tisdag 17 maj 2011

Free Will and Finite Precision Computation 3

This is a continuation of Free Will 2:

So can we make it rain tomorrow by leaving the car window open? Can the flap of a butterfly in Brazil set up a tornado in Texas?

How can we tell? Well we have already answered this question: Take away the butterfly and observe tornados anyway, close the window and observe rain anyway. Or let the butterfly flap and observe no tornados, open the window and observe no rain.

Evidently we are talking about a big effect (tornado, rain) from a small cause (butterfly, car window), which is only possible if the system under consideration is unstable. Why? Because the definition of an unstable system is that a small cause can have a big effect.

If the effect of any small causes is small, then the system is stable. Most of the systems we can observe are (more or less) stable, because unstable systems tend to break down or explode into non-existence.

Is the weather unstable? Well, we say that the weather is unstable when changes are unpredictable, and we know that this is often the case. How unstable can then the weather be?
Can it be so unstable that the flap of butterfly can cause a tornado? Probably not.

We expect that sufficiently small perturbations cannot change the major features of the weather and cause a tornado. This means that it is irrelevant whether a butterfly flaps or not, or if we leave the car windows open or not, as concerns tornados and rain.

If we accept that small causes do not change major features, that is, that we are dealing with a (more or less) stable system as a typical system which we may be confronted with, then we could say that we could leave certain little things to be determined by chance, by throwing a dice:
• It would not change anything essential.
• It would save us time for essentials by avoiding getting drowned into pedantry.
• In fact, it would be necessary to not get bogged down by details.
• In other words, we would have to act with finite precision in order to not get stuck on the spot at a specific point in time.
• Time is advancing and so we have to advance as well and thus we have to take decisions with finite precision only, because we have no time to do everything with infinite precision.
We are now approaching the question of free will. Can we do anything we could of think of doing? No, our abilities are limited, but within these limits we would say that we have some form of free will.

We could decide what to study at the university, with whom to engage, how to dress, what to eat, what to say, but all these decisions could fit into some form of master plan for our life, which we probably should search for if we don't have any. Our free will would not be entirely free but subordinate to a master plan, which we may have chosen by free will or inherited from our parents, spouse or friends or society. So maybe our free will as concerns big things is not that free, as if the main pattern of our life largely is predetermined.

We could still argue that we have a free will to decide little things, what movie to see, what to have for dinner et cet, but we could also say that we will only spend limited time on these issues to find the "optimal solution". We could even use a dice to decide if we cannot easily make up our mind or come to some agreement with somebody.

But you would not like Luke Rhinehart decide big things, such as getting divorced or not, by throwing the dice, because that would quickly ruin your life.

In short, you would act with finite precision and feel that you have some form of free will in particular for little things, possibly exercised using dice, while you may feel that the main path of your life (or at least other people's lives) is more or less predetermined.

This corresponds to something between full determinism (no dice) and full indeterminism (all dice) as a for of finite precision determinism (dice only for little things). In other words, a free will which is not completely free, but not completely unfree either:
• a finite precision free will.
PS Suppose Tom wants to show Harry that he has a free will. Consider the following conversation:

Tom: Look, I can decide to lift my left arm or my right arm according to my own free will.

Harry: How you decide to lift the left or the right? Do have some predetermined preference?

Tom: Of course not, then it would not be free will.

Harry: OK, but if you are completely neutral, how are going to decide?

Tom: Let me think...should I lift the right arm...or should I lift the left...what could be a good reason to lift the right arm...instead of the left...well, I cannot really decide...I need more time...
but even so I don't how to choose while staying fully neutral...

Harry: Can I offer some help? What about flipping a coin?

Tom: Flipping a coin? Yes, that must be the only possibility which is completely neutral, without any perdetermined predjudice for right and left. That's what I will do to not get held up by this silly test...

4 kommentarer:

1. I don't get this finite precision arithmetic business.

As best I can make out, you seem to be saying that to get the right answers, we have to use something like 16 bit floating point mathematics. And that will result (I think) in lumpy numbers as they're rounded up or down to whatever precision we're using.

Is the universe like that? When the Sun calculates how much the Earth has to move in its orbit, does it use finite precision math? Is there an inherent 'lumpiness' in nature that results in it looking like energy comes in packets (quanta)?

It makes sense that us humans can't calculate the numbers to infinite precision, because we don't have the time to do so. We have problems even with numbers like Pi, which can and does fill whole books. But does it make sense that the universe isn't any better than us humans, and sort of just gives up and says, 'Well, it's somewhere around there, plus or minus yet another decimal point.

I suspect I've gotten hold of completely the wrong end of the stick here. Maybe you can point me at a bit of finite precision arithmetic which shows exactly why it's so much better than infinite precision arithmetic (apart from not taking quite as long to work out).

2. I view physics as a form of analog computation, ultimately based on quantum mechanics which by e g Heisenberg's uncertainty relation, has finite precision.

Finite precision is real, infinite precision is not.

3. I still don't get it.

I can understand that some ways of calculating things are more precise than others. I can understand also that infinite precision is unattainable by man, and is in that sense 'unreal', and that finite precision is something that can be done, and is therefore 'real'.

Suppose that I wish to find what half of some number is, but my mathematical precision is restricted to integers, because I can only count on my fingers. So if I want to find what half of 27 is, I come up with 13. In fact I'm pretty sure that it's bigger than 13, but less than 14, but I have no way of expressing this. Is this the point where I must roll a dice, and decide the matter that way? Rolling the dice, sometimes I'll find that the answer is 13, and sometimes it will be 14. And so all numbers are made up of things we are sure about (e.g. 13) and things we are not sure about (e.g. maybe it's 14). We can use our finite precision mathematics (i.e. our fingers) to work out the important bit (13), but we have to throw dice to find the less important bit that is beyond the limits of our precision.

Am I on the right track? Or am I hopelessly lost?

4. Yes, you are on theright track! You can go on, if you want, or decide to quit, by free will. What do you choose?