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Shut up and calculate: an extreme mathematical universe

159 points by Phithagoras 9y ago

fizixer 9y ago
Two points: - What he describes in the abstract is nothing new. Part of your work as an applied mathematician is to take mathematical models developed by others, e.g, in the form of systems of differential equations, and treat them as absolute truths (for the sake of computation; essentially you create "a mathematical bubble" for yourself and choose to live in it), study the consequences, and present the results. So his claim of "I advocate" is disgusting plagiarism/credit-snatching of a well-known technique. - What he actually advocated is total bullshit as far as physics is concerned. You can live as long as you want in the bubble of your mathematical models. But if you have to connect your math to reality, you have to make your math subordinate to empirical data, and if the data disagrees, your shiny mathematical model is wrong, period. Also don't forget what George Box said "All models are wrong. But some are useful".
krylon 9y ago
It is remarkable how sufficiently advanced physics / cosmology sounds to the layperson like a conversation between two teenagers who smoked weed a little while ago. (My favorite: https://en.wikipedia.org/wiki/One-electron_universe https://en.wikipedia.org/wiki/One-electron_universe) I do not say that to belittle physicists. Not at all, I have the utmost respect for what they do. But as they dig deeper and deeper, what they come up with sounds like the universe is a pretty strange place to be. Personally, I am okay with that idea, because, if I look closely enough, I have been living all my life in a world that does not really make sense. But it is really strange, nevertheless.
westoncb 9y ago
It baffles me that people ignore the fact that they have subjectivity when making these claims. It makes a certain amount of sense since science itself aims to remove any taint of subjectivity in its practice, so folks who want to be thoroughly 'scientific' in other aspects of their lives (e.g. their personal philosophies of reality) carry over over this ideal. But it's also a contradiction since science scientific methodology demands we check our results empirically: the fact that you have subjectivity (i.e. that qualia exist) and what it's like, is the most direct empirical fact we have access to and which needs reconciling in any discussion of the universe's constitution. I think people sort of push this point aside either by saying that qualia/subjectivity could be generated by math (or a computational process), and that's what really matters. But we have no examples of abstract math generating anything; physical processes generate things and we use math to influence their behavior—but that's it! And furthermore, even if math generated subjectivity—it's still there in all of its un-mathematical obviousness at the end of the day. I've yet to see an argument in favor of math having higher ontological status than this which wasn't every bit as unrestrainedly imaginative and unfounded as Plato's 'ideal forms'. I would expect a thoroughgoing adherent to scientific methodology to instead say something on the lines of, "this is outside the domain of science to judge one way or the other; the most we can say is the idea that reality is made of math is one untestable hypothesis among many whose probabilities of correctness each approach zero by dint of their many peers."
- whatshisface 9y ago
  I know of a nice argument against simulationism that I think applies here: If simulated universes can be "real," then we're assuming that there's a mapping between patterns of electrons in a computer (or provable theorems, if you're brave enough), and the real universe they represent. (Say, interpreting two numbers that may be stored very far apart on the chip as the positions of two particles that are, in the universe, very close together.) The problem that this introduces is that there are no well-motivated ways that we could decide which mappings are "real," and could be "lived in." For example, we might claim that the state of the simulation must be map-able in polynomial time to some charts and pictures that a human could interpret - but an inhabitant wouldn't care. So, since all mappings are equally motivated as far as an inhabitant is concerned, allow me to choose one that maps this grain of sand to a universe the same as ours. Well, there you go. Our world in a grain of sand.
  - drdeca 9y ago
    Not to say that I think that the world is a simulation or anything like that (I don't), but I think that computational complexity and related things probably do induce a natural notion of "naturality" on the different mappings. Even if there might not be a single most natural mapping, I think it makes sense to say that some mappings are more natural than others. I think there is a natural sense which is independent from human perception and intent, in which a computer running, say, a graph coloring algorithm, is doing that more than a waterfall is. Even if there is a bijection between a collection of initial states of the waterfall with the input of program, that fits together with one between the outputs and the results of the waterfall.
    - whatshisface 9y ago
      Computational complexity is an insufficient motivator, because while in our current universe some problems are hard and others are easy, in the hypothetical universe there might be a law of physics that solves NP-hard problems. In fact, if the problem is right, you might even end up in a situation where they can't hook their NP-hard problem solvers together in a way that solves problems that we would consider polynomial.
usnigg 9y ago
I am a theoretical physicist and I think mathematics is just a tool. A language to describe concisely but always approximately what goes on out there in nature. A way to foramlize intuition and test it for logical inconsistencies. I personally think it is counter productive to focus so much on giving tools a deeper meaning. Sadly many a theoretical physicist seems to be drifting into that corner never to come back... The cure? Keep talking to your experimentalist friends. They usually keep their feet on the ground :-)
deleted 9y ago
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daveydog 9y ago
I don't think the idea that everything is math, is particularly original.
cs702 9y ago
As stated by the author, this is necessarily a speculative piece ("assuming that..."), but still worth reading -- as long as you keep that in mind. It makes no sense to criticize it or dismiss it for being speculative, as others have done here! By the way, if you enjoy reading this piece, you might also enjoy really interesting paper coauthored by the same person: "Why does deep and cheap learning work so well?," available at https://arxiv.org/abs/1608.08225 https://arxiv.org/abs/1608.08225 -- it posits that deep learning works well because of the particular structure of the laws of physics in the universe in which we happen to live.
kingkawn 9y ago
a Subset cannot have greater cardinality than the set it derives from. Subjective existence cannot be reduced to Math because it is our explorations in subjectivity that led to math’s invention and refinement
- kybernetikos 9y ago
  By refering to 'math's invention' you are already prejudicing the matter. It is precisely whether Maths is a subset of subjective experience that we invented or whether it has always been a superset of the universe our subjective experience is embedded in that we merely discovered is precisely what is at issue here. It's easy to get confused by the language used to frame things. I could say that language only has countably infinite possible statements, so it must therefore be impossible to use set theory which is learnt through and derives from language to analyse uncountably infinite sets, but it would be a bad argument.
- drdeca 9y ago
