Level IV Multiverse: Other Mathematical Structures and Physical Reality

Introduction

FIG. 7. Theories can be crudely organised into a family tree where each might, at least in principle, be derivable from more fundamental ones above it. For example, classical mechanics can be obtained from special relativity in the approximation that the speed of light c is infinite. Most of the arrows are less well understood. All these theories have two components: mathematical equations and words that explain how they are connected to what we observe. At each level in the hierarchy of theories, new words (e.g., protons, atoms, cells, organisms, cultures) are introduced because they are convenient, capturing the essence of what is going on without recourse to the more fundamental theory above it. It is important to remember, however, that it is we humans who introduce these concepts and the words for them: in principle, everything could have been derived from the fundamental theory at the top of the tree, although such an extreme reductionist approach would of course be useless in practice. Crudely speaking, the ratio of equations to words decreases as we move down the tree, dropping near zero for highly applied fields such as medicine and sociology. In contrast, theories near the top are highly mathematical, and physicists are still struggling to understand the concepts, if any, in terms of which we can understand them. The Holy Grail of physics is to find what is jocularly referred to as a “Theory of Everything”, or TOE, from which all else can be derived. If such a theory exists at all, it should replace the big question mark at the top of the theory tree. Everybody knows that something is missing here, since we lack a consistent theory unifying gravity with quantum mechanics.

Suppose one embraces the Platonist paradigm and avows that there truly exists a TOE at the apex of Figure 7 — and that we have merely yet to discover the correct equations. Then an unsettling query persists, as John Archibald Wheeler emphasised: Why these specific equations, and not others? Let us now delve into the notion of mathematical democracy, whereby universes governed by alternative equations are equally veridical. This constitutes the Level IV multiverse. First, however, we must assimilate two further concepts: the notion of a mathematical structure, and the proposition that the physical world may be one such structure.

What is a mathematical structure?

FIG. 8. Relationships between various basic mathematical structures (Tegmark 1998). The arrows generally indicate addition of new symbols and/or axioms. Arrows that meet indicate the combination of structures — for instance, an algebra is a vector space that is also a ring, and a Lie group is a group that is also a manifold. The full tree is probably infinite in extent — the figure shows merely a small sample near the bottom.

Many conceive of mathematics as a collection of stratagems assimilated in schooling for manipulating numbers. Yet, most mathematicians harbour a markedly different perspective on their discipline. They scrutinise more abstract entities such as functions, sets, spaces, and operators, and endeavour to prove theorems concerning the relations between them. Indeed, certain modern mathematics papers are so abstract that the sole numbers encountered within them are the page numbers! What affinity exists between a dodecahedron and a set of complex numbers? Notwithstanding the plethora of mathematical structures bearing intimidating appellations such as orbifolds and Killing fields, a salient underlying unity has emerged in the preceding century: all mathematical structures are merely special instances of one and the same entity: so-called formal systems. A formal system comprises abstract symbols and rules for manipulating them, stipulating how novel strings of symbols, termed theorems, may be derived from given ones, termed axioms. This historical development epitomised a form of deconstructionism, for it expunged all meaning and interpretation that had traditionally been ascribed to mathematical structures, and distilled out only the abstract relations capturing their very essence. Consequently, computers are now capable of proving theorems regarding geometry without possessing any physical intuition whatsoever concerning the nature of space.

Figure 8 delineates some of the most rudimentary mathematical structures and their interrelations. Although this family tree likely extends indefinitely, it illustrates that there is naught ambiguous about mathematical structures. They exist “out there” in the sense that mathematicians discover them rather than create them, and that contemplative alien civilisations would discern the same structures (a theorem holds true regardless of whether it is proven by a human, a computer, or an extraterrestrial).

The possibility that the physical world is a mathematical structure

Let us now assimilate the notion that the physical world (specifically, the Level III multiverse) is a mathematical structure. Although traditionally taken for granted by many theoretical physicists, this is a profound and far-reaching notion. It signifies that mathematical equations describe not merely some circumscribed aspects of the physical world, but all aspects thereof. It signifies that there exists some mathematical structure that is, in the parlance of mathematicians, isomorphic (and hence equivalent) to our physical world, with each physical entity possessing a unique counterpart within the mathematical structure, and vice versa. Let us consider some exemplars.

