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Space Science

Can the Multiverse Be Tested Scientifically? 147

astroengine writes: Physicists aren't afraid of thinking big, but what happens when you think too big? This philosophical question overlaps with real physics when hypothesizing what lies beyond the boundary of our observable universe. The problem with trying to apply science to something that may or may not exist beyond our physical realm is that it gets a little foggy as to how we could scientifically test it. A leading hypothesis to come from cosmic inflation theory and advanced theoretical studies — centering around the superstring hypothesis — is that of the "multiverse," an idea that scientists have had a hard time in testing. But now, scientists at the Perimeter Institute for Theoretical Physics, in Ontario, Canada have, for the first time, created a computer model of colliding universes in the multiverse in an attempt to seek out observational evidence of its existence.
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Can the Multiverse Be Tested Scientifically?

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  • by Rosyna ( 80334 ) on Sunday July 20, 2014 @04:39AM (#47493023) Homepage

    String theory is math, not science.

    • by jd ( 1658 ) <imipakNO@SPAMyahoo.com> on Sunday July 20, 2014 @05:12AM (#47493105) Homepage Journal

      It's testable, it's measurable, it's repeatable, it's capable of prediction. it's either the simplest model that meets these requirements AND produces correct predictions, OR it is not.

      Therefore it is science.

      Maths is a science, for the reasons given in the first line. Science is a mathematical system, because ultimately there is nothing there, just numbers. (See: Spinons and other quasiparticles.)

      • Re: (Score:2, Informative)

        by Anonymous Coward

        > Maths is a science
        > it's measurable, it's repeatable, it's capable of prediction

        Nope. Measure -1?

        Math isn't knowledge, it's a model. Models aren't science, they are a way to predict it.

        • Uh, yeah, we can measure -1. The charge of an electron. The distance along the x-axis that I travel when I walk one meter west. The effect on a wave when it encounters an identical one 180 degrees out of phase.

          And if negative numbers worry you, this will blow your mind: by all indications, the way things really work, at the quantum level, is unavoidably governed by complex numbers. Don't let that "imaginary" label fool you...those bastards are really real, too. Sorry.

          • Re: (Score:2, Informative)

            by Mr. Slippery ( 47854 )

            Uh, yeah, we can measure -1. The charge of an electron. The distance along the x-axis that I travel when I walk one meter west. The effect on a wave when it encounters an identical one 180 degrees out of phase.

            Not at all. None of those things "are" -1. They are observable phenomena that we tag with the human invention, the word/concept, "-1". Mathematics is not an aspect of objective observable reality, it is a language that we have found useful for describing our observations.

            • You're right of course. I took the AC to mean that negative quantities don't exist objectively, which is a different issue. I may have misunderstood.

            • by mark-t ( 151149 )
              I think you'll have a considerably harder time convincing yourself that negative numbers don't exist once you finally realize that it's subtraction that doesn't "really" exist.
              • Except subtraction existed long before we had the concept of negative numbers.

                Gronk the caveman knew if he had two deer, and gave one to Grue, he had one deer. He most certainly wasn't adding -1 deer.

                • by mark-t ( 151149 )
                  Zero wasn't thought of until a few centuries BCE... that doesn't mean it isn't less fundamental than multiplication, which was thought of long before it.
            • by Zero__Kelvin ( 151819 ) on Sunday July 20, 2014 @12:33PM (#47495139) Homepage
              An apple "isn't" 1 either. So by your definition nothing is real, which actually turns out to be true. See also Maya [wikipedia.org]. All science is a model. Our very perception is but a model. "essentially, all models are wrong, but some are useful" [wikipedia.org]. Is math useful? OK, then, it's as real as it gets.
              • by 7-Vodka ( 195504 )

                Your logic sounds flawed.
                All science are models, all maths are models, therefore all maths are science?

                Nay.

                • My logic sounds flawed because you have just perfectly expemplified my point. I stated something. You perceived me to have stated something completely different, and then made the mistake of thinking your flawed perception was reality. You then projected your inadequacy onto me. I in fact said nothing even remotely close to what you repeated back.

                  That being said, it has already been stated quite clearly in this thread: "It's testable, it's measurable, it's repeatable, it's capable of prediction." In o
                  • Too bad you needed words to make your point about math. Maybe English is even more science than math.
                    • two times seven is fourteen.
                      7477 6f20 7469 6d65 7320 7365 7665 6e20 6973 2066 6f75 7274 6565 6e2e

                      "Too bad you needed words to make your point about math."

