Nazzzgul666 Posted August 2, 2017 Author Posted August 2, 2017 You're refer... (*Clears throat*) Sorry. You're referring to the continuum hypothesis, which states that there is no cardinal number between <edited typo> |N and |R. Its independence of other axioms of set theory is notoriously hard to prove (like you said, it incapacitated Cantor himself), that's why I mentioned the halting problem. Halting problem for Turing machines is easily unprovable, by the same process I've shown below. The fact that |R is more "powerful" that |N is provable, assuming some form of AC is correct, by a process called Cantor's diagonal argument. Here it is, somewhat simplified for shortness. Let's assume that we can "count" real numbers, say, those ranging from 0 to 1. Let's "count" them and write them down, starting from first, in their decimal form. What we get is an infinite table: (1st real number) .1491775..... (2nd real number) .3523057..... (3rd real number) .14097154018.... (...) Now, let's take the diagonal of that table: 1st column in 1st row, n-th column in n-th row, etc. Let's change each number we've got to anything else (i.e. +1 mod 10, so that 9 becomes 0). What do we have? A decimal representation of some real number between 0 and 1. But it's not the first number that we've written down, since the its first decimal sign is different, because we've changed it. It's not the second, since it's 2nd decimal sign is different. Et cetera. So what we've got is a decimal number not in our table. Which cannot be, since we've written down them all! Ergo, our initial assumption is incorrect and real numbers cannot be counted, whichever method we use. With this true, the Continuum hypothesis states, that if a set cannot be counted (with natural numbers, i.e. computable) it must be at least as large as |R (the corresponding cardinal number is called continuum). Not sure if you're edit means you'd like to rewrite that, but i have the feeling we're getting closer. I'm familar with Cantors Diagonal argument, that's why i called |R "over-countable", what's the correct english word for that? Just uncountable? "Beyond countable"? I'm not familiar with the Continuum hypothsis, though. *edit: just checked Continuum Hypothesis in wikipedia, you're right, that's what i mean, just didn't know the name. It mentions Paul Cohen who showed that its independence is not "only" hard to prove, it's actually undecidable. And i was wrong about the time, it was already in 1960, no idea why i was sure it was in the last 20 years. I must have confused it with something else, and i'm a bit ashamed i forgot that it was Cohen. For the sociology: sure, there are a lot of factors. You have to create your model in that way that it's not (so very) vulnerable to changes in those factors. This can mean to use big numbers like Assimov did, it also can mean to avoid beeing exaggerated specific, or too far in the future. In a country of the size of germany, you can be pretty sure the numbers of people who go to work won't change that much. On a certain day/week there can be holidays, strikes, a big flu, whatever. But over the year there won't be much difference to last years. And if there is an event that really changes that, like... the flu is spanish flue, pretty much all predictions you make don't matter that much anymore. But that doesn't mean you shouldn't or can't use them. The point where i'm really wary is when somebody makes predictions about how things will be in 20, 50 or 100 years even on a large scale, there are too many possibilities. Or if they are much too specific like "growth rate will be 2.5%". Or if the samples are too small, for both studies to collect data and predictions. To say "well, we have those 10 women and 3 are pregnant, so we can say from ~40mio women in germany there are currently 12mio pregnant" is obviously bullshit. But if i take the whole population to collect data for the whole last year it's very likely that i can get very close with a prediction for next year. There are a lot of things that can happen during that year, big and small, but only few would make a big difference, and even combined it's not very likely.
