Mathematics & Theology

[LegionOnomaMoi]

You can use probability to look at the likelihood of a God - Dawkins did that in the God delusion.

How? What's the probability space?

 

[Reptillian]

How? What's the probability space?

Whatever it is, I'd imagine that its countably infinite... You'd need to assign a probability to every possible definition of god and then assign a probability to the compliment (the non existence of god) and show that its greater than the other elements in the space. Dawkins is a biologist though, so I'd imagine his arguement is more qualitative.

 

[LegionOnomaMoi]

Whatever it is, I'd imagine that its countably infinite... You'd need to assign a probability to every possible definition of god and then assign a probability to the compliment (the non existence of god) and show that its greater than the other elements in the space. Dawkins is a biologist though, so I'd imagine is arguement is more qualitative.

If his argument is qualitative, then how can it be based on probability? And if it's countably infinite, than the probability of any one thing happening is zero. And how do you assign probability values? Or rather, how did Dawkins?

 

[Reptillian]

If his argument is qualitative, then how can it be based on probability? And if it's countably infinite, than the probability of any one thing happening is zero. And how do you assign probability values? Or rather, how did Dawkins?

I'm guessing his arguement was more of a "the Bible predicts God does this and that and the other thing, those things aren't possible and we don't see evidence of them in nature, therefore God doesn't exist" kind of thing. How can they all be zero if the sum of all probabilities must equal one? There must be a countable infinity of zero values to counteract the other infinities and some events must have a non-zero probability. I don't know how one would show that the non existence slot has the largest probability though. I guess thats where the counter arguement to Pascal's wager comes in...since we can't decide which god is best, none of them must be. They must be equiprobable. Since there is an uncountable infinity of zeros already, the remaining non-zero gods must also have probability of zero and thus the non-existence of god must have a probability of 1. Therefore, god does not exist. The crux is whether you think all gods are equiprobable.

 

[LegionOnomaMoi]

How can they all be zero if the sum of all probabilities must equal one?

If there are an infinite number of possibilities, than the possibility of any one of these is 1 over infinity, or zero. In fact, given a normal distribution of values (a bell-curve), the possiblity of any one score is 0. It's a continuous distribution, which means there are an infinite number of points between 0 and 1, so any single score has a probability of 1/infinity, which again is 0.

 

[Reptillian]

If there are an infinite number of possibilities, than the possibility of any one of these is 1 over infinity, or zero. In fact, given a normal distribution of values (a bell-curve), the possiblity of any one score is 0. It's a continuous distribution, which means there are an infinite number of points between 0 and 1, so any single score has a probability of 1/infinity, which again is 0.

Only if they're all equiprobable is the the probability of any one of n possibilites 1/n. Besides, the limit as n approaches infinity of a constant (zero) is that constant (zero)...so when adding up probabilities you get 0+0+0+0...=0 which is decidedly not equal to one. One of the axioms of probability theory is that the sum of probabilities=1

 

[LegionOnomaMoi]

Only if they're all equiprobable is the the probability of any one of n possibilites 1/n.

Yes, but I didn't say 1/n. The gaussian curve is a continuous function. The probability range constitutes an infinite number of values between 0 and 1. The chance of selecting any single value is 0. That's true of any continuous function because the probability space is infinite. Given a continuous function with a distribution of X scores, the probability of selecting a single x score is 0.

One of the axioms of probability theory is that the sum of probabilities=1

Yes, because distributions are "scaled" so that will be true. However, 0 and 1 are the limits of the function, and the individual scores range over the entire interval of infinite points. Hence, an infinite probability space.

 

[Reptillian]

Yes, but I didn't say 1/n. The gaussian curve is a continuous function. The probability range constitutes an infinite number of values between 0 and 1. The chance of selecting any single value is 0. That's true of any continuous function because the probability space is infinite. Given a continuous function with a distribution of X scores, the probability of selecting a single x score is 0.


Yes, because distributions are "scaled" so that will be true. However, 0 and 1 are the limits of the function, and the individual scores range over the entire interval of infinite points. Hence, an infinite probability space.

The difference is that the Gaussian is an uncountable infinity.

 

[LegionOnomaMoi]

The difference is that the Gaussian is an uncountable infinity.

