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 pageok [Eugene Volokh, May 2, 2006 at 2:13am] Trackbacks "Nearly Infinite": An otherwise very good item that I read a while back refers to a certain set of things as "nearly infinite." It then pointed out, as evidence, that there were over 100 elements in that set. Now that's defining infinity down. Of course, even if there were thirty-seven googol elements, that would still be infinitely far from infinity. There's no such thing as "nearly infinite." The problem isn't like the asserted (but in my view overstated) problem with "more perfect" (as in "more perfect Union") or "more round." You can get materially more perfect or round than you were before. But so long as something is finite, it's not nearly infinite, no matter how much it grows. Yes, I know it's a figurative usage. Yes, I know that language isn't mathematics. The mathematician in me just feels entitled to an irrationally hyperliteralistic snit fit now and again. UPDATE: Reader John Dickinson writes, in an excess of practical good sense, "I think 'nearly infinite' just means that it has nearly the same consequence from the relevant perspective as if the thing were infinite." OK, OK, I suppose that's so. Still: Grrrr. (link) john dickinson (mail) (www): I think "nearly infinite" just means that it has nearly the same consequence from the relevant perspective as if the thing were infinite. For example: "What if I can't find another girlfriend?" "But there are a nearly infinite number of girls in the world!" 5.2.2006 3:44am (link) r4d20 (mail): I never realised that I had a "nearly infinite" amount of money. Practically, J. D. is right - depending on context it could easily mean simply "more than you could make use of in the course of your life" in much the same way that "it would take longer than the life of the universe" is a usual standard in declaring some time-dependent random event to be "impossible" ( contrary to the saying, a million moneys would take longer than the universe has been in existance to actually produce Hamlet). However, what kind context is such that 100 is "nearly infinite" on even this level? I'm putting my money of innumeracy for now. 5.2.2006 4:06am (link) DrewSil (mail): This is a general problem that occurs whenever people decide to use mathematics to describe reality. The language of mathematics gets extended beyond its original "proper" usage. As a scientist, and specifically a physicist, I support such usage. The phrase "almost infinity" has a precise and reasonable meaning when applied to any situation outside of pure mathematics. It means that if you were given an infinite amount of x, rather than the finite but large amount of x that you already posses, there would be no appreciable impact on your results. For example an amount of nuclear weapons sufficient to destroy the earth is almost infinite, as there would be no appreciable difference between having that many and having an infinite supply of nukes. We only have one planet to destroy. Taken to extremes a single doomsday machine that can destroy the universe is an almost infinite number of such machines. That a specific concept would be meaningless mathematically does not make it's usage incorrect in another context. Such usage can often convey important points, and allows people to better describe the world in which we live. Such usage only becomes a problem when it morphs into falacious mathematical reasoning, eg. one divided by almost infinity is almost zero. 5.2.2006 4:16am (link) HankP (mail): I think your problem, as expressed above by other posters, is trying to reconcile Plato's "realm of mathematics" to the real world. It can't be done, there are no perfect mathematical forms in our universe (that I know of), and you're just going to have to accept that people will misuse words outside of their mathematical meaning. I do remember reading a book called "1,2,3 ... Infinity" by George Gamow when I was young, as I recall it did a very good job of explaining mathematical and physical concepts in plain english. 5.2.2006 4:32am (link) John Armstrong (mail): DrewSil, you're right. The real problem is the converse. I can't count (figurative, not technical sense) the number of times I've seen a philosopher, psychologist, sociologist, historian, critic, or (sorry) lawyer argue using a technical term in its nontechnical sense and implicitly insisting that the two are the same. Alan Sokol did a wonderful job of skewering this practice, but mathematicians and physicists generally ignore critical theory's tempest-in-a-teacup of "Science Wars", while Theorists go to great lengths to pretend his hoax doesn't exist. So why does it matter? Because exactly this sort of critical theory, philosophy, and psychology underlie much of modern political science, policy analysis, and legal theory. How can we trust a think-tank run by disciples of Lacan? And what if -- God forbid -- some future administration is as guided by their positions as the current administration seems stacked with Straussians (n.b.: I am not advocating for or against Strauss' viewpoints. I am merely noting that an administration can be heavily influenced by a single school of policy.)