dnoah Posted August 2, 2017 Posted August 2, 2017 You're refer... (*Clears throat*) Sorry. You're referring to the continuum hypothesis, which states that there is no cardinal number between <edited typo> |N and |R. Its independence of other axioms of set theory is notoriously hard to prove (like you said, it incapacitated Cantor himself), that's why I mentioned the halting problem. Halting problem for Turing machines is easily unprovable, by the same process I've shown below. The fact that |R is more "powerful" that |N is provable, assuming some form of AC is correct, by a process called Cantor's diagonal argument. Here it is, somewhat simplified for shortness. Let's assume that we can "count" real numbers, say, those ranging from 0 to 1. Let's "count" them and write them down, starting from first, in their decimal form. What we get is an infinite table: (1st real number) .1491775..... (2nd real number) .3523057..... (3rd real number) .14097154018.... (...) Now, let's take the diagonal of that table: 1st column in 1st row, n-th column in n-th row, etc. Let's change each number we've got to anything else (i.e. +1 mod 10, so that 9 becomes 0). What do we have? A decimal representation of some real number between 0 and 1. But it's not the first number that we've written down, since the its first decimal sign is different, because we've changed it. It's not the second, since it's 2nd decimal sign is different. Et cetera. So what we've got is a decimal number not in our table. Which cannot be, since we've written down them all! Ergo, our initial assumption is incorrect and real numbers cannot be counted, whichever method we use. With this true, the Continuum hypothesis states, that if a set cannot be counted (with natural numbers, i.e. computable) it must be at least as large as |R (the corresponding cardinal number is called continuum). Not sure if you're edit means you'd like to rewrite that, but i have the feeling we're getting closer. I'm familar with Cantors Diagonal argument, that's why i called |R "over-countable", what's the correct english word for that? Just uncountable? "Beyond countable"? I'm not familiar with the Continuum hypothsis, though. *edit: just checked Continuum Hypothesis in wikipedia, you're right, that's what i mean, just didn't know the name. It mentions Paul Cohen who showed that its independence is not "only" hard to prove, it's actually undecidable. And i was wrong about the time, it was already in 1960, no idea why i was sure it was in the last 20 years. I must have confused it with something else, and i'm a bit ashamed i forgot that it was Cohen. (Math: pedantry) Not rewrite, just skip, because it was unnecessary. We were both having translation problems, it seems. For the sociology: sure, there are a lot of factors. You have to create your model in that way that it's not (so very) vulnerable to changes in those factors. This can mean to use big numbers like Assimov did, it also can mean to avoid beeing exaggerated specific, or too far in the future. In a country of the size of germany, you can be pretty sure the numbers of people who go to work won't change that much. On a certain day/week there can be holidays, strikes, a big flu, whatever. But over the year there won't be much difference to last years. And if there is an event that really changes that, like... the flu is spanish flue, pretty much all predictions you make don't matter that much anymore. But that doesn't mean you shouldn't or can't use them. The point where i'm really wary is when somebody makes predictions about how things will be in 20, 50 or 100 years even on a large scale, there are too many possibilities. Or if they are much too specific like "growth rate will be 2.5%". Or if the samples are too small, for both studies to collect data and predictions. To say "well, we have those 10 women and 3 are pregnant, so we can say from ~40mio women in germany there are currently 12mio pregnant" is obviously bullshit. But if i take the whole population to collect data for the whole last year it's very likely that i can get very close with a prediction for next year. There are a lot of things that can happen during that year, big and small, but only few would make a big difference, and even combined it's not very likely. (Sociology) Yes, and I totally agree that statistical analysis is perfectly viable within its limits and field of application. When I was talking about instability, I meant that a lot of systems/processes in our society are inherently chaotic and thus hugely unpredictable. And statisticians and analysts tend to be completely blind to that fact. And indeed, it is troubling when someone tries to apply statistics (or any other abstract theory) outside its limits, and even more so considering how many people tend to believe it rather than try and think for themselves. (More pedantry )So, why do I think that the game from your OP shouldn't be treated seriously? Well if it were to be [treated seriously], it could do with some adherence to the core principles of a scientific study. 1. Do people really trust each other less than in times of WWI? What we got is just one example, one photo, really, in a very specific environment. I tend to intuitively agree with that statement. But the scientific method says that I should cast aside personal feelings and look for measurable facts. I'm not a historian and really have very little idea as to what did the western society look like back in the day. Well, the credit market has certainly increased. Credit implies at least some level of trust, doesn't it? Hmm... 2. By the way, what *is* trust, exactly? Is it even quantifiable? Let's say, I have a battle comrade that I can trust with my life, but it just so happens that he is that sort of disorganized person that tends to constantly borrow small amounts of money, sometimes forgetting about it, which is annoying. So, is my life worth to me less than some lunch money? What about a corporate employee sharing insider information with someone he "trusts", for mutual benefit? This situation looks a lot like that game's model, but I doubt that's the example that game was intended for. Which leads to: 3. What exactly is the application of that game's model? What questions, if any, can it actually answer? Whether crime rate is correlated to the "level" of trust? Whether to expect another war or financial crisis? What will follow the possible decrease in communications due to the recent Internet censorship events or SJW phenomenon? Is the game theory applicable to any of those questions at all? I could go on, but the point is, this game is a good demonstration of some principles of game theory, but it does not have any practical application - without some context, at the very least.
