Wednesday, December 12, 2007

About determinism and tossing coins /2.

It is not really true that there is non-determinism (meaning "not a-priori determinable") about tossing a coin. On the contrary.
When i toss a coin i make so that from a position that it had with respect to the observers, it moves to another position. In order to do that i toss it in a certain way, expecting that it would fall about in a certain place, on one side or the other. In the previous post i gave some indication of things that can influence the motion, and so the result, bothering also mr. Coriolis. They were only examples, perhaps not even well hit, because i do not know if those causes really influence the motion, and even less, if they do, how much. But it is clear that the motion of the coin can be completely described by the external influences minutely formalizable. One can also ask if there is really any need to formalize them, instead of accepting the uncertainty of the result, but it is obvious that they are.

The reason we are sure they are formalizable can be found in Gödel's theorem about incompleteness of Arithmetics. In few words that theorem demostrates that a formal system (as Arithmetics is) can never be considered complete, which means that there is atleast one problem that does not fall in the formalization of that system, and so it is not solvable by mean of the rules of that system. Gödel's theorem so, says, between the lines, also other two things:
1) whatever can be expressed through that formal system, it is also formalizable, or it is possible to completely describe it with all the cause-effect relationships that insert it in a context.
2) that the formalization of a probelm outside the formal system presupposes then a formalization, which is the creation of a new formal system in which that new problem could be demonstrated.
In other words the Gödel theorem tells us that, observing the toss of a coin, i can ask myself "why does it fall on that side?", knowing that, even if we don't know the alswer of that question, that answer exists. In short, that question presupposes a search of a formalization of the events that influence the motion of the coin so that it falls on that side, but this search itself presupposes the hypotesis that the formalization exists.
In other words, answering to the question "why does it fall on that side?" with "there is no reason", not only we make the chain of cause-effect relationships vain, but also we make the question itself vain. Considering possible to pose oneself that question demanding that it makes sense, means trusting that, despite how difficult it could be, the answer exists. Who asks that question already knows (or believes) that the answer is deterministic.
So, it is not the chance that makes head or tail come out, in the sense that it is not a lack of rules that brings the result, unpredictable because given by a lack of rules. We can instead say that there are rules and they are well applied, but we do not have the technical capability to know them or, more simply, we don't feel like to investigate for them.

Then there is the problem that the observation can influence the production of the result.
Let's suppose that Werner Karl Heisenberg and i are in a closed room, sitting next to a table. Let's suppose that he has a coin in his hand and he tender to bet 100 euros on the result of the toss he's going to do. Let's suppose i bet on the side "head". The sound waves caused by my voice that pronounced that word generate a distribution of the molecula of air in the room that influences the motion of the coin so that it falls with result tail. If i was able to sharply compute this cause-effect relationship (or, in other words, to predict the behavior of all the variables involved, including the deterministic position of all the molecula of air in the room when i finish to pronounce the word "head", and so to predict that the result of the toss is tail), being motivated to win the bet, i wouldn't say "head", i would say "tail". But, my voice, pronouncing this word, generates a different distribution of the air molecula in the room, and such a distribution influence  the coin in a different way, and the result is head. Being able to predict also this cause-effect relationship, the only chance to me is not to bet at all.
Obviously, this example is not very realistic. I am not able to find out if really the pronunciation of the word "tail" insted of "head" could influence on the motion of the coin enough to determine the result of the toss, because actually the study on this experiment would be too complex also to somebody much more clever than me. Moreover, and above all, the problem is that the experiment is not repeatable. If i pronounce  the word "head" and the result is tail i could not propose Werner Karl to repeat the experiment with the same conditions, except my engagement to pronounce the word "tail". Because with all our good will, we wouldn't be able to recreate the same conditions. The coin would have been already tossed once. It would have fallen on the table, and as much the coin would have been clean there would have been some substances stuck on its surface which in part now moved on the table and on the Werner Karl's fingers, the air in the room moved and any molecula stopped in a different position. And even if we would  also be able to put back all those elements in their original conditions, the neurons contained in my and my friend's brains theirselves would have different connections. If we were able to put back also the neurons to their original place, the result would be that we loose the memory of the previous experiment and the conclusions that we found, so we wouldn't remember about my engagement to say "tail".
And if we couldn't re-create exactly all the same conditions of the previous experiment, perhaps, pronouncing the word "tail" as established, finally the result would be tail. This would make me win the bet, but with the result to distort the experiment.

