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Are you violating someone's rights when you're doing an animal? Note that animals don't have rights, as DavidOdden neatly put it, so this is not a valid argument at all. However, if an animal is someone else's property, this would be illegal (without the consent of the owner) and would be an issue of property violation. Not to mention how depraved of dignity (or perhaps I should say sex, but then if you are not esteemed much by anyone I doubt you can get any) you would have to be to engage in such an action. By the way, how did you come to think about this issue at all? What made you ask this question, neverborn?

valjean, your question was answered. Immoral means are no means to an end, and no end justifies its means. You are now jumping into a different story, which requires a broader context. To the extent that your previous question can be applied to it, I will only say that tax as a means to public education is immoral and unjustified. Why? Because once you've been robbed (nevermind whether through tax or otherwise), it wasn't just your money or posessions that were taken. There is no knowing what kind of fututre you've been robbed of, but it is certainly better than the one you will fall victim to now. Taxing is a robbery sactioned and performed by the government. So, tell me what sort of mentality considers this "common sense?"

Cake, Ah, I see. For some reason (I think because you said "Things like ends..." rather than "Terms like...") I thought you were talking in the sense of concrete ends and concrete means, in which case they aren't concepts. Given this context, your argument does make perfect sense (with the slight corrections DavidOdden provided). And, it is a good argument. In a nutshell: Evil is evil because A is A, and no amount of B that comes afterwards changes A into non-A.

I'm afraid I don't quite follow the logic of your argument. I stumble on the first sentence: Ends and means are concepts? What kind of concepts? How do these "concepts" relate to someone's intent?

Assuming that you mean that it is "good" ends which should somehow justify immoral means to such end, tell me first which immoral means ever led to what good ends? On the other hand, if your immoral means ever get you to some kind of end, how do you know that this is the end? Just because you have stopped utilizing the means that got you to this end? What if there are consequences to your actions which you cannot predict because you know nothing of the position you are in? Consider that the end is you becoming a doctor. Consider the immoral means to this end being faking a diploma and other neccessary papers. So, you become a doctor. Is this the end? How can it be if your problems are just about to begin? How are you supposed to treat a patient, since you don't know the first thing about medicine? What if someone finds out your papers are fake? The end is only reached when you have, or can have, complete control of the situation you are in and nothing can take away that control (except perhaps a disaster). What I'm trying to say is that immoral means lead to no ends. They only lead to perpetual torment and eventual failure.

I really liked "The Neverending Story." I've seen it a long time ago, but I thought that it portrayed very well what's going on with art today (when people with lack of imagination become self-proclaimed SF or fantasy writers, when many movies are made by templates and stereotypes, when music artists are made by "remaking" another artist's work, and when painters or sculptors are great if they create something that resembles nothing).

I am an amateur photographer and I have had the chance to have my photograps used by an artist as a motif for his painting. Comparing my photo and tha artist's painting, I have come to realize why indeed photography is NOT an art form. My photo was that of a large landscape, and I think it is one of the best photos I have ever done. However, looking at the artist's painting, I noticed how he had neatly removed some things. For example, the camera had recorded various shrubbery that the painter deemed could spoil the theme of his painting. By this simple decision, the artist had removed the unimportant and the inconsequential from his work in order to emphasize what he saw as important for his theme. The camera cannot do that. Yes, you can use various techniques to emphasize what you liked about the particular shot, but the imperfection, the unimportant is still there. The greatest artist includes in his work only that which he sees important, only that which emphasizes the subject of his work. Whatever you find in an artwork is important to its subject and the principles the author wanted to expose. This is why, when using a camera, you are not an artist, but rather a journalist. Whether you take shots of scenery you have no control over, or of the scenery you deliberately set up only for the photo, you are not making a work of art. There is always a shadow that doesn't have to be there, or a nuance that the artist would not want to include in his painting for some reason. As a photographer, you do not choose what to include in your photograph - whatever is in the scope of your objective lens goes in whether you want it or not. Thus, in an artwork, whatever you find inside was included because it is important; in a photograph, whatever you find inside is included simply because it is. I also suggest reading the last chapter of OPAR.

