Commenters minunderstand your problem and your argument for its solution. I take your problem to be “What could the probability of a mathematical proposition be besides its comparative likelihood of proof or disproof?
Perhaps the answer is that there are reasons besides proof to believe even a mathematical proposition. Empirical reasons, that is.
If we know that for some particular class of propositions, some number were proven false quite easily, and some number were proven true but only after hundreds of years and great struggle, and the rest have remained unproven, it seems intuitively reasonable to suspect that more of the unproven ones are true than a naive ratio would suggest. I don’t know how to make this rigorous though.
I take your problem to be “What could the probability of a mathematical proposition be besides its comparative likelihood of proof or disproof?
Except that the example he gives is not a mathematical one, but the question of free will. I have to wonder if the question of free will is not the real subject of the post.
No; free will was the question that got me thinking about it. It seems to me at first glance that the existence of free will could be disproven, but couldn’t be proven. Should that impact my belief in it?
(Short answer: No, because the payoff matrix for believing and not believing in free will is peculiar. But that peculiar case doesn’t affect the question I’m asking here. The free will question itself is not interesting to me.)
Commenters minunderstand your problem and your argument for its solution. I take your problem to be “What could the probability of a mathematical proposition be besides its comparative likelihood of proof or disproof?
Perhaps the answer is that there are reasons besides proof to believe even a mathematical proposition. Empirical reasons, that is.
If we know that for some particular class of propositions, some number were proven false quite easily, and some number were proven true but only after hundreds of years and great struggle, and the rest have remained unproven, it seems intuitively reasonable to suspect that more of the unproven ones are true than a naive ratio would suggest. I don’t know how to make this rigorous though.
Except that the example he gives is not a mathematical one, but the question of free will. I have to wonder if the question of free will is not the real subject of the post.
No; free will was the question that got me thinking about it. It seems to me at first glance that the existence of free will could be disproven, but couldn’t be proven. Should that impact my belief in it?
(Short answer: No, because the payoff matrix for believing and not believing in free will is peculiar. But that peculiar case doesn’t affect the question I’m asking here. The free will question itself is not interesting to me.)