1, 2, 3, Infinity

Is the universe finite or infinite?   That is, does it go on forever in at least one direction, or is there unlimited detail in at least one place?  This question has been asked as long as we’ve understood the concept of infinity, and many feel we are making substantial progress lately.  In particular, many cosmology papers consider the differing physical implications of infinite vs. finite space or time, and these papers are often reported on more widely as perceptibly changing our estimates about whether the universe is infinite.

But if we continue to frequently make perceptible progress on an ancient open question, that question cannot long remain open.  So the future will long remember our era as one that either settled the physical infinity question with 90% (or even 99%) confidence, or as one that was overconfident about its rate of progress.

I can see two kinds of evidence here.  First we gain evidence that the universe goes at least so far in particular directions, and has at least so much detail in particular places.  Second, when we compare finite versus infinite versions of our current best theories of physics and cosmology, one may appear to be conceptually simpler, while the other may seem more an awkward ad hoc modification.

But both of these forms of evidence seem to at best offer only very weak new evidence on physical infinity.  If, in the "1, 2, 3, infinity" spirit, we had a prior expectation that a universe larger than a certain size threshold was probably infinite, we should have long since passed that threshold.  Even a single continent that lasts a millennium is large enough to surprise most children.  And the modifications used to turn a finite theories into infinite ones, or vice versa, have so far required only modest changes to the conceptual simplicity of our best theories, with no clear pattern on which side tends to be favored. 

So it seems to me that overconfidence is the more likely explanation for recent perceptible changes in our expectations about whether the universe is physically infinite.  This question will likely remain open for a very long time, if it was not already mostly settled for rational minds a long time ago.  In our arrogance we think otherwise.

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  • http://profile.typekey.com/felix_typekey/ Felix

    Har, har. Just stand back and consider the statement, “I’m 99% sure the universe is infinite. Not 99.99999999999999999999999999999% sure, mind you, but, yep, 99%.”

    Oh, btw, does it count if the universe is infinite not in space, but in time?

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Felix, sure time counts as a direction in space-time.

  • http://amnap.blogspot.com/ Matthew C

    I think the popularity of infinite theories (in time, space, and universes), despite a lack of observable evidence, is largely due to the fine-tuning problem / anthropic principle. If you are going to be an atheistic materialist, you need all these infinities of unobservable universes / regions of the universe with an infinite random assortment of physical laws, or at least constants, in order to account for the characteristics of this one.

    For a similar reason the Everett Interpretation is practically de rigeur in certain circles, since that seems to be the main popular approach of thinking on how one might metaphysically postulate a “real-world” materialist ontology out of the observed unreality of quantum mechanics. Personally, “I have no need of that hypothesis”.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Matthew, you might need a very large universe to get what we see via selection effects, but an infinite universe is far far larger than what is needed for such effects – it is in fact infinitely larger than needed. Personally, I do have need of a “real-world” ontology.

  • http://manufacturedgods.blogspot.com/ LP

    Would you say ‘finite vs. infinite universe’ is basically a metaphysical question, not verifiable through scientific investigation? I think so, and I guess this is why, though the question will always remain open, it doesn’t present much difficulty for ‘rational minds’ anymore, as Robin notes.

    As far as picking the ‘conceptually simpler’ option for scientific purposes, this is a fine thing to do, as long as no one confuses it with ‘evidence’ for one or the other. Which hypothesis is simpler and more elegant (and therefore has an accompanying feeling of ‘truthiness’) changes, depending on what sort of data are available and what sort of math needs to be done.

  • rcriii

    I have two questions:

    Does “Substantial Progress” or “perceptible progress” have to mean that we are closer to an answer? Couldn’t it mean that we have discovered new possibilites/implications/tests, or eliminated some?

    Also, which cosmology is “conceptually simpler” (especially when you take into account the effort required to square one or the other with our observations and theories), finite or infinite? Actually, if I read you correctly Robin, neither really is.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    LP, I don’t find the distinction between “scientific” and “metaphysical” questions to be of much use.

    Rciii, by “perceptible progress” I had in mind a perceptible change in our degree of belief.

  • http://amnap.blogspot.com/ Matthew C

    You are right Robin, I should not equate “infinite unobservable universes” with “stupendously huge but finite number of unobservable universes”.

