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http://mathworld.wolfram.com/topics/Founda...athematics.html You could use this page for starters, if you are interested in mathematical logic. When you get familiar with the terms, you'll be able to find books you need yourself.

If you hold it does, then prove it. I claim that it doesn't, because claiming that it might would be embracing the arbitrary as probable. Yet if you are inconsistent in applying the axioms, there is no longer a system to talk about. There is a hodgepodge of statements, but that is not a system. Survived how? It's being remembered? So are the dead. Edit: Changed "claim" into "hold".

It is possible to define 2 in such a way. However, it is of no practical value. Just as in your example with the axioms. You can invent any kind of axioms and any kinds of rules. If you are consistent with your application of them, you can have any kind of system. 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. Too bad it doesn't work the same way with philosophy.

Am I allowed to make a copy of this essay so I don't have to read it off the screen?

When I introduced myself to Objectivism and when I began studying it seriously, I too was often surprised to find that my emotions were practically gone. I could remain cold-blooded to anything. If you have only recently began studying Objectivism, I may have an answer for you. If you have read the part of OPAR concerning how you program your emotions consciously, you will know that it is possible to consciously change what you feel in certain situations. This, however, takes time. It will not work immediately, you will gradually need to change your emotional response. Objectivism will make you change many of your emotional responses, depending of course on what your background is. So it is my theory that while this is hapenning, you may be going through a period of this "emotional numbness" when you are trying to change a lot and in little time, so your response to certain situations is somewhere in the middle - neither what you would feel before, nor what you think you should feel. However, it is a curious question: if you think should feel something in this situation about your loss, this implies that you think that you have lost something of value, in which case you should not have broken up with your boyfriend in the first place. You say that you are trying to remain friends. Maybe it is because of this that you feel nothing? Perhaps the value you saw in him was not worthy of love, but only of friendship? One way you can establish whether or not you should feel anything is try to think about what values your ex-boyfriend represented for you. Then ask yourself if you can access these values when you are bonded with him only through friendship.

Debate on this, as well as on any established fact, is superfluous. Someone who doesn't understand the proof and asks a question about it doesn't start a debate, he shows that he doesn't understand it. In other words, in order to understand it, he needs to learn the definitions and theorems leading to this proof. That something is debatable means that there are certain points which haven't been cleared. Mathematical proof eliminates all such points, which means debate is not only superfluous, but also impossible. I thought that by default we speak of things which are valid in a given context, and that it is the opposite - when we speak of things which are valid in any context - that needs to be stressed. If it is the other way around, I apologize.

Thanks for this. I was referring to this what you call "unique readability" as being unambiguous, as I have explained in subsequent posts, however, I'm not sure with how much clarity. It is rather difficult for me to study mathematical terms in Croatian, then talk about them in English.

These things can easily get lost or stolen. And if someone found it, they would have no trouble entering your house, given the functionality of this chip allows that, etc.

According to this, there is no contradiction with the identity axiom that I can see.

Consider the metaphor as an example. You are using language to describe something, but you are using the terms which mean something qute different. So somebody's emotions might suddenly become a wild ocean. This is an example of the language ambiguousness. In mathematics, there is no such things. Once the symbols are defined, they are used as defined and nothing else. Another thing with language is that even if you have clearly defined the terms you are using, and you explain something with them, you may not be understood in the way you mean what you are saying. Yes, this has to do with (not) defining your terms and if you define them you may be understood correctly. However, the thing about language, unlike mathematics, is that you don't define your terms. You take them to mean what you think they mean, and you explain things in your terms. Thus while speaking, you leave room for ambiguousness. And as Hal noted, this is not purely a sociological thing. This, however, is not to say that it is impossible to achieve clarity. To do that, however, you need to invest extra effort and - and this is very important - others need to make effort to understand you. This is why in spoken language, this is often an impractical thing to do and nobody will listen (except if you are lecturing students in class, or somebody explicitly asks you for a definition). You must make due with the words and their definitions at hand, and get your point across this way, then deal with possible misunderstandings later. This is the ambiguity in language I was referring to. It can be avoided, but there is no formalized method saying how to do this. Logic can help, but not give a definite answer on how to do it.

"12" isn't a mathematical statement. It is not a theorem or a formula or a definition, and certainly not an expression. It says nothing. It is merely a symbolic representation of an integer between 11 and 13. I can't tell you because you haven't defined these symbols.

Actually, the burden of proof is on the one claiming that thought can exist without a body. He's the one who's seen it! (or has he?)

Saying "12" doesn't make you a mathematician. As for "x" it is true that in different contexts it means different things, but in each and every context, a mathematician can find out exactly what that "x" refers to and when grasped fully and correctly, this "x" will deliver exactly the same message as the author intended. This is what I meant by unambiguousness. With language this is much harder to achieve.

