Can infinite quantities exist? A philosophical approach
Initially attracted to Less Wrong by Eliezer Yudkowsky’s intellectual boldness in his “infinite-sets atheism,” I’ve waited patiently to discover its rationale. Sometimes it’s said that our “intuitions” speak for infinity or against, but how could one, in a Kahneman-appropriate manner, arrive at intuitions about whether the cosmos is infinite? Intuitions about infinite sets might arise from an analysis of the concept of actually realized infinities. This is a distinctively philosophical form of analysis and one somewhat alien to Less Wrong, but it may be the only way to gain purchase on this neglected question. I’m by no means certain of my reasoning; I certainly don’t think I’ve settled the issue. But for reasons I discuss in this skeletal argument, the conceptual—as opposed to the scientific or mathematical—analysis of “actually realized infinities” has been largely avoided, and I hope to help begin a necessary discussion.
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 real 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 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: indistinguishable sets are identical. Properties determine sets, so you can’t define a proper subset of brutely distinguishable things.
3. Arguments against actually existing infinite sets.
A. Argument based on brute distinguishability.
To show that the existence of an actually existing 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 if infinite quantities exist, then relative frequency should equal probability. 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 actually realized infinities 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 and eternal, 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.
Seems like yet another confusion about the definition of “exist”, which you conveniently don’t give.
If you rephrase it as “Can infinite quantities be observed?” then the answer is negative. If you phrase it as “Can models with infinities in them fit the observations better then those without?”, then the answer is affirmative. If you are interested in a metaphysical answer, such as “Do numbers exist?”, then you have to be clear in what you mean by each term.
We have an apparently very deep philosophical difference here. Some “quite smart people” have offered different accounts of existence: Quine’s, that we are committed to the existence of those variables we quantify over in our best theory, comes to mind. My use of “exists” is ordinary enough that most any reasonable account will serve. I think the intuition of “existence” is really extremely clear, and we argue about accounts, not concepts. Existence is very simple
Maybe addressing your specific examples will clarify. “Can infinite quantities be observed?” as a meaning of existing. Clearly doesn’t mean the same thing. Whether something exists or it can be observed are two different questions, existence being a necessary but insufficient condition for observability. “Can models with infinities in them fit the observations better than those without?” Still not existence. There are instrumentalist models and realist models. (Realists will agree; some intrumentalists will consider all theory instrumental, but that’s another question.) There’s a difference between saying something predicts the data and saying that the model describes reality (what exists) even if the latter claim is justified by the former. “Do numbers exist?” There the dispute isn’t about existence but about numbers, and it’s only because we do have a clear intuition of “existence” that the question about numbers can arise. So, we get different theories about numbers, which imply that numbers exist or don’t.
So, even when it comes to numbers, I don’t think there’s much problem with the concept of existence. Sometimes one sees an unphilosophical tendency to treat problems regarding concepts as though they could be resolved by a mere choice of definition. Such flaws so easily corrected rarely arise in sophisticated thought. The question here is whether our intuition of existence implies that only the finite can exist. In analyzing an intuition, it rarely helps to start with a definition.
If that’s what you think, maybe you are on a wrong site then.
Something’s got to be primitive, and I can’t think of a candidate better than existence.
This is the worst answer possible, given that some quite smart people disagree on the meaning of this term, and it renders your post meaningless. Consider rereading Skill: The Map is Not the Territory for one possible answer.
If you’re going to dodge defining existence, please at least clarify your point by telling us which of these things “exist”:
a) irrational numbers
b) sets
c) postmodernism
d) the number of Langford pairings of length 100
e) negative numbers
f) quaternions
Moved to Discussion.
Reminds me of another question I read recently: “Has anyone really been far even as decided to use even go want to do look more like?” I may have better luck parsing your post if you chose to work on its formatting. Feedback you’ve received on LW in the past, to little avail. Avast! (?)
I’m also not sure about your apparently new concept of “brute distinguishability”, my only association is “et tu, brute?” which of course is historically inaccurate.
If you’re not sure of the “brute distinguishability” concept, I’ve conveyed something, because it is the main novelty in my argument.
