Thursday, 18 October 2012

Why God is mathematically improbable

Let's start with a simple mind experiment. If you're an atheist, ask yourself now how many times you've heard a theist say "prove to me that there's no god". If you're a theist, ask yourself how many times you've asked an atheist this very question.

An atheist will often say "there's no evidence that there's no god and therefore I think He probably doesn't exist" or words to that effect. A theist, on the other hand, is likely to say "absence of evidence isn't evidence of absence" or something along those lines.

I intend to show here (using simple mathematics) that god is indeed very improbable. Atheists will be probably disappointed with the figures we arrive at at the end. Don't be. Keep in mind at all times that this argument is designed to show that, even if we extend deliberate generousity to theist assumptions, the existence of a god is nevertheless statistically improbable.

Let us begin.

Some terms:

By way of introduction, let's make it clear that we will here be discussing the truth of propositions. The central question is what weight can be given to a proposition that is not supported by any evidence and how this impacts on the probability of God's existence.

In our discussion, a proposition is any claim that relates to a question of fact (as opposed to a normative value - eg we shouldn't kill - or preference). Thus, an example of a proposition is "The Sun is green". Another example of a proposition would be "The Sun is not green".

The second term I need to introduce is "Unevidenced Proposition" (UP). An unevidenced proposition is simply a proposition for which there is no evidence and against which there is no evidence. We could think of it as a hypothesis. Note that I am deliberately steering away from terms that have some conventional scientific usage. We are not here concerned with science (science is all about data and repeatable experimentation; things that don't apply to God for our purposes).


1. Equalizing the scale

Let's take an Unevidenced Proposition (UP) and consider the following three statements about that proposition. Note that I am deliberately speaking in general terms here without actually proposing a specific UP. We are trying to agree on rules regarding UP's generally. So, we are speaking of just any hypothetical Unevidence Proposition.

a) UP is more likely than not to be true;

b) UP is more likely than not to be untrue;

c) UP is equally likely to be true as it is to be untrue.

Remember that there is no evidence in favour of UP and there's no evidence against it. In these circumstances, which of the above three statements is true? I am going to ask us to agree that only (c ) is true. Some atheists (ironically) might object at this point. Don't worry, trust your DJ. And remember, we are going out of our way to be generous.

There is, however, a problem with the above. Take two Unevidenced Propositions: Rabbits exist on Pluto and Green Rabbits exist on Pluto. Clearly, we can't say that the probability of both propositions being true is equal. The Green Rabbits proposition contains an additional requirement. The odds (very small, I'd think) that there are ANY rabbits on Pluto must be greater than the odds that some of those rabbits are green. There might indeed be some weirdly adapted rabbits on Pluto and none of them green.

I propose to introduced a sub-class of Unevidenced Proposition. In fact, two subclasses: Simple Unevidenced Proposition and Compound Unevidenced Proposition.

Thus, A Simple Unevidenced Proposition (SUP) is a proposition for which and against which there is no known evidence and which doesn't depend on (consist of) any other unevidenced propositions.

A Compound Unevidenced Proposition (CUP), in turn, is a proposition for which and against which no evidence exists and which contains in itself another unevidenced proposition.

In this way, "There are rabbits on Pluto" would be a Simple Unevidenced Proposition (SUP)and "There are green rabbits on Pluto" would be a Compound Unevidenced Proposition (CUP).

So, what can we conclude about a, b and c above? For our purposes, it will suffice to restrict it to SUP's. Thus, a simple unevidenced proposition is equally likely as unlikely to be true. Once again, I am fully aware that the scientist in you may not be liking this suggestion. But again, I ask you to be patient and remember that we are being generous here. We should also remember that we are going to discuss the existence of God; arguably not something that falls within science in any event. It's not observable. God by definition resides outside our realm (for our purposes He will, anyway) and we can't really say that our observations of things inside the Universe allow us to extrapolate conclusions about things outside the Universe. This is part of the reason for my generosity here.


2. Let's do some maths

Since we are going to make probabilistic calculations, we need to put the above into some maths. We have agreed that the odds that a SUP is true are equal to the odds that it's not true. This means that we can comfortably assign the probability of 50% to every SUP. Thus P(SUP)=50% or 0.5.

3. Let's define God

There are many ways to define God. As a simple minimum, let's go with "non-material, timeless, omnipotent, omniscient, sentient, uncaused entity that created the Universe".

4. God is a Compound Unevidenced Proposition

There are a number of disctinct attributes of God. Each of those attributes is (in principle) possible to be possessed by a non-God being. It's only when we put all of them together that we have a true god. For instance, there may be lots of non-material or omniscient entities. Each of those aspects of God is in fact a Simple Unevidenced Proposition in itself. There's no evidence that a non-material being exists. There's no evidence that it doesn't either. There may be many or there may be none. But what about a being who is BOTH non-material AND timeless? Well, this now becomes a CUP - a Compount Unevidenced Proposition. Each of the two attributes is Simple Unevidenced (there's no evidence that timless OR non-material beings exist) and, in combination with each other, they form a Compound Unevidenced Proposition.

