Max Tegmark’s revised Mathematical Universe paper,1 accepted by Foundations of Physics, is out. The “initial conditions” section I’d criticized for being unclear earlier has been ironed out, but I still think the main thesis is full of holes. What follows is a more detailed take-down of the paper than my previous spiel.
To refresh our memories:
- The External Reality Hypothesis (ERH) states that there exists an external reality completely independent of humans.
- The Mathematical Universe Hypothesis (MUH) states that our external physical reality is a mathematical structure.
I can think of at least three senses of “independence” that could apply in this context, and Tegmark hops between all of them in constructing his argument. Reading the paper the first time six months ago, and re-reading it now, I still get the sense that he’s moving the target around.
So let’s lay out the different senses of “independence” that Tegmark uses at various points in the paper. The first is independence of acts of observation. That is, the world exists, and exists in such-and-such a state, independent of whether an act of observation occurs. That Tegmark is thinking of at least this sense of “independence” is shown by his remark that “adherents of the Copenhagen interpretation of quantum mechanics may reject the ERH on the grounds that there is no reality without observation” (section IIA).
The second sense of “independence” is independence of the kind of being that is observing and interpreting the reality. This is what Tegmark means when he says:
The ERH implies that for a description to be complete, it must be well-defined also according to non-human sentient entities (say aliens or future supercomputers) that lack the common understanding of concepts that we humans have evolved, e.g., “particle”, “observation” or indeed any other English words. (Section IIA)
And, if we infer from his usage of the term “baggage” that “baggage” is used to refer to anything that is not “independent of humans”, there is an implied third notion of “independence”:
As an extreme example of a “theory”, the description
of external reality found in Norse mythology involves a gigantic tree named Yggdrasil, whose trunk supports Earth. This description all on its own is 100% baggage, since it lacks definitions of “tree”, “Earth”, etc. Today, the baggage fraction of this theory could be reduced by describing a tree as a particular arrangement of atoms, and describing this in turn as a particular quantum field theory state. (Section IIB)
So we can tentatively infer that one aspect of Tegmark’s idea of “independence” is having a clear definition.2
To summarise, here are the three notions of what “independent of us humans” involves:
- Independent of acts of observation
- Independent of the kind of life-form or intelligence interpreting or observing the world
- Clearly definable (damn, don’t we wish we know how to clearly define ‘clearly define’!)
My intuition is that the sense of “independence” meant in the statement of the ERH is either of or both of the the first and second senses. The second sense is almost certainly included, since it is rather normal for scientists to think of themselves as examining an objective reality that is the way it is whether or not it happens to be aliens or humans examining it. I doubt the third sense is implied by the phrase “independent of us humans”, since something can be clearly defined by us without being independent of us, and unlike the first two senses, it is not a common interpretation of the phrase “independent of humans”. It is only the third sense, however, that will lead us to the MUH, and even then not without its own problems.
My previous post argued that the second sense of independence could obtain even if the world is not a pure mathematical structures. Now I will make the converse argument, that a pure mathematical structure need not satisfy the second sense of independence. Yes, “it is we humans who introduce these concepts and words” for entities like trees, cups, and so on. But it is also “we humans” who introduce concepts and words for entities like variables and functions. So what makes a mathematical variable more independent of human quirks than a tree? How do you teach someone what a variable is without recourse to metaphors in much-maligned “plain English”? (a point Karl Schroeder, commenting on my previous post made, as did Wittgenstein). And how would we know that if we can define something in terms of a mathematical structure, that aliens can do so as well? Have we ever tried to teach an alien the meaning of “variable” and “function”? The fact is, all we know is that it is quite possible to teach every human some form of mathematical reasoning, and we teach them this using ordinary language. You can’t begin to teach someone how to manipulate mathematical symbols without relating the symbols to more “concrete” concepts.
This can be demonstrated using one of Tegmark’s toy mathematical structures. Consider the group known as “addition modulo two”. It contains two elements, “0″ and “1″, and they satisfy the relations:
Now, can we “define” this structure without employing any of the “baggage” of ordinary English? Try to define what “element” is without ordinary language! Try to do it by just pointing, or some equivalent non-linguistic gesture (but gestures are human too, no?). How would you explain to an alien the different roles of the elements 0 and 1 and the operations + and =? (Clearly you have to distinguish between those two roles in order to get anything interesting out of the structure — you wouldn’t say that anyone understood what “addition modulo two” was if they did not understand how the operations and the elements differ. For one, if they interpreted the written relations as expressing a kind of geometrical figure indicating the relative positions of objects, you’d think they’d deeply misunderstood what “addition modulo two” is.)
