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Everything posted by Hal

  1. Actually the building in the original post has grown on me; it looks a lot less fragile once you see it as part of its environment. God its beautfiul.
  2. It depends on your goals; are you primarilly trying to communicate with people, or to impress them with your vocabulary/intelligence?
  3. edit: I decided to remove this post - snide anti-C++ comments probably wont help you to make a decision However, I would say that learning C++ is only worthwhile if you think you would be happy working in a job where you have to use C++. Personally I find that idea very uinappealing, but ymmv.
  4. This is not what Carnap said, in fact its in direct contradiction to his formalist view of mathematics as just being a formal system. I think youre confusing him with (eg) the later Wittgenstein. Carnap would have agreed that infinite only comes into mathematics when/if we introduce it into our symbolism, but I think he would have disagreed with most of the other claims youve attributed to him (most notably "this talk of a system misses the point", and "Math is not a system, really--rather, you should think of it as a linguistic, social practice"). What do you mean by 'proving Platonism'? What sort of thing would count as evidence for it? Anyway, its incorrect to say that the contraries of Platoism have proved contradictory. For instance, Formalism is perfectly consistent, its just pointless and ducks the interesting questions. Intuitionism is consistent, its just very complex and cant reconstruct all of classical mathematics. None of the more empirical positions which have become more popular recently (eg those espoused by Polya, Putnam, etc) are contradictory either (and imo they are a lot closer to the truth than any of the traditional approaches - I would recommend this as an excellent introduction to 'quasi-empiricist' philosophies. On a sidenote, while Objectivism doesnt have a philosophy of mathematics as such, I would tentatively suggest that this has more in common with what I see as being the 'spirit of Objectivism' than the traditional approaches do) -> means 'tends to', its used to denote the limit of a function/sequence. Its the value which the sequence is getting closer and closer to, as the variable obeys some condition. For instance, the function f(x) = 1/x tends to 0 as x tends to infinite, because as x gets bigger and bigger, f(x) gets smaller and smaller. Hence we write "f(x) -> 0 as x-> infinity". Similarly, we would write "f(x) -> infinity as x-> 0", since 1/x becomes larger and larger as x gets closer to 0.
  5. 0/x tends to 0 as x->0 (think about it in terms of epsilon-delta). The whole point of limits is that the function doesnt need to be defined at the point youre evaluating the limit (f(x) = 0/x is discontinuous at 0 where it has a removable singularity, but the limit at both sides is 0). But anyway, the way that infinity is treated within standard analysis is different from the way its treated within set theory (this is the distinction between potential and completed infinities, and why Cantor's work was so controversial). In the context of analysis, we would say something like "The fact that there are infinite integers means that the sequence x_n, where x_n takes integer values and x_n+1 > x_n for all n, has no limit". And this captures what we mean by unboundedness. But when doing set theory, we talk about the completed set of natural numbers, N. And this is not meant as a potential infinity. The Axiom of Infinity used in ZFC states, quite unambiguously, "There exists at least one infinite set". On a sidenote, you dont need the limit concept to do calculus - you can do it using infinities and infinitesimals like Newton and Leibniz did. Calculus was originally founded upon infinite(simal) numbers, and the reason why the limit method is taught in undergraduate analysis classes today is because of historical coincidence (a rigorous formulation of calculus using limits was found around a century before a rigorous formulation using infinitesimals), not one of logical necessity. Limits dont show us that infinity doesnt exist mathematically, they show us that we dont need to postulate infinity in order to do calculus (ie we can choose to include it in our system, or not). Of course, none of this means that 'infinity' exists in the real world, if this statement is taken to mean that you can 'infinitely many objects' (although you cant have -4 objects or 3+2i objects either). Actually, defining a point known as infinity is useful, and often done. You need it to construct the Riemann sphere for example, and its fundamental to projective geometry.
  6. I think the "arbitrary statements dont have truth values" thing is one of Peikoff's inventions, I havent read anything where Ayn Rand mentions it.
