Re: TIME - Achilles and the tortoise.

[Jack Uretsky <jlu@HEP.ANL.GOV>]

Hi all-
        I try not to miss opportunities to advertise my book (still in
progress ... but see my sneaky tactic in the October - I believe it's
October - AJP).  So here's some comments based on chapter 1.

On Sat, 30 Aug 2003, Michael Edmiston wrote:

> I read some of the things I was able to find about Lynds' assertions,
> and it doesn't make sense to me.  However, I do not have access to the
> original stuff, so I had to read summarized versions.
>
> I am mostly writing about the Achilles/Tortoise puzzle to see if others
> see this the same way I see it.
>
> 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.  Early in general physics we routinely solve
> this type of problem by finding the equations of motion for Achilles and
> the tortoise, and simple algebraic analysis or graphical analysis of
> these equations of motion show us who reaches the finish line first
> (depending on their respective velocities, the overall distance of the
> race, and the amount of head start given the tortoise).
_____________________________________snip_______________________________
         I agree with Michael's title that the relevant issue has to do
with "time", but I think that the relevant "paradox" is the paradox of the
arrow.  Zeno's argument, as passed on to us by Aristotle, was that an
arrow is at a particular point at each instant of time (OK, nitpickers,
make that a fixed point on an arrow is at ...).  Since it is <somewhere>
at each instant, it is therefore motionless at each and every instant, so,
obviously, if it is motionless at every instant it cannot be moving.
        I believe that a modern construct of this argument, slightly more
sophisticated, might go like this: If I want to talk about the velocity of
an arrow I need to talk about pairs of points and the time it takes to
travel the distance between them.  So if the concept 'velocity' involves
pairs of points, how can I ever talk about the velocity at a single point,
especially in those cases where the velocity keeps changing? Because if
the velocity keeps changing, then it is different at every point.
        The fact that Aristotle - not the world's dumbest citizen despite
the bad press in physics texts - had to struggle with this issue suggests
the appeal the argument must have made to to ancients.  The question I
find interesting is, why?  My proposed solution is that they were missing
two profound notions, the concepts of function and graph. Aristotle was
about 1300 years early for those concepts.  Using the concept "function"
to describe the arrow's flight, it was a relatively easy step - certainly
one taken by Galileo - to realize that velocity could also be
conceptualized as a function and that the two functions, position and
velocity, each had a value at every "point" of time.  The next steps,
relating one to the tangent points of the other came in a rush of a few
decades.
        Any way, that's my take on the issue.
                                Regards,
                                        Jack