Yes. If anyone were to challenge me to prove my assertion, prepending a useless long loop depending on input size at the beginning of a valid algorithm would be my answer.:)
Though I still have a problem with the other part of your post. Suppose I am allowed to assume the trivial algorithm exists; I cannot however assume anything about its complexity (that would be cheating). So, assuming that by "drastically different run time" I mean to say that they're not big-O equivalent, I am stumped when deciding how big to make the initial, useless loop; for, for any complexity that I might choose for that initial loop, it may be the case that the trivial algorithm already has at least that complexity.;)
I agree that "A reduces to B" is equally nonsensical; it's an artifact of a growing (and sometimes irritating) tendency to suppress passive clauses in English. The passive version, "A is reducible to B", is more in agreement with my reasoning, since it doesn't convey the idea that the reduction is performed by A.
My problem with viewing problems as "pages of code" is that there is no one-to-one correspondence between problems and programs for solving them. I would rather view an algorithm as a machine where certain problems fit, but others don't. (Imagine one of those things used to teach shapes to children.) "A is reducible to B" implies that, although A does not fit in any machine where B fits, there is a machine that can transform an instance of A into an instance of B ("bend its shape") so that it may be solved by any machine for B.
But of course no one has to have the same mental image that I do.:)
Nitpick: when I said "show me a problem" above, I mean of course a problem solvable with a known algorithm. Otherwise the GP might demand that I present him with two different algorithms for the Halting Problem, and I'll be stumped.:)
Ultimately it's a matter of semantics. But it doesn't sound right to say that a problem "solves" another. Problems don't solve anything; algorithms do (on a tangent: show me a problem, and I will present you with two different algorithms for it with dramatically different run times). On the other hand, "reducible" captures the notion that you can represent the reduced problem as an instance of the problem it reduces to.
Quite true. However, I do not know of a single problem in P whose optimal known algorithm is O(n^c) with c bigger than 4 or 5. (That, of course, is no proof that none exists.) Does anyone know if there is any mathematical insight for why it should be this way?
Yes, and you do realize that what the GP is saying is that, no matter how clever you are when assigning 5-tuples to flows, you'll never be able to support more than about 30,000 users per IPv4 address? It's called the pigeonhole principle.
And that figure of 30,000 assumes that each user needs a single connection (read: a single TCP session). Most browsers already use six TCP connections when fetching a single web page. In fact I would be surprised if, with today's popular applications, you were able to support more than a thousand users behind a single IPv4 address without denying connections to any of them. And still, those users would not be able to use p2p applications, host network games, or do anything else that requires the ability to receive a direct connection.
I find it curious that you are against "artificial bandwidth restrictions and bullshit like that", yet in favor of artificial scarcity of network addresses (which is what sticking with IPv4 means).
It could work as you describe, but not without some massive investment by the ISPs. That investment would be better made on IPv6 which is a definitive solution as opposed to the band-aid that NAT is.
and by absolutely not destroy them...they mean, hey, let's have a bit of peace, for now at least, we still don't recognize your right to exist, and hey we might try to annihilate you AGAIN in the future, like we tried to do in '48 and '53 and '67 and '73 and so forth...you know those wars when we original lost the land...but for now...hey let's all just get along.
Are you talking about Palestine or Israel? Because, you know, apart from the specific years you quoted, your discourse would totally apply to the other side of the conflict.
I'll take a stab at this. To begin with, one thing most people have been missing (or at least not stating explicitly) in this thread is the following fact: mathematical notation is a matter of convention. That is, the meaning of notation is not god-given, nor is it a matter of intuition; it derives from precise and agreed upon rules.
Here we are talking about representing real numbers in decimal notation. So what are the agreed upon rules for this notation? Let's consider two cases: first the case of a finite representation, and then the case of an infinite representation.
For a finite representation, the rule is the following: if the representation is in the form "0.abcd", the corresponding number is given by the formula:
0.1*a + 0.01*b + 0.001*c + 0.0001*d.
Of course, the rule above only works for representations between zero and one with only four decimal digits; but I'm sure you can figure out what to do with longer representations. (I do not want to post the general formula here so we do not get unnecessarily distracted by heavy notation.)
Now consider an infinite representation. Can we apply the same rule as for the case of a finite one? Well, we have a problem: since there is an infinite number of digits, we are not able to carry out the sum given by the formula above and arrive at a result in a finite amount of time. In other words, we have a "sum of an infinite number of terms", which mathematicians call a series.
