Can infinite quantities exist? A philosophical approach

metaphysicist

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Can infinite quantities exist? A philosophical approach

by metaphysicist

3rd Jan 2013

5 min read

-26

[Crossposted]

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.

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Can infinite quantities exist? A philosophical approach

by metaphysicist

3rd Jan 2013

5 min read

-26

[Crossposted]

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

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

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.

B. Argument based on probability as limiting relative frequency.

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.

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.

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metaphysicist13y00

Thank your for the astute response.

1.You say that the points are brutely distinguishable and later you say that they are indistinguishable, which nevertheless you hold to be different properties.

The points are brutely distinguishable, but the sets aren't.

2.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.

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.

3.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?

If infinite sets and brutely distinguishable elements exist, infinite sets with brutely distinguishable elements should exist.

4.What is your opinion about the set of all natural numbers? Is it finite or can't it be defined?

It is infinite, but it isn't "actually realized." (They don't exist; we employ them as useful fictions.)

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?

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.

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 occurrences) divided by (limit of the number of trials).

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?)

And emphasis is usually marked by italics, not red.

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.

prase13y10

If infinite sets and brutely distinguishable elements exist, infinite sets with brutely distinguishable elements should exist.

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 fo... (read more)

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