  A subset can have equal cardinality as the set it is a subset of, if the set is infinite. I don't think that subjectivity can be reduced to math, but reducing math to subjectivity doesn't obviously imply that subjectivity cannot be reduced to math, as far as I can see right now.
- wolfgke 9y ago
  > a Subset cannot have greater cardinality than the set it derives from. I am not completely certain in the subtleties of non-standard set theory, but at least if we replace the axiom of Choice (AC) by the axiom of Determinacy (AD) - which is perfectly fine, though it is less common to do mathematics in this system - we can partition the real line R by an equivalence relation that has more equivalence classes than elements of R. So at least we can partition R into more disjoint subsets than R has elements. Source: https://math.stackexchange.com/questions/1104028/what-are-disasters-with-axiom-of-determinacy/1148609#1148609 https://math.stackexchange.com/questions/1104028/what-are-di...
  - xelxebar 9y ago
deleted 9y ago
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Houshalter 9y ago
I can't even comprehend what the alternative viewpoint would be. The the universe isn't mathematical. A universe that doesn't obey any system of logical rules. A universe with true randomness might satisfy that literally. But random variables can be incorporated into a mathematical model. But what would it even mean for a universe to not be describable by any kind of math at all? I don't think such a concept is coherent or discussable.
- carvalho 9y ago
  To be a fully random sequence, a sequence would have to contain substrings of apparent predictable beautiful order. If it didn't, it would be more predictable (you could then discard ordered continuations), not unpredictable. > [Model-agnosticism] consists of never regarding any model or map of the universe with total 100% belief or total 100% denial. Following Korzybski, I put things in probabilities, not absolutes... My only originality lies in applying this zetetic attitude outside the hardest of the hard sciences, physics, to softer sciences and then to non-sciences like politics, ideology, jury verdicts and, of course, conspiracy theory. -- RAW A universe isn't a map, it is the territory. An ordered universe can still be mapped/modeled with maths. A universe that is impossible to be mapped/modeled and has no predictable patterns would be an impossibly boring universe indeed (Kolmogorov Random Universe). Perhaps our brains would evolve to think more in probabilities than absolutes in such a universe. Or they could simply not be allowed to exist (intelligence requires order).
omazurov 9y ago
The tale of the frog and the bird may be good for a less mature audience but here we can really go with a richer analogy: the two views correspond to how a virtual machine is seen from within and from the environment in which it runs. I prefer this analogy to the "abstract mathematical structure, an immutable entity existing outside of space and time"because it not only allows to comprehend how space and time could emerge within that VM and not only what it means to exist outside that emergent space and time, but also that that outside environment could have its own space and time and in turn be just another VM, totally beyond our reach but not inconceivable.
Bromskloss 9y ago
I couldn't go to his lecture in Stockholm the other day. Were any of you there and could tell us something about it? Edit: Hm, actually, the lecture was about dangers of AI, so not the same topic.
m3kw9 9y ago
Meaning everything can be predicted if you know the equation and have enuough processing power. Problem is doing the calculation in the same universe where you are also predicting it also disrupts the calculation. Something in the line of the uncertainty principle
iainmerrick 9y ago
I don't think any of this stuff is very meaningful or useful, but the title seems particularly cheeky. The whole point of that catchphrase "shut up and calculate!" was supposed to be "stop speculating about things we can't prove or disprove and wouldn't have any direct impact on us if we could; focus on crunching the numbers and figuring out what's effective." This whole "mathematical universe" thing seems like practically the definition of premature speculation! It's no more useful than ancient Greek philosophers speculating on the nature of the elements. Those other people over there drawing circles in the sand, now, they're getting somewhere.
daxfohl 9y ago
Wasn't Einstein'srelativity a product of intuition rather than of calculation? And, intuition of processes that far exceed the bounds that Darwinian evolution would have prepared us for? If we all just shut up and calculated, we'd still be doing Lorentz transforms near speed of light "just because they work", right?
usnigg 9y ago
http://backreaction.blogspot.ch/ http://backreaction.blogspot.ch/
js8 9y ago
I like the idea that the reality is just mathematics, I thought about it independently, but it gets weird really fast. I yet to have read this essay, but I had two thoughts: 1. Similar to Turing completeness, actual physical reality might not matter for the internal observer. All Turing-complete machines can calculate same things and observer from the inside of the machine cannot determine on what hardware the machine is implemented. Similarly, this can be case for observers within our universe (humans). So maybe there is, at a high enough level of abstraction, only one mathematically possible universe. 2. The mathematical theorems are shortcuts to computation. For example, I can add two numbers on a computer, or I can use a theorem, which will tell me only some property of the result, but avoids having to do the actual calculation. So if universe is just a computation, one could avoid actually computing it in certain cases by having a theorem that would tell them enough information they need. In other words, you don't actually need to simulate anything in order for objects in simulation to "exist". So there is no need for actual hardware on which the universe runs to exist. Addendum: So I read the essay, and I disagree that the multiverse is necessary. As I already stated in point 1, it can be the case that all possible mathematical realities are the same at high enough level of abstraction.
- hyperion2010 9y ago
  There is a seriously interesting question here about the information content of the theorem and whether it is equivalent to the Shannon limit on compressing the 'code' needed to run the computation. I bet someone published on this years ago but I haven't seen it phrased in quite the way you have it in point 2.
- carry_bit 9y ago
  A concrete example of #2 is the hashlife algorithm. Intermediate steps can be skipped in a region of space, yet since it implements the CA nothing inside can notice that the computation didn't actually happen.
- Barrin92 9y ago
  If you like the idea of thinking about physical space in computational terms give one of the classics a try, Calculating Space by Konrad Zuse.
gfodor 9y ago
Haven't read the paper but I'm assuming this is the same concept (likely more digestible for the layman) in his book Our Mathematical Universe, which I highly recommend!
lou1306 9y ago
I'm a philosophy noob, but isn't this very similar to Berkeley-style immaterialism?
- Koshkin 9y ago
  Not really. Berkeley's idealism contests the objectivity of reality, while here we are presented with another view of the foundation of the (objective) material world.
  - lou1306 9y ago
    This is true, but: > When we derive the consequences of a theory, we introduce new concepts — protons, molecules, stars — because they are convenient Couldn't matter itself be just another "human", "convenient" concept? Indeed the author writes: > our external physical reality is assumed to be purely mathematical Why should a purely mathematical reality also be material? Doesn't this lead to the logical conclusion that ideas, err, maths is "all that there is"? Sure, in this case ideas are objective and not subjective, but Berkeley's "mind of God" argument also leads to an objective, albeit immaterial reality, doesn't it?