Sorry, but I do not think this is a well written article. The formatting is strange and hard to read, and your points meander a lot. You also need to give coherent summaries a lot more. As written now, your post is hard to read, and I can’t quite tell what points you are really trying to convey. Please take this as constructive criticism, and work to make your post better.
Thank you for the criticism. I will indeed consider it. It may be that we have different theories of writing. Regarding our likely differences considering how to write, see my “Plain-talk writing: The new literary obfuscation.”
I don’t see how it can be accused of meandering. I’d be pleased to receive a personal note explaining.
metacomment:
You seem to be getting a decent response, but the formatting screams “internet wingnut at his coming out party” to me.
I think most of this worrying is dissolved by better philosophy of mathematics.
Infinte sets can be proven to exist in ZF, that’s just a consequence of the Axiom of Infinity. Drop the axiom, and you can’t prove them to exist. You’re perfectly welcome to work in ZF-Infinity if you like, but most mathematicians find ZF to be more interesting and more useful. I think the mistake is to think that one of these is the “true” axiomatization of set theory, and therefore there is a fact of the matter over whether “infinite sets exist”. There are just the facts about what is implied by what axioms.
If you’re worried about how we think about implication in logic without assuming set theory, perhaps even set theory with Infinity, then I agree that that’s worrying, but that’s not particularly an issue with infinity.
Then, on the other hand, you might wonder whether some physical thing, like the universe, is infinite. That’s now a philosophy of science question about whether using infinite sets or somesuch in our physical theories is a good idea. Still pretty different.
Aside: your specific arguments are invalid.
The indistinguishability argument, regardless of whether it’s good in principle, is incorrect. For infinite X, X and X’=X \union {x} are distinguishable in ZF. For one thing, X’ is a strict superset of X, so if you want a set (a “property”) that contains X but not X’, try the powerset of X. I’m not really sure what else you mean by “indistinguishability”.
In the relative frequency argument you do limits wrong. It can be the case that lim f(x) and lim g(x) are both undefined, but that lim f(x)/g(x) is fine.
Another mathematical point is that mathematical models involving infinite things can sometimes be shown to be equivalent to mathematical models involving only finite things. Terence Tao has written extensively on this; see, for example, this blog post. So quibbling about infinities is very much quibbling about properties of the map, not properties of the territory.
First, I second the other’s requests to define “exist”.
Second, I don’t understand the arguments.
“Let A and B be brutely distinguishable points. Define the sets M and N as M = {A, B} and N = {B}. N is a proper subset of M.”
What I have done wrong in the preceding quotation? It seems like something a mathematician could easily say.
You say that the points are brutely distinguishable and later you say that they are indistinguishable, which nevertheless you hold to be different properties.
Why are the sets indistinguishable? Although I don’t particularly understand what predicates you allow for brutely distinguishable entities, it seems possible to have X = set of all brutely distinguishable points (from some class) and Y = set of all brutely distinguishable points except one. It is, of course, not a definition of Y unless you point out which of the points is missing (which you presumably can’t), but even if you don’t have a definition of Y, Y may still exist and be distinguishable from X by the property that X contains all the points while Y doesn’t.
If the argument were true, haven’t you just shown only that you can’t define an infinite set of brutely distinguishable entities, rather than that infinite sets can’t be defined at all?
What is your opinion about the set of all natural numbers? Is it finite or can’t it be defined?
And how does the argument depend on infiniteness, after all? Assume there is a class of eleven brutely distinguishable points. Now, can you define a set containing seven of them? If you can’t, since there is no property to distinguish those seven from the remaining four, doesn’t it equally well prove that sets of cardinality seven don’t exist?
The frequentist definition of probability says that probability is the limit of relative frequencies, which is the limit of (the number of occurrences divided by the number of trials), which is not equal to (limit of the number of occurences) divided by (limit of the number of trials). Note the positions of the brackets.
The sequence {2n/n} = {2/1, 4⁄2, 6⁄3, 8⁄4, … } = {2, 2, 2, …} has an obvious well defined limit, even if the limits of both {2n} and {n} are infinite.
A note about formatting: consider not copying a text from a word processor or a web browser directly to the LW post editor. The editor is “smart” and recognises the original font size and type and grey background color and whatever else and imports it to the post, which therefore looks ugly. I’d suggest copying to a Notepad/gedit-style editor first which kills the formatting and then to LW. (And emphasis is usually marked by italics, not red.)
