# Is the universe just a mathematical construct?

Consider some continuous mathematical function. You can define it as a set of initial conditions and a set of rules to extend it to the rest of its domain. For instance, for a linear function, y = mx + b, "b" would be the initial condition and "mx" would be the "rule" that extends it to the rest of the domain. It doesn't make sense to ask "where" the function came from; it exists solely as a mathematical construct.

A hypothetical universe can be defined similarly - set some initial conditions, and a set of laws ("the laws of physics" for the universe) and the rest of the universe follows as the laws acting on the initial conditions. Again, most people wouldn't ask about its "origin"; it exists solely as a mathematical construct.

Most people tend to ask about the origin of the "real universe" though, treating it as more than just a mathematical construct. This seems as arbitrary as asking about the origin of a random function. The "real universe" seems no different from a hypothetical one, apart from the fact that we exist as part of the consequences of the laws acting on the initial conditions.

An equivalent restatement of this question would be "What evidence, if any, suggests the existence of this universe is different than mathematical existence?"

• I'd like to clarify the distinction between a mathematical universe, and a computable one. You gave your example of a continuous function "defined by a rule," which is a fairly decent nontechnical characterization of a computable function. Something that represents an algorithm. There are countably many such functions if your strings are finite and your alphabet is at most countable. But what if the universe is described by a mathematical function that doesn't happen to be computable? An algorithmically random function, a string of 0's and 1's that can not be compressed. Two different cases. – user4894 Jul 15 '14 at 2:21
• I suggest looking at Exotic R4 and noncommutative geometry, for help on candidates for "mathematical construct". In general, I think you need to struggle with the question of whether the mathematical construct is finitely definable; for this, see recursively enumerable set. RE shows up in Gödel's incompleteness theorems, via "effectively generated". – labreuer Jul 15 '14 at 4:09
• I think it makes sense to ask where the function came from since we both just implicitly asked it. As far as evidence goes: (1) The universe appears to have a beginning, but a mathematical formula could move time indefinitely into the past. (2) Quantum indeterminism would require functions giving probabilities. (3) A deterministic universe need not be mathematical; a mathematical model is susceptible to breaking with new data that does not fit in the model exactly. (4) The more data there is in the universe that needs explaining the more difficult it will be to find a math model. – Frank Hubeny Jan 24 '18 at 0:39

Is the universe just a mathematical construct?

Its probably worth beginning by saying that the question as constituted is one that can really only be asked in the tradition of logical positivism which takes ontology as a matter of convention; whilst this may be a useful manouever, or have useful ramifications I don't think it can in fact be true.

You can set up such a simulation of a universe on a computer and watch it evolve, and even perhaps evolve life. However, this is a simulation, we have no procedure to actually change the laws of physics (even if in the recently fashionable multiverse view, at least in Smolins take on it, there could be different universes with different laws).

The point is that the world is phenomenological; that is it acts on us; it is not simply structure; though that is a neccessary part of its constitution.

The "real universe" seems no different from a hypothetical one, apart from the fact that we exist as part of the consequences of the laws acting on the initial conditions.

Existence is a primary notion; however in medieval philosophy (Averroism) it was asked whether essence preceded existence (this then was religously framed, from the rationalist falsafa tradition in Islam); this in a sense is your question. Is the essence of the universe mathematical, and does essence precede existence?

What evidence, if any, suggests the existence of this universe is different than mathematical existence?

As the reformulation of the question shows essence is different from existence; for one to actually have a universe, one cannot merely posit a possible one (though people have). Sartre later turned this around, saying that essence suceeded existence; this is the attitude of contemporary physics: we live in a universe, let us see if we can extract its (mathematical) essence.

Following Sartre, and in another direction, this question is simply asking can there be other universes? In contemporary physics this option is fore-closed by Occams Razor - even though it is a respectable one - and has been source of highly speculative (physically based) cosmologies - the multiverse again.

