Today, the forgotten lecture in your math course. The
University of Houston's College of Engineering presents
this series about the machines that make our civilization
run, and the people whose ingenuity created them.
What engineer doesn't remember
dozing off in math class when the professor took up
existence and uniqueness? Those words have
a special meaning in math. Not all problems have
solutions, and, if they do, there might be more than one.
A problem with no solution isn't much use. And
more than one solution puts us in a quandary.
Engineering students grow impatient with those issues
because equations that reach their classrooms almost
always do have unique solutions. Suppose, for example, a
fast-moving car hits a turn. It either rolls over or gets
around the curve safely. It doesn't do both. We'd better
get just one solution when we solve the problem. It's
cold comfort if our math tells us it will both roll over
and get around safely. We need to know which will
happen.
Try another kind of accident. I have a long slender glass
rod -- say a yard long and an eighth of an inch in
diameter. I stand it, on end, on the floor -- then begin
pressing downward on the top. Nothing happens, so I press
harder. I finally reach a point where the rod suddenly
bows outward and breaks. If I set up the math correctly,
it'll tell me how much load the rod will take.
But that solution is not unique. There's another
solution. It has the rod bending into an S-shape instead
of simply bowing outward. More solutions also exist: a
triple bow, a quadruple bow, and so on. Since the fancier
failure modes take a lot more force, the rod usually
collapses by simply bowing outward. We trigger that
simple failure mode as we increase the load slowly. But
suppose we add a greater load all at once. Then those
fancy collapses actually can occur.
If you hold the glass rod by its end, high above a
concrete floor, then drop it, the load is applied so
suddenly that the rod will collapse in one of those wavy
shapes. I once did that. The glass broke into six pieces,
all the same length. The rod had been wavy as a snake
when it collapsed under an enormous shock loading.
That taught me a fine lesson about the limitations of my
work. Louis Pasteur once remarked that "Chance favors only the prepared mind."
If I can paraphrase that, I'll
say that rod taught me to expect surprise. Even
when we've polished and honed our means of prediction,
outcomes might still not be unique. Nature has all kinds
of alternatives up her sleeve, and we'd better be ready
for them.
Swiss mathematician Leonhard Euler solved the collapsing
rod problem in the eighteenth century. Determinism then
appeared to be the way of the world. Now the twentieth
century has given us first quantum indeterminacy and then
chaos theory. We struggle to create new descriptions of a
reality that's not so straightforward. We once could yawn
when math teachers began talking about existence and
uniqueness. Now our ears prick up. For we no longer live
in that simple world where only one outcome is thinkable.
I'm John Lienhard, at the University of Houston, where
we're interested in the way inventive minds work.
(Theme music)
For some discussion of mathematical existence and
uniqueness, see, e.g., Sneddon, I. N., Elements of
Partial Differential Equations. McGraw-Hill Book
Company, Inc., 1957, pp. 9 and 48.
For a very good account of the Euler column analysis,
see: Pippard, A.J.S., The Analysis of Engineering
Structures. London: Arnold, 1957; or Popov, E. P.,
Introduction to Mechanics of Solids. Englewood
Cliffs, NJ: Prentice-Hall, Inc., 1869.
Three modes of failure of a column. (The force needed
to cause a collapse rises as the square of n.)
The Engines of Our Ingenuity is Copyright
© 1988-2001 by John H. Lienhard.
