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 *e ^{iπ}=-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 🙂

Hi Christophe:This is an interesting post, I have added a link to it. I think Tegmark addresses the question of (un)computability somewhere in his paper, therefore he has several possible versions of ‘mathematical reality’.In my post I focused on the more philosophical questions and avoided the mathematical and physical stuff (well you know, you make the post too long, nobody will read it). I find the paper on the philosophical side very weak since e.g. the central argument (first page) that the ‘external reality hypothesis’ implies ‘the mathematical universe hypothesis’ evidently doesn’t hold without further assumptions about ‘mathematical structure’ or ‘human baggage’, which however is not clearly stated in the paper. I had a similar concern to yours about the length, namely, how do dimensionful constants come into mathematics? In a certain sense, I think it is the dimensions (mass, length, time…) that make things ‘real’. Best,B.

Hi and why,You write on the bottom of your blog-page:” Disclaimer: All ideas presented on this blog belong to Christophe de Dinechin. They do not necessarily represent the position of my employer. Copyright Christophe de Dinechin.”Do you mean that if I will discuss your ideas or what you write and I show you my ideas on your blog-site, you claim that my ideas presented on your blog belongs to you.What you claim implies that I am afraid to comment anything interesting and important on your blog.

Hi Ingvar,The disclaimer simply states that the blog is not a work for hire. As far as I know (but I know little), the usual rules of copyright are: without explicit copyright assignment, writings belongs to their authors. This includes comments on this blog or the contents of the blog itself.One exception in the US is work for hire, where material may belong to whoever paid for it. I thought it was necessary to make it clear that this blog is not paid for by my employer. That’s it. Nothing nefarious here, no intent to steal whatever idea you might want to discuss here.

I had a look at your TIM paper. My comments are 1. The title would be better as “A general framework for describing physical measurements”, as this is all that is provided. Theories are rather more specific than what is given, and make predictions – theories can either can be fitted or not be fitted into the particular framework given. Other frameworks are often less general, including, for example, ‘quantum’ logic (measurements represented by propositions), W*-algebras (measurements represented by algebras with a ‘complete’ set of idempotent elements), and convex set theory (states represented by convex linear measures) – see, eg, Hans Primas, “Chemistry, quantum mechanics and reductionism” (Springer), and Stanley Gudder’s “Stochastic quantum mechanics”, for overviews. However, more general approaches, at similar levels to your own, are the operational frameworks of Guenther Ludwig and of Charlie Randall and David Foulis – see, eg, the references at http://plato.stanford.edu/entries/qt-quantlog/ . 2. The paper is sincerely and honestly written, and often quite well written, and clearly the ideas have provided you with a personally satisfactory way of thinking about physical phenomena. However, there is often little more than this to be achieved by embedding known theories into a particular framework unless they (i) suggest better theories, and/or (ii) allow better ‘understanding’ of known theories. Point (i) is the important one, as point (ii) is quite subjective – for example, one person’s interpretation of QM is very often another person’s anathema! Since no better theories are in fact suggested, the content of the paper has only subjective significance – this is not necessarily a bad thing, but it does make it more of interest to philosophers rather than to mainstream physicists such as myself. Similar comments apply to the operational frameworks mentioned above. 