hunterrose

Hmm. That's not quite the objectivist ideal society. It'd be more appropriate to say that an ideal objectivist society would be one in which individuals are free to pursue their desires, with the notable restrictions on initiation of force. The enemy of such a system wouldn't be unearned rewards, but unitiated force. Yeah, your "system" sounds awfully strict Nothing'd be (legally) wrong with a person giving their money to a worthless relative in the ideal society. Even if preventing the unworthy from prospering were a valid goal, using force would negate whatever advantage was to be gained. The simple fact of economies of scale and large amounts of cash don't make entering a field impossible. It might make this "talented" person have to work harder than he wishes, but it's not monopoly. Monopoly only exists when it's impossible for competition, not merely difficult or even highly unlikely. The things you've said may make things difficult for the "talented," but nothing about meritocracy says that the talented should have their disadvantages forced away. Ultimately, it sounds like a bizarro entitlement.

Hrm. I'm a huge horror movie fan, but I haven't seen any previews that looked interesting lately. I might check this out this weekend.

I've been working on phrasing my thoughts as coherently as possible. I'll start off by saying that I do agree with the Objectivist virtues. Unfortunately, it seems to me that ethics are inevitably empiricist, even though I believe a non-empiricist ethics to be desirable. Hopefully, this example will be adequate to explain my misunderstandings: The Artful Dodger lives by exploiting man. Pickpocket, con artist, panhandling, it's all been his MO since his earliest memories. He has some qualms, among which he won't murder another person. A. Our Artful Dodger is being chased by thugs ( imagine that ) In his goal of escaping, he runs down an alley which forks in two directions. One direction has objects that provide hiding places, a staircase to higher floors and the roof of an adjacent building, and a low brick wall which can be jumped to get into a heavily wooded area, and exits into an open street. The other direction has 25 foot walls on each side and is a dead end. What differentiates the two choices? Wouldn't whatever you call this be inherent value? B. Oliver wishes to convince the Dodger that honesty is a virtue. "Sure it is... sometimes," the Dodger replies irreverently. Oliver replies that the Dodger will be better off if he adheres to the principle of honesty. "You're saying it's always against my interest to lie? What validates that, not sometimes, but always?" Oliver tells the Artful Dodger that honesty is the recognition of the fact that the unreal is unreal and can have no value. The Dodger looks at Oliver peevishly "... what validates that, and not just sometimes?" How does Oliver reply?

I'm going with that no-name actor. Galt was an enigma, so the movie Galt should be too. A no-name likely will carry less baggage and more mystique than the People magazine set.

Welcome I don't think being theist will cause any problem around here... as long as you don't try to push those beliefs on anyone I'd rather have a pro-capitalist theist on my side over an anti-capitalist theist anyday, so participate away

Despite everything else said, I agree that morality based on principles is superior to morality based on probability. I've read your articles ("Thinking About Principles" and "Why Act on Principle?") I agree with much of your position, but I still have a few questions: 1) How can the million dollars not be an inherent value? Food is an inherent value to (physical) survival, both long and short-term. The only way to thus say that food wasn't inherently valued would be if there were other things associated with it, that is, would make the food's value ambiguous and/or contextual. Wouldn't it then be that food has some inherent value, but that the actual value would be dependent on contexts of how it was gained, what was its use, etc? If food wasn't an inherent need, then what would justify putting emergencies outside of the context of principles like honesty? 2) By this, you don't mean that a person can't exist by robbing banks, or some other vice, right? I'm not one with precise definitions, but I'm "seeing" two points from your argument. Exist - as in be alive. Survive - as in thrive, life to a maximal amount. Are you saying that, while robbery won't necessarily prevent him from existing, it would be impossible for him to survive? Similarly for the use of "unreality," "faking reality," etc. My terminology isn't 100% accurate, but I hope you get my gist. 3) I was originally trying to make the point that robbery isn't a "one-time" or occasional act, but a mindset that'll continually, by the existence of such a set of values, impede your life/survival/whatever. Your claims of empiricism aside if this was derived from the nature of theft, couldn't a proper principle, with perhaps some other similarly derived validations, that such acts are always bad be derived? If so, you can fault me for stating secondary (and tertiary) reasons not to rob, but you couldn't call my original and prime argument empiricism. I never said that was the only reason. I also didn't tie everything together, but it wasn't the prime reason, either. Obviously, I still take offense at that

