Let's see that Turing machines can approximate the execution of NN :) That's why there are issues related to numerical precision, but the contrary is also true indeed, NNs can discover and use similar techniques used by traditional algorithms. However: the two remain two different methods to do computations, and probably it's not just by chance that many things we can't do algorithmically, we can do with NNs, what I mean is that this is not just related to the fact that NNs discover complex algorithms via gradient descent, but also that the computational model of NNs is more adapt to solving certain tasks. So the inference algorithm of NNs (doing multiplications and other batch transformations) is just needed for standard computers to approximate the NN computational model. You can do this analogically, and nobody would claim much (maybe?) it's running an algorithm. Or that brains themselves are algorithms.

Computers can execute precise computations, it's just not efficient (and it's very much slow).

NNs are exactly what "computers" are good for and we've been using since their inception: doing lots of computations quickly.

"Analog neural networks" (brains) work much differently from what are "neural networks" in computing, and we have no understanding of their operation to claim they are or aren't algorithmic. But computing NNs are simply implementations of an algorithm.

Edit: upon further rereading, it seems you equate "neural networks" with brain-like operation. But brain was an inspiration for NNs, they are not an "approximation" of it.

But the inference itself is orthogonal to the computation the NN is going. Obviously the inference (and training) are algorithms.

NN inference is an algorithm for computing an approximation of a function with a huge number of parameters. The NN itself is of course just a data structure. But there is nothing whatsoever about the NN process that is non-algorithmic.

It's the exact same thing as using a binary tree to discover the lowest number in some set of numbers, conceptually: you have a data structure that you evaluate using a particular algorithm. The combination of the algorithm and the construction of the data structure arrive at the desired outcome.

That's not the point, I think: you can implement the brain in BASIC, in theory, this does not means that the brain is per-se a BASIC program. I'll provide a more theoretical framework for reasoning about this: if the way to solve certain problems by an NN (the learned weights) can't be translated in some normal program that DOES NOT resemble the activation of an NN, then the NNs are not algorithms, but a different computational model.

This may be what they were getting it, but it is still wrong. An NN is a computable function. So, NN inference is an algorithm for computing the function the NN represents. If we have an NN that represents a function f, with f(text) = most likely next character a human would write, then running the inference for that NN is an algorithm for finding out which character it's most likely a human would write next.

It's true that this is not an "enlightening" algorithm, it doesn't help us understand why or how that is the most likely next character. But this doesn't mean it's not an algorithm.

We don’t have evidence that a TM can simulate a brain. But we know for a fact that it can execute a NN.