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    In "What is Evidence?", I wrote:

    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:

    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.

    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.

    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.

    If there are any Christians in the audience who know Bayes's Theorem (no numerophobes, please) 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?

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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.

    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".

    '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.
    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)

    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
    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.
    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.


    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.

    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.
    But what if you're hallucinating the increase in mental capacity and resulting discernment?
    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"?

    Wasn't that what Einstein said about QM?
    Almost. Eliezer is making a bad wordplay with what Einstein said.
    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.
    Do those studies have a placebo group?
    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.
    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.

    (Related: Religion's Claim to be Non-Disprovable)
    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.
    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.
    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.

    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).
    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"?
    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.
    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 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.
    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.

    I hate to break it to you, but if setting two things beside two other things didn't yield four things, then number theory would never have contrived to say so.

    Numbers were invented to count things, that is their purpose. The first numbers were simple scratches used as tally marks circa 35,000 BC. The way the counts add up was derived from the way physical objects add up when grouped together. The only way to change the way numbers work is to change the way physical objects work when grouped together. Physical reality is the basis for numbers, so to change number theory you must first show that it is inconsistent with reality.

    Thus numbers have a definite relation to the physical world. Number theory grew out of this, and if putting two objects next to two other objects only yielded three objects when numbers were invented over forty thousand years ago, then number theory must reflect that fact or it would never have been used. Consequently, suggesting 2+2=4 would be completely absurd, and number theorists would laugh in your face at the suggestion. There would, in fact, be a logical proof that 2+2=3 (much like there is a logical proof that 2+2=4 in number theory now).

    All of m... (read more)

    Verbal expressions almost certainly predate physical notations. Unfortunately the echos don't last quite that long.
    In your last paragraph you turn everything around and inexplicably claim that math is more primary than observation of reality, though you did a good job -- and one I agree with -- of pointing out the opposite in the previous part of the comment.

    When it was noticed in the 1800's that the perihelion of Mercury did not match what Newton's inverse-square law of gravity predicted, did we change the way math works? Or did we change our understanding of gravity?

    Math is the most fundamental understanding of reality that we have. It is the most thoroughly supported and proven aspect of science that I know of. That doesn't mean that our understanding of math can't be fundamentally flawed, but it does mean that math is the last place we expect to find a problem when our observations don't match our expectations.

    In other words, when assigning probabilities to whether math is wrong or Newton's Theory of Gravity is wrong, the probability we assign to math itself being wrong is something like 0.000001% (sorry, I don't know nearly enough math to make it less than that) and Newton's Gravity being wrong something like 99.999999%.

    See what I'm saying?

    Yup. I think we agree. My disagreeing post was a mere misunderstanding of what you were saying.
    After a few recent posts of mine it looks like I need to work on my phrasing in order to make my points clear. No harm no foul.

    Woah, I think that's a little overconfident...

    You're saying that in the mid nineteenth century (half a century before relativity), the anomalous precession of Mercury made it seem 99.999999% likely that Newtonian mechanics was wrong?

    After all, there are other possibilities.

    cf. "When it was noticed in the 1800's that the perihelion of Neptune did not match what Newton's inverse-square law of gravity predicted, did we change the way math works? Or did we change our understanding of gravity?" In this case we actually postulated the existence of Pluto.

    Similar solutions were suggested for the Mercury case, e.g. an extremely dense, small object orbiting close to Mercury.

    And that's leaving aside the fact that 99.999999% is an absurdly high level of confidence for pretty much any statement at all (see ).

    If I were a nineteenth century physicist faced with the deviations in the perihelion of Mercury, I'd give maybe a 0.1% probability to Newton being incorrect, a 0.001% probability to maths being incorrect, and the remaining ~99.9% would be shared between incorrect data /incomplete data/ other things I haven't thought of.

