Posted by: Alexandre Borovik | February 24, 2009

Elusive boundary of proof

This is one of the stories that came in response to my call for childhood stories.

Alexei Lisitsa:

I read about the Fundamental Theorem of Arithmetic when I was about 10 or 11. I felt very uncomfortable about the proof of the Theorem. The statement seemed so obvious that the proof appeared to be absolutely redundant, not adding anything to understanding and making me worrying that I am missing something. I would be then much happier rather to accept the Theorem as a self-evident axiom.

Alexandre Borovik:

Perhaps, there is indeed an intrinsic danger in proving obvious things. But, at the same age of 10 or 11, would you accept as obvious another, and closely related fact:

If a prime number divides a product of two integers, it divides one of them?

For example, 13 divides 46189 and 46189 = 209 * 221. Hence 13 should divide either 209 or 221. (Indeed, 13 * 17 = 221 and 11 * 19 = 209.)

A more general question: what was approximately the boundary where you started to feel that a proof could be useful? I feel that this elusive boundary is one of the most intriguing things in mathematical education.

Alexei Lisitsa:

I would consider the statement “If a prime number divides a product of two integers, it divides one of them” as equally  obvious  to FTA and  not requiring any  proof.

For me then, these statements were approximately at the same level of transparency as commutativity of multiplication: you don’t need to prove it, you just put coins into a rectangular shape on the desk and you see  that,  indeed,  multiplication  is commutative, it is a “law of nature”. Of course to get such a like  feeling on FTA (or your statement on product of two numbers)  you need more playing with numbers and that was what I was doing  at the time. The proof of  FTA appeared then as an attempt to reduce the obvious thing to the equally obvious things.  For the contrast, approximately at the same time I found the proof of irrationality of square root of 2 very exciting, it was a real proof!

The question about boundaries is a quite delicate one. In a sense I still have this strange feeling about the proof of FTA, but only when I think about it as about  the statement  on “proper”, or “real”, or “just”  natural numbers in a semi-formal setting. In the university, of course I learned on broader algebraic and logical context and it is then when I started to see that the proof of FTA does make a sense. One may have Euclidean and non-Euclidean rings, and FTA may have its explanation via more fundamental concepts. Or, you may fix some axiomatic system and ask whether FTA is derivable, or independent – not that I was thinking about particular status of FTA at the time, but at least I saw the framework where the proof of FTA does make a sense.

Alexandre Borovik:

And what would you say about an intentionally elaborate and beautitiful  proof of a self-evident statement, like a proof of a divisibility lemma in my old post Induction over prime numbers?

Alexei Lisitsa:

Concerning the proof of divisibility lemma by induction over prime numbers, if I saw it when I was a schoolboy I would definitely have the same reaction. The basis of induction is very similar to the statement, so, again, it is a reduction of “obvious” to “obvious”. And I still feel a bit uncomfortable about the proof now, because of some vagueness in the distinction between what is taken for granted and what is to be proven. Ideally, I would like to see an explicit list of principles (axioms) needed to carry out the proof.

Responses

1. I first saw the proof of the fundamental theorem of arithmetic when I was maybe 13, and I had a similar reaction. It wasn’t until several years later when I saw the example of unique factorization failing in Z[i] that I realized why it needed proof. My intuition told me the same thing in both cases, but intuition can be wrong.

2. Alexandre: concerning the proof of divisibility lemma by induction over prime numbers, if I saw it when I was a schoolboy I would definitely have the same reaction. The basis of induction is very similar to the statement, so, again, it is a reduction of “obvious” to “obvious”. And I still feel a bit uncomfortable about the proof now, because of some vagueness in the distinction between what is taken for granted and what is to be proven. Ideally, I would like to see an explicit list of principles (axioms) needed to carry out the proof.

3. Note that UF holds in Z[i], but it doesn’t in similar rings, such as Z[\sqrt{-5}].

4. Yes, that’s what I meant to say. Sleep has not been my friend this week.