by Max Tegmark
Figure 12.3: The 3-D Tetris clone FRAC embodies a mathematical structure where both space and time are discrete rather than continuous.
Going still more radical, there are many mathematical structures that do away with space and time altogether, so that there’s no meaningful sense in which anything is happening in them. Most of the structures exemplified in Figure 12.4 are of this type; there’s nothing going on inside the abstract dodecahedron, say, because this mathematical structure contains no time.
Figure 12.4: A computer program can automatically generate an ordered master list of all finite mathematical structures, where each one is encoded as a sequence of numbers. The table above shows a few examples, using the encoding scheme from my 2007 mathematical universe paper. The words and diagrams in the second column are redundant baggage, reflecting some ways in which we humans name and illustrate these structures.
Our Postal Code in the Level IV Multiverse
As we discussed in Chapter 10, a mathematical structure is simply a set of abstract elements with relations between them. To explore the Level IV multiverse more systematically, we can write a computer program that automatically generates a list of the mathematical structures that exist, starting with the simplest and progressing to ever more complex ones. Figure 12.4 shows ten entries from such a list, using an encoding scheme that I described in my 2007 mathematical-universe paper (http://arxiv.org/pdf/0704.0646.pdf). The details of the encoding don’t matter for this discussion, except that it has the nice property that every mathematical structure with a finite number of elements will appear somewhere on the list. So every one of these mathematical structures can be identified by a single number: its line number on this master list.
For finite mathematical structures, all relations can be described by finite tables of numbers, generalizing the idea of a multiplication table to other kinds of relations. For structures with very large numbers of elements, these tables get big and bulky, which generates large encodings that end up far down on the list. However, a small fraction of these very large structures embody an elegant simplicity that make them quite simple to describe. Consider, for example, the mathematical structure where the elements are the whole numbers 0, 1, 2, 3,…and the relations are addition and multiplication. It would be hugely wasteful to specify how multiplication works by writing out an enormous multiplication table for all pairs of numbers: even if we limit ourselves to the first million numbers, we’d need a table with a million rows and a million columns, containing a trillion entries. Instead, we teach our school kids merely a multiplication table for the first ten numbers, and then a simple algorithm for how to use this table to multiply bigger multi-digit numbers. We describe multiplication to our computers even more efficiently than to our kids: by representing the numbers in binary, we need to specify only a 2 × 2 multiplication table for zeros and ones and a short computer program that specifies how to use this to multiply arbitrarily large numbers.
A computer program is stored as simply a finite string of zeros and ones (a bit string), which can be interpreted as a single whole number written out in binary. This gives us an alternative way of encoding and enumerating the mathematical structures from Figure 12.4: we let each mathematical structure be represented by the number whose bit string is the shortest computer program whose functions define all the relations of the structure. Now structures will appear near the top of the list as long as they’re simple to describe, even if they’re huge structures in terms of their number of elements. The complexity-theory pioneers Ray Solomonoff, Andrey Kolmogorov and Gregory Chaitin have defined the algorithmic complexity (or just complexity, for short) of a bit string as the length in bits of its shortest self-contained description, for example a computer program that outputs the string. This means that our alternative master list ranks mathematical structures in order of increasing complexity.
A nice feature of this list is that it can also handle mathematical structures with infinitely many elements. For example, to define the mathematical structure of all whole numbers with addition and multiplication, we need to specify just the shortest computer program that can read in arbitrarily large numbers and add and multiply them—Mathematica and other computer algebra–software packages have precisely such algorithms. Mathematical structures involving infinitely many points on a continuum, such as spacetime, electromagnetic fields, and wavefunctions, can often be well approximated by finite structures that can be processed on a computer as well—indeed, this is how my colleagues and I perform most of our theoretical-physics calculations in practice.
In summary, the Level IV multiverse can be systematically mapped by enumerating mathematical structures with a computer and studying their properties. If we one day succeed in identifying which mathematical structure we inhabit, then we can refer to it by its number on the master list, and for the first time specify our address in the full physical reality, as whimsically illustrated in Figure 12.5. Different countries on Earth have different schemes of specifying addresses: for example, some use postal codes with numbers, some use postal codes with letters and some use no postal codes at all. Similarly, the way you write the most local part of your address will vary between mathematical structures: most have neither quantum mechanics nor inflation, and therefore lack Levels III, II and I, as well as planets, whereas others may contain other types of parallel universes that we haven’t even dreamed of.
Figure 12.5: To specify my address in the full physical reality, I need to list my location in the Level IV multiverse (my mathematical-structure number), in the Level III multiverse (my branch of the quantum wavefunction), in the Level II multiverse (my post-inflationary bubble), in the Level I multiverse (my horizon volume), and in our Universe. I’ve listed only finite numbers in this example, although there may be infinitely many members at all four levels, making my actual address involve numbers too huge to fit on an envelope.
Click here to see a larger image.
