Quote: Original post by frob
When you start citing biblical prophecy, and statements like "I know the end from the beginning", you can make a case that the religions propose determinism, but then you have the freedom to make choices and choose right and wrong, so you get the Freedom model.


Existential debates about random numbers can be so much fun. [smile]

In Judeo-Christian religions, the human freedom doesn't surpass God's ability to know/plan our entire existance beforehand. So, at least Catolicism is deterministc. Prophecy and predestination defends that.

But lets NOT turn this thread into a religious debate. I agree with you it's fun tough :)

its me the random guy i was brain storming lol random number is not stupid like i said before i just dont know you you would use random numbers but an AI guard that can run on uncourdinated paths is kinda cool if the AI can gaurd different parts of your kingdom at random and just choose a path or maybe just not go on a path at all that would be a great feature for mmos with gaurds that patrol the areas

Quote: Original post by Sneftel

Quote: Original post by Iftah
in probability saying "all possibilities have equal likelihood" means when the possibilities num is infinite all possibilities are Zero likelihood.

That's not true. One divided by infinity is not zero. (At least, as far as matters for summation.)

I havent learnt advanced probability (which requires measure theory) so maybe I got some things confused, but here is what I think:

First, you are wrong: 1 divided by inifinity is zero.
proof: it is obviously not negative and it must be smaller than any (non zero) positive number. Give me a counter example to prove me wrong.

I over-simplified: yes, I know you can give equal probability to continious numbers (i.e. between zero and one) and ask about the probability of a subset, but I was talking about a discrete infinite set. In the continuous world you need to enter the problem of measure ("size" of infinte sets), but the same problem exists with sets of infinite measure.

If you say I am wrong then please tell me this:
suppose there is a way to make an algorithm output a (real) number with each number having the exact same probability.

what is the probability of output "1" ? *zero* (because it must be less then every epsilon)
what is the probability of output which is an even number? *zero*
what is the probability of output between 0 and 1? *zero*
what is the probability of output which is a positive number? I am not sure about this one... zero? half? I think its zero because you can make a series of growing segments each with zero possibility so the limit is zero.

So even for a continious set of numbers if the set is of inifinte measure then you cant give equal possibility for each or you get zero for each.
(and whatever the algorithm will return it will be an error)

PS. a real computer algorithm will never work on an inifinite set because it has only finite memory (and for example a number with 10^10^10^10^10^10 digits is still a possibility, but it too much to hold in memory even for a computer the size of the universe). But even with inifinte memory I argue that you cant make an algorithm to give all numbers an equal possibility.

Iftah.

Quote: Original post by Iftah

Quote: Original post by Sneftel

Quote: Original post by Iftah
in probability saying "all possibilities have equal likelihood" means when the possibilities num is infinite all possibilities are Zero likelihood.

That's not true. One divided by infinity is not zero. (At least, as far as matters for summation.)

I havent learnt advanced probability (which requires measure theory) so maybe I got some things confused, but here is what I think:

First, you are wrong: 1 divided by inifinity is zero.
proof: it is obviously not negative and it must be smaller than any (non zero) positive number. Give me a counter example to prove me wrong.

Not quite.

The limit of 1/x as x approaches inf is zero.

1/inf is undefined by definition. Some people call it reciprocal infinity, but it is still not defined. Infinity is a concept and not a number, and by definition you cannot divide by it. Infinity itself does not exist, it is just the generic term for an unbounded quantity.

But since you've learnt advanced probability, you should know that already.

If you really need references for that, [google] for "divide by infinity". Check some of the more authoritative sites like mathworld.wolfram.com, or math usenet groups' FAQ sheets

Quote: Original post by frob
But since you've learnt advanced probability, you should know that already.

note I said I haveNT leant advanced prob. (I keep forgeting to put "'" in haven't).

Division by zero is undefined but you can extend the numbers to include infinity and define any number devided by it to be zero (once you include inifinity its the only possibility).
I agree that while inifinity is undefined then (obviously) division by it is undefined, but in some cases and in some places in math inifinity is a valid symbol to work with just as any other number.


edit: I read now about reciprocal infinity, I understand now that once you include infinity you need to include the reciprocal. my bad. I should have known as its a must for things like integral to work. Still I stand by my "theory" (for lack of other word) that you cant give equal probability to all numbers and expect an algorithm to work.

Iftah.

A couple of things that might (I say might) help the conversation! ;)

1) It isn't so much that quantum mechanics is about probabilities. It's simply that quantum wave functions have the properties of density functions with some equivalence to probability density functions. I seem to recall reading something recently showing where they diverge from each other...but for the life of me I can't recall it. 8(

2) By definition, probabilities are not defined for individual points in an infinite set. One can only consider the probability of a point as the limit of the probability of an event falling within a given volume defined as the neighbourhood of a point, as that neighbourhood size goes to zero. I.e.,

Pr(X=x) = lim  Pr(X>x-δ AND X<=x+δ)        δ->0

If I had to give a phenomenological definition of randomness, I'd prefer to go with: An event is random if, presuming all relevant knowledge regarding a domain, a mathematical equation cannot be written down that describes exactly which event would occur given an initial condition for the domain. A sequence is random if one cannot write down a mathematical equation describing exactly which event should follow any other from the domain. Hence we give rise to probabilities as a description of the likelihood of events and sequences.

Cheers,

Timkin

Quote: Original post by Timkin
An event is random if, presuming all relevant knowledge regarding a domain, a mathematical equation cannot be written down that describes exactly which event would occur given an initial condition for the domain.

That would be fair me thinks. That would render randomness and chance to what they are: "presumptions" :)

I too would say that a *pure random number* would be a number whose digits are uniformly distributed and is algorithmically incompressible. You cannot compute this number. Also one must walk careful when they talk about what is possible and what is not. We must never forget that we speak and work within and with constructs we built ourselves.

