In “What is Evidence?” I wrote:1

This is why rationalists put such a heavy premium on the paradoxical-seeming claim that a belief is only really worthwhile if you could, in principle, be persuaded to believe otherwise. If your retina ended up in the same state regardless of what light entered it, you would be blind . . . Hence the phrase, “blind faith.” If what you believe doesn’t depend on what you see, you’ve been blinded as effectively as by poking out your eyeballs.

Cihan Baran replied:2

I can not conceive of a situation that would make 2 + 2 = 4 false. Perhaps for that reason, my belief in 2 + 2 = 4 is unconditional.

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?

How to Convince Me That 2 + 2 = 3
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Do we consider it to be evidence in Christianity's favor that more people believe in it than Islam? Does the average IQ of adherents of a religious belief cause it to become more plausible to us?

In the interests of disclosure, I am an agnotheist who was baptized Catholic and raised mainline Protestant, so we're still waiting for Eliezer's requested comment.

1Sengachi
People can believe wrong things by the millions (Yay! the Earth is flat), that does not make it right. Stupid people can believe correct things and intelligent people can believe incorrect things. And if you were basing the belief system you believe in on average IQ, you'd go with atheism anyway. But none of these things are evidence. Something would be evidence even if everybody in the world disagreed. Relativity was true even when every single physicist (a group of rather educated people) just knew that Newtonian physic was truth.

People's belief in something is evidence for that thing in the sense that in general it's more likely for people to believe in a thing if it's true. Less Wrongers sometimes use the phrase "Bayesian evidence" when they want to explicitly include this type of evidence that is excluded by other standards of evidence.

One way to think about this: Imagine that there are a bunch of parallel universes, some of which have a flat Earth and some of which have a spherical Earth, and you don't know which type of universe you're in. If you look around and see that a bunch of people believe the Earth is flat, you should judge it as more likely you're in a flat-Earth universe than if you looked around and saw few or no flat-Earthers.

However, people's beliefs are often weak evidence that can be outweighed by other evidence. The fact that many people believe in a god is evidence that there is a god, but (I think) it's outweighed by other evidence that there is not a god.

See also "Argument Screens off Authority".

2[anonymous]
'People's belief in something is evidence for that thing in the sense that in general it's more likely for people to believe in a thing if it's true'. What's the evidence for this statement?
5Andreas Källberg
The overwhelming majority of all human beliefs are (trivially) true. Things like "If I drop a rock, it will fall down", "If I touch hot fire it will hurt", etc. and "I am sitting down", "I am typing on a keyboard", etc. The human brain has evolved to determine truths about the world around it, especially in cases where the knowledge directly affects survival chances ("Tiger is dangerous"), but also for cases where the knowledge could help indirectly (basically all human progress including first tool usage -- achieved especially through curiosity and strive to find truth). It fails catastrophically in some cases, but most of the time it does an excellent job.
20ctoDragon
Something to consider is that if you allow your beliefs to be influenced by the beliefs of others you are in danger of creating a feed back loop. When deciding what to believe based on what others believe you must rule out those who are simply following others as well

Certainly. The probability of Christianity having more followers than Islam is greater if Jesus rose from the dead and less if he did not.

It's not necessarily strong evidence of course. Disavowing Islam has enormous social consequences, so I would expect there to be a large number of Muslims in both the world where Muhammad received the Quran from Gabriel and the world where Muhammad hallucinated. But I still expect there to be more Christians if Jesus rose from the dead than if he did not.

IQ is only weakly correlated to rationality. A much better thing to do is to ask Christians why they believe. If you know the reasons a Christian believes, then the evidential weight of their reasoning will replace the evidential weight that comes from the fact that they believe.

The causal flow looks like this:

Reality --> Reason to believe -> Person believes

By d-separation, once you know a person's reasons for believing, the fact that they believe is no longer useful information to you.

