Can infinite quantities exist? A philosophical approach

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.

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.

If infinite sets and brutely distinguishable elements exist, infinite sets with brutely distinguishable elements should exist. Why? It doesn't follow.

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.

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?

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.

Can infinite quantities exist? A philosophical approach

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.)

  1. 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.

Can infinite quantities exist? A philosophical approach

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.

Can infinite quantities exist? A philosophical approach

some quite smart people disagree on the meaning of this 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.

Can infinite quantities exist? A philosophical approach

If you're not sure of the "brute distinguishability" concept, I've conveyed something, because it is the main novelty in my argument.

The noncentral fallacy - the worst argument in the world?

Where can i find out what "near-type" means here?

It refers to "near-mode," which is jargon in construal-level theory for "construed concretely." So in context, it means direct and involving personal experience, as opposed to reading or discussing abstractly.

Robin Hanson applies construal-level theory speculatively in numerous posts at Overcoming Bias. A concise summary of construal-level theory can be found in my posting "Construal-level theory: Matching linguistic register to the case's granularity.".

[Poll] Less Wrong and Mainstream Philosophy: How Different are We?

What state of affairs is "correspondence theory is true" congruent with?

The concept of scientific truth--the concept used by scientists--is the state of affairs some correspondence theories purport to be congruent with.

[Poll] Less Wrong and Mainstream Philosophy: How Different are We?

That's an excellent argument if it's the case that correspondence theory is not the sort of thing allowed to have truth values under correspondence theory. Why do you say it's not?

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