In fact I once had this sort of mathematical experience.
Somehow, while memorizing tables of arithmetic in the first grade, I learned that 11 - 6 = 7. This equation didn't come up very often in elementary school arithmetic, and even more seldom in high school algebra, and so I seldom got any math questions marked wrong. Then one day at university, I received back a Math 300 homework assignment on which I'd casually asserted that 11 - 6 - 7. My TA had drawn a red circle around the statement, punctuating it with three question marks and the loss of a single point.
I was confused. There was nothing wrong with 11 - 6 = 7. Why would my TA have deducted a point? Everyone knew that 11 - 6 = 7, because it was just the reverse of 7 + 6 = wait-a-minute-here.
Pen. Paper. I grabbed eleven coins and carefully counted six of them away. There were not seven of them left. I started writing down remembered subtraction problems. 11 - 4 = 7. 11 - 5 = 6. 11 - 6 = 7. 11 - 7 = 4. One of these sums was clearly not like the others. I tried addition, and found that both 7 + 6 = 13 and 6 + 7 = 13.
The evidence was overwhelming. I was convinced. Confused, yes—fascinated by where my error could have come from, and how I could have held onto it so long—but convinced. I set to work memorizing 11 - 6 = 5 instead.
It didn't entirely take. Twenty years later, the equation 11 - 6 = 7 still feels so right and familiar and uncontroversial that I've had to memorize 11 - 6 = stop. I know the answer is probably either 5 or 7, but I work it out manually every time.
I don't know if the American elementary curriculum is better than it was (I hope so) but this mistake is less likely to happen now. My niece in 2nd grade is learning different methods of 'knowing' arithmetic. They still memorize tables, but they also spend a lot of time practicing what they call 'strategies for learning the addition facts'.
For example..
11-6 = (10-6)+1 = 5 is the compensation approach.
and 11-6 = 10-5 is the equal additions approach.
They also spend a lot of time doing mental math. I'm impressed with how different things are, and hope that ...
In “What is Evidence?” I wrote:1
Cihan Baran replied:2
I admit, I cannot conceive of a “situation” that would make 2 + 2 = 4 false. (There are redefinitions, but those are not “situations,” and then you’re no longer talking about 2, 4, =, or +.) But that doesn’t make my belief unconditional. I find it quite easy to imagine a situation which would convince me that 2 + 2 = 3.
Suppose I got up one morning, and took out two earplugs, and set them down next to two other earplugs on my nighttable, and noticed that there were now three earplugs, without any earplugs having appeared or disappeared—in contrast to my stored memory that 2 + 2 was supposed to equal 4. Moreover, when I visualized the process in my own mind, it seemed that making xx and xx come out to xxxx required an extra x to appear from nowhere, and was, moreover, inconsistent with other arithmetic I visualized, since subtracting xx from xxx left xx, but subtracting xx from xxxx left xxx. This would conflict with my stored memory that 3 - 2 = 1, but memory would be absurd in the face of physical and mental confirmation that xxx - xx = xx.
I would also check a pocket calculator, Google, and perhaps my copy of 1984 where Winston writes that “Freedom is the freedom to say two plus two equals three.” All of these would naturally show that the rest of the world agreed with my current visualization, and disagreed with my memory, that 2 + 2 = 3.
How could I possibly have ever been so deluded as to believe that 2 + 2 = 4? Two explanations would come to mind: First, a neurological fault (possibly caused by a sneeze) had made all the additive sums in my stored memory go up by one. Second, someone was messing with me, by hypnosis or by my being a computer simulation. In the second case, I would think it more likely that they had messed with my arithmetic recall than that 2 + 2 actually equalled 4. Neither of these plausible-sounding explanations would prevent me from noticing that I was very, very, very confused.3
What would convince me that 2 + 2 = 3, in other words, is exactly the same kind of evidence that currently convinces me that 2 + 2 = 4: The evidential crossfire of physical observation, mental visualization, and social agreement.
There was a time when I had no idea that 2 + 2 = 4. I did not arrive at this new belief by random processes—then there would have been no particular reason for my brain to end up storing “2 + 2 = 4” instead of “2 + 2 = 7.” The fact that my brain stores an answer surprisingly similar to what happens when I lay down two earplugs alongside two earplugs, calls forth an explanation of what entanglement produces this strange mirroring of mind and reality.
There’s really only two possibilities, for a belief of fact—either the belief got there via a mind-reality entangling process, or not. If not, the belief can’t be correct except by coincidence. For beliefs with the slightest shred of internal complexity (requiring a computer program of more than 10 bits to simulate), the space of possibilities is large enough that coincidence vanishes.4
Unconditional facts are not the same as unconditional beliefs. If entangled evidence convinces me that a fact is unconditional, this doesn’t mean I always believed in the fact without need of entangled evidence.
I believe that 2 + 2 = 4, and I find it quite easy to conceive of a situation which would convince me that 2 + 2 = 3. Namely, the same sort of situation that currently convinces me that 2 + 2 = 4. Thus I do not fear that I am a victim of blind faith.5
1See Map and Territory.
2Comment: http://lesswrong.com/lw/jl/what_is_evidence/f7h.
3See “Your Strength as a Rationalist” in Map and Territory.
4For more on belief formation and beliefs of fact, see “Feeling Rational” and “What Is Evidence?” in Map and Territory. For more on belief complexity, see “Occam’s Razor” in the same volume.
5If there are any Christians reading this who know Bayes’s Theorem, might I inquire of you what situation would convince you of the truth of Islam? Presumably it would be the same sort of situation causally responsible for producing your current belief in Christianity: We would push you screaming out of the uterus of a Muslim woman, and have you raised by Muslim parents who continually told you that it is good to believe unconditionally in Islam.
Or is there more to it than that? If so, what situation would convince you of Islam, or at least, non-Christianity? And how confident are you that the general kinds of evidence and reasoning you appeal to would have been enough to dissuade you of your religion if you had been raised a Muslim?