Monday, August 11, 2014

Black holes can't keep secrets

At first, math seemed to show that anything that enters a black hole, is lost forever. Later, it seemed that black holes evaporate, but the secrets remain lost. But maybe it is not so.

My new video at the FQXi contest is called Can a black hole keep a secret?, and can be seen and rated at http://fqxi.org/community/forum/topic/2205:

http://fqxi.org/community/forum/topic/2205

To rate my video or those of my competitors, click "rate this video". You will be required to enter an email address to avoid duplicate votes. Then press "go" and vote.

You can check and rate other videos at FQXi Video Contest - Spring, 2014. You can submit your own video until August 22.

In the previous post, named The puzzle of quantum reality, in theaters near you, and at FQXi, I mentioned my other video, and my son's.

On youtube, my videos can be watched with subtitles, in English or in Romanian:
The puzzle of quantum reality
Can a black hole keep a secret?

Thursday, August 7, 2014

The puzzle of quantum reality, in theaters near you, and at FQXi

I made a 7 minutes video introducing some puzzling aspects of quantum mechanics to a general audience. At the end it contains a proposed view which, at least to me, makes the things clearer, so I hope it can help others too.

http://fqxi.org/community/forum/topic/2183

To rate my video or those of my competitors, click "rate this video". You will be required to enter an email address to avoid duplicate votes.

I also compete against my son, whose video is at http://fqxi.org/community/forum/topic/2176:
http://fqxi.org/community/forum/topic/2176

You can check and rate other videos, ranging from fun to informative, at FQXi Video Contest - Spring, 2014. You can submit your own video until August 22.




A confused sleeping beauty 2

This post contains a small twist of the original experiment discussed in the previous post, A confused sleeping beauty. The new version doesn't require putting anyone to sleep and removing her memories, because we replace memory removal with lack of information.

Confusing Sleeping Beauty without erasing her memory


Sleeping Beauty is no longer required to sleep, but she may still need to sleep, to remain beautiful.

Consider the following settings:
- We toss a fair coin.
- If it lands heads, we will ask once Sleeping Beauty her belief for the proposition that the question landed heads.
- If the coin lands tail, we ask her twice.
This is similar to the original experiment, but instead of erasing her memory, we just do the following:
- Before asking her any question, we toss the coin a large number of times.
- Then we ask Beauty, but not in the same order in which we tossed. For example, when we toss a coin, if it landed heads, we write down a question and don't ask it yet. If it landed tails, we write down two questions, and don't ask them yet. Then we shuffle the questions and we ask Beauty one at a time. We take care to keep track for each question to which toss is connected.
To prevent the possibility that she adjust her estimates by counting counting the number of heads and tails about which she was already asked, we don't tell her whether she guessed or not, until the end of the experiment.

We see that the most rational answer she can give is 1/3. On the other hand, of course she knows that the probability that when the coin was tossed it landed heads is 1/2.

Tuesday, July 29, 2014

A confused sleeping beauty

The Sleeping Beauty problem

 A recent post by Sean Carroll reignited a debate about the "Sleeping beauty problem".
This is a simple problem of probabilities, involving tossing a coin. But for some reason, it seems to be no agreement about its solution.


Consider the following experiment:
- On Sunday, put Sleeping Beauty to sleep.
- Toss a fair coin.
- We are interested to ask Sleeping Beauty the question 
Q. What is your belief now for the proposition that the coin landed heads?
- If the coin comes up heads, wake up Sleeping Beauty on Monday and ask her the question. Then drug her to forget that awakening.
- If the coin comes up tails, wake up Sleeping Beauty both on Monday and Tuesday and ask her the question. Each time drug her to forget that awakening.
- In both cases, don't forget to wake her on Wednesday and end the experiment.

If you have trouble convincing a Beauty to let you put her to sleep and drug her, you can try your luck with people who already have very short memory, like Lucy Whitmore from "50 First Dates", Leonard from "Memento", or Dory from "Finding Nemo". Or you can make the experiment with Dory from "Finding Nemo". 

You can also perform the experiment with Dory from "Finding Nemo".

Those thinking they know the answer are mainly in one of two camps: halfers, who think she should answer 1/2, and thirdirs, who think she should answer 1/3. Thirdirs say that when Beauty is waken and interviewed, she thinks she can be in one of three situations. Since only in one of the cases the coin turned up heads, the answer must be 1/3. Halfers say that this answer is wrong, being probably caused by drug abuse, and since the coin is fair, the answer should be 1/2. There is nothing that can provide new information to Sleeping Beauty, so this answer should remain 1/2.

I will not detail here the debates still ongoing on the net, and the articles which are written about this. I just want to explain why I think that this debate is based on different understandings of the question.

Another experiment

Consider the following experiment.
- Prepare a large box, in which you can put apples and oranges, without seeing its content.
- Toss a fair coin.
- If the coin comes up heads, put one orange in the box.
- If the coin comes up tails, put two apples in the box.
- Repeat this many times.
- At the end, randomly extract a fruit from the box. Unless the experiment took too long, the fruits are not yet rotten, so you can extract a fruit.
- Then answer the following questions:
1. What is the probability that the fruit you will extract was introduced after the coin landed heads?
2. What fraction of the total times the coin was tossed, it landed heads?

