If you are certain that 100+98=198, but not that 99+99=198, some unfortunate accident has befallen the rule that if (A+1)+B=C, A+(B+1)=C.
The “unfortunate accident” may be that you simply didn’t have enough time to apply the rule. If you have limited time, you have to stop somewhere. There doesn’t have to be anything magical between steps N and N+1, it could merely mean that the timer went off at step N.
Just to expand, this example looks deceptively suspicious to humans because the proof is so obvious, even to us, and can be done in a single line using that one rule. For simple arithmetic like this, a robot would be able to prove it immediately after being asked whether 99+99=198 with hardly any time or effort. The procedure is so simple that, in practice, it would be efficient to delay generating the proof until it’s asked for. Many human readers will mentally prove it while reading the sentence without even noticing that it took any work to do so. It’s so fast that there’s no need to guess the answer probabilistically beforehand.
So I’m not sure this example is good for an intuitive illustration that “some unfortunate accident” is going on. It’s very strange to run out of time in a procedure that takes so little time to complete, so it’s hard to reason about the appropriateness of guessing at circumstance.
Maybe a better example would be, having already proved Pythagoras’ theorem, to prove that the body diagonal of a unit cube has length sqrt(3). This might take most humans a little while to think about, and if someone ran a stopwatch and said you have five seconds to decide whether it’s true or not, the idea of guessing P=0.5 for something you don’t have time to prove might be easier for people to understand. =)
The “unfortunate accident” may be that you simply didn’t have enough time to apply the rule. If you have limited time, you have to stop somewhere. There doesn’t have to be anything magical between steps N and N+1, it could merely mean that the timer went off at step N.
Just to expand, this example looks deceptively suspicious to humans because the proof is so obvious, even to us, and can be done in a single line using that one rule. For simple arithmetic like this, a robot would be able to prove it immediately after being asked whether 99+99=198 with hardly any time or effort. The procedure is so simple that, in practice, it would be efficient to delay generating the proof until it’s asked for. Many human readers will mentally prove it while reading the sentence without even noticing that it took any work to do so. It’s so fast that there’s no need to guess the answer probabilistically beforehand.
So I’m not sure this example is good for an intuitive illustration that “some unfortunate accident” is going on. It’s very strange to run out of time in a procedure that takes so little time to complete, so it’s hard to reason about the appropriateness of guessing at circumstance.
Maybe a better example would be, having already proved Pythagoras’ theorem, to prove that the body diagonal of a unit cube has length sqrt(3). This might take most humans a little while to think about, and if someone ran a stopwatch and said you have five seconds to decide whether it’s true or not, the idea of guessing P=0.5 for something you don’t have time to prove might be easier for people to understand. =)
Yup, that’s the idea :)