Yes, I agree: the kind of "deep learning calculus" you're talking about does not exist today.
> ... I get the feeling that most deep learning papers simply describe the most successful of a series of knob-turning experiments. "We turned this knob here and performance went up by 0.05%!"
While guessing and tinkering is necessarily part of it, for the most part we're not talking about "random knob-turning" experiments. Design decisions are usually motivated by (a) mathematical intuition (e.g., if we have reason to believe the high-level abstractions to be learned have 'spatial meaning,' we might try using convolutions) and/or (b) numerical evidence (e.g., if gradients backpropagated during training are exploding or appear to be too high in a particular group of layers, we might try clipping gradients or using lower learning rates in that group of layers during training.)
> ...the whole field seems to me (as an outsider, of course) to be much more interested in finding what works, than understanding why it works.
That characterization is not fair, in my view.
There's a growing list of things we understand. For example: We understand why and how convolutions work. We understand why and how LSTM and GRU cells work. We understand why and how ReLUs backpropagate gradients so well. We know that dropout reduces to a scaled form of L2 regularization in linear models. We understand why SELUs are self-normalizing (in fact, they're constructed to be self-normalizing). We have a good intuition as to why initializing recurrent-layer weights with identity matrices works so well. Our understanding as to why residual layers work so well in really deep networks has improved considerably. Our understanding as to why SGD works so well in non-convex problems with many local minima has improved considerably. We have a good inkling as to why square-error and Wasserstein losses are less susceptible to 'mode collapse' for training generative adversarial networks.
These are just some examples I can rattle quickly, off the top of my head, in no particular order. This short list doesn't really do justice to how much our understanding of what works in deep learning and why has improved over, say, the past five years.
That said, we're still in the early days of AI, and there's a lot of work to be done to improve our understanding of it. Like Chris Olah, I believe (but can't prove, of course) that we've stumbled into something new and profound.[0][1]
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[0] For example, there appear to be deep connections between the symmetries that define the physical laws of our universe and the ability of deep learning models to work so well with what I like to call "ludicrously high dimensional" data. The fact that certain simple convnets are equivalent to certain Renormalization Group transformations is evidence of the existence of these connections, I think (but can't prove).
[1] As another example, to the best of my knowledge there is no good, formal, precise way of measuring (let alone discussing!) the kind of "hierarchically decomposable structure that is highly robust to loss" that is prevalent in natural high dimensional data -- the kind of data for which deep learning works so well. Simply put, measures like Shannon entropy or Kolmogorov complexity don't tell us much about the "hierarchical complexity" of, say, objects in a photo. The fact that such a measure doesn't exist is evidence that we've stumbled into something new and profound, I think (but can't prove).