Theory on framework issues

Tuesday, January 1, 2013

19.0. Can infinite quantities exist?

1. The actuality of infinity is a paramount metaphysical issue.

Some major issues in science and philosophy demand taking a position on whether there can be an infinite number of things or an infinite amount of something. Infinity’s most obvious scientific relevance is to cosmology, where the question of whether the universe is finite or infinite looms large. But infinities are invoked in various physical theories, and they seem often to occur in dubious theories. In quantum mechanics, an (uncountable) infinity of worlds is invoked by the “many worlds interpretation,” and anthropic explanations often invoke an actual infinity of universes, which may themselves be infinite. These applications make actual infinite sets a paramount metaphysical problem—if it indeed is metaphysical—but the orthodox view is that, being empirical, it isn’t metaphysical at all. To view infinity as a purely empirical matter is the modern view; we’ve learned not to place excessive weight on purely conceptual reasoning, but whether conceptual reasoning can definitively settle the matter differs from whether the matter is fundamentally conceptual.

Two developments have discouraged the metaphysical exploration of actually existing infinities: the mathematical analysis of infinity and the proffer of crank arguments against infinity in the service of retrograde causes. Although some marginal schools of mathematics reject Cantor’s investigation of transfinite numbers, I will assume the concept of infinity itself is consistent. My analysis pertains not to the concept of infinity as such but to the actual realization of infinity. Actual infinity’s main detractor is a Christian fundamentalist crank named William Lane Craig, whose critique of infinity, serving theist first-cause arguments, has made infinity eliminativism intellectually disreputable. Craig’s arguments merely appeal to the strangeness of infinity’s manifestations, not to the incoherence of its realization. The standard arguments against infinity, which predate Cantor, have been well-refuted, and I leave the mathematical critique of infinity to the mathematicians, who are mostly satisfied. (See Graham Oppy, Philosophical perspectives on infinity (2006).) 

2. The principle of the identity of indistinguishables applies to physics and to actual sets, not to everything conceivable.

My novel arguments are based on a revision of a metaphysical principle called the identity of indistinguishables, which holds that two separate things can’t have exactly the same properties. Things are constituted by their properties; if two things have exactly the same properties, nothing remains to make them different from one another. Physical objects do seem to conform to the identity of indistinguishables because physical objects are individuated by their positions in space and time, which are properties, but this is a physical rather than a metaphysical principle. Conceptually, brute distinguishability, that is, differing from all other things simply in being different, is a property, although it provides us with no basis for identifying one thing and not another. There may be no way to use such a property in any physical theory, we may never learn of such a property and thus never have reason to believe it instantiated, but the property seems conceptually possible.

But the identity of indistinguishables does apply to sets of existing things (actual sets): indistinguishable actual sets are identical. Properties determine actual sets, so you can’t define a proper subset of brutely distinguishable things.

3. Arguments against actual infinite sets.

A. Argument based on brute distinguishability.

To show that the existence of an actual infinite set leads to contradiction, assume the existence of an infinite set of brutely distinguishable points. Now another point pops into existence. The former and latter sets are indistinguishable, yet they aren’t identical. The proviso that the points themselves are indistinguishable allows the sets to be different yet indistinguishable when they’re infinite, proving they can’t be infinite.

B. Argument based on probability as limiting relative frequency.

The previous argument depends on the coherence of brute distinguishability. The following probability argument depends on different intuitions. Probabilities can be treated as idealizations at infinite limits. If you toss a coin, it will land heads roughly 50% of the time, and it gets closer to exactly 50% as the number of tosses “approaches infinity.” But if there can actually be an infinite number of tosses, contradiction arises. Consider the possibility that in an infinite universe or an infinite number of universes, infinitely many coin tosses actually occur. The frequency of heads and of tails is then infinite, so the relative frequency is undefined. Furthermore, the frequency of rolling a 1 on a die also equals the frequency of rolling 2 – 6: both are (countably) infinite. But when there are infinitely many occurrences, relative frequency should equal the probability approached in a finite world. Therefore, infinite quantities don’t exist.

