aleph_0

No, the mathematician's claim is not that .999... = 1 is an approximation. The mathematician claims that .999... is identical to 1. As someone else explained before, this is a naming issue. Mohammad Ali = Cassius Clay and likewise .999... = 1. They are two names for the same thing. Eiuol's proof is, to me, the most obvious proof. As for your argument, Amaroq, long-division isn't a tool the pure mathematician uses. It is a method for calculating things near enough to the answer. Sometimes it even gets you exactly the right answer, but not always (for instance, divide pi by any rational and you'll never reach the end), and so it is not acceptable in a mathematical proof. You can use heuristic arguments if you like, in order to feel more at ease with the notion that .999... = 1, but the fact is that this identity is true merely as a product of the formalization we give to mathematics. To take another instance, we could define a homomorphic operation * over the integers, such that 0*0 = 0, 0*1 = 1, 1*0 = 1, 1*1 = 0. Then 2 = 2*0 = 1*1*0 = 1*1 = 0, so 2 = 0. But this is just a result of some formalism that we chose. (In fact, it is the familiar formalism of the integers mod 2. That is, you repeat your numbers up to 2. For the integers mod 3, you only deal with the numbers 0, 1, 2. So 0+1 = 1, 1+1 = 2, 3 = 1+2 = 0 [mod 3], and 2+2 = 1 [mod 3]. Does this mean, in any metaphysical way, that 2 = 0 or 3 = 0? No. It just means that we picked a formalism that is useful for some task and live with the deductive consequences of the formalism.)

Is there an alternative diagram, or are you looking for any representation? You can use quantificational logic to express the same concepts, and with much greater expressive power.

Shalom. Yesh chavrim ysraelim poh. (Hopefully that didn't come out retarded. Four years since I took Hebrew.)

Right, I'd like to reinforce D'kian's point by saying that none of those quotes imply that Rand was talking about physiognomy, but each of them could (and should) be interpreted as talking about an intuitive grasp of somebody's behavior and expression. Moreover, keep in mind that this is her fiction, so people's physical appearance are going to be tailored to their role in the story--beauty on the inside is generally going to reflect beauty outside, except in cases like Roark's construction worker friend, who was probably made ugly at least in part as a statement to the audience that you don't need to be beautiful to be good. This should not be taken to mean that this reflects how the world actually is rather than how it could and should be.

The movie may not be funny, but this sure is:

Stossel ∈ {Good things}

Not YouTube, but in the same spirit. This is the creator of the Space X project, it's extremely entertaining and inspiring. http://www.hulu.com/watch/95407/foratv-sci...te-space-travel [Edit: I just got to the part where he says that 16 teams are working to put the first private rover on the moon!]

I believe the same logic could make these points about any form of art: Why do you need a concretization of your values on a piece of canvas? Does that paint actually make you realize what you valued all along? What makes it aesthetic? There are people who prefer bare walls, are they all wrong? I like to have the reminder, because when I see the images it invokes my pride and a feeling of dedication. That makes it aesthetic.

My tats are aesthetic and serve as an eloquent reminder of the principles and goals I have dedicated myself to embody. If I had more money, I'd certain have more ink.

http://www.youtube.com/watch?v=X2EDtxEumFI Hilariously stupid teen. His glasses are famous.

Tom, we've been over too much of that to rehash--I'll refer to the previous conversation. As for Kant, the thing-in-itself is that which causes the events we experience, but which is not themselves those events. It is what our perception is about. I don't claim to agree with Kant, but so goes the thesis. And whether right or wrong, the fact is that he did contribute to the conversation. I believe, on this account, he was right.

I believe Brian was using that topic to gain information about what mathematics is. Those aren't the only two options, and very few philosophers and mathematicians have ever thought that mathematics is mere symbol manipulation (only Hobbes, and arguably some of the 19th and 20th century philosophers have, to my knowledge). Only slightly more philosophers have ever thought that mathematics was grounded in observation, so there is a vast and varied field of philosophy of mathematics which you are not even aware of. I don't understand your interpretation of Kant. Is the noumenal world a world--is it any part of reality? If so, then the senses tell us about reality, just not the thing-in-itself directly. I don't recall Kant having a particular abhorrence for Aristotelianism. Greenberg even interprets Kant as a post-Hume Aristotelian. Kant added that human means of understanding are part of the description of any known facts. That was entirely original.

