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Amanojack comments on The two meanings of mathematical terms - Less Wrong
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Okay. I have several sources of skepticism about infinite sets. One has to do with my never having observed a large cardinal. One has to do with the inability of first-order logic to discriminate different sizes of infinite set (any countably infinite set of first-order statements that has an infinite model has a countably infinite model - i.e. a first-order theory of e.g. the real numbers has countable models as well as the canonical uncountable model) and that higher-order logic proves exactly what a many-sorted first-order logic proves, no more and no less. One has to do with the breakdown of many finite operations, such as size comparison, in a way that e.g. prevents me from comparing two "infinite" collections of observers to determine anthropic probabilities.
The chief argument against my skepticism has to do with the apparent physical existence of minimal closures and continuous quantities, two things that cannot be defined in first-order logic but that would, apparently, if you take higher-order logic at face value, suffice respectively to specify the existence of a unique infinite collection of natural numbers and a unique infinite collection of points on a line.
Another point against my skepticism is that first-order set theory proper and not just first-order Peano Arithmetic is useful to prove e.g. the totalness of the Goodstein function, but while a convenient proof uses infinite ordinals, it's not clear that you couldn't build an AI that got by just as well on computable functions without having to think about infinite sets.
My position can be summed up as follows: I suspect that an AI does not have to reason about large infinities, or possibly any infinities at all, in order to deal with reality.
I reject infinity as anything more than "a number that is big enough for its smallness to be negligible for the purpose at hand."
My reason for rejecting infinity in it's usual sense is very simple: it doesn't communicate anything. Here you said (about communication) "When you each understand what is in the other's mind, you are done." In order to communicate, there has to be something in your mind in the first place, but don't we all agree infinity can't ever be in your mind? If so, how can it be communicated?
Edit to clarify: I worded that poorly. What I mean to ask is, Don't we all agree that we cannot imagine infinity (other than imagine something like, say, a video that seems to never end, or a line that is way longer than you'd ever seem to need)? If you can imagine it, please just tell me how you do it!
Also, "reject" is too strong a word; I merely await a coherent definition of "infinity" that differs from mine.
Yes but it doesn't matter. The moon can't literally be in your mind either. Since your mind is in your brain, then if the moon were in your mind it would be in your brain, and I don't even know what would happen first: your brain would be crushed against your skull (which would in turn explode), and the weight of the moon would crush you flat (and also destroy whatever continent you were on and then very possibly the whole world).
But you can still think about the moon without it literally having to be in your mind.
Same with infinity.
I can visualize the moon. If I say the word "moon," and you get a picture of the moon in your mind - or some such thing - then I feel like we're on the same page. But I can't visualize "infinity," or when I do it turns out as above. If I say the word "infinity" and you visualize (or taste, or whatever) something similar, I feel like we've communicated, but then you would agree with my first line in the above post. Since you don't agree, when I say "infinity," you must get some very different representation in your mind. Does it do the concept any more justice that my representations? If so, please tell me how to experience it.
We refer to things with signs. The signs don't have to be visual representations. We can think about things by employing the signs which refer to them. What makes the sign for (say) countable infinity refer to it is the way that the sign is used in a mathematical theory (countable infinity being a mathematical concept). Learn the math, and you will learn the concept.
Compare to this: you probably cannot visualize the number 845,264,087,843,113. You can of course visualize the sign I just now wrote for it, but you cannot visualize the number itself (by, for example, visualizing a large bowl with exactly that number of pebbles in it). What you can do is visualize a bowl with a vast number of pebbles in it, while thinking the thought, "this imagined bowl has precisely 845,264,087,843,113 pebbles in it." Here you would be relying entirely on the sign to make your mental picture into a picture of exactly that number of pebbles. In fact you could dispense with the picture entirely and keep only the sign, and you would successfully be thinking about that number, purely by employing its sign in your thoughts. Note that you can do operations on that sign, such as subtracting another number by manipulating the two signs via the method you learned as a child. So you have mastered (some of) the relevant math, so the sign, as you employ it, really does refer to that number.
Well I agree that I can think just with verbal signs, so long as the verbal sentences or symbolic statements mean something to me (could potentially pay rent*) or the symbols are eventually converted into some other representation that means something to me.
I can think with the infinity symbol, which doesn't mean anything to me (unless it means what I first said above: in short, "way big enough"), and then later convert the result back into symbols that do mean something to me. So I'm fine with using infinity in math, as long as it's just a formalism (a symbol) like that.