It happens that there is a whole theory of series, and that theory allows us to calculate the sum of a series without actually performing an infinite number of additions. The sum may be finite or infinite (it may be surprising to you that the sum of an infinite number of non-zero terms can be a finite number, but I will ask you to accept this as fact; I would explain it to you, but that's tangential to this discussion).
So, when we have an infinite representation of the form "0.abcde..." (the digits go on), we may imagine it to be the result of the "sum"
but in fact the corresponding number is obtained by considering the series made up of the terms in the sum, and determining the sum of the series.
So, when we have the infinite representation "0.999...", the corresponding number is given by a series whose first few terms are
0.9, 0.09, 0.009, 0.0009,...
and it turns out (according to the theory of series) that the sum of this particular series is 1.
Of course, the interesting and tricky part is knowing how to determine the sum of a series; but that really is not my point here. My point is that the fact that 0.999... = 1 rests on agreed upon rules on which number corresponds to a given decimal representation, and those rules assign the same number to "0.999" as they do to "1". Maybe this is not the answer you were expecting, because you would like an intuitive explanation that would make you feel it makes sense, instead of having to accept it as being simply the result of the application of seemingly arbitrary "agreed upon rules". Other people in this thread have tried, to some degree of success, to provide such intuitive explanations, and I welcome them. I'm merely making the point that, even though intuition plays an important role in mathematics, the meanings we ascribe to mathematical symbols are a matter of convention, not intuition.
I think your problem is that you have trouble seeing the difference between a number (the abstract concept) and a representation of a number (a concrete matter of notation). You are assuming that two different representations (namely, "0.999..." and "1") necessarily correspond to two distinct numbers. However, that is not the case in the base-10 positional system for representing real numbers.
I think we should switch to Anarchism, where the market is completely uncontrolled because there is no controlling entity so that there cannot possibly be any of that corruption that's so prevalent in government. Where nobody will ever be struck down by The Man and we can all live in huts and grow our own food without being bothered by anybody but the gangs of people out to steal all our shit.
You're pretty naïve if you belive that markets are under control only if there is a designated formal entity in control of them. Anarchism, like Communism and the Free Market, is just another unattainable utopia.
The real choice is between having a de jure government (which may sometimes not be the de facto government, but at least lets you influence its behavior through periodic elections), or having a de facto (not de jure) government "by the powerful, for the powerful" in which the rest of the people have no say at all. I certainly know which alternative I prefer.
Ken's "attack" is easily broken provided you have access to an already existing C compiler without the backdoor. Of course, if you're paranoid enough you might think that perhaps the backdoor is present in every existing C compiler...
Not only that, but arguably a heuristic cannot be considered an algorithm since it is not guaranteed to solve the problem --- in the same way that most people would not call shuffle sort a sorting algorithm.
So what do you do when you want to produce a 1kg weight to use in physical experiments? Do you take 3.40812408 gazillion photons with a 550.9455543 nm wavelength and place them on one of the plates of a scale?
It does not suffice that you define the kilogram rigorously. The definition must also suggest a practical way to produce a 1 kg specimen in the lab (up to a given degree of precision), which yours doesn't.
Incidentally, the reason why the silicon sphere thing hasn't yet replaced the International Prototype Kilogram is that it still gives less precision than the IPK, despite all the shortcomings of the latter (or so I read lately).
I am discussing the truth value of the proposition put forth by the original poster: that "atheists stake their eternal future on the presumption that God does not exist" and (not said but implied by the poster) god-believers somehow do not. I presented an argument showing that the proposition is false. My argument is not affected in any way by personal beliefs (either yours, mine, or anyone else's) or the intricacies of particular religions, which is all your last post refers to. Your remarks, though relevant and even interesting in other contexts, do not have a bearing on anything I said previously.
You seem to be too focused on your own personal beliefs to analyze this matter from an unbiased, rational point of view. I'm not inviting you to give up on your beliefs; however, in order to have a rational conversation, you must at the very least be able to abstract from them (which does not mean ceasing to believe in them). Otherwise, you will soon find you cannot have any meaningful conversation with people with different beliefs than yours (and then you probably wonder why they do not seem to respect your own).
