The difficulty of finding out what we don't know

A snippet of a long Wired article "Scientific Data Has Become So Complex, We Have to Invent New Math to Deal With It"

Topological methods are a lot like casting a two-dimensional shadow of a three-dimensional object on the wall: they enable us to visualize a large, high-dimensional data set by projecting it down into a lower dimension. The danger is that, as with the illusions created by shadow puppets, one might be seeing patterns and images that aren’t really there. (italics mine)

“There’s a terrible math joke that topologists can’t tell the difference between their rear end and a coffee cup because the two are topologically equivalent,” said Benjamin Recht, a computer scientist at the University of California, Berkeley who says it is so far unclear when TDA works and when it might not. The technique rests on the assumption that a high-dimensional big data set has an intrinsic low-dimensional structure, and that it is possible to discover that structure mathematically. Recht believes that some data sets are intrinsically high in dimension and cannot be reduced by topological analysis. “If it turns out there is a spherical cow lurking underneath all your data, then TDA would be the way to go,” he said. “But if it’s not there, what can you do?” And if your dataset is corrupted or incomplete, topological methods will yield similarly flawed results.