*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?

## 8 comments:

Hi Gordon

There are some factual inaccuracies in what you've written here. Firstly:

"The set of rational numbers essentially contains those real numbers which can be given a finite decimal expansion, such as 23.45786."

Every schoolboy knows this is not true. A rational number is defined to be a number that can be written a/b, where a and b are integers. There are rational numbers that cannot be given a finite decimal expansion. A very well-known example is 1/3, which is 0.33333 recurring - forever. This inaccuracy undermines your argument and also undermines your credibility somewhat!

Secondly:

"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."

Well yes, it is theoretically possible to construct real numbers from decimal expansions, but only if you allow the number of terms to tend to infinity. Other constructions exist too, that depend on a similar argument. You need to state what this means in practical terms. For me, it is basically saying, you can obtain real numbers from discrete (or analog from digital) if you permit an infinite amount of digits. In other words, you can get analog from digital as long as it is already analog

to all intents and purposes! I guess that is the point of the essay contest though: if you can only measure down to a certain length scale, what can you say for sure about the underlying structure?Yeah, you're right on the first point, but I felt it would make for an excessively convoluted blog post if I had to explain the relationship between rational numbers and numbers with a finite decimal expansion!

On the second point, you seem to equate analog with infinite in the final step of your argument, but note that the rationals are already an infinite set, albeit a countably infinite set. The elements in an infinite sequence are always countably infinite, so the argument is that you can get an uncoutable infinity (the reals) by taking countably infinite sequences of elements from a countably infinite set (the rationals).

You can also, of course, just take the power set, (the set of all subsets), and you're guaranteed a new set of greater cardinality.

Perhaps we should define what we mean by 'analog' and 'digital'.

To me, analog means that you can examine the information at any level of detail. For a given level of detail, you can always examine the information at a lower level of detail. Think film versus digital photography.

Digital, to me, means representing information using a finite number of digits. For example, as any computer scientist will tell you, floating point numbers are just a digital approximation of real numbers using a finite number of digits.

If you are going to base your argument on allowing the digital reality to be represented with a countably infinite number of digits, rather than a finite number of digits, then please explain your premise for this assumption! :)

In this context, I'm happy to equate 'digital' with either the discrete or the computable. Note carefully the definition of computable functions and computable numbers. There is, for example, a countable infinity of computable numbers; they have the same cardinality as the set of rational numbers.

The Foundational Questions Institute provides the following background to its question: "While classical physics...is based on real numbers with a continuous set of values, quantum mechanics indicates that certain physical quantities can take only a countable set of discrete values."

My purpose was thus to demonstrate how an uncountably infinite cardinality (the reals) supervenes on a countable infinite cardinality (the rationals), and how the continuum can therefore be said to supervene upon the discrete. I'm

notarguing that the infinite supervenes on the finite.Hello,

Nice article Gordon. You propose an interesting argument for overlaying a continuous system on a discrete one.

Douglas - I think you are mixing up the notion of discrete with the notion of finite. Also, I don't think you should be so aggressive in attacking the author's credibility, especially when it was clear that he was being intentionally vague in order to avoid complicating his argument. As it happens the detail he has brushed over does not undermine his argument -and this comes back to your misinterpretation of 'discrete' as 'finite'.

Thanks Amin.

Doug is an old friend, so I think he was being 'tongue-in-cheek' aggressive!

In a nutshell the problem revolves around the remainder or the rounding error. Discrete implies a finite number of decimal points and, thus, rounding error. Now, consider the question: if there is rounding error built into the fabric of the universe then how does it not spin out of control, destroying itsefl? If "uncertainty" an iron law which keeps the rounding error in check then uncertainty must be perfectly plastic, continuous, seamless, dynamic, without gaps of any kind, i.e., without rounding errors. What does this mean? It means where ever uncertainty exists it is computed/calculated continuously to an infinite number of decimal points.

Thanks Peter.

Why do a finite number of decimal points imply a rounding error? A finite number of decimal points only implies a rounding error if the variable being measured is actually continuous. If the variable being measured is discrete, then a final decimal expression can be exact.

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