But here is one reason why I want to object to the "realist" interpretation of infinity via this argument that it's just a formalism and has no physical or experiential interpretation, besides "way big enough": The Christian god, for example, is supposed to be infinite this and infinite that. This isn't intended - AFAIK - as a formalism nor as an approximation ("way powerful enough"), but as an actual statement. Once you realize this really isn't communicating anything, theological noncognitivism is a snap: the entity in question is shown to be a mere symbol, if anything. (Or, to be completely fair, God could just be a really powerful, really smart dude.) I know there are other major problems with theology, but this approach seems cleanest.
*ETA: This needs an example. Say I have a verbal belief or get trusted verbal data, like a close friend says in a serious and urgent voice, "(You'd better) duck!" The sentence means something to me directly: it means I'll be better off taking a certain action. That pays rent because I don't get hit in the head by a snowball or something. To make it into thinking in words (just transforming sentences around using my knowledge of English grammar), my friend might have been a prankster and told me something of the form, "If not A, then not B. If C, then B. If A, then you'd better duck. By the way, C." Then I'd have to do the semantic transforms to derive the conclusion: "(I'd better) duck!" which means something to me.
To know reality we employ physics. Physics employs calculus. Calculus employs limits. Limits employ infinite sequences. Does that pay enough rent?
I did say I'm fine with using infinity in math as a formalism, and also that statements using it could be reconverted (using mathematical operations) into ones that do pay rent. It's just that the symbol infinity doesn't immediately mean anything to me (except my original definition).
But I am interested in the separate idea that limits employ infinite sequences. It of course depends on the definition of limit. The epsilon-delta definition in my highschool textbook didn't use infinite sequences, except in the sense of "you could go on giving me epsilons and I could go on giving you deltas." That definition of infinity (if we'll call it that) directly means something to me: "this process of back and forth is not going to end." There is also the infinitesimal approach of nonstandard analysis, but see my reply to ata for that.
If statement A can be converted into statement B and statement B pays rent, then statement A pays rent.
Your original definition:
Is a terrible one for most purposes, because for them, no matter how big you make a finite number, it won't serve the purpose.
Also, meaning is not immediate. Your sense that a word means something may arise with no perceptible delay, but meaning takes time. To use the point you raised, meaning pays rent and rent takes time to pay. Anticipated sensory experiences are scheduled occur in the future, i.e. after a delay. The immediate sense that a word means something is not, itself, the meaning, but only a reliable intuition that the word means something. If you study the mathematics of infinity, then you will likewise develop the intuition that infinity means something.
The epsilon delta definition is meaningful because of the infinite divisibility of the reals.
Unlike your original definition, this is a good definition (at least, once it's been appropriately cleaned up and made precise).
Only if the mathematical operation is performed by pure logically entailment, which - if a meaningless definition of infinity is used and that definition is scrapped in the final statement - it would not be. We will just go on about what constitutes a mathematical operation and such, but all I am saying is that if there is a formal manipulation rule that says something like, "You can change the infinity symbol to 'big enough'* here" (note: this is not logical entailment) then I have no objection to the use of the formal symbol "infinity."
*ETA: or just use the definition we agree on instead. This is a minor technical point, hard to explain, and I'm not doing a good job of it. I'll leave it in just in case you started a reply to it already, but I don't think it will help many people understand what I'm talking about, rather than just reading the parts below this.
For example? Although, if we agree on the definition below, there's maybe no point.
That's why I said "could potentially pay rent."
Looks like we're in agreement, then, and I am not a finitist if that is what is meant by infinite sequences.
But then, to take it back to the original, I still agree with Eliezer that an "infinite set" is a dubious concept. Infinite as an adverb I can take (describes a process that isn't going to end (in the sense that expecting it to end never pays rent)); infinite as an adjective, and infinity the noun, seem like reification: Harmless in some contexts, but harmful in others.
I'm really surprised at the amount of anti-infinitism that rolls around Less Wrong.
"Infinity-noncognitivist" would be more accurate in my case (but it all depends on the definition; I await one that I can see how to interpret, and I accept all the ones that I already know how to interpret [some mentioned above]).
It's not just you. There was just recently another thread going on about how the real numbers ought to be countable and what-not.
From your post it sounds like you in fact do not have a clear picture of infinity in your head. I have a feeling this is true for many people, so let me try to paint one. Throughout this post I'll be using "number" to mean "positive integer".