I took my response to at least give a first step in answering your second question as well--we find it trustworthy because it is the very method by which we think. To try to doubt it, or think of cases in which it is wrong or to think in terms that violate mathematics, is meaningless. Based on that, I would propose that people with mental defects of that sort just don't have the tools necessary to talk about their being wrong. I mean, this would involve something other than carelessness, right? So we couldn't just say that when they compute 3*5 they just lose track of the numbers. It would have to be that they somehow can't understand 1+1 = 2, that they don't have the concept of quantity. It also wouldn't be enough if they have something systematic. If their addition always gave 1+1 = 2, 2+1 = 2, 2+2 = 3, 3+1 = 3, 3+2 = 4, 3+3 = 5, 3+4 = 6, ... This would actually be a working, if cumbersome, system of mathematics (of course, you wouldn't necessarily have all the same additive properties, and this operation would not be addition). But it might still be used for counting and performing some algebraic operations which are as useful as addition. What we are interested in this case must be that they really cannot understand mathematics in principle, so when they go to compute 3*5 and get 17, they cannot then go to the apples and consistently get 15, because that would require that they know how to count the apples, which is the same principle of mathematics which is involved in computing 3*5.

That's an extremely good, sophisticated question. My first thought in response is that there is probably a problem with the question. Is the hypothetical situation even meaningful? What would it be for mathematics in our heads to not work in practice? For instance, human beings are notoriously bad at calculating probabilities. There's a paradigm example of a person who witnesses a hit-and-run and identifies the car as green. 80% of the cars in that town are blue and 20% are green, but eye witnesses are 90% accurate at identifying the color of cars. What are the odds that the car was green? Most people who have not had a class in probability choose 90%, because that's how reliable witnesses are, and of those who don't pick 90% they usually pick some other percentage based on notably poor probability reasoning. It seems evolution decided we didn't have a particularly strong need to calculate probabilities, and yet our ability to do algebra (high school as well as abstract algebra) is--I think--remarkably strong. It takes some training and practice, but humans largely seem able to absorb and use it naturally. So does that mean that our heads have something wrong with them, that we calculate probabilities wrongly? Only in a superficial way, because obviously, some very intelligent people were able to find out the correct probability analysis. The apples example is better. But then, what would our actions amount to? We do 3*5 and get 17, then get three sets of five apples and find 15. So we know something's wrong. We may re-count the apples or re-do the arithmetic, depending on which we are least confident about. So we go back to 3*5, but then how do we calculate this? Well it's 5+5+5, but then how do we calculate this? Well it's (1+1+1+1+1)+(1+1+1+1+1)+(1+1+1+1+1). So how do we calculate this? By thinking out the numbers in order, 1+1 = 2, 2+1 = 3 ... = 15. But then we have 15, unless we lose our place, but then that's not really the math going wrong but just our clumsiness. We could have made the same mistake with the apples. We could have had in our heads that 3*5 = 15 and yet count out three sets of five apples and get 17. Perhaps a more bone-cutting example would be 1+1 = 2. Could we imagine getting this wrong, or it actually being true? I don't think so. I think it is just meaningless to talk about mathematics being wrong, or even to think outside of the terms of mathematics. To even engage in a conversation about mathematics you must assume some mathematical knowledge, so you get caught in a kind of circularity when you try to then use that language to talk about it being false or unworkable.

I can't trace 100,000,000 x 5 to the perceptually self-evident because I've never seen (and known that I've seen) 100,000,000 of anything. You have to work purely from the concept of addition (and recursion), which you may come to understand by means of learning the language, but we've had this conversation a million times over and there's no point it flaring it back up because I find the whole thing pretty threadbare. Philip Kitcher argues that math is an empirical science, he might have a good argument, but I know he's often just ego-bloated and stubborn so I'm not sure if I would find it worth-while looking into it. But his is the only argument to that effect that I've ever suspected of being well-formed.

Depends on how much econ background you have. I found "Theory of Money and Credit" extremely rewarding, but a significant portion of it I couldn't follow because I haven't studied much econ.

My math texts do not depend on experience in the same way, but I don't consider that a refutation of the text. At the end of "The silent decade" in the wiki article on Kant, it gives some details about this, as well as citations. I've also found other, slightly more embellished articles on the internet about this, but I can't remember where. A Google search using the information from the wiki article would probably turn some up.

First, let's not forget that Kant was largely over-looked when he published the Critique. It only gained steam when, in a debate between two other philosophers, one of them referenced Kant's work. That debate became embroiled and drew the people involved to read Kant, which then became a sensation. So naturally, there would not be refutations of the whole corpus immediately after its publication--not to mention the fact that it's a massive book, that cuts deep into philosophy, and so requires a very careful study to understand let alone methodically refute. Schopenhauer largely attacked Kant, but you're probably looking for a refutation in the anglophone tradition. I can't think of anything off the top of my head, though Problems from Kant by van Cleeve is famous and might be critical.

Note: I was just nit-picking and never intended for it to be a full-blown debate.