In the interests of disclosure, I am an ex-Christian who spent a year learning Arabic because I believed that God was calling me to be a missionary to Muslims. When I learned Bayes theorem, I attempted to use... (read more)

2wizzwizz4
But Jesus isn't a randomly-selected human. He already had followers before being executed by the state, so shouldn't we be using the probability of a randomly-selected religion/cult leader rising from the dead? (Not that that's much different.) Though I'm not sure we have enough information to use Bayes' rule properly here. P(Person rose from the dead | Person is God) = 1, and we'll assume¹ that P(Person is God | Person rose from the dead) = 1 so that we just need to consider "person rose from the dead"… okay, never mind, I just got it wrong. Your argument holds. --- ¹: even though that's a simplification from a theological point of view; the argument could be made rigorous by your particular denomination simply by making this statement specific enough to be correct
1kremlin
Does his argument hold? Because I had the same intuition as you, that a "random person raising from the dead" isn't the comparison to make here, but I can't fully articulate what the right comparison to make would be.
2wizzwizz4
I think the bigger problem with the argument is: if Jesus rose from the dead for the same reasons that a randomly-selected human would, then Jesus is just an arbitrary human who got lucky, and mainstream Christianity is false. So if you're invoking that to try to estimate whether Christianity is true or not, you're clearly asking the wrong question.

I am a jew (born and raised). I can easily imagine that if I were raised in the muslim world to a muslim family that I would be a muslim today. However, were I born to a christian family (and perhaps this is simply my inner biases talking) I suspect that I would have been attracted to various aspect of the Jewish religion which are not present (or not nearly as strong) in christianity, like the idea of a "contract with God".

In full disclosure, I do not continue to call myself a Jew because I believe the Torah to be more likely than any other mainstream religious text, but because I find the ethical framework to be superior.

-5yaro

To apply the same reasoning the other way, if you aren't a Christian, what would be a situation which would convince you of the truth of Christianity?

The Second Coming? An opportunity to have a chat with the Lord Himself? An analysis of a communion wafer revealing it to, in fact, be living human flesh? It's seriously not that hard to think of these.

Which is more likely "God exists" or "I just hallucinated that" For the third one, probably that He exists, for the second one, definitely hallucination, for the first, I'm not sure.

Second one: depends. I was kind of assuming that you have some way of verifying it, like you ask Him to create something and someone who wasn't there later describes some of its previously determined properties accurately without being clued in. First: you'd need a massive global hallucination, and could use a similar verification method.

That seems accurate. Remember that a single person can hallucinate that someone else verified something, but this has low prior probability.

3MTGandP
Given my current mental capacities, I think that any "proof" of God would be more easily attributed to hallucination. However, it should still be possible for God to prove His existence. If He is omnipotent, then he can increase my mental capacity to the extent that I can distinguish between divine intervention and a hallucination of divine intervention.
3MaxNanasy
But what if you're hallucinating the increase in mental capacity and resulting discernment?
4MTGandP
It may be theoretically possible to increase my mental capacity in some way such that I can distinguish mental capacity from hallucination. I cannot conceive of how that would be done, but it may be possible. P.S. I love when people reply to comments that are two and a half years old. It feels like we're talking to the past.

I once conducted an experiment in which I threw a die 500 times, and then prayed for an hour every day for a week that that die consistently land on a four, and then threw the die 500 more times. Correlation was next to zero, so I concluded that God does not answer prayers about dice from me.

Haven't you ever heard the saying, "God does not throw dice games"?