    I do find the question of an infinite vs. enormously stupendously huge but finite number of unobservable universes / domains less interesting than whether or not all postulated, hypothesized and forever unobservable entities actually exist — regardless of whether there are a googleplex of them or an infinite number. I’m afraid that caused me to misinterpret the gist of your question, my apologies.

  • Roy Haddad

    I lean towards “not infinite” for aesthetic reasons: if space and time were not quantized, I thinkk even the position of a single particle would contain an infinite amount of information.

  • joe

    If space is finite, but expanding at an increasing rate, as I believe we do have evidence of, then at any one point in time, the universe might be finite, but may increase in size infinitely. In this situation, is the universe infinite or finite?

  • http://jamesdmiller.blogspot.com/ James D. Miller

    Robin,

    You are suffering from overconfidence bias in assuming that the universe exists. Surely there is some chance that the universe is of size zero.

  • Hopefully Anonymous

    James, I didn’t read Robin as assuming that the universe exists. Think of the resources saved by not typing “apparently” after every statement that he can then use to maximize our mutual odds of persistence.

  • http://www.scottaaronson.com Scott Aaronson

    Robin, again and again in science we see discoveries that (1) don’t bring us any closer to answering to a particular question, in the specific sense of shifting our probabilities in the right direction, but (2) so completely transform how we think about the question that it’s hard to imagine an eventual answer not building on those discoveries. An analogy would be if you were struggling for decades to decode some ancient papyrus, and then at some point you realized you were holding the papyrus upside down. You still can’t guess better than before what the papyrus actually says — but doesn’t your realization represent “progress”?

    With regard to the finitude vs. infinitude of the universe, would you agree that, while human beings have been asking this question for thousands of years, we’ve only really understood what we meant by it for the last 90? (Namely: treating spacetime as a pseudo-Riemannian 4-manifold, is the manifold compact?)

    If you agree to that, then I’m going to press my advantage, and point to more recent discoveries in cosmology that I think similarly transformed our view of the question. What about the Big Bang, or dark energy, or the fluctuations in the cosmic microwave background? Conditioned on the finiteness question someday being answered, what would you say the probability is of the answer not building on these discoveries?

    Incidentally, conditioned on the finiteness question being answered, it’s of course much easier to imagine the answer being “finite” than its being “infinite.” But again, even if all we can ever do is make statements like “if the universe is finite, then it must be at least 1,000 times bigger than the Hubble radius,” to my mind that still represents some sort of progress on the original question. You might tell me that the difference between one Hubble radius and a thousand Hubble radii is not something an ancient Greek could readily appreciate. And I would respond: exactly! 🙂

  • http://profile.typekey.com/sentience/ Eliezer Yudkowsky

    James, if the universe doesn’t exist, it would still be nice to know whether it’s an infinite or a finite universe that doesn’t exist.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Scott, if we ever answer a hard question, there is a good chance we will use our best related theories of that time to do so. So to the extent we are improving our best related theories, we are making “progress” toward that hard problem. But that is about the weakest sense of “progress” you could find. And it would not perceptibly change our expectations about the answer.

    So while I agree that the concept of a compact manifold is conceptual progress, it is only the weakest possible progress toward answering the physical infinity question. I do think that an ancient Greek could understand one versus one thousand Hubble radii, if translated into their unit system.

  • Hopefully Anonymous

    Can anyone recommend or provide good explanations for why our brains perceive things as continuous and can conceptualize infinity? Also, is the concept of infinity a historical innovation (I think I read about a science/history book that claimed that recently), or like language, does it seems to be prehistorical.

  • http://amnap.blogspot.com/ Matthew C

    You are suffering from overconfidence bias in assuming that the universe exists.

    Please. No one is arguing that this universe does not exist.

    Some people question:

    1) Do the stupendous number of unobservable, untestable anthropic-unfriendly alternate universes / domains postulated by selectionist accounts of the actual observable universe we do see actually exist?

    2) Given the findings of quantum mechanics, does it make sense to talk about a physically “real” universe? Everett’s Many Worlds Interpretation of quantum mechanics postulates a physically real “waveform” that underlies the observable physical universe, but it makes no new predictions over any other version of QM. The alternatives to a “physically real” universe include Bostrum-style simulation, monistic idealism, “shut-up and calculate-ism” etc., which fit quite nicely with the experimental findings of QM, and do not require a hypothetical “physically real” waveform which brings nothing to the table except preserving the old ideas of a “physical” universe. I prefer to take the actual findings of QM as a given rather than attempting to explain them in terms of some kind of imagined “underlying” reality that might be more comfortable for our macroscopic, evolutionarily-derived ideas of how things ought to be — but delivering no new predictions.

