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About Nodrog

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  1. Why 'String' Theory

    Well, I would say that a unification of seemingly unrelated mathematical phenomenon is (obviously inconclusive) evidence that they could be onto somethng - there's only so much that can be attributed to sheer coincidence. It would hardly be the first time that physicists were guided more by mathematical intuition than by physical results (Dirac being a prime example). As far as I know, attempts are being made to grind some testable predictions out of the string theory machine, such as the emergence of supersymmetry at higher energy levels that can currently be produced. I dont know nearly enough to comment on this specifically, but if some predictions can be found, then I would class it as being physics rather than maths. It doesnt really matter to me whether their hypothetical entities exist - the level of underdeterminism present at that small a scale is currently too great for me to believe any specific claims of what 'actually exists' down there. All that's important is whether the theory is coherent, and can predict new things.
  2. Why 'String' Theory

    Sorry, I should have specified what I meant by 'non-trivial'. As far as I know, string theorists claim that one of the main appeals of their theory is that it unifies a lot of highly unexpected mathematical results, in a way that not many people would have expected. I also understand that said mathematical results are highly difficult and esoteric, and largely outwith the mathematica experience of most physicists (ie non-theoreticians). Assuming this is true, the opinions of these physicists would seem irrelevant, since they arent actually in a position to properly assess the claims being made.
  3. Why 'String' Theory

    What percentage of these physicists would you say have a mathmatically non-trivial understanding of string theory?
  4. What to call the enemy

    I dont think that Islam is intrinsically, in terms of the content of relevant 'holy books', more violent than Judaism; the Torah (Old Testement) is hardly a book filled exclusively with peace and love. Christianity is in the same boat - look at past history for examples of how the religion can be used to endorse mindless violence and barbarism. It's not primarilly about the religions themselves - its about the attitude of the majority of practioners. Most modern Christians are relatively apathetic towards their religion; they might attend Church on Sunday and vote against gays and pornography, but they dont have anywhere near the fanatical devotion that modern Islamists do. If Judeo-Christians cared more about religion, you might see a lot more abortion doctors getting murdered, and similar atrocities.
  5. Essay writing

    I think it depends on the person; I often treat writing as a form of thinking, and do my best reasoning while writing things down. Others will generally try to get things straight in their head before putting their thoughts onto paper. I suspect this has implications for how people actually write. I think people like me will write an essay by producing a series of rough 'first drafts', all written/revised in a single sitting, and getting progressively better. People who like to 'think t out first' will write incrementally - a rough plan, then the actual essay produced over several days/weeks, paragraph by paragraph. This is just a theory though.
  6. That's not my job!

    It depends on the job; its hard to judge without context. If its something that you think people might be doing as a career, then yes, that is a bad attitude to have. But if its just something where most people will be temporary workers looking for a bit of cash on the side,(eg fastfood, or other jobs which are primarilly filled with college students), then I dont really blame them. If I was working harder than others at a job I was only doing for the money, I would expect to be paid more than them. If this didnt happen, then I'd finish my work and do something more fun (eg browse the internet or make a phone call). Why would you want to do the job of 2 people yet only get paid for the work of 1, unless you either a) enjoyed the job, or had aspirations of promotion? It's the management's fault for not having some kind of performance based pay, not the workers.
  7. Justice Scalia, and Originalism

    I think thats a really bad amendment and dont think it should have been included. The problem is that it could be interpreted to mean almost anything - is there a 'right to healthcare' that exists though wasnt enumerated in the Constitution? What about a right to education? The Constitution gives no definition of what rights are, so if you want the 9th Amendment to be invoked to justify the creation of non-written rights you believe should exist, you can hardly complain if others use it to justify non-written rights you believe shouldnt exist.
  8. Justice Scalia, and Originalism

    I'm not American but I have a passing interest in American law, stemming from my respect for the principles the society was founded on. I've noticed that on several occasions, Justice Scalia has come in for criticism from both Objectivists (not on this forum) and libertarians. I find this confusing because, based on my limited information, he is the Judge that seems most concerned with objective law and original intention. In particular, I've just finished reading a transcript of his speech here where he gives a brilliant and principled account of his approach to Constitutional interpretation, with which I agree entirely. So, are there any problems with Scalia that I'm unaware of? Do any people share the hostility towards him which I've seen elsewhere, and if so, could you tell me why?
  9. Gödel's Incompleteness Theorem