Thank your for the astute response.
The points are brutely distinguishable, but the sets aren’t.
No predicates besides brute distinguishability govern it. Entities that are brutely distinguishable are different only by virtue of being different.
The sets that differ but for one element differ because their cardinality is different. This is how they differ from the infinite case.
If infinite sets and brutely distinguishable elements exist, infinite sets with brutely distinguishable elements should exist.
It is infinite, but it isn’t “actually realized.” (They don’t exist; we employ them as useful fictions.)
To make the cases parallel (which I hope doesn’t miss the point): take 7 brutely distinguishable points; 4 more pop into existence. The former and latter sets are distinguishable by their cardinality. When the sets are infinite, the cardinality is identical.
This doesn’t seem relevant to actually realized infinities, since the limit becomes inclusive rather than exclusive (of infinity). The relative frequency of heads to tails with an actually existing infinity of tosses is undefined. (Or would you contend it is .5?)
Are my aesthetics off? I’ve decided that unbolded red is best for emphasizing text sentences, the reason being that it is more legible than bolding, centainly than italics. If you don’t use many pictures or diagrams, I think helps maintain interest to include some color whenever you can justify it.
Color seems increasingly used in textbooks. Perhaps its a status thing, as the research journals don’t use it. But blog writing should usually be more succinct than research-journal writing, and this makes typographical emphasis more valuable because there’s less opportunity to imply emphasis textually.
Why? It doesn’t follow. (As a trivial case, imagine that there are only two brutely distinguishable things in the world.) (Assuming that by “infinite sets with brutely distinguishable elements” you mean “set with infinitely many b.d. elements”.)
Also, you say that sets are distinguishable whenever there is a predicate which applies to one and doesn’t apply to another. That is, X and Y are distinguishable iff for some P, P(X) and not P(Y). Right?
But then you argue as if the only allowed predicates were those about cardinality. To closely follow your example, let’s denote X = “the former set containing infinitely many b.d. points” and Y = “the latter set containing all those points plus the additional one which ‘popped into existence’”. Then we have a predicate P(Z) = “Z is a subset of X”, and P(X) holds while P(Y) doesn’t. What’s wrong here?
Your aesthetics are incompatible with most of the readers. You’ve got quite a lot of negative responses to your formatting, not a single positive response (correct me if I am wrong), yet you still persist and speculate about status reasons. Even if it were true, I’d suggest taking the readers’ preferences more seriously, if you want the readers take you more seriously.
To me, coloured text really doesn’t seem more legible than bold or italics. Moreover I like when a website has a unified colour scheme which your colours break. All violations of local arbitrary design norms are distracting; the posts aren’t art, therefore aesthetics shouldn’t trump practical considerations. But if you really that much insist on using colours for emphasis (but consider there may be colourblind people reading this), please at least use the same font and background colour as everybody else.
I just found it curious: I’ve addressed typography issues in a blog posting, “Emphasis by Typography.”
I have to say I’m surprised by your tone; like you’re accusing me of some form of immorality for not being attentive to readers. This all strikes me as very curious. I read Hanson’s blog and so have gotten attuned to status issues. I’m not plotting a revolution over font choice; I’m only curious about why people find Verdana objectionable just because other postings use a different font.
The argument concerns conceptual possibility, not empirical existence. If actually existing sets can consist of brutely distinguishable elements and of infinite elements, there’s nothing to stop it conceptually from being both.
You have located a place for a counter-argument: supplying the conceptual basis. But it seems unlikely that a conceptual argument would successfully undermine brutely distinguishable infinite elements without undermining brutely distinguishable elements in general.
You can distinguish the cardinality of finite sets with brutely distinguishable points. That is, if a set contains 7 points, you can know there are seven different points, and that’s all you can know about them.
Sorry for the perceived tone, it wasn’t my intention to accuse you of anything immoral (although I think you aren’t being much attentive to readers, but that is hardly immoral). I was mainly trying to say that violating the local aesthetic code against the disagreement of everybody who cares to voice their opinion is instrumentally bad. Even if you think that your style makes communication easier, the disagreement of others is a strong piece of evidence that it doesn’t, at least with the LW audience.
