At a lecture in 1939, Paul Dirac said that “pure mathematics and physics are becoming ever more closely connected”. He went on to say that the two subjects might unify, with “every branch of pure mathematics then having its physical application”.

Dirac’s prognosis was, and remains, highly speculative. Today, there is no question of a unification of these fields. Techniques from pure mathematics are used in economics, engineering and finance, but there’s no sense in — nor reason for — these fields becoming one.

Dirac’s sentiment rankles with pure mathematicians because it suggests that physicists regard mathematics more as a tool with which to study the natural world than as a discipline in its own right. Such a view can be a barrier to fruitful collaboration. But when mathematicians and physicists do attempt to solve problems on equal terms, the results can be sublime — as we have seen in the physics of materials and in topology, a branch of pure mathematics that studies shapes and how they are arranged in space.

Mathematicians and physicists working in these fields have made lasting contributions to understanding the quantum Hall effect, which was discovered during a transformative experiment 40 years ago1,2. How they achieved this holds lessons for the way in which disciplines — and not only those in the physical sciences — could more successfully engage with each other on common problems.

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