    However, I agr... (read more)

    For well-established math, sure. We certainly will look for experimental mistakes, unnoticed observables (e.g. the hypothesized planet Vulcan to explain Mercury's deviation from Newtonian gravity), and better theories in about that order. However for less well established mathematics at the frontiers we do consider the possibility that we've made a mistake somewhere. Off the top of my head the biggest example I can think of was von Neumann's proof that hidden variables were inconsistent with quantum mechanics, which was widely believed and cited at least into the 1980s, despite the fact that David Bohm published a consistent hidden variables theory of quantum mechanics in 1952. I'm curious if anyone can recall a case in which an experimental result led us to realize that a previously accepted mathematical "fact" was incorrect. Here's a whole gallery of math which we were later proven to be mistaken about.
    At what point are there two plus two things, and at what point are there four things? Would you not agree that a) the distinction itself between things happens in the brain and b) the idea of the four things being two separate groups with two elements each is solely in the mind? If not, I'd very much like to see some empirical evidence for the addition operation being carried out. English is so firmly grounded in the physical reality that when observations don't line up with what our english tells us, we must change our understanding of reality, not of english. I hope the absurdity is obvious, and that there are no problems to make models of the world with english alone. So, do you find it more likely that math is connected to the world because we link it up explicitly or because it is an intrinsic property of the world itself?
    On the other hand...
    Elezier, do you believe that someday humans could create an AI and put that AI in a simulated enviroment that accurately simulated all the observations humanity made until now? If you do, what further observations would that AI have to make to arrive at the belief that they were created by an intelligent entity?
    If we assume that humanity has gained access to effectively infinite computing power, and has put AIXItl or something similar into a copy of the universe, simulated at whatever level unifies quantum mechanics and gravitation into a coherent, leakproof framework, AIXItl would have an extremely small belief that it was inside a simulation. Only if the simplest unification of quantum mechanics and gravity turns out to be "we're in a simulation," would a hyperintelligent AI in a perfect simulation of our universe come to the belief that it's in a simulation. So, the epistemically perfect AI would come to an incorrect decision. This does not imply a flaw in its method for forming beliefs; it merely implies the tautology that there is no way to find out what there is no way to find out.
    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 consis

    "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?"

    -And Jesus said unto them, Because of your unbelief: for verily I say unto you, If ye have faith as a grain of mustard seed, ye shall say unto this mountain, Remove hence to yonder place; and it shall remove; and nothing shall be impossible unto you. - Matthew 17:20

    If mountains moved when Christians told them to, every time, and no one else could effectively command mountains to move, I think most of us non-believers would start going to church.

    Alternatively, if the world looked like it was designed and regulated by a loving being, it would help. That might not promote Christianity specifically, but it would be a much better start than what we actually see.

    Having seen that verse in several translations, it reads to me as a primitive admonition against belief in belief.  (Which matches up with his criticism of praying or fasting as a publicity stunt instead of because you actually hope to accomplish something.) Consider:  If it were a point of Christian faith that a particular mountain should be torn down and cast into the sea, and people really believed in their religion instead of just believing that they believed... well...  Even with just picks and shovels there aren't many mountains that would survive the wrath of 2.3 billion people for very long.  And without careful study of the circumstances, it would seem like something of a miracle that some massive army of workers just spontaneously organized and did such a mighty task without there being a king or some other authority figure forcing them to do it. Basically, lots of the things that ancient religions attributed to "God" or "The power of Faith" are very real phenomena that they simply couldn't explain, and the fact that we can now explain them (at least a little better) doesn't necessarily render the old practical advice on how to make use of them worthless.  There are often better sources for it that are more clearly stated for the modern mind, but there can also be some value to knowing that the thing you are studying has been known about since the dawn of recorded history and that your ancestors were not, in fact, total fools.

    I am confused by this discussion. Are we talking about integers or things?

    Analytic truths may or may not correspond to our situations. When they don't correspond, I guess that's what you all are calling "false." So, if we're engineers working on building a GPS system, I might say to you, "Careful now, Euclidean geometry is false."

    Similarly, quantum physicists on the job might say, "Watch out now, two and two isn't necessarily four."

    I'm thinking of this excellent blog post I came across last week:

    ...Consider a basket with 2 apples in it. Now toss in 2 more apples. Examine the basket, and you will find (surprise!) 4 apples. However, you cannot prove a priori that there will be 4 apples in the basket. It is an empirical question, albeit a trivial one, whether baskets of apples (which are physical things) behave in the same manner as the non-negative integers under addition (which is an abstract logical construct).
    This distinction might seem hopelessly pedantic at first, but you can easily go astray by ignoring it. For example, many people naively expect photons to behave in the same manner as integers under addition, but they don’t. “Number of photons” is not a conserved quantity in the way that “number of apples” is; photons can be created/destroyed, one photon can be split into two, et cetera....

