Friday, 13 December 2013

The Universe is Made of Mathematics

Max Tegmark
A couple of years ago I was reading Anathem by Neal Stephenson and a number of ideas started to click into place for me. Without going into the novel too much (spoilers), it prompted me to think about the nature of reality and why it might be that the universe exists.

Over the course of a few sleepless nights, it all came together and it seemed too make so much sense that I could entertain no doubt: the universe was what I thought of as a "Platonic algorithm".

To my knowledge at the time, this idea had never before been proposed, and a bit of half-hearted searching at the time turned nothing up. I became more convinced of my idea over time and started developing an interest in philosophy so as to learn how to communicate and argue for the idea.

As my research ramped up, I inevitably came across Max Tegmark's Mathematical Universe Hypothesis (MUH), and with a curious mixture of disappointment and vindication realised that it was essentially the same idea.

Nevertheless, Tegmark's idea was still relatively little known, so I decided I would devote some time to building up a rigorous argument with a view to perhaps writing a book on the topic some day. That is actually the single most important reason I started this blog, as many of the arguments I have outlined in my posts will serve to support the MUH.

I have now learned (courtesy of Massimo Pigliucci's blog Rationally Speaking) that Tegmark is in fact about to publish the popular science book that is so sorely needed. I guess I can put aside that ambition for now.

But it's also time for me to put this blog to the use it was originally intended for - enough beating around the bush!

Perhaps my hesitancy to address the subject arises out of the fact that it at first seems completely mad: I am utterly convinced that the universe is made of mathematics and that the concept of physical reality is incoherent.

Now let me try to explain why.

Why the Abstract Feels Physical

I will freely admit that this at first seems like a meaningless platitude to say "the universe is mathematics", akin to statements such as "God is Love". Massimo Pigliucci quite rightly stresses on his recent blog post that we are not claiming that reality is best described by mathematics -- the claim is literally that the fundamental substance of reality is mathematical structure.

As he pointed out, it's hard to see how this can be. How can mathematics possibly explain substance? Abstract concepts such as mathematics are surely entirely different from physical stuff, and to claim the two are the same seems to be perverse.

From Pigliucci's description, Tegmark's answer to this criticism seems rather weak. He insisted that electrons are mathematical objects having mathematical properties, but apparently failed to provide a convincing reason not to regard them as physical objects having physical properties which can be described mathematically. I can see why Pigliucci feared he may have been making a category mistake.

I would take another tack. I would argue that the intuition that mathematics cannot be the stuff of, well, stuff, arises from the false belief that this is a physical universe (and indeed that the physical universe is a concept that even makes sense).

But before I elaborate on this, let me state plainly that I believe the argument in favour of the mathematical universe rests on three crucial premises.

1) All mathematical objects exist abstractly and independently of minds (mathematical Platonism)
2) The mind is a computational process (The Computational Theory of Mind or CTM)
3) The universe behaves according to laws of physics which are expressible mathematically (metaphysical naturalism)

I have made arguments in favour of all of these premises previously on the blog (Platonism, CTM, Naturalism), so for now I am going to assume them to be true.

How can I seriously doubt that the universe is physical? Two powerful analogies help to explain this.

One hypothesis gaining currency in recent years is the idea that the universe is not physical but a computer simulation, as argued rather interestingly by philosopher Nick Bostrom. If Bostrom is right, the physical universe may not exist and the fundamental stuff of the universe could be information, i.e. the bits flowing through a computer program.

Another idea, explored in the book Sophie's World and elsewhere, is the idea that our world may not physically exist because we may be fictional characters living in a fictional world being described by some author. From the point of view of a fictional character, the world seems real, so none of us can really know for sure that we are not such a character.

In both of these ideas, the universe is not real (not physically at any rate), but this fact is forever hidden from us.

The MUH is quite similar in many ways. The crucial difference is that it removes the dependence on a greater reality. Unlike the computer simulation idea, we need no external hardware to support us. Unlike the characters in a novel, we are genuinely conscious. Unlike both, there is no creator, no programmer or author.

We have no need of a programmer or author, no cosmic computer or reader. The bedrock of our existence is mathematical Platonism, and unlike all other explanations this is sufficient as an ultimate cause. Mathematical objects are not created and need nothing to sustain them. They exist necessarily of their own nature.

Let's first consider the idea that the universe is a simulation. If the physics of the universe is computable (as it seems to be), then it could certainly be simulated by a computer of sufficient power, even if such a computer would be unfeasible to construct in this universe. It therefore follows that in principle it is entirely possible that it and we ourselves are simulated, virtual entities (although if you doubt the Computational Theory of Mind then this does not follow as we are evidently conscious).

