Tegmark’s Mathematical Universe

Max Tegmark Our Mathematical Universe: My quest for the ultimate nature of reality Allen Lane 2014

10 Physical Reality and Mathematical Reality

The Mathematical Universe Hypothesis

External Reality Hypothesis (ERH): there exists an external physical reality completely independent of us humans.

Mathematical Universe Hypotheses (MUH): Our external physical reality is a mathematical structure.

[The ERH] isn’t too controversial … .

Reducing the Baggage Allowance

If … reality exists independently of humans, then for a description to be complete [it] must be expressible in a form free of any human baggage  … .

[A Theory of Everything] must be a purely mathematical theory … .

[But] is it actually possible to find such a  description?

Mathematical Structures

To answer this question, we need to take a closer look at mathematics.

Modern mathematics is the formal study of structures that can be define in a purely abstract way, without any human baggage. …. [We] don’t invent mathematical structures – we discover them, and invent only the notation for describing them.

In summary …

  1. The ERH implies that “a theory of everything” … has no baggage.
  2. Something that has a compete baggage-free description is precisely a mathematical structure.

Taken together, this implies the Mathematical Universe Hypothesis  … . Everything in our world is purely mathematical – including you.

(Footnote: Gordon McCabe has argued that the term universal structural realism should be used for my hypothesis that our Physical Universe is isomorphic to a mathematical structure.)

What is a Mathematical Structure?

Baggage and Mathematical Structures

Mathematical Structure: Set of abstract entities with relations between them.

Symmetry and other mathematical properties

[The] Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from the properties of its ultimate building blocks but from relations between those building blocks.

12 The Level IV Multiverse

Why I believe in the Level IV Multiverse

Mathematical Democracy

[There is] a fourth level of parallel universes … corresponding to different mathematical structures.

How the Mathematical Universe Hypothesis implies the Level IV Multiverse

[The] MUH says that a mathematical structure is our external physical reality, rather than being merely a description of.

Limits of the Level IV Multiverse: Undecidable, Uncomputable and Undefined

How well defined do mathematical structures need to be to be real … ? There’s a range of interesting options … :

  1. No structures (i.e., the Mathematical Universe Hypothesis is false).
  2. Finite Structures. …
  3. Computable structures … .
  4. Structures with relations defined by computations that aren’t guaranteed to halt … .
  5. Still more general structures, such as one involving a continuum where typical elements require an infinite amount of information to describe.

The Transcendent Structure of Level IV

Figure 12.6: The arrows indicate the close relations between mathematical structures, formal systems and computations. [These may be] all aspects of the same transcendent structure, whose nature we haven’t fully understood.

If a formal system describes a mathematical structure, mathematicians say that the latter is a model of the former.

This structure (perhaps restricted [in some way]) exists “out there” in a baggage-free way, and is both the totality of what has mathematical existence and the totality of what has physical existence.

Implications of the Level IV Multiverse

Symmetries and Beyond

… symmetries are … simply properties of the mathematical structure … .

The Illusion of Initial Conditions

[The MUH] leaves no room for such arbitrary initial conditions … .

The Illusion of Randomness

Regardless of whether anything seems random to an observer, it must ultimately be an illusion … .

The Illusion of Complexity

[The] whole can sometimes contain less information than…  one of its parts!

Initial Conditions Reinterpreted

[The question] isn’t fundamentally about our physical reality, but about our place in it.

Randomness Reinterpreted

[Randomness] is simply how it feels when you are cloned: you can’t predict what you’ll perceive next if there’ll be two copies of you perceiving different things.

How Complexity Suggests a Multiverse

Most physicists hope for a Theory of Everything … that can be specified by enough bits to fit in a book, if not on a T-shirt … .

Testing the Level IV Multiverse

The Mathematical Regularity Prediction

[The MUH] explains the utility of mathematics for describing the physical world as a natural consequence of the fact that the latter is a mathematical structure, and we are simply uncovering this bit by bit.

I know of no other compelling explanation for [the trend for physics to keep uncovering new mathematical regularities] than that the world is completely mathematical.

My Comments

As a mathematician, I certainly agree with the notion that we need to take a closer look at the relevant mathematics, and that mathematical modelling is significant. But the hypothesis itself seems less compelling than, for example, the accounts of Boole and Kant.

Boole argues that any sufficiently held belief that is not directly testable is not indirectly falsifiable, since we can always modify less firmly held beliefs to save our most firmly held beliefs. And Tegmark’s proposed underpinnings for a ‘theory of everything’ seem to me an example of such a belief. The calim is that the hypotheses leads to some predictions which, if fulfilled, would at least lend credibility to the hypothesis. But the key here is that Tegmark does not recognize any other explanation for such predictions.

Before Copernicus it was thought that the earth was the centre of the universe, with planetary motion described by epicycles on epicycles. The expectation was that as measurements were refined these models could be refined indefinitely. And so they could. But we do not think this good grounds for accepting the theory, since we now have alternative theories. In the same way, Tegmark’s lack of imagination is not a good argument for a theory.

