yoginka: (thinking)
Пару лет назад написала немного о книге Сасскинда (http://yoginka.livejournal.com/314675.html). Но тогда поленилась отметить один мелкий момент о вероятностях, который мне сразу понравился. А сегодня был спор с Ж., и я вспомнила о нем, поискала и нашла это место в книге. Пусть оно будет здесь тоже:
(The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics)
"Before closing this chapter, I want to come back to the thing that troubled Einstein so deeply. I don't know for sure, but I suspect that it had to do with the ultimate meaningless nature of probabilistic statements. I have always been mystified by what they actually say about the world. As far as I can tell, they don't say anything very definite. I once wrote the following very short story, originally included in John Brockman's book What We Believe but Cannot Prove, that illustrates the point. The story, "Conversation with a Slow Student,'' is about a discussion between a physics professor and a student who just can't get the point. When I wrote the story, I was thinking of myself as the student, not the professor.

Student: Hi Prof. I've got a problem. I decided to do a little probability experiment - you know, coin flipping and check some of the stuff you taught us. But it didn't work.

Professor: Well I'm glad to hear that you're interested. What did you do?

Student: I flipped this coin 1,000 times. You remember, you taught us that the probability to flip heads is one half. I figured that meant that if I flip 1,000 times I ought to get 500 heads. But it didn't work. I got 513. What's wrong?

Professor: Yeah, but you forgot about the margin of error. If you flip a certain number of times then the margin of error is about the square root of the number of flips. For 1,000 flips the margin of error is about 30. So you were within the margin of error.

Student: Ah, now I get it. Every time I flip 1,000 times I will always get something between 470 and 530 heads. Every single time! Wow, now that's a fact I can count on.

Professor: No, no! What it means is that you will probably get between 470 and 530.

Student: You mean I could get 200 heads? Or 850 heads? Or even all heads?

Professor: Probably not.

Student: Maybe the problem is that I didn't make enough flips. Should I go home and try it 1,000,000 times? Will it work better?

Professor: Probably.

Student: Aw come on Prof. Tell me something I can trust. You keep telling me what probably means by giving me more probablies. Tell me what probability means without using the word probably.

Professor: Hmmm. Well how about this: It means I would be surprised if the answer were outside the margin of error.

Student: My god! You mean all that stuff you taught us about statistical mechanics and Quantum Mechanics and mathematical probability: all it means is that you'd personally be surprised if it didn't work?

Professor: Well, uh ... If I were to flip a coin a million times I'd be damn sure I wasn't going to get all heads. I'm not a betting man but I'd be so sure that I'd bet my life or my soul. I'd even go the whole way and bet a year's salary. I'm absolutely certain the laws of large numbers - probability theory - will work and protect me. All of science is based on it. But, I can't prove it and I don't really know why it works. That may be the reason why Einstein said, "God doesn't play dice." It probably is.

From time to time, we hear physicists claim that Einstein didn't understand Quantum Mechanics and therefore wasted his time with naive classical theories. I very much doubt that this is true. His arguments against Quantum Mechanics were extremely subtle, culminating in one of the most profound and most cited papers in all of physics. My guess is that Einstein was disturbed by the same thing that bothered the slow student. How could the ultimate theory of reality be about nothing more concrete than our own degree of surprise at the outcome of an experiment? I have shown you some of the paradoxical, almost illogical, things that Quantum Mechanics forced on a classically wired brain. But I suspect that you are not entirely satisfied. Indeed, I hope you are not. If you are confused, you should be. The only real remedy is a dose of calculus and submersion in a good Quantum Mechanics textbook for a few months. Only a very unusual mutant, or a person brought up in an extremely peculiar family, could naturally have the wiring to understand Quantum Mechanics. Remember, in the end even Einstein couldn't grok it."

Date: 2016-01-14 08:21 pm (UTC)From: [identity profile] evgeniirudnyi.livejournal.com
Мат. статистика предполагает существование случайных величин. В данных рамках есть ненулевая вероятность выпадения миллиона решек подряд.

В физике однако многие решения которые выражаются через экспоненциальную функцию. На этом пути полезно помнить, что например exp(-at)никогда не достигает нуля (a - константа, t - время). Соответственно на вопрос, когда заканчивается процесс, описываемый exp(-at), нет прямого ответа. Ответ возможен только при волевом решении - выборе некоторого малого значения, которое достигает эта функция.

Date: 2016-01-15 06:01 pm (UTC)From: [identity profile] yoginka.livejournal.com
Мне понравилось здесь то, что я выделила жирным шрифтом. Сама всегда так чувствовала. При этом на аксиоматическом и чисто математическом уровне по мне теория вероятностей не хуже остальных. А вот с физическим и философском смыслом всегда были нелады. Приятно, что я не одинока, и даже в хорошей компании оказалась.

Date: 2016-01-15 06:25 pm (UTC)From: [identity profile] evgeniirudnyi.livejournal.com
Однако решение exp(-at) у вас похоже таковых проблем не вызывает. В чем разница?

Date: 2016-01-15 10:18 pm (UTC)From: [identity profile] yoginka.livejournal.com
https://en.wikipedia.org/wiki/Probability_interpretations
Здесь обрисованы интерпретационные проблемы.

Date: 2016-01-15 10:21 pm (UTC)From: [identity profile] yoginka.livejournal.com
Я хотела сказать, что дело не в закономерностях, не в математике, а в том, как все это интерпретировать.

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