? So how can people get away with these "intellectual impostures"? To spout this "fashionable nonsense"? Because there is a fundamental lack of basic reasoning skills among the general populace. Gone are the days when Lincoln advocated "cold, calculating, unimpassioned reason". Now training in rationality is the province of mathematicians and lawyers, and neither is exactly the most revered archetype around. Yes, Professor Volokh may be somewhat pedantic to harp on the imprecise use of the term "infinite", but it's a pedantry that comes with a commitment to clarity in exposition and rhetoric -- a pedantry I think we could do with an awful lot more of around here. 5.2.2006 4:43am (link) zarevitz (mail) (www): What about the "necessary and proper" clause? necessary means indispensable, not convenient, and and just adds a second requirement, it's not an alternative. However... 5.2.2006 7:50am (link) David Chesler (mail) (www): Hank beat me to Gamov. The title refers to an anecdote in which the two greatest mathematicians in all of Hungary decide to hold a contest to see who is the best. They decide that whoever can name the highest number will be the winner. The first says "One", the other says "Two", the first responds "Three". The other ponders the matter for a long while and then concedes. (I hadn't thought about the joke for a while, probably since I was an undergraduate and a roommate suggested that the title meant that a system with degrees of freedom that allow more than three possibilities may as well allow infinite possibilities. Besides the first meaning of the joke, there is a second irony that Hungary produced a surprisingly large number of great mathematicians during the 19th century, but now some third of a century since I first read the book, I'm wondering if there is a third meaning, that the mathematicians realized contest was unwinnable.) Aleph-naught bottles of beer on the wall, aleph-naught bottles of beer, if one of those bottles should happen to fall, aleph-naught bottles of beer on the wall... 5.2.2006 8:11am (link) ed in texas (mail): Keep it up and you'll arrive at "nearly perfect". (Or "absolutely relative", for that matter.) 5.2.2006 9:16am (link) logicnazi (mail) (www): Ohh c'mon I'm sure we can define a situation where it is reasonable to call something nearly infinite. Lets say we have some space and put something like a measure (not really...violates additivity amoung other things) on the space, i.e., something we can call an estimation of 'size' of sets from that space. Furthermore suppose this sorta-measure has the property that a set has sorta-measure greater than or equal to 1 iff the set is infinite. In this case I think it might be very reasonable to call a set with sorta-measure .99 almost infinite. Sure it isn't almost infinite in terms of counting but so what? I bet there are actual natural real world examples where something like this actually comes up. It just all depends on what you mean by infinite. 5.2.2006 9:28am (link) logicnazi (mail) (www): Last sentene should say what you mean by 'ALMOST infinite'. 5.2.2006 9:28am (link) Ron Hardin (mail) (www): Infinite'' doesn't necessarily even mean large.'' When the value of human life is said to be infinite, it means (in the original) that you can't calculate with it, not that it's big. It means without bounds,'' such as you'd need to nail it down precisely enough. It refuses to be nailed down. So the question of the value of human life has to be approached anew each time, from some other approach entirely, for the case at hand. This squares in fact with the pragmatic result that the value of human life doesn't seem in fact to be particularly huge. Nevertheless it's said to be infinite. Mathematics is a specialization of ordinary language, not a model for ordinary language. 5.2.2006 9:49am (link) Eric Rasmusen1 (mail): ...First ask the listener: "What is Epsilon?" Suppose he says, "Epsilon equals .01″. Then you can say that any number greater than "Infinity minus .01″ is "almost infinite". That includes Infinity, of course. Whether a number such as "infinity minus .009″ is different from "infinity" is a question too hard for me. for more in this vein, see http://www.rasmusen.org/x/2006/05/02/1158/ 5.2.2006 10:35am (link) Eh Nonymous (mail) (www): The distinction you're looking for is between "nearly infinite" (which has a clear everyday meaning, but not a well-defined mathematical meaning) and "effectively infinite" - which even a lawyer or a mathetmatician should be able to understand. 