5. Let's calculate God's odds - P(G)

a) the odds that a non-material entity exists must be 50%

b) what are then the odds that this entity is also a timeless entity (or that an entity who is both exists)? In order to calculate these odds, we use the "AND" probabilistic operator. Thus, P(A AND B)= P(A) * P(B)

Our odds that a non-material and timeless entity exists are therefore .5*.5=.25=25%

c) what are the odds that an entity exists who is non-material, timeless and omnipotent? Once again, this is an unevidenced proposition and once again it receives the same treatment. We end up with .25*0.5=0.125=12.5%

d) and omniscient? 0.125*0.5=0.0625

e) sentient 0.0625*0.5=0.03125

f) uncaused: the above * 0.5 = 0.015 or 1.5%

g) creator of Universe: 0.75%

Now, I may have missed some definitional elements of God that some of us might like to include. Perhaps there are some who might disagree with some of the ones that I did include. But that's not important. It will not change the result overly dramatically. What is important is that, BASED ON the "no evidence for or against entitles us to equal odds" principle (which in itself is of course pure generosity towards theism!), the odds that God exists are only 0.75%.

And this confirms intuitively. There are huge numbers of hypothetical entities (probably infinite!) that we could in principle conceive of. Assuming a finite reality, only a finite number of them actually exist. Hence, the odds of correctly guessing out (without evidence) that a particular creature with particular attributes exists must be very low indeed.

Atheists know this. This is why they feel a little weird when a theist demands evidence that there's no god. To us, lack of evidence that a god does exist is a very prudent reason to start with the working assumption that no god exists indeed.

Comments are welcome




  1. I'm not sure I understand your argument very well. Do you mind putting it in premises?

  2. 1. An unevidenced proposition is a proposition there's no evidence for or against.
    2. The odds that any unevidenced proposition is true is 50%.
    3. The odds that any TWO propositions are BOTH true is the product of the odds that each of them is true.
    4. The odds that any number of unevidenced propositions are ALL true is the product of their respective propabilities.
    5. But the probability that an unevidenced proposition is true is 50% (see 2).
    6. Therefore, the probability that n unevidenced propositions are all true is 0.5^n.
    7. The definitional properties of God are: non-material, timeless, omnipotent, omniscient, sentient, uncaused being that created the Universe
    8. There's no evidence for a being with any of those properties.
    9. There's no evidence against a being with any of those properties.
    10. Therefore the existence of a being with any one of those properties is an unevidenced proposition.
    11. There are 7 of these unevidenced propositions in the definition of God.
    12. Therefore, the probability that God exists is P(G)=0.5^7=0.0075=0.75%

    1. Well I responded to this on twitter, but I do not think I made my response very clear. I basically have two objections. 1 - The argument is self-defeating. 2 - The premises are false. I will explain both of them.

      1 - The argument is self-defeating.
      Based on the reasoning of the argument, unevidenced propositions have a 50% chance of validity. So, you go on to explain, when we combine unevidenced propositions, their combined truth probability decreases. The problem with this is: **we have several premises in your argument that are a priori, or unevidenced propositions.** From this it follows by the logic proposed in the argument, that if we combine these unevidenced propositions, their combined truth probability decreases. But if that is the case, then we can conclude similarly in regard to your argument, that you conclusion is mathematically improbable.

      2 - The premises are untrue.
      This idea that the odds of an unevidenced proposition being true is 50% seems undefended to me. Can you show the method that you used to gather this statistic? It seems made up to me. There are many propositions for which there can be no evidence, but we are rational to accept them. Let me list a few.

      "Evidence is a valid form of reaching conclusions." To try to provide evidence for this statement would be reasoning in a circle.
      "There are other minds other than my own."
      "The universe was not created five minutes ago with an appearance of age."
      "There are physical materials."
      "I am not dreaming."

      The last four are all statements for which there can be no evidence. But let's apply the reasoning of this argument. That would mean that the probability that all of these unevidenced propositions were true would be .5^5, or a 0.03125% probability that this series of unevidenced propositions are true.

    2. Richard...

      1. What unevidenced propositions in my argument? Be more specific.

      2. "Evidence is a valid form of reaching conclusions" is axiomatic. If we don't accept that, we're not talking at all.

      "There are other minds than my own" is equally axiomatic. We all ASSUME that the world outside of our mind exists. But if it doesn't, then there's no god out there either.

      "The universe was not created five minutes ago with an appearance of age". As above. We assume this because otherwise we can't work on the basis of evidence to start with. And again, if you refuse to accept evidence as a basis for conclusion then we have nothing to discuss.

      "I'm not dreaming" again, same thing.

      What you're really done is taken TWO axioms, put them in different words (to make it look like there are more) and claimed they are unevidenced propositions.

      Let me ask you this question:

      If you have no evidence that X exists and none that X does not exist, what's more likely? That X exists or that X doesn't exist? With no prior knowledge, what likelihoods will you assign to these propositions?

      Do you disagree that the more conditions a proposition contains, the less likely it is to be true? I can't imagine you would disagree with that.