Nazzzgul666 Posted August 2, 2017 Author Posted August 2, 2017 (Sociology) Yes, and I totally agree that statistical analysis is perfectly viable within its limits and field of application. When I was talking about instability, I meant that a lot of systems/processes in our society are inherently chaotic and thus hugely unpredictable. And statisticians and analysts tend to be completely blind to that fact. And indeed, it is troubling when someone tries to apply statistics (or any other abstract theory) outside its limits, and even more so considering how many people tend to believe it rather than try and think for themselves. (More pedantry )So, why do I think that the game from your OP shouldn't be treated seriously? Well if it were to be [treated seriously], it could do with some adherence to the core principles of a scientific study. 1. Do people really trust each other less than in times of WWI? What we got is just one example, one photo, really, in a very specific environment. I tend to intuitively agree with that statement. But the scientific method says that I should cast aside personal feelings and look for measurable facts. I'm not a historian and really have very little idea as to what did the western society look like back in the day. Well, the credit market has certainly increased. Credit implies at least some level of trust, doesn't it? Hmm... 2. By the way, what *is* trust, exactly? Is it even quantifiable? Let's say, I have a battle comrade that I can trust with my life, but it just so happens that he is that sort of disorganized person that tends to constantly borrow small amounts of money, sometimes forgetting about it, which is annoying. So, is my life worth to me less than some lunch money? What about a corporate employee sharing insider information with someone he "trusts", for mutual benefit? This situation looks a lot like that game's model, but I doubt that's the example that game was intended for. Which leads to: 3. What exactly is the application of that game's model? What questions, if any, can it actually answer? Whether crime rate is correlated to the "level" of trust? Whether to expect another war or financial crisis? What will follow the possible decrease in communications due to the recent Internet censorship events or SJW phenomenon? Is the game theory applicable to any of those questions at all? I could go on, but the point is, this game is a good demonstration of some principles of game theory, but it does not have any practical application - without some context, at the very least. I'd like to start with 2. "What is trust?" which is the most important question, or at least it needs some kind of defintion for anybody researching this. It may work for us here and now without one, but for scientific studies it's pointless to talk about something undefined. I can always say it's true or false or both if it isn't clear if trusting somebody just means that i have some hope another human won't kill me, or if trust means that i can jump off a bridge and he'll catch me. Or something between. I consider the question if my life is more or less worth than money (or how much money) rather philosophical and not that much important in most cases, usually sociologists don't deal with such alternatives. But that there are at least different levels of trust and where they draw the lines between those levels has to be mentioned at any study imho. 1. I think that's actually true, just like you. But i would disagree that especially after 2008 credit amounts are related to trust (the debtors) . Remember, the reason that governments had to give money to banks or at least give guarantees was because banks didn't trust each other enough to give (short term) loans. The problem wasn't a lack of money in general. And before that they didn't trust in the debtors either, they had some trust in their mathematical models, they had some trust that they can sell the debts to somebody else before it'll all blow up, and partly they fucking knew that it will blow up and gave the credits anyways and sold them. And the banks buying this stuff had some trust in the other banks, that they wouldn't be that much fucked up or at least it wouldn't be that bad. The reason that there is a bigger amount of credits is that there are bigger amounts of money. I'm pretty sure in a time where (not all, but some) deals were sealed by handshake, trust even in credits was much higher than today. This not necessarily a question to the answer if there is less trust, but i'm sure credits in general are no valuable variable here. One could say, trust was replaced by contracts. So if you want to use credits or any business, you can't count any deal that has a contract. If you make your deal with a handshake, you have to have trust, i think that would be valuable. But due to the nature of those things... statistics are probably hard to get. 3. I think it's not meant to answer questions, maybe that's what you mean with "not serious". But imho, sometimes questions are more important than answers and i think it can do a good job in questioning premises like "you have to cheat to win". And here i take it quite serious, although it's not capable of answering that question. Like many parts of math, it has little practical value on it's own, but that doesn't mean somebody couldn't take it and find a valuable application for it - or giving it context, like you said. Number theory isn't my favorite, but even before the internet started i wouldn't thought of it as something "not serious", just because there was no practical use. But of course, yes. The game is meant as a show case for game theory, and every question out of that is just a bonus. I think that i like it that much is because it did a good job on getting some boni, but it won't answer any questions.
carlfatal Posted August 3, 2017 Posted August 3, 2017 Only hopping in to say, that I definitely like reading your thougths here. I found this little game, cause fefe linked to it, and well, here at LL seems to be the right place to discuss such things.
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