However all computer programs are mathematical structures, and as such, Platonism holds that all computer programs exist in the abstract. It follows that even if no computer could ever be built to run the simulation of the universe, this program exists as a mathematical structure, and within this structure can be found perfect descriptions of all the objects in our universe, including our minds, our thoughts, etc.

But now, if we start thinking about an abstract mathematical object containing all of our thoughts and inner experiences and determining our life stories, it seems to me that we are getting close to the idea that we may all be characters in a work of fiction.

Unlike fiction, what happens in our universe is determined by mathematical laws and these have the regularities necessary to support life, brains and real intelligence. I don't believe characters in fiction are actually conscious because their actions are not determined by internal genuine thought processes. There is consciousness there, but it is not in the minds of the characters but in that of the writer. In contrast, our consciousness is our own and is the product of our own brains.

But like the lives of characters in a novel, our lives and our thoughts are all mapped out and there for the reading. We are implicit in the structure of this mathematical object, and in principle our stories could be discovered by simulating it on a powerful computer. This is just like the way we can explore mathematical objects such as the Mandelbrot set with computers.

We are the conscious characters in a cosmic narrative determined by no author but the laws of physics. Our lives exist even if no simulation is run, just as the stories of fictional characters continue to exist even while nobody is reading the novel and just as the Mandelbrot set has always existed, even before it was discovered.

So think of our universe and our life stories as being something like an enormous fractal structure arising from some simple mathematical rules. It's a beautiful, amazing, complex, surprising thing, but it needs no creator or sustainer.

The Incoherence of the Physical Universe Hypothesis

Far more than being merely a plausible account of reality, I view the Mathematical Universe Hypothesis as being necessarily true, because it reveals the incoherence of the concept of a physical universe.

Given that the universe obeys mathematical physical laws (naturalism), there must be a mathematical object (given Platonism) which perfectly describes the universe and which contains within it structures analogous to all objects within the universe, including ourselves.

What's more, given the computational theory of mind, those structures corresponding to our minds are necessarily conscious. As such, even if there is such thing as a real, physical universe, there must also be an isomorphic (having precisely the same form or structure) abstract non-physical universe. There is a physical you and an abstract you, and both have exactly the same experiences and neither has any way of knowing which universe they find themselves in.

In fact, there is no observer anywhere who can distinguish between the physical and the abstract universes. There is nothing we can say about the physical universe that is not also true about the abstract universe, except for the fact that it is physical.

So let's unpack that. What does physical mean? In everyday speech, it is used to distinguish between objects we can interact with directly (such as rocks) and objects we can only think about and discuss (such as numbers). For something to be physical it must be present at some time and place within the universe, and for something to be abstract it must exist outside of space and time.

So what do we mean when we claim that a universe is physical? It exists in time and space? But it doesn't -- it contains time and space within it.

Ok, so let's say that the universe is a special case, that it is physical not because it is inside spacetime but because it contains spacetime.

So now let's consider a hypothetical abstract universe other than our own and ask ourselves whether it is physical. (For illustrative purposes let's say that universe is Star Wars but let's assume that the universe is not a work of fiction but a mathematical object in much the same way that I'm claiming that our universe is a mathematical object).

This universe is not present within the spacetime of our universe, so from that point of view it is not physical. It does contain its own spacetime, so by our broader definition that would make it physical.

But it's just a made-up universe! Our instinct is that it ought not to be considered physical. So let's say that only universes which contain the spacetime of our own universe are physical.

So now, only our own universe can be physical, by definition. This seems to be a rather unsatisfactory result, because there are very good reasons (such as the anthropic principle) to believe there may be other universes. We want that option to be open to us, if only so we can discuss the possibility.

It's also a patently subjective definition of physical. From the point of view of an observer in this universe, the Star Wars universe is not physical. But from the point of view of Han Solo, our universe is not real.

It seems to me that the only way out of this mess is to realise that the application of the concept of physicality to a universe is a category mistake. Physicality as a concept only makes sense within the context of a given universe.

For example, you would no doubt regard yourself as physical but Luke Skywalker as abstract. However, Luke Skywalker is physical to Han Solo, while Han would consider you to be non-existent. There is no objective, universe-agnostic way to say that you are really physical but Luke Skywalker is not.

So given that physicality as applied to universes seems to be incoherent, and given that physicality is the only (completely undetectable) property that distinguishes the mathematical universe from the physical universe, it seems to me that the only sensible conclusion is that only the mathematical version of our universe exists. This accounts for the existence of the universe, fine-tuning and everything we observe.

The Physical Universe Hypothesis is therefore unnecessary, redundant and incoherent.

Reasons to Believe

I have explained why I think the Mathematical Universe Hypothesis follows necessarily from naturalism, Platonism and the computational theory of mind, however there are plenty of people who are skeptical of some or all of these propositions. Nevertheless, I think there are independent reasons to find the MUH plausible.

