Gaming philosophy conundra

[Lovewyrm]

Originally Posted by Daigoro

That is the point of rolling dice though. You could just as well throw d6's at your GM if you wanted that as a game mechanic.

It might be a sphere which holographically displays its result above it when it comes to rest. I would also presume it to give integer results, so that two very close results could be compared in finite time.

But my original point was that rolling d∞'s would have the paradoxical result that the probability of an outcome was determined by the order of the dice rolls, whereas for finitely sided dice the probability is independent of the order.

Well, swordfighting with long sticks is something I can get behind.
Infinity is not, especially not math infinity since that's fake anyway.
d∞ is probably just a d3 anyway, either you win or you lose or you draw.
It's infinite so you either go below, above or the same.
Why would the order matter unless this is some weird not really infinity thing?
Kind of a rethorical question, I probably wouldn't understand the reasoning anyway because it's gonna be some math trick that humans have invented to think they can deal with infinity.

Like the illusion of functional programming even though one cannot escape imperative programming in computing because that's just how computers work (at least the ones I know).
It's all imperative in the end.

[whswhs]

Originally Posted by Lovewyrm

Well, swordfighting with long sticks is something I can get behind.
Infinity is not, especially not math infinity since that's fake anyway.
d∞ is probably just a d3 anyway, either you win or you lose or you draw.
It's infinite so you either go below, above or the same.
Why would the order matter unless this is some weird not really infinity thing?

Well, that I understand. The first roll must come up as some positive integer. For any positive integer, there are finitely many positive integers that are less than or equal to it; but there are infinitely many positive integers that are greater. So the probability of the second roll landing on a greater number is vastly higher than the probability of its landing on a number that is not greater; you can treat them as effectively 1 and 0.

As for "fake," in both mathematics and RPGs, we start out by making assumptions, which we agree to pretend are true. Within the context of the game, calling them "fake" is not a valid move; to do so is to refuse to play the game. In this case, the game is exploring the concept of a die in the shape of a regular polyhedron with infinitely many sides.

One of my players likes to tell the story of a campaign he was in, where the GM said to the players that their characters had walked into a room whose floor was tiled in octahedra. One of the players said, "It's not possible to tile a surface in octahedra." And the GM said, "That's right. Roll for SAN loss."

[Anthony]

Originally Posted by whswhs

Similarly, on a spherical surface, the area occupied by each rationally numbered point seems to go to zero. The fact that those points do not make up the entirety of the spherical surface may not make a difference.

The problem is that you don't wind up with a fair die (where 'fair' is defined as 'equal probability for every result') doing that.

[whswhs]

Originally Posted by Anthony

The problem is that you don't wind up with a fair die (where 'fair' is defined as 'equal probability for every result') doing that.

It's not obvious to me that that's the case. It seems as if the probability of rolling any number is 0, and 0 = 0 = 0.

[Anthony]

Originally Posted by whswhs

It's not obvious to me that that's the case. It seems as if the probability of rolling any number is 0, and 0 = 0 = 0.

You're talking about infinities now. 1/aleph-null and 2/aleph-null are both equivalent to zero, but non-equal.

[whswhs]

Originally Posted by Anthony

You're talking about infinities now. 1/aleph-null and 2/aleph-null are both equivalent to zero, but non-equal.

But is it necessary to assume that you would have such a case arise?

[malloyd]

Originally Posted by whswhs

As for "fake," in both mathematics and RPGs, we start out by making assumptions, which we agree to pretend are true. Within the context of the game, calling them "fake" is not a valid move; to do so is to refuse to play the game. In this case, the game is exploring the concept of a die in the shape of a regular polyhedron with infinitely many sides.

There's a reason mathematicians treat these sorts of things as limits these days rather than infinities or infinitesimals. I suspect but can't prove that if you build a proper model of constructing an polyhedron of n faces and let n go to infinity that something explodes mathematically in a way it doesn't for continuous infinite variation for a sphere.

I more than half suspect it's going to get you back to the axiom of choice - you know the one that allows you to cut a sphere into two identical spheres.... The math joke is "the Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" - the joke being that's pretty much true by inspection on the superficial definitions, and you can prove all three are restatements of each other.

[TGLS]

Originally Posted by whswhs

It's not obvious to me that that's the case. It seems as if the probability of rolling any number is 0, and 0 = 0 = 0.

In any continuous random variable, the absolute likelihood of rolling any given value is zero. But it's still possible to calculate the probability or rolling higher or lower than any given number.

For example, consider a continuous uniform distribution from 0 to 1. The absolute likelihood of generating value 0.8 is zero, while the probability of rolling less than (or less than or equal to) is 0.8. This can be generalized to p(x)=(x-a)/(b-a), where a is the lower bound of the distribution, b is the upper bound, and x is the upper bound of values we're testing for.

Plug in a=0, b=infinity, and you get p(x)=x/infinity=0. So if you know what the GM rolled, the chance you roll less than whatever the GM rolled is zero. Of course, this is probably wrong for two reasons:
1) Both die rolls are independent events, thus knowing what the GM rolled can have no effect as to whether you roll higher than what the GM rolls.
2) This is coming from arbitrarily plugging in infinity into a formula.

So let's take it from a different angle of attack. Dice of the form 1dX have an average of (X+1)/2, a standard deviation of sqrt((X^2-1)/12)), and the odds 1dX>1dX is (X-1)/(2X). As X approaches infinity, the average and stdev approach infinity, and the odds 1dX>1dX approaches 50%. So save yourself some time searching from your d∞ and toss a coin instead.

[Agemegos]

Originally Posted by malloyd

There's a reason mathematicians treat these sorts of things as limits these days rather than infinities or infinitesimals.

And have done so at least since my freshman maths classes, forty years ago.

[Daigoro]

Originally Posted by TGLS

Of course, this is probably wrong for two reasons:
1) Both die rolls are independent events, thus knowing what the GM rolled can have no effect as to whether you roll higher than what the GM rolls.
2) This is coming from arbitrarily plugging in infinity into a formula.

I might add something along the lines of:
3) There's a difference between probability and possibility. It might be infinitely improbable to roll below the GM's target, but it's not impossible. In other words, probability theory is a flawed model of reality.