The single and cogent argument was made in my first post [ (1) in post #21 ], and it wasn't empiricism Living by driving a car is less "risky" than living by robbing banks Being a driver has a better chance for survival than being a robber, but the robber still might luck out. Either you misinterpreted my use of which lifestyle is better, or you were wrong about living by driving being no less "risky" than living by robbery

Well! Which of your validations was different from mine? What are the differences between your argument and mine? Haven't you misinterpreted me? Were you disagreeing with me because I used the validations instead of the principle first? I observed instances that validated my principle. And while I didn't state the principle, my instances all lead to that. As he didn't want the principle, but why the principle applied, I gave the validating instances.

If you want my stance in such a format, it'd be: Man is rational, so don't do it. From man's rationality, I derived that man will act to prevent you from immorality, before, during, and after the immoral actions. I used some other things too, derived from identity, knowledge being contextual, and the nature of values. I could've included a lot of other stuff, but I believe that was sufficient. Aren't your positions derived from something similar? I also didn't use "unpredictable" as you're using it, at least as a primary justification. And I certainly didn't justify morality by saying that moral actions simply carry more statistical probability of success than immoral actions. The problem with questions such as "is robbing a bank moral if I can get away scot-free?" is that they attempt to take away the context of reality. Unfortunately, the problem with the answers is that you can only say yes, say no, evade, or give a contextually dependent answer i.e. yes in these circumstances, no in those circumstances. Which should I have given? What grounds would I have to say "no" to such a moral question, when morality presumes men are rational, and identity exists? Of course, I could've evaded an answer by digressing, ignoring hypotheticals, attacking his moral status, claiming to not know, giving an undefined maybe, etc. I imagine you still have disagreements with my stance, though, and I'd like to hear how this position is faulty. What immorality does my stance allow? As far as it goes, I have some disagreements with your position, too.

Oh, stop fear-mongering I hate to distract you from efforts to analyse Potter with my questions and statements about the nature of Objectivism. I didn't realize Rowling's books were so important. I apologize for affending anyone, especially if someone would tell what was wrong in my comments (as IntolerantMan is too busy piecing together the significance of the Half-Blood Prince.) Isn't Objectivism based on objective reality? Doesn't it have the same relationship as science, in that both are derived from objective existents? Did Rand fabricate principles, or derive principles from objective reality? As IntolerantMan has used my quote, I can only imagine one of these statements was either false, or not to be asked. Personally, I think nitpicking over whether such things are "created" or "discovered" is irrelevant, but if I'm wrong, I'm wrong. Let me know which ones were prohibited or incorrect, and I'll make sure it doesn't happen again.

You never know; you might learn something Here's a part of my position. The reason examples aren't "arguments" is that they don't apply across the whole system. I.e. "You might get caught" isn't an argument, because sometime people don't get caught, and thus the example doesn't apply in those cases. If the example did apply across the whole, then you would be able to derive an ironclad principle from this ironclad example, right? Likewise, if some of the examples derived from a principle aren't ironclad examples, then the principle can't be ironclad either, can it? We could pick straws at whether stating ironclad examples is acceptable, or whether such examples have to be distilled into the principles inherent to the examples before justifying the immorality of robbing, but it's not that crucial IMO. I could just as easily say "don't rob the bank because A is A and existence exists." The examples can be derived from that principle, but still, such a statement couldn't be expected to sway anyone... Thanks for the blog! I've never been interested in reading daily routine or political party zealotry blogs, but I do like the idea of Objectivism blogs. Can you believe this is my first time looking at a blog? Anyway, I found some relevant points in the Sept 4, 1, and Aug 30 posts. I didn't have time to check the pertinent links under Best of Anger Management, but I will when I get back home. Of course, that begs the question... what are these objective life-based moral principles, and how does illicitly receiving, say, a million dollars in an undetectable way, violate these principles?

