Number Theory: Why always elementary proofs?

[James MC]

I find that in number theory, number theorists (and mathematicians more generally) generally prefer elementary proofs over any other kind of proof. Am I right about this? If so, why is this? Is this something to do with the content of number theory itself? Thanks!

 

[mathman]

I don't think the preference applies only to number theory. Simpler proofs are easier to understand and offer better insight into what is going on.

 

[jgens]

Simpler proofs are easier to understand and offer better insight into what is going on.

I am not under the impression that "elementary" means simpler here. The prime number theorem is usually taken as an example of this phenomenon. Elementary proofs (due to Erdos) exist, but the proofs using complex analysis are considerably simpler and easier to understand.

 

[James MC]

I am not under the impression that "elementary" means simpler here. The prime number theorem is usually taken as an example of this phenomenon. Elementary proofs (due to Erdos) exist, but the proofs using complex analysis are considerably simpler and easier to understand.

That's correct. Indeed the fact that proofs using complex analysis are simpler and easier to understand makes it even more curious as to why number theorists (as far as I can tell) prefer elementary proofs (i.e. proofs based on the Peano axioms). That's why I wonder whether it has something to do with the content of number theory itself?

 

[Therodre]

Au contraire!
I'd say that, generally speaking, number theorist as well as algebraic geometer (and... come to think of it, almost all mathematicians) prefer conceptual proofs, that give a good understanding of the situation, rather than a clever trick, which can be nice of course, but sometimes hides the deep meaning of a situation.
To illustrate this let me quote silverman, from his "arithmetic of Elliptic curves".
"It has been the author's experience that "elementary" proofs (...) tend to be quite uninstructive.(...)
But little understanding come from such a procedure".
I could of course also quote Grothendieck, and the famous "rising tide" philosophy.

Although, there is something of a challenge, in finding an elementary proof of a difficult theorem. And number theorists may like that (as much as the next mathematician), but this is just the icing on the cake. A conceptual and comprehensive proof is always favored.

 

[Mandelbroth]

I find that in number theory, number theorists (and mathematicians more generally) generally prefer elementary proofs over any other kind of proof. Am I right about this? If so, why is this? Is this something to do with the content of number theory itself? Thanks!

Two main reasons.

First, if you have an elementary proof, it's easier to understand to a wider number of people and it's more aesthetically pleasing. Plus, you get to do less work. Everyone wins.

Second?

"Fix an acapuchamahta in the Finklestein sipplidoodle to satisfy Wingledingle's Dingbat Lemma. Then, we can proceed by the Galloping Rocinante Postulate, allowing for the pancatootle to be silly. Thus, the pancatootle's silliness implies the existence of a Grand Poobah winklidinkle, completing the proof that any prime integer is prime."

If you can understand any of that, I think you should consult a doctor about a very serious disorder known as "bat**** crazy." Long, intricate proofs are similar. No one will want to read your proof! (Unless the result is important.)

Edit: To clarify, specialized proofs can be helpful. But, if you're attempting to understand something, you probably don't want to learn an entirely new subject.

 

[jgens]

First, if you have an elementary proof, it's easier to understand to a wider number of people and it's more aesthetically pleasing.

As mentioned earlier in this very thread, elementary proof has a very specific meaning here that has little to do with simplicity. Again appealing to the standard example, there are elementary proofs of the prime number theorem, but none of them are as simple nor understandable (IMHO) than the ones utilizing complex analysis.

Long, intricate proofs are similar. No one will want to read your proof! (Unless the result is important.)

The elementary proofs in number theory are often the long intricate ones that are almost impossible to motivate.

 

[Mandelbroth]

As mentioned earlier in this very thread, elementary proof has a very specific meaning here that has little to do with simplicity.

I didn't say elementary proofs were simple (in humor, however, there seem to be only two ideals on the subject matter here ). I was attempting to point out that you don't have to learn as many things to understand an elementary proof.

I once read an introduction to mathematical writing (I don't remember where) that gave two basic rules for mathematical writing.

1. Be kind to the reader.
2. Be kind to the editor.

I think it violates rule number one to throw in a large number of ideas that the reader wouldn't know. That's what I meant by "easier to understand to a wider number of people." I would argue that it's aesthetically pleasing for the same reason a portrait made by a computer isn't as valuable as one made by finger painting (assuming that both produce the same image). Extending the metaphor, the caveat is that it takes a lot more work when we are finger painting.

Edit:

The elementary proofs in number theory are often the long intricate ones that are almost impossible to motivate.

I will concede this point. I was thinking of a particularly gruesome proof of the Fundamental Theorem of Algebra and not in this context. Most number theory proofs using only elementary techniques are long.

 

[jgens]

I was attempting to point out that you don't have to learn as many things to understand an elementary proof.

This is arguable. For example I would wager that you (along with most of the lay-public and even many practicing mathematicians) are likely unfamiliar with the asymptotic estimates used to provide an elementary proof of the prime number theorem. More than that most are probably unfamiliar with the specific techniques used to derive these estimates. On the other hand, the complex analysis needed for the proof is something every undergraduate knows, and while the average layperson clearly lacks this knowledge, they also lack the background knowledge needed for the elementary proof!

I think it violates rule number one to throw in a large number of ideas that the reader wouldn't know.

Again I would argue more people are familiar with the techniques from complex analysis than are familiar with the "elementary" methods used in these proofs.