It doesn't actually matter if the sample space is a denumerable infinity or not. 1/infinity is still zero.

 

[PolyHedral]

It doesn't actually matter if the sample space is a denumerable infinity or not. 1/infinity is still zero.

The inverse of a value that increases without bound is an infinitesimal value. Depending on precisely what number system you're using, this might not equal zero.

 

[Reptillian]

The inverse of a value that increases without bound is an infinitesimal value. Depending on precisely what number system you're using, this might not equal zero.

That and the axioms of probability theory ensure that the sum of the probabilities must be equal to one. So the pigeon hole principle guarantees that at least one value must have a value of 1 or less. Right?

Do you see any problem with the probabilistic arguement above that I gave earlier for the non existence of god? It seems to be convincing to most non believers. The "pigeonhole" they seem to choose is the value corresponding to the non existence of god.

Personally I think some gods are more probable than others, thus the values aren't equiprobable. So I think there is likely a distribution of probabilities between various deities (or the compliment) and it likely isn't possible to use probability theory to make a definative decision.

 

[LegionOnomaMoi]

The inverse of a value that increases without bound is an infinitesimal value. Depending on precisely what number system you're using, this might not equal zero.

In probability, it is: "Given a value for the parameters of the distributions, mu, and signma^2, the curve shows the rlative probabilities for every value of x. In this case, x can range over the entire real line, from -infinity to +infinity. Technically, because an infinite number of values exist between any two other values of x (ironically, making p(x=X)=0, for all X), the value returned by the function f(x) does not reveal the proability of x, unlike the Poisson distribution above (as well as any other discrete distributions). Rather, when using continuous pdfs, one must consider the probability for regions under the curve....The probability that x equals any number q is 0 (the area of a line is 0)."
from Lynch, S. (2010. Introduction to Applied Bayesian Statistics and Estimation for Social Scientists. Springer: NY.

 

[LegionOnomaMoi]

That and the axioms of probability theory ensure that the sum of the probabilities must be equal to one. So the pigeon hole principle guarantees that at least one value must have a value of 1 or less. Right?

See my post immediately above. Probability functions which are continuous cannot be defined at a single value (or rather, they can be but that value is always zero).

 

[Reptillian]

See my post immediately above. Probability functions which are continuous cannot be defined at a single value (or rather, they can be but that value is always zero).

Aye, but my suggestion was that the deity distribution would be countably infinite, not continuous. So intstead you have a set with an infinity of terms which can be labelled. {god 1, god 2, god 3,....god n} so the sum of probabilities {p(god 1), p(god 2),....p(god n)} and the compliment p(no god) must be equal to 1 right? So at least one element in the set or the compliment must have non zero probability. If n were infinite and all the gods were equiprobable, then it stands to reason that each element in the set has p(x)=0 so the compliment p(no god)=1 right?

For the continous distributions, the area under the curve must be one so there would be a range of values corresponding to the non existence of god...possibly? In example, gods of type 1.75 through 5.6 on the number line don't exist? IDK, just trying to adapt the finite model to the infinite case.

 

[no-body]

Well, God is utterly simple and not made of parts so to talk about God in regards to properties (assuming you mean matter here) is a mistake IMO.

At the very least you could say Plato was onto something with his theory of forms.

Behold: the dangers of thinking God is a book, building, dogma, etc. One thinks God is utterly simple and explainable because to say otherwise is to say that ones belief might be debatable.

As if the ultimate being in the universe could be understood or analyzed in any way.

 

[LegionOnomaMoi]

Aye, but my suggestion was that the deity distribution would be countably infinite, not continuous. So intstead you have a set with an infinity of terms which can be labelled. {god 1, god 2, god 3,....god n} so the sum of probabilities {p(god 1), p(god 2),....p(god n)} and the compliment p(no god) must be equal to 1 right? So at least one element in the set or the compliment must have non zero probability. If n were infinite and all the gods were equiprobable, then it stands to reason that each element in the set has p(x)=0 so the compliment p(no god)=1 right?

I don't see how. If you combine and denumerable infinite set with any other denumerable set you still get a denumerable probability space. And the probability of selecting any one thing from the union of those sets is still 1/infinity or zero.


