The Foundational Questions Institute is currently inviting submissions for an essay contest entitled Is reality digital or analog?. Now, this is essentially the same thing as asking whether the physical universe is discrete or continuous. Whilst general relativistic cosmology represents the universe to be continuous, some aspects of quantum theory are discrete, and many people working in quantum gravity clearly expect the universe to be discrete at a fundamental level.
Let's take a step back, however, and consider the question in more general terms. In particular, given that the physical universe has many levels of structure, and given that the theories which describe different levels of structure can possess radically different properties, is it even possible to know whether the universe is discrete or continuous at a fundamental level?
Before going a little further, it's necessary to introduce a philosophical concept called supervenience. This is a useful concept relating different levels of structure because, unlike allied concepts such as reduction and emergence, its definition is generally agreed upon and fairly uncontroversial. Supervenience, then, holds that any change in the states or processes at a higher level of structure, must correspond to a change in the lower level states or processes. Supervenience asserts that a lower-level state or process uniquely determines the higher-level state or process, and that there is a many-one mapping between the lower-level states/processes and the higher-level states/processes.
Now, suppose on the one hand that there is a fundamental level of structure, and the fundamental level of structure is continuous. Discrete structures can clearly supervene upon such a continuous substratum. As an intuitive example, just think of the manner in which a chess board is defined upon a continuous plane. Given a continuous substratum, one can divide it up into discrete, contiguous 'chunks', assign a discrete set of possible states to those chunks, and then define a finite set of rules by which those states change from one discrete time-step to the next. A more formal notion of such a discrete system is a cellular automaton. Even if we found that space-time is a cellular automaton at some level, it is possible that such a discrete level of structure is merely supervening upon a more fundamental continuous substratum.
This seems to be reasonably intuitive, but does the converse also hold? If we suppose that the fundamental level of structure is discrete, is it possible that continuous structures can supervene upon it? Well, in one sense, this is already well-known to be true: for example, solids and liquids are known to consist of discrete collections of atoms and molecules, yet because such systems consist of large numbers of discrete entities, they can be conveniently and approximately represented as continuous systems, described by continuous fields such as those representing pressure, stress, density, internal energy, velocity etc.
Yet this is merely a form of approximate supervenience; we know that solids, for example, are really crystalline atomic lattices, or polymer chains, and that continuum solid mechanics is merely a handy tool with a limited domain of applicability. Is it possible, however, that a continuous level of structure could exactly supervene upon a discrete substructure?
Let's think about this in more abstract, mathematical terms. The set of real numbers is said to possess the cardinality of the continuum. There is an infinite number of them, and they cannot be placed in one-to-one correspondence with the set of 'whole' numbers (1,2,3, etc), hence the continuum is said to be uncountably infinite. Within the real numbers, however, there are discrete subsets, such as the set of integers (...-2,-1,0,1,2,...), and the set of rational numbers. The set of rational numbers essentially contains those real numbers which can be given a finite decimal expansion, such as 23.45786, or a recurring infinite expansion. Numbers such as pi, which cannot be given a finite or recurring decimal expansion, are real numbers, but not rational numbers.
Now, given the set of rational numbers, the set of real numbers can be obtained from them by simply taking the limit points of all sequences of rational numbers. In other words, those real numbers such as pi, which require an infinite non-recurring decimal expansion, can be seen as the limit of an infinite sequence of rational numbers, each member of which has a finite or recurring decimal expansion. One says that the set of rational numbers is (topologically) dense in the set of reals.
Defined in this sense, the set of real numbers, a set with the cardinality of the continuum, clearly supervenes upon the set of rational numbers, a discrete set. Any change from one real number to another entails a change in the sequence of rational numbers with which it is associated. If we suppose that the fundamental level of structure in the physical world is discrete like the set of rational numbers, then it is clearly possible for continuous structures to supervene upon discrete substructures, and for the supervenience to be exact.
This example opens up a more general question for the ontology of mathematical physics: If the physical world objectively possesses a mathematical structure, then it presumably follows that it also possesses any substructure of that structure; however, does it also follow that the physical world possesses any superstructure within which that structure can be embedded? The answer to the latter question is surely 'no', for by taking a disjoint union of structures, one can embed the structure of the physical universe within a superstructure to which it is totally unrelated. The crucial additional condition which needs to be added is that of supervenience, and I propose the following:
Any structure which can be constructed from the apparent structure of the physical world, and which supervenes upon that structure, must also be said to physically exist.
The example considered above, in which one structure is densely embedded inside another, can be seen as one of the tightest supervenience relationships it is possible to define!
So, in conclusion, it seems that discrete structures can supervene on continuous structures, and conversely, continuous structures can supervene on discrete structures. Given this fact, it seems impossible to establish what the fundamental cardinality of the universe is, unless one can also ascertain that the fundamental level of structure (if there is one) has been reached. And how could we know that?