The Structure of the Level IV Multiverse
It’s interesting to study the Level IV multiverse. If we adopt the popular formalist definition of mathematics as “the study of mathematical structures,” then studying the Level IV multiverse is what mathematicians do for a living. For a physicist like myself who believes in the Mathematical Universe Hypothesis, studying it is also tantamount to exploring the ultimate physical reality and searching for our place in it. And, conveniently, it’s easier to explore the Level IV multiverse than any of the lower multiverses or even our own Universe, because it doesn’t require rockets or telescopes—merely computers and ideas! I’ve therefore had lots of fun over the years writing computer software to perform the sort of mathematical-structure tabulation and classification that we have just talked about.
When doing this in practice, one encounters an enormous redundancy. There are very many different ways of writing a computer program that performs any given calculation, and there are similarly huge numbers of equivalent ways of describing finite mathematical structures by tables of numbers—corresponding to, for example, different ways of ordering/labeling the elements. As we discussed in Chapter 10, a mathematical structure is an equivalence class of descriptions, so the master list should contain each mathematical structure only once, specified only by that one of its many equivalent descriptions that is the shortest.
For any two mathematical structures, you can define a new one by combining all their elements and relations. Many structures on the master list are of this composite type, and when studying the Level IV multiverse, it makes sense to ignore them. This is because there are no relations connecting the two parts, which means that a self-aware observer in one of its parts will be forever unaware of and unaffected by the existence of the other part, so she can just as well act as though the other part didn’t exist—or wasn’t part of her mathematical structure. The only way in which composite structures may perhaps matter is if they enter into the solution of the measure problem, altering the likelihood you’d assign to living in differen
t mathematical structures. Because a composite structure is more complicated to describe, it will typically occur much farther down the master list than its parts, which might give it a lower “measure.” Indeed, for any finite number of structures in the Level IV multiverse, there’s also a single composite structure extremely far down the master list that contains all of them.
Although the different mathematical structures in the Level IV multiverse aren’t connected in any physically meaningful sense, there are many interesting relations between them at the meta-level. For example, we just discussed how one can be the combination of others. Another example is that one structure can in a sense describe another: the elements in the first structure can correspond to the relations in the second, and relations in the first can describe what happens when you combine relations in the second. In this sense, the twenty-four-relation “cube-rotations” structure from Figure 12.4 is described by what mathematicians call the “rotation group of the cube,” a structure having twenty-four elements corresponding to all possible rotations that leave a perfect cube looking unchanged. Many different mathematical structures share these cube symmetries and thus have some claim to being the cube—for example, the structures whose elements correspond to cube faces, corners or edges and whose relations specify either how rotations rearrange these elements or who’s neighboring whom.
Limits of the Level IV Multiverse: Undecidable, Uncomputable and Undefined
How big is the Level IV multiverse? For starters, there are infinitely many finite mathematical structures. As infinitely many as there are numbers 1, 2, 3,…to be precise, since we just saw that they can all be tabulated in a single numbered list. But how many infinite mathematical structures does the Level IV multiverse contain, each of which has an infinite number of elements? We saw that some infinite structures can also be defined and included in the master list together with the finite structures, by using computer programs to define their relations. However, embracing infinity opens a Pandora’s box of ontological problems. To see this, consider a mathematical structure where the elements are the numbers 1, 2, 3,…which includes the three relations (functions) in the following list, rules that take numbers as input and compute a new number according to the definitions listed:
1. P(n): Given a number n, P(n) denotes the smallest prime that’s greater than n.
2. T(n): Given a number n, T(n) denotes the smallest twin prime that’s greater than n (a twin prime is a prime number that has a prime number as next-nearest neighbor; 11 and 13 are examples of twin primes).
3. H(m, n): Given two numbers m and n, H(m, n) equals 0 if the mth computer program on the master list of all computer programs will keep running forever when fed the bits of n as input, and H(m, n) equals 1 if the program instead halts after a finite number of steps.
Does this structure qualify for membership in the Level IV multiverse, or is it not sufficiently well defined? The first function, P(n), is a piece of cake: it’s easy to write a program that starts checking whether the numbers following n are prime and stops as soon as it’s found one, and we’re guaranteed that this program will halt after a finite number of steps because we know that there are infinitely many primes—a fact that Euclid proved over two thousand years ago. So P(n) is an example of what we call a computable function.
The second function, T(n), is trickier: it’s again easy to write a program that checks each number following n to see whether it’s a twin prime, but if you plug in a number n greater than 37568016956852666669 − 1 (the largest twin prime known as of my writing this), there’s no guarantee that the program will ever stop and deliver an answer, because despite the best efforts of our most talented mathematicians, we still don’t know whether there are infinitely many twin primes. So for now, we don’t know whether T(n) is a computable function and hence rigorously defined, and it’s debatable whether a mathematical structure containing such a sloppily specified relation qualifies as well defined.