    Eliezer is right; numbers are first an abstraction of the world around us. There are a vast number of possible abstractions; the reason we have been so very interested in numbers, compared to all the other possible abstractions, is that numbers happen to describe the world around us. It need not have been so.

    What's an example of another possible abstraction?
    Other possible abstractions: * groups * rings * modular arithmetic * sets * simple computer programs (like these cellular automata) These abstractions describe aspects of the world around us too, just not counting macroscopic objects like apples and earplugs.
    Yes it does need be so. Precisely because numbers are an abstraction of the world around us, an abstraction which we as wonderful human beings have advanced into a more and more sophisticated abstraction for many years, they reflect (if that is the right word) the world around us. It is not "the unprecedented success of math," but of man.

    "A priori reasoning" takes place inside the brain; which is to say, any particular form of "a priori reasoning" is part of a simple physical process unified with the empirical questions that we are reasoning about. It is no great surprise by selecting the right form of "a priori reasoning" we can manage to mirror the outside world. Inside and outside are part of the same world.

    When you think about mathematics, your thoughts are not taking place inside another universe, though I can see why people would feel that way.

    As Within, So Without? [ducks the rotten tomatoes]

    The truth of an arithmatic equation and the truth of the content of a religion like Islam or Christianity are really not comparables at all. Within the domain of mathematics, "two plus two" is one definition of "four". Conversely, "four" is one definition of "two." (In a sense these truths are tautalogical.)

    The Greeks noticed that mathematics is a field of knowledge that can be developed entirely in the mind. The manipulative objects that we use to teach children basic arithmetic operations are not actually the subje... (read more)

    Eliezer: When you are experimenting with apples and earplugs you are indeed doing empirical science, but the claim you are trying to verify isn't "2+2=4" but "counting of physical things corresponds to counting with natural numbers." The latter is, indeed an empirical statement. The former is a statement about number theory, the truth of which is verified wrt some model (per Tarski's definition).

    Gray Area, if number theory isn't in the physical universe, how does my physical brain become entangled with it?

    Rozendaal, sounds like you bought into one of religion's Big Lies.

    You seem to be using the word 'religion' when you are more specifically talking about Platonism, right?

    Let me take another crack at this...

    I do not believe any situation could ever convince Eliezer that 2+2=3.

    If he proclaims "two and two makes three," then he must be talking about something other than the integers. You cannot be mistaken about the integers, you can only misunderstand them. It's like saying "some women are bachelors." You are not mistaken about the world, you've merely lost your grasp of the terminology.

    Lee B, Gray Area: what if you had a proof that 2 + 2 = 3, and, although you seem to recall having once seen a proof that 2 + 2 = 4, you can't remember exactly how it went?

    Integers are slippery in a way that apples and poodles are not. If you say something unconventional about integers, you cease to talk about them. --- Does anyone disagree with that?

    (1) Peter de Blanc asks what happens when I cannot follow a proof properly. I count that as a failure of rationality rather than an instance of being mislead by evidence. That is not, I think, what Eliezer intends when he says "convinced."

    (2) If I observe some trick and say, "wow, two and two makes three," then I am dropping the integer system and adopting some other. My "wow" is the same one that we all said when we learned that Euclidean geometry doesn't hold in our universe.

    Lee, the situations I talked about for convincing me that "2 + 2 = 3" could only actually occur if 2 + 2 actually equalled three within the realm of the integers. This is right and proper: why should I allow myself to be convinced by something that would not be valid evidence?

    I do not, therefore, ever expect myself to actually encounter any of these situations, because I currently believe that 2 + 2 = 4.

    If I expected to encounter such evidence in the future, the expectation of my probable future probability estimates must equal my present probab... (read more)


    Eliezer: "Gray Area, if number theory isn't in the physical universe, how does my physical brain become entangled with it?"

    I am not making claims about other universes. In particular I am not asserting platonic idealism is true. All I am saying is "2+2=4" is an a priori claim and you don't use rules for incorporating evidence for such claims, as you seemed to imply in your original post.

    A priori reasoning does take place inside the brain, and neuroscientists do use a posteriori reasoning to associate physical events in the brain with a priori reasoning. Despite this, a priori claims exist and have their own rules for establishing truth.

