See Wikipedia on the RSA problem and in particular Breaking RSA may not be equivalent to factoring - which actually shows it's a lot less likely than you might think that anyone is going to prove breaking RSA equivalent to integer factorization.
No cryptosystem that can be cracked given unbounded computing power can currently be "proven secure" since we're a long way from showing that any useful problem is computationally hard. The best you can do is show that it's hard given some assumption.
Not wishing to be rude, but what caused such incredible overconfidence that you would say
I believe RSA can only be cracked by prime factorisation with the same certainty that I believe the primes are infinite. The only scenario in which they are false is one in which I am insane.
without personally understanding a proof to that effect? I do personally understand a proof that certain Rabin-based cryptosystems are as hard as integer factorization to break, and I'm still less confident of their theoretical security than I am of the infinitude of primes.
EDITED TO ADD: I should put a massive caveat on my assertions about Rabin before anyone gets the wrong impression: the proof that the cryptosystem is as hard as factoring depend on what is known as the "random oracle model", which is a very useful but unrealistically strong assumption about our hash functions.
The cause of my overconfidence was a combination of exaggeration, arrogance and thinking I had a proof to the effect. As I said above, I had read that the problem was easy and so when I found a 'proof' I assumed it to be correct.
It was, as you point out, a huge error. I will aim never to repeat it.
So, the FBI allegedly arranged for a number of backdoors to be built into the OpenBSD IPSEC stack. I don't really know how credible this claim is, but it sparked a discussion in my office about digital security, and encryption in general. One of my colleagues said something to the effect of it only being a matter of time before they found a way to easily break RSA.
It was at about this moment that time stopped.
I responded with something I thought was quite lucid, but there's only so much lay interest that can be held in a sentence that includes the phrases "fact about all integers" and "solvable in polynomial time". The basic thrust of my argument was that it wasn't something he could just decide an answer to, but I don't think he'll be walking away any the more enlightened.
This got me wondering: do arguments that sit on cast-iron facts (or lack thereof) about number theory feel any different when you're making them, compared to arguments that sit on facts about the world you're just extremely confident about?
If I have a discussion with someone about taxation it has no more consequence than a discussion about cryptography, but the tax discussion feels more urgent. Someone walking around with wonky ideas about fiscal policy seems more distressing than someone walking around with wonky ideas about modular arithmetic. Modular arithmetic can look after itself, but fiscal policy is somehow more vulnerable to bad ideas.
Do your arguments feel different?