Stubborn Highlanders

On the other hand, wi' enough Scotch, a broccoli fudge sundae might be a bonnie fine thing....

On the other hand, wi’ enough Scotch, a broccoli fudge sundae might be a bonnie guid thing….

Now that we’ve recapped, let’s explore the free will of immortals again.

Going back to here, let’s examine the following scenario:

Connor MacLeod has solemnly vowed never to eat a broccoli fudge sundae.  Though we don’t and probably can’t, know that the universe is, in fact, eternal, for purposes of the discussion we’ll assume it is.  Further, we’ll assume that Connor has the means to planet hop in case Earth ever becomes uninhabitable.  Thus he will truly live forever, not just until the universe’s heat death.  Thus, for Connor, “never” means never through all eternity.

Now, in probability, the likelihood of something occurring is between zero (it cannot occur–more on this later) and one (it must occur–this needs to be unpacked, too, but we’ll hold off for just a bit).  Things that must occur (bachelors must be single, two plus two must be four, a finite whole must be larger than any of its parts) are necessary truths with a probability of one (mathematically, P=1); things that cannot occur (a bachelor being married, two plus two equaling 85, a finite part being bigger than its associated whole) are logical contradictions with a probability of zero (P=0).  Everything else is a possibility of greater or lesser likelihood (high likelihood of the sun rising tomorrow, P>0.9999999999 ; low likelihood of Klingon invasion, P<0.000000001).  It’s important to note that a probability higher than zero, no matter how small, is still a possibility (this, too, is an important point).  The sun almost certainly will rise, but it might not; Klingon invasion is vanishingly unlikely, but it could occur.

If we assume that Connor keeps his word, forever, this appears to be equivalent to saying  that the probability that he eats a broccoli fudge sundae is zero (mathematically, P(B)=0, where B means “Connor does eat the sundae,” and P(B)=0 means “the probability of Connor eating the sundae equals zero).  This, is indeed puzzling, though.  How can even an immortal say with one hundred percent certainty that he will never, never, ever eat such a sundae?  One can imagine (especially if you’re a scriptwriter for Highlander) hypotheticals that might tie the hands of even an immortal and force a change of mind.

I think the issue here is what we mean by “probability”, “can”, “will”, “can’t”, and “won’t”, and how we define probabilities of one and zero.  In short, we’re discussing the interpretation of probability.

I said above that something with P=0 cannot occur, and with P=1 must occur; but I’m not sure that’s quite right.  Some things are by definition impossible–married bachelors or parts bigger than a finite whole, for example.  Such things certainly have a probability of zero, but this seems like a trivial restatement of their impossibility.  Likewise, some things, by definition–e.g. that a bachelor is unwed–must be so and thus have a probability of one.  This, too, seems tautologous.  To assign probabilities to such things seems redundant.

Normally, when we talk of such things, we mean something like this, from the first Wikipedia link above, my emphasis in boldface:

If an event is sure, then it will always happen, and no outcome not in this event can possibly occur. If an event is almost sure, then outcomes not in this event are theoretically possible; however, the probability of such an outcome occurring is smaller than any fixed positive probability, and therefore must be 0. Thus, one cannot definitively say that these outcomes will never occur, but can for most purposes assume this to be true.

The converse would be true about sure not to happen vs. almost sure not to happen.

Thus, to pull out an old chestnut, the molecules of air in this room move randomly.  In principle, every single molecule of air in the room, being in the gaseous state, could all bounce into a single corner of the room at the same time, leaving the rest of the room a vacuum in which I would suffocate.  This does not violate any law of physics, and while it is almost certain not to occur, “almost certain” is not quite the same as “certain”.

However.  We’re talking about an infinite time-scale.  Hypothetically, given an infinite amount of time, anything that can happen in a given universe will happen, sooner or later.  Sooner or later all the air particles in some room somewhere will all simultaneously jump into one corner.  It’s like the cliche of the monkeys that over a sufficiently long time will randomly type a Shakespeare play.  True, the chances of this are small; note the quote from the linked article, my emphasis:

Ignoring punctuation, spacing, and capitalization, a monkey typing letters uniformly at random has a chance of one in 26 of correctly typing the first letter of Hamlet. It has a chance of one in 676 (26 × 26) of typing the first two letters. Because the probability shrinks exponentially, at 20 letters it already has only a chance of one in 2620 = 19,928,148,895,209,409,152,340,197,376 (almost 2 × 1028). In the case of the entire text of Hamlet, the probabilities are so vanishingly small they can barely be conceived in human terms. The text of Hamlet contains approximately 130,000 letters.  Thus there is a probability of one in 3.4 × 10183,946 to get the text right at the first trial. The average number of letters that needs to be typed until the text appears is also 3.4 × 10183,946, or including punctuation, 4.4 × 10360,783.

As the paragraph says, these numbers can “barely be conceived in human terms”.  If you said that this was, in years, a trillion trillion trillion…and then said “trillion” about 2700 more times–that many times the age of the universe–well, that still doesn’t really adequately convey the size.  However, these numbers are insignificant, practically nothing, compared with the vast sweep of infinity.  Given an infinite time, sooner or later the monkey will indeed produce Hamlet, however many universes may be born, live, and die in the interim.

Thus, there seems to be a distinction between things that have a probability of zero because they could not happen on any time scale in any universe (the married bachelor again), and things that have a zero probability, or very close to it, on a finite time scale, but which become inevitable on an infinite time scale (the monkey and Hamlet).  Connor MacLeod’s decision never to eat a broccoli fudge sundae is certainly not in the first category.  There is no logical (as opposed to aesthetic) reasons for it to be impossible or contradictory.  That would seem to land it square in the second category.  However, if this is the case, it seems that no matter how stubborn Connor is, he’ll eventually cave in.  We’ve already said that given unlimited time, anything that is logically possible, that is, possible in principle–anything that can happen, which eating a broccoli fudge sundae certainly is–will happen.  Some unique set of circumstances (perhaps too much Scotch!) which will compel Connor to change his mind will eventually come to pass.

This, however, would seem to contradict the ability of Connor to hold firm; which would imply a lack of free will; which we insist he actually has.  So what gives?  What’s even worse is another factor that seems to call into question Connor’s free will; but eleven hundred words is more than enough for this post, so we’ll look at that next time.

Part of the series “You Pays Your Money and Takes Your Chances: Free Will

Posted on 15/01/2014, in Christianity, metaphysics, philosophy, religion, theology and tagged , , , , , , , , , , , , , , . Bookmark the permalink. 10 Comments.

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