  • Stuart Armstrong

    Given the findings of quantum mechanics, does it make sense to talk about a physically “real” universe?

    Yes. Translate “the size of the universe” into a broad collection of observables (built up from momentum, position, and other such observables). Then add a consistency requirement on these observables, to get our intuitive feeling of “size”. Then our observations fall into several categories:
    1) Our observables are consistent, and the universe has a definite size
    2) Our observables are consistent, and the universe has a definite minimum size, but no observed maximum
    3) Our observables are inconsistent; size is not a well defined concept for our universe

    For the moment, all our observations tend to show 2).

  • Stuart Armstrong

    But if we continue to frequently make perceptible progress on an ancient open question, that question cannot long remain open.

    I feel that the question “is the universe infinite?” is not a well posed scientific question. As far as I know, there’s no decent choice of prior on the size of the universe. Since every piece of data is consistent with a finite universe, it’s no surprise that the question is still open – different priors get different results, and we can’t choose among them.

    So we can get answers to “is the universe infinite, for this choice of prior?”, but not to the general question. That general question is probably metaphysical – in that it can’t be answered by science, only a priori or based on taste.

    Other issues involved: how do we know that size will always make sense in the universe (see my previous comment), and if it does, it may be modeled on other fields such as the hyperreals, or other more pathological sets.

    So it seems to me that overconfidence is the more likely explanation for recent perceptible changes in our expectations about whether the universe is physically infinite.

    I agree totally.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Stuart, whether or not the question is well post “scientifically” (whatever that means), it seems well-posed conceptually.

  • http://manufacturedgods.blogspot.com/ LP

    Stuart: you said what I was trying to say, with greater elegance. Thanks!

    Robin: Conceptually, yes, the question is well-posed. It asks a reasonable question about a concept we can grasp, if not articulate that well. But questions where no scientific validation is possible — questions that can’t be answered by looking around at the physical universe — are metaphysical: literally, ‘above physics.’ This doesn’t mean there’s anything necessarily wrong with asking them and debating them, but it does mean that they can’t be discussed in terms of scientific evidence or scientific progress.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    LP, you use the word “scientific” as if you thought it had more than a vague suggestive definition. It doesn’t.

  • Stuart Armstrong

    Rephrasing, with duplicates thrown in.

    Imagine an infinite line of boxes, each containing a ball. They represent the different sizes the universe can be. Most of them are black, and one of them (the actual “size of the universe”) is white – or there may be no white ball, in which case the universe is infinite.

    You arrive, and want to establish the size of the universe. However, before even arriving, you are secretly duplicated, and your duplicate placed in cold storage.

    You set up a reasonable prior estimate as to the size of the universe, “P”. Assume you have set P(infinity) = p, with 0 < p < 0.5. Then you start opening boxes, one by one. There will come a number "X" where after opening X boxes and seeing only black balls, you conclude the universe is more likely to be infinite than finite. At that moment, your duplicate is released from cold storage. The situation he sees is the same as what you had at the beginning - a infinite series of boxes, with possibly a white ball in them. Since he uses the same reasoning as you, he sets the prior "P" on the size of the universe. So while you have concluded, to your satisfaction, that the universe is most likely infinite, your duplicate argues it has probability p < 0.5 of being infinite. Who is correct? (note: the p<0.5 statement is not needed for this argument, but makes the example more elegant) Ways round this argument: if we have exterior evidence that fixes "P" for you but gives a different initial prior for your duplicate. For example, if you could open all the boxes in finite time (say an hour), then you can give them a cumulative probability based on the time taken to reach that box. And set the probability of infinity to the length of a second, arbitrarily. If your duplicate does the same, then you will agree. Or you could find that white ball. But in our universe, that hasn't happened. I think this addresses the issue: But if we continue to frequently make perceptible progress on an ancient open question, that question cannot long remain open.
    For those who set up their priors early, the question would no longer be open. For those who only set up their priors recently, the question would remain very open. This may explain why people feel differently about the issue.

    Stuart, whether or not the question is well post “scientifically” (whatever that means), it seems well-posed conceptually.