5.2.2006 10:58am (link) Anderson (mail) (www): Is it really that much worse than "more unique"? 5.2.2006 11:26am (link) abb3w: But how can you call your hyperliteralistic snit fit "irrational" if it only happens a fraction of the time? =) I suppose one could make a reasonable case for some set of objects being figuratively characterized as "nearly infinite", if it was of large cardinality, and the removal of real-world limiting factors would result in an infinite set. EG: "possible DNA sequences" might reasonably be characterized as nearly infinite... being restricted in practice by (among other things) the limit of around 10^80 baryons in the universe. 5.2.2006 11:32am (link) John Armstrong (mail): logicnazi, measure theory doesn't do it. There's no topology on the $\sigma$-algebra of measurable sets that doesn't make the ones of measure 1 dense in the whole. Every set is "nearly infinite". The problem with these mathematical uses of "infinity" is that "infinity" is shorthand for giving up. The closest Greek term to this sense was "apeiron", which literally means "a mess". In order to apply rigorous tools to the infinite, you need to rigorize the notion of the infinite, which generally turns it into what Cantor termed the "transfinite". To try to inject mathematical rigor into the original context, "nearly infinite" should be replaced as "effectively infinite" (as Eh Nonymous suggests), or by verbiage connoting unboundedness rather than the messy "infinite". 5.2.2006 11:48am (link) Another Anon (mail): Oh, come on. No one has mentioned how novelists do infinity? Prachett: One, two, many, lots. Adams (from mememory, not an exact quote): There's an infinite number of worlds. Not all of them are inhabited, so there's a finite number of inhabited worlds. Any finite number divided by infinity is near enough to zero as makes no difference. Therefore the population of the entire universe is zero. What you get when you apply mathematical concepts to the real world: nonsense. 5.2.2006 12:01pm (link) Swimmy: Economists consider 100-year bonds to be "nearly infinite" and calculate their present value the same they would for truly infinite bonds. Considering what making the expiration that large does to the function, it's perfectly reasonable. 5.2.2006 12:08pm (link) BobH (mail): Can a quantifiable number approach infinity asymptotically? 5.2.2006 12:48pm (link) Silicon Valley Jim: Economists consider 100-year bonds to be "nearly infinite" and calculate their present value the same they would for truly infinite bonds. Considering what making the expiration that large does to the function, it's perfectly reasonable. It's reasonable in the sense of giving an answer that is close to the mathematically correct answer, but I'd say that it's unreasonable relative to the additional effort of calculating their actual present value, which is trivial (the effort, not the actual present value). For some reason, the discussion of "nearly infinite" brings to mind an old math joke: 1+2=4 for sufficiently large values of 1. 5.2.2006 12:49pm (link) JosephSlater (mail): Anderson: "Somewhat unique" is the one that gets me. Oh, and people that misuse "literally," e.g., the sportscaster the other day who claimed that a player "literally turned on a dime." 5.2.2006 12:51pm (link) David Matthews (mail): logicnazi, I think what you're looking for is an "outer measure." To get technical about it, let the underlying set be Z, the positive integers, consider the ring of all possible subsets of Z (set-theoretic definition of ring, not the algebra definition of ring....), and define the outer measure, M, of any set to be n/(n+1) for any finite set, where "n" is the number of elements in the set, and 1 for any infinite set. (Outer measures need only be finitely SUB-additive, not additive, i.e. M(evens U odds) = 1 <= M(evens) + M(odds) = 2, since both sets are infinite.) Then you could talk about finite sets being within some particular bound of infinity, if 1 - bound < M(set) < 1. Of course, this is a lot of mathematical baggage just to say that finite sets are practically infinite if they have more than, say, 200 elements, or whatever. Personally, when I'm looking at a stack of 80 beginning algebra assignments to grade, 80 is practically infinite, while when I'm looking at days until the start of fall semester, 80 is practically zero. 5.2.2006 1:17pm (link) Mahlon: The unfortunate thing is that, with time, the meaning of "infinity" will morph into something like "a really big number to which we can't really count but which we think is out there somewhere." Imprecise usage is rampant. God made dictionaries so we can learn the meaning of words. Too bad he didn't pass them out with hair and teeth. English is being degraded. How low can we go? 5.2.2006 1:29pm (link) Paul Gowder (mail): Yea, what bobh said: wouldn't "nearly infinite" mean "approaching infinity asymptotically?" Eugene, you did computer stuff, so you probably know more calculus than everyone else in this virtual space combined, surely "nearly infinite" can be given meaning that way? In fact, doesn't calculus have infinite limits? I know I vaguely recall such a thing from some explorations... 