4. The nonexistence of actually realized infinite sets and the principle of the identity of indistinguishable sets together imply the Gold model of the cosmos.

Before applying the conclusion that actual infinite sets can’t exist, together with the principle of the identity of indistinguishables, to a fundamental problem of cosmology, caveats are in order. The argument uses only the most general and well-established physical conclusions and is oblivious to physical detail, and not being competent in physics, I must abstain even from assessing the weight the philosophical analysis that follows should carry; it may be very slight. While the cosmological model I propose isn’t original, the argument is original and as far as I can tell, novel. I am not proposing a physical theory as much as suggesting metaphysical considerations that might bear on physics, whereas it is for physicists to say how weighty these considerations are in light of actual physical data and theory.

The cosmological theory is the Gold model of the universe, once favored by Albert Einstein, according to which the universe undergoes a perpetual expansion, contraction, and re-expansion. I assume a deterministic universe, such that cycles are exactly identical: any contraction is thus indistinguishable from any other, and any expansion is indistinguishable from any other. Since there is no room in physics for brute distinguishability, they are identical because no common spatio-temporal framework allows their distinction. Thus, although the expansion and contraction process is perpetual, it is also finite; in fact, its number is unity.

The Gold universe—alone, with the possible exception of the Hawking universe—avoids the dilemma of the realization of infinite sets or origination ex nihilo.

(Edited July 25, 2013: Clarified in Section 2, last paragraph and other places, that the identity of indistinguishables applies to actual sets.)


  1. Hi,
    Sorry for the delay, I read the post and let it swirl around in my head for a few weeks. My main question is this:

    Does not the singularity, at the "start/end" of every universal cycle, itself represent inifinity?

    Another question, which I thought I ought develop, but failed to, is:
    Does Zero, represent the infinitessimal? If something can be infinitessimally small, doesn't that provide an argument of sorts for the inifinite?

    At which point I began to imagine a singularity as an infinite amount of nothing (infinitessimally small), existing for no (infinitessimally small) time, and I had no idea where to go with that...

    1. I wholeheartedly agree that countenancing actual infinitesimals commits one to actual infinities. I develop this idea in 19.2 ( )

      You make the excellent point that if the universe collapses into a singularity, the theory has reintroduced the infinities and infinitesimals it sought to avoid, resulting in the contradictions that I think in fact show that such "quantities" are incoherent. The logical conclusion is that the universe doesn't collapse into a singularity. Since positing singularities isn't something to welcome, this is all to the good metaphysically, although I don't know whether it is physically tenable.

      Zero and infinitesimal quantities are distinct concepts: to say there is really zero of something is to say it simply doesn't exist. A singularity involves infinitesimals because the universe must (somehow) exist at the same time as it has no real-number dimension.

      What your point brings home and that I hadn't thought of is that the Gold universe has the metaphysical advantage over the Hawking universe of potentially avoiding singularities and associated infinitesimal quantities. If this analysis is correct, it's a decisive advantage, but of course it must be squared with the physical evidence.

  2. I read the article on infinitesimals yesterday, after I posted the above comment, I'll definitely have to read that one a few times before it sinks in.

    A minor point, I think current observations are against a cyclical universe as to the best of our knowledge the rate of expansion (of the universe) is accelerating.

    There also seems to be a healthy debate as to whether a singularity has to be an actual infinity or not, I think this has to do with what happens when we apply relativistic laws to objects on the planck scales of length and mass, and planck time.

    1. But is planck time relevant if there's an underlying hidden-variables determinacy to q.m? I think we've discussed that before and tentatively concluded it probably isn't.

    2. If a quantum interpretation, with hidden variables were true (which I think you said may not be in line with the philosophy of science, and would entail infinities) planck time could be an epistemoloigical limit. Still, that isn't established, so if it is an ontological limit, what happens to time dilation as you approach light speed? As I understand if person B travels at light speed, and we sit person A in our stationary frame of reference then person A observes zero time passing for person B. That view depends on time being continuous, as I understand it. Though I don't know what difference it would make if rather than zero time passing a single quantum of time passed, I'm guessing it makes a difference to whatever physical laws govern objects like singuarities.

