# Probabilistic Interpretation of Hume's Problem of Induction

I am a mathematician who has recently started delving into philosophy and I had a thought today. From what I understand, the scientific method can be described in the following terms.

Suppose we have an experiment E to run with pre-conditions P to be assured are in a certain state, then after the pre-conditions are set, we observe an outcome O. Then for each additional experiment E' with the same pre-conditions with the same outcome O, the estimated probability of the causal relationship between the pre-conditions and the outcome O increases. By the law of large numbers the sample mean should converge to the true mean, where if we assign 0 to outcome O not occuring, and 1 to outcome O occurring, if causation is consistent and unwavering as in a physical law, then the sample mean should equal to the true mean, which is 1.

Then my question is: how could one distinguish between a world in which the probability of a causation in some scenarios not occurring is non-zero. For example if the probability that after I turn on the heat on my stove to the boiling temperature of water and put a pot of water on, the probability the water will not boil is non-zero in some world A, albeit very very small, and is zero in some world B. Then presumably in world A, strange things should occur a non-zero number of times, but catching a rare probabilistic event in the act so to speak may never occur in the lifetime of humanity.

Again I am not a trained philosopher so maybe my question has false assumptions/problems, but any insights into this much appreciated.

• Problem of Induction only implies all empirical inductive knowledge/laws are uncertain inherently, since there's no strict non-fuzzy logical causal justification. If both observations in world A & B are same for your boiling water case (like for a stone to cross a wall), then either you change your pot size (reduce probability multiplication effect, like reduce a stone to a particle) to retest or study further about its components to identify world A from world B. Pure reason cannot help much based on such a simple setup. Mar 21 at 22:33
• I am not quite sure how this relates to Hume's problem of induction. Hume's point was that even when we observe a perfect correlation between A and B we never observe the causing of B by A. Indeed, it can be a case of common cause, as with thunder and lightning, or of parallelism for mathematical reasons, say. So the "probability of causation", if one can make sense of it, is not something that can be extracted from how frequently occurrences of B follow occurrences of A. Causation is a category that we attach holistically, when we have theoretical reasons to postulate a causal law. Mar 23 at 4:20
• @Conifold, right. That makes sense. Mar 23 at 19:20
• @Conifold small correction suggested for your above last statement "we have theoretical reasons to postulate a causal law". I think it's better to replace "theoretical" to "empirical" or "inductive". Regarding Hume's problem of induction, there's no theoretical deduction involved, only empirical induction as manifested in science and most practical knowledges. Only when we use logic or math to further apply causal laws or push them to their logical end, we do things theoretically. Mar 24 at 4:10
• @DoubleKnot We typically postulate a causal rather than just a phenomenological law when it serves purposes such as simplification and unification of the theory. It is beyond empirical induction, but takes place at the genesis stage of the theory, logically prior to its application. This is essentially a modernized version of Kant's solution to Hume's problem. Mar 24 at 4:29