“Well-Defined”: More Examples

In this recent post, I explained the metamathematical concept of well-definition, and I made the claim that this notion is pervasive in the organization of mathematical and nonmathematical ideas.  I gave a few examples, but barely scratched the surface.  So a few more won’t hurt.

1. The slope of a line.
The slope of a (nonvertical) line is defined to be $m = \frac{y_2-y_1}{x_2-x_1}$, where $(x_1,y_1), (x_2,y_2)$ are two distinct points on the line.  This might seem to depend on the choice of the points used to compute, or perhaps on the order in which they are taken, but in fact it does not.  This is well-defined.

Indeed we can turn this idea around to give us another way of getting at the definition of a line.  The idea is that if you have any set of (at least two) points in the plane, the quantity $m = \frac{y_2-y_1}{x_2-x_1}$ is well-defined if all the points on a line, and ill-defined if they don’t.  So we can define the concept of slope of an object as above, with the understanding that it doesn’t always make sense.  Then lines are precisely the objects that actually have a well-defined slope.  We know that slope is a fundamental property of a line, but here we have another way of getting at how fundamental at is: one way to define line-ness is by having a slope!

Note: the above paragraphs are officially about nonvertical lines, if we handle slope in the traditional numerical way.  See this post for a way to put all lines in all directions on the same footing.

2. Power of a point.
This is one of those basic facts that no one seems to learn in high school geometry, more’s the pity.

Let $O$ be any circle of any size, and let $P$ be a point in the same plane; it doesn’t matter whether $P$ is inside, outside, or on the circle.  Draw a line through $P$ which intersects the circle.  (If $P$ is on or inside the circle, this could be any line through $P$ whatsoever, while if the point is outside, then some lines would miss the circle entirely; don’t choose one of these.)  If $A,B$ are the intersections of the circle with the line (where $P=A$ if $P$ is on the circle and $A=B$ if the line is tangent to the circle), then we say the power of point $P$ is the product of the distances $PA$ and $PB$.

Now this really doesn’t look well defined.  The definition superficially depends on the choice of the line through $P$, and unless the point happens to be the center of the circle, changing the line can increase and decrease $PA$ and $PB$ arbitrarily (within a certain range).  But by some kind of geometric miracle (actually by similar triangles), it is well-defined.  Moving the line to make one intersection point further away will always make the other intersection point closer, just “closer enough” to exactly compensate for the increase in the other factor.

(This definition is actually a little bit off.  For technical reasons, for points inside the circle we say the power is the negative of the product of the lengths above.  Lots more information available on wikipedia.)

3. Conflating words and things, again.
I already gave the example of how properties of words are usually not well-defined properties of the objects they refer to.  The grammatical gender example which I chose is alien to our English language, but of course there are lots of other properties of words.  If I draw two pictures and ask whether they rhyme, that doesn’t actually make sense. Depends on the choice of label. Cat and hat rhyme, sure, but calico and cap don’t.

And we can’t save this situation the same way we saved the situation for numerators and denominators.  There we eliminated the dependency on choice by identifying a canonical choice, the lowest terms representation of the fraction.  But when an object has many representations, having a single canonical one is an unusual thing, rather special to the fraction situation.  I don’t believe there is a canonical word that describes each object.

Questions like this appear all the time on small children’s school worksheets!  Ill-defined questions like this bothered me as a child, and they still do.  When I was asked, as a child, to “circle all the pictures that start with G” (or whatever), I would make it a personal challenge to find an interpretation where I could circle all the pictures.