    - Koshkin 9y ago
      1) But convenience is objective! (It is not convenient to sit in a chair that is hanging upside down.) Which is why matter, even if it is seen as a "convenient" mental construct, is still objective. 2) Materialism and "objectivism" are closely tied together. Mathematics consists of abstractions (or models), and those reside, in particular, in our mind; being "convenient", these abstractions are therefore reflections of what lies outside, and so - almost by definition - the objective world itself, matter, cannot possibly be built from math. (Besides our mind, "abstractions" can also reside in other, inanimate, objects; regardless, this still means that some part of matter merely reflects, to a degree, some other part).
    - otakucode 9y ago
      >Sure, in this case ideas are objective and not subjective, but Berkeley's "mind of God" argument also leads to an objective, albeit immaterial reality, doesn't it? No, it leads to a subjective inability to know anything at all. It's a difficult philosophical view to talk about, because the people espousing it can not possibly actually believe it and usually don't even understand it. The non-existence of a shared, knowable, objective reality means that, even in a totally solipsistic model, inductive reasoning can not be applied. You can not assume that attempting to breathe in the next moment will extend your life rather than extinguish it. You can not conclude that the air in your lungs will not suddenly become ants. You certainly can not leap to the extensively complex chain of reasoning necessary to believe that between this sentence and the next the English languages will turn into French. It introduces aggressive, pervasive, impossible-to-dismiss inconstancy in all things, including ones self. A true believer in a subjective universe would remain still in their bed, doing nothing, until they died. Any action whatever beyond automatic biological functions instantly betrays their belief that they know their body, know the environment they are in, and know how the mental activity necessary to cause the action will elicit a change in the environment which is not likely to destroy them. And that is a thing they can never know in a subjective universe. The 'mathematical universe' on the other hand is simply discussing how the shared, knowable, material universe we live in came to be or is best understood. I happen to think it is factually wrong on a few points (the author just off-hand claims that if we had a big enough supercomputer we could exactly calculate the future development of the universe which is fundamentally untrue for one) but it certainly doesn't amount to subjectivity.
1001101 9y ago
The author expands on this concept in his 2014 book (~7 years after the publication of this paper), Our Mathematical Universe: My Quest for the Ultimate Nature of Reality. I've read it, and I would highly recommend it to my fellow HN'rs. I found his multiverse classification scheme [1] very compelling and something that finds its way into my worldview and a lot of my thinking. [1] https://en.wikipedia.org/wiki/Multiverse#Max_Tegmark.27s_four_levels https://en.wikipedia.org/wiki/Multiverse#Max_Tegmark.27s_fou...
danharaj 9y ago
Which mathematics?
- Koshkin 9y ago
  The one of homotopical type.
aj7 9y ago
That was always Feynman’s attitude, obscured by the strong human layer he ran.
- Koshkin 9y ago
  Didn't Feynman think that the world was essentially void?
jcmoscon 9y ago
Math works because reality works. Not the opposite. Reality is bigger than math. Math is just a tool that models reality.
- kaffeemitsahne 9y ago
  ..and with one simple statement, HN user jcmoscon forever put an end to this age-old debate.
  - Koshkin 9y ago
    Still, if you think about it, that statement is very deep (and is probably true).
  - Avshalom 9y ago
    Well no, Samuel Johnson and a good sized rock put an end to it centuries ago, jcmoscon is just trying to remind people.
- hackinthebochs 9y ago
  On the contrary, reality works because reality cannot contain contradictions. But the set of possible structures without contradiction is a superset of what is physically realizable. And mathematics is just discovering and cataloging the set of structures without contradiction. So reality works because math works.
  - kaiwen1 9y ago
    In _Road to Reality_, Roger Penrose has a diagram that illustrates the Platonic world of mathematics fully encompassing the physical world, and the physical world fully encompassing the mental world (since minds are physical), and the mental world fully encompassing the Platonic world of mathematics. Although each encompasses the other in its entirety, each depends only on a small part of its predecessor. Penrose says "...the Platonic world may be the most primitive of these three, since mathematics is a kind of necessity, virtually conjuring its very self into existence through logic alone."
  - deleted 9y ago
vorg 9y ago
The author asks how we could use the mathematical equations which describe all of reality to compute the frog’s view of the universe — our observations — from the bird’s view. Wouldn't we also need to include our self-awareness in those observations, which would make the computation impossible according to Turing's halting theorem, or is it Godel's?
- jxy 9y ago
  The basic result from Turing/Godel is that you don't need any self-referential. As long as it is an enough powerful formal system, it is always incomplete from day one. There will always be some mythical things you cannot compute. There will always be some universes you cannot prove their existence or nonexistence.
jcelerier 9y ago
> For example, a sufficiently powerful supercomputer could calculate how the state of the universe evolves over time without interpreting what is happening in human terms. Hasn't it been understood for a long time that quite a bit of things were entirely non-deterministic ? eg radioactive decay. You could compute a possible state, but the universe is a sequence of state spaces.
bitL 9y ago
Alright, why not go one step further and make Universe computational instead? (i.e. state is important in von Neumann-style instead of meaningless in classical math-style; or algorithms vs formulas)
- eref 9y ago
  Tegmark's Mathematical Universe Hypothesis (MUH) is actually pretty much a computational theory of the universe. It basically makes the assumption that there exists somewhere the simplest process imaginable: One that counts through and runs all mathematical formulas which includes all programs, which, in turn, includes all (computable) universes. This assumption explains why our universe is oddly complex: If all universes exist, then a weird one such as ours merely exists because all universes do. This line of reasoning is also called the "anthropic principle". I think, Tegmark does not talk about computations per se because the hypothesis is agnostic about what kind of computer our universe is. It could, for example, be a geometric computer in which the position of objects can be determined with infinite precision (i.e. using the real numbers). Such a computer would be strictly more powerful than a Turing machine or equivalent (i.e. all computational models that can be described and run inside a Turing machine). Enumerating the space of all formalisms (i.e. the domain of mathematics) is as agnostic as you can be.
  - gfodor 9y ago
    I didn't take Tegmark's argument to be a computational theory. I took it to be an inductive one based upon the trend of the physics over the last decades towards the more and more generified multiverse theories. Ultimately, you land on a model where all consistent mathematical representations of a reality exist. At that level of abstraction, there is no where to go: it is the limit of the induction. There's a rather satisfying sense of self-evidence about it, but that doesn't mean it's falsifiable or true. IIRC there are no arguments about computation or what underlying medium(s) are involved, it seems like a separate question.
