A somewhat more philosophical post than I usually do... I wrote this up during an exchange w/ a theologian differentiating two ways we can gain knowledge.
To a theologian:
Let's start with mathematical truths -- I think that they are true and in a very real sense constitute knowledge. Knowledge about something that is abstract, however. One of the features of abstract subjects, in my view, is that it is quite possible to make real progress in them without new inputs. Euler, an amazing mathematician, placed in a shielded box and given enough time could make real progress in mathematics.
I mean this to contrast methods of inquiry about external reality. When it comes to external reality -- how stars work, the rules that govern the motions of the planets, etc., a physicist in a box could make lots and lots of models and guesses and arguments... but, unless he re-analyzes data from before he came into the box, he won't make progress in physics.
For example, Einstein, with the data of Mercury's weird orbit and the results from Michelson, could be put in a box, and with enough thinking could come up with General Relativity.
However, Einstein, placed in a box before the first experiments in quantum mechanics were done, would never end up concluding quantum mechanics. He might, given infinite time, detail hundreds of thousands of possible physics on small scales that includes our modern conception of QM, but he would be in no position to choose one from the others with any confidence at all. He might even pick what he thinks is the most beautiful physics at small scales, but what counts is not beauty or arguments, per se, but whether his physics matches reality.
That brings me to what I think is our main point of conflict between our approaches.
I fully acknowledge that progress can be made in all areas of mathematics and much of philosophy from internal reflection and argument without any new data coming into a person's head. When a person thinks that this abstract knowledge can tell us something specific about external reality, the existence of God for example, they are simply making a category error.
Can theology reason out the attributes of what would be a perfect being? Sure. Can abstract reasoning alone tell us something about external reality? Possibly... although I'm having a hard time thinking of an example where it's happened before.
Keep thinking about the difference between math and physics. If you wander too far away from direct measurement and experience, you can do a lot of work, and convince yourself of quite a lot... although you won't learn much about external reality.
I'm not expecting this to be unanimously agreed upon -- comments?