3. I have little to remark on the relativity sections, as my only disagreements are at the ‘philosophical’ level – eg, I would have given Einstein a little more credit for physical justification of GR (principle of equivalence, and the recognition of curvature effects in SR from a mass rotating about a fixed point); writing down the local Minkowski metric or geodesic equation does not give them any physical significance per se; the two-slit experiment is a poor justification for curved spacetime; and the existence of the Planck length could just as well be used to argue against instead of for scale invariance of physical laws. 4. I think that the quantum sections are a little misleading to a careless or non-expert reader(though not intentionally so!), both in terminology and presentation – the arguments given to introduce linear operators and wavefunctions and the like have little substance, and do not lead even to the usual Hilbert space description. The best one can say is that standard QM fits into your general framework, but that very little of it is implied by this framework. Some specific remarks: 4a. One can certainly, and trivially, rewrite measurement probability distributions p=(p1, p2, …, pn), which live on the n-simplex, as “unit vectors”, psi=(psi1, psi2, …,psin), which live on the surface of an n-ball (where psi1=sqrt[p1], etc). However, there is no reason whatsoever in the TIM framework to postulate that measurements and evolution are linear with respect to psi, rather than being linear with respect to p (or indeed with respect to any any other function g=(g1, g2, …, gn) of the probabilities – eg, g1=exp[p1]). And it is merely ex post facto analysis in the light of QM to suggest such postulates are even reasonable! One needs something else to ‘derive’ a physical significance for linearity in psi (eg, Kaehler manifolds). In the absence of this something else, it would seem far more reasonable, for example, to postulate that the evolution and operators should be linear with respect to the probability vector p ! 4b. Even when one formally writes down a vector psi as above, it only inhabits a real Hilbert space, rather than the complex Hilbert space needed for QM (eg, to explain the physical double-slit experiment, the Aharanov-Bohm effect, the singlet state, etc). Hence, while the formulas written down “look” quantum mechanical, much more is needed. 4c. The notation M^|psi> for general nonlinear operators, and M^|psi>=m|psi> for eigenvectors, is misleading. One does not have, for example, the property(M^ + N^) |psi> = M^|psi> + N^|psi>in general, even though the notation suggests it. It would be more accurate to represent nonlinear operators as functions mapping the sphere to itself, eg, M(psi), and to refer to vectors satisfying the property M(psi) = m psi as ‘fixed points up to scaling’ or similar. 4d. The QM of entangled systems (more than one particle) is not easily explainable by postulating that collapse is limited to the speed of light, and does not appear to be compatible with recent experiments by Gisin et al (eg, http://prola.aps.org/abstract/PRA/v63/i2/e022111 and http://lists.paleopsych.org/pipermail/paleopsych/2006-April/005093.html,). For example, suppose Alice and Bob live in spacelike separated regions and share a singlet state (or ensemble thereof), and each decides to measure spin in directions a or b, at random. The probability for Bob to measure spin -up is 1/2 in any run. Now, if collapse is at the speed of light, then his result cannot depend on whether Alice has found spin up or spin down for her measurement – it can only depend on local variables in his vicinity. But in fact Alice and Bob find perfect correlations when they happen to make the same choice of measurement direction – which cannot be explained via such local variables (Bell’s theorem). So, whatever entanglement is, its effects cannot easily be explained, and indeed are exacerbated in strangeness, by restricting collapse to the speed of light. I apologise for corresponding anonymously, but after a friend pointed me to your paper and said that you were keen for feedback, I thought this might be helpful as a once-off thing.