                      Your problem seems to be that you don't understand what does and does not constitute a word, and how they relate to math.

      • by Pino Grigio ( 2232472 ) on Sunday July 20, 2014 @06:05AM (#47493251)
        No, it's a computer model. A compute model is often (in engineering for example) a conceptual representation of real entities. However in many cases the model is more a conceptual representation of the biases and assumptions of the people who made it, being unreal in that sense. It isn't science and math isn't science either.
        The idea that ultimately there's nothing here indicates (though you might not know it) the presence of what Cairns-Smith called "the bomb in the basement of modern physics" and the difference between the thing in itself and the thing as it appears. Physics is good for the latter but has nothing to say about the former.
        • This hinges on if you believe in one of several mathematical universe hypotheses, or not. As for Cairns-Smith's "bomb" the "feelings" argument presented for it by Cairns-Smith becomes meaningless if feeling, and emotions are simple mathematical constructs of the brain.
          • "Let's be blunt, it's a nasty game" (says The Hitchhikers Guide to the Galaxy), "but then anyone who has been to any of the higher dimensions will know that they're a pretty nasty heathen lot up there who should just be smashed and done in, and would be, too, if anyone could work out a way of firing missiles at right angles to reality."

        • by DRJlaw ( 946416 )

          No, it's a computer model. A compute model is often (in engineering for example) a conceptual representation of real entities. However in many cases the model is more a conceptual representation of the biases and assumptions of the people who made it, being unreal in that sense. It isn't science and math isn't science either.

          But it is. Both.

          You've confusing hypothesis with observation. This does not purport to be observation. This is an element of the hypothesis -- identifying what sort of tests and obse

      • by zr ( 19885 )

        Math most certainly isn't a natural science, since it doesnt depend on nature for evidence.

        Math most certainly is a science since it does depend on evidence same as any other discipline based on the scientific method.

        I modded Rosyna's commend insightful because while technically somewhat inaccurate, she (or he) is making an insightful point that string theory is at this stage of its development is much more about math than about empirical evidence.

        • Math does not depend on evidence. Math depends on logical proof. There's evidence that Goldbach's Conjectore (that all even numbers > 2 can be represented as the sum of two primes) is true, but there isn't a proof.

          Science doesn't depend on logical proof, but rather evidence. We've had very good mathematical models that failed under some circumstances. In science, people would be treating Goldbach's Conjecture as if it were true (if it actually mattered in some science).

      • No it's not testable. I don't know where you get the idea that it is.
      • >Maths is a science

        Um, no. There's a reason why you get a BA in Math, not a BS.

        Math is an exemplar of a priori thinking. You can literally do math in your head by just picking some starting axioms and deriving from there, with no reference to the outside world.

        Science is an exemplar of a posteriori thinking. You make empirical observations about the world, generate hypotheses, and see if the evidence matches the model.

        • "Maths is a science"

          Um, no. There's a reason why you get a BA in Math, not a BS.

          Well, I'm going to call BS on that one.

          I know numerous people with a BS in Mathematics. In fact, I have one. I've only seen the mathematics department as part of the science department, and I don't know anybody with a BA in mathematics.

          If you are claiming mathematics isn't a science, then I'm going to say you're full of it.

          You know what's not a science but uses a lot of math? Economics, which is 3 parts ideology and 1 part ma

          • I don't know anybody with a BA in mathematics.

            Hi! There are a few of us, but most math bachelor's degrees are BSs.

            I don't classify mathematics as a science. Coming to conclusions in mathematics is very different from coming to conclusions in science.

          • >You know what's not a science but uses a lot of math? Economics, which is 3 parts ideology and 1 part math.

            It sounds to me like you're running on three parts ideology and one part math.

            Economics is actually very much a science! They make empirical studies of the world, and test them to see if they hold up.

            Math is very much not a science.

            • Economics is actually very much a science! They make empirical studies of the world, and test them to see if they hold up.

              No, they make wooly models about how they believe the economy works, perform math which has terrible assumptions and overly huge margins of error, and pass it off as objective fact.

              How you interpret economics is dependent on how you want to believe economics works. It is not an objective science in any sense of the word.