    3. As far as I know, hidden variable interpretations don't necessarily introduce infinite quantities. Did I once claim otherwise? The problem I anticipate with embedding an "ontological" planck time in a Hawking cosmology is that this re-introduces the creation ex nihilo that it is one of the objectives of the Hawking view to avoid.

    4. On the idea that Hidden Vavriable theories might introduce infinities, I think it was this comment, in "continuing discussions on randomness..." that gave me the idea:

      *198, 3rd Aug 2012, 10:49pm Stephen R. Diamond
      - Joseph, on quantized time: Am I correct in thinking that for Bohm quantization is purely epistemic; that the Bohm view, speaking ontologically, substitutes continuous variables for quantized variables. For some reason I'd assumed this, but I don't recall why -

      Sorry if I've got the wrong end of the stick, or read into it to much, and apologies for taking a while to reply, the prospect of trawling through that many posts put me off a bit!

      But it seems without an ontological quantum of time we are left two options as I see it, a) time is not passing, or b) time is composed infinitesimally small units, which may or may not be the same, I am having a lot of trouble understanding that infinitesimals are different from zero, Peter Hurford's argument that 0.333.... x 3 = 0.999.... = 1 doesn't help!

    5. I can think of two potential stumbling blocks as you try to understand infinitesimals: the difference between actual and potential infinitesimals and the incoherence of actual infinitesimals.

      If time is continuous (which I think must be assumed to avoid origination _ex nihilo_) then it is potentially infinitesimal: it can be subdivided without limit. But that doesn't mean it _is_ ever infinitesimally subdivided. The discussion of Zeno's Paradox in 19.2 might help here.

      If I'm right, actual infinitesimals (and actual infinities) are incoherent: self-contradictory. Then, you can't expect really to "understand" them. The fact that actual infinitesimals are indistinguishable from zero is my reason for rejecting them! (I think this conclusion is much more widely acceptable than my rejection of actual infinities.)

      But how actual infinitesimals are _conceptually_ distinct from zero can be brought home by thinking in terms of the definitions from Euclidean geometry, although Euclid didn't speak in those terms, which hadn't yet been invented. Consider the Euclidean concept of a line segment, which has length but no width. Is it's width conceived as zero? I don't think so because: 1) with literally zero width, there would be literally nothing there, as there's supposed to be; 2) if you lay out an infinite number of line segments side by side, you get a plane; and 3) if they were zero width, you couldn't lay them out side by side: they'd be on top of one another.

      So, infinitesimals have some of the properties of zero and some of positive numbers. But (I contend) these properties are contradictory if posited as _realized_ (rather than potential).

      In Peter's example, .999.... is infinitesimally smaller than 1. But clearly, it isn't 1, as it would be if infinitesimals were simply zero. The mathematics are consistent, but their realization is impossible (I contend); there can be no .999.... in the real world.

    6. Joseph,

      I now think the argument from infinitesimals is the best argument against actual infinities. Peter's example provides what for some might be an easy way to understand the argument. .999... can't exist because it's difference from 1 is infinitesimal. The _same_ conclusion about .999... can be drawn for the fact that the 9s, represented as an infinite series, can never be completed, never truly represented except by the metaphor conveyed by the ellipsis. Infinitesimals and infinity are inseparable, as this example illustrates, but to complete the argument, you have to consider unbounded infinities--.999... being bounded--by 1.

      I recall a quote from Einstein that might help clarify his position on actual infinities in cosmology: 'The only two things that are infinite are the universe and human stupidity, and I'm not sure of the first.' Although he embraced the Gold model for a time, he later favored models, I think, that allow for infinite space-time. As a piece of gossip, I seem to remember that he was romantically involved with a physicist who believed in the infinitude of the universe with a religious fervor.