Firstly, and perhaps most importantly, it explains why the universe exists. It does not tell us what caused the Big Bang, or if the Big Bang had a cause at all, but it does explain why there is something rather than nothing, without appeal to a creator or any other unsatisfying ultimate cause. This echoes my rebuttal to the Kalam Cosmological argument.

For a creator God, we are left to ask who created the creator - but if the universe is a mathematical object, it needs no creator (on Platonism at least), so this is a very satisfying answer to that eternal question. It has always existed and will always exist outside of space-time as a mathematical construct.

It also provides a powerful explanation for fine-tuning. As discussed previously, this universe seems to be perfectly calibrated to support life. Much attention in discussions of fine tuning is focused on the physical constants such as the charge of the electron or the speed of light, etc, but very rarely is it asked why are the equations and the laws of physics themselves of the form they are. Why could the universe not be completely different, not just having different constants but having completely unrecognisable physical laws?

If all possible universes exist, then we have our answer. Our universe is fine-tuned because it is one which has the ability to support conscious thought selected from an infinite multitude of mathematical structures, most of which are lifeless.

Better than other multiverse hypotheses, where it is proposed that there might be a great number of universes, the MUH posits that all possible universes exist, not merely a great many. This is actually simpler because, as Tegmark says (and as explained previously on this blog), there are no free parameters. We have no reason to wonder why universe X exists but universe Y does not. If all universes exist, nothing is arbitrary. We have an ultimate explanation of everything.

There's something attractive to me about the idea that all universes exist. After all, what's to stop any given possible universe from existing? It is not subject to the laws of other universes. No law of this universe can prevent another entirely causally disconnected universe from existing. Even if there is a multiverse with its own meta-laws (e.g. the String Theory multiverse), what's to stop another multiverse with different meta-laws from existing? Even without asserting that the universe is made of mathematics, it seems to me to be perfectly sensible to propose that all possible universes exist. Why not?

Finally, the idea that the universe is literally a mathematical object explains what physicist Eugene Wigner called "the unreasonable effectiveness of mathematics" in modelling and predicting the natural world, often leading physicists to empirical discoveries they could not otherwise have made. If the universe really is a mathematical object, it is hardly surprising that mathematics should be effective in describing it.

There are objections to the Mathematical Universe Hypothesis of course, and I intend to return to them in time.


  1. I like the argument against the universe being physical - It somewhat reminds me of the arguments against P-zombies, except the thing being argued against is physical rather than immaterial.

    I think physicality actually does have a lot of the same problems as dualism, especially in terms of where it all comes from and how abstract mathematics is able to affect it. Are there "grooves" somehow that matter follows? Grooves in what?
    Where do the elementary physical constituents of the universe come from? They can't really be made of anything themselves, or else they wouldn't be fundamental. What's directing them to take the form that they do, or keeping them from diverging into another, arbitrary form? It seems to me they'd almost need to be something abstract themselves, to avoid an infinite regress.

  2. Another great entry, Disagreeable. Thoughtful and informative as all your other posts are.

  3. This is excellently clear. I'm curious about your statement "Physicality as a concept only makes sense within the context of a given universe." What's your view, given all this, on Lewis's distinction between 'actual' vs. 'possible' worlds vis à vis 'physical' vs. 'abstract' universes in this sense? Are they equivalent, or do you have a distinction in mind?

    1. Hi Ryder,

      I'm not an expert on Modal Realism but it seems to be very close to the MUH, differing perhaps only in not really having a rigorous account of what is possible, not recognising that worlds are not only akin to mathematical objects but actually mathematical objects, and not really having an account of why the possible worlds exist.

      As such, I think Modal Realism is essentially correct, but it's not the whole story, and doesn't really work on its own.

      I think the parallel you draw between actual/physical and possible/abstract is very astute, but it depends on what we mean by physical.

      Until we adopt a standard definition of what "physical" means in the context of the MUH, it remains a somewhat ambiguous (and potentially incoherent) concept, whereas "actual" is perfectly well defined in Modal Realism.

      Lewis seems to very specifically interpret "actual" as meaning our universe, whereas I'm not so sure what "physical" means.

      The physical universe could be interpreted to mean the actual universe in Lewis's sense, but on the other hand it also seems reasonable to me to describe as physical all universes which have conscious observers who perceive them to be physical, while other universes are only abstract.

      But "physical" can also be used to refer to the hypothesis that the universe is not abstract, that the MUH is false.

      So, to me, it's a problematic term, as is the distinction between physical and abstract (on the MUH they are the same thing from an objective point of view). I think "actual/possible" is more useful going forward once the MUH has been explained and adopted.

