Dr Max Tegmark looks like a very interesting person to discuss with. He is clearly a very credible scientist with a knack for explaining to the layman (that’s me), but he also indulges in what he calls “bananas” theories of everything.
I find his mathematical universe article fascinating. It is well researched and well argumented. However, I do not subscribe to most of the ideas presented there. Below are some of my objections to the reasoning in the article.
How do we find equivalent mathematical systems?
On page 4, Dr Tegmark writes a particular mathematical structure can be described as an equivalence class of descriptions, so that there is nothing arbitrary about the mathematical structure itself. If you read the article, however, you will realize that this is only illustrated with some very small, finite, mathematical structures. Mathematicians know that going from finite to infinite is usually fraught with peril.
On page 25, this is made more explicit, since Dr Tegmark writes that there is a simple halting algorithm for determining wether any two finite mathematical structures are equivalent (emphasis mine). Even ignoring the fact that finite mathematical structures are (in the present state of knowledge) less interesting as far as physics is concerned, I would have liked a reference here, because I’m not even sure what algorithm he is referring to.
How do we define mathematical systems without “baggage”?
Similarly, also on page 4, Dr Tegmark writes that the number 4 is well defined. I sent him email about this, and I am very curious to hear his response. In my opinion, the number 4 has a number of widely different definitions. It can be the result of counting, i.e. 1+1+1+1. It can be the product of 2 by 2. It can be the surface of a square of side 2. It can be the first composite number, the number of sides of any polygon with only square angles, or whatever else. Many of these definitions depend on some particular mathematical axioms. For example, the surface and square angles properties only hold in Euclidean geometry. On a sphere, you can build a triangle with three square angles.
This means in my opinion that “modding out the baggage”, to take Dr Tegmark’s expression, is in reality impossible, because the axioms of the mathematical theory have to be part of the baggage. In the case of a physics theory, there is one big central axiom, something like “this theory represents our universe”. To me, Dr Tegmark’s article does not prove at all that such an axiom can be modded out.
As an aside, in the theory of incomplete measurements (TIM), instead of trying to get rid of all axioms, I decided to see what could be considered axioms of physics. I proposed to define a measurement process as a) a physical process, b) with known input and output, c) giving consistent (repeatable) results, d) that depend only on their input, e) that impact only their output, and f) with a symbolic interpretation for changes in the output. These are more postulates than axioms, because it is a choice we make (a “baggage” as Dr Tegmark would call it) to isolate some physical processes among all possible ones and call them measurements. I believe that there is only one real axiom in the TIM, which is experimentally, there are measurements (at least to a degree of approximation that satisfies us). It is noteworthy that, if my reasoning is correct, the so-called axioms of quantum mechanics can be derived from this set of measurement postulates.
What does “defined” mean?
Even if I were to accept that “4 is well defined”, this does not mean that this applies for example to any real number. The evil spectre of Skolem’s paradox comes back to haunt us. Since the number of propositions you can write with a mathematical system made of finite sequences of symbols taken in a finite set is demonstrably countable, and since the set of real numbers is demonstrably not countable, there will be real numbers that are not the subject of any mathematical proposition. Of course, all real numbers are “subject” of a proposition like x=x, but this cannot in any way be interpreted as defining a particular real number. I can define 4 using a finite set of symbols, e.g. 1+1+1+1. I can even define some irrational numbers the same way, for example eiπ=-1 is one possible definition of π. But there will be real numbers that will not be defined that way, in any meaningful sense of “defined”.
What does scientific (or falsifiable) mean?
Dr Tegmark is clearly aware of this problem. He notices on page 11 that an entire ensemble is often much simpler than one of its members. We can describe the set of real numbers, even if there are real numbers we cannot describe. This is relevant to the discussion of a “theory of everything”, however, in a way that he did not resolve in the article, at least, not in a way intelligible or satisfactory to me. Is a theory that predicts other universes than the observable one still a theory of physics? In particular, is a theory that predicts all possible universes or mathematical structures (which seems to be Dr Tegmark’s ultimate objective) still a scientific theory, since by construction it cannot be falsified by any physical experiment?
Is there a preferred length in the universe?
On page 6, the article states there appears to be no length scale “1” of special significance in our physical space. I would personally have thought that the Planck length qualifies. Actually, as I wrote in section 3.5 of the TIM, the fact that hbar appears in physics equations at all is the best argument I can think of justifying Laurent Nottale’s scale relativity, leading to a “non-galilean” composition of scales law, as well as to impassable scales of nature playing with scale a role similar to c for translation. I do not agree with everything Nottale wrote, but the appearance of a non-dimensionless constant in a fundamental equation is worth explaining.
Did Dr Tegmark reformulate “God”?
In conclusion, I find that Dr Tegmark’s formulation for the “theory of everything” (TOE) is written in such a way that its existence itself belongs to belief more than to science. The TOE is “defined” largely by non-proven and, in my opinion, non-provable properties. It describes all possible universes (or multiverses), covers all possibilities, including identical copies of you if you look far enough (page 14), it leaves no room for initial conditions (page 10), it is eternal (page 18). And since we live within it, it is not unreasonable to state that it “creates” us at any instant.
I find it somewhat ironic that all these attributes are the very same attributes that the main monotheist religions gave to God. The theory of Dr Tegmark looks a lot like it could be described as I am that I am. One property is missing, however: these religions consider God as a person, a sentient being. Actually, the “self-aware” property is not entirely missing, since the article does mention “self-aware substructures”. But I did not see a discussion of whether the mathematical universe itself would be self-aware…
Is the mathematical universe self-aware?
That is an interesting topic. I am self-aware, but I cannot prove that you are. You being self-aware is a preferred hypothesis for me, based on what Dr Tegmark calls a “consensus view”, and recursively I believe that you believe that I am self-aware. None of this, however, addresses the question of whether, generally, a collection of people can be called self-aware. What about a collection of self-aware beings combined with some “inert” stuff? Further still, can you call “self-aware” a universe that contains self-aware beings capable of reflection about the universe?
Really, I can’t help but be amazed at how smart the guy was, who originally wrote “I am that I am” in the Bible. Thirty centuries or so later, it still takes us about 28 pages by a respected scientist to reach more or less the same conclusion. For some reason, I am ready to bet that this is not the way Dr Tegmark himself would put it 🙂