Dear Anonymous,I really appreciate the time and thought you put in your comment. This is exactly the kind of feedback I was looking for.1. On the “theory” vs. “framework” aspect, I had not thought of this. I believe that you may be right, at least if we see this as a theory of physics. On the other hand, seing this as a theory of measurements, wouldn’t you say that it enables some interesting predictions on the properties of measurements? In other words, isn’t is a legitimate “theory of measurements” (per the title), without necessarily being a “theory of everything”?2. You are entirely right that I was first looking for a personally satisfactory way of thinking about physics. I found it embarassing to be reluctant to tell my kids about quantum mechanics because even if I understood how it worked, I could not explain why it might work. Now, I feel that I can explain the basics with much less intellectual dishonesty or ad-hoc deus-ex-machina stuff than before. Maybe I’m entirely wrong and it’s just another case of “ignorance is bliss” ;-)The TIM is not just philosophy, though, it includes new tools (or so I think). That much should be interesting even to mainstream physicists, don’t you think? Now, it turns out that explaining the new tools and why they seem compatible with existing knowledge already took about 30 pages. So I thought that any far-fetched predictions I might want to offer would have to wait for follow-up articles. Basic length constraints…Don’t get me wrong, there are predictions in the TIM, and I hint at some of them in the article you read. For instance, that space-time is only a manifestation of electromagnetic interactions, or that the correct normalization of the wave-function is discrete. I don’t think that this is considered mainstream knowledge today. Of course, these ideas may just as well turn out to be very wrong. And that’s why I consider them as “predictions”: if these statements are wrong, it will be possible to prove it with an experiment, and that will invalidate all or part of the TIM. Either way, we learn something.There are more predictions to come. I’m spending a little time here and there on two follow-up articles, but I’m a bit reluctant do expose them to an anonymous reader in a public forum 😉 What I can tell for now is that the first one will be about bats and jetfighters, and the second one about what happens when you consider perturbations of X, XY, XYZ and XYZT, and why it’s so strange that this seems to look like precisely the forces we know.3. I’m a great admirer of Einstein. I have always been amazed by how far forward looking and intellectually honest he was. His remarks that Gaussian coordinates are required by his theory without having a real justification (ref [8]) is a good illustration.But we may disagree on whether something like the principle of equivalence can be considered a satisfying “physical justification” today (as I believe you implied). Similarly, if by the “recognition of curvature effects in SR” you refer to something similar to reference [12], then I discuss this in the subsection about “Non-Euclidean geometry”, and I consider the deduction to be logically wrong. The conclusion is true, but cannot be deduced from the premise. Specifically, one cannot deduce that a 4D space is curved from the existence of a curved 3D surface, for the same reason that having a curved 2D surface like a sphere does not imply that the 3D space is curved. Maybe I misunderstood what you were referring to?You are right that the two-slit experiment alone is not a good justification for a curved space-time. But I used it only as one element in a demonstration that we just don’t know how to build an Euclidean geometry with light rays. Nothing more is implied than that.Regarding scale invariance, your comment made me realize that the corresponding paragraph was misleading. What I indended to say was that there seems to be a scale invariant (Planck’s length) just like there is a speed invariant (the celerity of light). I did not argue in favor of the scale invariance of physical laws in general. Rather to the contrary, any such scale-invariant physics would require a serious re-thinking a la Nottale. Opinions on Nottale’s work are highly contrasted, but at the very least, he identified a key issue.4. All your remarks are entirely legitimate. My objective was not to get Hilbert formalism at any price, but rather to answer the question: how is it possible that such a formalism works at all? So I started with conditions which I think are more “primitive”, and deduced something that looks enough like Hilbert that one might think: “Ah, that’s why it works”.4a. The lack of linearity is precisely the point 🙂 I find linearity in psi in QM just a little bit too ad-hoc for my taste. That’s why I prefer to *construct* an operator $\check{M}$ that makes eq(28) look like QM, much like for example eq (12) is constructed to look like a time. Following eq (28), we have linearity in psi by construction, but that does not mean linarity in psi is considered a general case. On the contrary, the idea is to a) highlight the required conditions, b) verify that we can reuse the QM tools.4b. The fact that Hilbert spaces in QM are complex-valued whereas probabilities are real-valued is a relatively frequent objection, and a somewhat subtle one. I am not sure that my answer will convince you 😉 If you read this blog, you should realize that I do not see real numbers, or even integers, as valid in physics a priori. For any law using an integer