      In any event, he seemed to consider the question _essentially_ empirical, and in that I respectfully disagree.

    7. I read this article, and am currently mulling it over, I thought you might find it interesting:

      And I think you touched on the point earlier, that it is possible to come up with mathematical systems where 0.999... is represented.

      I think I'd have to abandon my preference for de broglie-bohm interpretations if the infinitesimal division of time were not possible and instead place a small bet on Ghirardi-Rimini-Weber theory, interestingly truly random events would then exist, and determinism would be somewhat undermined.

      A minor thought that occured, string theory is probably a hidden variable theory, have to check that.

    8. >I think I'd have to abandon my preference for de broglie-bohm
      >interpretations if the infinitesimal division of time were not

      If I can check whether I understand you, are you saying that if time were quantized (instead of infinitely divisible: _potentially_ infinitesimal) that would disfavor hidden variables? Makes sense.

      But to be clear, I take my argument against actual infinities to be an argument *for* the strictest determinism. Repudiating infinity, you can't, for cosmological reasons, have ontic probabilities (I claim). (If you admit ontic probabilities, you get infinitely many cycles in an oscillating model.)

      >And I think you touched on the point earlier, that it is possible to
      >come up with mathematical systems where 0.999... is represented.

      It's really no big deal. Leibnitz originally developed the calculus with infinitesimals. Limits came later, and some claim Leibnitz's way is actually better pedagogically. I think looking at the calculus translated into the language of infinitesimals is philosophically useful—it's the approach I take.

    9. Yes, that is what I am saying, I think quantised time would favour a non-hidden variables interpretation, though as for me the "measurement problem" is the biggest problem with quantum interpretations that leaves Ghirardi-Rimini-Weber as my next favourite.

      I am confused that you've said you've rejected ontic probabilities. I remembered you'd said:

      " *173 - Stephen R. Diamond says:
      1 Aug 2012, 4:47 pm  
      1. Bohm doesn’t have objective probabilities. Even the epistemological limits apply only as approximations under specific conditions. But even if you want to call these limits objective, the point is that propensity theory make them ontological."

      Does this reflect a change of heart regarding propensity theory, ontic probabilities, or both, or have I really screwed up and the answer is neither?

      Also I am a bit perplexed by:
      "(If you admit ontic probabilities, you get infinitely many cycles in an oscillating model.)"

      As in the above article you said:

      "Thus although the expansion and collapse process is perpetual and eternal, it is also finite, it's number is unity"

      I thought this allowed infinitely many indisitinguishable cycles, so in this case, I've clearly misunderstood.

      I just wanted to add that I am amazed by Cantor, hyperreal numbers, transfinites etc! They seem like a massive deal. I honestly wish my mathematical understanding was better, it seems like a beautiful world! Enough gushing.

    10. It represents a change of heart. I hadn't invented this argument yet; I had no basis for rejecting ontic probabilities, and I think the majority (but short of "consensus") of physicists accept ontic probabilities.

      My Leibnitzian argument is really the most novel proposal. Oscillation precludes infinity because there's really only one. Counter-intuitive, but coherence would satisfy me.

      For this to work, each oscillation must be *precisely* identical. Otherwise, you can distinguish them, and there are infinitely many. Ontic probabilities would make the cycles different from one another.

    11. Fair enough, may I ask what, if any, quantum interpretation you prefer currently? I agree with you that in favouring either the MWI or Copenhagen Interpretation, which seem to be the favourites, Physicists are currently supporting ontic probabilities. I now understand why your exactly repeating universe would not allow ontic probabilities.

    12. I don't know enough about the alternatives to have an opinion. Some variant of Bohm, I should think, or else something nobody's yet thought of.

  3. My apologies, I have but scanned this article, but I immediatly noted that the author claims the radius of a universe, which follows the gold model, is infinite:

    1. It looks like this is a different Gold model--essentially the Hoyle steady-state approach. I came upon Gold's oscillating universe discussed in a book by philosopher Huw Price, "Time's Arrow and Archimedes' Point: New Directions for the Physics of Time" (1997) []. Now somewhat dated scientifically but still extremely interesting. I haven't found it discussed elsewhere but haven't really looked. Hoyle conceived of the universe as infinite temporally and spatially.