    Theres an infamous paradox in the philosophy of knowledge, called the Liar paradox. This involves the sentence "This statement is a lie" - is it true, or false? If it's true then it becomes false, and if its false, it becomes true. Godel's incompleteness theorem lets us construct Liar paradox-type sentences within formal systems. now: Short Answer We can essentially construct a statement within our formal system which says "I am not provable within this formal system". An outside observer can 'see', this statement must be true. But it is impossible to prove it within the formal system (since if you could prove it, it would become false). Hence it is a true statement which cannot be proved, so the formal system is incomplete. Long Answer Godel discovered that there is a way to represent statements in a formal system by natural numbers. This is called Godel Numbering - every (valid) legal combination of basic symbols has its own unique number. For example, in my MMM system above, the string 'MMM' would have a unique number, and so would the string 'NMNN'. Since every theorem of the system now has a unique number, we can 'talk about' other theorems of the system within the system. So for instance, if the theorem 'NMNN' has the number 67, we can construct another theorem which says essentially "Theorem 67 is provable within this system". If a statement has a proof then that proof will have its own Godel Number. So assume the proof of Theorem 67 ('NMNN') is theorem 127. We can now construct a formula, m(x), which takes a single argument x, and returns true if and only if theorem x has no proof within our formal system. So, m(67) is false because we said theorem 67 is proved by theorem 127. The string 'MM' is not a valid theorem of our system, so if 'MM' had the Godel number 17, then m(17) would be true since it has no proof. Since m(x) is a theorem of our system, it has its own Godel Number. Lets call this 'g'. So g is a name for 'm(x)', just like 67 is a name for 'MMM', and 17 is a name for 'MM'. Now, consider m(g), ie our above formula with itself as an argument. Since m(67) said "theorem 67 is not provable within this system", it follows that m(g) says "m(g) has no proof within this system". Or in other words, m(g) says "m(g) has no proof within this system" - ie it is 'talking about itself'. m(g) is now the equivalent of our Liar Paradox statement. Just like the liar paradox said "This statement is a lie", m(g) says "I have no proof in this formal system". Again, an outside observer can 'see', that m(g) must be true. But it is impossible to prove it within the formal system (since if you could prove it, it would become false). Hence m(g) is a true statement which cannot be proved, so the formal system is incomplete.
  10. Gödel's Incompleteness Theorem

    There seem to be a lot of very bad arguments made involving evolution, many involving the Naturalistic fallacy. From the top of my head: The weak should be left to die because Darwin proved that survival is the strongest of the fittest (social Darwinism, one of the driving forces behind Nazism and eugenics in general) Evolution shows there that is no point to life - everyone is just here to eat, sleep, reproduce and die Free will doesnt exist - everyone is controlled entirely by their genes Being promiscuous is a good thing because Darwin showed that those who reproduce most have the highest chance of passing on their genes. And so on.
  11. Epistemology Exercise #2

    Wouldnt this also apply to pretty much every hero in ancient mythology, including those in the Iliad who were largely controlled by gods and didnt really have much in the way of choice either? Wouldnt it also commit you to the view that heroism didnt really exist until the concept of free-will was first introduced, several centuries after the birth of Christ?
  12. Gödel's Incompleteness Theorem

    Indeed; GIT is probably rivalled only by quantum mechanics and evolution as the most widely abused scientific notion of the 20th century. As well as the absurd attempts to use it in order to attack 'systems' which have no relation to mathematics (such as law, and the axioms of Objectivism), GIT also enjoyed a small degree of popularity within artificial intelligence for a while, where some people (notably Roger Penrose) tried to invoke it in order to show that a computer could never fully model human reasoning. GIT is a relatively interesting result, but one which has a very small domain of applicability.
  13. Gödel's Incompleteness Theorem

    Oops, made a mistake. 3) should say "Apply rule 2 to the leading N giving MNMM"
  14. Gödel's Incompleteness Theorem

    Mathematics is treated as being a formal deductive system by some approaches. A formal deductive system works as follows - you have a list of axioms, represented as strings of symbols. Then you have a list of rules, that allow you to change certain symbols into other symbols. A string is called a 'theorem' of the system if you can produce it from the axioms by applying the rules of the system. This kind of thing is a lot easier to explain by example: Cionsider this basic formal system: Axiom 1: MMM Axiom 2: NNN Rule 1: M -> NN      (ie replace any 'M' with 'NN') Rule 2: N -> M        (ie replace any 'N' with 'M') In other words, we are allowed to start with either the strings MMM or NNN, and we can produce further strings from these by replacing an M with NN, or replacing an N with M. Now, we can prove that the string MNMM is a theorem of this system by deducing it from the axioms as follows: 1) Start with Axiom 1: MMM 2) Apply rule 1 to the leading 'M', giving NNMM 3) Apply rule 2 to the leading 'N', giving NMNN We can also produce the string NNM as follows: 1) Start with Axiom 2: NNN 2) Apply rule 2 to the last N, giving NNM So the strings NMNN and NNM are theorems of our formal system. However the string MMSM will not a theorem, since there is no way to produce an 'S' from the rules we are allowed to use. Now, all of mathematics can be represented by a sufficiently complex formal system similar to this (only with a lot more axioms and rules of inference). The benefits of using formal systems was because it was thought to make proofs more 'objective' - rather than appealing to 'mathematical intuition', we could write down a proof as being a step by step transformation of one symbolic string into another, and the deduction could then by checked by (eg) a computer. For this approach to work, there needs to be a way of deciding for ANY string of symbols whether that string is a theorem of the system (ie whether it can be deduced from the axioms). Assuming that we can do this for any string of symbols, we say the system is 'complete' (although I have slightly simplified). A similar property of a formal system is 'consistency'. This means that given your starting axioms, it is not possible to show that a given string is and is not a theorem of the system - ie the system contains no contradictions. Godels incompleteness theorem states that for ANY consistent formal system that is powerful enough to allow the formalisation of arithmatic, there will be at least one statement which is true but unprovable within that system - ie it cannot be shown from the axioms of the system that this string either is or is not a theorem. This string will essentially 'say' (in a formal language) "I am not provable within this formal system". In other words, no sufficiently powerful formal system can be both complete and consistent. As you might expect, the GIT result had very serious implications for the project of formalising mathematics.
  15. Quotations and punctuation marks

    I always go outside if it doesnt belong to the quoted part because that makes more sense to me, and I would encourage others to do the same since thats the only way to change the standard convention