A driver leaves home and drives along a road with a railway crossing, and a train will pass the crossing at some time.
analysis/buridan New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
A driver leaves home and drives along a road with a railway crossing, and a train will pass the crossing at some time. Let t0 be the time she leaves home and t = (time she crosses) - (time the train passes). If t is a continuous function of t0, it is negative for early t0 and positive for late t0, so by the intermediate value theorem some t0 gives t = 0 and she is killed on the crossing. Does this "Buridan's Ass" argument show that there is always a nonzero chance of death through indecision at a crossing? [ Editor: problem statement reconstructed from the posting. ]
Show solution
Remember the old story about the donkey ("Buridan's Ass") which was
stuck exactly halfway between two bales of hay, and thus found it
impossible to decide which one to head for? Well, it turns out you
can make this into a real paradox without too much trouble.
The situation is as follows. A person leaves home, and is driving
a car along a road which has a train crossing. There is a train
coming at some stage in the near future.
Let t0 = time of leaving home.
Let t = (time person crosses crossing) - (time train passes crossing).
Consider t as a function of t0. For t0 relatively low, t will be negative
(the person will cross the crossing before the train). For t0 relatively
high, t will be positive (the train will pass the crossing first).
But, assuming that nature is continuous (leaving quantum mechanics aside
for now), t must be a continuous function of t0 (when all other variables
are held constant). By the Mean Value Theorem, there must therefore be some
value of t0 such that t=0. So, for some value of t0, the person must get
killed on the crossing!
This is quite contrary to our intuitions. We tend to think "well, if she
gets there a long time before the train, she'll cross. Otherwise, she'll
give the train the benefit of the doubt and wait." But the above result
implies that there must be some point of indecision, in between, where
she cannot make the decision in sufficient time, and ends up being run over.
(To be even more counter-intuitive...the result still holds even if there is
a gate which comes down 3 minutes before the train!) The point is:
continuous world => continuous brain => continuous results. Of course, the
probability of a crash is very low, but it is non-zero (the train has a
non-zero width). So there is a nonzero probability of death through
indecision _every time_ YOU come to a crossing...if you happen to be at
just the wrong point. It holds whether you are drunk or sober!
So much for our intuitions of brains as reasonably ideal objects. Of
course, there are some counter-arguments, such as arguing from the
presence of other random factors (I don't think these arguments are
compelling: they don't destroy the possibility of death, they just
distribute it more widely. If you like, we can hold all of these
factors constant), or arguing from quantum physics (whether QM implies
that the world is discrete or continuous is still unsettled. Either
way, I would be dissatisfied if the only solution to this paradox was
dependent on the contingent fact of QM).
In fact, there are a reasonable number of deaths every year at crossings.
We tend to attribute these to unusual causes, such as poor warning devices
or drunkenness. This result implies that even in a perfect world, we
should expect some such accidents. Maybe the "paradox" is responsible
for some of these deaths!
-- Dave Chalmers <dave@cogsci.indiana.edu>
[ Editor: the step used is the intermediate value theorem, which the
posting calls the Mean Value Theorem. ]