I'm going to have to give a full reply tomorrow, but I'll say that you I believe you've misinterpreted me, DPW You also haven't said why robbing a bank would be wrong... especially without referring to disliked empirical examples.

Natch! I don't think it affects my argument, though. Interesting point about romancing. It certainly wasn't classed with the "rational" actions, as I used them previously in # 23. Apparently I was being imprecise, but romancing would still be a "rational" action, because consensual acts don't have this unpredictable threat that robbing a bank would. Robbing a bank is unpredictable and unreliable not just because you can't be assured you'll get away. It's an unreliable means to support life particularly because of what they'll do after they find out. If I get rejected by a girl, it's not likely she's going to hold it against me (unless she's Kathy Bates ), and even far less that she's going to think of me as dangerous to her life and thus a threat to her. Obviously, nature isn't going to react vindictively toward you. But robbing the bank? That'd leave some long-term consequences, unlike romancing and science. Robbing a bank is immoral, primarily (IMO) because when you commit such acts, the interaction between you and the victim isn't over. This is why initiation of force against other men is the primary vice it's the most dangerous thing you can do to your own life Now, if we ignore the possibilities of this retribution, then I'd stick with the prior point about applying your mind toward nature (and consensual acts!) is more reliable than the chance to successfully rob the bank. And if you ignore the risks of getting caught in the act or failing, then I go back to once you establish robbing banks in particular reward/risk scenario as a value, what would possibly dissuade you from doing it again when that same reward/risk scenario reappears? Those are all secondary, but valid in and of themselves IMO, arguments to the long-term consequences/retribution/reaction argument. And if you still want to exclude something, well, you can eventually exclude enough stuff to make robbery "rational." Every violation of another's rights (why the quotes?) has risks that are more unpredictable than nature itself. With nature, the worst you can do is fail. You might get hurt as part of that failing, but once the action is over, nature isn't going to hold a grudge against you. The worst unpredictability of robbing a bank is the threat to you even if you do get away with the money. The victim begins searching, organizes with others, etc. Have you ever seen Ransom, with Mel Gibson? Or read The Count of Monte Cristo? The Punisher? Batman? And I won't even get into d'Anconia.... Even if you ignore such "extreme" people, the fact that people organize government and police in the first place is an indication of that unpredictability and it's threat against every use of immoral force. The defensive action of rational beings can't safely be predicted as to how they'll react, how long they'll pursue you, or what kind of changes your robbery and their apparent need to better protect themselves will have.

Well, it's not that I don't believe it; it's just that I don't think you can prove 0.999... = 1 by such methods. It seems reasonable, but still seems to depend on accepting another infinite expression. Inevitably, using these methods, you have come to a point that you say it's "obvious", but not proven, that 0.999... = 1, 8.999... = 9, or 0.333... = 1/3. That's not to say your proof on #53 is the same. I do acknowledge and agree with all the other proofs, particularly the ones based on the principles behind (real) numbers. Thanks to you, LauricAcid, and many others that have shown them

I don't think you'll be satisfied by my answer Let's assume that this bank robbery is a "one-time" action, and at all other times only rational values will be pursued. Rational actions have risk/reward levels too, as you mentioned. But when you decide to drive a car, you can know how the car works, what conditions increase your danger, how to repair it, etc. The brakes aren't going to choose to malfunction, principles of combustion don't warp out of boredom. Blowing a tire doesn't change the probability or conditions of blowing a tire at any other time. None of this applies to irrational acts. But let's assume that a guard's break times, the bank president's efforts to mark bills, and choice and timing of security system upgrades are just as predictable and dependable as nature. We'll consider the case where, even in this one irrational act, depending on your ability hide your acts from volitional, unpredictable men can be calculated with scientific precision. Why is it wrong to commit a bank robbery that has no negative consequences, on rational principle? It's not. Take away the increased risk of depending on unpredictable men instead of predictable nature, the detrimental effect of your fellow man making proactive efforts to stop you, others finding out and reacting, the possibility of failure, the limits of any amount of wealth (what amount of money would be "enough" to an Objectivist??)..... i.e. the capacity of men to be rational and the possibility of failure? I suppose this question is just hypothetical. mrocktor, the bank robbery would be irrational because such conditions can't exist, but if enough conditions existed, robbery would be just as rational as any virtuous act. Violating other people's doesn't always lead to less overall value for any single instance, but knowing when those "not always" are at hand would require some faculty that'd make the robber superior to us like men are superior to animals. That's probably not satisfactory, but really, are any of these conditions even possible?

