially this: neuroscientific studies indicate that mathematical thought takes place in individual brains, principally in non-language centers. Yet mathematical thought is communicated by language, diagrams, gestures, and specialized marks and symbols. How does the brain transfer mathematical thought to a communicative medium and, conversely, how does the brain translate communicative media into non-language, mathematical, brain regions?
Sometimes the communicative stream from a top flight mathematician who is thinking original thoughts – presumably utilizing the brain regions identified by Amalric & Dehaene – is so intense, that one wonders by what mechanism the brain stimulates the communicative flow. As a somewhat extreme example, consider the snippet of Terry Tao’s talk “Controlling the de Bruijn-Newman constant” at the Perspectives on the Riemann Hypothesis, Bristol 2018, meeting, from the 50 minutes mark to the end of the video:
Terry’s communicative efforts are evident in the last 3 minutes of the video: he repeatedly looks at his expert audience and smiles. This is in marked contrast to the first few minutes of the video where he does not look at his audience and seems to write merely to set the stage for what is to follow.