On 08/30/2003 11:26 AM, Michael Edmiston criticized
Zeno's "paradox", saying:
> I would say the tortoise won by distraction.
I agree with the sentiment ... but even that is giving
too much credit to the losers. The tortoise didn't win.
Achilles wasn't distracted. Only Zeno was distracted.
I would have said that this "paradox" merely illustrates
the maxim
No matter what you are doing,
you can always do it wrong.
In this case, at the primary level of analysis, we are
required to answer the yes/no question, will Achilles
overtake the tortoise? For extra credit, we can calculate
the point at which A will overtake T.
At the meta-analysis level, we may ask what are the
better or worse algorithms for carrying out the primary
analysis. Here goes:
-- There exists an efficient algebraic algorithm: one
linear equation in one unknown.
-- There exists a less-efficient algorithm that calculates
the standings of the racers at 23 irrelevant intermediate
points.
-- There exists an even-less-efficient algorithm that
calculates the standings at 137 irrelevant intermediate
points.
-- Zeno takes the booby prize by discovering an infinitely
inefficient algorithm that calculates the standings at
an infinite number of irrelevant intermediate points.
As a point of elementary logic: The existence of
inefficient algorithms does not preclude the existence
of efficient algorithms. This is where Zeno's "logic"
went spectacularly wrong.
> I have periodically argued with local math professors that we do not
> need an infinite series to "solve the puzzle" because there isn't any
> puzzle in the first place.
Indeed. It is disappointing that the topic could
even come up.
> Therefore it amazes me there are many web sites that claim the
> Achilles/Tortoise puzzle continues to baffle scientists and
> mathematicians.
I'm amazed and appalled.
> And it bugs me that math professors are teaching students that an
> infinite series is the way to solve the puzzle, when simple algebra
> and the equations of motion clearly describe the race and the
> results.
Indeed.
> Am I missing something here?
No.
=======================
As another example that illustrates the same point:
I once read a math book that advocated calculating the
inverse of a matrix using Cramer's rule. Fortunately
I mentioned this to a real mathematician (one with his
head screwed on properly) and he disabused me before
any harm was done.
Compared to Gaussian elimination, Cramer's rule is
spectacularly bad. (It's not infinitely bad [as Zeno's
algorithm is], but pretty bad. It is more laborious
and much less accurate.)
In computer science, there is an entire subfield devoted
to analyzing the efficiency of algorithms. By way of
illustration, consider the sorting algorithms listed at http://www.wikipedia.org/wiki/Sort_algorithm
Computer scientists a sense of humor about this, as
exemplified by bogo-sort, described at http://www.catb.org/jargon/html/B/bogo-sort.html
... although I'm not sure that bogosort is the
"archetypical perversely awful algorithm"; Zeno's
algorithm is incomparably more perverse.
(Perhaps the contest should be restricted to algorithms
that get the right answer eventually, in which case
Zeno's algorithm is so perverse that it is disqualified
from the list of perverse algorithms.)
==============
You can assign just-for-fun homework to your class:
make up your own "hall of shame" of awful algorithms:
Zeno's algorithm
Bogo-sort
Cramer's rule
... how many more can you come up with?
but don't spend too much time on it; time is much
better spent understanding good algorithms.
================================
As a tangent that can be pursued a very long way:
At the opposite end of the spectrum from Zeno (who
attacked a trivial problem) we can consider questions
that are answerable but for which no efficient
algorithm exists. Some problems are easy, some
problems are hard (nonlinear optimization), some
are provably uncomputable (Turing's halting problem),
and some are provably undecidable (Gödel).