Welcome

I don't know the declarations are wrong until they're proven wrong. However, I'll try to be more conciliatory. Okay, I think I've got answers to all my questions on the "0.999... = 1" topic except this one. Aren't the "one third" and "9x = 9" "proofs" similar to the "terrible [rationales] they give in 10th grade algebra to keep your orientation?" I accept all of the other answers, but these two proofs can't be sufficient, can they? 1/3 = 0.333... 3 * (1/3) = 3 * 0.333... 1 = 0.999... Doesn't this rely on the fact that 1/3 = 0.333...? I can't declare it but what answer do you give to someone who considers that as dubious as 0.999... = 1? "I'm the expert, shut up?" The "9x = 9" proof seems similar. 9.99... - 0.999... = 8.999... right? Now if I get 8.999... as an answer, that doesn't get me any further than I started out at, it seems. If you're saying I commited an error in getting 8.999... as an answer, what was the error? Don't misinterpret me: I understand the other proofs; I can't declare these two former proofs to be inaccurate, but they seem to be "orientation" proofs.

Interesting question. I'm not sure it is good to distinguish between Politeness and politeness. There are many non-Objectivist meanings of justice, and it seems that the only reason to declare any meaning as "Justice" is one's personal values. That said, there are different connotations for virtues and vices, and it may be best to be specific about what you mean, jrs. Politeness could be considered as a virtue (if it's well defined,) but it wouldn't be one of the fundamental virtues. It'd probably be hierarchically equivalent to the virtue of Smiling at Strangers. Both can facilitate one's life in certain contexts, but those contexts are extremely narrow, whereas the fundamental virtues (of which justice is one) apply almost everywhere. Actually, I think your definition (or use) of politeness ("demands that one not call unnecessary attention to the faults of others") seems accurate, though it obviously lives or dies based on what is "unnecessary." But to answer the questions. 1) The way to reconcile opposing virtue is (for Objectivists) to remember that a virtue is only a virtue in its capacity to be beneficial to a man's life. If only one is beneficial in a situation, only that one is a virtue in that particular scenario. If both are exclusive, yet both beneficial to man's life, I suppose you'd choose the one that were of greater value to life in that situation. 2) There is a polite way of dispensing justice. 3) Must Objectivists be impolite? Certainly not! That sounds like making rudeness a virtue. However, politeness would be an exceedingly slight virtue in most contexts for an Objectivist, so politeness can be superseded by a load of other virtues. It's pretty far down on the totem pole.

*hunterrose proceeds to attack* I think you are right as to your answer, but not your reasons. It is at the expense of someone's else's life because you're taking the lifeblood (the productive work) of another person. Self-preservation is not immoral, but of course we must be careful by what we define as "self-preservation." You would have to restitute the victim of your theft, and choosing to die would be immoral IMO, as iouswuoibev (nice name ) said. Virtues are only virtues in the context that they are means of maximizing one's life. But doesn't this eliminate the concept of emergency? There is a thin line between emergency and carelessness catching up with you, but I'm not sure this situation falls into the carelessness category, let alone that one should be obligated to die because they have bad credit As studentofobjectivism said (though with different rationale) A person has no right to claim the life of another person in any situation and this includes emergencies. My position is that, even though this would be a violation of another's rights, it isn't immoral. That's not to say that emergencies justify any violation of another's rights, though.