I never said I was wrong earlier, I only said that I thought WI_Rifleman's proof was invalid. This doesn't mean I thought that 0.999~ != 1. Proven it for myself? What in the world does that mean? Are you now going to tell me that logic is arbitrary and that my proof "may be true for me but not for others?" Statements such as "let [symbol1] mean [some operation], and [symbol2] mean [some other operation]" are exactly what make mathematics unambiguous. By saying this, the author has made certain that he will be understood in exactly the way he wants to be understood. It is exactly the contextuality which leads to practical unambiguousness. I say practical because inventing and applying a new symbol which has never ever been used in mathematics for every [some operation] or [some other operation] is impractical.

I second splitting the thread, but I suggest locking the thread containing real analysis, as it is clear now that indeed in mathematics 0.999~ = 1. I proved it by correctly applying the mathematical theorems, as well as several people before me (and among them is WI_Rifleman, thread starter, to whom I now apologize, because his proof is in fact valid, contrary to the claims I have made earlier). Those who disagree that 0.999~ = 1 should study mathematics. Debate on this is superfluous, as this has been established as fact. To Hal: Mathematics is unambiguous. When there is a mathematical formula, or a symbol, it can only mean one thing and nothing else. Language we use, however, is a different story. This is a well known fact. I really don't see what's the point in your bringing this up, or the point of your whole argument for that matter.

Here's a mathematical proof that 0.999~ = 1. Let f(x) = Sum[n=0 to infinity](x / 10^n). When x = 9/10 => f(x) = 0.9 + 0.09 + 0.009 + ... = 0.999~. Also, f(9/10) = 9/10 * Sum[n=0 to infinity](1 / 10^n) Since 1/10 < 1, I can use Maclaurin formula to calculate the sum of the above series. Maclaurin formula states that: Sum[n=0 to infinity](y^n) = 1 / (1 - y) for each y < 1. In the above series, y = 1/10. So, f(9/10) = 9/10 * 1 / (1 - 1/10) = 9/10 * (1 / (9/10)) = 9/10 * 10/9 = 1 Now we have: f(9/10) = 0.999~ and f(9/10) = 1 therefore, 0.999~ = 1 Edit: This post earned me a promotion to "Advanced Member."

To my knowledge, quantum theory has trouble explaining gravity. Thus, no matter how much "horsepower" you have, you will always get nonsensical results.

Right. Using other notations (binary, octal, hexadecimal) shows this, but they have different limitations.

I always think of pure mathematical logic as something that does not take time into account. The apple clearly changed through time, as you have taken a bite out of it. However, the law of identity states that an apple is an apple, not that an apple will always be an apple and its properties won't change. Another thing, a professor at college told me that fuzzy logic deals with things like the following: a number approximately equal to 2 + a number between 3 and 3.5 = a number approximately equal or greater than 5 and less or approximately equal to 5.5 Edited to quote my professor.

You mean the law of identity? You don't have to doubt it! A theory is a man's attempt to describe some aspect of reality in such a way as to be able to apply his knowledge of it. That man is able to describe certain phenomenons and apply them in technology, does not mean that he is absolutely correct in his description of them. It only means that his theory is good to some extent and to this extent he can apply it. Consider classical physics. It is useless in the quantum realm, yet it is good enough to explain simple mechanics. Quantum mechanics, on the other hand is useless if you want to explain some macroscopic phenomena. Why? Because A is not A? No! It is because theories are not good enough to respect the law of identity in every context. They are as they are because we don't yet know any better. People once thought the Earth was flat, yet there was evidence to the contrary. A wise thing to do when one encounters a contradiction is to check one's premises, not begin to doubt the law of identity. It's not that Earth is sometimes flat and sometimes round, but it is round, but so big it looks flat on the surface. Moreover, the law of identity is basis of all proof. You say that scientists consider it a proven fact that electrons defy the law of identity. So, by utilizing the law of identity on trying to prove a theory, they came to a contradiction and thus have established that the law of identity is flawed? Something just doesn't click into place there. Isn't it by far more likely that the theory is incorrect? My advice to them is: check your premises. One of them is wrong. What's wrong, however, isn't easy to answer. I've been in a situation many times in life, especially when doing mathematics, when everything seemed to go fine and everything seemed to click into place, only the end result was a complete nonsense. I didn't know what was wrong, until I checked my premises. I once discovered that I have applied a certain theorem improperly, tearing it out of the context in which it was defined and applying it in some other context. However, when the grounds you walk on are new and yet unknown, finding where you turned a wrong alley can be much more difficult. So difficult, in fact, that you begin to think that the streets change.

