An uncountable countable set
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From: David Marcus on 1 Nov 2006 19:44 cbrown(a)cbrownsystems.com wrote: > David Marcus wrote: > > When the mental picture differs from the math, you > > rely on the math to draw conclusions. This is not the way most people > > think. Most people don't use logic at all. > > I think this is an unduly pessimistic point of view (I'd say it's a > mixture); and it certainly doesn't follow that people cannot /learn/, > to their benefit, to think /more/ logically. > > Fo example, the most demonstrably successful form of psychotherapy we > have so far is CBT (cognitive behavioral therapy), which (put very > simply) asks that we learn to use our logical capacities to examine our > self-assessments, in light of our actual experiences. > > The Enlightment was all about the premise that, with some effort, > everyone can benefit from developing this capacity. It's a skill which > can be developed in anyone (albeit to differing degrees); just as > reading and writing are beneficial skills we can all develop which are > not neccessarily "innate". Just because you are pessimistic doesn't mean you are wrong! :) I would certainly be happy to discover that many people really can think logically and that people can learn to think more logically than they do. Unfortunately, it is only through experience that I've developed my current "pessimistic" perspective. When I was young, I assumed that people thought logically--after all, that's how I thought. I think you underestimate how powerful non-logical thinking can be. I suspect it is more than enough for a person to use CBT. Are there any posters to sci.math that you would like to take as test cases for whether people can learn to think more logically? > > If you have a mental picture of a vase that is full of water and you > > start to let water drain out, then the amount of water in the vase is > > obviously decreasing. The fact that the mathematics says that the amount > > of water in the vase before noon is always infinite just doesn't agree > > with the mental picture. > > > > You can't really have a mental picture of an infinite vase full of an > > infinite amount of water, so I doubt anyone really has such a picture, > > unless they've overlaid the math onto their mental picture, and learned > > to think mathematically/logically. > > I don't know that I agree; I think that the common mental image of "day > after day, forever" can be developed into a mental image of "an > infinite amount of something". That's why Woody Allen's joke is funny: > "Eternity is a long time, especially towards the end." It evokes two > inconsistent mental images at once (which is a common source of humor). I'll go along with that. Time can be imagined to be endless. But, that's where Tony gets "noon doesn't exist". In the real world, we can't squeeze an infinite number of actions into a finite time. For the vase, we can build a mental model where the n-th removal occurs at time n. Now, time goes to infinity, so noon never arrives. > My mental image corresponds to a dishroom where I once worked spooning > out grub. There was a stack of plates pushed up by a spring; and no > matter how many plates I pulled off the top of that damn thing, there > was always a next plate at the top of the stack. Not too hard to > imagine that the plates went on "forever". When did the stack become empty? If never, then noon never arrives. > > > He chooses not to consider whether this is consistent with the fact > > > that by his own lights, the amount of water in the vase in the second > > > scenario at any time t is always less than the amount of water in the > > > vase at that same time in the former scenario. So, by his "infinite > > > induction", if b_n < a_n for all n, then lim b_n < lim a_n; and we have > > > a contradiction. > > > > He's not using logic at all. He probably doesn't agree that b_n < a_n. > > After all, the mental pictures don't show this. > > Sure they do - at any /given/ time before noon, there's at least as > much water in the "start full" vase as the "start empty" vase, and then > some to boot; so that's more water. Not a hard mental image to > construct at all. See below... > > In one case, you've got > > a vase filling with water. In the other, you've got a full vase that is > > emptying. Clearly the vase that is emptying can't always have more water > > than the vase that is filling! > > But that is a /different/ mental image - the former was a mental image > /at/ a time; the latter is a mental image of /during some times/. Which > image we find "more compelling" is something that we learn to choose > (whether by some logical method, or some other method). Given a puzzle, the first step is to produce a mental image. Then we look at the image to get the answer to the problem. Once we have the mental image, we don't go back to the problem. The mental image is all the info we have! For the original problem, the mental image is an empty vase that fills by noon. For the modified problem where all the balls are put in first, then balls are removed, the mental image is a full vase emptying by noon. So, we have two mental images. Now, you ask whether one vase always has more water than the other. Obviously not! One starts full and becomes empty, while the other starts empty and becomes full. As for how Tony chooses a mental image, he reads the problem and accepts whatever mental image forms in his mind. Logic is not used. Now, if you could ask a problem which would generate a mental image of two vases at a particular time, maybe you could get a different answer. > > > But that is not a problem; because "those are different situations - > > > it's an obvious obfuscation of the original problem". > > > > > > Consistency - it's not everyone's cup of tea! > > > > If you aren't using logic, then you don't even understand what > > consistency is. On the other hand, I think he is being consistent with > > the mental pictures. The problem is that the problem is intentionally > > non-physical, so you can't create an accurate mental picture without > > embracing the mathematics. > > The problem is that his various mental images are not /mutually/ > consistent; but he doesn't see that as problematic, because he only > entertains one mental image at any given time. I think he entertains both of them. However, he doesn't critically compare them