              And it never has been.

              Increasingly, some economists are starting to

      • In what way is String Theory testable or measurable?

        As far as I understand, it's complete conjecture, exists outside of anything you could actually test ... and therefore fails the science test.

      • String theory is testable, eh?

        So, how'd the tests come out?

        Crickets. All I hear are crickets.

    • Math is a science.

      • by pigiron ( 104729 )

        No. Math is an abstraction. Science is about measuring and explaining actual phenomena. Science may use math to make approximate models of physical reality but is not reality itself.

        • by Anonymous Coward

          Math is one field of applied philosophy. If you try to make it into something else you are doing yourself a disservice.

    • by ledow ( 319597 )

      My university had a school of mathematical sciences, a school of physical sciences, and a school of computer sciences.

      If you think that all three are not only completely separate but also not interchangeable in places, then you haven't been taught enough science (of any kind) for an opinion to have much worth.

      As a hint, I'm not a physicist. I flunked the physics module that I was required to do as part of my Mathematics & Computer Science degree. I have no need to defend physics. But saying that a th

  • No (Score:4, Insightful)

    by Anonymous Coward on Sunday July 20, 2014 @04:56AM (#47493055)

    Can the Multiverse Be Tested Scientifically?

    You can test specific hypotheses related to how the parts of a multiverse might interact, but no you can't test the general concept of a multiverse since there's nothing inherent to it that requires any detactable phenomena.

  • Yes, quantum suicide [wikipedia.org]. The idea is if you attempt to kill yourself, your consciousness persists only in the subset of universes where the attempt fails, and can become justifiably suspicious that, in its own experience, every effort prove ineffective.

    However, I think it is a bit small minded to use this only to test the muiltiverse hypothesis. Given that it's true, why not build a huge robust death chamber which you activate based on, e.g., whether or not you win the lottery, whether or not you quantum tun

    • by jd ( 1658 ) <imipakNO@SPAMyahoo.com> on Sunday July 20, 2014 @05:14AM (#47493113) Homepage Journal

      Wrong multiverse theory. And, indeed, wrong experiment. In fact, the wrongitudinal level of your post is so extreme that it should really be on K5.

      • While the idea was developed with a mind toward the quantum multiverse, the result is effectively the same for any multiverse model which allows for infinite universes. It doesn't really matter whether they are the branching kind or the spatially separated kind.

    • If I built a similar chamber and got a different lottery number, would I be guilty of causing your 'death'?

      • According to the theory, you could *both* win the lottery, because the universe would split into two copies.

        • That's objective and not the subjective outcome though. In each of our worlds one would be rich and the other would be dead.

    • by Patch86 ( 1465427 ) on Sunday July 20, 2014 @06:09AM (#47493263)

      Correct me if I'm wrong, but this would only "prove" the existence of (that variety of) multiverse in a very small subset of universes.

      So, let's say I try to poison myself with a pill from a bottle containing 99 cyanide pills and 1 sugar pill. There is a 99% chance I'll die, and a 1% chance I'll live. So in 1% of all universes, I live. I repeat the expriement multiple times, until only 1 in 1 million universes has a surviving me in. That means that in 0.0001% of universes, a very smug version of me is winning a Nobel prize for proving the existence of the multiverse. In 99.9999% of universes, I am dead and nothing has been proven except that I really shouldn't be allowed access to the lab's supply of cyanide pills.

      • by Anonymous Coward

        I repeat the expriement multiple times, until only 1 in 1 million universes has a surviving me in. That means that in 0.0001% of universes, a very smug version of me is winning a Nobel prize for proving the existence of the multiverse

        No. All you'll have proved, even in those 0.0001% of universes, is that every so often a one in a million chance pays off.

        • by Anonymous Coward

          Yeah, suppose I have a million sided dice (impressive but maybe it's a virtual dice, anyway we're all convinced it's a fair dice) and I say "I'm going to roll exactly 1,000,000 on this dice", so I roll it and it comes up 1,000,000. You might think "I wish I had luck like that" or you might think "it's obviously rigged somehow" but you're probably not going to think "wow, that proves there's a multiverse!". You can make a bigger dice, 1 trillion sides or whatever, it still doesn't demonstrate that there's a

    • The problem is that you would probably have to lock the complete universe into your death chamber, because Occam's razor says that the universe has *one* soul.