  7. edit: sorry, I misread your post
  8. I'm not sure whether youre saying that the piece of paper is a waste, or that college is a waste. If the latter, then I would strongly disagree. The purpose of education doesnt have to be 'getting a job' - learning interesting new things can be an end in itself.
  9. I dont know much about physics so I could be wrong, but I'm fairly sure that QFT doesnt specify a quantum physics interpretation - its just a mathematical framework within which physical theories can be constructed. Any interpretation of quantum physics which was comptaible with the standard mathematical QM formalism would by definition be compatible with QFT, because QFT is built on the QM axioms, not the physical interpretation of these. Leaving aside the more complex question of spacetime and just concentrating on space, it would be perfectly coherent to say that space is curved. This is still compatible with the idea of space being a relational concept, because saying that space is a relation doesnt imply that Euclidean geometry describes the physical world. A non-Euclidean geometry would still have its physical distances defined as 'relations between objects'.
  10. These dont stop human spammers, only automated bots. And even then they dont work, since you can generally write pattern recognition programs which can answer the CAPTCHA's correctly anyway. CAPTCHAs are like copy-protection schemes in computer software, or fullbody searches at nightclubs - they just irritate legitimate users, while doing very little to prevent the thing they are ostensibly trying to prevent.
  11. I would say that the question of whether unattached sex is enjoyable is slightly different from whether prostitution is enjoyable. The problem with prostitution is that you know you havent conquered your partner in any significant way - they are just with you because you are paying them. In a sense, its an expression of your own powerlessness; an admission that you can only gain this person via something external to you rather than through any essential part of your character*. In contrast, even if you were just having meaningless (non-commercial) sex with a one night stand, the person you were with would still want to be with you - maybe she just saw you at the bar and thought you were hot or whatever, but she still wants you, rather than just the $100 youre giving her at the end. In my opinion, unattached sex can still have the elements which make sex fun, whereas prostitution generally cant, hence the 2 things need to be considered seperately. edit: talking about sex without the mental parts/conquest isnt like talking about bumper cars or pool, because youre removing the element which actually creates most of the enjoyment in the first place. To use a somewhat dubious analogy, its like the difference between playing poker for money, and playing for matchsticks. Sure, its possible to just use matchsticks, but really - whats the point? Once youve removed the part of the game which actually generates the excitement - the idea of winning/losing something tangible - youve defeated the purpose of playing and you'd be better off just having a game of chess instead. A perhaps even more dubious example would be playing through a computer game using "God" mode, or some other kind of cheat which gives you infinite health - sure, you get to finish the game, but is there going to be any real feeling of satisifaction here? * the obvious counterargument to this is that a person's earnings are an aspect of their character, and that prostitution can actually be framed in the language of conquest - the prostitute user is showing that because of their ability to generate wealth, they are able to sleep with women who they could not otherwise obtain. And this is admittedly an expression of power. You could also argue that a women being attracted to you mainly because of your physical appearence wouldnt involve anything 'essential' about you either, but this seems more dubious.
