I suspect the claim “All beliefs are experimentally testable” is either vacuous or false. Our evidence for most of mathematics is deductive, not empirical. But it would be very strange to say that I don’t have beliefs with substantive content about, say, the the Fundamental Theorem of Algebra.
You might say that mathematical investigation is a kind of experiment—but in that case one wonders what causes for a belief aren’t experiment. Is any evidence whatsoever ‘experiment’?
[I]t would be very strange to say that I don’t have beliefs with substantive content about, say, the the Fundamental Theorem of Algebra.
I am, nonetheless, willing to bite this bullet. You do not have beliefs about the FTA; you have opinions on the usefulness of the definitions which imply it. Moreover, your phrase “deductive evidence” is an oxymoron. Deduction is not evidence, it is tracing the consequences of definitions; definitions are not beliefs. All theorems are in some sense tautologies, that is, they are inherent in the axioms. So a “belief about” a theorem is actually a “belief about” the axioms, and this is precisely what I’d like to forbid.
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.
You do not have beliefs about the FTA; you have opinions on the usefulness of the definitions which imply it.
This is false as a psychological description of my personal state of mind. I don’t know the precise definitions that entail the FTA and I certainly don’t know a proof. (In particular, I don’t think I could give you a correct construction or definition for the real numbers.) I believe in the theorem because I’ve seen it asserted in trustworthy reference works. Somebody somewhere might have beliefs about the theorem that were tied to their beliefs in the definitions, but this doesn’t describe me. I can believe the [deductive] consequences of a claim without knowing the definitions or being able to reproduce the deduction.
Here’s a related example with a larger bullet for you to chew on. Suppose I have a (small) computer program that takes arbitrary-sized inputs. I might believe that will work correctly on all possible inputs. Is that a belief or not? It can be made as rigorously provably correct as the FTA.
When I say “the program is correct”, I am not saying “it is useful to construe the C language and the program code in such a way that...”. I’m making an assertion about how the program would behave under all possible inputs.
Beliefs about computer programs might feel more empirical than beliefs about theorems, but they are logically equivalent, so either both or neither are beliefs, it seems.
Please observe that one of the possible inputs to your computer is “A cosmic ray flips a bit and turns JMP into NOP, causing data to be executed as though it were code”. In other words, your proof of correctness relies on assumptions about what happens in the physical computer. Those assumptions are testable beliefs, just like the intuitions that go into geometry or the FTA.
I suspect the claim “All beliefs are experimentally testable” is either vacuous or false. Our evidence for most of mathematics is deductive, not empirical. But it would be very strange to say that I don’t have beliefs with substantive content about, say, the the Fundamental Theorem of Algebra.
You might say that mathematical investigation is a kind of experiment—but in that case one wonders what causes for a belief aren’t experiment. Is any evidence whatsoever ‘experiment’?
I am, nonetheless, willing to bite this bullet. You do not have beliefs about the FTA; you have opinions on the usefulness of the definitions which imply it. Moreover, your phrase “deductive evidence” is an oxymoron. Deduction is not evidence, it is tracing the consequences of definitions; definitions are not beliefs. All theorems are in some sense tautologies, that is, they are inherent in the axioms. So a “belief about” a theorem is actually a “belief about” the axioms, and this is precisely what I’d like to forbid.
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.
This is false as a psychological description of my personal state of mind. I don’t know the precise definitions that entail the FTA and I certainly don’t know a proof. (In particular, I don’t think I could give you a correct construction or definition for the real numbers.) I believe in the theorem because I’ve seen it asserted in trustworthy reference works. Somebody somewhere might have beliefs about the theorem that were tied to their beliefs in the definitions, but this doesn’t describe me. I can believe the [deductive] consequences of a claim without knowing the definitions or being able to reproduce the deduction.
Here’s a related example with a larger bullet for you to chew on. Suppose I have a (small) computer program that takes arbitrary-sized inputs. I might believe that will work correctly on all possible inputs. Is that a belief or not? It can be made as rigorously provably correct as the FTA.
When I say “the program is correct”, I am not saying “it is useful to construe the C language and the program code in such a way that...”. I’m making an assertion about how the program would behave under all possible inputs.
Beliefs about computer programs might feel more empirical than beliefs about theorems, but they are logically equivalent, so either both or neither are beliefs, it seems.
Please observe that one of the possible inputs to your computer is “A cosmic ray flips a bit and turns JMP into NOP, causing data to be executed as though it were code”. In other words, your proof of correctness relies on assumptions about what happens in the physical computer. Those assumptions are testable beliefs, just like the intuitions that go into geometry or the FTA.
Clearly it’s false. Plenty of human beliefs are non-testable, some even self-contradictory. LP did not claim anything of the sort.