From: MoeBlee on 1 Nov 2006 19:57 Lester Zick wrote: > On 31 Oct 2006 15:22:10 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >Lester Zick wrote: > >> On 30 Oct 2006 17:45:48 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > >> > >> >Lester Zick wrote: > >> >> According to MoeBlee mathematical definitions require a "domain of > >> >> discourse" variable such as IN(x) and OUT(x). > >> > > >> >I did not say that. Regarding a particular definition I gave, I > >> >explained to you that the variable ranges over the domain of discourse > >> >of any given model. I didn't say, in general, that definitions > >> >"require" variables (some defininitonal forms do require variables, but > >> >not all definitions do), nor did I suggest using variables in the > >> >mindless way you have done in certain examples you've posted that > >> >misrepresent the actual definition I gave. > >> > >> Then exactly why did you go apeshit over the issue, Moe? > > > >Because you kept representing that I said what I did not say, you > >idiot. > > And you know that how, Moe? Because the material you deleted backs you > up? Or because the material you deleted doesn't back you up? Deleted material? I haven't deleted any posts. And the fact that I haven't included every previous quote in every of my posts is not an issue unless you show some lack of inclusion that did materially distort what you said. I'm not going to commit to including all accumulated previous quoted posts in each of my replies. And I may trim to certain quotes and portions of previous posts to emphasize the scope of my particular interest in what I am replying to. If you claim a material distortion, then go ahead and show what you think has been distorted, if you like. Meanwhile, what backs me up are the posts in the threads, including your own posts in which you mangled my formulas and what I had said, which (as far as I know) are not deleted by you and remain just as they were posted. Now, since you're whining about wanting to move on, then do it. Or don't and continue to whine about it. But no matter what you do, you always manage to make yourself into an utter fool and boor doing it anyway. MoeBlee
From: MoeBlee on 1 Nov 2006 20:08 Lester Zick wrote: > I'm(x)just(x)interested(x)for(x)the(x)record(x)to(x)which(x)axiomatic(x) > assumptions(x)of(x)truth(x)mathematikers(x)can(x)appeal(x)to(x)for(x) > justification(x)of(x)their(x)otherwise(x)unjustifiable(x)speculations(x). Maybe if I have the time and inclination, I'll post some axiomatizations. But I won't do so under the presumption that they are as you describe "assumptions of truth" as YOU define 'truth'. And, as I said, as far as any such posting being for your benefit - your "interest" as you say - I don't see the point, since you are not familiar with the basic mathematical notation in which the axioms are stated with suitable precision (and I am wary of less precise, less formal, statements since they too easily can become fodder for misunderstandings due to crank oversimplifications and crank strawman captiousness). I suggest first that I would post a specification of a syntax for a formal first order language for set theory. Then perhaps we could move on to an axiomatization of the first order calculus, identity theory, and then different set theories. MoeBlee
From: cbrown on 1 Nov 2006 20:58 David Marcus wrote: <snip> > I've been trying to run the discussion in reverse (although with limited > success). I've trimmed back my notion of what "success" constitutes in this endeavor to, um, infinitesimal levels. :) > Suppose we start with this math: > > For j = 1,2,..., let > > a_j = -1/floor((j+9)/10), > b_j = -1/j. > > For j = 1,2,..., define a function f_j: R -> R by > > f_j(x) = 1 if a_j <= x < b_j, > 0 if x < a_j or x >= b_j. > > Let g(x) = sum_j f_j(x). What is g(0)? > > I would think that people would agree that g(0) = 0, even if they don't > agree that this is the math that is equivalent to the English of the > balls and vase problem. Yep; it's really not something that most people have a problem with, provided of course that they think it's worth examining at all. Just like most people, when they actually consider that it's perfectly clear that every ball placed in the vase is also removed from the vase before noon, "get it": regardless of how counterintuive it might seem at first glance, the vase /must/ be empty /at/ noon. > But, if they don't agree this is the equivalent > math, my question to them is how would they translate this math into an > English problem of a vase, balls, and time? I'd like to see how the > result would differ from the standard balls and vase problem. I think most of those who agree with the math but disagree that it's a model of the problem do so on physical grounds; it can't be modelled, because it simply cannot be /done/; so it's 'pointless' to try to model it; it's "just an absurd fantasy". (E.g., HdB, WM, and although you haven't apparently encountered him yet, the episodically active David Petry). > > On the other hand, Tony said that g(0) = oo. Right; and I predict that that will be the primary assumption upon which he will base his response to your argument. I went round and round and round and round with Tony regarding the "staircase" problem, in which, to make the point that the lim f(g(n)) is not always the same as f(lim g(n)), I gave an absurd "proof" that the length of the diagonal of a unit square is 2. (A google of "chas diagonal" in sci.math brings you to a random but representative section of the discussion under the thread "Calculus