I think it is silly to give so much credit to something that is called a theory, to begin to question an axiom. A theory is like a work in progress. It tells you something about what you want to know, but it is not a definite answer. A theory is not a 100% precise description of reality. Much of it is probably wrong, especially when it contradicts the axiom so blatantly. We only keep it and use it because it helps us do what we need to do, whatever we need to do.

I don't know much about zoning laws in the US, but I have come up with what I see as a good system of acquiring land. I wrote about it on my blog. Tell me what you think. The posts are here: Part one - mostly about owning land and utilizing it in Croatia http://source.thinkertothinker.com/?p=4 Part two - the free Capitalist system as I imagine it http://source.thinkertothinker.com/?p=6

You are working with a premise that he will actually be asked about it. What if he doesn't? Why is it not in his own self interest to kill a man, regardless of whether or not the society will judge him for it? I think I have an answer to that. Let's say you kill once in your whole life and it is not a justified course of action. You hide this and you do not speak of it. Such an act is not easily forgotten. Try to think of a situation when you will trying to teach your children for example, the principles of individual rights. Will you be able to tell them about it, without deep within yourself feeling like a hypocrite? Even if you don't ever teach anyone about the principles you have violated, it is still an issue. Whatever you accomplish in your life, you will always know within yourself that the means to your getting there was killing a person. It is not through your own effort. When it was tough you decided to remove those who were making it tough, rather than compete them as an honest person would do. You will never be able to reach the kind of happiness that you could if you did not allow yourself this serious moral breach. You killed a man, and by doing so you have destroyed your own life. And if this person was not someone in your way to such "success," then you acted to satisfy a temporary whim and that says a lot about your integrity (i.e. you have none), and it is doubtful that you will in this case ever accomplish anything.

I think there is really more to this. While reading Branden's article, he mentioned some of his patients coming to him and asking certain questions and he mentioned the answers he gives them. What I think he does is that he indoctrinates his patients with his own pseudo-objectivism under false pretenses: his patients probably don't even know (much) about his break with Ayn Rand and he exploits that. I mention this only as a possibility though, not as a fact that I've established. But judging from his article, I would put that forward as something more than a possibility: a probability.

Most of what I wanted to say about Branden's article was already said by AisA. While Branden has a point in saying that people need to learn how to be rational, this is nothing new. It isn't simple to know how to make sense of things, how to approach problems in order to solve them. He tells a lot of falsehoods hoping to show that Ayn Rand rejected that. Also, notice how he calls benevolence, mutual helpfulness and mutual aid principles, and supporting life, assisting and alleviating suffering virtues. I found that to be very irresponsible of him, as it is clear what virtues are and why. Principles on the other hand should be practical (as in possible to apply in practice), and mutual helpfulness or mutual aid are not. I find this sentence very dubious: What am I as the reader supposed to conclude from that? On the subject of repression, I can tell many things, however, I will just quote a few sentences he wrote, to show that it's all nonsense: Forgetting for a moment that "the overwhelming majority of adolescents" didn't develop the level of reasonable thinking even close to that of Howard Roark, my qusetion to his last sentence is, why should he? Yes, he has been expelled from school, but as he says later, he should have left sooner, because there was nothing new he could have learned. The mere fact that he was expelled changes nothing in his life, as he is competent enough - even without the diploma - to design his own buildings. He is the victim of injustice, Branden says, but is he really? Think about it. In a free society, every school and every college chooses whom to teach and whom not to teach. Why should it be unjust that he was expelled? Because he was a great architect? It's the school's loss, really. He is misunderstood. So what, I ask? Should he be miserable because nobody understands that a building should be the image of it's architect's soul? Branden says that Roark finds others puzzling and incomprehensible. Really, why should he be concerned about their motives? He is alone and has no friends. The most that can be said about that is that it's sad that there are no people of value to such a great man. The questions Branden raises are all trivial and have no connection to Howard's goal in life, or his happiness. Now, irrational adolescents such as Branden correctly notices make "the overwhelming majority" might be bothered by this to the extent which is irreparably self-destructive. But Howard is an ideal man, not an average adolescent. Another thing Branden does not understand is the difference between moral depravity - or "evil" - and an honest mistake. He shows this in the part he entitled "Encouraging moralizing," but I can't find the exact sentence where this becomes obvious. Anyway, he says something to the effect of that telling a person what he's done is immoral or irrational is bad because that will induce the feeling of guilt and whatnot, and that this wrongdoing will most likely then be repeated. On the contrary, however, if this wrongdoing was due to real moral depravity, the wrongdoer will most likely be angry at you or whatever their whim tells them, while if this wrongdoing was the result of an honest error, the result will be an apology and the confirmation of your judgement. Most likely, the response will be "you're right, I did do wrong, I'm sorry, I won't do it again" or something to that effect. Once again, it shows why it is virtuous to be just.