There is no such thing as "not taking a chance" in this matter. Either you do believe in gods (one or more of them) or you do not believe in any. In either case, you will either be rewarded, punished, or disappear into nothing. But none of your choices invalidates any of these possibilities (even if you're an atheist: there might be a god who is benevolent enough to send you to heaven even if you did not believe in him while alive).
The point is not which choice is best; the point is that there is not enough information to make a choice that is certainly better than the others. And this is what the original poster would have you disagreeing with.
You are either trolling me or missing my point. I'll assume the second hypothesis is true.
What christian theology says is irrelevant to my argument. Just conceive an alternate god whose decision process will send you to hell if the christian god would send you to heaven in the same conditions, and vice-versa. You cannot rule out the possibility that either of these gods exists, just as you cannot prove they do.
So, by behaving in accordance with what the christian god deems to be worthy of inhabiting heaven after you die, you are not ensuring anything about your afterlife. It may well turn out that the christian god does not exist but the alternate god does, and by leading such a life you earn yourself an eternity in hell.
As with Pascal's Wager, your theoretical god is not "as likely to exist" as a different god. It's unknown (and many would say unknowable) how likely either of them are. So, you're right in that both gods are in the same boat -- nobody knows how likely it is that either exists. It's incorrect to go from that to "they're equally likely to exist".
I didn't mean "as likely as" in the statistical sense; I just meant that neither can be shown to exist (or to not exist, for that matter).
In any case, the fact that you cannot ascribe probabilities to the existence of every conceivable god does not alter my point: with some probability (possibly zero) you will go to hell; with some probability (possibly zero) you will go to heaven; and with some probability (possibly zero) you'll simply cease to exist in any physical or metaphysical sense. You just don't know what the probabilities are, so you cannot affirm a christian is better off than a muslim or a hinduist or a satanist or an atheist or whatever, as Pascal does with his wager. Pascal would only be correct if 1) all existing gods were benevolent and 2) each of them would never punish you for believing in other god than him.
(I realise you're not countering my argument; I'm just clearing things up for others.)
Since we are talking about unprovable matters, I could also postulate there is a god that will send you to hell for being a nice person. This god is as likely to exist as yours. In fact, as long as there is more than one religion in the world, there are potentially many gods, who, once you die, will send you straight to hell for not believing in them. Surely, by being a practitioner of religion X, you are staking your eternal future too?
Slashdot Mirror
Yes. If anyone were to challenge me to prove my assertion, prepending a useless long loop depending on input size at the beginning of a valid algorithm would be my answer. :)
Though I still have a problem with the other part of your post. Suppose I am allowed to assume the trivial algorithm exists; I cannot however assume anything about its complexity (that would be cheating). So, assuming that by "drastically different run time" I mean to say that they're not big-O equivalent, I am stumped when deciding how big to make the initial, useless loop; for, for any complexity that I might choose for that initial loop, it may be the case that the trivial algorithm already has at least that complexity. ;)
I agree that "A reduces to B" is equally nonsensical; it's an artifact of a growing (and sometimes irritating) tendency to suppress passive clauses in English. The passive version, "A is reducible to B", is more in agreement with my reasoning, since it doesn't convey the idea that the reduction is performed by A.
My problem with viewing problems as "pages of code" is that there is no one-to-one correspondence between problems and programs for solving them. I would rather view an algorithm as a machine where certain problems fit, but others don't. (Imagine one of those things used to teach shapes to children.) "A is reducible to B" implies that, although A does not fit in any machine where B fits, there is a machine that can transform an instance of A into an instance of B ("bend its shape") so that it may be solved by any machine for B.
But of course no one has to have the same mental image that I do. :)
Nitpick: when I said "show me a problem" above, I mean of course a problem solvable with a known algorithm. Otherwise the GP might demand that I present him with two different algorithms for the Halting Problem, and I'll be stumped. :)
Ultimately it's a matter of semantics. But it doesn't sound right to say that a problem "solves" another. Problems don't solve anything; algorithms do (on a tangent: show me a problem, and I will present you with two different algorithms for it with dramatically different run times). On the other hand, "reducible" captures the notion that you can represent the reduced problem as an instance of the problem it reduces to.
Quite true. However, I do not know of a single problem in P whose optimal known algorithm is O(n^c) with c bigger than 4 or 5. (That, of course, is no proof that none exists.) Does anyone know if there is any mathematical insight for why it should be this way?
