Imagine a hypothetical universe where 1 million days have gone by.
Imagine another where 1 billion days have gone by.
Assume that in each case, one has found no direct evidence of god operating in the world. Is this longer period of time of absence further evidence against god?
Should one feel more confident in god’s non existence in the second compared to the first scenario?
It is impossible to provide a meaningful answer to your question without first having a clearer hypothesis about the nature of god and especially about the nature of god's interaction with the Universe. It is possible, for example, that having made the Universe, god decided to have nothing more to do with it, so you can wait as long as you like and never find evidence. Or perhaps god set a timer to remind him/her to check on the Universe every 1,000,000,000,000,000,000,000,000,000,000,000,000 years, so your propose intervals are too short to make a difference. Or perhaps god always acts in stealth mode, making interventions that are faultlessly disguised as natural causes. And so on ad nauseam. So the relationship between time and probability depends upon one's underlying assumptions about god.
If your hypothesis was that god reveals herself to the peoples of the Earth by appearing in a fiery chariot every million years, then you could build a probability distribution which determined the likelihood of it being true as a function of time. The likelihood would have dwindled to zero after a million and one years if god had not appeared beforehand.
Well, to try out my math skills (I need practice, a helluva lotta practice), let's define the function F(t) as taking the input time elapsed (t) since some arbitrary point on the timeline and giving the output the probability of truth (about anything) being discovered and let us suppose the function F describes a direct proportionality relationship between t and F(t).
So, F(1 billion years) > F(1 million years) i.e. the probability of discovering the truth about God (whether he exists/not) should be greater for 1 billion years than 1 million years.
My worry/concern is the nature of a possible intermediary function G(t) we need to work out such that F(G(t)) is our composite function linking the probability of discovering the truth to time elapsed.
Me two sikkas.
Let H be the hypothesis that God exists and ~H the negation of that hypothesis. A Bayesian would say that the probability of H being true is given by Bayes rule:
P(H|X) = P(X|H)P(H)/P(X) = P(X|H)P(H)/[P(X|H)P(H) + P(X|~H)P(~H)]
where P(H) is your prior belief in the existence of God and X represents our observations of God. However, we will instead look at the odds ratio - the ratio of the probability of God's existence and non-existence,
P(H|X)/P(~H|X) = P(X|H)P(H)/[P(X|~H)P(~H)] = P(X|H)/P(X|~H) * P(H)/P(~H)
The second term in the final expression is called the Bayes factor and it basically tells us how much the observations affect our relative belief in the existence and non-existence of God.
If it describes a God that regularly makes appearances - say "Q" in Star Trek the Next Generation, then P(X|H) and P(X|~H) will be very different and not observing "Q" (at least once a series) will give reasonable evidence that "Q" doesn't exist. In this case, the Bayes factor will be substantially less than one. In this case, absence of evidence is to some degree evidence of absence.
Now, if our hypothesis of God describes a God that rarely makes an appearance then P(X|H) and P(X|~H) are going to be very similar in the case of the absence of evidence and the Bayes factor will be close to 1. In this case, our posterior belief in God is basically just a restatement of our prior beliefs and absence of evidence is not evidence of absence.
For most theological beliefs, we will be in the latter case, and pretty much all evidence based arguments for and against the existence of God will be just restating our prior beliefs, and probably not a good use of time.
This is essentially formalising @MarcoOcram's answer (+1)
Wonderful question, thanks. If you believe God performs miracles (and hasn't politely withdrawn from the world), then absolutely, the lack of miracles for a long time would amount to better evidence He does not exist.
If I claimed to be able to walk on water (or recite pi to ten places) but haven't yet today vs forty years, the latter is better evidence that I am wrong, especially if people keep asking me to.
I don't know what besides miracles would class as evidence, rather than arguments, for God.
There is no evidence for the absence of God. It is logically impossible to prove a negative.
The burden of proof is on the theist side. The absence of God is the default assumption until the presence of God is proven.