- We can’t appeal to the “human” origins of ordinary language to claim that mathematics is the only language independent of humans, because mathematics has human origins too.
- Mathematical concepts seem “well-defined” to us because we can seemingly attain broad agreement, broader than in any other discipline, about the meanings of mathematical entities. But, when it comes down to it, we still have to “define” very basic concepts like variables and mappings in terms of ordinary language. (Someone like Wittgenstein would say that we never define them in the usual sense of “define”.)
- That mathematics provides, to us, the most “well-defined” descriptions available, doesn’t imply that mathematics will appear “well-defined” to other forms of intelligence. After all, the only forms of intelligence that mathematics has been ‘tested’ on are ourselves and artificial intelligences constructed by us for the very purpose of doing mathematical computations.
The inescapable link between mathematics and ordinary language suggests a more profound reason for why the MUH cannot be true, or at least for why it is impossible for a scientific theory to describe the world as a mathematical structure devoid of “baggage”. According to any reasonable formulation of what a scientific theory is, a scientific theory must provide the possibility of verifying statements made by the theory — of checking if the predictions made by the theory are correct. There is no way we can do that without relating the symbols in a mathematical description to actual observations of reality. Suppose we have this complete, accurate Theory of Everything, and it’s just a mathematical structure. How will we know, then, what regularities we find actually confirm that structure? We have to relate the elements and operations in the structure (like +,-,0,1) to observable entities, and the TOE cannot tell us how to do so. Which part of the mathematical TOE tells us this is what you should observe, if by definition the TOE cannot mention anything that actuallyrefers to something non-mathematical; if the TOE consists merely of “abstract entities with relations between them”? How will we ever know we’ve found the right thing if the theory is unable to have implications of the type “if you do X and Y, you will see pattern Z on your oscilloscope?” Even if it is true that a purely mathematical TOE describes reality, we have no idea of knowing that that is the case, because such a description would have no empirical consequences. Although you might want to call it a Theory of Everything, it wouldn’t be a scientific theory. Tegmark writes in Section VIIIB that under the MUH, the correspondence rules linking the mathematical domain to the empirical domain can in principle be derived from the mathematical structure itself. But if the mathematical structure is to contain “no baggage”, it is hard to see how one can derive correspondence rules from it. A self-contained mathematical structure is logically distinct from any empirical assertions — how can a purely abstract structure that says nothing about what its elements and relations represent logically imply anything about the things we actually observe? The most you can deduce from such a structure is a further network of mathematical theorems and relations, but these also do not say anything about what actually obtains.
How does all this impact on Tegmark’s argument? He summarizes his argument his way:
- The ERH implies that a “theory of everything” has no baggage.
- Something that has a baggage-free description is precisely a mathematical structure.
Taken together, this implies the Mathematical Universe Hypothesis formulated on the first page of this article, i.e. that the external physical reality described by the TOE is a mathematical structure.
I have already laid out my case for why the ERH means “baggage” only in the sense of “dependent on acts of observation” and “dependent on the kind of intelligence observing and interpreting it”. So step 1 in the argument is referring to that kind of baggage. I have argued in this post that step 2 is wrong because we have no reason to think that mathematical concepts are human-independent, since so far they’ve been couched only in human languages. My previous post on this subject argued for the falsity of step 2 from another perspective: the possibility that there exist human-independent entities and concepts that are not mathematical.
On a final note, Tegmark admits in Section IID that his argument relies on structural realism, and in particular on ontic structural realism as first laid out by James Ladyman.3 It’s only because he assumes ontic structural realism that he is entitled to refer to scientific theories as descriptions of reality. So really it’s more like ERH + ontic structural realism = MUH, which is a lot less pithy. And I’m not sure how many scientists would subscribe to ontic structural realism anyway.
 Tegmark, M. The Mathematical Universe. arXiv:0704.0646v2 [gr-qc]
 It might be obvious to some readers that there are significant problems with assuming that the only clear definitions are reductionistic, physicalist definitions, but this blog post looks like it’s getting messy enough without bringing those issues in, so we’ll let that pass.
 Ladyman, J. What is Structural Realism? Studies In History and Philosophy of Science Part A, Vol. 29, No. 3. (September 1998), pp. 409-424.