March 21, 2002
What is the main subjective difference between mathematics as taught in school and experienced by non-mathematicians, and mathematics as practiced by mathematicians?
It is the use of proofs. I claim that common sense includes rigorous logical reasoning, and that it is possible for anyone of ordinary intelligence to follow and appreciate real proofs, if they are presented properly. If this were taught consistently at an early age, many more people would learn to like mathematics!
Of course, the right definition of proof to use here is "a completely convincing argument," rather than the definition learned in school geometry courses involving axioms and rules of inference. The proofs I am talking about are not formalized like that, but they are still real proofs.
The ideal proof for a non-mathematician should have the following features:
1) There should be no prerequisites (the subject matter is something she already knows about and the proof shouldn't need any definitions or depend on any known results).
2) The conclusion should not be obvious.
3) The argument should give insight into why the theorem is true.
It's OK if the proof is long or tricky as long as you "tell the story" properly and there are beautiful insights along the way.
Here are some of my favorite examples (what are yours?). The proofs are intended to be demonstrated by a person, not read from a page.
PROOFS IN GEOMETRY:
G1) The square on the hypotenuse of a right triangle is the sum of the squares on the two sides. (You do have to define some terms here, but the proof is a pretty picture:
G2) Every polygon can be cut into pieces which can be pasted together to make a square. Here is a verbal version of the proof, though it works better with pictures:
1) Draw diagonals to break the polygon into triangles.
2) Cut each triangle along the altitude to the longest side and along the perpendicular bisector of this altitude. Then rotate the two pieces that don't touch the long side around to make a rectangle.
3) Show how any rectangle can be transformed into one with a base up to twice as wide by drawing a line from the upper left corner to a point on the lower half of the right side, sliding the top piece down and to the right until the lowest point is level with the base, and moving the triangular peak left on top to fill the hole at the bottom.
4) Adjust all the rectangles repeatedly as above, until they are as wide as the widest one; then stack them all on top of each other to make a new rectangle.
5) Turn the new rectangle so it is standing on its shorter side, and adjust it repeatedly as above until it is a square.
PROOFS IN ARITHEMETIC:
A1) There is no largest prime number (usual proof).
A2) No integer square is twice another square. (Do the usual proof, then apply it to prove that the length of the diagonal of a unit square is not a fraction.)
A3) Every integer factors uniquely into primes. (This can be presented effectively to 8th graders, but it will need a whole class period because it is subtle; it is important to begin by showing that it's not obvious, using an example like 91 x 187 = 119 x 143).
PROOFS IN TOPOLOGY:
T1) Knots don't cancel--if two real knots are tied successively in a piece of string and the ends are attached, the string can't be unknotted. (There is an amazing visual proof, recommended only if you have prepared a lot of pictures.)
But my all-time favorite "theorem for everyone" is one in combinatorics:
C1) The Marriage Theorem: Given a collection of men and women, each of whom has a preference ordering for the members of the opposite sex, there is a stable way to pair them off. This means that there will be no "elopements"--no possible man-woman pair who prefer each other to their current partners.
See the new challenge below.
OLD CHALLENGE. I have noticed that it takes me longer to run around a loop near my house in one direction than the other; one direction has a long slow rise and a steep downhill, where the reverse direction has a long slow downslope and a steep climb. Which direction is faster?
ANSWER. The winning response comes from Joe DeVincentis: "I'd guess that the direction with the steep downhill is faster, because you probably compensate automatically for a very gentle slope in order to maintain a steady running speed, while the steep climb in one direction slows you down considerably in one direction."
Another way of putting this is that gentle uphills and downhills are in the range where you can still run at an efficient pace; while you waste energy climbing a steep hill (due either to slowing down and making more strides, or to expending energy at an unsustainable rate) but can make use of the gravitational energy on a downhill as long as it is not too steep to run down.
NEW CHALLENGE. Prove the Marriage Theorem (C1 above) in a way anyone can understand. Extra credit for a proof which treats men and women symmetrically, or for an example showing that an endless cycle of elopements is possible if the matchmaking is bad enough (or for a proof that no endless cycle is possible).
Copyright 2002, Frank Morgan.
Send answers, comments, and new questions by email to Frank.Morgan@williams.edu,and JoeShipman@aol.com, to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.