Actually it is unthinkable to consider everything (and with everything i mean really everything: what is included in the universe and the laws that control its behavior), because it is too complex. And so, it is better to rely to easier considerations accepting the fact that one cannot give a reliable answer to the toss-a-coin problem, but atleast it can be said that there is 50% of probability to win.
It is unthinkable to consider everything also for another reason, which is that the fact to think everything itself influence its behavior.
Let's put it in this way: let's suppose that one day human beings, even if not clever enough to consider the whole universe atom by atom and to determine so the behavior of any event, they are clever enough to build a super-super-computer with a super-super-software able to do it. This machine would be able of any forecast.
If i asked to that machine the result of the toss that Werner Karl is going to do, the machine would be able to tell me the answer. Even better, i wouldn't even need to ask it, because the machine would be able to forecast also my question. So, sitting in front of the terminal i would read the word "head" that is the answer to the question that i didn't ask but that the machine knows i would have. Knowing before the answer to my question, Werner Karl and i would know the result of the toss, which would make useless to really toss it. But if we don't toss the coin, asking the result of the toss would make no sense, therefore i wouldn't ask the machine the answer, and so the machine, forecasting it, wouldn't have any reason to give an answer. But if it happens in this way, Werner Karl and i would have an uncertainty about the result, we would decide then to procede with the toss, and we would like to know the result before tossing it, and so we would go back to the idea to ask the machine, and this is a vicious circle difficult to exit. Cool eh?

A machine like that would generate another logical problem. A program that consider a so huge amount of data would need the support of a huge hardware. How much hardware? Well... it's difficult to give an answer to this question because, let's suppose to fix at a certain point an amount that represent the number X of atoms to be considered. And let's suppose that that amount of atoms would be completely described by an amount of hardware made by a certain amount Y of atoms. Since the machine belongs to the universe too, and its behavior influence the behavior of the X atoms under analysis, at the end, to consider everything, the machine would have to describe also the behavior of the elements that constitute the machine itself, or, in other words, it has to work on an amount of data of X+Y atoms, and not X. But the hardware of this new model of machine cannot be made by a number Y of atoms, it must be made by a number a little bigger: let's say Y'. Which means that the complexive number of the atoms domain of the machine is not X+Y, but X+Y', which means that the hardware of the machine must be made by a number a little bigger of atoms, let's say Y''... In conclusion we should deduce that a machine capable to describe the whole universe must be atleast big as much as the universe is and so, its task restricts just to describe itself.
Actually we must say that a machine like that already exists. It is the universe itself, which is able to work and give the right results as effects of causes upon exact rules that work on exact datas. Infact we don't need a computer that tells us the result of the toss of a coin. It's enough that we toss it and look the result. If the result is head it means that the very complex calculation of that result generates the anwer "head", if it is tail, the answer "tail". And we are sure that the rules for which the result of the calculation would have been generated in that way have been perfectly respected, however complex they would be.


rowena said...

In your discussion here, that is why I felt that I had a chance of winning the red Mini D in the contest. Even if the odds might say differently, there is always the chance of the result to happen in my favor.

As for the super super computer. Uhm, that's almost like hypothesizing on a god-like creature. That is, if you believe in the idea of a god as an all-knowing entity. Should a machine ever come into existence, the word 'guess' would become obsolete!

tychecat said...

Hmmm.. Fun and games with Chaos Theory.
There are actually three possibilities when you toss that coin -it may land heads, it may land tails, or it may land on its edge (in which case nobody wins) but there are certain givens: it MUST do one of these three things but under ideal conditions, you cannot predict which for any SINGLE toss.
If it lands heads ten times in a row, I suggest you bet on heads for the next toss- conditions are not ideal.
Thanks for your comment at Socrates_cafe - please check us out more often and make your usual splendid contributions - we haven't had nearly the arguments since you stopped commenting
BTW As you probably know my comment about wearing clean sox at the airport was a not-so-subtle dig at the ridiculous idea that passengers might conceal bombs in their shoes or between their toes or some place- hence everyone de-shoes before boarding.