> "“I can just grasp its truth by thinking.” But if that’s true of any non-natural fact, then NR must be true!"
How does that follow? I think we can grasp truths just by thinking. But I don't think that our grasping truths involves non-natural causes shifting atoms around in our brains. Rather, I think that grasping truths is an epiphenomenal process: there are neural underpinnings that (together with the psychophysical bridging laws) give rise to our conscious understanding or "grasp" of various abstract or otherwise non-physical truths. To count as knowledge, the connection has to be non-chancy in the right kind of way. But (as I argue in Knowing What Matters) beliefs can be reliable/non-chancy in this way without needing to be literally caused by their truth-makers. A kind of structural isomorphism to mathematical facts can reliably yield mathematical knowledge, for example, without needing the numbers themselves to do the causal work. It's neither magical nor mysterious that computers can reliably do arithmetic, after all. We're different in that when our brains do arithmetic, it produces in us some *conscious understanding* of the mathematics that is (presumably) missing in computers. But I don't see any basis for thinking that introspection on this process reveals non-mechanistic causes operating on our brains.
When you grasp a truth, the truth must make a difference to your grasping. When you see that something is true, you are identifying properties in virtue of which it is the case. But if you do that, then that it is the case is making a difference! I don't think computers have beliefs, much less justified ones, and while they can get reliable answers to math problems, that is because we have specifically designed them to do that. If God designed our faculties to be accurate, then this could give us justification for believing non-natural facts. But if it's just a coincidence that our beliefs line up with the facts, then that undermines justification.
Surely it's not the merely possible world itself that changes atoms, but rather the psychological projection of that possible world (which is itself a movement of physical particles) that changes atoms. In the same way that my thinking about Gandalf can influence my behavior, although the non-existent Gandalf himself can't influence my behavior.
Re: 1, anyone inclined to accept a more physicalistic view of the mind probably wouldn’t be too impressed by this sort of introspective evidence. You wouldn’t necessarily expect a physical system to have the kind of complete, infallible introspective access the argument seems to assume.
Re: 4, I’ve never understood why we’re supposed to assume that human mathematical cognition is complete in the way Gödel showed no consistent formal system can be.
I've come across a paper called 'Grasping the third realm's written by John Bengson that I think you'll find interesting. I'm still trying to wrap my head around it, but he offers an account of how we can come to know things like non-natural facts just by thinking. He puts forward a constitution model.
> "Many suppose, for example, that to generate knowledge, a belief-forming process has to be safe, such that the one who affirms it would believe the truth in nearby possible worlds, or sensitive, in that it has to be dependent on the truth of the proposition. But both of these are undermined by denial of NR."
This isn't right. Any belief in a necessary truth is maximally "safe", no matter its cause. There is no possible world at all (let alone nearby one) in which one holds *that* belief falsely.
Perhaps you're thinking of assessing a coarse-grained process like "believing what seems intuitive" rather than specific beliefs (or any more fine-grained characterization of the process that builds in something about its substantive starting points). But then I think it's that coarse-grained approach, not the rejection of NR, that is really responsible for the skeptical results. At any rate, one certainly should not combine epiphenomenalism with such a coarse-grained epistemology.
To bring out why the coarse-grained approach is the more fundamental problem, consider that even on your account, it must be very easily possible that someone might have unreliable intuitions (presumably you think those who disagree with you are not reliably intuiting the truth?). So "believing what seems intuitive" cannot possibly be a *generally reliable* process. At best, it can only be reliable for those lucky few who actually find their beliefs are causally responsive to the non-natural truths. But this doesn't seem all that different from my view, on which it's only reliable for those lucky few whose beliefs happen to line up (non-causally) with the non-natural truths. Given that introspection alone can't identify whether you're one of the lucky few or not, the sheer fact of philosophical diversity shows that philosophical knowledge is only possible if knowledge is compatible with (a certain kind of constitutional) luck.
> "“I can just grasp its truth by thinking.” But if that’s true of any non-natural fact, then NR must be true!"