XOR Probability", if you're feeling interested or even just excessively bored). Part of the argument was an (increasingly painstakingly detailed) definition of "the limit of a sequence of curves", which gave the limit of a sequence of progressively finer staircases in R^2 traversing opposing corners of the unit square as being the diagonal of the unit square. Each staircase has length 2; but of course (I said, spreading my arms and smiling encouragingly) the length of the diagonal of the unit square obviously /cannot/ be 2, so therefore... Or can it? To my dismay, Tony then latched onto the conclusion that I had therefore actually /proven/ that 2 /is/ the length of the diagonal line (or at least of some sort of diagonal-line-like-thing), and from this went on to deduce that my argument also proved that points in R^2 contain "directions", and that there are "infinitesimal line segments" causing the limit to be something (indistinguishably!) different from what I had, after all, explicitly /defined/ it to be, and so on through a thickening quagmire of ever vaguer redefinitions. At which point, good liberal that I am, I engaged in a strategy of "cut and run"; while Tony continued to develop his "theory" until he could claim "Mission Accomplished!". Moral of the story: if Tony has decided that g(0)=oo, he will find some creative way to deduce that there is an error your argument that g(0)=0, typically by redefining the terms of your assumptions (such as: there are infinite naturals that you haven't considered, so you have neglected f_oo(x), f_(oo+1)(x), etc. in your sum g). That's why I claim that Tony works backwards, mathematically: he assumes his conclusions, then deduces the definitions of the terms in the problem so that they then support these conclusions. It /is/ a logic, of sorts; and seems to be the type of "mathematical puzzle" that Tony enjoys. Cheers - Chas
From: Tony Orlow on 1 Nov 2006 23:27
Mike Kelly wrote: > Tony Orlow wrote: >> Mike Kelly wrote: >>> Tony Orlow wrote: >>>> Mike Kelly wrote: >>> <snip> >>>>> Now correct me if I'm wrong, but I think you agreed that every >>>>> "specific" ball has been removed before noon. And indeed the problem >>>>> statement doesn't mention any "non-specific" balls, so it seems that >>>>> the vase must be empty. However, you believe that in order to "reach >>>>> noon" one must have iterations where "non specific" balls without >>>>> natural numbers are inserted into the vase and thus, if the problem >>>>> makes sense and "noon" is meaningful, the vase is non-empty at noon. Is >>>>> this a fair summary of your position? >>>>> >>>>> If so, I'd like to make clear that I have no idea in the world why you >>>>> hold such a notion. It seems utterly illogical to me and it baffles me >>>>> why you hold to it so doggedly. So, I'd like to try and understand why >>>>> you think that it is the case. If you can explain it cogently, maybe >>>>> I'll be convinced that you make sense. And maybe if you can't explain, >>>>> you'll admit that you might be wrong? >>>>> >>>>> Let's start simply so there is less room for mutual incomprehension. >>>>> Let's imagine a new experiment. In this experiment, we have the same >>>>> infinite vase and the same infinite set of balls with natural numbers >>>>> on them. Let's call the time one minute to noon -1 and noon 0. Note >>>>> that time is a real-valued variable that can have any real value. At >>>>> time -1/n we insert ball n into the vase. >>>>> >>>>> My question : what do you think is in the vase at noon? >>>>> >>>> A countable infinity of balls. >>> 1) It's not clear to me what you mean by that phrase but I'll assume >>> the standard definition. Still, the question remains of which balls you >>> think are in the vase? Does every natural number, n, have a ball in the >>> vase labelled with that n? >> Conceptually, sure. > > Yes or no? What is the set of balls in the vase at noon? Which balls > are in the vase and which are not? > >>> 2) How come noon "exists" in this experiment but it didn't exist in the >>> original experiment? Or did you give up on claiming noon doesn't >>> "exist"? What does that mean, anyway? >> Nothing is allowed to happen at noon in either experiment. > > Nothing "happens" at noon? I take this to mean that there is no > insertion or removal of balls at noon, yes? Well, I agree with that. > But what relevence does this have to the statement "noon does not > exist"? What does that even *mean*? > > When you've been saying "noon doesn't exist", you actually mean to say > "no insertion or removal of balls occurs at noon"? > > How about this experiment, does noon "exist" in this experiment : > > Insert a ball labelled "1" into the vase at one minute to noon. > > ? > >> They both end up with countably many balls in the vase at noon. > > For now, I am going to try to restrict myself to discussing this new > experiment, because I want to understand what "noon doesn't exist" is > supposed to mean. And, again, your answer is ambiguous. I asked which > balls are in the vase at noon, not the cardinality of the set of balls > in the vase at noon. I then asked whether "noon exists", not whether > anything "happens" at noon. Please try answering the questions people > actually ask; it aids in communication. > >> The experiment's stated sequence logically precludes that the vase become empty. > > It logically precludes that balls without a finite natural number on > them get added to the vase, but that doesn't seem to bother you. Ho > hum. > > <snip more stuff about original experiment> > The iterations of insertion and removal are specified as such, and their times specified, such that the number of balls is a function of time, discontinuous for finite n or t, but constant for n, and constantly exponential for t. It is true that: 1) the vase contains balls, is thus non-empty, at every time before noon. 2) No removals occur at noon. 3) The vase can only become empty, after having contained balls, though removal of balls. |