Unfortunately the writer appears not to have understood how science works and therefore the article ends up being profoundly misleading. There are several areas of misunderstanding and it may be instructive to look at each in turn.
First of all there’s the notion that science only studies the predictable. In fact, science begins with observation of phenomena. These phenomena may not appear at all predictable at first. Over time, some regularities may become apparent. Behind these regularities are likely to be general rules that provide some statistical probability of something occurring. The degree of precision depends on many factors; we would regard it as absurd to say that science can’t tell us about an individual raindrop in the middle of a monsoon and therefore science has a big gap that renders the whole enterprise dubious. As we learn more about how storms form and develop, meteorological models become less imprecise. But we don’t need precision down to the level of individual raindrops to tell the people of Delhi that a major downpour is on its way in the next three hours.
Next there’s the question of emergent properties. It makes sense to study a system at the level of emergence appropriate to what you’re trying to learn about that system. So for example a whale consists of billions of cells, and the Brownian motion within each cell is what causes much of the internal cellular mechanisms such as ribosomes and proteins to move around. But if we want to understand the migratory patterns of Blue whales we don’t try to account for the movement of each molecule inside each of the whale’s cells. Instead we look at a different level of abstraction, in this case the whale and its pod.
Far from “most of the universe being chaotic… and we focus only on the small amount of structure there is” the reality is that the precise opposite is in fact true. The universe is astonishingly ordered. We don’t see gravity behaving in a wide variety of ways at different times and in different places; we don’t see the speed of light varying at random; we don’t see any of the fundamental forces bouncing around stochastically. What we do see is consistency everywhere we look.
The fundamental confusion seems to arise from the fact that as we look at more complex systems we need more complex mathematics to model them. Quantum systems are more complex than the systems of classical physics and thus we needed to develop more sophisticated mathematics. But that is not at all the same as saying that we are moving towards a mathematics of total chaos and disorder. The universe is not at all in a state of “total chaos.” If it were, everything would be very different indeed everywhere we look. Quantum systems, for all their complexity, still obey clear laws. Were this not the case it would be very difficult to build microprocessor chips these days, and building quantum computing devices (even the primitive early ones we’re building today) would be impossible. Even at the quantum level, nature is not arbitrary.
The author is basically claiming that the universe is in fact mostly random and rule-free and we’re only studying those few phenomena that happen to obey certain types of rule. That’s the same as saying we walk down the road with objects slamming into us at every moment but we only choose to notice those that are painted red. This is, to put it mildly, an “interesting” philosophical position and in fact does not follow in any way from the author’s meander through number theory. It is in fact unclear from where in the argument the author derives his position as it appears early on as a claim and is thereafter not at all either (a) derived from some known set of facts, nor (b) demonstrated independently. It is merely a claim that is asserted; a claim that is in fact at odds with everything we see around us at all scales of investigation.