Friday, June 25, 2010

Infinite processes in the real world

A long time ago, in ancient Greece, one of the philosophers asked a simple yet very important question - is matter infinitely divisible? He of course formulated the question in a much more intuitive way: what will happen if you take a stick and break it in half, than take one of the halves and break it in half again and so on. Thinking about this problem, he concluded that at some point we will not be able to continue breaking the stick. According to him, after a finite amount of time we will reach an indivisible component of matter. He named this indivisible component "atom".
As with any new idea, there were those who believed in it and those who concluded that this idea is wrong. Likely for both sides, there were no means to actually check it so they could argue as much as they wanted.

Even though we are much more advanced today we still don't know the answer to this problem. Ironically we have discovered particles which we named atoms only to find out that they can be split apart as well only a few years later. Although, to be really precise, we need to remember that the problem can be formulated as the "atom" being the basic component of a specific type of mater. In other words, one possible understanding of the problem is that it asks to find a "part" that if divided further looses the recognizable properties of the object we started with. If we formulate the problem in this way, then there are indeed such "atoms" - molecules.

At this point you are probably wondering what is this about and how is it connected to infinity. To understand this lets look on a somewhat famous paradox - the Thomson lamp. Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose a being able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of all these progressively smaller times is exactly two minutes.
So, in the end, is the lamp on or off?

It turns out that there is no clear answer to this problem. While we know the state of the lamp at any time during the process, we cannot tell what is the state at the end. Now lets return to our original problem. Lets suppose for a second that "atoms" don't exist. With this in mind we can take the being from the lamp paradox and instead of it toggling the switch we will make it break sticks in half. Since there are no atoms, the process doesn't end before two minutes pass. But what do we have after two minutes?

In this case it is rather simple to look on the problem mathematically. Lets substitute the stick for the line [0,1]. The whole process can be described then as just a limit of [0,2^(-n)] when n goes to infinity. The limit is a single point, so that would mean that we will get a "particle" with size and mass equaling zero. However, that would suggest that the matter is build from particles with zero mass, and this is a rather bizarre conclusion.
The only possible result we can get from this line of thought is that if such a being actually exists then there are "atoms". However, if there is no such being then we cannot say anything.

While I would like to finish this post with at least a partial solution to the problems I presented, there is no solution as far as I know. There is, however, a funny "solution" to the Thomson lamp paradox. Lets assign numbers to the states of the lamp - 1 and 0. If we do this then the state of the lamp after n steps is: 1-1+1-1+...+(-1)^n.
Therefore, if we take the limit when n goes to infinity, we will get the state of the lamp after two minutes. So lets see what the limit is.


As you can see, after two minutes the lamp is half on. :)

Thursday, June 17, 2010

End of the Semester

Today I went to the last lecture of this semester. As it is somewhat typical with last lectures, the professor talked about interesting problems that are somewhat above the scope of the course. If only those problems didn't tend to be more complex that what can be explained in a 45 minute lecture... Luckily, this is of little importance. While the problems discussed were interesting, I rather spend my time working on staff that is more relevant to me now. With the semester finally over, I now have tests to worry about, but I should also have plenty of time to write new posts. Actually, I have a few posts already in the making, I just need some time to actually finish writing them. With the semester over, I finally have time to do so.

To be honest, this year was for some reason really difficult for me. I never was good with making timetables for myself, so I ended studying till I was too mentally tired to do anything else. While I am pretty sure that I managed to do well in all of my courses, I barely kept up with my activity on the net. Both this blog and my stumble upon blog were not active most of the year. Hopefully next year will go in a more normal fashion.

In other news, I am considering to close my Windows Live account. It is not very useful to me, and I noticed that I am getting a lot of spam from it. Initially I opened it in order to have access to free online storage for my files. However, I cannot say that I am satisfied with the service, and therefore I will likely close this account. To be honest, I sometime think about closing my Facebook account as well, but it is slightly better than Windows live. And what is more important is that I can login to other sites using my Facebook account.

I have also started a little project. About two weeks ago, I got an invite to Dropbox. Basically, it is a file sharing site, but it has two features that make it nearly perfect for my uses. Firstly, Dropbox integrates into the desktop. That is instead of having to upload your files to the site manually, all you need to do is to put the files in a specific folder on your computer and Dropbox will upload them to the web and then sync them with your other computers.
Secondly, and much more importantly for me, Dropbox officially supports Linux and works well on it. I tried to find other similar services, but they all either don't support linux or they worked horrible. This even includes Ubuntu One (at least the version I tried about half a year ago).
The only downside it has is that they only give 2GB to free users. However, it is possible to get more space by inviting others - go to the site if you are interested in details, I am pretty sure that this policy will not last for a long time so there is no point to write much about it.

Right now I am using Dropbox to backup and share some video files (documentaries about dinosaurs and other scientific topics) that I have collected. I never cared much about video quality, so I encode the video files in low quality (all the important details are still there) add subtitles and then upload them. In one case, I managed to compress 3 hours into 300MB. As long as I watch them on the computer display, it is perfectly fine.