    2. I agree, Aang ;) The problem with defining the 'physical' is that it could just as easily be interpreted as matter in contrast to energy, which given their equivalence is arbitrary; if we include all forms of energy and matter, then we've nothing to define what 'physical' is other than 'that which follows the set of possible rules for physics in our universe'.

      And as you pointed out, the vagueness of the former meaning grants a bit of liberty for the definition, and it seems appropriate to just make the definition of physicality "that which follows the set of possible rules" -- but then we're off to the races as far as the possible/actual distinction is concerned.

      Still, this leaves us with something of an uncomfortable claim, that what counts as physical is arbitrary. Applying a maxim that physicality requires observers, I think will work well, but then it seems to me that we've locked ourselves into the weird argument that physicality is instantiated by observation, and fundamentally so. Weird as in that claiming "observation only affects the possible universe it's part of" begins to look suspect, without some inter-possible-world relation holding between different observations (which sounds suspiciously like a very strong metaphysical claim). Fundamentally weird in that a 'dead' possible universe with viable physical laws identical to our own but carrying no possible observers are ... well, logically impossible, despite the weaker physical viability holding.

      For instance, if I imagine a different big bang, in some other universe, of sufficiently greater force that the initially generated constituents of matter move far apart enough from one another, quickly enough, that they never interact, then no observation, in the simplest sense, can possibly happen for that situation. No possible observation means no logically possible "physicality" by the prior definition. So a dead universe becomes logically impossible, despite being mathematically constrainable. Problem?

      At least, I'd presume that situation is tractable in math, but I'm not a physicist. If you know any that could answer that question I'd be delighted.

    3. "Fundamentally weird in that a 'dead' possible universe with viable physical laws identical to our own but carrying no possible observers are ... well, logically impossible, despite the weaker physical viability holding."

      I think you misunderstand me a little. I'm not suggesting that there is anything ontologically different about dead universes or that the act of observation changes the observed (Heisenberg notwithstanding). I'm suggesting that physicality is in the eye of the beholder, so without a beholder it's a concept that just doesn't apply.

      It's like asking whether the dead universe is beautiful if nobody ever sees it, or whether a tree that falls in the forest with nobody to hear makes a sound.

      By the way, I'm Aang as kind of a joke. My avatar is the Avatar. I like metahumour. :)

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  5. Hi Disagreeable,
    After our discussion at Rationally Speaking, I decided to check out your blog. Fascinating and thought provoking writing. I find the MUH interesting.

    I've written myself that we are all information, patterns of patterns of whatever ultimate reality is, whether that ultimate reality be elementary particles, strings, branes, field excitations, or mathematical objects. I'm open that it could be structure all the way down.

    The idea that all universes are real is interesting. The Star Wars universe exists, if nothing else as patterns in our thoughts, but since we're patterns ourselves, what right do we have to be uppity about it? Except perhaps that our patterns have been around longer and their patterns wouldn't exist without ours.

    Here's a question though, is the universe mathematical, or is mathematics simply patterns we've discovered about the universe?

    1. Hi Patterns,

      "I've written myself that we are all information, patterns of patterns of whatever ultimate reality is,"

      Precisely my view, which is why I liked your handle :)

      "The Star Wars universe exists, if nothing else as patterns in our thoughts"

      Yes, it exists.

      But if it's only as patterns in our thoughts, then there is no universe where Han Solo and Luke Skywalker are actually conscious and perceive their world to be real. That's one potential reason to be uptight about it.

      I suspect there is no universe that's really just like Star Wars (with conscious creatures) because I doubt that that universe really has an internal logic that makes sense. For one thing, I find it doubtful that creatures that are so much like ourselves could exist in a universe with such radically different physics as to allow FTL travel, lightsabers, force powers etc.

      Extending this a bit, I'm skeptical of the physical reality of any work of fiction, even those set in our own world, because I am not sure how accurately a human author can emulate a world so as to correctly conceive of a series of events that could actually happen in a universe based on our laws of physics. For example, I worry that there exists no brain state that would produce quite *this* response to *that* stimulus.

      Ultimately, I'm open-minded but I would be cautious about assuming that all (or any) fictional worlds exist.

      "Here's a question though, is the universe mathematical, or is mathematics simply patterns we've discovered about the universe?"

      I think the universe *is* a mathematical object, in just the same way as the Mandelbrot set or Conway's Game of Life.

      Mathematics is certainly not simply patterns we've discovered about the universe because much of mathematics is abstract and has no known practical application to modelling events in the universe.

    2. Thanks. I agree completely on Star Wars. Like all fictional worlds, it's our world with a few modifications to make it more exciting. Of course, on close inspection, those modifications are often incoherent, even if they're fun.

      The question is, is the Star Wars universe's existence in our minds, its patterns, less "real" than our patterns, even if they are patterns built on top of and interlaced with ours?

