Artificial vs naturalness

Question

I don't know if it is appropiate to ask this, but I'm a little bit confused when mathematicians say a problem is artificial or has artificial methods. That one would prefer what is natural. I quite can´t see the difference. Perhaps you can help by an example.

Answer 1

This is an interesting question, but since neither "natural" nor "artificial" have a mathematical definition, the meaning of these notions is somewhat opinion-based. Nevertheless, let me try to give some examples to illustrate these notions.

The missing brick. Suppose you have a well established mathematical theory but that some conjecture would make the theory even nicer if it were true, this conjecture is usually considered a natural question. Famous examples include the missing cases of the Generalized Poincaré conjecture and of the Burnside problems. Generalizations also fit into this category. If you have a result on fields of characteristic $0$, asking whether a similar result holds for any characteristic would likely be considered a natural question.

The appealing theory. If you happen to have a convincing path for future research in mathematics, then the problems you encounter along this path are usually promoted to the rank of natural problems. A prominent example is Langlands program.

Now, what about artificial problems? I know this will shock a lot of people, but a very famous example is Fermat's Last Theorem. This was at least Gauss' opinion, who wrote in 1816:

I confess that Fermat's Last Theorem, as an isolated proposition, has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.

See also this comment from Glenn H. Stevens in the Scientific American

The other problem is that Fermat's claim has always felt, well, marginal. It is hard to connect the Last Theorem to other parts of mathematics (...)

As you can see, being isolated, not connected to other results, seems to be the characteristic of artificial questions.

P.S. To avoid a heart attack to FLT's fans, let me explain why FLT is nevertheless considered a major result in mathematics: if the problem itself was isolated, its solutions were not. Quoting again Glenn H. Stevens

In fact, if one looks at the history of the theorem, one sees that the biggest advances in working toward a proof have arisen when some connection to other mathematics was found.