A mathematician who won the Fields Medal, the Abel Prize, and the Wolf Prize spends an hour on the equals sign.
This is Pierre Deligne at the Institute for Advanced Study, September 2018, speaking at the memorial conference for Vladimir Voevodsky. The title is a question: what do we mean by equal.
Deligne proved the last of the Weil conjectures. He holds every major honour mathematics gives. He chose to talk about the symbol every schoolchild learns first.
The problem is real. Two objects can be the same in a way that carries information about how they are the same, and set theory throws that information away. Voevodsky spent his last years rebuilding foundations around exactly this.
Watch the board. Set theory's definition of equality on one line, homotopy groups underneath, and the gap between them opening as he writes.
A research engineer I know rewatched the setup twice and said type systems stopped feeling like an implementation detail.
Free on YouTube from the Institute for Advanced Study, chalk only.
Equality looked settled for 2,000 years. It was a placeholder.