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Can randomness be defined?

Can randomness be defined? Since I believe that the word means the absence of order, the proving of randomness involves proving a negative. What, then is the meaning of the terms random number and random sequence? Does the designation of any events e.g. radioactive decay, as random mean anything other than "uninfluenceable by any known agency" and/or "not showing any regularity discernible by humans"?

Someone who knows more about the physics than I do would need to answer the latter question, but I believe that certain physical phenomena are supposed to be random in a much stronger sense that just "unintelligible to humans". There is supposed to be nothing other than probabilistic facts about when and how the phenomenon might occur. Even God cannot know in advance when and what is to happen, for there is simply no such fact to be known.

There is a sophisticated mathematical theory of randomness. See the Wikipedia entry on Kolmogorov complexity. It describes a way, in terms of informational complexity, to characterize when a sequence of 0s and 1s is random.

Someone who knows more about the physics than I do would need to answer the latter question, but I believe that certain physical phenomena are supposed to be random in a much stronger sense that just "unintelligible to humans". There is supposed to be nothing other than probabilistic facts about when and how the phenomenon might occur. Even God cannot know in advance when and what is to happen, for there is simply no such fact to be known. There is a sophisticated mathematical theory of randomness. See the Wikipedia entry on Kolmogorov complexity . It describes a way, in terms of informational complexity, to characterize when a sequence of 0s and 1s is random.

What if a person could indefinitely predict the outcome of a coin flip?

What if a person could indefinitely predict the outcome of a coin flip? I understand that's not much of a question; but I want to know what that would mean in terms of either that single person, or the universe in general. If it happened tomorrow, what happens next?

I'm not sure I understand the question: Is it this? Let's say we have a fair coin, C, and that it is going to be flipped once every minute starting at noon tomorrow. Now let's imagine a person Fred. Fred is about to have an amazing streak of luck. Each minute, he is going to call "heads" or "tails", and Fred is going to get it right every time, and he will continue to do so for as long as he keeps going. I don't see anything impossible about that. Maybe it's unlikely, but it's clearly possible, and nothing much seems to follow.

Now we don't have Fred making an infinite prediction here. So maybe we should change the story. Let's say that Fred says that the first flip will be even, every even flip will be followed by an odd one, and every odd one by an even one. And now let's suppose that, for as long as we keep flipping the coin, Fred turns out to be right. We can even suppose that we flip the coin forever, and Fred is always right. Again, that's clearly possible, if very unlikely, but nothing much seems to follow.

But perhaps what you are wondering is this. Let's go back to the original case. Suppose Fred has been right for several days. So he decides to start making some wagers. He offers you two-to-one odds on the next coin flip: If he's right, you pay him $2; if he's wrong, you get $4. Do you take the bet? Is it rational for Fred to offer such odds? There are clearly two sorts of responses here. One is that Fred's past success is no guarantee of his future success: He's just been lucky, and his odds of getting it right the next time are even, as always, so you should take the bet, and it's irrational for Fred to offer it. The other response is that something's clearly up with Fred, and I ain't goin' there.

Something about this problem strikes me as similar to Newcombe's Problem. Imagine the following experiment. Fred is wealthy a psychology professor, and he offers you the following choice. He puts two boxes before you. The right box, he tells you, contains $100 more than the left box. You may either take just the money in the left box or you may take the money in both boxes. Obviously you take the money in both boxes, right? Well, there's a catch. Before putting the boxes down, Fred has carefully considered what he thinks you will do. If he thought you would take both boxes, he put no money in the left box and $100 in the right box. If he thought you would take only the left box, he put $10,000 in it and $10,100 in the right box. He has given this little test many times before. He has always been right about what he thought the person would do. So everyone who has taken two boxes has gotten $100, and everyone who has taken only the left box has gotten $10,000.

What do you do? Are you a one-boxer or a two-boxer? As I believe Robert Nozick was the first to mention, everyone seems to be absolutely sure what it is right to do. But they don't all agree. Some are very sure you should one-box, and others are equally sure you should two-box. Me? I haven't a clue.

I'm not sure I understand the question: Is it this? Let's say we have a fair coin, C, and that it is going to be flipped once every minute starting at noon tomorrow. Now let's imagine a person Fred. Fred is about to have an amazing streak of luck. Each minute, he is going to call "heads" or "tails", and Fred is going to get it right every time, and he will continue to do so for as long as he keeps going. I don't see anything impossible about that. Maybe it's unlikely, but it's clearly possible, and nothing much seems to follow. Now we don't have Fred making an infinite prediction here. So maybe we should change the story. Let's say that Fred says that the first flip will be even, every even flip will be followed by an odd one, and every odd one by an even one. And now let's suppose that, for as long as we keep flipping the coin, Fred turns out to be right. We can even suppose that we flip the coin forever, and Fred is always right. Again, that's clearly possible, if very unlikely, but nothing much...