Forget about Tegmark’s 4 levels. The stages of truth I can remember are:
• Old Greeks saying "We only see a faint reflection of reality", i.e. we have observation, and that's flawed.
• Old Chinese saying "All we have is observation. Reality is observation, and observation is a function of the human form" which is a most interesting thing. They state that sense is inherently limited by our being. Excellent.
• Descartes saying "to know stuff, you must have doubt. Knowledge is developed by doubt" which means testing: the scientific method. Which he didn't invent, but put on a logical footing. And also founding it all on "I think, therefore I exist".
• Karl Popper saying that the essential property of what's knowable is what can be tested, questioned. This continues from Descartes and quite a few more in between including Kant obviously who's really cool but illegible.
When I think about 'absolute' truth, I am coming from a perspective that our 'reality' is necessarily constructed, broken down/separated into 'things', defined and labelled - in a way that is necessarily at odds with how these 'things' really are in an interdependent 'cosmic' whole. Science, for example, is a conventional prism through which we observe, construct and make sense of our experience or 'reality'. There are things that we agree are true by convention. For example - we might agree that the object on my desk is a "cup". It is labelled "cup". But it has no intrinsic "cupness" in and off itself - from its own side. It is dependently originated upon its materials and production in an infinite regression of causes and conditions. We can't closely examine the object and find an intrinsic quality of cupness apart from the qualities we conventionally ascribe to it. I am coming to this from a Buddhist perspective but I find echoes of it in Bohr's statement “Everything we call real is made of things that cannot be regarded as real.” By "non-dual" I mean that 'reality', ultimately, has no intrinsically existent opposites (which are really only understood by reference to what they are opposed to) like good/evil, light/dark, subject/object, self/other, God/Devil, etc., etc. I suppose you could say that 'reality' is indeterminate in a quantum sense, and not 'classical'. The laws of physics contain within them a few constants (e.g., mass of the electron, magnetic permeability of space, etc.). The values of these constants seems to be arbitrary, and they can only be determined by experiment. This bothers some (not many) physicists, particularly because small changes in the values of any of these constants completely changes the universe (e.g., if Planck's constant determines fusion efficiency in stars; change is little and stars don't form). One solution to this is to speculate that there are infinitely many possible universes, each with its own random set of values for these constants.
In the 1990's I was still in college, and this stuff interested me a lot at the time (still does). When the multiverse theory was applied to help explain the apparent wave-light pattern of photons in the two-slit experiments, evidenced by the interference pattern observed during the two-slit experiment, I wrote a paper to my physics’ Professor, José Maria de Quadros e Costa. In the paper I described the following experiment.
One way to test the existence of other universes is to run a long series of slit experiments. Some will be one-slit experiments; some will be two-slit experiments. Let's say we choose to run 50% one-slit experiments, and 50% two-slit experiments. If the order of the series is determined by change, say, by fair dice:
1 1 slit
2 1 slit
3 1 slit
4 2 slit
5 2 slit
6 2 slit
We can expect that, when we roll a 1, 2, or 3, the pattern will reflect a single slit. If we roll a 4, 5 or 6, we can expect the interference pattern associated with a 2 slit experiment. That's the control experiment. While we expect a random order of the experiments, we also expect nothing strange. We should see the interference patterns during each 2 slit experiment, and none when we perform the 1 slit experiments. Now, if I create a quantum number generator, and the 'lnformation leaking among multiverses' explanation of the interference patterns is correct, then our expectations are different. Let's say you have a box with radiation detection on two sides. Quantum theory expects that in some multiverses, a given radioactive decay event will happen in some universe, but not others. When the detector on the left side of the box goes off, run a 1 slit experiment. When the detector on the right side goes off, run a 2 slit experiment. If multiverses exist, and if the quantum information leaking across multiverses hypothesis is correct, and if we have correctly created a quantum random number generator, we should always see the interference pattern, even when we run a 1 slit experiment. This is because our other selves in 1/2 of the other multiverses will be running a 2-slit experiment while we are running the 1-slit experiment. While this experiment may test the existence, or absence of multiverse, it can only detect multiverses if the leaking information hypothesis is correct. Moreover, this experiment would not inform on the existence or absence of a deity. Especially the all-powerful kind like the flying spaghetti monster, who should, being almighty, be able to change all of the experiments in all multiverses to single slit experiments with his noodly appendages.
I fondly remember many after-class discussions with him regarding physics’ topics such as this. Alas, Quadros e Costa is already in one of the Multiverses.
Bottom-line: I don’t like Tegmark’s approach to the Measurement Problem. If the universe is Mathematical because of the Everettian Theory, it all seems rather stupid to say the least. “The Many Worlds of Hugh Everett III - Multiple Universes, Mutual Assured Destruction, and the Meltdown of a Nuclear Family” by Peter Byrne, “What is Real - The Unfinished Quest for the Meaning of Quantum Physics” by Adam Becker, and “The Emergent Multiverse - Quantum Theory According to the Everett Interpretation” by David Wallace are much better treatments on the nature of reality.
2 stars for Mad Max’s personal histories in Physics.