    - eref 9y ago
      I am unsure now whether he describes the set of mathematical objects by enumeration and evaluation. He definitely describes the set of universes this way, as being “run”, so it is a computation. But you are right that he never discussed the substrate that they are run on. He assumes the existence of that enumeration to be axiomatic, like an uncaused cause, somewhere in the space of all mathematical objects that simply exist. I guess, if we had a proof that all non-computable formal systems are inconsistent then we could also skip one layer and assume the enumeration of all programs as axiomatic in order to be maximally agnostic.
lainon 9y ago
Here's something more philosophical concerning the mathematical universe hypothesis: http://shelf1.library.cmu.edu/HSS/2015/a1626190.pdf http://shelf1.library.cmu.edu/HSS/2015/a1626190.pdf
Santosh83 9y ago
An interesting article addressing this from Sabine Hossenfender: http://backreaction.blogspot.in/2007/10/mathematical-universe.html http://backreaction.blogspot.in/2007/10/mathematical-univers... Her more recent article (review of Max Tegmark's book) also comments on the same subject: http://backreaction.blogspot.in/2017/11/book-review-max-tegmark-our.html http://backreaction.blogspot.in/2017/11/book-review-max-tegm...
- westoncb 9y ago
  This (the universe is 'made' of math) seems to me like the most egregious error in the general outlook of contemporary scientifically-minded people—and one I made myself for many years. If you really consider the issue closely, you'll see it depends on not overlooking certain dark corners of it which you can sort of ignore and assume to not contain contradictions—but they're there! And once you see them, it's hard to unsee them: what in the world would it mean for a desk to 'made' of mathematics—it's nonsensical. I think programmers are especially susceptible to this mistake because they're used to using mathematical descriptions to generate things, but even there, the idea that the math is doing any generating is a bias of our perspective since we're focused on the math; what's actually going on is a physical process involving silicon and plastic and metals etc., which can also be described mathematically, but the leap to being math is an unfounded step which could probably be predicted by the fact that that's where our focus lies. It also begins to look kind of silly when you consider how human math is: it's an outgrowth of an evolved mental faculty. Cognition is certainly impressive relative to the other things we and various animals are able to do—but it's probably still in the same category of our other faculties like the olfactory, visual, tactile etc.—and unlikely to be involved in the literal fabrication of the universe.
  - gfodor 9y ago
    Have you read the paper or the author's book? Because you're distilling a rather rich and refined argument down to a simplistic description. The author doesn't argue the universe is "made" of math.
    - 9y ago
otakucode 9y ago
A purely mathematical universe immediately brings to mind a question: Since chaotic systems evolve in a way provably intractable to calculation, how could any system based solely upon calculation address, well, almost everything that exists?
- klank 9y ago
  That's an odd definition of chaotic systems you are using. It's not so much that chaotic systems can not be calculated but instead that enough precision of starting conditions can't be obtained to ensure an accurate result (for example, highly non-linear systems). The double pendulum system is a perfect example where computation is simple yet the system is still chaotic.
  - otakucode 9y ago
    I think it's an odd definition you are using. Chaos depends upon the sensitivity to those initial conditions, not what those initial conditions ARE. You do not need actual physical objects to see this. Conjecture that a double-pendulum system starts in a perfectly real-number-valued mythical reality where spacetime is perfectly continuous (a fiction). Now predict what state the system will be in at a bunch of 'close together' states. Then try to do some 'closer together' states. See if you can ever get them to line up. Do it on pencil and paper, of course, since you want to capture the full real-number-infinity effect. Of course, you can't write down the real-number result with infinite precision, can you? You'll have to chop it off at some point. That's OK. That's what error bars are for. Just keep going and keep track of your error. Oops, you'll find that your error grows larger than the prediction itself with remarkable speed! THAT is the chaos. It doesn't matter if you measured the starting conditions with literal infinite perfect precision. The error bars on your predictions are going to explode like a busy beaver function. You're not going to catch that. Nothing can. (I imagine the actual physical limit would be that your predictions can be accurate within a distance equivalent to the radius of a sphere expanding at the speed of light from the initial position but I haven't actually read anything suggesting that, it just seems sensible.) Then you've got stupid simple cellular automata like rule 31 1D systems. Totally discrete. Totally basic. No precision even needed anywhere. And yet, you can't predict it. Not without accumulating an error greater than the state in the system itself if you skip even the slightest act of computation necessary to evolve the system as a whole from one step to the next.
followmeon 9y ago
If our reality consists of mathematical programs and the multiverse theory holds, then the theory of everything is a simple incrementing binary counter. Every possible binary string is ran on a Turing machine in a different universe. The halting problem (endless loops) is sidestepped by being contained to a single universe. Also, apparently our universe got lucky as we get to live in an ordered universe, and not dissolve into nothingness due to a bug in our universe's binary code. Of course, like "normal" mathematics, "reality" mathematics should also be incomplete. Furthermore, there is no way to know everything of the universe one is a part of, as this requires at least 1 bit more than there are available bits in the universe. But this still allows for the possibility of the universe itself being an inference machine. Edit: Then again, I believe that mathematics is a mental construct, mathematics does not exist without brains, or if it really exists external to human minds, it exists in all possible forms: Humans do not discover maths as a whole, they cast a local net of subjective observation over it, and keep the things that make sense to our brains and are consistent within the current framework. Again, this does not preclude the possibility that our mathematical reality exists inside the mind of a giant supercomputer.
- agumonkey 9y ago
  can we rollback that tape ?
  - shruthijose 9y ago
    it somewhat more complicated
- Houshalter 9y ago
  I think you've confused the representation with the computation. Sure maybe that binary number could represent a universe state in some encoding. But there's no computation happening to it. It's static. >I believe that mathematics is a mental construct, mathematics does not exist without brains Brains evolved through natural selection to survive in the universe we exist in. Ideas similarly "evolved" to be good at modelling the universe. If mathematics didn't exist outside of our head, it would be useless. If there are two apples on the ground, and two more fall of a tree, then there are four apples on the ground. This is true whether or not a human is there to observe it and count them.
skc 9y ago
Theoretical physicists are my favorite people. They all sound slightly insane and I love it.