      On mathematics, I have to admit to not having much knowledge of higher order math (I barely made it through calculus in college), so please excuse if these questions are hopelessly ignorant.

      Don't all mathematics eventually have to relate to real world objects? Is mathematics ultimately something more than shortcuts to arithmetic and geometry? If not, then what reference do we have to know whether those objects are valid or not?

      What would you see the difference being between the universe being mathematical and mathematics representing the patterns of the universe?

      These are actual questions I have, not ones I'm asking for rhetorical purposes. No worries if they are outside the scope of your post.

    3. "The question is, is the Star Wars universe's existence in our minds, its patterns, less "real" than our patterns, even if they are patterns built on top of and interlaced with ours?"

      I would say so, because if that's all it is, then there is no such thing as "The Star Wars Universe", there is only the mental model I have, the mental model you have, the mental model George Lucas has etc. These are all different and do not refer to one object, and although each of these is a real entity in itself, it is perhaps not easily separated from the rest of the mental baggage we carry around with us.

      "Don't all mathematics eventually have to relate to real world objects?"


      "Is mathematics ultimately something more than shortcuts to arithmetic and geometry?"


      "If not, then what reference do we have to know whether those objects are valid or not?"

      Logical analysis. A mathematical object is valid if it is self-consistent. That's really all that's required.

      "What would you see the difference being between the universe being mathematical and mathematics representing the patterns of the universe?"

      I interpreted "the universe is mathematical" to mean the universe is made of or described by mathematics. I intepret "mathematics represents the patterns of the universe" to mean that mathematics is limited to describing and modelling the real world. I would disagree with because some mathematics has no obvious relationship to the real world.

      I think many mathematical objects have no or little relation to the real world. I don't think there is any structure in the real world that looks like The Mandelbrot Set. I don't think there is any system in the real world that behaves like Conway's Game of Life.

      Nevertheless studying these abstract structures might indirectly help us to make analogies with the real world. Mandelbrot helped us to understand the concept of fractals, which are crudely manifested in nature in the structure of fern leaves, for example. Conway's Game of Life serves as an analogy to help us see how complex and interesting behaviour can spontaneously arise from systems driven by simple rules.

    4. On Star Wars, doesn't the fact that we, two minds, can discuss it mean it that it has some trans-mind existence? Otherwise, what are we discussing? Granted, its existence is different than a tree's, since the tree will still be there even if absolutely no one contemplates it. Star Wars is a pattern that only exists in minds, but how different is that from democracy, socialism, or mathematics?

      Thanks for the link. After my last comment, I realized that mathematics, even if it is founded on real world relations, could still extrapolate into structures with no real world correlates. It might be more accurate to say that these structures are mathematically possible whether or not they are physically possible. But now I'm wondering what that means for the MUH. Wouldn't the MUH require that all mathematical structures exist somewhere?

    5. "On Star Wars, doesn't the fact that we, two minds, can discuss it mean it that it has some trans-mind existence?"

      Good point. We could say it's a family of related structures. My structure is similar enough to yours to allow conversation, but it's still not quite the same as a formal mathematical structure which is very precisely defined. In the latter case we really are talking about the same thing.

      With Star Wars, we can imagine areas where it's no longer clear if we're talking about Star Wars any more. For some people, Episodes 1-3 don't count. Most people would discount the Ewok christmas special thing. What if I make up my own story in the Star Wars universe, but don't use any characters or locations from the canon, or even refer to the fact that it is Star Wars. Is that still Star Wars? Etc, etc, etc.

      It's a vague concept so it's hard to pin down what we mean when we debate whether it exists. Insofar as it can be described precisely, it exists. While it remains vague or ambiguous, I'm more hesitant to say it's a real thing.

      "Granted, its existence is different than a tree's, since the tree will still be there even if absolutely no one contemplates it."

      I actually think that if it exists at all it exists independently of minds. If it can be defined precisely, then that precise definition constitutes a mathematical structure (as mathematics is in a way the study of precise, formal definitions), and that structure exists Platonically. In this view, George Lucas didn't create it but discovered it. But then all acts of creation are actually acts of discovery.

      "Wouldn't the MUH require that all mathematical structures exist somewhere?"

      Yes, but not in this universe, and not necessarily in any other either. They exist in the ensemble of all possible mathematical structures. Some few of these will contain analogues of our time and space, so we might be tempted to call them universes. Some very few will contain replicators which we might be tempted to call life. Some very very few will contain conscious observers.

      But the ensemble also contains such structures as the Mandelbrot set. We would not be tempted to call that a universe because it has no time and nothing happens there. It exists in the ensemble of mathematical objects, but whether it exists "somewhere" depends on what you mean by a "where". It doesn't exist as an object within a structure we would recognise as a universe, and it contains no conscious observers to regard it as physical or as existing in a real physical place.

