Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

Do the proofs you find in your textbook or in the literature look contrived to you?

A mathematical fact is usually stated as a Theorem together with a proof. The theorem is a mathematical assertion and its proof is a demonstration that the assertion can be derived from simpler mathematical principles. The principles and the demonstration that they imply the theorem must be accepted by a sufficient number of other mathematicians with recognized expertise in the area.

To a newcomer in the area, the proof often does look contrived or magical (and also often beautiful precisely because it appears so magical). But you have to remember that the published proof omits all tribulations the individual went through who came up with the statement and its proof in the first place. The discovery of a theorem usually proceeds in stages. The mathematician has a hunch, makes a guess, finds a counterexample, refines the guess, tries to prove the guess, fails, modifies the guess, finds a counterexample ... the process goes on and on.

When trying to figure out something it is not clear what assumptions are justified, possible, or appropriate. Textbook problems of the type "Show that ..." are unrealistic in that regard. The truth and its proof are not discovered separately, rather their development is inextricably intertwined. The final result is published with a proof as short and slick as possible, and the result looks like magic. However, in reality it is the result of a lot hard work. It looks natural to the individual who conceived it since she turned over every last pebble in the vicinity of this piece of mathematics, but it's bewildering to the student who is new to the area.

If you study that area, after a while the proof will look natural, and soon you'll be able to come up with similar proofs yourself!

Fine print, your comments, more links, Peter Alfeld, PA1UM