5.2.2006 1:34pm (link) logicnazi (mail) (www): Armstrong, Yes, I realize that an actual measure will not do it. This is why I called it a sorta-measure in my original post and parenthetially remarked that it would violate additivity. Sorry if I was unclear but was pretty sleepy when I posted. Matthews, Yes, an outer measure might be one class of measure like things that have this property. I always forget what restrictions an outer measure must obey (once I took my qual that was one of the things that fell out of my brain). But just defining some arbitrary outer measure doesn't really give the justification for using nearly infinite that I was aiming at. What I was trying to say but apparently in my tiredness ended up saying very poorly is that there are plenty of situations where something like a measure, i.e., a notion of size of a set, comes up. There are plenty of notions like this that aren't really measures at all (but since they all come up in fields I don't do, i.e. not recursion theory none spring to mind). All I was trying to say is if some natural notion of size like this came up and in some situation sets were greater than a certain 'size' iff they were infinite then we might reasonably call sets that were very close to that size as almost infinite. I mean the vague thought I had in mind was that we are looking at subgroups of some big infinite group. Now we assign a 'size' to each subgroup by counting up the number of generators of that subgroup (if I am screwing up my group theory terminology the smallest n s.t. an n element set of that subgroup generates the entire subgroup). Now without actually doing too much group theory on the question it seems plausible that we could find *some* group so that every finite subgroup could be generated by less than 201 elements while every subgroup requiring 201 or more generators is actually infinite. In this case it would seem reasonable to call a subgroup that needed 200 elements to generate it almost infinite. If some group theory fact I'm forgetting makes this imposible just take this sorta idea and apply it somewhere else in math. Apparently the mention of measure just servered to confuse people. All I meant it to do is give a flavor of how we should think about the function, i.e., something giving a 'size' to subsets. 5.2.2006 1:42pm (link) logicnazi (mail) (www): Also when I used to be a physics major there were plenty of situations that would come up that would justify some finite amount being called 'almost infinite'. For instance consider an electron trapped in some potential well. If the energy barrier provided by the potential well is so huge compared to the energy scales the electron is at (though in human terms still small) then we might call the potential barrier an almost infinite potential barrier since we can approximate this situation very well by an electron in an infinite well. Alright so perhaps it would be a bit better to call it an effectively infinite potential well. -- Basically the reason I am objecting to this idea that it is absurd to talk about being 'almost infinity' is that it seems to all be a matter of perspective. I mean if we are comfortable saying (for real numbers) that y is almost 0 if |y|< epsilon then why not just say that z is almost infinity if |1/z|< epsilon (and if you want + infinity z is positive). It all just depends on the application you are looking at things for. Not that I am disputing whatever particular case you saw which was probably eggregious. 5.2.2006 1:57pm (link) Sanjay (mail): So, one day I walk into a room and my neurobiologist wife is staring at a book, puzzled. It says, the number of connections between neurons in your brain, is larger than the number of particles in the universe. She is thoroughly confused. Not only is that not true, it is so obviously not true, that how could anyone have thought it? And it turns out that that particular mistake shows up all over in serious publications. You can trace it back to its origin --- they're really trying to say something different from what they are saying (they mean to say something about the number of possible connections, although that rests on some questionable ways of thinking about neurons). But, God, it's funny as hell to see it in print. 