  12. I dont think the point at infinity has anything to do with either trichotomy or the axiom of choice. In mathematics, the term 'infinity' is used in various different ways ("completed infinite sets", "infinite as a limiting concept", etc). The 'infinity' in question when doing set theoretic stuff like the axiom of choice involves the cardinal/ordinal numbers of sets. However the 'infinity' involved in the idea of 'point at infinity' is different - its a purely geometric/algebriac notion. All youre essentially doing is adding an 'extra' point to the real numbers (or complex plane) via an extension and introducing some new axioms for it, similar to what you do when youre (eg) constructing the hyperreal numbers, or defining a plane with 2 origins (if you havent studied any abstract algebra, the previous sentence probably wont make sense). And formally speaking, this doesnt have anything to do with transfinite ordinals/cardinals, other than that the word 'infinity' is used in both cases (and of course, we could choose to call it something else). In other words, having a point at infinity is perfectly compatible with ZFC. edit: oops, it isnt a field extension since R with the point at infinity isnt a field I've never actually seen the construction so I'm not sure what details are involved in practive (although I should have guessed that since it obviously isnt going to have a multiplicative inverse :/))
  13. I probably rambled on too much about the AoC so I'll be brief with this. The problem I have with saying 'mathematical objects are just concepts' is that it fails to explain what we actually do when we do mathematics. An example will hopefully clarify what I mean. There was a time when mathematicians wanted to know whether continuous nowhere-differentiable functions (CNDFs) existed (dont worry about what these things are, its just a random example). So, they spent quite a lot of time trying to find one. Now, these people were not looking to see whether the 'concept of a CNDF" existed. This wouldnt make sense; they already knew that the concept of a CNDF existed - they had this concept inside their skull! They knew what CNDF's were, and they could define them quite easily, they just didnt know whether any actually existed (compare to having having the concept of a unicorn, but not knowing if unicorns exist. We know the concept of a unicorn exists, but that is not the question here). The mathematicians were not asking a conceptual question, so saying that CNDF's 'only exist as concepts' seems to missethe point - the concept of a CNDF would still exist regardless of whether CNDF's had mathematical existence (the idea of a triplet of integers satisfying Fermat's Last Theorem exists, even though there is no such triplet). Saying that the concept of '4' is even, would be like saying that my concept of 'unicorn' has a horn. But this is obviously absurd - the concept is inside my skull, probably represented as a neuronal pattern in my brain.
  14. I think it comes down to intuitive plausibility. There are obviously infinitely many natural numbers (whether you want to define this in terms of potentiality or whatever, and yes I realise I'm glossing over the difficulties), so an axiom of infinity seems reasonable. But the axiom of choice (and its equivalents) have very little intuitive content. In a sense, there are independent grounds for accepting AI, but we only accept AC because we like the things we can do with it.
  15. I havent done much set theory (other than the basics you need for doing actual math) so someone else might correct me on this, but I'm fairly sure that you dont need the AoC to show that the cardinality of the reals is greater than the cardinality of the rationals. I'm 95% certain that the proof you have given in this link, for instance, doesnt invoke the AoC at any stage. I'll try to explain why, although its pretty abstract and confusing. Youre assuming, in order to deduce a contradiction, that you can find a 1-1 correspondence the reals and the natural numbers. Now, you want to construct a real number that isnt on the list, by selecting one decimal place from each of the listed numbers. But you dont need the AoC to do this, because you are giving an explicit rule for choosing the decimal places (youre saying "pick the nth decimal place from the nth number on the list"). You only need the AoC if youre trying to make a choice without an explicit rule (eg, if you wanted to say "pick a random decimal place from each real number"). To use a fantastic example given by Bertrand Russell, if you had infinitely many pairs of shoes, then you wouldnt need the Axiom of Choice to select one shoe from each pair, because you can give the explicit rule "always pick the left shoe". But if instead you had infinitely many pairs of socks, then you would need the AoC to select a sock from each pair, because the socks in a pair are indistinguishable hence you cant give an explicit rule for making the choice - youre now saying "just pick one - I dont care which". One of the key points here is that once you've chosen your choice set of socks/shoes, you _know_ which shoes are in the shoe set (it will be all the left ones), but youve no idea which particular socks are in the sock set, since the choice was arbitrary/random. This is why the AoC is fundamentally non-constructive - you cant explicitly specify the set you produce with it, all you can say that it exists. You dont know _which_ socks got chosen, you only know _that_ some got chosen. And this lies at the heart of the controversy over the Axiom of Choice. If I have infinitely many sets of integers, then I can explicitly construct a set which contains one integer from each set, by giving the rule "choose the smallest integer from each set". And this doesnt need the AoC. Its an entirely constructive process, because I know which integer from each set will have been chosen (the smallest). But if I were choosing one real number from infinitely many sets of reals,I can no longer say "pick the smallest real number from each set", because there isnt guaranteed to be a smallest real number (eg, the set of all reals greater than 0 doesnt contain a smallest element). So if I'm not able to actually specify a way to make the choice, and I cant name a single element of the resulting choice set, why am I justified in assuming that such a chioce set exists? Well, this is where I invoke the AoC, and say that it just does. And again, like the sock example, I've no idea which particular real numbers will make up this set, because the choice was by definition arbitrary/random. Note that this is intimately connected to the Well Ordering Theorem (which is equivalent to the axiom of choice) - if the reals can be well ordered, then I can specify an explicit rule for making the choice by saying "pick the minimal real number from each set, where 'minimal' is defined by the well ordering", and this is now guaranteed to exist. But here this just moves the problem back one level, because the Well Ordering Thereom is itself non-constructive - it just says that a well-ordering exists, without saying what this well-ordering is. So again, we have no idea which real numbers actually got chosen by the choice function (and you need the AoC or something equivalent to prove the well-ordering theorem is true). edit: Here's a good clarification of what the AoC is and isnt saying. As it hints at various points, the controversy ultimately boils down to what 'exists' means in a mathematical context.