A century ago, when classical physics still reigned supreme, many scientists believed that physical space was isomorphic to the mathematical structure known as R3 : three-dimensional Euclidean space. Moreover, some surmised that all forms of matter in the universe corresponded to divers classical fields: the electric field, the magnetic field, and perchance a few undiscovered ones, mathematically corresponding to functions on R3 (a handful of numbers at each point in space). In this view (subsequently proven incorrect), dense clumps of matter, such as atoms, were simply regions in space where certain fields were potent (where certain numbers were large). These fields evolved deterministically over time according to some partial differential equations, and observers perceived this as entities moving about and events transpiring. Could, then, fields in three-dimensional space constitute the mathematical structure corresponding to the universe? Nay, for a mathematical structure cannot change — it is an abstract, immutable entity existing outside of space and time. Our familiar frog perspective of a three-dimensional space wherein events unfold is equivalent, from the bird perspective, to a four-dimensional spacetime wherein all of history is contained, ergo the mathematical structure would be fields in four-dimensional space. In other words, were history a cinematographic film, the mathematical structure would not correspond to a single frame thereof, but to the entire videotape.

Given a mathematical structure, we shall assert that it possesses physical existence if any self-aware substructure (SAS) within it subjectively, from its frog perspective, perceives itself as living in a physically veridical world. What would, mathematically, such an SAS be like? In the classical physics example above, an SAS, such as yourself, would be a tube through spacetime, a thickened version of what Einstein referred to as a world-line. The location of the tube would specify your position in space at different times. Within the tube, the fields would exhibit certain complex behaviour, corresponding to storing and processing information regarding the field-values in the surroundings, and at each position along the tube, these processes would give rise to the familiar yet mysterious sensation of self-awareness. From its frog perspective, the SAS would perceive this one-dimensional string of perceptions along the tube as the passage of time.

Although our example illustrates the notion of how our physical world can be a mathematical structure, this particular mathematical structure (fields in four-dimensional space) is now known to be the incorrect one. Subsequent to realising that spacetime could be curved, Einstein doggedly sought a so-called unified field theory wherein the universe was what mathematicians term a 3+1-dimensional pseudo-Riemannian manifold with tensor fields, but this failed to account for the observed behaviour of atoms. According to quantum field theory, the modern synthesis of special relativity theory and quantum theory, the universe (in this case, the Level III multiverse) is a mathematical structure known as an algebra of operator-valued fields. Here, the question of what constitutes an SAS is more subtle (Tegmark 2000). However, this fails to describe black hole evaporation, the first instance of the Big Bang, and other quantum gravity phenomena, ergo the true mathematical structure isomorphic to our universe, if it exists, has not yet been discovered.

Mathematical democracy

Now suppose that our physical world truly is a mathematical structure, and that you are an SAS within it. This signifies that within the Mathematics tree of Figure 8, one of the boxes is our universe. (The complete tree is likely infinite in extent, ergo our particular box is not one of the few boxes from the bottom of the tree that are shown.)

In other words, this particular mathematical structure enjoys not only mathematical existence, but physical existence as well. What of all the other boxes in the tree? Do they, too, enjoy physical existence? If not, there would be a fundamental, unexplained ontological asymmetry built into the very heart of reality, bifurcating mathematical structures into two classes: those with and those without physical existence. As a means of resolving this philosophical conundrum, I have suggested (Tegmark 1998) that complete mathematical democracy obtains: that mathematical existence and physical existence are equivalent, such that all mathematical structures exist physically as well. This constitutes the Level IV multiverse. It may be regarded as a form of radical Platonism, asserting that the mathematical structures in Plato’s realm of ideas, the Mindscape of Rucker (1982), exist “out there” in a physical sense (Davies 1993), casting the so-called modal realism theory of David Lewis (1986) in mathematical terms akin to what Barrow (1991; 1992) refers to as “π in the sky”. If this theory be veracious, then, given that it possesses no free parameters, all properties of all parallel universes (including the subjective perceptions of SASs within them) could, in principle, be derived by an infinitely intelligent mathematician.

