Bullying the Integers

This is something people with only basic exposure to cryptography and security are confused about a lot.

To them "breaking RSA" and "breaking something using RSA" sound essentially the same.

Systems are only as strong as their weakest part.

Nobody bothers attacking basic cryptographic primitives like AES, or RSA, or SHA-1 in practice. These are standardized, simple, have extremely straightforward interfaces, very well researched, and with enormous level of public exposure. Even theoretical weaknesses found occasionally tend to have little practical relevance.

What gets attacked is everything else. Implementations bugs, protocol corner cases, differences between assumptions and actual requirements, side channels, human factors, and very few people looking at it ever. They are highly complicated, messy, with no clear boundaries, and every system starts essentially from scratch.

When P(system is secure) = P(system is secure|AES is secure) P(AES is secure)

You want to focus all your effort on P(system is secure|AES is secure). Both protection effort, and backdooring effort.

RSA is somewhat between the two. It's somewhat more complex, and requirements for securely using RSA are a lot more complicated than for securely using AES. The entire public key crypto is built on far weaker foundations than private key crypto. There's very little hard number theory involved.

What about Shor's algorithm? Is this not at least in theory a easy way to break RSA. Only the practical barrier of building a quantum computer capable of cracking practical encryption remains.

Shor's algorithm was found/invented after RSA so for close for two decades there was no known theoretical solution. That did not mean that it was not possible however.

Many of the people who thought it was impossible to break RSA in polynomial time were arrogant, they thought they understood the world much better then they actual did. Many of the people who thought that it was obvious that RSA would one day be solved in polynomial time were also arrogant. You are right it is not possible to decide the answer one way or another, nature can not be bullied. It is however often necessary to take actions before you can not for sure one way or another. So your colleague probably internally transformed his best guess in to a 'fact' so he could act on it. Or maybe it transformed from guess in thought to an utterance of fact out of subconsciousness tendency because it sounds more convincing to most of the audiences he/she has talked to.

To my knowledge Shor's algorithm is currently accepted as correct the exact requirements for a practical quantum computer that can solve today's RSA encryption is still being explored.

I wouldn't be excessively surprised if someone found a regularity in a common pseudorandom generator algorithm that could be exploited to narrow down the search for the prime numbers used for RSA keys.

This would fall under "breaking some implementations of RSA" and not "breaking RSA", but is close enough for practical purposes that your colleague might be right (or at least, not be in a seperate cast-iron category of "wrong", especially if you also consider quantum computing arguments).

The basic thrust of my argument was that it wasn't something he could just decide an answer to

If the question was whether prime factorisation was likely to become easy, then probably you'd be justified in saying, essentially, "you don't get to have an opinion!". But since RSA is only an implementation, not a pure essence of mathematics, it might be vulnerable in ways we don't know about yet. It wouldn't be the first time. (Of course your interlocutor might not have intended this interpretation.)

I think this is a good example of a common case, where our certainty concerning ideal objects like mathematics can blind us to the existence of uncertainty in the real world. If someone designs an AI tomorrow and provides a proof of its friendliness, should we implement it?

This question nicely straddles the dichotomy between taxation (which has policy consequences and is somewhat subjective) and modular arithmetic (which has no policy consequences, and "can look after itself").

This seems to be intuitionist.

For practical purposes the rejection of induction seems trivial. For all real problems I will only ever be concerned about a finite number of numbers. I can prove the numbers in the particular problem I am working on are all fitting whatever criteria I need them to fit without relying on induction. I only need induction if I want to think that infininity means something beyond just Really A Whole Lot, or Really Really Big.