The quick answer I am often given is, "Yes, of course hearing a drum's shape is possible. All that is needed is a sufficiently large dataset of sounds associated with drumhead shapes (for supervised learning) or indeed even without association to the shapes (for unsupervised learning) and use of an effective training algorithm and validation methodology. Once a model has been trained on the data, it will infer the shape of a drum from any recorded spectrum it is given."
This answer is wrong, and it is the reason Kac's famous question merits being revisited in the context of today's AI to solve complex problems. In the 1990s, mathematicians finally proved that it is, in fact, not possible to hear the shape of a drum, or at least not uniquely.9 This is because drumheads exist of different shapes that produce exactly the same sound, or in mathematical terms, are isospectral.”
That seems entirely unfair. The question is if you hear the shape of a drum. The actual answer is yes. But he says this answer is wrong, because you can’t uniquely hear the shape of the drum.
I’m trying to decide if this goalpost moving is a better metaphor for this full piece, or for much of the analysis of AI in general.