  16. There just obviously isnt a well-ordering of the real numbers, therefore the Axiom of Choice cant possibly be valid But on a slightly less flippant note, I'm ambivalent towards the AoC. Philosophically speaking, I agree with the constructivists - talking about mathematical objects 'existing' even though we havent managed to find them yet is nonsenscal, and sounds more like theology than science. But at the same time, some of the results which follow from the AoC are incredibly sexy, and the non-constructive proofs are often very elegant (the non-constructive proof for the existence of Hamel bases almost blew my mind the first time I saw it, and some the things you can do with the Baire Category theorem are very cool). Its annoying, because intuitively, I feel that some of the theorems which follow from the AoC _should_ be true (eg Baire Category, Hamel bases, countable union of countable sets being countable), but I also feel that some should be false (well-ordering/Vitali theorems are 2 that come to mind). Its really a case of deciding whether being able to prove some very nice theorems justifies bringing in a set of extremely dubious philosophical assumptions, along with some counterintuitive results. Personally I'm still undecided, although I tend to lean towards pragmatism here :/ The worst part about it is that it tends to leave me with a feeling of fundamental uncertainty. When you use the AoC to prove that X exists, I always find myself wanting to ask "yes, but does it ACTUALLY exist?". Would it actually be possible, for instance, to find a well ordering of the reals and write down an explicit definition of it? Well, the AoC tells us that there must be one 'somewhere', but what ultimate grounds do we have for believing the AoC is true? Similarly, the AoC tells us that we will never be able to find a vector space without a basis. But why cant we do this? What if we did? I dont think I could ever have the same degree of confidence in a result which required the AoC as I could in a result which was proved by more 'standard' methods, just because there always remains the possibility that the AoC itself might not be true (whatever 'true' means here).
  17. Diversity isnt an intrinsically terrible idea, just write (eg) that you think its good that poor people who werent able to attend quality high schools have a chance to go to university. You can say that affirmative action should be based on social class rather than skin colour, which has the benefit of being both true, and letting you avoid supporting racism. If you need to focus on racial issues, you can talk about how (eg) children of immigrants can often be more dedicated that native students, and mention how Asians have a repuation for working hard and so on. I doubt you could write a good argument why diversity/AA is bad in 100 words anyway (I mean heck, this post is over 100 words long), so if you go down that route I suspect you'll end up with a bad essay. I think youd need at least 500-1000 to make proper arguments, so youre really looking at platitudes here.
  18. Why is someone living at home less likely to be pursuing a fruitful career? Cant you do both?
  19. What does living with your parents have to do with striving for success? I dont know about America, but the reason more young people are living at home in Britain is due to the overinflated prices in the housing market. $250000 minimum for a cramped one-bedroom flat in a poor area of London? I dont think 'laziness' is the primary factor here.
