- The Mathematical Universe
He also has a more easily digestible version on the arxiv, titled “Shut Up and Calculate”. Since the guy is quite entertaining, it is worthwhile to check out the recording of his colloq at PI, and the talk at the Many Worlds at 50 conference.
Let me first roughly summarize what Tegmark says. To begin with there is the underlying assumption: Reality exists. This is a rather vague statement that nevertheless many of us seem to share, so I am willing to accept it . Tegmark then formulates what he calls the ‘External Reality Hypothesis’:
ERH) “There exists an external physical reality completely independent of us humans.”
And says this ERH implies the ‘Mathematical Universe Hypothesis’
MUH) “Our external physical reality is a mathematical structure.”
He justifies this by saying mathematics is the right tool to get rid of the ‘baggage’ of our human existence, and is therefore the way to approach reality independent of human tainted thought:
- “So here is the crux of my argument. If you believe in an external reality independent of humans, then you must also believe in what I call the MUH: that our physical reality is a mathematical structure. In other words, we all live in a gigantic mathematical object – one that is more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names like Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced theories. Everything in our world is purely mathematical – including you.”
MAT) “Whereas the customary terminology in physics textbooks is that the external reality is described by mathematics, the MUH states that it is mathematics.”
He further explains
B) “A mathematical structure is […] abstract entities with relations between them.” And that MUH applies “With a sufficiently broad definition of mathematical structure".
Conscious beings like us appear within the mathematical universe as a specific kind of self-aware-substructure (SAS), that however has to cope with the problem of not having an overview on the whole universe. He calls that the ‘frog view’ and the ‘bird view’.
2. Think Maths
Indeed, I think of my job as describing reality with the use of mathematics. Therefore I find Tegmark’s hypothesis a bit disturbing. Despite (or maybe just because of) this, I find it also interesting. However, there is a good reason why I would say mathematics describes reality rather than it is reality.
To use a well known example, think of us as frogs in a cave with a fire where we can see only the shadows of ‘real’ things on the wall, like e.g. dancing Gods. Some of us frogs try to describe and understand these shadows, and we find drawings quite useful. After centuries of practice we get really good with the drawing. Since all the frogs sit on slightly different places, their drawings used to depend on their perspective. But after a lot of croaking they find a way to get rid of this froggy baggage. A couple of bright frogs even come up with interesting theories that make their drawings better descriptions of reality. E.g. they might invent some kind of interaction between the dancing Gods that explains a lot of seemingly odd behavior.
Now one of the frogs says these drawings are not descriptions of reality, they are reality. Quack, quack, quack, go the frogs. Obviously that can’t be. We can draw a lot of things that never appear on the walls.
Similarly, my problem with MUH is that I can observe a ‘real’ electron. I can think of it and describe it as a complex-valued wavefunction, a state in an ‘abstract’ Hilbert space. Now take away the electron. I can still think of and describe a complex-valued wavefunction. But that doesn’t make the electron ‘real’. At least I have the impression there is some kind of difference. If MUH is correct this could mean
- Either our thoughts are not real. Which I could try to read out of Tegmark’s sentence “We humans can imagine many things that are mathematically undefined and hence do not correspond to mathematical structures”. This however would mean that the whole idea of the reality being mathematical is based on something not being real and mathematical, i.e. our thoughts. Which doesn’t give much credibility to hypothesis that mathematics is all of reality.
- Or our thoughts are real. Then they are mathematical structures. But since human thought so far has to my best knowledge never produced an electron, our mathematical thoughts can only be very bad and insufficient descriptions of reality . Which is kind of depressing to begin with, and in addition doesn’t give much credibility to the hypothesis that mathematics is all of reality either. (Saying that the electron I think of might exist in some other part of the multiverse doesn’t soothe me either since I actually want to describe reality here.)
Bottomline: MUH is founded on our ability to think of mathematical structures. Yet circumvents to explain the ‘real’ difficulty, namely the relation between thoughts and reality. This relation however is essential to the credibility of the hypothesis to begin with.
2. Word Worries
The statement MAT that external reality is mathematics is empty without explaining what mathematics is. One could equally well say whatever reality is, let's just call it mathematics. The formulation B is more concise. However, here the problem is shifted into the word ‘abstract’:
- ab·stract (ăb-străkt', ăb'străkt') adj.
1. Considered apart from concrete existence: an abstract concept.
2. Not applied or practical; theoretical. See synonyms at theoretical.
3. Difficult to understand; abstruse: abstract philosophical problems
The open question remains whether there is ‘real’ mathematics (drawings of shadows) and ‘pure’ mathematics (drawings without shadows). To show that MUH has to follow from ERH, it is necessary but not sufficient to claim that mathematical structure is ‘with no baggage whatsoever’. One needs to know it is the only thing that can exist independent of human baggage.
I further have a problem with the reasoning of ERH implied in the word ‘independent’. We humans exist. If physical reality is a mathematical structure i.e. MUH holds, our existence has to follow from it – probably as kind of an emergent substructure. If it did not follow, physical reality was different, namely without us (you belong to my external physical reality). Thus, physical reality is not independent of us humans. Ergo, from MUH follows not ERH or from ERH follows not MUH.