Suppose that there is a distinction we can draw between certain types of numbers and other types of numbers. For example, we could make a distinction between "primes" and "non-primes". A standard way to communicate the fact that we have drawn this distinction is to say that there is a "set of all primes". This language need not be construed as meaning that all primes together can be coherently thought of as forming a collection (though it often is construed that way, usually pretty carelessly); the key thing is just that the distinction between primes and non-primes is itself meaningful. In the case of primes, the fact that the distinction is meaningful follows from the fact that there is an algorithm to decide whether any given number is prime.
Now for "infinite": A set of numbers is called infinite if for every number N, there exists a number greater than N in the set. For example, Euclid proved that the set of primes is infinite under this definition.
Now this definition is a little restrictive in terms of mathematical practice, since we will often want to talk about sets that contain things other than numbers, but the basic idea is similar in the general case: the semantic function of a set is provided not by the fact that its members "form a collection" (whatever that might mean), but rather by the fact that there is a distinction of some kind (possibly of the kind that can be determined by an algorithm) between things that are in the set and things that are not in the set. In general a set is "infinite" if for every number N, the set contains more than N members (i.e. there are more than N things that satisfy the condition that the set encodes).
So that's "infinity", as used in standard mathematical practice. (Well, there's also a notion of "infinity" in real analysis which essentially is just a placeholder symbol for "a really large number", but when people talk about the philosophical issues behind infinity it is usually about the definition I just gave above, not the one in real analysis, which is not controversial.) Now, why is this at all controversial? Well, note that to define it, I had to talk about the notion of distinctions-in-general, as opposed to any individual distinction. But is it really coherent to talk about a notion of distinctions-in-general? Can it be made mathematically precise? This is really what the philosophical arguments are all about: what kinds of things are allowed to count as distinctions. The constructivists take the point of view that the only things that should be allowed to count as distinctions are those that can be computed by algorithms. There are some bullets to bite if you take this point of view though. For example, the twin prime conjecture states that for every number N, there exists p > N such that both p and p+2 are prime. Presumably this is either true or false, even if nobody can prove it. Moreover, presumably each number N either is or is not a counterexample to the conjecture. But then it would seem that it is possible to draw a distinction between those N which satisfy the conclusion of the conjecture, and those which are counterexamples. Yet this is false according to the constructive point of view, since there is no algorithm to determine whether any given N satisfies the conclusion of the conjecture.
I guess this is probably long enough already given that I'm replying to a five-year-old post... I could say more on this topic if people are interested.
I think you mean, 'determine that it does not satisfy the conclusion'.
I think my original sentence is correct; there is no known algorithm that provably outputs the answer to the question "Does N satisfy the conclusion of the conjecture?" given N as an input. To do this, an algorithm would need to do both of the following: output "Yes" if and only if N satisfies the conclusion, and output "No" if and only if N does not satisfy the conclusion. There are known algorithms that do the first but not the second (unless the twin prime conjecture happens to be true).
You're pointing to a concept represented in your brain, using a label which you expect will evoke analogous representations of that concept in readers' brains, and asserting that that thing is not something that a human brain could represent.
The various mathematical uses of infinity (infinite cardinals, infinity as a limit in calculus, infinities in nonstandard analysis, etc.) are all well-defined and can be stored as information-bearing concepts in human brains. I don't think there's any problem here.
It looks like we agree but you either misread or I was unclear:
I'm not asserting that the definition of infinity I mentioned at the beginning ("a number that is big enough for its smallness to be negligible for the purpose at hand") is not something a human brain could represent. I'm saying that if the speaker considers "infinity" to be something that a human brain cannot represent, I must question what they are even doing when they utter the word. Surely they are not communicating in the sense Eliezer referred to, of trying to get someone else to have the same content in their head. (If they simply want me to note a mathematical symbol, that is fine, too.)
I also agree that various uses of concepts that could be called infinity in math can be stored in human brains, but that depends on the definitions. I am not "anti-infinity" except if the speaker claims that their infinity cannot be represented in anyone's mind, but they are talking about it anyway. That would just be a kind of "bluffing," as it were. If there are sensical definitions of infinity that seem categorically different than the ones I mentioned so far, I'd like to see them.
In short, I just don't get infinity unless it means one of the things I've said so far. I don't want to be called a "finitist" if I don't even know what the person means by "infinite."