Wednesday, April 29, 2009

Wednesday Math, Vol. 69: Multiple Definitions

In the 20th Century, there was a major push in mathematics for precise definition. Ever since the Greeks were working on geometry, way back before the birth of Christ, math has used language more precisely than most other disciplines, but over the past one hundred years, there was an emphasis on getting rid of any ambiguities that might exist in understanding of what a particular mathematical object was and the best way to define it, which in math usually means the most concise.


Consider, for example, a parallelogram. You might remember this definition from when you took geometry.

1. Parallel side definition: A parallelogram ABCD is a four sided polygon where AB is parallel to CD and AD is parallel to BC.

This definition is most common because it uses the idea of parallel sides, which of course is where the name parallelogram comes from.

There are actually three more definitions as well.

2. Equal side length definition: A parallelogram ABCD is a four sided polygon where the length of AB is equal to the length of CD and the length of AD is equal to the length of BC.

3. Equal opposite angle definition: A parallelogram ABCD is a four sided polygon where the measures of angle A is equal to the measure of angle C and the measure of angle B is equal to the measure of angle D.

4. Supplementary adjacent angle definition: A parallelogram ABCD is a four sided polygon where the sum of the measure of any two adjacent angles is 180 degrees, which is shortened in math to say adjacent angles are supplementary.

The thing about each of these definitions is that if you take any one as the given statement, you can prove the other three are also true. As far as geometric proofs go, they aren't that hard. The facts to use are:

1) The interior angles of an four sided polygon must add up to 360 degrees
2) The angles created by two parallel lines and a line that traverses both lines create eight angles with two possible measures, either x degrees or (180-x) degrees, and
3) a diagonal from one corner to the opposite creates two triangles, and if the four sided object is a parallelogram, those triangles are congruent.

20th Century precision would change these definitions very little, only making sure that the four points involved and the four line segments that connect them have to lie on a single plane. 20th Century conciseness would stipulate that the statement of the definition is exactly one of the four, since when any one is stated, the others can be proven directly.

Precision in mathematical definition is the discipline's great strength. Even though there are four definitions of a parallelogram given here, you would never see mathematicians argue about whether an object is a parallelogram or not depending on which definition was used.

The study of law prides itself on rigorous definition, but compared to mathematics, it is still ambiguous to a ridiculous level. We can see this in the arguments currently raging about what constitutes torture.

1 comment:

Mathman6293 said...

I haven't been around much lately because I have been teaching my students about parallelograms.