Induction and the problem of miracles
There is a curious problem with miracles that arises if our only way of knowing the world is inductive. Since very careful deduction using mathematics in science is a feature of the modern world, we might call our inductive problem with miracles, the pre-scientific problem of miracles.

Suppose we have pulled 1 trillion blue marbles from a bag, and nothing else. By the arguments given here - the probability that the next marble is other than blue is roughly 1 trillion to 1.

Now suppose someone testifies that he pulled a red marble. Let's also suppose we find him 99% reliable. We would then update our probabilities. We would reject his claim, but rather than believing the odds of a non-blue marble are 1 trillion to 1, we might now only see them as 10 billion to 1, a 100 fold increase based on our witness.

But while we do not believe the witness’s specific claim, we know that 1 trillion marbles have been pulled. We also believe the odds of non-blue marbles are 10 billion to one. Should we now believe it is probable that in the history of marbles 100 non-blue marbles have been pulled? No. If the claim of our witness is true, then we should think there is one red per trillion, and if false, then there are no reds per trillion. But our witness is rather reliable, so we should estimate 1 red per trillion. The paradoxical result is that while we doubt his specific claim, the idea of a red marble in the entire history of marbles becomes quite reasonable, maybe even probable.

Once careful deduction from other evidence enters the picture, this changes. Instead of 1 trillion to 1, we may be able to set the odds of a non-blue marble at a trillion trillion to one. This, even though we have only ever observed 1 trillion marbles. Now if we get testimony of a red marble, we doubt not only the specific claim, but also the idea that such a thing could happen in the history of marbles, since the odds are now 1 trillion to 1 against it having happened, even in the whole history of marbles. This then is the post-scientific analysis of miraculous claims.

The problem is not that science proves miracles can't happen. As I point out on my page on
truth and knowledge, science is not the last word on truth. Because science does not consider certain forms of evidence like testimony, personal experience, and other non-public and non-reproducible data, science may get an answer that is correct, by the rules of its game, but consideration of that other data may or may not change the picture of truth as shown by science.  Science does not generate absolute proof, and so it can not absolutely exclude miracles.

What science can do, however, is create such high degrees of justified certainty, that other forms of evidence simply can not change the result. When very trusted testimony and personal experience are weighted against the results of science, science may win.

How can a religion, like Christianity, that depends on the idea of a central miracle, deal with this problem? On my
evidence and testimony page I speculate that maybe one could try to consider the person giving testimony holistically and trust them as a whole person, rather than trying to evaluate the details of their credibility. However, some would argue that this is an error. Even if we feel we can't trust the person with 100% certainty, how can we rule out the possibility that they are deceived themselves?

I've heard some Catholic scholars say that they don't believe a video camera would have captured the miracles described in the bible. Rather, they say God put the idea of the miracles in the minds of the evangelists. The message is the important thing for them, not the truth claims. For more conservative Christians this comes too close to saying the whole thing was just made up. I have one possible suggestion that partially answers this problem, although I doubt many will find it a compelling answer.

Science can only generate near certainty regarding finite questions. For example if we have drawn 1 trillion blue marbles in a row, it is highly probable that the next marble is blue. But how certain are we that the next 1 trillion marbles will all be blue? Our justified certainty drops to around 66%. How certain can we be that all marbles are blue? The answer is that we have no certainty at all. Our sample of 1 trillion marbles tells us nothing about an infinite number of marbles. We can say there is a 50% chance that all marbles are blue, and a 50% chance that they are not, just because these are the only two possibilities, but this is just a statement of our ignorance.

If we analyze the claims of miracles in the bible critically, as finite events, in light of scientific knowledge, we may conclude we can not have a justified belief that they actually happened. However, if we convert this to an infinite question the situation changes. Suppose we ask if a specific miracle could have happened anywhere in an infinite universe or even in other universes or other dimensions if they exist? Now we can no longer say the miracle is unlikely. Our answer now becomes, "I don't know". We can have a 50% justified believe that the miracle happened somewhere. And then we could add that God just made sure the message got here to us, via the minds of the evangelists. As I point out on my page on
faith and evidence, if we can show that two possibilities are equal in terms of evidence, we can be justifies in picking one over the other based on criteria other than evidence, like utility, or preference. This is a justifiable use of faith. Note also, that Occam’s razor really does not apply here. Occam’s razor can tell us that all else being equal the simplest theory is the best one for predicting finite events. “Marbles are blue” is the best theory. However, when the question becomes infinite, there is no best theory. Finite experience is irrelevant.

But one could object here that any story we create then has a 50% chance of being true somewhere, and perhaps more importantly all religions can make the same claim. The miraculous events of Islam also have a 50% chance of being true. How could one justify picking one idea out of this endless range of ideas? This is where personal experience and faith in the testimony of others can come into play. If our own experience, or our faith in the testimony of others about their experiences, tells us that a certain set of beliefs leads to a life correctly led, then this might allow us to select that set of beliefs as guiding principles for life. The actual truth claims need not be a barrier. We can say that there is a 50% chance they happened somewhere, and we choose to believe they did, based on faith. Where they happened is not important. That they did happen, and that we got the message is important. And, after all, if they happened somewhere, why not here?
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