David Deutch - The Fabric of Reality

In virtual-reality terms: no physically possible virtual-reality generator can render an environment in which answers to non-computable questions are provided to the user on demand. Such environments are of the Cantgotu type. And conversely, every Cantgotu environment corresponds to a class of mathematical questions (‘what would happen next in an environment defined in such-and-such a way?’) which it is physically impossible to answer.

Although non-computable questions are infinitely more numerous than computable ones, they tend to be more esoteric. That is no accident. It is because the parts of mathematics that we tend to consider the least esoteric are those we see reflected in the behaviour of physical objects in familiar situations. In such cases we can often use those physical objects to answer questions about the corresponding mathematical relationships. For example, we can count on our fingers because the physics of fingers naturally mimics the arithmetic of the whole numbers from zero to ten.

The repertoires of the three very different abstract computers defined by Turing, Church and Post were soon proved to be identical. So have the repertoires of all abstract models of mathematical computation that have since been proposed. This is deemed to lend support to the Church-Turing conjecture and to the universality of the universal Turing machine. However, the computing power of abstract machines has no bearing on what is computable in reality. The scope of virtual reality, and its wider implications for the comprehensibility of nature and other aspects of the fabric of reality, depends on whether the relevant computers are physically realizable. In particular, any genuine universal computer must itself be physically realizable. This leads to a stronger version of the Turing principle:

(for physical computers simulating each other)

It is possible to build a universal computer: a machine that can

be programmed to perform any computation that any other

physical object can perform.

It follows that if a universal image generator were controlled by a universal computer, the resulting machine would be a universal virtual-reality generator. In other words, the following principle also holds:

(for virtual-reality generators rendering each other)

It is possible to build a virtual-reality generator whose repertoire

includes that of every other physically possible virtual-reality

generator.

Now, any environment can be rendered by a virtual-reality generator of some sort (for instance, one could always regard a copy of that very environment as a virtual-reality generator with perhaps a very small repertoire). So it also follows from this version of the Turing principle that any physically possible environment can be rendered by the universal virtual-reality generator. Hence to express the very strong self-similarity that exists in the structure of reality embracing not only computations but all physical processes, the Turing principle can be stated in this all-embracing form:

It is possible to build a virtual-reality generator whose repertoire includes every physically possible environment.

This is the strongest form of the Turing principle. It not only tells us that various parts of reality can resemble one another. It tells us that a single physical object, buildable once and for all (apart from maintenance and a supply of additional memory when needed), can perform with unlimited accuracy the task of describing or mimicking any other part of the multiverse. The set of all behaviours and responses of that one object exactly mirrors the set of all behaviours and responses of all other physically possible objects and processes.

This is just the sort of self-similarity that is necessary if, according to the hope I expressed in Chapter 1, the fabric of reality is to be truly unified and comprehensible. If the laws of physics as they apply to any physical object or process are to be comprehensible, they must be capable of being embodied in another physical object — the knower. It is also necessary that processes capable of creating such knowledge be physically possible. Such processes are called science. Science depends on experimental testing, which means physically rendering a law’s predictions and comparing it with (a rendering of) reality. It also depends on explanation, and that requires the abstract laws themselves, not merely their predictive content, to be capable of being rendered in virtual reality. This is a tall order, but reality does meet it. That is to say, the laws of physics meet it. The laws of physics, by conforming to the Turing principle, make it physically possible for those same laws to become known to physical objects. Thus, the laws of physics may be said to mandate their own comprehensibility.

Since building a universal virtual-reality generator is physically possible, it must actually be built in some universes. A caveat is necessary here. As I explained in Chapter 3, we can normally define physically possible process as one that actually occurs somewhere in the multiverse. But strictly speaking, a universal virtual-reality generator is a limiting case that requires arbitrarily large resources to operate. So what we really mean by saying that it is ‘physically possible’ is that virtual-reality generators with repertoires arbitrarily close to the set of all physically possible environments exist in the multiverse. Similarly, since the laws of physics are capable of being rendered, they are rendered somewhere. Thus it follows from the Turing principle (in the strong form for which I have argued) that the laws of physics do not merely mandate their own comprehensibility in some abstract sense — comprehensibility by abstract scientists, as it were. They imply the physical existence, somewhere in the multiverse, of entities that understand them arbitrarily well. I shall discuss this implication further in later chapters.

Now I return to the question I posed in the previous chapter, namely whether, if we had only a virtual-reality rendering based on the wrong laws of physics to learn from, we should expect to learn the wrong laws. The first thing to stress is that we do have only virtual reality based on the wrong laws to learn from! As I have said, all our external experiences are of virtual reality, generated by our own brains. And since our concepts and theories (whether inborn or learned) are never perfect, all our renderings are indeed inaccurate. That is to say, they give us the experience of an environment that is significantly different from the environment that we are really in. Mirages and other optical illusions are examples of this. Another is that we experience the Earth to be at rest beneath our feet, despite its rapid and complex motion in reality. Another is that we experience a single universe, and a single instance of our own conscious selves at a time, while in reality there are many. But these inaccurate and misleading experiences provide no argument against scientific reasoning. On the contrary, such deficiencies are its very starting-point.

We are embarked upon solving problems about physical reality. If it turns out that all this time we have merely been studying the programming of a cosmic planetarium, then that would merely mean that we have been studying a smaller portion of reality than we thought. So what? Such things have happened many times in the history of science, as our horizons have expanded beyond the Earth to include the solar system, our Galaxy, other galaxies, clusters of galaxies and so on, and, of course, parallel universes. Another such broadening may happen tomorrow; indeed, it may happen according to any one of an infinity of possible theories — or it may never happen. Logically, we must concede to solipsism and related doctrines that the reality we are learning about might be an unrepresentative portion of a larger, inaccessible or incomprehensible structure. But the general refutation that I have given of such doctrines shows us that it is irrational to build upon that possibility. Following Occam, we shall entertain such theories when, and only when, they provide better explanations than simpler rival theories.