• Are you assuming that the universe's essence must be discovered before it (the essence) can exist? Heidegger see its possibility as its essence: "Surely in order for something to be 'actual' and to be able to be 'actual,' it must first be possible." - Does existence precede quiddity? – Chris Degnen Jul 15 '14 at 8:31
• @Degnan: I'm undecided; Heideggers view is the traditional Aristotelian/Averrosian one as far as I can discern; but on the whole, I'd probably go along in that direction. – Mozibur Ullah Jul 15 '14 at 8:47
• @ Mozibur - Yes, it make a lot of difference if quiddity is taken as possibility rather than an 'understanding' (of whatness) that may be drawn by analysis, after the fact of encountering something's existence. The latter seems to be Rev. Joseph Owens position, noted here. – Chris Degnen Jul 15 '14 at 9:40
• @degnan: My understanding of Quiddity wouldn't be possibility but Whatness; but existence then seems to then be inherent in quiddity; if existence precedes quiddity it seems then to theorising about bare existence and how does this differ from possibility or essentia? – Mozibur Ullah Jul 15 '14 at 9:55
• @ Mozibur - for a thing to be possible its essence must therefore be set (even in the absence of an observer), and then it may or may not become actual. An observer might encounter the existing thing and subsequently deduce its essence, but they could then say the essence must have been prior to its existence in order to give it possibility. (Essence as comprising real predicates, and possibility as possible, specific, real/non-contradictory arrangement being pretty much the same thing.) – Chris Degnen Jul 16 '14 at 7:51

I cannot imagine what the universe could possibly be other than a mathematical construct, nor do I see any problem in assuming that it is.

We consider things real, rather than hypothetical, when they are part of the physical universe. But how can we make this distinction for the universe itself? If the universe is everything, then it can't be part of anything, so what makes it real or not? Any mechanism that could actualize the universe can instead be considered part of the universe, leading back to the original conclusion.

Even if you can imagine such an actualizing mechanism, why believe in it? What does it explain? How is a real universe different from a hypothetical one, to the observers inside it? In the complete spacetime description of a hypothetical alternate universe, you can find beings talking about the apparent reality of their universe, in all the same philosophical detail as us. Any argument we make for the reality of our universe, they make as well. And we don't have to simulate their universe or anything to make that true, it's just a mathematical fact about their universe.

This is a beautifully simple and self-evident idea, once you wrap your head around it. It provides a pleasing answer to the question "why does the universe exist?", which becomes no more mysterious than "why do the integers exist?" The universe doesn't need to exist in any physical sense, it just needs to work mathematically, and include you. That's enough to explain any observation you can ever make about it.

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?

• +1 One way sensations are distinct from a mathematical object is that they are subjective while the mathematical object is objective. By turning the subjective sensations into a formula one has lost the subjective experience. – Frank Hubeny Jan 24 '18 at 0:34
• Newman's problem is an excellent argument in support of the MUH. There is no way to objectively define when a mathematical structure has been "instantiated", since any arrangement of matter can be interpreted to represent any mathematical structure. It follows that our experience of the universe cannot depend on any such instantiation. – jedediah Jan 24 '18 at 8:35
• To your overall point, sensations in the MUH could be described roughly as "symmetries between parts of the universe, due to physical laws and the structure of brains". If you find smoke near people in the universe, you will find the smell of smoke in the brains of those people. It's fundamentally no different than any other mathematical description of a physical system. – jedediah Jan 24 '18 at 8:58

You all forget that we only see what the mind perceives and logically interprets. For us all, it is the iterations of neurons in the gray matter, nothing more, interpreted by them and translated into some purported meaning. Is it an illusion or reality? We can't know because we are bound to our own individual perceptions. We are slaves of survival. More than likely reality is something altogether different. For us humans, it is a construct of our minds evolved for survival and that is unreliable except as it relates to survival. We are primitive creatures, after all. Pray math will save us from our delusions, but as Godel pointed out, mathematical proof may also be an illusion. Yes, that it is a possible construct is unquestionable, but it suggests a higher being or higher being reasoning? But, even ancient alien civilizations may not know the answer, although they may pretend knowledge of the ultimate reality as we do. At least, from experience, I can say they have mastery of non-linear time and he (or she or it) showed me the evidence.

• If you have references to those taking a similar view this would be your opportunity to share them with the reader and guide the reader to getting more information. In the process the references would support your answer. – Frank Hubeny Feb 27 at 11:15
• Nowhere said Gödel such a thing. This is a clear abuse of Gödel's theorem. – Jishin Noben Feb 27 at 22:34