    6. If I understand correctly, the MUH asserts that mathematics is a superset of the universe (or all universes). If so, the only way that we could ever falsify that theory would be to discover aspects of the universe that are not mathematical. Of course, we could never eliminate the doubt that it's simply an aspect we don't understand the mathematics for yet.

      I'm going to have to spend some time digesting this. I can still see mathematics being grounded in patterns in our universe, even if some of the structures built on top of those foundations don't exist in the universe.

      Thanks for a fascinating discussion!

  6. [Hi, this is Steve Morris from WordPress. I'm commenting using my business ID as I can't login using my WordPress ID on your blog.]

    Well, this is a very interesting article and I agree with much of it.

    Ultimately, I believe the universe was created out of "nothing". However, our word nothing doesn't really explain what I mean - we casually use nothing to mean the opposite of something. But, for instance, Atheism isn't the opposite of belief in God - it's simply the absence of a belief in God. So when I say that the universe was created out of nothing, I mean that it really is nothing, but an interesting kind of nothing. I think this is probably very similar to your underlying belief.

    You're right that if the universe was a computer simulation, then we (probably) couldn't tell. So therefore the universe must (probably) behave as if it were a computer simulation, whether or not any computer actually exists.

    I think that one problem I see is with your assertion that the universe obeys mathematical physical laws. What if at the deepest level it doesn't? What if the behaviour merely approximates mathematical laws at the scale of subatomic particles and at energies and timescales that we can currently investigate? What if at the deepest level it's just chaos or something equally unpalatable? At this point we can't be certain.

    Also there's a problem with creation (Big Bang), but that problem already exists in physics and presumably has the same solution.

    My main problem is with your very first hypothesis: 1) All mathematical objects exist abstractly and independently of minds (mathematical Platonism).

    I don't think that mathematical objects do exist abstractly. I think that they exist only as concepts that must be rendered in the physical universe. In your platonic view, there could be (must be) a "god" (infinite number of gods) floating around in non-space outside the universe (existing as abstract concepts).

    I have written about this in an entirely clumsy way here:

    You can see me thinking through the question and finally arriving at an answer in the second part here:

    What you seem to be saying is that a mathematical construction can itself be a physical universe. But how? Can 1 + 1 = 2 be a physical universe? Not a very interesting one, but is there a certain level of complexity or completeness necessary for a mathematical structure to form a universe?

    Still, I'm attracted by this idea that a mathematical construction can be (is the only kind of) reality. I like the idea that all possible universes exist, although I strongly dislike the anthropic principle as a means of explaining our own universe.

    Suppose we accept this hypothesis that a universe is a mathematical object that contains spacetime, mass, energy, etc as mathematical structures. Question - what are the rules for constructing such an object? Is our universe the only mathematical object that can exist in this way, or are others possible? Could there be different types of universes, or only one, or many copies of the same thing? Until you can answer such questions I don't think you have a theory, just the start of a theory.

    Assuming everything you say is correct, then your theory potentially explains everything in physics. The area of uncertainty then moves to mathematics itself. Where does it come from? What is this Platonic mathematics that seems to have some kind of deep reality? What about the incompleteness and fundamental problems associated with attempts to pin down mathematical axioms( Godel, Turing, etc)?

    I'm not expecting answers here - just thinking out loud!

    1. Hi Steve,

      Great to hear from you.

      I enjoyed reading your thoughts on the existence of ideas, though ultimately I disagree. I think there is no fact of the matter on whether mathematical ideas exist, there are only different interpretations of "existence" and different ways of thinking about it.

      There are a couple of points I would make, however. You seem to equate ideas to their physical representations, but this does not seem to account for what is the same about the concept "2" in my mind and the same in yours, never mind the same concept stored in a database or as exemplified by two objects. All these are radically different physical configurations of matter, yet they all seem to represent the same thing. I find it hard to see how we can think about this situation coherently without accepting that that thing has some kind of existence.

      We also need to be able to account for how two mathematicians can hit upon the same mathematical object independently, e.g. Newton and Leibniz independently inventing calculus. While there may be ways to construe this as creation, it seems to me more natural to think of it as discovery.

      Finally, most of your examples in the "Afterthoughts" post seem to me to establish only that mathematical objects do not physically exist. But nobody is claiming they are. Abstract existence is of a different sort than physical existence, so it doesn't much trouble me if nobody has written a particular number before. I do think it ought to trouble you, however, if you can't tell the difference between the act of creating a new number and discovering a number that was previously created by somebody else.

      These thoughts are covered in greater detail on my blog post here.

      Ultimately, I think it's much easier to discuss the MUH if we first accept mathematical Platonism, but the MUH is tenable even without it. It just means that this universe doesn't really exist. This seems perverse, but if your definition of existence is so narrow as to exclude mathematical objects then I think it's reasonable - the concept of physical existence doesn't really apply to universes as I explain in this post. It's perhaps not much weirder than the idea that we are all characters in a novel or virtual people in a simulation.