Actually, it's not an iff. If P=0 then P=NP no matter the value of N. ;)
Yes, and you do realize that what the GP is saying is that, no matter how clever you are when assigning 5-tuples to flows, you'll never be able to support more than about 30,000 users per IPv4 address? It's called the pigeonhole principle.
And that figure of 30,000 assumes that each user needs a single connection (read: a single TCP session). Most browsers already use six TCP connections when fetching a single web page. In fact I would be surprised if, with today's popular applications, you were able to support more than a thousand users behind a single IPv4 address without denying connections to any of them. And still, those users would not be able to use p2p applications, host network games, or do anything else that requires the ability to receive a direct connection.
I find it curious that you are against "artificial bandwidth restrictions and bullshit like that", yet in favor of artificial scarcity of network addresses (which is what sticking with IPv4 means).
It could work as you describe, but not without some massive investment by the ISPs. That investment would be better made on IPv6 which is a definitive solution as opposed to the band-aid that NAT is.
Regret is a standard term in economics (esp. game theory). If anyone is to take the blame over a poor choice of noun, it certainly isn't Google.
Are you talking about Palestine or Israel? Because, you know, apart from the specific years you quoted, your discourse would totally apply to the other side of the conflict.
I'll take a stab at this. To begin with, one thing most people have been missing (or at least not stating explicitly) in this thread is the following fact: mathematical notation is a matter of convention. That is, the meaning of notation is not god-given, nor is it a matter of intuition; it derives from precise and agreed upon rules.
Here we are talking about representing real numbers in decimal notation. So what are the agreed upon rules for this notation? Let's consider two cases: first the case of a finite representation, and then the case of an infinite representation.
For a finite representation, the rule is the following: if the representation is in the form "0.abcd", the corresponding number is given by the formula:
0.1*a + 0.01*b + 0.001*c + 0.0001*d.
Of course, the rule above only works for representations between zero and one with only four decimal digits; but I'm sure you can figure out what to do with longer representations. (I do not want to post the general formula here so we do not get unnecessarily distracted by heavy notation.)
Now consider an infinite representation. Can we apply the same rule as for the case of a finite one? Well, we have a problem: since there is an infinite number of digits, we are not able to carry out the sum given by the formula above and arrive at a result in a finite amount of time. In other words, we have a "sum of an infinite number of terms", which mathematicians call a series.
It happens that there is a whole theory of series, and that theory allows us to calculate the sum of a series without actually performing an infinite number of additions. The sum may be finite or infinite (it may be surprising to you that the sum of an infinite number of non-zero terms can be a finite number, but I will ask you to accept this as fact; I would explain it to you, but that's tangential to this discussion).
So, when we have an infinite representation of the form "0.abcde..." (the digits go on), we may imagine it to be the result of the "sum"
0.1*a + 0.01*b + 0.001*c + 0.0001*d + 0.00001*e + ...
but in fact the corresponding number is obtained by considering the series made up of the terms in the sum, and determining the sum of the series.
So, when we have the infinite representation "0.999...", the corresponding number is given by a series whose first few terms are
0.9, 0.09, 0.009, 0.0009, ...
and it turns out (according to the theory of series) that the sum of this particular series is 1.
Of course, the interesting and tricky part is knowing how to determine the sum of a series; but that really is not my point here. My point is that the fact that 0.999... = 1 rests on agreed upon rules on which number corresponds to a given decimal representation, and those rules assign the same number to "0.999" as they do to "1". Maybe this is not the answer you were expecting, because you would like an intuitive explanation that would make you feel it makes sense, instead of having to accept it as being simply the result of the application of seemingly arbitrary "agreed upon rules". Other people in this thread have tried, to some degree of success, to provide such intuitive explanations, and I welcome them. I'm merely making the point that, even though intuition plays an important role in mathematics, the meanings we ascribe to mathematical symbols are a matter of convention, not intuition.
I think your problem is that you have trouble seeing the difference between a number (the abstract concept) and a representation of a number (a concrete matter of notation). You are assuming that two different representations (namely, "0.999..." and "1") necessarily correspond to two distinct numbers. However, that is not the case in the base-10 positional system for representing real numbers.
To be pedantic, that is only true if f(x) is continuous. But of course, in the situation at hand it is.
You're pretty naïve if you belive that markets are under control only if there is a designated formal entity in control of them. Anarchism, like Communism and the Free Market, is just another unattainable utopia.