Wrap ups LauricAcid, did you mean if 0.999repeating and 1 are real numbers, that there is a real number z, such that 0.999repeating < z < 1? I'm learning as I go on this and came across the fact that there are an infinite number of real number in between any two real numbers. Obviously this would prevent my concept {1 minus an infinitesimally small amount} from existing separate from one. I'm assuming that there's likely some long chain of mathematics that depends on these definitions, so it's necessary that my number not exist for consistency's sake. Or something like that. That being said, I'm still finding it hard to come to grip with the whole thing Does this then mean that the infinitesimally small doesn't exist, even conceptually, wheras the infinitesimally large does exist conceptually? Could we say that the answer to the function of 1/x equals 0? Not just the limit, but the actual function? The red bird example assumes the "red bird" phrase were the only scientific way to refer to a bird that was red. I still stand by the fact that you can't prove that 0.999... equals its limit without relying on [all such numbers equal their limit.] That is, the "proofs" of the premise seem dependent on the premise being true I'm assuming, based on what I'm reading and LauricAcid's said, that [all such numbers equal their limit] is a property of some subset of numbers. Better late than never, eh? Learn something new everyday

Finally! Last analogy: Suppose a scientist defined the phrase "red bird" to scientifically mean a cardinal (the bird.) Thus, any time you use the scientists' term "red bird," you'd be referring to a cardinal. Furthermore, suppose scientists defined any phrase involving a bird and its red color to mean a cardinal. You'd now have no way of referring to other red birds, e.g. a red hummingbird. Though the red hummingbird would be objectively considered as much a red bird as the cardinal, according to science, only the cardinal would be considered a "red bird." In fact, if terms involving red and bird were the only way a bird could be referred to, there would be no scientific phrase to refer to the red hummingbird! That's the situation this question falls into. There is no mathematical way to represent the number { 1 minus an infinitesimally small amount} even though this would be a real number. As terms such as 0.999repeating are defined as the limit, the answer doesn't even need any proof (all of which rely on using the unprovable definition) anyway. Now, there may be a reason this definition is mathematically necessary, but I haven't seen it. Does anyone know if 0.999repeating has to be defined as equalling one? 3. See my previous remarks about how you regard this problem. Moreover, not only are you claiming that there is another context, without limits, in which to address this matter, but now, despite that you've been given information that would disabuse you of your misconceptions, you are, ironically, inspired to claim something even stronger, and even more silly, which is that ANY evaluation of this problem in terms of limits is an incorrect one. How ridiculous. You don't even have a coherent theory to present, yet you arrogate that the existing mathematical theory, which dissolves this problem in an instant, is irrelevant. Who made you the determiner of what mathematics may and may not be brought in to correctly address a mathematical question? Such arrogance. There is another context from which to evaluate the term 0.999repeating. That'd be the definition of 1 minus an infinestimally small part. No limit necessary. Have I missed any misconceptions? I'm not sure I said any evaluation of the problem using limits is incorrect. My thoughts on this have been a work in progress, so it's possible, though. What I mean is that, in the way limits have been used for solutions, you get the desired answer because you define certain terms to be limits, whether they have an alternate interpretation or not. I've tried to present my theory coherently, if I haven't, my bad. This theory you say dissolves my point by arbitrarily defining a term to mean one of its possible meanings. In other words, you're right because the math Powers that Be have decided in your favor, not because your answer is verifiable by proof. You can bring in limits to the equation, but once you define 0.999repeating to equal one, or something similar, the "solution" is self-fulfilling: 0.999repeating equals one because we say so. The only way to prove that 0.999repeating is necessarily the limit of 0.999repeating, is to use calculations that involve the questioned limit. You have to rely on 0.999repeating being a limit to "prove" 0.999repeating is a limit. Self-fulfilling mathematics. First, you don't know what you're saying in the context of infinite decimal expansions, when you talk about infinite sums that are not defined as limits. That may have been true, but is there any reason why 0.999repeating has to be defined as one, especially considering there is no way to prove 0.999repeating equals one?? Or I could just be wrong