    It is. But what would constitute evidence for its answer, and to what extent, is not well-posed conceptually. For normal questions this isn’t so much of a problem, because extra evidence adds information. But here an extra black ball leaves the set-up exactly as before. As LP says, the question feels “metaphysical”.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Stuart, you are making many assumptions that you are not making clear to me, but the comments of this blog post are not the right place for long explanations of your analysis.

  • http://profile.typekey.com/sentience/ Eliezer Yudkowsky

    So while you have concluded, to your satisfaction, that the universe is most likely infinite, your duplicate argues it has probability p < 0.5 of being infinite. Who is correct?

    Stuart, this looks to me like classic Jaynesian Mind Projection Fallacy – if we are uncertain about a phenomenon, that is a fact about our state of mind, not a fact about the phenomenon. Probabilities are states of uncertainty (the universe itself is either one way or another), and states of uncertainty have no place outside of some particular individual’s mind. Where different individuals have different evidence, there is not anything the least bit surprising about them having different probabilities.

  • current grad student

    Stuart, I think that you have to be careful approaching this question from a bayesian point of view. Evidence against your estimated size of the universe cannot constitute evidence that the universe is infinite…only that under your prior, you think it is more likely that the universe is infinite. In your situation, evidence against the universe being finite can also be viewed as evidence against your modeling assumptions for the distribution of the size of the universe under the hypothesis that the universe is finite. Instead, any new evidence should simply be taken into account in your posterior to portray how large you believe the universe is.

    No finite amount of evidence should be able to constitute evidence that the universe is infinite… and I believe therein lies the dilemma with addressing this question with probability or statistics.

  • Stuart Armstrong

    Where different individuals have different evidence, there is not anything the least bit surprising about them having different probabilities.

    Here the two duplicates have the same evidence – simply the second duplicate judges some of the evidence irrelevant (the extra boxes already opened). The question is whether two situations – (situation A) and (situation B = situation A + one extra box outside situation A known to contain a black ball) – should be treated the differently. In the absence of some extra structure differentiating A and B, I don’t see why this should be.

    In fact, I can make the two situations precisely equivalent by adding an infinity of opened boxes containing black balls at the beginning (no longer equivalent with the size of the universe). Now the situations facing the duplicates are precisely the same, so they should come up with the same prior (at different times) and hence disagree (note: this is only true if the probability of the universe being infinite – of there being no white ball – is taken as non-zero).

    This is not so much a Jaynesian Mind Projection Fallacy, as an argument claiming there are no consistent priors for deciding the question “is the universe infinite?”.

    Stuart, you are making many assumptions that you are not making clear to me, but the comments of this blog post are not the right place for long explanations of your analysis.

    Can you give links to relevant websites? I’m not particularly enamoured with the results of my analysis, and would be quite keen to see it shown wrong.
    I made many assumptions about how the box model relates to the “size of the universe” model, yes. There are many ways of doing that, but I don’t think that really matters.
    Assumptions for the box model itself: two isomorphic situations should be given identical priors, and Initial Situation is isomorphic to Initial Situation + one extra box known to contain a black ball.

  • current grad student

    “simply the second duplicate judges some of the evidence irrelevant (the extra boxes already opened). ”

    There seems to be something fundamentally flawed in your argument if one is deeming evidence irrelevant. If one person judges the universe to be infinite and the other says “no, the evidence you used to make that determination is irrelevant and I’m not going to take it into account,” then why is it surprising that they have different answers?

    “This is not so much a Jaynesian Mind Projection Fallacy, as an argument claiming there are no consistent priors for deciding the question “is the universe infinite?”. ”

    There is something wrong with your claim on a deeper level. When we can’t even decide what would consitute proof of an infinite universe (perhaps impossible ?), then it seems especially silly to decide the question from the bayesian point of view. In your example, any evidence that you are taking as evidence for the universe being infinite is also evidence against your assumption of the distribution of the size of the universe in the case that it is finite.

  • Nick Tarleton

    Stuart, you’re saying that since (the boxes) and (the boxes A hasn’t opened) have the same cardinality, the boxes A has opened are irrelevant, right? This may be true, but it’s not like the size-of-the-universe situation, because in the size-of-the-universe situation, the boxes aren’t identical to each other – each one is numbered with a possible size of the universe. So A and B certainly see distinct evidence.