      • Occam's Razor says no such thing. What, prey tell, is simple about the idea of one soul? How does it get distributed? Does it grow or shrink with the life and death of living being? There's nothing simple about the proposition it all. If we assume at least one soul then Occams's Razor calls for a one to one correlation of living beings and souls. Indeed, absent the assumption that there is at least one soul, Occam's Razor, if it said anything at all on the subject, would say that there are zero souls.
    • a rather foolish test even if the multiverse exists, since you may simply die in all universes. Over one hundred billion humans have existed since 50,000 B.C. but almost all of them are dead. Most humans are dead. One more going into the normal state of most humans doesn't matter and seems very likely.

      • by Altrag ( 195300 )

        Actually, thanks to our exponential growth explosion over the last couple hundred years, there's more humans _alive_ today than in all previous history. Meaning there has been less than 14 billion humans to ever live.

        Pretty sure most of them still eventually die in all universes though, unless there's a universe where humans are legitimately immortal and not just statistically unable to kill themselves.

    • Seems no different to flipping a coin until you get heads. When you flip heads, stop.

      Not quite sure how it provide evidence of anything, though. Someone who flips 10 tails in a row either a) has found themselves in a 10-tail universe with a probability of 1/1024 or b) happens to have flipped 10 tails in a row in the one universe which exists, also with a probability of 1/1024.

  • Multiverse theory (Score:5, Informative)

    by jd ( 1658 ) <imipakNO@SPAMyahoo.com> on Sunday July 20, 2014 @05:08AM (#47493093) Homepage Journal

    There are many multiverse theories and they can all be tested.

    Many Worlds: The theory that there are no real "probability waves" in QM, merely overlapping realities that diverge at the time the "waveform" collapses.

    This is an easy one. Entangled particles operate using the same physics as wormholes. If one of the entangled pair is accelerated to relativistic velocities, say in a particle accelerator, they will not exist in the same relative timeframe. It would seem to follow that if Many Worlds is correct, one of the particles will be entangled with multiple instances of the other particle, which would imply that every state would be seen at the same time. If the options are left spin and right spin, you'd see an aggregate state of no spin even if no spin isn't a physical possibility. And seeing something that doesn't exist either means you're in a Phineas and Ferb cartoon or Many Worlds is correct.

    Foam Universe: This is the sort described in the article.

    Yes, impact studies are possible, but they're only meaningful if you have enough data and you can't possibly know if you do. You're better off trying to make a universe, preferably a very small one with a quantum black hole at the throat of the bridge linking this universe to that one. What you will observe is energy apparently vanishing, not existing in any form - mass included, then reappearing as the bridge completely collapses.

    Orange Slice Universe: This conjectures that multiple, semi-independent, universes formed out of the same big bang and will eventually converge in a big crunch.

    It doesn't matter that this universe would expand forever, left to its own devices, because the total mass is the total mass of all the slices. Although they are semi-independent, they interact at the universe-to-universe level. In this scheme, because there's a single entity (albeit partitioned), leptons cannot have just any of the theoretical states. The state space must also be partitioned. Ergo, if you can't create a state for an electron (for example) that it should be able to take, this type of multiverse must exist.

    Membrane-based Universe: This postulates that universes are at an interface between a membrane and something else, such as another membrane.

    However, membranes intersecting with the universe are supposed to be how leptons are formed, in this theory. The intersection will be governed by the topology of the membranes involved (including the one the universe resides on), which means that lepton behaviour must vary from locality to locality, since the nature of the intersections cannot vary such as to perfectly mirror variations in the shape of the membrane the universe is on. Therefore, all you need to do is demonstrate a result that is perfectly repeatable anywhere on Earth but not, say, at the edge of the solar system.

    • What happens when two multiverse theories collide?
    • "There are many multiverse theories and they can all be tested."

      Ha ha. You fooled a few people I guess.

      "and will eventually converge in a big crunch."

      OK, test the big crunch, I dare you.