    2. Got to be quick, Saturdays are unfortunately rather busy, but here is another document, which by page 3 or 4 again states that an Einstein Steady State/Static Universe has an infinite radius:

    3. The article makes the point that if the universe is expanding and unbounded, it is infinite under General Relativity. If I'm not mistaken, this is well-accepted.

    4. did I get the correct model this time? I admit I'm finding it hard to find the exact model we are discussing. Is an unbounded universe not infinite?

    5. An unbounded expanding universe is infinite. But the Gold Universe isn't unbounded. If the Gold model isn't accessible, probably the closest things are the current cyclical models. If the universe eventually collapses, it's not unbounded (to my understanding).

    6. The first article was the one I got when I searched for current cyclical models, though they may already be old hat, it does seem that the Gold Model was abandoned as it failed to make various predictions/ some of it's predictions were falsified, which is of course not to deny that a variant may aswer those concerns. As neither of us is a Physicist I guess we will have to settle on a "maybe".

  4. Identity of indistinguishables is refuted by noncomputable numbers. Noncomputable numbers are thos real numbers that are random in the sense of information theory: their shortest description is as long as the number themselves. We know noncomputable numbers exist, because without them the real line would be full of holes. There are only countably many computable numbers; and uncountably many noncomputables. In short, almost all real numbers are noncomputable.

    Now, a noncomputable number can not be uniquely characterized by any finite-length string of symbols. In other words it can not be characterized by any property. Given any two noncomputable real numbers, there is no property that distinguishes them.

    Elsewhere on this blog the author states that sets are characterized by their properties. But this is false. A mathematical set is entirely characterized by its elements. If sets were only characterized by properties, then there could only be countably many sets; for the reason that there are only countably many properties. But we know from the work of Cantor in the 1870's that there are uncountably many different sets of natural numbers. However, most of these sets can never be distinguished by properties; only by their elements.

    This falsifies most of the content of the present post; and most of the other opinions of the author on this blog regarding infinity. The author would be well advised to study some elementary set theory to learn what a set is.

    1. Thank you for the lucid account of noncomputable numbers, but your argument against the claim that it requires properties to define a set of *existing things* suffers from the hypermathematical tendency in the philosophy of infinity that the essay bemoans You won't engage the philosophical issues regarding infinity if you refuse to countenance the nonmathematical concept of "existence" (a term is used in a different sense in mathematics) and if you insist that because something is mathematically possible, it's metaphysically possible. When I wrote "The identity of indistinguishables does apply to sets," I hoped the context would convey that I meant a set of actually existing things; otherwise, why my disclaimers distinguishing the mathematics of sets from the philosophical treatment of existence? The identity of indistinguishables is debated as a principle of metaphysics, not a principle of logic or mathematics, where as you point out, it doesn't hold.

      The basic problem with importing a mathematical argument concerning real numbers into a metaphysical argument about reality is that numbers are inventions, fictions; they don't exist, although many mathematicians think they do: Cantor certainly did, Hilbert didn't. Mathematics won't tell you whether its constructions exist (or are properly interpreted as purporting to exist or as possibly existing). In effect, the essay's argument is also an indirect argument against the existence of numbers, although the fictionalist account of mathematics also rests on other grounds.

    2. Clarified in the text that the identity of indistinguishables applies to sets of existing things rather than sets in general. Thanks again for your comment.

  5. If there were aleph_0 natural numbers, i.e., more than are in any finite initial segment (FISON) {1, 2, 3, ..., n} of |N, then, as a consequence, no FISON could contain all numbers. So the natural numbers must be dispersed over many FISONs. That is mathematically impossible by the inclusion monotony of the infinite sequence of FISONs. Further it is disproved by the fact that there are no two FISONs containing more that each of them.

    Regards, WM


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