The first question is from Math; the second is from theology/philosophy.
I have a set of N elements. Each element can make a decision in favor of A or against it. The probability of each element making decision is given as P(A). If enough elements make decision A, the set will make decision A. If not, it won’t. The proportion of elements necessary to make a decision is given as R. I want to define P(S), the probability of the set making decision A, in terms of N, P(A) and R.
For example: In a parliament of 100 members, at least 75 members have to vote "yes" for a law to pass. The probability of each member voting "yes" on the given law is 30%. What’s the probability that the parliament will vote "yes" on the law? In this problem, N = 100, P(A) = 0.3, and R = 0.75, and we need to find P(S).
According to David Hume, there is a fundamental problem in our understanding of the world. We define "nature" as a series of causes and effects. But what does causality mean? When we say that A causes B, what exactly do we mean? It seems that we mean that A must necessarily lead to B. But if we try to explain the nature of causality, we can do two things:
1) Say that due to its properties, A always leads to B. I.e., say: “What do you mean, what is the relationship between A and B that results in A causing B? The relationship is that whenever A happens, B necessarily happens.” But this doesn’t really explain why or how it is that A results in B; it just means that in our observation, A has always been followed by B. But why does this mean that A necessarily leads to B? Maybe A leading to C (or remaining A) is a low-probability event that has never been observed? In that case, A can be said to be highly correlated with B, but not be its “necessary cause”.
2) Define A in terms of a number of sub-events and do the same with B. So, say that A really consists of A(1) and A(2) and A(3) which lead, respectively, to B(1), B(2) and B(3), which comprise B. But then, we are back to the original question: what does it mean that A(n) leads to B(n)? We haven’t really explained the causality, we just increased the power of our microscope and zoomed in.
So, if you read a little more David Hume and think about it yourself, it doesn’t seem that we have a satisfactory answer to the question “What exactly does it mean that A necessitates B?” All our answers will either involve defining the causal relationship of A and B in terms of sub-relationships of their elements, or saying that A and B are strongly correlated as per our observations. But correlation, as everyone knows, is not causation. At least, it shouldn’t be. [This paragraph wasn’t in the post originally, but I realized that just because my explanation of David Hume’s problem with causality makes sense to me, it doesn’t mean it makes sense to everyone. For a better treatment of the problem, listen to this lecture by Rabbi Gottlieb.]
Philosophers have argued whether Hume really thought that “causation is reducible to pure regularity” (source) or not, but that’s not really interesting to me.
What is interesting is that in the 20th century, scientists have discovered Quantum Mechanics, which postulates that if you go “turtles all the way down”, to the level of sub-atomic particles, the microscopic events cannot be said to cause each other with absolute certainty. They are at most bound by a probability. So, A cannot be said anymore to lead to B with 100% certainty. It has a probabilistic relationship with B — and with a number of other outcomes, all of which comprise A’s “probability cloud”.
The reason why in the macroscopic world, we observe things as seemingly causing each other is that microscopic probabilities "pile up", such that even though it is probable that one day you’ll let go of an apple, and it will float upwards, the chance of that happening is so low that you’d have to wait for a whole bunch of lifetimes of the Universe before you observe it. So, yes, causality is regularity, but a very-very strong regularity. (So, ice dissolving in a hot bath is still a law of physics, but it is a probabilistic law. Then again, the probability is so high that you can very well put your money on that happening.)
Now, my question is: can we really have such concepts as “nature” and “miracles”?
We could say that either G-d created the laws of nature and let the world run its course (sometimes interfering in a miraculous way) or that He creates every single instance of the world.
I will address the first possibility very briefly. Or, rather, I will quote Rabbi Gottlieb (who is quoting Rav Dessler): “If all you can tell me about cause-and-effect is that ‘it always happens’, and you agree that G-d is the source of the world, aren’t you left with the idea that He makes every single thing happen directly?” (If you don’t follow the logic, keep reading.)
But, according to our tradition, Hashem creates the world every single instance of time. So, that’s not a big chiddush. One could still say that most of the time, He creates the world according to a pattern that He set up Himself (and such events are defined as “natural”), and in other times, He makes an exception to that pattern (i.e., making “miracles”).
So, we can define two modes of Hashem’s relationship with the world: a “natural” one, whereby He creates the world according to a pattern, and a “miraculous” one, whereby He creates [an aspect of] the world outside the pattern.
But then, let’s go back to our question of causality.
Before, when we believed in such a concept as A leading inevitably to B, we could say that whenever A → B happens, that’s nature, and whenever A → *** happens, that’s a miracle. But now that our conception of inevitable relationship between A and B has collapsed and been replaced with “when A happens, it leads to B with high probability”, how do we differentiate between a natural event and a miraculous one? Should we set an arbitrary cut-off point in probability of the event happening, where higher than this probability is “nature” and lower is a “miracle”?
Furthermore, before, we could say that certain events happen by hashgacha klalis, general Providence of G-d, i.e., they come from G-d, but rather “automatically”, according to a pattern He set up, and some happen by hashgacha protis, specific Providence, with His “personal attention”. The natural events’ outcome was “certain”, their nature sealed and pre-ordained by G-d. The miraculous events’ outcomes, however, required G-d’s individual decision which differed from moment to moment. (And I mean here exactly the same event, under exactly same conditions.)
But now that we know that every time A happens, its outcome must pass through a “purple fuzz” of probability (as Rabbi Gottlieb calls it), should we not say that every event demands personal attention from G-d, since its outcome is not sealed at all?
To put it in a different way: before, when we believed in causality, we could say that each event could either find a cause in another event (in this case it was deemed “natural”), or it could be said to have no cause in the world (according to the “laws of nature”), in which case it was happening ex nihilo (by definition) and was “miraculous”.
But now, with every event not really having a cause and only having a certain correlation with another event preceding it in time, all events are happening ex nihilo! All events have to be said to stem from G-d, in a “miraculous”, hashgacha-protisdik way!
Reading this sicho, we discover:
[T]here is a difference of opinion between the approach of the Rambam (and the others who follow the approach of Chakirah, Jewish metaphysics) and the Baal Shem Tov.
According to the Rambam, G‑d’s involvement in the particulars of the future of any being other than the righteous is “passive”. He has created the natural order, and He has deemed that the natural order control the fate of these entities.
In contrast, according to the Baal Shem Tov, every element of existence and every slight change that occurs regarding it depends on G‑d’s will and desire, as it is written: “I will conceal My face.” The intent is that only the inner (p’nimiyus) expression of His providence is hidden. Thus a person can convince himself that his difficulties “find him”, that they are part of the natural order or a function of circumstance. In truth, however, every aspect [of his life] is being controlled by Divine providence.