Second, there is another concept of what 0.999repeating can mean. On the other hand, you just asserted that .999... not= 1, and you haven't even give a proof; moreover, you haven't even given a definition of .999... that makes sense... The definition of 0.999repeating could essentially be what Bryan said in post #18: the quantity of one with an infinitesimally small part taken away. This quantity cannot be equal to one, as you are subtracting a positive non-zero value from one. If x and y are defined such that x > 0 and y = 1 - x then (for any x) y < 1 But real numbers are not infinite sequences, while they are limits of infinite sequences. ...if you are speaking outside the context of limits, then you haven't said in what sense an infinite sequence is a real number. All real numbers aren't necessarily limits of infinite sequences, if I'm right. From what I can offhand dig up, any number is a real number if it is between the values of negative infinity and positive infinity. E.g. 0.9 < 0.999repeating < 1.5 Let's pretend you have 1 cup of coffee. You take the smallest sip of it that you possibly can. You now have less coffee in the cup than you had before. We'll say that you now have .999~ cups of coffee. That there is no smallest possible sip is exactly why .999... = 1. The poster has it backwards: .999... exists all right, as it is the limit of a sequence and that limit is 1; and this relies upon, rather than contradicts, that there is no "smallest sip" or smallest real number to subtract from 1. Self-correction to #95: The proof does not even rely upon there not being a smallest positive real number but only upon the fact that for any positive real number, there is a smaller positive rational number. That this "smallest possible sip" isn't measurable or possible in any physical capacity doesn't eliminate the concept, any more than the concept of infinity could be eliminated by 'A is A.' If if they both have no physical (i.e. matter) implementations, they are still both valid concepts for our intents. The problem with your corrected proof: For any positive real number x, there is a smaller positive rational number y. By this I take it you mean there is a rational number z, such that 0.999repeating < z < 1 I have no problem with z < 1. However, that does not prove that for any positve real numbers x and y, there is a positve rational number z such that, x < z < y. Quite simply, any positive rational number < 1 is going to be less than or equal to 0.999repeating (as I'm defining it), but not greater than.