As aesthetic preferences are usually difficult to explain or even describe, I probably cannot provide a deep reason why your style is unwelcome. Few particular things I find annoying:
I like stylistic unity, so when everything on the site is one style and one post is another style (yet all the surroundings are original), it is the one post which I perceive odd by default
too many fonts; you have a normal LW headline, then a link in some serifed font, then one paragraph in the standard LW sans-serif font, then the rest of the article in another sans-serif font (or perhaps the same but different size)
the section headlines are larger than the main headline (aaargh!)
blank line missing between sections 1 and 2, other sections are separated by blank lines
the grey background, which is only slightly different from the standard white background and (aargh again) surprisingly missing in the first paragraph and at some (but not all) blank lines; the most annoying thing is that it forms visual boxes which unite the body of a section with the headline of the following section
the red emphasis; on first reading I tend to interpret red not as emphasis, but rather as a text marked for further revision or deletion in a draft
Originally I thought that these “features” accidentally arose when you had copied the text from elsewhere. Now when you are defending the typography, I am curious whether you really have a reason for them all, especially no. 5.
Suppose I assigned each of the points different spacetime coordinates. Now another point pops into existence at a different spacetime coordinate. The two sets of points are distinguishable because one of them has a point at a spacetime coordinate that the other one doesn’t. I don’t see the contradiction here.
Um. Why?
In general, I agree with the other comments that this post is unclear and not well-written. In particular, I agree with shminux’s comment that your definition of “exist” is unclear. You should taboo it.
As an abstract entity, infinity has been quite useful in describing the real world. Perhaps the question would be easier to answer if it were rephrased with “exist” tabooed.
What makes you think there’s some equivocation in my usage of “exists”? (Which is where taboo is useful.) If I were pushing the boundaries of the concept, that would be one thing. I’m not taking any position on whether abstract entities exist; what I mean by exist is straightforward. If the universe has existed for an infinite amount of time, the infinity is “actually realized,” that is, infinite duration is more than an abstract entity or an idealization. If I say, the universe is terribly old, so old we can approximate it by regarding it as infinitely old, then I am not making a claim about the actual realization of infinity.
My first thought upon reading this: You need to read Cantor, who showed that infinite sets can still have different cardinalities; it is entirely possible that some infinity N divided by some other infinity M is a finite quantity. You’re trying to define infinity as a single thing; it’s entirely possible for there to be multiple infinities.
Or, in other words, infinities are far, far messier than you’re presenting them here, and your proofs are relatively naive in regard to the work that has already been done on the subject. I would suggest starting with Cantor; his is hardly the last word, but as old as it is, it is one of the best words. Kronecker, a contemporary of Cantor, is a decent read for the infinity-skeptic side of the equation, but didn’t have much substantive to offer.
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My recollection is that The existence of infinite sets is neither proven nor contradicted by the basic operation available in set theory. You need an additional axiom to get them, and if you want it the additional axiom is consistent with the prior ones.
This means that if the above argument were formalized, there must be a step which effectively assumes no infinites. Likely in a definition of a term.
It’s not clear to me that MWI has infinite worlds rather than merely ‘a mindbogglingly huge number, so large one may reasonably approximate it as infinite for most purposes’.
Worlds aren’t fundamental in MWI; they don’t show up the equations. It’s just a loose, non-technical term you can apply to a subset of the wavefunction. But under most common definitions of a “world”, there are “uncountably many” in the same sense that there are “uncountably many” (sub-) line segments within a given line segment.
I’m aware of that. I mean that there are fewer effectively different worlds than that. Like, if you consider a scattering event—a major source of decoherence and thus new world formation.
With which other particle will this particle’s next interaction (over some minimal threshold of impulse) occur, cut off by time X in some reference frame (e.g. our frame, right now)? (setting aside the fact that particles don’t have individual identities, since that would decrease the world count)
This is a stronger question than that asked by decoherence, but you can already see we’ve narrowed it down to something that seems like picking out of a list—even if the universe is infinite, the number of particles in a truncated future light cone is finite.
Of course you’re doing this a huge number of times, and each one creates a huge number of alternatives, but that’s just making an unimaginably huge finite number. See?