    2. On to your specific points:

      "Ultimately, I believe the universe was created out of "nothing"."

      I don't have any strong beliefs about how this universe came to be, because the idea of it coming to be is a scientific question which assumes time. This question is best answered by cosmologists.

      On the MUH, from the point of view of an observer outside the universe, the universe did not come into being at all. It exists eternally/timelessly in its entirety as a mathematical object. like a parabola that has no beginning and no end. (You could view the Big Bang as analogous to the point where the slope of the parabola is zero.)

      "I think that one problem I see is with your assertion that the universe obeys mathematical physical laws. What if at the deepest level it doesn't?"

      If you really think that's a possibility, it only downgrades the MUH from a logical necessity to a scientific hypothesis, which would probably please Tegmark as he views it as science. He predicts that no such non-mathematical behaviour will ever be observed. My point of view is that anything that cannot be defined mathematically even in principle is incoherent, and so impossible. I have hinted at some of these ideas here:

      "your platonic view, there could be (must be) a "god" (infinite number of gods) floating around in non-space outside the universe (existing as abstract concepts)."

      Sure. I don't see that as a problem. Just as on quantum mechanics I could suddenly teleport to Mercury but don't expect to because it is extremely improbable. Any theory that implies that anything can happen necessarily implies that profoundly weird things can happen, but that's not a problem as long as there are infinitely more prosaic things that can happen which drown them out.

      >Can 1 + 1 = 2 be a physical universe?<
      I think it would be perverse to call it one because it is so simple and has nothing analogous to time and space and certainly could not support consciousness. "Universe" is not really a well defined term. Everything that exists is a mathematical object. Some of these we may choose to call universes because they resemble ours in some way, but which we choose to call universes and which we don't is purely a matter of taxonomic convention. The same thing goes for what we consider to be physical.

    3. "although I strongly dislike the anthropic principle as a means of explaining our own universe."

      Why? How else would you explain fine tuning?

      "what are the rules for constructing such an object?"

      There are no rules. Any mathematical object that is well-defined and consistent exists. If it resembles what we think of as a universe we can call it a universe.

      "Is our universe the only mathematical object that can exist in this way, or are others possible?"

      Others are possible.

      "Could there be different types of universes, or only one, or many copies of the same thing?"

      There can be many different types. There cannot be many copies of the same thing in my view because any two mathematical objects which are identical in every way are the same object.

      "Where does [mathematics] come from?"

      It doesn't come from anywhere. It exists necessarily, as long as we adopt the Platonist view that regards existence as equivalent to mathematical consistency.

      "What is this Platonic mathematics that seems to have some kind of deep reality?"

      There is no Platonic mathematics. There is only mathematics. Platonism is an ontological attitude towards it, not a branch of it.

      >What about the incompleteness and fundamental problems associated with attempts to pin down mathematical axioms( Godel, Turing, etc)?<

      What about them? These are often raised to criticise Platonism and the MUH in particular, but as far as I am aware no coherent argument has been made which use them to effectively criticise either. Godel in particular shows that we cannot prove that any particular mathematical system is consistent (at least not without using some other system), but that does not mean that it is not consistent. It only means we can't know it for sure. As long as it is possible for consistent mathematical structures to exist, then the MUH is fine.

  7. Hi, it's Steve Morris again, replying anonymously, as Blogger and I really don't seem to be on speaking terms.

    Thanks for answering all my questions. I'm happy with most of your answers.

    I still remain unconvinced by Platonism however. I will read your article on this and see if I have any new thoughts.

    My dislike of the anthropic principle is based on the resulting arbitrariness of the universe. The history of science is that arbitrary facts are explained when deeper understanding is obtained. Why does a day last 24 hours? Because of the rate at which the Earth is rotating. Why does water boil at 100C? Because of the physical properties of water molecules. Why is the speed of light the value we observe? Because of some reason we don't yet know. OR just because. I don't like "just because." If we have been satisfied with "just because" 2,000 years ago, we would still know nothing.

    I think this is perhaps the most bothersome aspect of MUH. Why is our universe like it is? Just because.

    Perhaps that's true. I think we need to find out a lot more before we can say for sure. At this point, MUH seems to be an interesting theory that has a lot going for it.

    1. Hi Steve,

      You can ask "why" endlessly, but to avoid problems of infinite regress, eventually you have to hit either logical necessity or something completely arbitrary.

      The MUH is an answer from logical necessity. Only our vantage point in the multiverse is essentially arbitrary, not the multiverse itself. The impulse to find more satisfying explanations has been seen in the past, for example Kepler's Platonic Solid model of the Solar System.

