I think that would count as a straightforward belief on Rolf’s view, because you can test it empirically. We have to qualify that by saying you ‘can in principle’, since testing for that (especially a chocolate cake) would be impractical, but that’s no problem for the meaningfulness criterion.
Which of course invites the question of whether I believe I can test it empirically.
Alternatively, one could ask whether I believe that a piece of chocolate cake didn’t spontaneously materialize in the asteroid belt at 2:15pm EST on April 11 2012 and float there for fifteen minutes before dematerializing… unless we want to say that the fact that I could have tested it last year counts as “can test it in principle.” Which I suppose is no sillier than anything else.
Regardless, I’m more interested in the second question. On Rolf’s view, why do (e.g.) my opinions on the usefulness of the definitions which imply the Fundamental Theorem of Algebra feel so much like beliefs about the Fundamental Theorem of Algebra?
Because the intuitions on which those opinions are based, actually are beliefs: They arise from your experience with subdividing things. The fifth axiom of Euclid’s geometry may be a more helpful example here; it is the one which states that given a line and a point not on it, exactly one line may be drawn through the point which never meets the first one. By denying this axiom we get Riemannian or hyperbolic geometry. Now, of the three versions of the axiom, which one does it feel like you believe? I rather strongly suspect it is the Euclidean one; there’s a reason it took two thousand years to formalise the other kinds of geometry. Yet they all describe the behaviour of actual straight lines in different circumstances.
On Rolf’s view, why do (e.g.) my opinions on the usefulness of the definitions which imply the Fundamental Theorem of Algebra feel so much like beliefs about the Fundamental Theorem of Algebra?
Because maintaining all these levels of indirection is hard.
Do I similarly not believe that there isn’t a piece of chocolate cake floating in the asteroid belt?
Cuz it sure feels like I believe there isn’t a piece of chocolate cake floating in the asteroid belt.
And if I don’t believe that proposition, what is the thing I’m doing to it that feels so much like belief, and why does it feel so much like belief?
I think that would count as a straightforward belief on Rolf’s view, because you can test it empirically. We have to qualify that by saying you ‘can in principle’, since testing for that (especially a chocolate cake) would be impractical, but that’s no problem for the meaningfulness criterion.
Which of course invites the question of whether I believe I can test it empirically.
Alternatively, one could ask whether I believe that a piece of chocolate cake didn’t spontaneously materialize in the asteroid belt at 2:15pm EST on April 11 2012 and float there for fifteen minutes before dematerializing… unless we want to say that the fact that I could have tested it last year counts as “can test it in principle.” Which I suppose is no sillier than anything else.
Regardless, I’m more interested in the second question. On Rolf’s view, why do (e.g.) my opinions on the usefulness of the definitions which imply the Fundamental Theorem of Algebra feel so much like beliefs about the Fundamental Theorem of Algebra?
Because the intuitions on which those opinions are based, actually are beliefs: They arise from your experience with subdividing things. The fifth axiom of Euclid’s geometry may be a more helpful example here; it is the one which states that given a line and a point not on it, exactly one line may be drawn through the point which never meets the first one. By denying this axiom we get Riemannian or hyperbolic geometry. Now, of the three versions of the axiom, which one does it feel like you believe? I rather strongly suspect it is the Euclidean one; there’s a reason it took two thousand years to formalise the other kinds of geometry. Yet they all describe the behaviour of actual straight lines in different circumstances.
(nods) That’s an excellent answer. Thank you.
Because maintaining all these levels of indirection is hard.