    I can imagine a world in which the mathematics we have developed is not useful, or in which commonly assumed axioms are false in that world. However, "The Pythagorean Theorem is a theorem of Euclidean geometry" is still true even if you're living on a sphere. If I say "I cannot be convinced that 2 + 2 = 4", I mean something like "I cannot be convinced that S(S(0)) + S(S(0))) = S(S(S(S(0)))) is not a theorem of Peano arithmetic."

    On the religion issue: I'll accept as divine any entity that can consistently reduce the entropy of a closed, isolated system, and will demonstrate this ability on demand. ;)

    If the entity is manipulating the system, then it's not closed or isolated anymore, is it? The system of entity+system could be isolated, but to know if the entropy of that was reduced you'd need to know the internal entropy of God. But if he can produce infinite neg-entropy, then his internal entropy is a meaningless concept.

    I am not making claims about other universes. In particular I am not asserting platonic idealism is true. All I am saying is "2+2=4" is an a priori claim and you don't use rules for incorporating evidence for such claims, as you seemed to imply in your original post.

    Please explain the miraculous correspondence to apples and earplugs, then.

    I confess that I'm also not entirely sure what you mean by "a priori" or why you think it requires no evidence to locate an "a priori claim" like "2 + 2 = 4" in the vast space of po... (read more)

    Mathematical claims do require justification. They even require stronger justification than empirical claims: mathematical proof. As Doug S explained, the proof that 2+2=4 is 2+2 = 2+(1+1) = (2+1)+1 = 3+1 = 4 QED. (Using the definition of 2, the associativity of +, the definition of 3 and the definition of 4 in that order). Empirical claims, such as "2+2=4 is related to earplugs or apples" do not require proofs, but they do require evidence.
    What an interesting argument... but I know of at least one religion that would tend to disagree with this anti-definition of God.
    Actually, does the Bible ever say that God is ontologically distinct from creatures, in any such way? I've read very little of it myself, but based on what I've heard I would expect that the early Old Testament might not include such distinctions (and basically portrays God in a similar manner to polytheistic deities, just with all their power concentrated in one entity). Obviously there's plenty of lines about how great God is, but some of that could be seen as moralizing rather than making factual claims. (although I do imagine that God having a boss screaming at him probably contradicts a lot of factual statements made in some holy books. So suppose that God is a construct of mundane physics in another universe, but is either the only sentient entity in that universe or the only one with any power in that universe.) If an entity existed which is capable of doing every act undertaken by God as described in any holy book, and which did in fact undertake every action undertaken by God as described in one specific holy book (like the Bible or the Old Testament), then that holy book could certainly be said to be "true" in some very important sense, would it not?
    Some scholars of religion have claimed that a straight-forward reading of the early Old Testament suggests it is better described as henotheistic than monotheistic.

    2+2=4 is a truth about mathematics. It is not a truth about the world.

    Truths in the world have no bearing on mathematical truths. While we learn mathematics from observations about the world, it is not from observation that mathematics derive truth. Mathematicians do not test theories empirically; such theories would become the domain of physics or biology or the like. Thus, the only evidence one could infer 2+2=3 from would be misleading mathematical evidence.

    Since 2+2=4 is so simple, there are not too many people who could be effectively mislead in this ... (read more)

    There is an example that often floats around, where it is 'proven' mathematically that 1=2 (or some other such equality, by the principle of explosion it doesn't really matter). The trick is that at some step in the proof, a non obvious division by zero occurred.
    I imagine it's the same proof that makes 2+2=5. There is a point in the proof where the correct result is obviously 0=0 (though never explicitly written), yet it continues as though it didn't happen. It's an example of making the problems so complex that you make a mistake, it's not a valid proof. The proof for 2+2=4, incidentally, is almost 400 pages long. The simplistic versions most use take for granted many things for granted (like + and = and 2) that the actual proof does not.

    Eliezer: I am using the standard definition of 'a priori' due to Kant. Given your responses, I conclude that either you don't believe a priori claims exist (in other words you don't believe deduction is a valid form of reasoning), or you mean by arithmetic statements "2+2=4" something other than what most mathematicians mean by them.

    (some arguments)
    Don't care. If you can reverse entropy, you might as well be a god. If some alien gives me technology to reverse entropy, then A God Am I.