You don't have to respond to criticism. What you have to respond to, if you hold that you are right, is an argument against your position, for which you think is flawed in some way. Simply stating that laissez-faire is "idealistic [sic] or impractical" because it doesn't account for something vaguely described as the "real world" does not constitute an argument. I put a [sic] near "idealistic," because ideals are practical - at least Objectivist ideals are. In any case, that laissez-faire does not account for something in the "real world" is gibberish. It, in fact, does - and more so than any other conceived social system - which is why each and every one is destroying itself today.

I'm writing a novel for children (at about the age when they go to primary school), and it's theme is related to knowledge and the sciences. The main character is called Marco and the time when the plot happens is distant future (which is somewhat similar to Anthem, only I focus on other things). Marco goes on a journey and on it he finds the library (which the characters call the book-house), and it is managed by "The Librarian." Enjoy: "The Librarian" It is I who's the one to manage this place And nothing inside did I ever displace. This shelf contains the books of the past And that one the tales that were written to last. Philosophy is always on a separate shelf And that my dear, I tell you myself It's very wise to keep it from physics away But still it is close to lyrics today. And the members have always been filling my book Or I never allowed them round shelves to look. If you should now be member thirty and three I'll let you inside the book-house for free. (Marco interrupts:) Say thirty and three, that must be a lot; So please can you tell me how much numbers you've got? (The Librarian replies:) Mathematics is a very, very difficult subject And it never has been of my studies an object. But Marco my dear, to read you must learn In order for all this knowledge to earn. I can teach you, my dear But look over here To do that you see, In my book you must be.

I have a computer game called "Freedom Force" which seems to be based on Freedom Force comics. I've never heard of a Freedom Force comic, but according to what I saw in the game, it was published by "Irrational Comics." I can't tell you about the comics, but the game is a lot of fun - what you do is you take a group of superheros and you fight off the bad guys, who are often typical collectivists. There is also a small amount of self-sacrifice being portrayed as a virtue, but not as much so as to spoil the game, or even the comics - if they really exist. Do be careful not to confuse it with Marvel Comic's Freedom Force.

It's called "potential energy." Like with the example of gravity - there is a gravitational force that is pulling you towards the center of the Earth, and there is the force of the ground which stops you from falling. So, you have potential energy and there is even a formula to calculate it. So, if the ground miraculously disappears beneath you, your potential energy will turn into kinetic energy and you will fall in the direction of the force of gravity.

And just how would that be moral?

Photons are energy. Therefore, I'm thinking in the lines of energies canceling out when two light beams interfere destructively. However, I'd rather first revise some interference equations before submitting this as my final answer.

Yes and I know that. However, it is empirical (or better yet, intuitive) to draw a conclusion from only the premises I mentioned in my last paragraph (using of course, no formal method). I don't know how much of a mathematician Felix is, so I thought it better not to get too deep into mathematics in my reply to him. Heh, but then I had to answer for it to someone who knows mathematics, and you know the rest of the story.

It was an error! I don't know how much more explanation you need to understand that. Thanks for the correction. I don't know if you read that post in particular, but I mentioned that I need to translate all mathematical terms from my language into english before I can use them. This is very delicate, especially because dictionaries usually don't translate mathematical terms into mathematical terms. As for this particular term, I must have misread it on Mathworld. You are constantly losing context. Look back! I mentioned that Felix' sets are both discrete because I was pointing to my error in the usage of the term density! Discrete sets aren't dense. Instead of speaking of density, I should have been speaking of cardinality, for the term is valid for both - dense and discrete sets - and if I did that, then there would have been no confusion. As for my original idea, of what exactly I meant about R being denser than Q. Q is a proper subsed of R. This means that R has all the members of Q, and some more. Square root of 3 is an example. Moreover, between each two members of Q there are infinite members of R. Remembering these two premises, it is a purely empirical (not mathematical) conclusion that R is denser than Q.

No! If it was, who knows what computers would be like today. Possibly several decades behind in technology. I couldn't stand that! Yes. You can see my original definition of what it means for a set to be denser than another set, is in fact the same as the definition which tells which set has greater cardinality. I made a loophole somewhere as I tried to explain infinite sets without using infinity, and I should have used cardinality instead. Set S proper subset of X is descrete in X if every x element of S has a neighborhood U subset of X, such that S interjection U == {x}. So, both of Felix' sets are discrete in R. If set S is discrete, then it is certainly not dense, so I misapplied the term "density" in my previous posts, which is what I was trying to get across.