159 points by Phithagoras 9y ago

Two points: - What he describes in the abstract is nothing new. Part of your work as an applied mathematician is to take mathematical models developed by others, e.g, in the form of systems of differential equations, and treat them as absolute truths (for the sake of computation; essentially you create "a mathematical bubble" for yourself and choose to live in it), study the consequences, and present the results. So his claim of "I advocate" is disgusting plagiarism/credit-snatching of a well-known technique. - What he actually advocated is total bullshit as far as physics is concerned. You can live as long as you want in the bubble of your mathematical models. But if you have to connect your math to reality, you have to make your math subordinate to empirical data, and if the data disagrees, your shiny mathematical model is wrong, period. Also don't forget what George Box said "All models are wrong. But some are useful".

krylon 9y ago

It is remarkable how sufficiently advanced physics / cosmology sounds to the layperson like a conversation between two teenagers who smoked weed a little while ago. (My favorite: https://en.wikipedia.org/wiki/One-electron_universe https://en.wikipedia.org/wiki/One-electron_universe) I do not say that to belittle physicists. Not at all, I have the utmost respect for what they do. But as they dig deeper and deeper, what they come up with sounds like the universe is a pretty strange place to be. Personally, I am okay with that idea, because, if I look closely enough, I have been living all my life in a world that does not really make sense. But it is really strange, nevertheless.

westoncb 9y ago

It baffles me that people ignore the fact that they have subjectivity when making these claims. It makes a certain amount of sense since science itself aims to remove any taint of subjectivity in its practice, so folks who want to be thoroughly 'scientific' in other aspects of their lives (e.g. their personal philosophies of reality) carry over over this ideal. But it's also a contradiction since science scientific methodology demands we check our results empirically: the fact that you have subjectivity (i.e. that qualia exist) and what it's like, is the most direct empirical fact we have access to and which needs reconciling in any discussion of the universe's constitution. I think people sort of push this point aside either by saying that qualia/subjectivity could be generated by math (or a computational process), and that's what really matters. But we have no examples of abstract math generating anything; physical processes generate things and we use math to influence their behavior—but that's it! And furthermore, even if math generated subjectivity—it's still there in all of its un-mathematical obviousness at the end of the day. I've yet to see an argument in favor of math having higher ontological status than this which wasn't every bit as unrestrainedly imaginative and unfounded as Plato's 'ideal forms'. I would expect a thoroughgoing adherent to scientific methodology to instead say something on the lines of, "this is outside the domain of science to judge one way or the other; the most we can say is the idea that reality is made of math is one untestable hypothesis among many whose probabilities of correctness each approach zero by dint of their many peers."

whatshisface 9y ago
I know of a nice argument against simulationism that I think applies here: If simulated universes can be "real," then we're assuming that there's a mapping between patterns of electrons in a computer (or provable theorems, if you're brave enough), and the real universe they represent. (Say, interpreting two numbers that may be stored very far apart on the chip as the positions of two particles that are, in the universe, very close together.) The problem that this introduces is that there are no well-motivated ways that we could decide which mappings are "real," and could be "lived in." For example, we might claim that the state of the simulation must be map-able in polynomial time to some charts and pictures that a human could interpret - but an inhabitant wouldn't care. So, since all mappings are equally motivated as far as an inhabitant is concerned, allow me to choose one that maps this grain of sand to a universe the same as ours. Well, there you go. Our world in a grain of sand.
- drdeca 9y ago
  Not to say that I think that the world is a simulation or anything like that (I don't), but I think that computational complexity and related things probably do induce a natural notion of "naturality" on the different mappings. Even if there might not be a single most natural mapping, I think it makes sense to say that some mappings are more natural than others. I think there is a natural sense which is independent from human perception and intent, in which a computer running, say, a graph coloring algorithm, is doing that more than a waterfall is. Even if there is a bijection between a collection of initial states of the waterfall with the input of program, that fits together with one between the outputs and the results of the waterfall.
  - whatshisface 9y ago
    Computational complexity is an insufficient motivator, because while in our current universe some problems are hard and others are easy, in the hypothetical universe there might be a law of physics that solves NP-hard problems. In fact, if the problem is right, you might even end up in a situation where they can't hook their NP-hard problem solvers together in a way that solves problems that we would consider polynomial.

usnigg 9y ago

I am a theoretical physicist and I think mathematics is just a tool. A language to describe concisely but always approximately what goes on out there in nature. A way to foramlize intuition and test it for logical inconsistencies. I personally think it is counter productive to focus so much on giving tools a deeper meaning. Sadly many a theoretical physicist seems to be drifting into that corner never to come back... The cure? Keep talking to your experimentalist friends. They usually keep their feet on the ground :-)

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daveydog 9y ago

I don't think the idea that everything is math, is particularly original.

cs702 9y ago

As stated by the author, this is necessarily a speculative piece ("assuming that..."), but still worth reading -- as long as you keep that in mind. It makes no sense to criticize it or dismiss it for being speculative, as others have done here! By the way, if you enjoy reading this piece, you might also enjoy really interesting paper coauthored by the same person: "Why does deep and cheap learning work so well?," available at https://arxiv.org/abs/1608.08225 https://arxiv.org/abs/1608.08225 -- it posits that deep learning works well because of the particular structure of the laws of physics in the universe in which we happen to live.

kingkawn 9y ago

a Subset cannot have greater cardinality than the set it derives from. Subjective existence cannot be reduced to Math because it is our explorations in subjectivity that led to math’s invention and refinement

kybernetikos 9y ago
By refering to 'math's invention' you are already prejudicing the matter. It is precisely whether Maths is a subset of subjective experience that we invented or whether it has always been a superset of the universe our subjective experience is embedded in that we merely discovered is precisely what is at issue here. It's easy to get confused by the language used to frame things. I could say that language only has countably infinite possible statements, so it must therefore be impossible to use set theory which is learnt through and derives from language to analyse uncountably infinite sets, but it would be a bad argument.
drdeca 9y ago
A subset can have equal cardinality as the set it is a subset of, if the set is infinite. I don't think that subjectivity can be reduced to math, but reducing math to subjectivity doesn't obviously imply that subjectivity cannot be reduced to math, as far as I can see right now.
wolfgke 9y ago
> a Subset cannot have greater cardinality than the set it derives from. I am not completely certain in the subtleties of non-standard set theory, but at least if we replace the axiom of Choice (AC) by the axiom of Determinacy (AD) - which is perfectly fine, though it is less common to do mathematics in this system - we can partition the real line R by an equivalence relation that has more equivalence classes than elements of R. So at least we can partition R into more disjoint subsets than R has elements. Source: https://math.stackexchange.com/questions/1104028/what-are-disasters-with-axiom-of-determinacy/1148609#1148609 https://math.stackexchange.com/questions/1104028/what-are-di...
- xelxebar 9y ago