Okay, my short answer is that 0.999repeating does equal 1, but it can't be independently proven to equal 1. First, (most?) solution problems: The answers derived (from 9.99repeating - 0.999repeating) were 9, 8.999repeating, and 9.000...00099. If you solely look at the '9' equation, you get your desired answer. On the other hand, since arranging the terms differently gives you answers that don't look equivalent, or at least haven't been proven to be equivalent, you can't necessarily say '9' is the answer without mathematically consolidating the divergent alternate answers. Okay, here I go. As my prior example showed (post # 102 ) "arranging" the series in different ways leads to apparently diverging answers. LauricAcid implied then that the ways I was arranging the series might be sensible, but ignored the fact that his answer involved an "arrangement" itself. Take two examples: the solution that led to an answer of "9", and the solution that led to an answer of "8.999repeating." We'll evaluate these two equations at "steps," where "steps" means that we take the first remaining digit off of the first operand, subtract the first digit from the second operand, and add that to our answer up to that point. For the "9" solution, we'll take out the "9" of the first operand so that the equation goes from the "simple" question of [9.99repeating - 0.999repeating] to the much simpler [9 + 0.999repeating - 0.999repeating]. "9" solution: x = 9 + 0.9 - 0.9 = 9 = 9 + 0.09 - 0.09 = 9 = 9 + 0.009 - 0.009 = 9 This will obviously give us our desired answer. On the other hand "8.999repeating" solution: x = 9 - 0.9 = 8.1 x = 8.1 + 0.9 - 0.09 = 8.91 x = 8.91 + 0.09 - 0.009 = 8.991 -> 8.999repeating. At best the answer merely "appear" different. However, if these two solutions are equal, that'd have to be proven. If you can't resolve the 8.999repeating, you can't resolve the 0.999repeating. On the other hand, if there is some satisfactory way of resolving that, feel free to contribute it. How do you prove the 8.999repeating answer is wrong? [sum of 9/10^x, x from 0 to infinity] - [sum of 9/10^x, x from 1 to infinity]. (@first x values) 9 - 0.9 = 8.1 For x = 0, there is no evaluation. For x = 1, this is 9 + 9/10 - 9/10, which is 9. LauricAcid's saying here that the proper way to evaluate the equation is not to evaluate the answer by the first digit of each operand like I did above, but by evaluating the answer at the same x for each equation, that is, line up the subtraction to subtract values at the same decimal place. Thus, x = 9 - 0.9 = 8.1 would be inaccurate, because I'm subtracting the value of the tenths place from the value of the ones place. His example relies on taking the 9 out of the first series, and then proceeding to operate on similar decimal places. While doing it this way is convenient, it may not be best, especially if doing it other ways leads to different answers. Depending on how you arrange the series, you get different answers. Then you should wonder whether the "ways you arrange" the series even make sense. Who took 9 arbitrarily out the series? And what do you even mean? Suppose we have two woodcutters who are cutting wood from their individual blocks, and putting their cutting into their individual baskets. The baskets are set on a scale such that the scale's value reads as the difference between the weight of the first woodcutter's basket and the second's. The first woodcutter A starts out with a block of 10 kilograms. The second woodcutter B starts out with a block of 1 kilogram. With each cut, a woodcutter cuts off 9/10 of the remaining block, and puts it in his basket. We'll assume that the woodcutters are cutting at the same consistent rate... but that we can allow one or both cutters to put a number of cuts into the basket before we evaluate them at synchronized cuts. If we just allow both cutters to put wood in the baskets without allowing any freebies, you get the 8.999repeating solution. If you allow A to put a cut in the basket, and then synchronize the woodcutters, you'll get your self-fulfilling solution of 9. You can have A put in a million cuts before synchronizing, the scale will always read a bit more than 9. You can have B put in a million cuts before synchronizing, the scale will always read a bit less than 9. Because you arranged the way you evaluate, it appears that the scale should read nine, which it does if you prearrange it. It's greatly simplifying to have the cutters putting an equal mass of wood into their baskets between each evaluation, but that doesn't mean it's the right way to go about. However you want to consider it, you have to work a little harder to get this clean and tidying answer of 9. If it's possible.

Whoa. Who stole your Rearden Metal? I don't know what I did or said to offend you, but that wasn't my intent. I'm sorry you misinterpreted me. I'm working on a more fleshed out rationale, but even without seeing the argument, I think you are exaggerating somewhat. Nonetheless, proof is in the pudding, so to work I go. You might find it interesting

Depending on how you arrange the series, you get different answers. Take [ 9 + {sum of 9/10^x, x from 1 to infinity} ] - [sum of 9/10^x, x from 1 to infinity]: (@first x values) 9 + 0.9 - 0.9 = 9 (@second x values) 9 + 0.09 - 0.09 = 9 ad infinitum = 9 That does equal nine. But you arbitrarily took the 9 out of the series, and that has an effect on the answer. The better way to do it would be to not arbitrarily take values out of the series. At any rate, if putting the 9 outside is valid, it should still give the same value as taking nothing out, or taking any other arbitrary value out. [sum of 9/10^x, x from 0 to infinity] - [sum of 9/10^x, x from 1 to infinity]. (@first x values) 9 - 0.9 = 8.1 (@second x values) 8.1 + 0.9 - 0.09 = 8.91 ad infinitum to 8.999...9991 = 8.999repeating [9.99 + {sum of 9/10^x, x from 3 to infinity} ] - [sum of 9/10^x, x from 1 to infinity]: (@first x values) 9.99 + 0.009 - 0.9 = 9.099 (@ second x values) 9.099 + 0.0009 - 0.09 = 9.0099 ad infinitum to 9.000...00099 Granted, the limit for 9 8.999repeating 9.000...00099 = 9 for all of these. But the problem isn't a limit one, and none of those numbers equals the others. On a similar note, the limit of 0.333repeating = 1/3, but 0.333repeating doesn't equal 1/3, I believe. I too could have mistaken, though