There is a challenge proponents of the mathematical universe hypothesis (MUH) must overcome. It is not clear how sensations are mathematical. Otherwise called "qualia" or "phenomenal experiences," sensations seem to be distinct from a mathematical object, but I do not know in what sense they are distinct. Let me try to explore this issue.
Perhaps we wish to deny sensations exist, but this is an initially very implausible thesis (though I'm sympathetic with it), and so the initial plausibility of MUH takes a hit. If instead we are realists about sensations, then I have at least eight specific puzzles/objections to offer. They will vary in strength as we go along. The first three come from a positive answer to the question: Are all mathematical structures sensations? These are not obvious, so bear with me.
(1) «Newman's problem.» In 1928, a man named Newman argued in response to an article by Russell that, for any set with n members, the set can be given any mathematically possible relational structure by specifying some criteria for the relation between the set's members. The idea is that any n-member set instantiates all n-member relational structures. (Short proof: if two members of this set are not related in some structure x, we can create another structure in which they are, by way of the property of "not being related in structure x.") I am not sure whether n needs to be natural, rational, real, or otherwise. At any rate, a few questions surface: Are all these mathematical structures being simultaneously instantiated by any collection of objects in a mathematical universe? Are all of the corresponding sensations ontologically juxtaposed, as it were? Furthermore, does this not lead to complete panpsychism, in which rocks and socks alongside jocks are conscious?
(2) «Higher-order infinities.» If we accept all mathematical structures are sensations, then we must accept that in the infinitely long series of infinities (which is so big it has no specific cardinality — the number of infinities is bigger than any kind of infinity) each ordinal corresponds to a sensation (perhaps a different one each time). This is not impossible and I cannot make anything out of it, but it gives food for thought.
(3) «Concreteness.» This is an unarticulated objection. Sensations seem in some sense concrete (as do physical objects in general), whereas mathematical objects seem in some sense abstract. This inarticulated seeming was expressed by Stephen Hawking when he wondered "what is it that breathes fire into our equations." That is, what makes them concrete? What implements their abstract structure?
Given these three puzzles, we may wish to reject that all mathematical structures are sensations, then. Now only a subset of all mathematical structures are sensations, those which hopefully satisfy some rule-based constraints. (That is, they are not randomly selected from the set of all mathematical structures.)
Take Giulio Tononi's thesis about sensations (qualia, consciousness, phenomenal experience) surfacing whenever a system acquires a certain level of "informational integration" (which he defined precisely, but which I do not understand well). This is merely an example, but I will use it throughout for ease of exposition. Accepting or rejecting Tononi's constraints (on which mathematical structures are sensations) will still leave us with the following five additional difficulties.
(4) «Brute facts.» What determines which mathematical structures are sensations, and which are not? Suppose Tononi's formulae are true. Is there any mathematical explanation to why these, and not others, are the bridging laws between «non-sensational mathematical formulae» and «sensational mathematical formulae»? Well, can there be a mathematical explanation for a metaphysical fact like this one? It does not readily seem so. But then the explanation would have to be non-mathematical. Yet, is the mathematical universe hypothesis (MUH) consistent with the existence of non-mathematical explanations? Or should we accept that there are crucial features of reality, like this one, which admit of no explanation — which are basic, in some sense? This does not get the MUH theorist off the hook. First, because the one of the appeals of the MUH was to eliminate "unexplained" features of reality: the intrinsic nature of objects, which weren't accounted for in our best physical theories. Second, because even if there were brute facts, it seem this would be a non-mathematical brute fact. But is a world in which there are non-mathematical brute facts a mathematical world?
(5) «Universe explosion.» Do only some mathematical structures 'exist'? What is the criterion for mathematical structures existing, and what explains the validity of this (apparently non-mathematical) criterion? Or is the MUH theorist committed to the thesis that all possible mathematical structures (and thus universes) exist? (Even those inconsistent with current quantum and string theories.)
(6) «Implicit dualism.» Are sensational mathematical structures different KINDS of entities from non-sensational ones? I am not keen on the literature on supervenience, property dualism, and substance dualism, but this may be a form of dualism about the ontology of mathematical entities. MUH theorists may not have realized this, if this is the case. (If it's not dualism, then we still have to suppose the existence of non-mathematical metaphysical facts such as "supervenience relations" and "bridging laws" between the two kinds of structure.)
(7) «Vagueness.» Things in mathematics may happen gradually. Take Tononi's equation, which as I have said I scantly understand. I suppose systems can increase their degree of "information integration" very gradually. When does sensation begin? This is a sorites paradox situation. Is there a threshold of integration beyond which things are more and more conscious, but before which things are perfectly unconscious? Since we are not nominalists about sensation, we cannot give the standard reply to the sorites paradox: that the answer is merely conventional. When does a bunch of salt crystals become a heap of salt? There's no answer, it's conventional. (Some argue.) This is nominalism about heaps of salt. Yet, we have denied nominalism about sensations and accepted their objective reality. Perhaps, then, we must accept there being a threshold, i.e. a very minute change in information integration that changes a system from non-sensational to the minimum possible level of sensationality. Does this sound acceptable?
(8) «The Levinas explanatory gap.» What explains the fact that this rather than that mathematical structure is this rather than that sensation? To give an example of the problem, what makes it so a certain mathematical structure feels like a sound, instead of feeling like a taste? What makes, let's say, a 40Hz vibration feel like the way we hear a low, deep bass voice, whereas a 20.000Hz vibration feel like the way we hear a sharp, piercing soprano voice? (Can there be any mathematical explanation for this? Is it a brute fact?)
These are the eight objections I have managed to think up. There must be some interesting literature on MUH, and if anyone would recommend to me a few articles on this I would be happy to read on as time allows. What do you folks think?