Comments on Max Tegmark’s Hierarchy of Reality

I’m in the middle of reading Max Tegmark’s recent book Our Mathematical Universe, which is (so far, I’m about halfway through) mostly about the idea that it’s possible the simplest (or most natural) interpretation of quantum mechanics directly leads to the conclusion that multiple universes must exist. I just finished reading an interesting “excursion” chapter in which he discusses the nature and perception of reality, and I would like to make some comments on it because it differs from my own work on the subject.

(my essay on this topic can be found here.)

Tegmark breaks reality into three pieces, and it will be easiest to see what’s going on if I show you the actual figure in the book (this is shamelessly stolen from Tegmark, and all credit is his. If it turns out he’s not ok with this, I hope he’ll let me know!)

Tegmark's Hierarchy of Reality, from Our Mathematical Universe (Although I'm assigning the word "hierarchy" to it for my own devious purposes)

Tegmark’s Hierarchy of Reality, from Our Mathematical Universe (Although I’m assigning the word “hierarchy” to it for my own devious purposes)

The idea here is that our perception of reality (“Internal Reality”) is governed by our senses, like sight and touch and smell. We interact directly with a version of reality which we can all agree on called “Consensus Reality”, and that consensus reality is a result of something which is abstractly true, “External Reality”. In the book he makes the point that to determine the fundamental “theory of everything”, we don’t need to actually understand human consciousness, because that’s explicitly separated from consensus reality by our own perceptions.

While there certainly are elements to this hierarchy that I like, I actually think making these divisions is pretty arbitrary. I can easily ask my physics I students questions which will break “consensus reality” but stay in the realm of classical physics. For instance, I recently asked someone “what is the acceleration of an object in projectile motion?” and they responded “in the direction of motion”, indicating the parabolic path. Ok, I asked a well-defined mathematical question and received an (incorrect) response that left the bounds of mathematical rigor, but it was about classical physics, and therefore solidly in Tegmark’s “consensus reality”. The student’s level of analysis was not high enough to understand that “acceleration” does not mean “velocity” (or whatever else they might have thought I meant), but it was within *their* consensus reality.

What am I driving at? Perhaps the reality we can all agree on is not mathematical, but only descriptive in nature. For instance, the student and I can both draw pictures of how an object moves in projectile motion because we’ve seen real-life objects move in projectile motion. On the other hand, if mathematics is objectively “right” then I can prove some versions of consensus reality incorrect (“The day is 24 hours long”). Of course, no one would really say “the day is 24 hours long” is *wrong*, just that if you define the day with respect to the background stars, you get something a little bit shorter.

So even if we split off the “perception of reality” piece from our hierarchy of reality, we still end up with some rather arbitrary definitions of reality, from purely mathematical up to descriptive. This suggests that reality should be viewed as a continuum, with no clear boundaries between abstractly true and subjectively true, which all occur at different levels of detail. So what can we use to determine which level we are talking about? I’ve called such a thing the axiom of measurement, and you can check out the link in the first paragraph if you want to read the original essay.

The idea is that in order to determine a standard of “truth”, we need a standard of “measurement”. I can verify the statement “objects in projectile motion move in parabolic motion” as long as I use a measurement tool which is not accurate enough to see the effects of air resistance. That defines our “consensus reality”. But once I build a better tool, I can prove our consensus reality wrong, which requires us to redefine it at each moment for each measurement. Thus we have a natural scale for truth, defined experimentally by whatever apparatus we available.

For me, the bonus with this approach is that you know when things are true; they are true when you know an experiment can confirm them. What you lose is the concept of absolute truth, but it’s easy to argue that the concept of absolute truth has brought us nothing but trouble anyway!

(just as note, I think we necessarily lose absolute truth because we would have to be able to say “we will never design an experiment to prove this wrong”, but I don’t think we will ever be able to do that. Can anyone imagine an experiment to prove that 1+1 is not 2? I think it might strain the logical system I’m working in. Anyway, more thought on this is required).

Of course, I’m really not trying to be super-critical of Tegmark, I actually like some of his analysis. But, I think his splitting here is someone on this side of homo-centric, since it includes human perceptions at all levels (after all, we didn’t even know about his transition between quantum and classical reality until ~100 years ago. I worry about a definition of reality which shifts in time!). If we include the experimental apparatus into the very definition of our theoretical model, we achieve consistency without having to worry either about either cognitive science or a shifting consensus of reality.


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