Kant notoriously thought that humans were somehow hard-wired to believe that Euclidean Geometry was good physics, which Tegmark does not. But if we were somehow hard-wired to have certain beliefs (such as in epicycles) then one would predict that as we uncovered anomalies we would continue to develop theories in line with our wiring, at least until there was an alternative that we found more credible. Following Kant, I would rather take Tegmark’s line of argument  as pertaining to the development of physics within the paradigm which he follows, without assuming that his paradigm is somehow ‘true’ or necessarily sustainable. In this sense I find it illuminating.

If we follow ancient practice, we may take the MUH as a hypothesis, with a natural antithesis that our conception of reality is at best a social construct to derive a synthesis:

Mathematical structures are socially constructed.

Many social scientists I know seem to take this view, but let me take this as  new hypothesis and contrast my antithesis:

Mathematical structures, in so far as they are logical, are logical in the sense of mathematical logic.  Commonplace beliefs about mathematical structures, on the other hand, are socially constructed.

My synthesis is that Tegmark’s views are socially constructed, rather than logic. So lets inspect his logic. (In which, I suspect, he is only making explicit what others seem to implicitly assume.)

I agree that, ideally, mathematical structures should be ‘baggage-free’, and it seems to me they are. I accept that a ‘theory of absolutely everything’ with baggage would not meaningfully be a theory of everything, since it would leave the baggage to be explained. But I am not clear that physicists are actually seeking such a theory. Rather, they seem to me to be seeking a theory of everything other than the logic on which it rests. From the point of view of mathematical logic, the choice seems clear: you cant have a ‘theory of everything’ that at the same time can be shown to be complete and consistent. So it must have some ‘baggage’ in the choice of the logic. Tegmark has gone for mathematical structure over mathematical logic. I would go for the converse: it seems to me a choice, which (making use of Boole) I can maintain as well as Tegmark can maintain his. But what are the implications of such a difference?

My own view is that, ideally, good theories (including physics) should be consistent with the relevant mathematics, including mathematical logic, and should not go beyond what can be demonstrated. But the current social reality seems to be that we demand theories of domains (such as economics, biology and physics) that are complete enough to be ‘actionable’ and at the same time ‘free of baggage’. From a mathematical perspective we would seem to need something very like the Mathematical Universe Hypothesis as social construct to support the belief that a particular theory meets those aims. But that is quite a different thing from supposing that these theories do meet those aims. Indeed, if such theories were even in principle possible then it seems to me that mathematical logic, and hence the whole of mathematics, would then have to be regarded as merely a social construct. Either way, the complete ‘theory of everything’ would be a social construct.

A difficulty with mathematical logic is that it denies the possibility of idealised knowledge, and so some people prefer to think of things like the MUH as a social construct that they can believe in. Thus while all knowledge may be a social construct, they at least ‘know’ that their constructs are actually true. This seems to be because they think that logic gets in the  way of ‘justified’ action. But it only bars justifications of their idealised kind: other kinds may be possible.

Tegmark notes that what matters in his view is ‘the relation between building blocks’. It seems to me that no amount of observing a computer system from the outside will tell you what its actual building blocks are. But you can nevertheless say something about its relations. The whole notion of a computer rests on the idea that application developers only need to understand the ‘virtual building blocks’ provided by the operating system, not the underlying building blocks. These virtual building blocks are modelled by mathematical structures, but one reason we have so many software updates is because the models are only ever ideal, with no precise correspondence to reality. If this is true for our engineered constructs, how can we deny it for models that are, in effect, a sophisticated form of curve fitting?

It seems to me that whenever we find a bug in an operating system our models of all the affected applications are wrong, and hence no basis for the use of such applications. But in practice even when we know are models are wrong we still expect many of the relationships between applications to be just as good as ever they were. That is, we can and do rely to a large extent on the relationships even when we know that the functionality of the applications is expressed in terms of structures that do not actually correspond to reality. Something like the same situation seems to pertain in any engineering field, and even in observational fields, such as physics.

The synthesis, then, is that we can always use mathematical models, as long as we treat them logically, as models of our current understanding of reality, not as models of reality. Does this make a difference? We don’t lose all confidence in our computers every time a new bug is discovered: we are still able to carry on with models that we know are wrong, because we know that they are still good to support necessary action with certain caveats. Thus I propose that:

While many important aspects of the real world do not correspond in any meaningful way to any possible mathematical structure, effective action in the real world (as distinct from those that are purely socially constructed) depends on modelling as an on-going activity consistent with mathematical logic.

I aspire to be able to support this view. Like Tegmark, I don’t think that it has been falsified and don’t see any alternative. But please note I only said ‘consistent with mathematical logic’, and recognize that a great many people (including some that Tegmark cites) were effective before mathematical logic was even ‘a thing’, and many who are very critical of the baleful influence of (pseudo) mathematics on contemporary life continue to be effective. Its just that I think their actions justified in so far as they happen to be consistent with the appropriate logic, whatever the thinking – logical or otherwise – behind their actions. But am I making an important distinction to Tegmark’s views?

Dave Marsay

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