5.2.2006 3:24pm (link) Jonah Gelbach (mail) (www): so long as something is finite, it's not nearly infinite, no matter how much it grows Well, I think bobh and Paul Gowder are on the right track (though I don't think CS folks are famed so much for their calc/real analysis skills as for their discrete-math chops, but as an economist I could be wrong on that). In any case, most statistical research (both the good and the bad) that is done these days involves appeal to precisely the idea that sample size can be treated as infinite as long as it is large "enough" and/or can be thought of growing unboundedly. The reason is simple: most estimators (including sample means, for those of you who think I'm delving into measure theory-like esoterica) that most people use involve sample averages of smooth functions of data. Such estimators are generally (except in pathological cases) what's called asymptotically normal. "Normal" refers to the probability distribution of the estimator---some people call the density function associated with the normal distribution a bell curve. In this context, "asymptotic" means that the sample is large enough that it might as well be treated as being infinite. Asymptotically normal estimators are random variables whose distributions become arbitrarily close to some normal distribution (which one depends on the probability limit of the random variable as well as its limiting variance, which sometimes have to be defined with some care). If you're familiar with the idea that statistical significance requires a sample mean to be at least two standard deviations away from zero (actually that's just one definition of significance), then you are implicitly familiar with the use of the kind of result I just described. This result is generally referred to as "the" Central Limit Theorem (quotes since there are multiple CLTs, each of whose appropriateness depends on the data generating process). A CLT lets us act as if estimators---like sample average wages, or fraction of 1Ls in the law school whose LSAT score topped 170---can be treated as normally distributed, even when it's easy to show that the estimators have discrete distributions (as is true with both average wages and fractions with scores above a cutoff level), whereas the normal is continuous. CLTs imply that the more the sample size grows, the closer the sample mean's distribution is to an appropriate normal distribution. "Closer" in this context is a term of mathematical art that I won't pursue further here. The advantage of this result is that it justifies using rules like when the mean differs from zero by more than 2 units of (estimated) standard deviation, then we can reject with 95% confidence the null hypothesis that the true population mean equals zero.Now, you might worry about the fact that the CLT explicitly involves taking the limit of the estimator's distribution as the sample size grows without bound (the mathematical term is "diverges", Paul Gowder). To paraphrase Eugene, no sample size ever actually is infinite, so how can we use the CLT? Well, suppose that the sample size is so great that any deviations between the true (cumulative) distribution (function) and the normal are so small that they affect only digits far to the right of the decimal point. Well, then using the normal approximation and the true distribution will virtually never lead to different conclusions about the population mean. So here's a working definition of "almost infinite" in the context of large-sample statistical theory (which, by its very name, might actually be called the study of "almost infinite" sample sizes):When the sample size is large enough that a given rejection rule would not be affected by replacing the normal approximation with the true distribution function, then the sample size is almost infinite.This definition, which is analogous to (a special case of?) more general suggestions in the comments above, has one disadvantage: whether or not the sample size is almost infinite will depend on the level of precision chosen for the rejection rule (e.g., do we use 2 standard deviations, 1.96, or even a more precise figure?). In other words, "almost infinite" has a contingent definition. But in practice that won't much matter, especially if there are clear conventions in place. And anyway, lots of mathematical concepts are contingent (for example, a faster convergence rate for an estimator requires that we have a large "enough" sample size to have any bite). Anyway, the reason I bring up this specific example is that the whole field of large-sample stats is implicitly founded on the (presumed) appropriateness of the concept of "almost infinite". And every one of you who has ever written an empirical paper -- about law, econ, law &econ, physics, sociology, demography, etc -- has most likely appealed to these concepts, knowingly or not. 5.2.2006 4:16pm (link) Jacko: English is being degraded. How low can we go? Forever. Ha ha ha! 5.3.2006 12:39am (link) Bruce Lagasse (mail): Silicon Valley Jim: "For some reason, the discussion of "nearly infinite" brings to mind an old math joke: 1+2=4 for sufficiently large values of 1." Or for sufficiently small values of 4. 5.3.2006 4:32pm pageok pageok