In that case I have probably answered your original question here.

I don't consider these mistakes to be no problem at all. What I meant to say is that the existence of these noise errors doesn't reduce the reasonabliness of me going around and using logical reasoning to draw deductions. Which also means that if reality seems to contradict my deductions, then either there is an error within my deductions that I can theoretically find, or there is an error within the line of thought that made me doubt my deductions, for example eyes being inadequate tools for counting pebbles. To put it more generally: If I don't find errors within my deductions, then my perception of reality is not an appropriate measure for the truth of my deductions, unless said deductions deal in any way with the applicability of other deductions on reality, or reality in general, which mathematics does not.

It's not as if errors in perceiving reality weren't much more numerous and harder to detect than errors in anyone's faculty of doing logical reasoning.

I think the point is that mathematical reasoning is inherently self-correcting in this sense, and that this corrective force is intentionistic and Lamarckian - it is being corrected toward a mathematical argument which one thinks of as a timeless perfect Form (because come on, are there really any mathematicians who don't, secretly, believe in the Platonic realism of mathematics?), and not just away from an argument that's flawed.

An incorrect theory can appear to be supported by experimental results (with probability going to 0 as the sample size goes to \infty), and if you have the finite set of experimental results pointing to the wrong conclusion, then no amount of mind-internal examination of those results can correct the error (if it could, your theory would not be predictive; conservation of probability, you all know that). But mind-internal examination of a mathematical argument, without any further entangling (so no new information, in the Bayesian sense, about the outside world; only new information about the world inside your head), can discover the error, and once it has done so, it is typically a mechanical process to verify that the error is indeed an error and that the correction has indeed corrected that error.

This remains true if the error is an error of omission (We haven't found the proof that T, so we don't know that T, but in fact there is a proof of T).

So you're not getting new bits from observed reality, yet you're making new discoveries and overthrowing past mistakes. The bits are coming from the processing; your ignorance has decreased by computation without the acquisition of bits by entangling with the world. That's why deductive knowledge is categorically different, and why errors in logical reasoning are not a problem with the idea of logical reasoning itself, nor do they exclude a mathematical statement from being unconditionally true. They just exclude the possibility of unconditional knowledge.

Can you conceive of a world in which, say, ⋀∅ is false? It's certainly a lot harder than conceiving of a world in which earplugs obey "2+2=3"-arithmetic, but is your belief that ⋀∅ unconditional? What is the absolutely most fundamentally obvious tautology you can think of, and is your belief in it unconditional? If not, what kind of evidence could there be against it? It seems to me that ¬⋀∅ would require "there exists a false proposition which is an element of the empty set"; in order to make an error there I'd have to have made an error in looking up a definition, in which case I'm not really talking about ⋀∅ when I assert its truth; nonetheless the thing I am talking about is a tautological truth and so one still exists (I may have gained or lost a 'box', here, in which case things don't work).

My mind is beginning to melt and I think I've drifted off topic a little. I should go to bed. (Sorry for rambling)