      Not only did this prove incorrect, but it turned out there were eight planets, not six, and furthermore that there are countless other solar systems out there, rendering the attempt to find a neat explanation for the structure of ours to be completely misguided.

      It seems to me that your view is like Kepler's, that we might in principle be able to discover deep physical truths about the universe from our armchairs because there is some underlying explanation for why it must be the specific way it is. Tegmark, on the other hand, says that this is impossible, because there are many logically consistent ways the universe could be. Where Kepler would take out his pencil and attempt to find geometric rationalisations for his limited understanding of the solar system, Tegmark would not believe anything without the experimental evidence afforded by telescopes, experiment etc. I think this is the healthier, more skeptical attitude, and it is more likely to lead to correct beliefs about reality.

  8. That the Universe is completely described by mathematics is indeed an old idea, however Pythagoras and Galileo did not provide enough arguments why it is so! It is more like a postulate in their philosophy. It is easy to say that the Universe is mathematical but we need epistemological and ontological basis for such claims. I saw in your website the link to ontic structural realism (OSR) so I guess you are familiar with it. OSR indeed provides good arguments about the underlying invariant structure of all our theories.

    I think Immanuel Kant is the first philosopher to provide the strong arguments why the Universe is described by mathematics. If you are familiar with history of philosophy, Kant reacted to the famous debate between Rationalists (Leibniz, Descartes, Spinoza) and Empiricists (Locke and especially David Hume). Rationalists claim that the source of knowledge is reason and innate ideas, while empiricist claim that the source of knowledge is experience through the senses. Both are right from their perspective. Kant said that to speak about innate ideas in our mind which ground mathematics, metaphysics (a priori knowledge) as rationalists did is lazy business. David Hume has shown that everything comes from experience but he had problems with establishing mathematics on firm ground because maths speak of experience a priori. He could not explain how mathematics is possible! Kant tried to defend this a priori knowledge (mathematics, theoretical physics) and so-called synthetic a priori judgments. That's why I have used Kant to model our cognitive framework (and the Universe as it appears to us) as a quantum computer defined on a grid of cells. I claim that this grid is invariant structure within which all our thoughts, knowledge and theories originate. The structure OSR seeks.

    Kant had influenced such mathematicians as Henri Poincaré and David Hilbert. In philosophy of mathematics Kant belongs to intuitionist school. It is also interesting to study the logicist school, that is Frege, Russell. I know that you are involved with FQXi. I claim that we will not understand ultimate reality unless we view everything as a system of mathematics, theoretical physics, philosophy of science and cognitive science. Cognitive science is of absolute importance in understanding ultimate reality because all our thoughts about the world originate in our brain. I know that you come from strictly scientific background but philosophy of science, philosophy of mind cannot be left out if you want to understand the ultimate reality.
    It seems that you have buried the philosophy of corporeal nature. This is the true purpose of proper metaphysics of corporeal nature - to assist mathematics and physics. They should go together. It does not matter that people did not know about the Higgs boson or the mathematical description of general relativity 200 years ago. What Kant and Hegel knew is fundamentals - how our knowledge about the world in general is possible. If you know the roots of your knowledge, the epistemological basis of mathematics and physics, everything else is just details. To understand ultimate reality we must understand how we understand things in the first place! That is, we must have the picture of our cognitive faculties in general. This yields the big picture of the Universe how it appears to us.
    That's why I took Kant who asked and provided answers in his work to the questions: ''how is mathematics possible?''. ''How is physics possible?''. ''How is metaphysics as science possible?''.

    1. Hi Darius,

      Thanks for commenting. Sorry I took so long to respond.

      Firstly, the MUH is more profound than OSR or the preceding mathematical interpretations of the universe, as I understand them. If the MUH is true, then it also entails the existence of a mathematical multiverse and it explains why the universe exists at all. Furthermore, from the argument I have presented, it seems to be logically necessary if naturalism, Platonism and computationalism are true (each of which I also think is logically necessary).

      I don't see most of the philosophical arguments you mention as being that centrally tied to the MUH. The MUH has very little to do with epistemology.

      I am not at all affiliated with the FQXi. Where did you get that idea? I didn't even know what it was until I googled it.

      I think that philosophy of mind is important to the MUH, but only to establish the truth of computationalism (which I argue is required for the MUH to be plausible), another view I take a very strong interest in.

      I have no problem with epistemology. I think that contemplating how and what we know is a worthwhile pursuit. It's just not one that greatly interests me right now and I don't think it is any more crucial for analysis of the MUH than it is for any other academic matter of interest. Mathematicians don't really need philosophy of mathematics, physicists don't really need philosophy of physics, and though I may be claiming to know things about the universe, I don't think I need a very solid grounding in epistemology to do that.