[exprofmaddy]

Some differential equations that model physics admit singularities and multiple solutions. Therefore, functions are not the most general way of describing relations. Functions are a subset of relations.

Although "non-deterministic" and "stochastic" are often used interchangeably, they are not equivalent. Probability is applied analysis whose objects are distributions. Analysis is a form of deductive, i.e. mechanical, reasoning. Therefore, it's more accurate (philosophically) to identify mathematical probability with determinism. Probability is a model for our experience. That doesn't mean our experience is truly probabilistic.

Humans aren't exceptional. Math modeling and reasoning are human activities.

[red75prime]

> Some differential equations that model physics admit singularities and multiple solutions.

And physicists regard those as unphysical: the theory breaks down, we need better one.

[exprofmaddy]

For example, the Euler equations model compressible flow with discontinuities (shocks in the flow field variables) and rarefaction waves. These theories are accepted and used routinely.

[red75prime]

Great. A useful approximation of what really happens in the fluid. But I'm sure there are no shocks and rarefactions in physicists' neurons while they are thinking about it.

Switching into a less facetious mode...

Do you understand that in context of this dialogue it's not enough to show some examples of discontinuous or otherwise unrepresentable by NNs functions? You need at least to give a hint why such functions cannot be avoided while approximating functionality of the human brain.

Many things are possible, but I'm not going to keep my mind open to a possibility of a teal Russell's teapot before I get a hint at its existence, so to speak.