Theory and Emergence in a Deterministic System

January 18, 2009

A cellular automaton is, in a clear sense, the simplest of universes. As such, it is a useful setting to apply and test ideas from philosophy. The cellular automaton universe (CAU) seems the least allowing for emergence (without considering trivial universes, e.g., an empty universe), since every phenomenon follows from the evolution rule. I will argue, here, that in some sense language–and therefore theory and thought–organizes and communicates structure which is emergent, even in a cellular automaton universe.

Context-dependent definitions
Suppose our CAU contains a large number of collections of contiguous on cells, which are roughly the same size. Suppose these groups move in different directions and at different rates and that their design and the rule of evolution is such that when two collide either they are both destroyed (every cell is turned off) or they bounce off of each other. In this setting we could define a term particle for these groups of cells which appear to move. How do we define particle so that we can make unambiguous statements about the nature of particles? We might specify size: particles are between 1 and 10 billion cells. We could specify shape, say in specifying the size of the boundary, or in specifying the ratio of longest diameter to shortest diameter. Of course, there is necessarily some structural requirements for the property that particles bounce or mutually annihilate, but it may not be clear what exactly these requirements are. Even without these requirements known, we consider it meaningful to observe the bouncing and annihilating of particles.

If, on the other hand, our CAU had every possible bounded arrangement of cells, each an infinite number of times, (e.g., a random initial state on an infinite space) it would be extremely tedious to have a theory about large groups of cells and in fact it would not be of any use, since for any two differing bodies there would be many bodies which were intermediate (a geodesic might be a path of single flips which never flips a cell twice) and different macro behavior would necessarily come down to the difference of a single cell. In that setting a macro theory is pointless. In a more limited CAU, as the first described here, a macro theory is useful and meaningful.

We might notice that roughly half of the particles are hollow, that is, have a cavity of off cells near their center, and that particles are only annihilated when exactly one is hollow.

When a hollow particle and a non-hollow particle collide, the two are annihilated, otherwise a collision results in the particles bouncing off of each other. (1)

Are the terms of this statement shorthand for statements about individual cells? They are not. To reiterate, in translating this statement in terms of individual cells, one necessarily gets a ridiculously cumbersome statement which contains no more information than the rule for evolution, which we may imagine is a very small piece of information.

What happens if the cells are too small to be observed or detected? Certainly here the best we can do is make empirical statements, such as (1) above. We are unable to discern two arrangements which differ by a single cell, but frequently we don’t need to, because a vast set of arrangements is absent from our universe. In our universe we recognize some objects as existing on a continuum, such as photons, but do not concern ourselves with the continuum connecting any two objects, such as a continuum connecting photons to protons, because such continua do not exist in nature. With the absence of said continua, nature suggests particular objects for naming. We have a name for a species, say horse, and we do not have names for each intermediate object, say between horses and cows. In hypothetical contexts, the proposition “X is a horse” is fuzzy, not well defined. Yet in the context of animals on earth, this proposition is perfectly well defined.

Things are further complicated if you imagine the scientist and the audience to be immense arrangements of cells. Supposing our universe is a CAU, the human eye cannot detect a single cell, by a long shot, but recognizes a discrete alphabet on this page. The human ear is never hit with two identical sounds, yet discerns discrete phonemes of speech from extremely complex and subtle patterns of changing air pressure. The brain is in two distinct states every two utterances of the same word and every letter ever printed is a different configuration of ink on a different surface of paper. It’s a bit of a miracle we discern discrete structures at all.

What is the observation that some particles are hollow and some are not, if it is not shorthand for arrangements of individual cells? We’ve said it is somehow dependent on the observable arrangements of cells, it must also be dependent on the available variety of people and the fact that each one is affected similarly by viewing a picture of a hollow cell and by holding an aerobie frisbee, that some abstract quality is registered in both cases. In this way we can view the quality of being hollow not just as a simple property–albeit fuzzy–about a particular physical object, but in fact an extremely complex property of affecting the human brain in a particular way. Abstractly we can say hollow is a property of shape; contrasted with solid; it describes objects which are lacking a large part of their interior. In any of these definitions we need not make the mistake that those terms are well defined solely in terms of the physical form of the object. We can recognize that these terms, too, are meant to reference the experience of the reader, and that although there is a relationship between solid and hollow, it only makes sense to define hollow in terms of solid if the reader has had some experience with solid, and can conceive of its opposite.

A theory that has simple words with simple relationships (e.g., of hollow and solid particles), which is deconstructed into a much finer physical system (e.g., cells of an automaton) only in an incredibly complex and intractable way (e.g., by including the nature of human observation, consciousness and language itself, each having to be further described in terms of cells) is emergent in some sense, is it not?

The objection could be raised that “emergence” is just a point of view, a priori as valid as its contender: “all effect is the cause of the rule of evolution.” I don’t disagree that recognizing “emergence” is merely a point of view. Yet science is in the business of choosing an appropriate point of view. And the people interested in science are extremely complicated, physically. To these people, some layers of cause and effect are easy–easy to understand, with limitations. From these a foundation is lain, from which to understand the other layers. So “emergence,” as far as I understand, is not just the state of things being incomprehensibly complex, but also the fact that from incomprehensibly complex systems, simple structure can emerge.