What do you think about debates about which axioms or rules of inference to endorse? I’m thinking here about disputes between classical mathematicians and varieties of constructivist mathematicians), which sometime show themselves in which proofs are counted as legitimate.
I am tempted to back up a level and say that there is little or no dispute about conditional claims: if you give me these axioms and these rules of inference, then these are the provable claims. The constructivist might say, “Yes, that’s a perfectly good non-constructive proof, but only a constructive proof is worth having!” But then, in a lot of moral philosophy, you have the same sort of agreement. Given a normative moral theory and the relevant empirical facts, moral philosophers are unlikely to disagree about what actions are recommended. The controversy is at the level of which moral theory to endorse. At least, that’s the way it looks to me.
What do you think about debates about which axioms or rules of inference to endorse? I’m thinking here about disputes between classical mathematicians and varieties of constructivist mathematicians), which sometime show themselves in which proofs are counted as legitimate.
First-, or possibly second-order predicate logic has swept the board. Constructivism is just a branch of mathematics. Everyone understands the difference between constructive and non-constructive proofs, and while building logical systems in which only constructive proofs can be expressed is a useful activity, I think there are not many mathematicians who really believe that a non-constructive proof is worthless. There is some ambivalence toward such things as the continuum hypothesis and the axiom of choice, but those issues never seem to have any practical import outside of their own domain.
I think there are not many mathematicians who really believe that a non-constructive proof is worthless.
One reason for this is that, generally speaking, non-constructive proofs can in fact be embedded into constructive logical systems. I think there is more controversy about what formal foundations we should endorse for mathematics. ZF(C) is good enough as a proof of concept, but type-theoretical and category-theoretical foundations seem to be better in terms of actually doing formalized mathematics in the real world.
I think there are not many mathematicians who really believe that a non-constructive proof is worthless.
One reason for this is that, generally speaking, non-constructive proofs can in fact be embedded into constructive logical systems.
While that may be a reasonable justification for what mathematicians do, I think it is false as a historical claim about what caused mathematicians to do what they did. Mathematicians settled on their foundations (“No one can expel us from Cantor’s Paradise,” 1926) before they understood power and limits of constructive methods.
I’m curious if you are making a practical claim or a formal one.
What do you think about debates about which axioms or rules of inference to endorse? I’m thinking here about disputes between classical mathematicians and varieties of constructivist mathematicians), which sometime show themselves in which proofs are counted as legitimate.
I am tempted to back up a level and say that there is little or no dispute about conditional claims: if you give me these axioms and these rules of inference, then these are the provable claims. The constructivist might say, “Yes, that’s a perfectly good non-constructive proof, but only a constructive proof is worth having!” But then, in a lot of moral philosophy, you have the same sort of agreement. Given a normative moral theory and the relevant empirical facts, moral philosophers are unlikely to disagree about what actions are recommended. The controversy is at the level of which moral theory to endorse. At least, that’s the way it looks to me.
First-, or possibly second-order predicate logic has swept the board. Constructivism is just a branch of mathematics. Everyone understands the difference between constructive and non-constructive proofs, and while building logical systems in which only constructive proofs can be expressed is a useful activity, I think there are not many mathematicians who really believe that a non-constructive proof is worthless. There is some ambivalence toward such things as the continuum hypothesis and the axiom of choice, but those issues never seem to have any practical import outside of their own domain.
One reason for this is that, generally speaking, non-constructive proofs can in fact be embedded into constructive logical systems. I think there is more controversy about what formal foundations we should endorse for mathematics. ZF(C) is good enough as a proof of concept, but type-theoretical and category-theoretical foundations seem to be better in terms of actually doing formalized mathematics in the real world.
While that may be a reasonable justification for what mathematicians do, I think it is false as a historical claim about what caused mathematicians to do what they did. Mathematicians settled on their foundations (“No one can expel us from Cantor’s Paradise,” 1926) before they understood power and limits of constructive methods.
I’m curious if you are making a practical claim or a formal one.