Rule-based mathematics, or deep learaning [sic], might never yield a real artificial intelligence.

It is in my opinion misleading to reduce deep learning (DL) to rule-based mathematics. Sure, software can be reduced to complicated if-then-else statements, but that misses the point: a deep learning network is a layered artificial neural network, which in itself evaluates functions. It is of course a mathematical model and it is implemented by software, but it distracts from the fact that a DL network is a very sophisticated model. After all, mathematicians have shown that a finite linear combinations of sigmoidal functions can approximate any continuous function of real variables (with support in the unit hypercube). A less technical discussion of so-called universality theorems can be found here.

The main difference between a simple rule-based model and one backed by neural networks (but also simpler machine learning algorithms) is that in a rule-based model you have to program all the branches, whereas in ML algorithms the algorithm determines the weights and relative importance of these weights, which in turn lead to 'branches', if you will. Someone still has to write an implementation of these algorithms, but by and large you don't consider, say, an optimization algorithm to be a collection of if-then-else statements. In the same way, few people think about programs as on/off of transistors.

At the moment, DL is mostly empirical. What is really lacking is a deep understanding of the inner workings. There seem to be a few ideas from physics that have parallels in deep learning (e.g. spin glasses and the renormalization group), but what these research articles do not show is the actual why. It's interesting that there may be a connection between deep learning and theoretical physics, but that does not really prove anything, just that the mathematics used to describe both is similar or even the same. Moreover, a lot of these theoretical comparisons use simplified networks (e.g. RBMs) and not the ones you would actually see in, say, Google DeepMind.