Using Emmy Noether's Ideas to Prove All Numbers Interesting

A great article about Emmy Noether recently appeared in the New York Times. My colleague Jon Rosenshine noticed it too. We wondered whether her ideas were accessible to eighth and ninth graders. So I did a little research and discovered that Noether is responsible for Noetherian induction, which is a proof technique used in algebra. Though it is a very powerful technique in higher mathematics, in it's most simple form it can be used to prove facts encountered in middle and high school. For example, one can use Noetherian induction to prove that every positive number greater than 1 is either prime, or can be decomposed as a product of prime numbers.

To illustrate how Noetherian induction works, let's instead prove the following pseudo-theorem:

All whole numbers are interesting.

This is a rather odd statement. But let's see if it makes any sense. Start with 0. Certainly 0 is an interesting number. It represents the absence of something. No other number has this property which shows that it is interesting. 1 is also quite interesting for many reasons. For example, it is the only number that cannot be expressed as a sum of smaller positive numbers, and also it is the only number that when multiplying other numbers does not change them (e.g. 1*3 = 3, 1*9=9, and so on). 2 is also very interesting. It is the smallest prime number, though other reasons make it interesting too. 3 is interesting as it is the smallest odd prime number. 4 is interesting, as it is the only positive number that is the product and sum of the same number ( 2+2 = 2*2 = 4 ). I could go on but proving individually that every number is interesting will make this blog infinitely long, and that's a problem. So instead, let's use the Noether's powerful induction technique to prove it.

The proof goes as follows:

Consider the set of all non-interesting numbers, and call that set S. Noether's induction principle, applied to whole numbers says that any non-empty set of whole numbers must contain a smallest element. If S were non-empty, apply Noether’s induction principle to find a minimal element, call it u (u for un-interesting). But don’t you think that u -the smallest un-interesting number is pretty darn interesting? After all, it is the smallest amongst all un-interesting numbers, which makes it quite unique, and therefore interesting. Thus we have arrived at a contradiction since we have found a number that is both interesting and un-interesting and our original supposition that S is non-empty must be false. I.e., all numbers must be interesting.

What makes this is a pseudo-proof is that there is no precise notion of what interesting means and the proof is appealing to the reader’s emotion, but this basic proof method can be used to give real proofs of real theorems such as the prime number theorem mentioned above (that theorem is called the fundamental theorem of arithmetic).