Determinism vs. Indeterminism

Ultimately I think the difference between deterministic and indeterministic laws is not significant.

If a physical law is deterministic then under it's influence Event A will "cause" Result X 100% of the time.

Why does Event A always lead to Result X? Because that's the law. There is no deeper reason.

If a physical law is indeterministic, then under it's influence Event B will "cause" Result Q, R, or S according to some probability distribution.

Let's say that the probability distribution is 1/3 for each outcome.

If Event B leads to Result R, why does it do so? Because that's the law. There is no deeper reason.

Event A causes Result X 100% of the time.

Event B causes Result R 33.3333% of the time.

Why? There is no reason. That's just the way it is.

Determinism could be seen as merely a special case of indeterminism...the case where all probabilities are set to either 0% or 100%.

So even if we are in an universe with indeterministic laws, this doesn’t have any major impact on what metaphysical conclusions we arrive at. Even assuming indeterministic physicalism, there are still initial conditions and there are still laws - the laws just have an intrinsically probabilistic aspect.

These probablistic laws are like the rules of a card game that includes a certain amount of randomness...for instance, requiring occasional random shuffling of the deck. But the number of cards, the suits, the ranks, and the rules of the game themselves are not random...those aspects are determined.

Similarly, in quantum mechanics using the Schrodinger equation, the evolution of the wavefunction describing the physical system is taken to be deterministic, with only the "collapse" process introducing an indeterministic aspect.

But as with the card example, the impact of this random aspect is limited in scope. No matter how random it gets, it doesn't change the rules of the game. No matter how randomly the deck is shuffled, you still only ever have 52 cards, 4 suits, and 13 ranks. The randomness is constrained by the deterministic aspects of the game.

Another example of constrained indeterminism are computer programs that use randomness, for instance the Randomized Quicksort. No matter what pivots you randomly select, the algorithm is still going to correctly sort your list. At worst, it will take longer than usual. Because the randomness of the pivot selection is constrained by the context provided by the deterministic aspects of the program.

The same goes for our universe in the indeterministic case. The randomness of indeterminism only increases the probability of the existence of conscious life that discovers something true about the underlying nature of the universe *IF* the initial conditions and the non-random aspect of the causal laws allow for this to be the case.

Is it possible that our causal laws are such that any given starting conditions (with respect to the distribution of energy and/or matter) eventually lead to conscious life that knows true things about the universe?

Here we return to our analogy of the quicksort algorithm, which can start with any randomly arranged list and always produce a sorted list from it.

Note, though, that the quicksort algorithm is a very, very special algorithm. If you just randomly generate programs and try to run them, the probability of getting one that will correctly sort any unordered list is very low compared to the probability of getting a program that won't do anything useful at all, or sorts the list incorrectly, or will only correctly sort lists with special starting orders, or sorts the list but does so very inefficiently.

Equivalently, if you just randomly chose sets of causal laws from a list of all possible combinations, the probability of selecting a set of laws that can start from almost any random arrangement of matter and from that always produce conscious life that perceives true things about the laws that gave rise to it must also be very low.

Infinity and Probability

How about this:

Lets assume we have an infinitely long array of squares. And a fair 6-sided dice.

We roll the dice an infinite number of times and write each roll's number into a square.

When we finish, how many squares have a "1" written in them? An infinite number, right?

How many squares have an even number written in them? Also an infinite number.

How many squares have a number OTHER than "1" written in them? Again, an infinite number.

Therefore, the squares with "1" can be put into a one-to-one correspondence with the "not-1" squares...correct?

Now, while we have this one-to-one correspondence between "1" and "not-1" squares set up, let's put a sticker with an "A" on it in the "1" squares. And a sticker with a "B" on it in the "not-1" squares. We'll need the same number of "A" and "B" stickers, obviously. Aleph-null.

So, if we throw a dart at a random location on the array of squares, what is the probability of hitting a square with a "1" in it?

What is the probability of hitting a square with a "A" sticker?

The two questions don't have a compatible answers, right? So, in this scenario, probability is useless. It just doesn't apply. You should have no expectations about either outcome.

BUT. NOW. Let's erase the numbers and remove the stickers and start over.

This time, let's just fill in the squares with a repeating sequence of 1,2,3,4,5,6,1,2,3,4,5,6,1,2,3,...

And then, let's do our same trick about putting the "1" squares into a one-to-one mapping with the "not-1" squares, and putting an "A" sticker on the "1" squares, and a "B" sticker on the "not-1" squares.

Now, let's throw a dart at a random location on the array of squares. What is the probability of hitting a square with a "1" in it?

What is the probability of hitting a square with a "A" sticker on it?

THIS time we have some extra information! There is a repeating pattern to the numbers and the stickers. No matter where the dart hits, we know the layout of the area. This is our "measure" that allows us to ignore the infinite aspect of the problem and apply probability.

For any area the dart hits, there will always be an equal probability of hitting a 1, 2, 3, 4, 5, *or* 6. As you'd expect. So the probability of hitting a square with a "1" in it is ~16.67%.

Any area where the dart hits will have a repeating pattern of one "A" sticker followed by five "B" stickers. So the probability of hitting an "A" sticker is ~16.67%.

The answers are now compatible, thanks to the extra "structural" information that gave us a way to ignore the infinity.

In other words, you can't apply probability to infinite sets, but you can apply it to the *structure* of an infinite set.

If the infinite set has no structure, then you're out of luck. At best you can talk about the method used to generate the infinite set...but if this method involves randomness, it's not quite the same thing.