Perhaps, but that's not their message. They may or may not believe that they have no right to change a person's healthcare plan, but believe they have the right to provide whatever the hell they're providing. Whether they believe they have the right to take a person's healthcare plan away from them has not been discussed. I disagree. Let's say that I make some statement, you misunderstand me, and I say, "Oh no, I wasn't saying that I'm going to stop you from giving your speech on Tuesday," and you fire back, "You don't have any right to stop me!" That's not apt--that's overreaction to something I never said I had a right to do in the first place. I think that's the general message of liberals at large, not specifically this administration, and I don't think that message is communicated particularly by this reform. Now if the statement were, "You have no right to use my tax dollars for other people's healthcare," or even "This reform is a threadbare first-step in destroying all private healthcare, which is unlawful, immoral, and a fundamental perversion of the function our government is supposed to perform," I would find that content perfectly apt. But it's like the government announced, "We're not saying x" and the guy responds by saying, "So you're saying x, and you have no right to!" I agree that they have no right, and I agree that they probably think they have the right, but the way of coming around to the topic is what seems off to me. That's all I was pointing out.

Nobody said the other molecule caused the property, nor did I say in the original example that the heavy object caused the property. Only if I'm talking about a concept and not a property. Even on ObjectivismOnline.net, I expect to discuss the subject matter, rather than the nearest convenient subject such that it conforms to Objectivist language and doctrine. Temperature is a property and not a concept. A thing that is hot is not also conceptually hot. There is, however, a concept of temperature, if you want to split hairs. In any case, it seems I've got you down for one rejection of emergentism (in the strong sense).

Emergence is also about causal power--properties, I believe, are effectively descriptions of causal power. (To say that something is wet will communicate that if you touch it, it will have a particular texture.) So it is also about external relations. For instance, the supposed emergent properties of a molecule, perhaps that of volatility, will cause it to--perhaps--release energy when in contact with another molecule. But if you want to talk about such super-entities, I still wouldn't mind talking about that entity which is a man and the thing he is lift. I believe in unrestricted composition. I think this implies that you disagree with emergence as complexity theorists present it (the strong thesis), and I believe I disagree with it as well. The strong thesis would imply that the property of life, if it is emergent, is something new and unique which springs out of the more fundamental properties, but which cannot be reduced to them--and so any description of a living organism in terms of its cells, no matter how perfect, will never fully describe the organism. I believe I have been working completely within the frame of properties rather than concepts. So you determine the location of the center of mass of a grain of sand to an incredible degree. But it's not perfect. So you guess as to its exact degree and, unbeknownced to you this guess gives the exact location. The point is just to look at an example of something which is not emergent, to help determine what emergence is, so we assume CM for the sake of conversation. I don't know what you mean by "introduce". It sounds like you think it's a persistent misconception that QM creates, rather than describes, uncertainty. As if the theory came along and all of the sudden there is uncertainty in the world, like saying the magic words "quantum mechanics" brought it into being. Which I doubt anyone holds. Otherwise it seems like most people (informed enough to discuss the matter) realize that QM's claim is that micro-particles do not posses sharp positions and velocities.

I don't understand. One man is a single entity, and of him you predicate the ability to life a heavy object. I don't understand this either, but that is perhaps because I am very uneducated about biology. Let us try a thought experiment, though: Suppose you completely describe a living organism, except that you make no explicit reference to life. Instead, you just talk about chemicals and their bonds, with such detail that when you finish your description, you have told everything that need be said to specify some particular organism. What then does talk about "life" contribute? Is this just something which is useful for humans to think about, so that they can abstract about general behaviors of living things? Or is it that, instead of just being something useful for limited human intelligence, you are actually still missing something in your description if you haven't talked about life? For instance, say that you want to predict the future behavior of the thing, and you try to do it completely in terms of its chemical description without reference to "life". Will you be able to talk about all the things that it can or will do, without talking about it in terms of life? Or does life provide a NEW property, with NEW causal powers, which are not contained just in the properties of the chemicals? Here again, it seems like talk of temperature is just a way for human to conceputalize more basic physical behavior. If you gave a complete description of the position and velocities of the molecules, you would not need to talk about "temperature" except that it's more convenient. But it doesn't actually add information, once you know all of the information about the smaller parts. I've been getting that kind of response from some scientists, but I don't understand why. We'll omit conversation about free will for the sake of simplicity and not needlessly making a subject more controversial than it already is. Let's just talk about inanimate matter, maybe organisms without a central nervous system. But let us suppose that you get very, very precise measurements of an initial condition. Then you do a bunch of guessing in order to make it more precise, but it just turns out that you're guesses are completely and perfectly accurate, even into the infinite decimal expansion. It seems like, in this case, you are able to use your theory to predict every event. Even though you didn't use a reliable method of measurement, you still got the numbers that give a perfect description. Naturally, QM may throw this off, but I'm just talking within the assumptions of Newtonian mechanics.