It would be nice if my first response could have been that - but it couldn't. For starters, I thought the formula is come from your imagination. I've never seen it or heard about it and I couldn't have concluded that it has to do something with computing, especially because my response was to Capitalism Forever and I haven't read your previous posts. Haha! That was great, thanks. Indeed, even a negative assertion must have some basis. However, this doesn't mean you can prove it beyond all doubt. A positive assertion, however, can be proven - beyond all doubt. I have given you the exact definition of this. If you don't understand, consider this: square root of 3 is a member of real numbers, but not a member of rational numbers. Yet, it is larger than 17/10 and smaller than 18/10, larger than 173/100 and smaller than 174/100, etc. It is here somewhere, in between of all the rational numbers. For a set to be dense means that there are infinite elements of that set between any two given elements of that set (so when I was speaking of natural numbers and their squares, I should have said that they are equally rare, although not even this is the correct mathematical terminology - I should in fact be speaking of cardinality - so consider this a correction. Note that the two sets are both discrete).

Yes, but there is no need here to go beyond standard ordering. There really isn't (except that the terminology isn't what a mathematician would use), if you are familiar with mathematical theorems concerning this. However, the context of my response to Felix (which I kept in my response to Hal) also included that he is misunderstanding the concept of infinity. Therefore, I was trying to explain to him that there isn't anything "magical" in it, by trying to explain his confusion (remember the sets he suggested) without much involving infinities. The problem with infinity is that when many people hear it, they tend to blank-out on it, and/or they drown in mysticism. Exactly what I said - compare infinities. If you have two infinite sets, you can't count how many elements each one has (like you would do with finite sets) and then compare the result. You must instead prove that there is/isn't a bijection from one infinite set to another to prove that one has/hasn't a greater cardinality than the other. You mean most people who studied higher mathematics? I didn't say that mathematical logic and set theory weren't important. I only said, and apparently I was wrong, that 2 = {{}, {{}}} has no practical value. I have never seen it, and I studied both - logic and set theory - and I applied both. When I saw it in your post, I thought it was some kind of a joke. Still, I may encounter it in my further studies.

Yes, but what you are comparing is not how big they are, but rather how dense they are. Both the set of real numbers and the set of rational numbers spread into infinity both ways. You can't ask how far do they go into infinity, because infinity means non-ending. There is no end to either rational or real numbers. What you can ask is how dense they are. It is because of density that the cardinality of real numbers (which is more dense) is greater. For set A to have greater density than B means that you can't form a bijection (transformation one-to-one and onto) from B to A. So a set with greater density has greater cardinality. For the two sets proposed by Felix, there IS a bijection from one to another. It is f(x) = x^2. So, this means same density and same cardinality, even though it may seem that the set containing the squares is not as dense as the set of natural numbers, for there are numbers in between. However, between 4 and 9, there are a finite number of natural numbers. Between any two closest squares, there are finite number of natural numbers. Finity is negligible in comparison (for lack of a better word) to infinity. So, both sets are equally dense and have the same cardinality, which is why they are "of the same magnitude" as Felix claims. When you are comparing the cardinality of two infinite sets, you are basically comparing their density, not their size, and when you say that one infinite set's cardinality is greater than the other's, you are saying that its density is greater, that there is no bijection from the second to the first. In effect, you are not comparing infinities. There is no mathematical theorem that will allow you to compare infinities.

Your choice of words doesn't add up to your argument, Felix. Do you perhaps have some basis for this claim? I submit that the term existed before it was imported into the field of mathematics. There is no "infinite amount" of anything. Your problem is that you are trying to grasp the concept of infinity as something finite; something that ends. Infinite is that which has no end! Again your notion of infinity being finite. There is no "magnitude" of infinity, and you can't compare that which is infinite. There is no magic to it either. What doesn't end, doesn't end. You clearly don't know much about mathematical infinity... ...hence this statement. However, there is one curiosity which I would like to know about before saying more: what do you mean by "reasonably used" here? If you are suggesting that it was his work that led him to depression, perhaps you have some proof to back this up?

If you really don't think that 1/3 = 0.333~ then try dividing 1 by 3 on paper. If you really think that 1/3 != 0.333~, then you expect that in your calculation you might find that it is something else. Do you still not believe me? Keep dividing. Still don't believe me? Keep dividing. Ad infinitum. Eventually you'll give up and admit that indeed 1/3 = 0.333~ and subsequently that 1 = 0.999~. If you have studied higher mathematics, you may find my proof very useful. It's in post #53.