1mszegedy
Wasn't that what Einstein said about QM?
4JoshuaZ
Almost. Eliezer is making a bad wordplay with what Einstein said.
6MixedNuts
I wouldn't expect a deity to answer that sort of prayer. You're not being sincere, just trying to test them, which many canonically find annoying because it shows mistrust; you don't need that die to land on a four; it suggests you'd use prayer to lowly ends (e.g. "Let me score a touchdown" rather than "Please solve world hunger"); it gives an easily publishable result, which no deity would characteristically accept - if they didn't want to be discreet they'd still be doing showy miracles. Studies where you pray to cure cancer or something are much stronger evidence.
2Lethalmud
Do those studies have a placebo group?
7Rixie
I read about a study like that, in which Christians prayed for people to recover from cancer. There was barely any difference between the patients that weren't prayed for, the patients that were prayed for and knee that they were being prayed for, and the patients that didn't know that they were being prayed for.
3Tintinnabulation
I recall the same study - and I seem to remember that the patients who knew they were being prayed for did a bit worse.

You're not being sincere.

Actually, if you run the test, you are. Given that you'd have changed your mind if it had gone the other way, of course.

5Document
(Related: Religion's Claim to be Non-Disprovable)
0descent
conversely, as a born pedant american christian who has raised countless prayers in the absolute good faith of childhood, god should know that only needlessly statistical tests would ultimately save me, and that any measurable manifestation of the divine would immediately cause me to pledge my life and the highest degree of propaganda / violence I could affect to any awful cause that (s)he could imagine. Unfortunately, YHWH turns away every chance he has at my safely partitioned acolytic fervor. Old testament lord was not above showy miracles, but so much changes between the two that I have a hard time even seeing it as an allegory or reformation. I can only imagine that it was a pretty steep reform.
0AlexanderRM
There's no situation which would convince me that Christianity had a 100% probability of being true, because the idea that the entire scenario since I first encountered evidence of Christianity being true was a hallucination or that I was a Brain-in-a-Vat could never be disproved, but I can easily imagine scenarios that could make me raise my estimated probability of Christianity much higher, to 50%, 90%, perhaps higher. If I were teleported into an alternate world where world history and the like seemed more consistent with Christianity being true, I could easily envision my probability ranking to as high as my current one for Atheism, to the point that I would act based on the assumption that it had a 100% probability.
2jeronimo196
God showing up and granting all humans Wolverine's healing factor would be evidence he exists. Providing a good explanation of why he permitted disease in the first place might convince me he is not as evil as described in the Bible. Edit: Aliens playing god would still be far more likely, but the above scenario would be evidence in favour of the god hypothesis.
3Marion Z.
I've always quite liked Scott Alexander's answer to the problem of evil. It is absolutely useless as a defense of Abrahamic beliefs in the real world, but is relatively satisfying to an atheist wondering how that question might theoretically be answered by a true god. In case you're not familiar, the basic idea is that God did create a perfectly good universe full of a near-infinite number of consciousnesses experiencing total bliss at all times - then decided that he wanted more net good to exist, so he made a universe which was almost exactly the same as the first but with one incredibly minor detail changed - making it just slightly less than maximally perfect. So on and so on, because to create an identical universe is not really to create one at all.  After some absurd number of universes, we arrive at ours (this explanation requires that you believe that our universe has more net happiness than suffering, which is admittedly just taken on faith). Ours is definitely closer to balanced between perfectly good and perfectly evil than not, but it still is more good and thus worth creating. He also implies that people who experience more suffering than happiness in their individual lives might be p-zombies, but I find that to be incredibly weird and have always left it out of explanations to people who might possibly feel that they have had a bad life.

The core issue is whether statements in number theory, and more generally, mathematical statements are independent of physical reality or entailed by our physical laws. (This question isn't as obvious as it might seem, I remember reading a paper claiming to construct a consistent set of physical laws where 2 + 2 has no definite answer). At any rate, if the former is true, 2+2=4 is outside the province of empirical science, and applying empirical reasoning to evaluate its 'truth' is wrong.