How does that follow? I think we can grasp truths just by thinking. But I don't think that our grasping truths involves non-natural causes shifting atoms around in our brains. Rather, I think that grasping truths is an epiphenomenal process: there are neural underpinnings that (together with the psychophysical bridging laws) give rise to our conscious understanding or "grasp" of various abstract or otherwise non-physical truths. To count as knowledge, the connection has to be non-chancy in the right kind of way. But (as I argue in Knowing What Matters) beliefs can be reliable/non-chancy in this way without needing to be literally caused by their truth-makers. A kind of structural isomorphism to mathematical facts can reliably yield mathematical knowledge, for example, without needing the numbers themselves to do the causal work. It's neither magical nor mysterious that computers can reliably do arithmetic, after all. We're different in that when our brains do arithmetic, it produces in us some *conscious understanding* of the mathematics that is (presumably) missing in computers. But I don't see any basis for thinking that introspection on this process reveals non-mechanistic causes operating on our brains.
When you grasp a truth, the truth must make a difference to your grasping. When you see that something is true, you are identifying properties in virtue of which it is the case. But if you do that, then that it is the case is making a difference! I don't think computers have beliefs, much less justified ones, and while they can get reliable answers to math problems, that is because we have specifically designed them to do that. If God designed our faculties to be accurate, then this could give us justification for believing non-natural facts. But if it's just a coincidence that our beliefs line up with the facts, then that undermines justification.
Surely it's not the merely possible world itself that changes atoms, but rather the psychological projection of that possible world (which is itself a movement of physical particles) that changes atoms. In the same way that my thinking about Gandalf can influence my behavior, although the non-existent Gandalf himself can't influence my behavior.
Re: 1, anyone inclined to accept a more physicalistic view of the mind probably wouldn’t be too impressed by this sort of introspective evidence. You wouldn’t necessarily expect a physical system to have the kind of complete, infallible introspective access the argument seems to assume.
Re: 4, I’ve never understood why we’re supposed to assume that human mathematical cognition is complete in the way Gödel showed no consistent formal system can be.
I've come across a paper called 'Grasping the third realm's written by John Bengson that I think you'll find interesting. I'm still trying to wrap my head around it, but he offers an account of how we can come to know things like non-natural facts just by thinking. He puts forward a constitution model.
I forgot to add that he also talks about worries pertaining to accidental beliefs
> "Many suppose, for example, that to generate knowledge, a belief-forming process has to be safe, such that the one who affirms it would believe the truth in nearby possible worlds, or sensitive, in that it has to be dependent on the truth of the proposition. But both of these are undermined by denial of NR."
This isn't right. Any belief in a necessary truth is maximally "safe", no matter its cause. There is no possible world at all (let alone nearby one) in which one holds *that* belief falsely.
Perhaps you're thinking of assessing a coarse-grained process like "believing what seems intuitive" rather than specific beliefs (or any more fine-grained characterization of the process that builds in something about its substantive starting points). But then I think it's that coarse-grained approach, not the rejection of NR, that is really responsible for the skeptical results. At any rate, one certainly should not combine epiphenomenalism with such a coarse-grained epistemology.
To bring out why the coarse-grained approach is the more fundamental problem, consider that even on your account, it must be very easily possible that someone might have unreliable intuitions (presumably you think those who disagree with you are not reliably intuiting the truth?). So "believing what seems intuitive" cannot possibly be a *generally reliable* process. At best, it can only be reliable for those lucky few who actually find their beliefs are causally responsive to the non-natural truths. But this doesn't seem all that different from my view, on which it's only reliable for those lucky few whose beliefs happen to line up (non-causally) with the non-natural truths. Given that introspection alone can't identify whether you're one of the lucky few or not, the sheer fact of philosophical diversity shows that philosophical knowledge is only possible if knowledge is compatible with (a certain kind of constitutional) luck.
You are completely 100% right about this, and I have thus corrected it.