The real choice is between having a de jure government (which may sometimes not be the de facto government, but at least lets you influence its behavior through periodic elections), or having a de facto (not de jure) government "by the powerful, for the powerful" in which the rest of the people have no say at all. I certainly know which alternative I prefer.
Ken's "attack" is easily broken provided you have access to an already existing C compiler without the backdoor. Of course, if you're paranoid enough you might think that perhaps the backdoor is present in every existing C compiler...
For details see this thread:
http://lwn.net/Articles/321218/
Not only that, but arguably a heuristic cannot be considered an algorithm since it is not guaranteed to solve the problem --- in the same way that most people would not call shuffle sort a sorting algorithm.
So what do you do when you want to produce a 1kg weight to use in physical experiments? Do you take 3.40812408 gazillion photons with a 550.9455543 nm wavelength and place them on one of the plates of a scale?
It does not suffice that you define the kilogram rigorously. The definition must also suggest a practical way to produce a 1 kg specimen in the lab (up to a given degree of precision), which yours doesn't.
Incidentally, the reason why the silicon sphere thing hasn't yet replaced the International Prototype Kilogram is that it still gives less precision than the IPK, despite all the shortcomings of the latter (or so I read lately).
I declare this thread officially Godwin'ed!
I am discussing the truth value of the proposition put forth by the original poster: that "atheists stake their eternal future on the presumption that God does not exist" and (not said but implied by the poster) god-believers somehow do not. I presented an argument showing that the proposition is false. My argument is not affected in any way by personal beliefs (either yours, mine, or anyone else's) or the intricacies of particular religions, which is all your last post refers to. Your remarks, though relevant and even interesting in other contexts, do not have a bearing on anything I said previously.
You seem to be too focused on your own personal beliefs to analyze this matter from an unbiased, rational point of view. I'm not inviting you to give up on your beliefs; however, in order to have a rational conversation, you must at the very least be able to abstract from them (which does not mean ceasing to believe in them). Otherwise, you will soon find you cannot have any meaningful conversation with people with different beliefs than yours (and then you probably wonder why they do not seem to respect your own).
There is no such thing as "not taking a chance" in this matter. Either you do believe in gods (one or more of them) or you do not believe in any. In either case, you will either be rewarded, punished, or disappear into nothing. But none of your choices invalidates any of these possibilities (even if you're an atheist: there might be a god who is benevolent enough to send you to heaven even if you did not believe in him while alive).
The point is not which choice is best; the point is that there is not enough information to make a choice that is certainly better than the others. And this is what the original poster would have you disagreeing with.
You are either trolling me or missing my point. I'll assume the second hypothesis is true.
What christian theology says is irrelevant to my argument. Just conceive an alternate god whose decision process will send you to hell if the christian god would send you to heaven in the same conditions, and vice-versa. You cannot rule out the possibility that either of these gods exists, just as you cannot prove they do.
So, by behaving in accordance with what the christian god deems to be worthy of inhabiting heaven after you die, you are not ensuring anything about your afterlife. It may well turn out that the christian god does not exist but the alternate god does, and by leading such a life you earn yourself an eternity in hell.
I didn't mean "as likely as" in the statistical sense; I just meant that neither can be shown to exist (or to not exist, for that matter).
In any case, the fact that you cannot ascribe probabilities to the existence of every conceivable god does not alter my point: with some probability (possibly zero) you will go to hell; with some probability (possibly zero) you will go to heaven; and with some probability (possibly zero) you'll simply cease to exist in any physical or metaphysical sense. You just don't know what the probabilities are, so you cannot affirm a christian is better off than a muslim or a hinduist or a satanist or an atheist or whatever, as Pascal does with his wager. Pascal would only be correct if 1) all existing gods were benevolent and 2) each of them would never punish you for believing in other god than him.
(I realise you're not countering my argument; I'm just clearing things up for others.)
What, you expect the same people who take works of fiction written thousands of years ago literally to read Hawking's words in a metaphorical light?
Since we are talking about unprovable matters, I could also postulate there is a god that will send you to hell for being a nice person. This god is as likely to exist as yours. In fact, as long as there is more than one religion in the world, there are potentially many gods, who, once you die, will send you straight to hell for not believing in them. Surely, by being a practitioner of religion X, you are staking your eternal future too?