The principle of indifference

In several previous posts I have applied the principle of indifference, albeit I did not call it by that name.

• In the post The eerie silence I wrote this:

The probability of the existence of extraterrestrial intelligence is 50%. As we know nothing, this is equivalent to throwing a coin, and if it comes up heads, we say that we are alone; if tails, that we have company.

• In the post Relativism in science? I said this:

If we have no reason to assume that a theory is true or false, its probability should be close to 0.5... Theories about which we have no information, for or against, with a probability between 0.4 and 0.6. I will cite the existence of extraterrestrial intelligence, the possibility of building strong artificial intelligence, or the various theories of the multiverse.

Martin Gardner

The principle of indifference can be useful in certain circumstances, but it can also lead to absurd results if misapplied. In his book Aha! Gotcha: Paradoxes to Puzzle & Delight, Martin Gardner offers the following examples of the misuse of this principle:

• Is there life on Titan, the largest of Saturn's satellites? As we know nothing, either for or against, we should assign both possible answers a probability of 50%. But now let’s consider these two questions: Are there microorganisms on Titan? Is there animal life on Titan? We still know nothing, so we could assign a probability of 50% to both possible answers to these questions. But then, what is the probability that there is neither animal life nor microorganisms? Are we tempted to answer 25%? (0.5×0.5). If we do, we’ll make a blatant mistake, because these two questions are not independent of each other, so their joint probability is not equal to the product of their probabilities.
• Will there be an atomic war before 2030? Let’s say the probability is 50%. Now let’s answer ten additional questions: Will there be an atomic bombing on the United States before 2030? On Russia? On the UK? And so, we list 10 countries. If we assign the answers to these questions a probability of 50% and use it to calculate the probability that none of them is bombed, we’d get 0.5¹⁰=1/1024. Then the probability that at least one of those countries will be bombed before 2030 would be 1023/1024: almost certainty. But we are making the same mistake as in the previous example: the answers to the ten questions are not independent and their probabilities shouldn’t be multiplied.

Blaise Pascal

Blaise Pascal applied the principle of indifference to the existence of God in his famous wager, about which I wrote another post. In his Pensée 233, he says this:

God exists or He does not exist. Which side shall we take? Reason cannot decide... Let's weigh the gain and loss, if we assume that God exists. Let's consider both: if you win, you win everything; if you lose, you lose nothing. So you must wager, without doubt, for His existence.

Pascal argues that if we apply the principle of indifference to the existence of God, we should assign a probability of 50%. But his argument does not stop there: he applies to both possibilities the first known example of game theory and breaks the equilibrium according to what we can gain or lose by adopting each answer.

We can argue against Pascal's phrase Reason cannot decide, which is equivalent to applying the principle of indifference to the existence of God, even if we interpret that the word reason means scientific reason. In my book in Spanish ¿Es compatible Dios con la ciencia? Evolución y cosmología I pointed out that although science does not offer proof of the existence of God, it does provide inklings. I estimate that these inklings unbalance the equiprobability of the response, and by adding other sources of knowledge, apart from science, the total result for me is close to 100%.

The same post in Spanish

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Manuel Alfonseca