(Just a long comment that doesn't directly answer the bold question, but does [I hope] address an implicit question.) What I've found fascinating in thinking about your answer from that other thread are the differences between intuition and proof.
Some of the benefits of intuition are:
- It gives us a sense of what is right and wrong, without the need for long complicated proofs. This can significantly shorten the time needed for learning, or the time needed when looking for new ideas, and proofs.
- Similarly, intuition can also tell us when something surprising is happening. It helps us know when to double-check that a proof didn't take a wrong turn.
- It can help us develop a simpler model of the world, and a way to explain known facts to others.
Some of the weaknesses of intuition are:
- It often doesn't give us the greater information included in a full proof.
- It might rely on a single model of a situation, rather than multiple models.
- Intuition might be "just wrong". (Though, those intuitions guided by proofs are often more robust.)
- Even when intuition is right, it is not always possible to convince others that it is helpful/useful/right without a proof.
We've all gone through the process of refining our intuitions. I thought David's example of the 2D beings learning about winding numbers in higher dimensions was perfectly apropos, because I can think of similar "aha!" moments where my intuition had to change. This other question about counterexamples in algebra contains quite a few of them.
Now, the fact that our intuitions differ is a good thing! It means we look at problems in different ways, which can help guide us to new, surprising solutions. Thus, just because I may not (currently) find "orientation-preserving" as intuitive as you, that's not a flaw in either of us. Moreover, the existence of algebraic proofs (whether natural or uniform or not) in this case tell me that your intuition (whatever it is) is a good one!
In the case of the ultrafinitists who question the consistency of PA, we'll really only ever know if their intuition is correct if they eventually find a provable flaw in the fabric of mathematics. My intuition says they won't find one. They, at least, have a hope of proving their intuition is correct.