The third function, H(m, n), is even more nefarious: the computer-science pioneers Alonzo Church and Alan Turing established that there’s no program that can compute H(m, n) for arbitrary input numbers m and n in a finite number of steps, so H(m, n) is an example of what we call an uncomputable function. In other words, there’s no program that can determine which other programs will eventually halt. Of course, any given program will either halt or it won’t, but the catch is, just as with the twin primes, that you might have to wait forever to find out. The Church-Turing discovery of uncomputable functions is closely related to the discovery by the logician Kurt Gödel that some theorems of arithmetic are undecidable, meaning that they can neither be proved nor disproved in a finite number of steps.
Should a mathematical structure be considered well defined even if it contains a relation like H that can’t be evaluated even on an arbitrarily powerful computer? If so, its structure can only be known to an oracle-like entity that in some sense has infinite powers and is capable of performing a truly infinite number of computational steps to get an answer. Such structures would never show up on the master list we discussed earlier, which covers only structures definable by normal computer programs, not ones requiring infinite oracle powers.
Finally, let’s consider one of the most popular mathematical structures of our time: that of the so-called real numbers, such as 3.141592…, whose decimals go on forever. They form a continuum, and to specify even a single generic one, we need to list an infinite number of decimals, that is, an infinite amount of information. This means that conventional computer programs are hopelessly incapable of processing them: the problem isn’t merely performing an infinite number of computational steps on a finite input as for the H-example, but that of inputting and outputting an infinite amount of information.
Alternatively, Kurt Gödel’s work might make us worry that the MUH makes no sense with infinite mathematical structures because our Universe would be somehow inconsistent or undefined. If one accepts the mathematician David Hilbert’s dictum that “mathematical existence is merely freedom from contradiction,” then an inconsistent structure would not exist mathematically, let alone physically as in the MUH. Our standard model of physics includes everyday mathematical structures such as the whole numbers and real numbers. Yet Gödel’s work leaves open the possibility that everyday mathematics is inconsistent, and that a finite-length proof exists within number theory itself, demonstrating that 0 = 1. Using this shocking result, every other syntactically correct claim about whole numbers could in turn be proven to be true and mathematics as we know it would collapse like a house of cards.
All such uncertainties about undecidability and inconsistency apply only to mathematical structures with infinitely many elements. Are infinities, undecidability and potential inconsistency really inherent in the ultimate physical reality, or are they merely mirages, artifacts of our playing with fire and using powerful mathematical tools that are more convenient to work with than those that actually describe our Universe? More specifically, how well defined do mathematical structures need to be to be real, i.e., to be members of the Level IV multiverse? There’s a range of interesting options for which structures qualify:
1. No structures (i.e., the Mathematical Universe Hypothesis is false).
2. Finite structures. These are trivially computable, since all their relations can be defined by finite look-up tables.
3. Computable structures (whose relations are defined by halting computations).
4. Structures with relations defined by computations that aren’t guaranteed to halt (may require infinitely many steps), like our H-example.
5. Still more general structures, such as ones involving a continuum where typical elements require an infinite amount of information to describe.
The Computable Universe Hypothesis
An interesting possibility is the Computable Universe Hypothesis (hereafter CUH) that option 3 is the limit and more general structures are disqualified:
Computable Universe Hypothesis (CUH):
The mathematical structure that is our external physical reality is defined by computable functions.
By this we mean that the relations (functions) that define the mathematical structure can all be implemented as computations that are guaranteed to halt after a finite number of steps. If the CUH is false, then an even more conservative hypothesis is the Finite Universe Hypothesis (FUH); that option 2 is the limit: our external reality is a finite mathematical structure.
I find it interesting that, as we discussed at the end of the last chapter, closely related issues have been hotly debated among mathematicians without any reference to physics. According to the finitist school of mathematicians, which included Leopold Kronecker, Hermann Weyl and Reuben Goodstein, a mathematical object doesn’t exist unless it can be constructed from whole numbers in a finite number of steps. This leads directly to option 3.
According to the CUH, the mathematical structure that is our physical reality has the attractive property of being computable and hence well defined in the strong sense that all its relations can be computed. There would thus be no physical aspects of our Universe that are uncomputable/undecidable, eliminating the concern that the work of Church, Turing and Gödel somehow makes our world incomplete or inconsistent. I don’t know exactly what properties our physical reality has, but I’m confident that these properties exist in the sense of being well defined: I have no doubt that nature knows what it’s doing!
Many authors have puzzled over why our physical laws appear relatively simple. For example, why does the standard model of particle physics have such simple symmetries as those we call SU(3) × SU(2) × U(1), requiring only the 32 parameters from Chapter 10, when most alternatives are much more complicated? It’s tempting to speculate that the CUH contributes to this relative simplicity by sharply limiting the complexity of nature. By banishing the continuum altogether, perhaps the CUH may also help downsize the inflationary landscape and resolve the cosmological measure problem, which as we discussed in the last chapter is in large part linked to the ability of a true continuum to undergo exponential stretching forever, producing infinite numbers of observers.