    More than that, you’re neglecting arguments based on physical law. If the physical laws we discover require the universe to be finite or infinite, well, then we know. Even if they don’t, they can still inform our choice of prior. (And, IIRC, the best arguments for the infinity of the universe are based on physical law as opposed to just observing the minimum size of the universe.)

  • http://manufacturedgods.blogspot.com/ LP

    Robin,

    Hmm. “Scientific” here means, “in accordance with the principles of the scientific method.” The scientific method is fairly well known: observe some phenomena, form a hypothesis about them, do some experiments to validate or invalidate the hypothesis, repeat. But actually, experiments are generally designed to invalidate the hypothesis, because ‘positive’ results might just mean that we haven’t checked in the one area of the universe that would invalidate the hypothesis.

    So here’s the problem: let’s say we form the hypothesis, “The universe is finite.” To verify this, we need to find a boundary somewhere, so we do experiments for a thousand years to look for one, but don’t find one. Does this invalidate the hypothesis? No, because maybe we haven’t looked in the right place yet. So instead, we form the hypothesis, “The universe is infinite.” Again, to verify this, we need to see if there are boundaries to the universe anywhere. We search for a million years and don’t find anything — does this validate the hypothesis? No, for the same reason. And a hypothesis that can’t be validated or invalidated by evidence isn’t scientific, because it’s not subject to the scientific method.

    Nick: While there are physical laws in the universe, we don’t know any. We only know models that tend to accurately describe observed phenomena. Since the models shift over time (sometimes dramatically), does this mean we ought to change our views on the size of the universe, along with these changes in model? If so, this was also my point in my original comment: pick whatever makes the math (modeling) work out better.

  • http://profile.typekey.com/snej/ Jens Alfke

    This post, and its discussion, are pretty absurd in that they make no reference to, nor do they correspond with, any current cosmological science. It’s pretty telling that, beyond referring to “many cosmology papers”, no one here has actually mentioned any specific papers.

    You folks can philosophize all you like about the scientific method, but among actual cosmologists (at least non-fringe ones) there’s no debate about the finite-ness of the universe. The Big Bang theory requires a finite universe that grew from a point singularity, at a finite rate, a finite time ago; and that theory has been so incredibly strongly confirmed by observation (like COBE) that it’s nearly universally (sic) considered correct.

    There were, in the past few decades, questions of the age of the universe, and whether it will keep expanding forever (meaning time is potentially infinite); both of those seem to have been mostly settled by now.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Jens, you are completely misinformed about modern cosmology.

  • Nick Tarleton

    Jens, to amplify: in an infinite Big Bang cosmology, space can still expand even if the universe is and always has been infinite. (To extend the classic analogy, think of an infinitely large loaf of raisin bread being baked. The raisins will still move apart from each other as the loaf expands.) Any region in the universe would have been much smaller in the past.

  • Aaron

    James, if the universe doesn’t exist, it would still be nice to know whether it’s an infinite or a finite universe that doesn’t exist.

    Are you a Catholic Jew or a Protestant Jew?

  • Stuart Armstrong

    each one is numbered with a possible size of the universe. So A and B certainly see distinct evidence.

    Of course. But the numbering doesn’t really help; lets assume the boxes are numbered in powers of two, in meters, starting at 2^0 = 1. After opening ten boxes, the next box is 2^10 m. Let me define a new unit, M = 2^10 m. Then the situation is the same as before.

    Of course, m and M are distinct – but we have no way of preferring one of them, apart from arbitrary choices.

    If the physical laws we discover require the universe to be finite or infinite, well, then we know. Even if they don’t, they can still inform our choice of prior.

    That used to be my argument. Turns out it’s pretty circular – it depends on the physical laws being the same everywhere – including in the unobserved portion of the universe. Getting that sort of result, all the way out to infinity, is at least as hard as showing an infinite universe in the first place.

    LP, I pretty much agree with your analysis. I’d say that “the universe is infinite” is half of a scientific statement – it’s disprovable if it’s wrong. But if it’s right, then there does not exist an experiment that can distinguish between it and its inverse “the universe is finite”.

  • Hopefully Anonymous

    How big is the observable universe in light years? How big is the suggested universe in light years? What is the tenable size (and also the observable size) of the universe in terms of lights years? quantized units of space-time? mass? particles? galaxies? stars? Planets? From our best knowledge. Inquiring minds want to know.