    • Not really (Score:5, Informative)

      by aepervius ( 535155 ) on Sunday July 20, 2014 @06:33AM (#47493319)
      "This is an easy one. Entangled particles operate using the same physics as wormholes. If one of the entangled pair is accelerated to relativistic velocities, say in a particle accelerator, they will not exist in the same relative timeframe. (SNIP)"

      That's a misunderstanding of entanglement. There is not per see communication between the particle. When you have an entangled particle there is not one "communicating" the other that it is getting observed. What happens is that *both* particle form a single system with the specific property that when the spin of one particle is measured , the other particle has the anti spin state. Using all sort of relativistic trick on one particle will not do anything whatsoever because there is no communication to the other particle therefor frame of reference do nothing whatsoever.

      I dislike the analogy because it does not represent the true nature of QM entanglement , but think of this : you have a red ball and a yellow ball. Put one in a packet at random, keep another one hiddden in a safe on earth. Then send the packet at c speed somewhere. Openning the safe 10 years later will reveal the color of the safe ball and by extent the color of the packet ball no matter how far and that despite not being in the same frame of reference and 10 light years away.
      What happens here in entanglement is similar. There is no "teleportation" at c speed of the state of one to the others. Read up on bell's inequality violation.
      • "This is an easy one. Entangled particles operate using the same physics as wormholes. If one of the entangled pair is accelerated to relativistic velocities, say in a particle accelerator, they will not exist in the same relative timeframe. (SNIP)" That's a misunderstanding of entanglement. There is not per see communication between the particle. When you have an entangled particle there is not one "communicating" the other that it is getting observed. What happens is that *both* particle form a single system with the specific property that when the spin of one particle is measured , the other particle has the anti spin state. Using all sort of relativistic trick on one particle will not do anything whatsoever because there is no communication to the other particle therefor frame of reference do nothing whatsoever. [. . .]

        I'm mostly an untutored observer in the domain of Quantum Mechanics, and even I could see the beginner's flaw in the thought experiment for the Many Worlds model.

        My best guess is that the OP, understanding the concept of quantum entanglement does not involve communication between particles, is trolling everybody (or just having some fun).

        My worst guess is that the OP has gotten so old the OP's mental faculties are slipping and so the OP has achieved subjective immortality [wikipedia.org] but neglected to acquire eternal me

  • 1.could there be at least one multiverse with a God? 2. how about an MV where entropy decreases? 3. finally, one where Ilsa stays with Rick?
    • by Snard ( 61584 )

      I guess in multiverse #2, Isaac Asimov's story "The Last Question" would have been quite different.

  • Just use a piece of fairy cake.
  • Nope (Score:3, Interesting)

    by countach ( 534280 ) on Sunday July 20, 2014 @05:53AM (#47493219)

    I doubt that there is any possibility to observationally test such a thing, and even if some weird experiment can be devised, I doubt it would really do more than hint at, rather than prove other universes. After all, by definition these other universes are not part of ours, so we can't get at them.

    But let's just assume for a minute what is likely, that it can never be proven... Will the pointy headed boffins admit that it is not science, its... well.... something akin to religion really. About as scientific as any religious belief. In which case, shouldn't we really stop the whining between the scientific and religious factions? The scientists must admit that certain things could well be true that they can't prove, and that such things are worth talking about in the same breath as "real science", i.e. the things that pretty much everyone admits is true because it is science.

    Next time some pseudo intellectual proclaims "that's not science", just remember... neither is a lot of stuff that gets published under the name of science, and which nobody seems to complain about.

    • by JustNiz ( 692889 )

      >> After all, by definition these other universes are not part of ours, so we can't get at them.

      I'm not clear how your conclusion necessarily follows from your statement. I'll agree that we probably can't get to them just by changing physical location within our own universe, but that's about it.

    • by znrt ( 2424692 )

      shouldn't we really stop the whining between the scientific and religious factions?

      no.

      look: if there are scientists (as you say) with blind faith in unprovable beliefs, then they're not being scientific but religious. in this case you are asking to stop the whining between religious factions, and that would be for them to decide. i guess they won't, not because they're religious but because they're factions. religion is ok if (and only if) one limits it's application strictly to oneself.

      what these guys are speculating may seem weird but is the effort of theorizing possible explanations fo

    • Your second paragraph seems confused. If something can't be tested by observation, it isn't science. It still may be true, and people may believe it (two different statements). I'm not aware of many scientists that disagree with that. Pain is an example of something that is subjective but real. We can't test to see if soft tissue injuries hurt, but doctors use pain as a diagnostic measure. Try philosophy: the logical positivist position is that anything you can't determine with objective evidence is

  • by K. S. Kyosuke ( 729550 ) on Sunday July 20, 2014 @06:00AM (#47493241)
    Don't ask in the headline! You've ruined it for the scientists; now it can't be tested scientifically anymore.
  • You ask Dr. Walter Bishop.