Quote: Original post by Iftah
edit: I read now about reciprocal infinity, I understand now that once you include infinity you need to include the reciprocal. my bad. I should have known as its a must for things like integral to work. Still I stand by my "theory" (for lack of other word) that you cant give equal probability to all numbers and expect an algorithm to work.

Iftah.

Careful now, you just made a very, very complicated and messy statement whose reality will take years to unravel. You cant just stick infinites and infinitesimals into your number system. and even after having done so, without expecting very large consequences. Look into Nonstandard anaylsis to see what I speak of.

Also, see: Law of Large Numbers

Quote: Original post by frob

Quote: Original post by Iftah

Quote: Original post by Sneftel

Quote: Original post by Iftah
in probability saying "all possibilities have equal likelihood" means when the possibilities num is infinite all possibilities are Zero likelihood.

That's not true. One divided by infinity is not zero. (At least, as far as matters for summation.)

I havent learnt advanced probability (which requires measure theory) so maybe I got some things confused, but here is what I think:

First, you are wrong: 1 divided by inifinity is zero.
proof: it is obviously not negative and it must be smaller than any (non zero) positive number. Give me a counter example to prove me wrong.

Not quite.

The limit of 1/x as x approaches inf is zero.

1/inf is undefined by definition. Some people call it reciprocal infinity, but it is still not defined. Infinity is a concept and not a number, and by definition you cannot divide by it. Infinity itself does not exist, it is just the generic term for an unbounded quantity.

But since you've learnt advanced probability, you should know that already.

If you really need references for that, [google] for "divide by infinity". Check some of the more authoritative sites like mathworld.wolfram.com, or math usenet groups' FAQ sheets

Not quite. :p

Numbers are naught but concepts with agreed upon properties and behaviours. Thus numbers are simply bounded concepts. :) Next you assume that we are all operarting within a system with an archimedean property, what you say is only true within such. Next, within the extended real number line, x/infinity = 0. On the hyperreal number line, x/Infinite = infinitesimal. In my opinion the extended number line is inferior to Non standard analysis. I can assure you that within such, dividing x by an infinite number is perfectly well defined.

[Edited by - Daerax on January 20, 2006 11:44:17 PM]

Anonymous Poster, certainly non contextual hidden variables do not exist in quantum mechanics. Anyone who tells you otherwise is lieing to you.

Quote: Original post by frob
But we haven't even begun to get into the Determinism of the Universe question yet.

[wink]

Even in quantum mechanics where it is a world of probabilities, we don't have the ability to PROVE that the results are random, since it would require states outside the Universe. (See Godel's and other people's work for that.)

Everything you say here is untrue. First it does not make sense to talk of states in terms of the universe (and certainly not states outside of it! Whatever that is). Next Godel's work says nothing on this and finally, we can certainly prove whether there exists randomness within quantum mechanics. At the moment we are just not sure how, although there exist hints. So contrary to what you will often hear, quantum mechanics did not sound the death toll for determinism, the jury is still out as to whether the universe is or is not deterministic.

Quote: Original post by Iftah
PS. a real computer algorithm will never work on an inifinite set because it has only finite memory (and for example a number with 10^10^10^10^10^10 digits is still a possibility, but it too much to hold in memory even for a computer the size of the universe). But even with inifinte memory I argue that you cant make an algorithm to give all numbers an equal possibility.

Iftah.

Your last statement is unclear and is in contrast with the first half of your post. But a computer can certainly work with infinite sets. A simple method to mind is that any enumerable set with a method of enumertation is certainly open for precise working even when being infinite.

You might also be interested in Chaitin-Kolmogorov Randomness and Kolmogorov complexity. Also the below link links to an interesting paper that was unavailable when I last tried to get it. Google has a cache for it though. Hopefully its restored, its quite good.

Quote: Original post by Daerax

Quote: Original post by Iftah
PS. a real computer algorithm will never work on an inifinite set because it has only finite memory (and for example a number with 10^10^10^10^10^10 digits is still a possibility, but it too much to hold in memory even for a computer the size of the universe). But even with inifinte memory I argue that you cant make an algorithm to give all numbers an equal possibility.

Iftah.

Your last statement is unclear and is in contrast with the first half of your post. But a computer can certainly work with infinite sets. A simple method to mind is that any enumerable set with a method of enumertation is certainly open for precise working even when being infinite.

I haven't read (yet) the links you posted, they seem interesting and I will read them later.
I just wanted to clarify my previous statement,
a real world computer (with finite memory) will never be able to work on infinite sets because it has only finite number of states (2^N where N is the number of memory bits). The number of states while being very large is still finite so you cannot realy talk about holding a "real number" in memory, only a finite subset of them.
Thus, you can maybe make an algorithm to give an equal probability to any real number the computer can show (since there are only a finite number of them).
Note - the enumeration only helps to a theoretic machine (from computation theory) with infinite memory.
A real world computer can't work with infinite sets even if they are enumerable - for example the natrual numbers, one cannot make a real world computer program that gives equal probability to the numbers between 1 and 10^10^10^10^10^10 simply because there arent enough atoms in the universe to build enough memory to represent these numbers. I mean, you could maybe write the program but no one will be able to run it.
In short -
There is no point talking about making a real world computer algorithm that gives equal probability to any number, simply because the (real world) computer cant even distinguish between any number.

But, what about a theoretic computer with infinite memory?
I am saying that one cannot give an equal probability to each number in the true (infinite) real numbers even if the memory constraint is removed.

Why? simply because that would result in absolute zero probability for each and every result.


Iftah.