n, you can find a large enough n where the law obviously breaks.Instead, in the TIM, I take the entirely opposite approach. What is the physical structure? What do the mathematical equations derived using the TIM methodology look like? If they look like x^2+y^2=d^2, then I can do a vast simplification by considering x,y,d as both infinite and continuous, use real numbers, and consider this an Euclidean geometry. But that’s only an approximation. In reality, my coordinates are not infinite, they are not continuous. In short, there are no real numbers in physics. Much less complex numbers.So what I’m saying is that using complex numbers is a simplification (e.g. to deal with the roots of polynoms). For me, it is enough that the equation looks quantum mechanical for complex numbers to be legitimate tools, as long as all the “output” of the equations remains real (and we know that’s the case in quantum mechanics), or more precisely, something that I can map back to the physical measurement and its associated calibration.If you don’t trust this line of reasoning, try to remember how complex numbers appear in the equations describing simple RLC oscillators… Do you think they are some sort of “natural” entity? Or are they a tool that makes equations easier to manipulate? And aren’t we ultimately simply facilitating the manipulation of equations that we could solve in completely different ways? What matters ultimately is the relationship I establish between the measurement results. If I want to use complex numbers, transcendental functions, or anything else, that’s really just a choice of tool, not something “natural”.Another example of how complex numbers emerge in some specific situations is given in the subsection “Trajectory measurements”. In that case, looking at the double slit experiment, it should become clear that complex numbers are only a way to sum probabilities that are not independent from one another. It’s the same in the Aharonov-Bohm (it’s also a phase difference, and that’s why it’s useful to use complex numbers), and in the singlet state case (since by construction this is a correlated state). That’s the most frequent reason for complex numbers to appear in QM.4c. I think that you are saying that the notation M|psi> implies linearity in psi. That is definitely not the intent in the paper, and it’s explicitely stated several times, for example in the first sentence of subsection “linearized operator”, which states that $\hat{M}$ is a possibly non-linear operator. Even if I write M(psi), this is misleading, as M(psi) is a functional notation, and the article points out a case where this does not behave like a function (i.e. it’s physically random, and therefore not even injective).4d. I believe that you refer to the paragraph “Normalization of the wave function”. If so, I precisely argues that collapse is a function of actual measurements that must be compatible with how we transmit information (i.e. at sublight speeds).Let me describe things more precisely. Let’s say Alice picks X or Y alternatively, and Bob picks always X (but Alice does not know). What QM predicts is that we will see a 100% correlation each time Alice picks X, and a 0% correlation each time she picks Y. Of course, we will not be able to establish that correlation until we have collected the measurement results of both Alice and Bob, so it will happen at a space-time location where their light cones starting at measurement events intersect. Before that, measurement results will appear random both for Alice and for Bob.The point in my paragraph is that the condition that makes Alice and Bob’s particles entangled is itself something that had to propagate sub-light. For example, the two particles are the result of a single disintegration. So the normalization condition on the wave functions of Alice and Bob particles can only be computed along the light cone starting at that initial correlating event. Therefore, it contains points that are in the past for both Alice and for Bob. And when we finally establish correlation again, this is done by comparing the data that comes back from Alice and Bob, so that correlation can only be established in Bob and Alice’s future. Neither in Bob’s view nor in Alice’s view is it true that the collapse is instantaneous.Yes, it is strange. I’m not postulating that collapse is limited to the speed of light, only that we cannot do any correlation on the results of measurements above speed of light, and that we must therefore normalize within this set of constraints.ConclusionThis was indeed really helpful, and I regret you decided to make this a once-off thing. I regret it all the more that this may suggest that helping a guy like me might be considered dangerous for your career. Isn’t that bad?Anyway, if one day you change your mind, please refer to yourself as the Helpful Anonymous, and I will know who you are 😉 And please rest assured of my gratitude for simply taking the time.

I came across this this wonderful post and truly high-level discussion by accident. (Oh well, maybe not quite.) So often, on-line statements on physical theories, their foundations and interpretations will rapidly degenerate into mindless shouting. So congratulations.While Tegmark is not new to me, this article was.I’ll first have to wade through it AND through your subsequent discussion if I am to make any meaningful comment on this. Which I do hope to be able to do. God and physics (mainly time) permitting. Meanwhile just thanks again for this fine contribution. – Marc