Okay but if I honestly believe that all natural numbers are interesting and thought of this proof as pretty validly matching my intuitions, what does that mean?
Unless you turn “interesting” into something rigorously defined and precisely communicated to others, what it means is that all natural numbers are {some quality that is not rigorously defined and can’t be precisely communicated to others}.
I guess I feel that even if I haven’t defined “interesting” rigorously, I still have some intuitions for what “interesting” means, large parts of which will be shared by my intended audience.
For example, I could make the empirical prediction that if someone names a number I could talk about it for a bit and then they would agree it was interesting (I mean this as a toy example; I’m not sure I could do this.)
One could then take approximations of these conversations, or even the existence of these conversations, and define interesting* to be “I can say a unique few sentences about historic results surrounding this number and related mathematical factoids.” Which then might be a strong empirical predictor of people claiming something is interesting.
So I feel like there’s something beyond a useless logical fact being expressed by my intuitions here.
Okay but if I honestly believe that all natural numbers are interesting and thought of this proof as pretty validly matching my intuitions, what does that mean?
Unless you turn “interesting” into something rigorously defined and precisely communicated to others, what it means is that all natural numbers are {some quality that is not rigorously defined and can’t be precisely communicated to others}.
I guess I feel that even if I haven’t defined “interesting” rigorously, I still have some intuitions for what “interesting” means, large parts of which will be shared by my intended audience.
For example, I could make the empirical prediction that if someone names a number I could talk about it for a bit and then they would agree it was interesting (I mean this as a toy example; I’m not sure I could do this.)
One could then take approximations of these conversations, or even the existence of these conversations, and define interesting* to be “I can say a unique few sentences about historic results surrounding this number and related mathematical factoids.” Which then might be a strong empirical predictor of people claiming something is interesting.
So I feel like there’s something beyond a useless logical fact being expressed by my intuitions here.