The human brain is generally understood to be a finite computational system. I think this is something we can basically agree on. Its volume, energy consumption, mass, and number of components are all finite, and it is basically discrete. From a logical point of view, the output of a neuron is probably binary, or perhaps ternary. Even if we disagree about this, from the perspective of quantum mechanics, the bandwidth between neurons and between the lower-level constituents of the brain is finite.
So we have reason to believe that the brain is not stronger than a Turing machine. If the brain were stronger than a Turing machine, we would be very surprised. It is at least meaningful to study physics under the assumption that the computational power of the brain is equivalent to that of a Turing machine.
Now let us consider the physical world. The physical world is first of all the object of consciousness whose physical basis is the brain; in the sense of human cognition, there is no physical phenomenon outside consciousness. We can use different words to describe the relation between consciousness and the physical world, but the substance does not change. One way to put it is that consciousness knows physical phenomena, and the sum of physical phenomena is the physical world. Another way to put it is that the physical world is generated in consciousness. I usually use the language of generation, but effectively, these two are no different: there is no physical phenomenon that affects our cognition but does not appear in consciousness. All physical phenomena, by definition, appear in some consciousness. Therefore we can say that the world is the consciousness-world. This term refers to the same thing as the experiential world or the physical world.
Of course, I am not required to assert that the physical world is an appendage of consciousness. I simply do not discuss that question. Is it possible that there exists some "object" independent of consciousness, and that the world in our consciousness is its projection? Probably. But whether it exists or not has no effect on the structure of how we know the world. Whether we believe physical phenomena are known by consciousness or generated by consciousness, we can state the following claim with confidence:
A physical phenomenon must satisfy the structure by which it can be known/generated by consciousness before it can appear in consciousness. That is, the physical world is not generated arbitrarily. It is generated according to the structure of consciousness; it has been filtered through that structure. Logically speaking, no physical phenomenon can fail to satisfy the structure of cognition and still be known. This sentence is a tautology in logic, but it is substantively important: the physical world must conform to the cognitive structure of consciousness. A physical world that does not conform to the cognitive structure of consciousness cannot be known by us, and therefore effectively does not exist.
Then how do we characterize the cognitive structure of consciousness? Some of it is introspective and belief-like, such as causality. We cannot know a world that violates causality. This is exactly the way Kant and Einstein approached physics: instead of asking what physical phenomena are like, we ask what kind of physical structure is cognitively possible.
Buddhism calls phenomena "dharmas." A dharma is something that conforms to determinacy, a most general form of causality. In this sense, Buddhism treats causality as the essence, and indeed the very body, of the phenomenal world. What kind of phenomenal world is possible? Buddhism's answer is: dharma.
On the other hand, we can treat this question formally. If we assume that the physical basis of consciousness is not stronger than a Turing machine, then we can say: in principle, the physical world must be simulable by a Turing machine. Behind this statement is a principle of indistinguishability: the knowing/generation of physical phenomena in consciousness is the result of computation by the human brain as a Turing machine (whether or not this computation is related to the "thing-in-itself"). Therefore the physical world must be simulable by a Turing machine. Otherwise, we would have to believe that the brain is essentially stronger than a Turing machine, which is very hard for us to accept.
The first consequence of this claim is that the physical world is finite. All of its infinities cannot be trivial infinities, because trivial infinity cannot be simulated by a Turing machine. For example, suppose a physical phenomenon is a line on a two-dimensional plane. For us to know it is equivalent to a Turing machine being able to generate it with finite resources in finite time. It must therefore be compressed from infinite space into finite space, which means it must be describable by an equation with finite parameters.
At the same time, this physical world must also reject infinite-precision measurement. That is, in general you cannot take an exact x and measure its y. Doing so would mean that across the domain of x you have judged no for infinitely many points, and this is still not simulable by a Turing machine. Therefore the form of measurement (that is, purposeful knowing/generation) must be: take x, with error < delta(x); measure y, with error < delta(y). This error is not technical. It comes from the essence of the Turing-machine simulator: when it cannot analytically compress a physical phenomenon, its general method of compression is to quantize the basic dimensions. Of course, this is not a strict proof. What I actually want to say is that the simulator cannot allow trivial infinite-precision measurement, but I cannot produce the precise mathematical form, so I will put it this way for now.
I want to emphasize again that this essay is not trying to give a substantive physics. It is trying to study, through the cognitive structure of consciousness, what kind of physics can possibly be known, and what kind of physical world can possibly exist in consciousness. This is a Kant-Einstein approach.
Of course, some people may object: what if your understanding of consciousness and the brain is completely wrong? What if the brain really is stronger than a Turing machine? What if the material basis of consciousness is not only the brain? These are very good questions, and I would be very happy to discover that I am wrong about them. But the way we ask these questions is precisely the fundamental motive for writing this essay: the structure of the physical world is, at root, about the structure of consciousness. To ask for a physical world stronger than Turing-machine simulation is to ask for a consciousness stronger than Turing-machine simulation.