When I imagine putting two apples next to two apples, I can predict what will actually happen when I put two earplugs next to two earplugs, and indeed, my mind can store the result in a generalized fashion which makes predictions in many specific instances. If you do not call this useful abstract belief "2 + 2 = 4", I should like to know what you call it. If the belief is outside the province of empirical science, I would like to know why it makes such good predictions.
No, the real world does not work via Peano arithmetic. Your experiments with apples and earplugs are simply applications of conservation of mass and immutability of inanimate objects, and other such principles. Before you learned such things, you were thrilled with the game of peek-a-boo -- of how someone could cease to exist, and then appear out of nowhere.
Consider this experiment: Take 2 apples, cut them in half. Take 2 more apples, cut them in half. Put all together. How many apples do you have? The answer is not "4 apples", the answer is "8 half-apples". Furthermore, each individual apple remains the same apple as before (minus the effects of time), so that any differences in size, shape, coloration, bruising, etc would remain the same. Apples aren't numbers, and can't be substituted for each other.
The world abounds with examples where Peano arithmetic does not apply. Consider adding two speeds together -- they do not add via Peano arithmetic, such that there exists a speed X, such that 2X + 2X = 3X. If we're using naive multiplication, that speed is where X=c*sqrt(7/36). None of this changes my beliefs about Peano arithmetic -- it is necessarily true given its axioms, and its correspondence to the physical world is entirely coincidental. Certainly, if Peano arithmetic didn't correspond widely to real world problems, I would never have learned about it in high school and it might not even have been invented -- but it remains true all the same.
This all just means that my idea of truth is different than yours -- I think things can be true or false regardless of their predictive value. Specifically, I value statements of the form "If A, then (A worded slightly differently)" and think that almost all knowledge has that form. For example, "If the universe is consistent and objective, the scientific method will tend toward accurately describing the universe". Once you introduce inductive reasoning, even for something as trivial as stating "the universe is consistent and objective", then you introduce uncertainty -- you switch from binary true/false to likely/unlikely and accurate/inaccurate and predictive/non-predictive.
If you equate "scientific type truth" with other people's "actual truth" you will get into many pointless arguments. For example, you seem greatly offended that religious people sometimes disagree with science. For example, they will point out that science has always been wrong, and is probably still wrong. But they'll probably agree that modern science more accurately and more precisely predicts about our world. In fact, you'll probably get them to agree that science is perhaps the best method to make accurate and precise predictions. A statement doesn't need to be true to be useful for making accurate predictions, for example Newtonian gravity. So why equate "makes accurate predictions" with "truth"?
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?