Uh . Sorry, Eager_Logician, LauricAcid is absolutely right. I don't know why I suggested this site to a beginner. However, once you get started on the path of mathematical logic, this site will be a good reference.

My exact words were: "However, the longevity of a system based on arbitrary axioms is determined by its practical use - if there is none, it won't survive. It will be dismissed as junk." This statement holds. To be frank, I have never seen this formulation of natural numbers. I did study set cardinality, so cardinality of set {0} is 1 and of {0, 1} is 2, but the actual formulation of numbers 1 = {{}} or 2 = {{}, {{}}} I have never seen. And no, none of my textbooks has it. However, I doubt that this particular notation has much meaning in design of digital computers. As I said, I've been studying engineering and have even designed several chips myself and I never needed this. Not even when I wrote assembly programs did I need it. How so? I don't know how else you prove a point you were making, unless you are willing to type it all here, which I don't doubt you would were the circumstances right. If I was asked for proof of something on these boards (note that I'm only speaking of proving things here in the forums), I would provide a link to a trusted site, rather than type it all myself. So, my comment was only meant to say that you hadn't even bothered to do that. I grant you that your goal isn't proving anything, however, I am curious as to how you would have someone prove to you a nagative assertion? If I was still convinced that 2 = {{}, {{}}} has no practical value, how do you suggest I would go about proving it doesn't? It is impossible - which is why only positive assertions need proof. That is the way I understand it. I think (one of) our misunderstanding is in that we are not speaking in the same context. If you remember, I replied to Capitalism Forever about defining 2={{}, {{}}}. I was speaking about formal systems, but not in the context of formal systems. My first post was only intended as an interesting side-note.