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Houshalter 9y ago

I can't even comprehend what the alternative viewpoint would be. The the universe isn't mathematical. A universe that doesn't obey any system of logical rules. A universe with true randomness might satisfy that literally. But random variables can be incorporated into a mathematical model. But what would it even mean for a universe to not be describable by any kind of math at all? I don't think such a concept is coherent or discussable.

carvalho 9y ago
To be a fully random sequence, a sequence would have to contain substrings of apparent predictable beautiful order. If it didn't, it would be more predictable (you could then discard ordered continuations), not unpredictable. > [Model-agnosticism] consists of never regarding any model or map of the universe with total 100% belief or total 100% denial. Following Korzybski, I put things in probabilities, not absolutes... My only originality lies in applying this zetetic attitude outside the hardest of the hard sciences, physics, to softer sciences and then to non-sciences like politics, ideology, jury verdicts and, of course, conspiracy theory. -- RAW A universe isn't a map, it is the territory. An ordered universe can still be mapped/modeled with maths. A universe that is impossible to be mapped/modeled and has no predictable patterns would be an impossibly boring universe indeed (Kolmogorov Random Universe). Perhaps our brains would evolve to think more in probabilities than absolutes in such a universe. Or they could simply not be allowed to exist (intelligence requires order).

omazurov 9y ago

The tale of the frog and the bird may be good for a less mature audience but here we can really go with a richer analogy: the two views correspond to how a virtual machine is seen from within and from the environment in which it runs. I prefer this analogy to the "abstract mathematical structure, an immutable entity existing outside of space and time"because it not only allows to comprehend how space and time could emerge within that VM and not only what it means to exist outside that emergent space and time, but also that that outside environment could have its own space and time and in turn be just another VM, totally beyond our reach but not inconceivable.

Bromskloss 9y ago

I couldn't go to his lecture in Stockholm the other day. Were any of you there and could tell us something about it? Edit: Hm, actually, the lecture was about dangers of AI, so not the same topic.

m3kw9 9y ago

Meaning everything can be predicted if you know the equation and have enuough processing power. Problem is doing the calculation in the same universe where you are also predicting it also disrupts the calculation. Something in the line of the uncertainty principle

iainmerrick 9y ago

I don't think any of this stuff is very meaningful or useful, but the title seems particularly cheeky. The whole point of that catchphrase "shut up and calculate!" was supposed to be "stop speculating about things we can't prove or disprove and wouldn't have any direct impact on us if we could; focus on crunching the numbers and figuring out what's effective." This whole "mathematical universe" thing seems like practically the definition of premature speculation! It's no more useful than ancient Greek philosophers speculating on the nature of the elements. Those other people over there drawing circles in the sand, now, they're getting somewhere.

daxfohl 9y ago

Wasn't Einstein'srelativity a product of intuition rather than of calculation? And, intuition of processes that far exceed the bounds that Darwinian evolution would have prepared us for? If we all just shut up and calculated, we'd still be doing Lorentz transforms near speed of light "just because they work", right?

usnigg 9y ago

http://backreaction.blogspot.ch/ http://backreaction.blogspot.ch/

js8 9y ago

I like the idea that the reality is just mathematics, I thought about it independently, but it gets weird really fast. I yet to have read this essay, but I had two thoughts: 1. Similar to Turing completeness, actual physical reality might not matter for the internal observer. All Turing-complete machines can calculate same things and observer from the inside of the machine cannot determine on what hardware the machine is implemented. Similarly, this can be case for observers within our universe (humans). So maybe there is, at a high enough level of abstraction, only one mathematically possible universe. 2. The mathematical theorems are shortcuts to computation. For example, I can add two numbers on a computer, or I can use a theorem, which will tell me only some property of the result, but avoids having to do the actual calculation. So if universe is just a computation, one could avoid actually computing it in certain cases by having a theorem that would tell them enough information they need. In other words, you don't actually need to simulate anything in order for objects in simulation to "exist". So there is no need for actual hardware on which the universe runs to exist. Addendum: So I read the essay, and I disagree that the multiverse is necessary. As I already stated in point 1, it can be the case that all possible mathematical realities are the same at high enough level of abstraction.

hyperion2010 9y ago
There is a seriously interesting question here about the information content of the theorem and whether it is equivalent to the Shannon limit on compressing the 'code' needed to run the computation. I bet someone published on this years ago but I haven't seen it phrased in quite the way you have it in point 2.
carry_bit 9y ago
A concrete example of #2 is the hashlife algorithm. Intermediate steps can be skipped in a region of space, yet since it implements the CA nothing inside can notice that the computation didn't actually happen.
Barrin92 9y ago
If you like the idea of thinking about physical space in computational terms give one of the classics a try, Calculating Space by Konrad Zuse.

gfodor 9y ago

Haven't read the paper but I'm assuming this is the same concept (likely more digestible for the layman) in his book Our Mathematical Universe, which I highly recommend!

lou1306 9y ago

I'm a philosophy noob, but isn't this very similar to Berkeley-style immaterialism?