In any case, I don't see how any of this can say anything about whether god (whatever god might be) exists.

 

[Reptillian]

I don't see how. If you combine and denumerable infinite set with any other denumerable set you still get a denumerable probability space. And the probability of selecting any one thing from the union of those sets is still 1/infinity or zero.


In any case, I don't see how any of this can say anything about whether god (whatever god might be) exists.

That contradicts the axioms though...at least one element even of the countably infinite set, must be greater than zero in order for the probabilities to add up to one. Pigeonhole principle. N pigeons fly into k holes, must be at least N/k rounded up pigeons in some hole. Again that assumes each god is equiprobable. Imagine an infinite collection of probabilities {a,b,c,....} and that the sum of the terms must be one. What I'm saying is that requirement guarantees that some terms are one or less.

 

[LegionOnomaMoi]

That contradicts the axioms though...at least one element even of the countably infinite set, must be greater than zero in order for the probabilities to add up to one. Pigeonhole principle. N pigeons fly into k holes, must be at least N/k rounded up pigeons in some hole. Again that assumes each god is equiprobable. Imagine an infinite collection of probabilities {a,b,c,....} and that the sum of the terms must be one. What I'm saying is that requirement guarantees that some terms are one or less.

But a denumerable infinite set can still have arbitrarily small values and sum to 1.

Let's assume (as you say) each god is equiprobable. However, the set of gods is a denumerable infinite set. So, as this set must sum to 1, what is the probability of selecting one god? You are basically talking about a dirac function. Every slice of the probability space is arbitrarily close to zero, with a limit at zero, but the series integrates to 1.

Otherwise, how could the set be infinite and equiprobable?

 

[PolyHedral]

That contradicts the axioms though...at least one element even of the countably infinite set, must be greater than zero in order for the probabilities to add up to one. Pigeonhole principle. N pigeons fly into k holes, must be at least N/k rounded up pigeons in some hole. Again that assumes each god is equiprobable. Imagine an infinite collection of probabilities {a,b,c,....} and that the sum of the terms must be one. What I'm saying is that requirement guarantees that some terms are one or less.

In extended real number systems, there are values smaller than all real numbers, let nonetheless larger than zero.

 

[Reptillian]

But a denumerable infinite set can still have arbitrarily small values and sum to 1.

Let's assume (as you say) each god is equiprobable. However, the set of gods is a denumerable infinite set. So, as this set must sum to 1, what is the probability of selecting one god? You are basically talking about a dirac function. Every slice of the probability space is arbitrarily close to zero, with a limit at zero, but the series integrates to 1.

Otherwise, how could the set be infinite and equiprobable?

I'm losing sight of the forest through the trees here, I suppose thats one of the dangers of having a conversation stretched out over multiple days...

To reiterate (mainly to remind myself) the arguement I put forth was sort of a mathematical version of the counter arguement to Pascal's wager: Suppose there is a countably infinite set composed of elements {p(no god),p(god 1),p(god 2),p(god 3),...,p(god n)....} where p(x) is the probability of god x. We know that the sum of the elements in the set must equal one via the axioms of probability theory. We further assume equiprobability of gods since there is no apparent criteria for distinguishing between them. Since there is an infinity of gods and we're assuming equiprobability, the probability of any one god p(god 1), p(god 2),...etc. is the limit as n goes to infinity of 1/n. Since the probabilities must sum to 1, p(no god) must equal one.

So if I understand what you and Polyhedral are saying, its that all the terms (even the no god term) could be infinitessimal, yet still sum to 1? My concern is whether a countably infinite series of infinitessimal terms is the same thing as an integral over an uncountable infinity. (such as a segment of the real number line) I mean, theres no such thing as a p(god pi) or a p(god square root of 2). The Dirac funtion might be somewhat applicable. I'd suppose its like having a spike at each positive integer though with a value between 0 and 1.

In extended real number systems, there are values smaller than all real numbers, let nonetheless larger than zero.

Gaaah! Real analysis! Unfortunately, I've never had a course in that. I've had complex analysis though. The Peano axioms are about all I remember concerning number theory. So in extended number systems there's a number between 0 and 0.00000....?