  20. I'd say the only benefits are legal, but they shouldnt be underestimated. To be honest, it would be rational to agree to marry someone you were friends with as soon as you turned 18, just to have the bit of paper that lets you pay less tax. You can always get divorced later if you want to actually get married properly.
  21. For reference, here is the relevant passage from The Objectivist Ethics
  22. Just wanted to say that I started reading this thread with the opinion that leaking wireless connections was quite obviously an instance of using someone's property against their will, but David's arguments have convinced me otherwise. Thank you for opening my mind. edit: A good (slightly more technical) analogy might be FTPs on the internet. There are many individuals and companies out there who, for various reasons, allow public access to their FTPs. If, while browsing the internet, I find an FTP which is configured to allow access using without a password (eg a FTP which accepts anonymous logins on the standard port 21), I would be fully justified in assuming that the owner has set it up for public use, and using it for upload/download. Even if it turned out that the owner had just set his daemon up incorrectly and he meant to restrict access, this would not be my fault/responsibility - I shouldnt be expected to go to the effort of trying to locate an email address for the owner of every FTP I find, just to ensure that he knows how to use a computer properly. The same applies to proxy servers - if I encounter an anonymous proxy which doesnt require a password to use, I'm justified in assuming the owner intends it for public use. I mean just how far do you want take this? Its certainly possible that any given website you visit on the internet wasnt actually intended to be publically viewable - it could just be that the site owner has configured Apache wrong. How could you know this? Well, you couldnt, hence its safe to assume that its intended to be public. As a non technical analogy, its common for organisations in London to provide free newspapers on public transport. There is one major paper (the Metro), and several minor ones. Hence, if I were to encounter a pile of newspapers on the bus, it would be reasonable for me to assume they were free for the public, and take one. Now it could be the case that the owner had mistakenly left them on the bus and I was actually taking his property, but given that there is no way I could be expected to know this, my actions are justified.
  23. Hey, welcome to the forum! Well, firstly you have to remember that Atlas Shrugged was written in the 1950's, before there was clearcut evidence showing that smoking could cause long-term health problems. I remember reading an anecdote about how Ayn Rand quit smoking when her doctor suggested she should, but I dont know whether this is apocryphal. Secondly, its not correct to say that because something can have bad effects, you shouldnt do it. The question is really whether the risks are significant enough to outweigh the benefit you gain from the action. For instance, eating a hamburger isnt very healthy, but as long as you dont do it every day, there arent really going to be any long term effects so there's nothing wrong it. And at the other end of the scale, injecting heroin might be extremely pleasurable, but doing so isnt likely to be in your rational long-term self interest and hence it would be pure hedonism (and immoral). Smoking lies somewhere between the 2 extremes - yes, there are possible long term hazards. But are these sufficient to outweigh the pleasure a person can derive from smoking? I'd say thats a decision each individual would have to make for themself, taking all the known evidence into consideration. Ayn Rand obviously felt that smoking was worth it, but again, we do know more about the effects than we did 50 years ago, and she may well have believed different if she were writing AS today. Personally I think smoking is gross regardless of the health issues, but meh.
  24. Well I'm sure this is what they said, but I'm sceptical about whether it is true. Perhaps I'm being cynical here, but I would guess that their actual thought process would be something closer to "withdrawing from China would lose us money, and I'm sure we will be able to invent some semi-convincing rationalisation for our actions anyway to avoid offending our Western fanbase".
  25. I admit that, ironically, I was too hasty in assuming that my experiences generalised. Although I do enjoy some classical music, I dont (and doubt that I could) listen to it in the same manner that I listen to metal. But yes, theres no reason to assume that this applies to everyone. However, I do think that theres a qualitative difference between the experience of (eg) listening to a Beethoven piece in a concert hall and dancing to trance music at a rave, but I'm not sure how to put it into words (my use of a vague phrase like "sense of unity and full body/mind engagement" probably gave this away).
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