You might call that nitpicking on words. It is. But it brings out the obvious fact that humans can not reliably state a formulation of reality which is independent of human ‘baggage’. The idea that we can do so is, well, an illusion. I.e. not ‘real’. Call that cognitive bias or The Principle of Finite Imagination.
Bottomline: Mathematics being without human baggage does not imply it is the only thing independent of humans, and therefore must constitute external physical reality.
3. The Level V Multiverse
Tegmark distinguishes four levels of multiverses (see illustration on page12):
- Level 1: Regions beyond our cosmic horizon
Level 2: Other post-inflation bubbles (aka the multiverse)
Level 3: The Many Worlds of Quantum Physics
Level 4: Other Mathematical Structures
He argues that we should take Darwin seriously and take into account that evolution did not train us to understand the universe, and that a fundamental description of reality therefore ought to seem odd and unintuitive to us. Interestingly, the very base of his hypothesis is that we humans are indeed able to grasp the foundations of reality, and that mathematics is the end of this leveling.
That might be true.
But maybe we should take Darwin seriously and take into account that there is no reason to believe our brains must be capable to understand true ‘reality’. The fact that we find mathematics an extremely compelling and powerful tool does not mean there is not level beyond it. So how about
- Level 5: Beyond Mathematics
Can we possibly describe everything about the electron with mathematics? Can we ever be sure we describe everything about the electron? What is reality? The idea of loosing human ‘baggage’ with using mathematical calculus is nice. But it ignores the fact that my - and I believe also Tegmark’s brain - is entirely human. To date nobody has given a definition of mathematics that is not filtered through what our brains are capable to do.
Bottomline: We have no reason to believe that our brains are capable to grasp fundamental reality.
4: The Initial Conditions
Tegmark argues MUH resolves the problem with initial conditions. Mathematical structures just ‘are’, and since all of them ‘are’ ‘somewhere’ in the Level 4 multiverse, we don’t need initial conditions to describe reality:
- “The MUH leaves no room for ‘initial conditions’, eliminating them all together.”
I like to think of the the universe being very well described by mathematics. That must not necessarily be a description by an evolution equation. I can imagine a mathematical description that just ‘is’ (and time is an illusion anyhow). That is to say the fundamental law might just not be a differential equation.
However, us being frogs means we sit somewhere inside this universe. The question we will typically ask is not ‘where are we?’ – which would then allow us to explain everything around us – but ‘how do I get from here to there?’. That is we are asking for a differential equation, a propagator, an evolution law. We measure here and now, apply our law, and get there and then. We have done that for centuries, and it works quite well. It requires however initial conditions. So this procedure might fail if we’re up to describe something like the whole Froggyverse.
Instead, we could ask ‘where are we?’ and try to go beyond an evolution law. For this one doesn’t need to assume MUH. However, we might have gotten rid of the initial conditions, but instead we now need to specify our location. The argument is then the familiar one: since everything ‘is’ somewhere, one doesn’t need to specify anything. Just that instead of finding the Lagrangian for the TOE we now need to find a measure that explains why we are ‘where’ we are.
(See here for my Thoughts on the Anthropic Principle. I haven't changed my mind, and I have nothing to add.)
So to me it doesn’t solve any problem, it just puts the frog into a bird’s perspective. However, a different perspective can turn out to be useful nevertheless.
Bottomline: Interesting way to get rid of the initial conditions that could turn out to be useful, but does not depend on reality being mathematical.
Despite the impression that you might have gotten I like Tegmark’s hypothesis because it is quite minimalistic. Usually we deal with mathematics but only part of it is ‘applied’ to reality. We have maths, and we have a reality, both of which needs to be specified. If both was identical one could drop this additional complication. I also find the idea really neat that every time I make up some mathematical ‘relations’ between ‘abstract structures’, I am actually describing a real universe – somewhere.
However, I see my task as describing the universe around me. I have little doubt that for some while mathematics will remain useful for this, and I don’t think whether or not mathematics is real makes a practical difference . I just don't find the hypothesis that reality is mathematics well motivated, since it is evidently constructed by humans. We have no reason to believe that this approach is indeed independent of human baggage, neither do we have any reason to believe it is the only way to describe reality.
It is nevertheless an interesting hypothesis, and might indeed lead to some insights. Sometimes a change of perspective is all it takes to make frogs fly.
See also: Christophe's post on MUH
 I want to point out that the question whether or not reality exists is not subject of discussion, but a more or less plausible assumption. I.e. unless you can prove or disprove the existence of reality, please spare me the comments.
 I could imagine that self-aware structures do carry around local versions of maths space with simpler structures.
 Except for a headache possibly.
TAGS: MATHEMATICS, SCIENCE, PHILOSOPHY
Max Tegmark (2007). The Mathematical Universe Foundations of Physics, 38 (2), 101-150 DOI: 10.1007/s10701-007-9186-9