NOTE: Quote tags didn't work for some reason, so I used bold tags instead. It is for whomever makes an assertion to support that assertion. Rarely are things that simple. It is for whomever makes a positive assertion to support it. You made a claim that 2 = {{}, {{}}} has a practical application, I said it didn't. I think it more than clear who has the burden of proof. But of course, your whole post is everything but a response to mine. I have asked you a simple question, and all you can say about a concrete formula is to vaguely name the context it is used in. I'm a student of engineering and studying about the digital circuitry and how they are built, transistor by transistor, didn't require me to learn about 2 = {{}, {{}}}. In all my textbooks from mathematics to digital electronics, there is no such formula - and, mind you, I have studied mathematical logic AND set theory. The nearest that came to 2 = {{}, {{}}} was a set of empty sets. But it's another thing to assert as a unqualified generalization, as you did, that a particular formula does not have any practical value. You call my statement a sweeping generalization? I was referring to one concrete formula which I held, and still do, until you prove that I am wrong, to be a figment of your imagination. Your response in a footlong essay did not provide any proof to the contrary. How difficult is it to post a link to a page explaining it, if it is so important and so famous as you claim? You skipped responding to my illustration. You know very well that the implications when speaking of the practical value of a fabric are very different from when speaking of the practical value of a mathematical formula. It's not only that you can't make clothes out of a mathematical formula, in your "illustration" you are actually holding the fabric, so, unlike the formula we speak of, it CAN'T be a figment of anyone's imagination. And if I'm pointing to a formula on a chalkboard, and you say, "That formula has no practical value," then what is the basis of your assertion? None at all. I don't need a basis to not believe you. What is the basis of your assertion that it does? You seem to be lost about who has the burden of proof. When Cauchy invented his integral formula, he didn't ask around to have it proven to him that it had no practical value. In fact, had there not been proof of its validity, nobody would believe that this formula is so important. Your formula seems to be a definition of something. Therefore, a good example of what it's good for should prove that you are right and I am wrong. If you said, "That formula has no practical value," then a reasonable response is, "How do you know that?" No, the reasonable response is "Actually, it does." And then provide the proof. And to say that the formula might have practical value is not saying that the formula probably does, but only that it might. A god might exist. An invisible pink unicorn might exist. If these were my thoughts, then I would certainly say: "Your formula might have a practical value." As it is, it doesn't - not within the context of all my knowledge. No, you've committed a non sequitur. By saying that a certain thing might be true does not entail that one thinks that anything might be true. If I assert that the train might arrive late does not entail that I hold that two might equal four. And you are dropping context, as well as twisting my words. We weren't predicting future, we were talking about a mathematical formula. You claimed the proposition that the formula has no practical value. The burden of proof for that claim is on you for that proposition. If I claim that the formula does have practical value, then the burden of proof is on me for that proposition. And if I claim that the formula might have practical value, the burden, though it be far less, is also on me. 1. No, if I claim a negative, I don't have the burden of proof. Expecting me to provide proof for the negative assertion is like expecting me to provide you with proof of god's nonexistence. The burden of proof is on you, not me. 2. Yes, you do claim that the formula we speak of has practical value. 3. No, you don't claim that the formula might have a practical value, you claim it does have a practical value. We may not know of a practical use for a particular piece of fabric or formula on a chalkboard, but that does not entail that we might not ever know of a practical use for them, and especially does not entail that other people might not have a use for them. Practical examples of the fallacy of such thinking are all around us. I'll take my parents as an example. They are building a tennis court. I ask them why. They say it's because somehow, some day it may pay off. They build apartments at sea. I ask them why. They say it's because they are hoping to sell them. They borrow money to neighbors, they are building a house for my grandmother. I ask them again why do they do these things, as they make no sense? My dad owns a private company and is making applications for other companies. Yet they do all of the above, they are buying land, planting crops, raising fences. They say that these things are investments in the future, that somehow, some day these things might all pay off. They do these things even if the future of all these investments is uncertain. Not long ago, all of this cashed in. My parents almost went bankrupt. Now, the apartments they were building are left unfinished. Neighbors didn't return money. My old grandmother's house was torn down, but the new one wasn't built. The tennis court is almost ruined, but there is no money to renew it. What they did was that they overextended their investments into an uncertain future. Holding on to ideas only because they "might" work out is not good enough - not in business, not in mathematics, not in philosophy - not anywhere. On all decisions, you must reach certainty - something either is, or it is not. In a business venture, something either will be, or it won't be. Once you have reached this kind of certainty within the context of your own knowledge, only then can you make a decision - will you keep holding on to the idea, will you set it in motion, or will you abandon it. You don't deal with "mights" and "probabilities". This is true in science as well. Holding on to formulas that have no value is irresponsible if it is done for the sole reason that their application might one day be found. I could come up with a thousand useless definitions if I wanted to put my mind to it. If and when any defnition becomes necessary, it will be made, no sooner and no later. Even if it did exist previously, if someone happened to stumble upon it somehow, it didn't have any practical value. Imagine a Neanderthal discovering an oil field. What's it to him? Within the context of what he knows, the oil is valueless. Even more than that, it is a nuisance; junk. If you mean for me to show you a particular machine or piece of technology that uses that fiesign, then I don't know of such a thing, especially since I've never looked for such a thing. But I did give you an answer as to the overall usefulness of this formula as part of a mathematical theory and mathematical thinking that has fed, probably more than anything else has, into the invention of modern computing. Again, this is a vague answer. With this information, I can't even begin a search on google. Actually,b]k for John von Neumann and read through the thousands of pages about him in search for the formula, but why? I don't believe you to begin][...]Anyway, again, my initial point was not even in showing that the formula does have use, but rather to ask you how you had arrived at the conclusion that it does not. I believe I have answered that. Oh please, what a blus/b]e. Not only do I know what consistency is, but I can give you a rigorous mathematical form the principle, and discuss several important theorems about the relations among consistency, satisfiability, provability, cardinality, number systems, etc. Not that I'm such an expert on the subject, but I know enough about it to make your challenge foolish. Actually, I was asking from a purely philosophical standpoint. Nonetheless, it is nice when philosophy and natural sciences go hand in hand like this.

Are you joking? You need to brush up on those terms. Arbitrary and probable are both defined in a dictionary and I'm sure those definitions will do. To claim that something you don't know anything about might be true, you implicitly show that you are functioning on the faulty premise that anything goes; any figment of anyone's imagination might be true then, according to you. I never heard of 2 being defined as {{}, {{}}}, so I asked for a concrete example (proof) where it is practically applicable. You haven't done that, instead you inverted the principle of who has the burden of proof. Still, I have never heard that 2 = {{}, {{}}}, or this formula's practical applications. So when you are working within a formal system you can't make an error? Do you know what consistency is? It means that your axioms don't contradict each other, and that your formulas don't contradict with your axioms. You can be inconsistent at any point of building a system, not only during the axiomatization. And the application is...