Koshkin 9y ago
Not really. Berkeley's idealism contests the objectivity of reality, while here we are presented with another view of the foundation of the (objective) material world.
- lou1306 9y ago
  This is true, but: > When we derive the consequences of a theory, we introduce new concepts — protons, molecules, stars — because they are convenient Couldn't matter itself be just another "human", "convenient" concept? Indeed the author writes: > our external physical reality is assumed to be purely mathematical Why should a purely mathematical reality also be material? Doesn't this lead to the logical conclusion that ideas, err, maths is "all that there is"? Sure, in this case ideas are objective and not subjective, but Berkeley's "mind of God" argument also leads to an objective, albeit immaterial reality, doesn't it?
  - Koshkin 9y ago
    1) But convenience is objective! (It is not convenient to sit in a chair that is hanging upside down.) Which is why matter, even if it is seen as a "convenient" mental construct, is still objective. 2) Materialism and "objectivism" are closely tied together. Mathematics consists of abstractions (or models), and those reside, in particular, in our mind; being "convenient", these abstractions are therefore reflections of what lies outside, and so - almost by definition - the objective world itself, matter, cannot possibly be built from math. (Besides our mind, "abstractions" can also reside in other, inanimate, objects; regardless, this still means that some part of matter merely reflects, to a degree, some other part).
  - otakucode 9y ago
    >Sure, in this case ideas are objective and not subjective, but Berkeley's "mind of God" argument also leads to an objective, albeit immaterial reality, doesn't it? No, it leads to a subjective inability to know anything at all. It's a difficult philosophical view to talk about, because the people espousing it can not possibly actually believe it and usually don't even understand it. The non-existence of a shared, knowable, objective reality means that, even in a totally solipsistic model, inductive reasoning can not be applied. You can not assume that attempting to breathe in the next moment will extend your life rather than extinguish it. You can not conclude that the air in your lungs will not suddenly become ants. You certainly can not leap to the extensively complex chain of reasoning necessary to believe that between this sentence and the next the English languages will turn into French. It introduces aggressive, pervasive, impossible-to-dismiss inconstancy in all things, including ones self. A true believer in a subjective universe would remain still in their bed, doing nothing, until they died. Any action whatever beyond automatic biological functions instantly betrays their belief that they know their body, know the environment they are in, and know how the mental activity necessary to cause the action will elicit a change in the environment which is not likely to destroy them. And that is a thing they can never know in a subjective universe. The 'mathematical universe' on the other hand is simply discussing how the shared, knowable, material universe we live in came to be or is best understood. I happen to think it is factually wrong on a few points (the author just off-hand claims that if we had a big enough supercomputer we could exactly calculate the future development of the universe which is fundamentally untrue for one) but it certainly doesn't amount to subjectivity.

1001101 9y ago

The author expands on this concept in his 2014 book (~7 years after the publication of this paper), Our Mathematical Universe: My Quest for the Ultimate Nature of Reality. I've read it, and I would highly recommend it to my fellow HN'rs. I found his multiverse classification scheme [1] very compelling and something that finds its way into my worldview and a lot of my thinking. [1] https://en.wikipedia.org/wiki/Multiverse#Max_Tegmark.27s_four_levels https://en.wikipedia.org/wiki/Multiverse#Max_Tegmark.27s_fou...

danharaj 9y ago

Which mathematics?

Koshkin 9y ago
The one of homotopical type.

aj7 9y ago

That was always Feynman’s attitude, obscured by the strong human layer he ran.

Koshkin 9y ago
Didn't Feynman think that the world was essentially void?

jcmoscon 9y ago

Math works because reality works. Not the opposite. Reality is bigger than math. Math is just a tool that models reality.

kaffeemitsahne 9y ago
..and with one simple statement, HN user jcmoscon forever put an end to this age-old debate.
- Koshkin 9y ago
  Still, if you think about it, that statement is very deep (and is probably true).
- Avshalom 9y ago
  Well no, Samuel Johnson and a good sized rock put an end to it centuries ago, jcmoscon is just trying to remind people.
hackinthebochs 9y ago
On the contrary, reality works because reality cannot contain contradictions. But the set of possible structures without contradiction is a superset of what is physically realizable. And mathematics is just discovering and cataloging the set of structures without contradiction. So reality works because math works.
- kaiwen1 9y ago
  In _Road to Reality_, Roger Penrose has a diagram that illustrates the Platonic world of mathematics fully encompassing the physical world, and the physical world fully encompassing the mental world (since minds are physical), and the mental world fully encompassing the Platonic world of mathematics. Although each encompasses the other in its entirety, each depends only on a small part of its predecessor. Penrose says "...the Platonic world may be the most primitive of these three, since mathematics is a kind of necessity, virtually conjuring its very self into existence through logic alone."
- deleted 9y ago

vorg 9y ago

The author asks how we could use the mathematical equations which describe all of reality to compute the frog’s view of the universe — our observations — from the bird’s view. Wouldn't we also need to include our self-awareness in those observations, which would make the computation impossible according to Turing's halting theorem, or is it Godel's?

jxy 9y ago
The basic result from Turing/Godel is that you don't need any self-referential. As long as it is an enough powerful formal system, it is always incomplete from day one. There will always be some mythical things you cannot compute. There will always be some universes you cannot prove their existence or nonexistence.

jcelerier 9y ago

> For example, a sufficiently powerful supercomputer could calculate how the state of the universe evolves over time without interpreting what is happening in human terms. Hasn't it been understood for a long time that quite a bit of things were entirely non-deterministic ? eg radioactive decay. You could compute a possible state, but the universe is a sequence of state spaces.

bitL 9y ago

Alright, why not go one step further and make Universe computational instead? (i.e. state is important in von Neumann-style instead of meaningless in classical math-style; or algorithms vs formulas)

eref 9y ago
Tegmark's Mathematical Universe Hypothesis (MUH) is actually pretty much a computational theory of the universe. It basically makes the assumption that there exists somewhere the simplest process imaginable: One that counts through and runs all mathematical formulas which includes all programs, which, in turn, includes all (computable) universes. This assumption explains why our universe is oddly complex: If all universes exist, then a weird one such as ours merely exists because all universes do. This line of reasoning is also called the "anthropic principle". I think, Tegmark does not talk about computations per se because the hypothesis is agnostic about what kind of computer our universe is. It could, for example, be a geometric computer in which the position of objects can be determined with infinite precision (i.e. using the real numbers). Such a computer would be strictly more powerful than a Turing machine or equivalent (i.e. all computational models that can be described and run inside a Turing machine). Enumerating the space of all formalisms (i.e. the domain of mathematics) is as agnostic as you can be.
- gfodor 9y ago
  I didn't take Tegmark's argument to be a computational theory. I took it to be an inductive one based upon the trend of the physics over the last decades towards the more and more generified multiverse theories. Ultimately, you land on a model where all consistent mathematical representations of a reality exist. At that level of abstraction, there is no where to go: it is the limit of the induction. There's a rather satisfying sense of self-evidence about it, but that doesn't mean it's falsifiable or true. IIRC there are no arguments about computation or what underlying medium(s) are involved, it seems like a separate question.
  - eref 9y ago
    I am unsure now whether he describes the set of mathematical objects by enumeration and evaluation. He definitely describes the set of universes this way, as being “run”, so it is a computation. But you are right that he never discussed the substrate that they are run on. He assumes the existence of that enumeration to be axiomatic, like an uncaused cause, somewhere in the space of all mathematical objects that simply exist. I guess, if we had a proof that all non-computable formal systems are inconsistent then we could also skip one layer and assume the enumeration of all programs as axiomatic in order to be maximally agnostic.

