From my paper "Miracles and the Case for Theism."
It is true that in order for miraculous occurrences to play a role
in a case for theism, it must be the case that such events contradict
naturalistic expectations. But it does not follow that, from the point
of view of naturalism, these events have to be maximally improbable.
Other events that contradict naturalistic expectations to a greater degree
can be passed off as mere anomalies because no plausible theistic
explanation is available to tempt the naturalist to alter his beliefs about
the way the world works. Paul Horwich gives an account of what it is
for an event to be surprising that may shed some light on this matter.
He claims that it is necessary to distinguish between unlikely events
and surprising events, since many unlikely events do not surprise us.
If I were to flip a coin 100 times and get heads every time it would
surprise me, even though any other sequence of heads and tails would
be equally unlikely. What distinguishes surprising events from other
unlikely events is the presence of an alternative account of the circumstances
under which the event occurred, an account not previously
accepted, that would diminish the improbability of the event in
question. Thus in the coin-tossing case the possibility that the coin
might not be fair causes me to wonder if the world is in fact the way
I, who am accustomed to coins being fair, previously thought it to be.
This explains why it would not be surprising if Jones were to win a
lottery amongst a billion people, but it would be surprising if Smith
were to win three lotteries amongst a thousand people, even though it
is more probable that Smith should win his three thousand-person
lotteries than that Jones should win a billion-person lottery. This is
because the Smith case gives me reason to change my background
assumption about the fairness of the lotteries in a way that the Jones
case does not. Thus surprisingness, for Horwich, does not vary with
improbability, it varies with the degree to which events force us to
change our hypotheses about how things happen in the world. 21 In
cases where there is evidence that a miracle has occurred, it is the
combination of natural improbability and the availability of supernatural
explanation that makes the evidence surprising from the point
of view of naturalism, not the improbability alone. So perhaps we can
attribute Mackie's insistence that miracles are maximally improbable
for atheists to the fact that good evidence for miracles would be maximally
surprising for atheists; for persons with a naturalistic bent the
acceptance of miracles requires a thorough revision of their view of
the world. (Miracles would also surprise theists, if they were not expecting
God to act in the way he did and would find it necessary to
change their view about what God is like). However, as Horwich has
shown, surprisingness is not strictly a function of improbability; therefore
Mackie is mistaken in assuming that since miracles are maximally
surprising they must also be maximally improbable.
10 comments:
Hi Vic. I've a question:
If I were to flip a coin 100 times and get heads every time it would surprise me, even though any other sequence of heads and tails would be equally unlikely.
Is that bolded part correct? I'm no probability expert, but that strikes me as wrong. Isn't the whole point that 100 heads in a row is actually less likely than most other combinations (say, 60/40 or 55/45)?
cl,
On the assumption that the coin is fair and the further assumption that this assumption is not at all doubted, Reppert is correct.
Now, suppose we've tossed a coin 99 times and landed heads each time. We might then suspect that the coin is biased towards heads, since if we are not certain that the coin is fair we update the expected frequency of heads in accordance with this evidence.
There's a bloggingheads discussion between Andrew Gelman and Eliezer Yudkowsky which opens with this problem: http://bloggingheads.tv/videos/2296
"Is that bolded part correct? I'm no probability expert, but that strikes me as wrong. Isn't the whole point that 100 heads in a row is actually less likely than most other combinations (say, 60/40 or 55/45)?"
It's correct if you consider the order. As in, there are 100 different ways to flip 99 heads and 1 tail and those 100 ways have an equal chance of occurring. If you disregard order, then 100 and 0 heads are the least likely.
Zachary,
Thanks for the effort, sincerely, but... that was and this is still pure gibberish for me. I honestly think those guys are just talking out their backside, and yes, I realize many people esteem them as great minds. I really don't think "probability" is real or even applies in this case. I can flip a small dish and make it land heads every time. There's no "probability" there, it's all in the throw. Why should this change because the dish gets smaller?
Now, if we're talking dice, or picking a card out of a stack, the concept of "probability" makes a little more sense, but... I would love a clear explanation of how one can say 100 heads in a row is just as likely as 45/55.
cl,
I'll take a stab at it.
You may be mixing up two different things: a particular series of results of 100 coin tosses, and a general grouping of proportional results.
Think of it this way. Imagine instead of 100 heads in a row, you get 99 heads, 1 tails.
But there's 100 different ways to get that. (heads, tails, tails, [...] tails) or (tails, heads, tails, tails, [...] tails) or (tails, tails, heads, tails, tails [...] tails), etc.
There's a difference between talking about "I got series 1, which was tails, then 99 heads" which describes a specific series, and "I got 99 heads and 1 tails", which describes 100 different series.
Cl,
"I really don't think "probability" is real or even applies in this case. I can flip a small dish and make it land heads every time. There's no "probability" there, it's all in the throw. Why should this change because the dish gets smaller?"
You can always use historic results (repeat trials and then count the number of hits over n) to create a probability distribution for real-world applications. In that case, the probability distribution for your small dish would be different than the one for the coin. The probability distribution for someone who is trained to throw dice a certain way for craps would be different from the average user. It's still the same problem, just harder for the player to insert bias into the equation. However, for theoretical purposes, the assumption of there not being bias (for the instrument itself and in the throw) is included, but there are ways to test this assumption in applications.
I hate to say this, because this isn't a critique of the substance of your post. However the formatting is attrociously ugly. There's a paragraph separation between each and every line. There's more white space than text.
cl,
Perhaps I can shed a little light on the probability question.
If you toss a coin 100 times, there are many trillions of possible sequences. Only one of those sequences consists of all heads. There are many sequences that consist of a mixture of heads and tails, but if you specify the exact order of the sequence, there is only one way to get that, so its probability is the same as all heads. On the other hand if you don't care about the order, there are many ways to get some heads and some tails. So the probability of that is much greater than all heads.
I fixed it manually.
cl,
Sorry I dumped that on you. I got on my hobby horse and took off at a gallop. This can only be adequately explained and discussed if you've studied probability theory and some philosophy of probability.
"Now, if we're talking dice, or picking a card out of a stack, the concept of "probability" makes a little more sense, but... I would love a clear explanation of how one can say 100 heads in a row is just as likely as 45/55."
It isn't. Notice that there is only one sequence of outcomes corresponding to 100 heads while there are lots of sequences corresponding to 45 heads. Each of these particular sequences is equally likely given that the coin is fair.
The point of the thought experiment in equation is, "once I've tossed heads 100 times in a row, how likely is it that the coin is fair?" If the coin is fair, we should have expected the frequency of heads to be close to 50%. But the coin being heavily biased towards heads explains the results so far much better, so we might think that prob(heads)>1/2 when we start toss 101.
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