The answer to question 1 is of course 1/3, because 1/3 of the fruits are oranges, and oranges were placed in the box when the coin landed heads.

The answer to question 2 is of course 1/2, because the coin is supposed to be fair.

My claim is that thirdirs were actually answering question 1, and halfers were answering question 2.

The question Sleeping Beauty was asked can be seen as equivalent to both question 1 and question 2.

To see how it can be seen as equivalent to question 1, consider a combination of the two experiments. Say that Sleeping Beauty is not only asked the question, but also it is given a fruit to put in the box. If the coin landed heads, she will receive an orange, and if it landed tails, she will receive an apple. She will put them in the box, then she will be put to sleep and forget about the awakening. Say the experiment is ran a large number of times. At the end, she can just count the fruits, and she will find that 1/3 of them are indeed oranges, so she will know that indeed the answer to the question is 1/3. Asking her about her belief that the coin landed heads that time is the same as asking her about her belief that she will receive an orange.

It is true that for every time the coin landed tails, she gets two apples, while every time it landed heads, she gets only one orange. This is why some tend to understand the question as being actually question 2.

Removing the confusion

So the dispute between thirdirs and halfers is due to the fact that they interpreted the question differently, and consequently answered different questions.

Instead of asking Sleeping Beauty the question as originally stated, we could just ask her two questions:
Q1. What is your belief that this awakening occurred following an event in which the coin landed heads?
Q2. What is your belief that when the coin was tossed, it landed heads?


Tuesday, March 25, 2014

Impossibility theorems, a counterexample (the seven bridges problem)

In mathematics and physics there are some results called no-go theorems, or impossibility theorems. To name just a few: Euler's solution to the problem of the seven bridges of Königsberg, Gödel's incompleteness theorems, Bell's theorem, Kochen-Specker theorem, Penrose and Hawking's singularity theorems.

Research is an adventurous activity - you can spend years on researching a dead end, or you can stumble by luck upon something worthy without even knowing (for example, the discovery by Penzias and Wilson of the cosmic microwave background radiation). To avoid spending years looking in the wrong places, researchers use various guiding lines. Impossibility results are some of them, which are by far the most reliable. Other guidelines are following the trends of the moment (also dictated by the need to publish and receive citations), following the opinions of authorities in the field, reading only what they read etc. I personally consider misguided the idea of interpreting the results and filtering what you read and research by using the eyes of the authorities, no matter who they are. But it is understandable that they may seem the best we have, and that anyway the "mainstream" follows them, so if you want to fit in, you have to do the same.

What about the impossibility theorems, aren't they more objective than just fashion trends dictated by authority figures? Of course they are. However, they apply to specific situations, contained in the hypothesis of the theorems. Moreover, they rely on a mathematical model of reality, and not on reality itself. While I think that the physical world is isomorphic to a mathematical model, this doesn't mean that it is isomorphic to the models we use.

I will give just a simple example. Remember the problem of the Seven Bridges of Königsberg. It was solved negatively by Euler in 1735, and led to graph theory and anticipated the idea of topology. The problem is to walk through the city by crossing each bridge once and only once. Here is a map, which is of course an idealization:

Euler reduced the problem to an even more idealized one. He denoted the shores and the islands by vertices, and the bridges by edges, and obtained probably the first graph in the history:


Euler was then able to show immediately that there is no way to walk and cross each of the bridges once and only once (without jumping like Mario or swimming in the river, or being teleported!). The reason is that an even number of edges have to meet at the vertices which are not those where you start or end the trip. But there are no such vertices in the above graph, so all four have to be starting or ending vertices. But at most two vertices can be used to start and end, so the problem has negative answer.

This illustrates the main point of this article. The problem has a negative answer, but this doesn't mean that in reality the answer is negative too. The mathematical model is an idealization, which forgets one thing: that the Pregel river has a spring, a source of origin. If we add the spring to the map, we obtain a different problem:


This problem has a simple solution, which is obtained by "going back to the origin":


This is "thinking outside the box", literally, because you have to go outside the original picture box. I came up with this solution years ago, when I was in school and read about Euler's solution. Of course, it doesn't contradict Euler's theorem, because the resulting graph is different than the one he considered, as we can see below:


Hence, Euler's theorem itself tells us how to solve the problem associated with this graph. The problem is solved by the very theorem which one considers to forbid the existence of a solution.

The main point of this simple example is that even in simple cases we don't actually know the true settings in which we apply the no-go theorems, or we ignore them to idealize the problem. We are applying the no-go theorems in the dark, so perhaps, rather than being guidelines, they are blocking our access to the real solutions of the real problems. While most researchers try to avoid being in contradiction with impossibility theorems, maybe it is good to reopen closed cases from time to time.