lainon 9y ago

Here's something more philosophical concerning the mathematical universe hypothesis: http://shelf1.library.cmu.edu/HSS/2015/a1626190.pdf http://shelf1.library.cmu.edu/HSS/2015/a1626190.pdf

Santosh83 9y ago

An interesting article addressing this from Sabine Hossenfender: http://backreaction.blogspot.in/2007/10/mathematical-universe.html http://backreaction.blogspot.in/2007/10/mathematical-univers... Her more recent article (review of Max Tegmark's book) also comments on the same subject: http://backreaction.blogspot.in/2017/11/book-review-max-tegmark-our.html http://backreaction.blogspot.in/2017/11/book-review-max-tegm...

westoncb 9y ago
This (the universe is 'made' of math) seems to me like the most egregious error in the general outlook of contemporary scientifically-minded people—and one I made myself for many years. If you really consider the issue closely, you'll see it depends on not overlooking certain dark corners of it which you can sort of ignore and assume to not contain contradictions—but they're there! And once you see them, it's hard to unsee them: what in the world would it mean for a desk to 'made' of mathematics—it's nonsensical. I think programmers are especially susceptible to this mistake because they're used to using mathematical descriptions to generate things, but even there, the idea that the math is doing any generating is a bias of our perspective since we're focused on the math; what's actually going on is a physical process involving silicon and plastic and metals etc., which can also be described mathematically, but the leap to being math is an unfounded step which could probably be predicted by the fact that that's where our focus lies. It also begins to look kind of silly when you consider how human math is: it's an outgrowth of an evolved mental faculty. Cognition is certainly impressive relative to the other things we and various animals are able to do—but it's probably still in the same category of our other faculties like the olfactory, visual, tactile etc.—and unlikely to be involved in the literal fabrication of the universe.
- gfodor 9y ago
  Have you read the paper or the author's book? Because you're distilling a rather rich and refined argument down to a simplistic description. The author doesn't argue the universe is "made" of math.
  - 9y ago

otakucode 9y ago

A purely mathematical universe immediately brings to mind a question: Since chaotic systems evolve in a way provably intractable to calculation, how could any system based solely upon calculation address, well, almost everything that exists?

klank 9y ago
That's an odd definition of chaotic systems you are using. It's not so much that chaotic systems can not be calculated but instead that enough precision of starting conditions can't be obtained to ensure an accurate result (for example, highly non-linear systems). The double pendulum system is a perfect example where computation is simple yet the system is still chaotic.
- otakucode 9y ago
  I think it's an odd definition you are using. Chaos depends upon the sensitivity to those initial conditions, not what those initial conditions ARE. You do not need actual physical objects to see this. Conjecture that a double-pendulum system starts in a perfectly real-number-valued mythical reality where spacetime is perfectly continuous (a fiction). Now predict what state the system will be in at a bunch of 'close together' states. Then try to do some 'closer together' states. See if you can ever get them to line up. Do it on pencil and paper, of course, since you want to capture the full real-number-infinity effect. Of course, you can't write down the real-number result with infinite precision, can you? You'll have to chop it off at some point. That's OK. That's what error bars are for. Just keep going and keep track of your error. Oops, you'll find that your error grows larger than the prediction itself with remarkable speed! THAT is the chaos. It doesn't matter if you measured the starting conditions with literal infinite perfect precision. The error bars on your predictions are going to explode like a busy beaver function. You're not going to catch that. Nothing can. (I imagine the actual physical limit would be that your predictions can be accurate within a distance equivalent to the radius of a sphere expanding at the speed of light from the initial position but I haven't actually read anything suggesting that, it just seems sensible.) Then you've got stupid simple cellular automata like rule 31 1D systems. Totally discrete. Totally basic. No precision even needed anywhere. And yet, you can't predict it. Not without accumulating an error greater than the state in the system itself if you skip even the slightest act of computation necessary to evolve the system as a whole from one step to the next.

followmeon 9y ago

If our reality consists of mathematical programs and the multiverse theory holds, then the theory of everything is a simple incrementing binary counter. Every possible binary string is ran on a Turing machine in a different universe. The halting problem (endless loops) is sidestepped by being contained to a single universe. Also, apparently our universe got lucky as we get to live in an ordered universe, and not dissolve into nothingness due to a bug in our universe's binary code. Of course, like "normal" mathematics, "reality" mathematics should also be incomplete. Furthermore, there is no way to know everything of the universe one is a part of, as this requires at least 1 bit more than there are available bits in the universe. But this still allows for the possibility of the universe itself being an inference machine. Edit: Then again, I believe that mathematics is a mental construct, mathematics does not exist without brains, or if it really exists external to human minds, it exists in all possible forms: Humans do not discover maths as a whole, they cast a local net of subjective observation over it, and keep the things that make sense to our brains and are consistent within the current framework. Again, this does not preclude the possibility that our mathematical reality exists inside the mind of a giant supercomputer.

agumonkey 9y ago
can we rollback that tape ?
- shruthijose 9y ago
  it somewhat more complicated
Houshalter 9y ago
I think you've confused the representation with the computation. Sure maybe that binary number could represent a universe state in some encoding. But there's no computation happening to it. It's static. >I believe that mathematics is a mental construct, mathematics does not exist without brains Brains evolved through natural selection to survive in the universe we exist in. Ideas similarly "evolved" to be good at modelling the universe. If mathematics didn't exist outside of our head, it would be useless. If there are two apples on the ground, and two more fall of a tree, then there are four apples on the ground. This is true whether or not a human is there to observe it and count them.

skc 9y ago

Theoretical physicists are my favorite people. They all sound slightly insane and I love it.