Evidence for a Level IV multiverse

We have delineated the four levels of parallel universes in order of increasing speculativeness, so why should we credit Level IV? Logically, it rests upon two discrete assumptions:

Assumption 1: That the physical world (specifically our level III multiverse) is a mathematical structure.
Assumption 2: Mathematical democracy: that all mathematical structures exist “out there” in the same sense.

In a celebrated essay, Wigner (1967) contended that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious”, and that “there is no rational explanation for it”. This argument may be construed as support for assumption 1: here, the utility of mathematics for describing the physical world is a natural consequence of the fact that the latter is a mathematical structure, and we are simply uncovering this bit by bit. The various approximations that constitute our current physics theories are successful because simple mathematical structures can provide good approximations of how an SAS will perceive more complex mathematical structures. In other words, our successful theories are not mathematics approximating physics, but mathematics approximating mathematics. Wigner’s observation is unlikely to be based on fluke coincidences, for far more mathematical regularity in nature has been discovered in the decades since he made it, including the standard model of particle physics.

A second argument supporting assumption 1 is that abstract mathematics is so general that any TOE that is definable in purely formal terms (independent of vague human terminology) is also a mathematical structure. For instance, a TOE involving a set of different types of entities (denoted by words, say) and relations between them (denoted by additional words) is naught but what mathematicians term a set-theoretical model, and one can generally discover a formal system that it is a model of.

This argument also renders assumption 2 more appealing, for it implies that any conceivable parallel universe theory can be described at Level IV. The Level IV multiverse, termed the “ultimate Ensemble theory” in Tegmark (1997) since it subsumes all other ensembles, therefore brings closure to the hierarchy of multiverses, and there cannot be, say, a Level V. Considering an ensemble of mathematical structures does not add anything new, for this is still just another mathematical structure. What of the frequently discussed notion that the universe is a computer simulation? This idea occurs frequently in science fiction and has been substantially elaborated (e.g., Schmidthuber 1997; Wolfram 2002). The information content (memory state) of a digital computer is a string of bits, say “1001011100111001...” of great but finite length, equivalent to some large but finite integer n written in binary. The information processing of a computer is a deterministic rule for changing each memory state into another (applied over and over again), so mathematically, it is simply a function f mapping the integers onto themselves that gets iterated: n 7→ f (n) 7→ f (f (n)) 7→ .... In other words, even the most sophisticated computer simulation is just yet another special case of a mathematical structure, and is already included in the Level IV multiverse. (Incidentally, iterating continuous functions rather than integer-valued ones can give rise to fractals.)

Another appealing feature of assumption 2 is that it provides the only answer so far to Wheeler’s question: Why these particular equations, and not others? Having universes dance to the tune of all possible equations also resolves the fine-tuning problem of Section II C once and for all, even at the fundamental equation level: although many, if not most, mathematical structures are likely to be barren and devoid of SASs, failing to provide the complexity, stability, and predictability that SASs require, we, of course, expect to find with 100% probability that we inhabit a mathematical structure capable of supporting life. Because of this selection effect, the answer to the question “what is it that breathes fire into the equations and makes a universe for them to describe?” (Hawking 1993) would then be “you, the SAS”.

What are Level IV parallel universes like?

The means by which we employ, test, and potentially refute any theory is to compute probability distributions for our future perceptions given our past perceptions, and to compare these predictions with our observed outcome. In a multiverse theory, there is typically more than one SAS that has experienced a past life identical to yours, ergo there is no means of determining which one is you. To make predictions, you therefore must compute what fractions of them will perceive what in the future, which leads to the following predictions:

Prediction 1: The mathematical structure describing our world is the most generic one that is consistent with our observations.
Prediction 2: Our future observations are the most generic ones that are consistent with our past observations.
Prediction 3: Our past observations are the most generic ones that are consistent with our existence.

We shall return to the problem of what “generic” signifies in secMeasureSec (the measure problem). However, one striking feature of mathematical structures, discussed in detail in Tegmark (1997), is that the sort of symmetry and invariance properties that are responsible for the simplicity and orderliness of our universe tend to be generic, more the rule than the exception — mathematical structures tend to possess them by default, and complicated additional axioms etc. must be added to make them go away. In other words, by reason of both this and selection effects, we should not necessarily expect life in the Level IV multiverse to be a disordered mess.