  • OK, so they seek to find collisions . But there may be endless forms of universes that are incapable of collisions or having any transfer of information or energy between each other. Perhaps thoughts can create a universe that springs into existence in a sort of absolute elsewhere and continues on its own path.
    • by Altrag ( 195300 )

      More of a problem in this bubble universe idea of the multiverse is that even if it exists, its far more likely to be akin to particles in empty space rather than particles in a lattice as the video suggested -- that is to say, the chance that we would have been hit is probably extremely slim even if the underlying theory is correct.

      And an even bigger problem is.. if we find a multiverse outside of our universe.. then what's outside of the multiverse?

  • by motorsabbath ( 243336 ) on Sunday July 20, 2014 @09:52AM (#47494163) Homepage

    We need to get Michael Moorcock on the red courtesy phone in the lobby - stat!

  • by blincoln ( 592401 ) on Sunday July 20, 2014 @01:56PM (#47495577) Homepage Journal

    Ever since I read The Elegant Universe years ago, I've had a number of questions related to this (as I imagine many people have). This is the first time I've seen the topic discussed by professional scientists, though, as opposed to people like myself with a hobby interest in the subject or in science fiction (Alastair Reynolds makes use of it in one of the Revelation Space novels, for example).

    For the most part, it seems like String/M-Theory is very difficult (at best) to test using technology we have access to at present. But because it includes the idea of gravity being a force which can travel between branes, it's seemed to me and a few friends of mine that this would definitely produce some interesting effects in the real world.

    As the article discusses, there should be some subtle evidence of the effects of gravity from external sources on the large-scale structures of our own universe. I would think maybe even enough to at least partly explain "dark matter" and "dark energy", since those are basically the known matter in our universe behaving as if there were a lot more mass that we can't actually see (one set to hold relatively closely-spaced matter together, and the other to accelerate the expansion of the large-scale structures away from each other, if I understand correctly).

    A simple flatland-style analogy for "dark energy" might be that our universe is a sheet of paper which is intersected by a universe which is wrapped around into a tube shape or a torus. The gravity of the mass in that second universe pulls objects in our universe toward it, so for the part of our universe in the "eye" of the tube, they tend to accelerate away from each other. That's a vast oversimplification, but I'm not a physicist :).

    For "dark matter", the idea that's always stuck with me since reading The Elegant Universe is that maybe some/all of the most massive objects in our own universe - especially the black holes at the centers of galaxies - are caused by the same kind of cross-brane effect. If you have a bunch of matter clumping together in one brane/universe, and it exerts gravity which can cross into other branes, then it seems like it would create corresponding accretions of mass in other nearby branes. Basically, that what we perceive to be a roughly spherical/point object would effectively be the hyperdimensional equivalent of that same shape that would "pin" itself together across branes.

    Where I see this as becoming testable (and I could be wrong - again, I'm not a physicist) is that if this were the case, there should be examples of anomalous astrophysical objects and events, where the mass we observe does not line up with effects we also observe. For example, a stable neutron star suddenly flashing into a black hole when it passes too close (hyperdimensionally, of course) to a large mass in another brane. Another example might be a star or planet whose mass can't be reconciled with its observed size - e.g. maybe there is a planet the size of our moon, but which exerts gravity as if it were made entirely out of a material ten times as dense as uranium.

    I know that in the context of our own universe/brane, there's no way to pull matter out of a black hole (other than Hawking radiation), but assuming the "hyperdimensional singularity"-type thing I described above is accurate, would it be possible for the cross-brane components to separate (since they wouldn't actually be touching, just exerting gravity on each other)? If so, there might be even stranger observable effects, like neutron stars that "flash" into black holes, but then return to their former state when the mass in the other brane(s) is pulled too far away. IE they would "blink".

  • Comment removed based on user account deletion
  • by Anonymous Coward

    once you step too far outside the observable universe, there is simply a coke machine in the hall.

  • Somewhere there was a split between the multiverse that can and the multiverse that can't be demonstrated.

Every nonzero finite dimensional inner product space has an orthonormal basis. It makes sense, when you don't think about it.

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