0rkr1410
There are some points of view that sometimes do require mathematical statements to be dependent on reality (i.e. constructivism, actual versus potential infinity debate, etc). Sometimes it is intuitive to require mathematics to behave this way, i.e. 'natural' numbers are called that for a reason, and they better behave like the apples or I'm postulating a change in nomenclature. P.S. Ii seems to me the OP's wording wasn't precise enough. I can very well imagine a situation in which some basic addition would yield non obvious results (like addition inside modulo N number space).
3Will_Sawin
When I reason inside a fully axiomatized formal system, the axioms don't depend on reality, but the rules for manipulating symbols depend on ... something. You could define it as "if I perform these manipulations in reality, I will get this result" but what if performing the manipulations in different places gets different results? What if, when you applied the rule "(x+Sy) => S(x+y)" twice and the rule "(x+0)=>x" once, to "(SS0+SS0)", you got "SSS0" instead of "SSSS0"?
0rkr1410
I guess when one reasons inside a fully axiomatized formal system, this something the rules for symbol manipulation depend on is the set of axioms. Now I'm putting on my uneducated hat, so excuse me if this is heresy: Starting with the axioms you apply logic to formulate more specific rules (in this case the abstract is empirically falsifiable, since we're working on natural numbers). So, to arrive at SS0+SS0=SSS0, you'd have to venture outside the realm of reason I'm afraid.Tthat would maybe manifest itself as magic - getting 4 apples on the table during night, but 3 during day when you put 2 and 2 apples side by side. And could mean ability to produce something from nothing by clever arrangement of apples. and waste disposal would become easy :) In other words my opinion is it's not possible even as thought experiment unless you introduce some random factor from beyond the scope of axioms.
4Will_Sawin
well there's the special other thing, the reason you can't explain Peano Arithmetic to a rock, which is that axioms are static sequences of signals, but in addition you have these dynamics. Best source on this is Lewis Carroll http://www.ditext.com/carroll/tortoise.html These dynamics are contained within the structure of our thoughts, which is why they're preserved in a thought experiment. But we still have to actually check our thoughts, which are part of reality. Sorry if this wasn't very coherent.
1MarkusRamikin
Hm... not precise enough for what? I think we all know what was meant... unless Eliezer did a ninja edit after you posted ;) this seems to cover it: What you suggested, that's not a "basic addition" any more, is it?

I don't think this is at all the core issue.

Eliezer's original post stated that beliefs need to come from mind-reality entangling processes.

If math is a part of "reality", then Eliezer's point stands and empirical reasoning makes perfect sense.

If math is not a part of "reality", then we would expect it to influence nothing at all, including our beliefs. Or even suppose that knowledge came from somewhere and could influence belief but still did not otherwise correlate with reality: Then it would be irrelevant. This, of course, is not the case - as anyone who's ever used any mass-manufactured device as well as bridges and roads, should realize. Math DOES have utility in real life. And I daresay that if it suddenly stopped helping us reliably predict the load-bearing limit of bridges, we'd treat is as suspect and false.

The ACTUAL core issue remains that a belief that cannot be reversed is useless.

At any rate, if the former is true, 2+2=4 is outside the province of empirical science, and applying empirical reasoning to evaluate its 'truth' is wrong.

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.

To apply the same reasoning the other way, if you aren't a Christian, what would be a situation which would convince you of the truth of Christianity?

You'd have to fix all the problems in belief, one by one, by reversing the evidence that originally convinced me of the beliefs' negations. If the Sun stopped in the sky for a day, and then Earth's rotation restarted without apparent damage, that would convince me there was one heck of a powerful entity in the neighborhood. It wouldn't show the entity was God, which would be much more complicated, but it'... (read more)

If you do not call this useful abstract belief "2 + 2 = 4", I should like to know what you call it.

I call it "2+2=4 is a useful model for what happens to the number of earplugs in a place when I put two earplugs beside two other earplugs". Which is a special case of the theory "arithmetic is a useful model for numbers of earplugs under some operations (including but not limited to adding and removing)".

If the belief is outside the province of empirical science, I would like to know why it makes such good predictions.

The mathematical claim "2+2=4" makes no predictions about the physical world. For that you need a physical theory. 2+2=4 would be true in number theory even if your apples or earplugs worked in some completely different manner.