The mind has many faculties of perception, intuition, and so on that are beyond the faculty of rationalization or reason. Reasoning is very limited; it only relies on facts that are obvious enough to be known to the analytical mind, as opposed to facets of a situation that are known only liminally, and it cannot “process” or work in a holistic manner. Consider the fact that artificial neural networks, a type of computer artificial intelligence, don’t work via a process of rationalization at all and yet are able to identify patterns with amazing success, and they are modeled after biological brains. And that’s to say nothing of the possibility that mind and its intuition and such come from more sources than just the neurological! (That’s a whole other subject, though. See Richard A. Nichols III’s answer to Is there a relationship between heart transplantation and recipient*s emotions and personality?)
Either way, just try completely rationalizing your emotional interactions with other humans.. you’ll come to a dead end fast!
The other factor to consider is that the universe is not necessarily completely mechanistic or otherwise straightforwardly comprehensible as rationalization would require. Nature is probably way too amazing, mysterious, enigmatic, magical and ineffable to fit into the box that analytical reasoning would like to put it in.
Why? Because for any frame of mechanism or causality you can put it in, that frame is probably open-ended and connected to higher, more meta frames (or potentially imaginable frames/models) because there’s no principle that would limit it. The universe is like an onion with unlimited layers to peel away to get to the center, just like the answering question of why some facet of physics operates the way it does can only lead to more questions, and asking why the universe even came into existence requires answering within the context of some larger field of existence.
To illustrate the part about answering a question only leading to more questions, say you want to know why things fall to the ground instead of rise into space. You may figure out that masses attract other masses with a force proportional to their mass and inversely proportional to the square of their distance. Then the question is why does it do that? Say you answer that with relativity, combining gravitational attraction with the concept of inertia and postulating that mass-energy bends space-time around it, thus changing the “world line” or quickest inertial route from one place to another. Why does it do that? Nobody knows. Some people have attempted to reduce gravity to a side-effect of the electromagnetic force. Why does the electromagnetic force exist? Nobody knows. If somebody did know what causes it to exist, the next step could only be to ask why those things that cause it to exist are as they are.
Or let’s say we figure out why two substances when mixed together create a third, completely different substance after releasing some smoke. We may explain that with the idea of atoms and chemistry. Why are atoms made the way they are and why do they behave as they do? We could explain that with the Standard Model, but why is the standard model the way it is? Nobody knows. If somebody did know why the Standard Model is as it is, the next step could only be to ask why those things that cause it to be the way it is are as they are.
Say you want to know where the universe came from or why it exists. Well, now we have the Big Bang to explain it. But why did the Big Bang happen? We could postulate, for example, the Eternal Inflation model, or some other model of a multiverse, or maybe just some kind of proto-time and “random” quantum fluctuations that happened before the Big Bang, but why did those things exist? The questions never end.
This is all just to point out that the universe is necessarily a never-ending field of Russian dolls, and the mechanics each doll (inasmuch as they even are mechanical) contains hints or implications of the nature of its containing doll and the doll(s) it contains, etc. (Of course even the separation of the universe into discrete layers of dolls may itself be merely arbitrary, but the principle holds either way.)
To get back to the problem of whether the universe is even mechanistic or not (which the process of rationalizing it would desperately want it to be), science (as in, popular thinking among scientists) currently assumes that it is fully mechanistic, but there is no proof of this. We assume that it is because of the extreme efficacy of mechanical models in predicting and manipulating the world; however, these predictions are only effective in limited domains.
For example, we can very little predict specifically how someone will behave (even if you can, more or less, in some circumstances), how the weather will turn out, or which way a butterfly will fly in.. you can assume that this is all just the result of atoms bouncing around in a billiard-ball sense and its unpredictability is merely 100% the result of its immeasurable and incomputable complexity, but that would be begging the question: how do you know there are not non-mechanistic aspects to their behavior?
Also, if quantum mechanics has taught us anything it’s that events are fundamentally unpredictable, things just behave with relative predictability on the macroscopic scale when particles move in aggregate. We call quantum events “truly random”, but how do we know there is not a rhyme or reason to them that simply goes beyond our ability to model it, perhaps even our ability to rationalize about it?
Also, even if the universe is ultimately modellable with rationality, we’d be unlikely to have the proper rationality, or to use rationality in the proper way, to ultimately model it. That’s because rationalization carries with it many assumptions; it carries with it a certain worldview. For example, what if Berkeley’s idealism is correct and everything is ultimately mental, i.e. all there is are conscious beings interacting with each other? (See Entangling Conscious Agents, Donald Hoffman.) What if that means there are many principles of nature that thus work more like psychology, with all its inconsistency, its open-endedness, its free will, and its regular contingency upon interpretation, than clockwork? Common rationality would abhor this.
I say a lot of “what if”s and “probably”s, but I actually have no doubt that the universe is magical and not completely mechanistic; I just wanted to help lead others’ rationalities easily to my point of view by not confronting them with positive, likely objectionable claims, and the idea that the universe may be non-mechanistic, magical, psychological, or otherwise non-modellable to some degree is sufficient to raise doubt that we may not be able to rationalize everything.
“The man who listens to reason alone is lost; reason enslaves all whose minds are not strong enough to master her.” -George Bernard Shaw
“A mind all logic is like a knife all blade. It makes the hand bleed that uses it.” -Rabindranath Tagore