Why do we care about multiple proofs of the same theorem?

Question

I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in university. We will be doing a lesson on visual proofs of the Pythagorean theorem.

There are at least three that are easy and fun to do using some paper triangles. They'll be encouraged to vote on their favorite. But these students are bound to ask why do it three ways? Why do it 118 ways?

This post, this post, and others do a fine job discussing what makes two proofs different and how many proofs a theorem can have, but what I want to know is this: why do we care? Is it just a sense of nerdy joy in finding a different argument for the same theorem?

Answer 1

Worth noting here is that there is an additional level of certainty in the validity of a statement when you have multiple proofs of said statement. By having multiple ways to solve the same problem, all yielding the same answer, our surity of the statement increases dramatically. In a classroom setting you could perhaps bring up examples of "proofs" which were thought to be groundbreaking but were later proved wrong by subsequent results; on the other hand, you might also consider giving examples of when a theorem was questioned by the mathematical community but was confirmed by independent proofs, causing the theorem to be more widely accepted.

Originally From Comments:

There are plenty of examples. For the first case, perhaps start with this MathOverflow post. For example, that page lists the false result that

A convergent infinite series of continuous functions is continuous. Cauchy gave a proof of this (1821)...Five years later Abel pointed out that certain Fourier series are counterexamples.

A more trivial example is a tougher question. That same page lists the Euler's Formula $V\text{(ertices)}−E\text{(dges)}+F\text{(aces)}=2$ was long thought to hold for all polyhedra, and a proof was given that was initially accepted for many years until a counterexample was found.

As for the second case, immediate examples that come to mind include Cantor's Set Theory (though subsequent proofs were largely due to Cantor himself, with some strong avocation from Hilbert) as well as Godel's Incompleteness Theorem (see the page Rosser's Trick as well as Boolo's Short Proof ). Neither of these examples are fantastic, but I am sure a little research could definitely yield some better examples!

Answer 2

I'd say different proofs usually employ different techniques, which in turn might be applicable to different sets of other theorems. So the more proofs I know for one theorem, the higher the chances that I'll be able to adapt at least one of them to a similar (or maybe not so similar) theorem I'm trying to prove.

Furthermore, seeing several techniques applied to a single theorem also helps comparing the techniques, both in objective terms (e.g. is it constructive, is it algebraic) and subjective terms (e.g. do I consider the approach intuitive, can I see how the goal dictates the steps to reach it). If each technique were demonstrated using a different theorem, the differences in these theorems might overshadow the differences between techniques.

Answer 3

Claude Monet did a famous series of over 20 paintings, all of one or two haystacks, all in the the same field, with various types of light and weather conditions. Here's one example:

Here is another, for contrast:

A mathematical proof is much like a painting. There is an overall conclusion and many fine details, and one gets a real sense of beauty from nicely made ones.

Answer 4

The main motivation is that different proofs generalize to different situations.

In addition to that

1. Different proofs may require different tools and different theory. Sometimes, it is nice to have an "elementary" proof, which only uses simple tools, but maybe in a difficult or hard-to-understand way. Sometimes, another theory, once developed, might give certain result as an obvious corollary, but to really understand it one has to understand the entire theory.
2. Different proofs often have slightly different assumptions. They might be written in a standard form, but upon inspecting the proof and seeing what assumptions are actually used, some of them might be weaker than what is written down. This is often the case in analysis, where there are several different degrees to which something can be continuous or differentiable.

Neither of these concerns is easy to demonstrate before university-level mathematics, I fear.

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A didactic reason is that different people might understand or remember different proofs, especially if some are visual, others algebraic, and so on. A proof is always a way to reconstruct and remember a theorem.

Answer 5

For me this question brought to mind Bill Thurston's On Proof and Progress in Mathematics. There is value in being able to understand the same idea from different viewpoints (that is, the same person adopting different viewpoints, not just different people choosing their own preferred one).

Answer 6

Sometimes a different proof can provide more insight, or a different kind of insight.

As a relatively elementary example, you could think of the angle sum formula for sine.

The first proof I learned used computed the length of a certain line segment in two ways: using the law of cosines, and using the distance formula. After a lot of algebra, the formula falls out. However, I get very little intuition for why the result is true.

A proof I discovered much later uses the linearity of rotation as a map from $\mathbb{R}^2 \to \mathbb{R}^2$. The angle sum formulas for both sine and cosine are the result of matrix multiplication of rotation matrices. Not only is this proof more intuitive, it is so intuitive that I no longer remember the angle sum formulas: I just take a second to derive them each time.

Take home message: a proof is an explanation for why something works. Sometimes some explanations appeal to you more than others. So it is worth it to prove the same result in many ways.

Answer 7

It is better to solve one problem five different ways, than to solve five problems one way. —George Polya