    Eliezer: It sure seems to me that our evolution and culture constructed ethical attitudes are entangled with the world. By the way, I don't think that we agree at all about what "I find it quite easy to imagine" means, but of course, some words, like "I", are tricky. It might be more interesting to ask "what data could I give a soundly designed AGI that would convince it that 2+2=3?" For you and for sound AGI designs, I'd like to know what situation would be convincing regarding the proposition "beliefs should not resp... (read more)


    I'm neither Eliezer nor (so far as you know) an AGI, but I think (1) I couldn't be convinced by evidence that beliefs should not respond to evidence but (2) I could be led by evidence to abandon my belief that they should. (Probably along with most of my other beliefs.) What it would take for that would be a systematic failure of beliefs arrived at by assessing evidence to match any better with future evidence than beliefs arrived at in other ways. I think that would basically require that future evidence to be random; in fact that's roughly what "ran... (read more)

    "I'm not sure that I can actually imagine a world like that, though." A computer simulation with infinite processing power that runs a person from an initial state (the standard one is solely based on their genetics, but for your experiment we could use a brain download of a given human from partway through their life) through all possible sequences of sensory inputs.

    Mathematics is about logical patterns. A world in which you can be mistaken about such fundamentals as the value of 2 + 2 is not a world where you can put any trust in your logical deductions. As such, if you ever do notice such a slip, I suggest that the cause is likely to be something deeply wrong with you, yourself, and not that you are living in a computer simulation.

    The test of any religion is whether cultures believing it tend to thrive and improve the quality of their lives or not. The whole point of the word of God is that following it gives you... (read more)

    "The test of any religion is whether cultures believing it tend to thrive and improve the quality of their lives or not." Um, I'm pretty sure the test of a religion is whether or not the model of reality proposed by that religion corresponds with actual reality or not (sorry I'm not sure how to phrase this in terms of a "test", without assuming the validity of sensory input). This is particularly noticeable in the case of religions which claim afterlives, where any impact of earthly actions on our afterlife utterly outweigh any impact that a religion has on earthly conditions. The very idea of debating whether a religion improves our quality of life on earth only makes sense from an Atheist or Agnostic viewpoint, considering whether that religion can be used as a practical tool regardless of it's truth.

    Wikipedia on a priori: Relations of ideas, according to Hume, are "discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe".

    This points out clearly the problem that I have with "a priori". It is a fundamentally Cartesian-dualist notion. The "mere operation of thought" takes place INSIDE THE UNIVERSE, as opposed to anywhere else.

    To observe your own thoughts is a kind of evidence, if the spikings of your neurons be entangled with the object of your inquiry (relative to yo... (read more)

    What has this to do with Peano Arithmetic and a mathematical proof "PA proofs 2+2=4" which is merely a string of symbols? On the other hand, what has PA to do with reality of earplugs except the evidence that PA is a good model for them? There is no miraculous correspondence, there is in fact a lot of evidence that FALSIFIES 2 + 2 = 4, like if it is 11 o'clock and 3 hours pass, it is 2 o'clock, and you can pour one glass of water and one glass of water into one glass of water, not to mention the already mentioned photons. So 2 + 2 = 4 seems acutally to be true only when we "know what we are doing", when we are applying it "correctly". (and I am sure that in the world where 2+2 earplugs lead 3 earplugs, you may still find instances where 2+2=4 (like photons or whatever).) But applying "correctly" bears a lot of information about how and where you should be entangled with reality in order to claim 2+2=4. That information is the difference between pure and applied mathematics. Also that is why there are two meanings of 2+2=4 which seem to have been mixed up in some of the discussion above. And that is what is meant by "2+2=4 is true in (pure) mathematics independently on whether or not it is true in the reality [when applied]". Using "a priory" is misleading, here I agree. Of course it is also concievable that you wake up one morning and PA proofs 1+1=3 BUT 2 ear plugs + 2 ear plugs is still 4 earplugs! Isn't it? I'm getting confused... if 2 ear plugs placed next to 2 earplugs lead 3, then how can you reliably write more than 3 sybmols next to each other to give a proof of anything from PA? spooky
    I think you've pointed out an issue of semantics, not falsified 2 + 2 = 4. If you pour one glass of water into another glass of water, you have one glass of water—but " one glass", in that case, is qualitative and not quantitative; it's not math.
    In regards to Hume's interesting contributions to the question, I stumbled across this video a while back which I think will be interesting: (don't let the title throw you off; there is content within it).

    It is perfectly acceptable for me to say, "I can think of no encounterable situation that would transform the terminal value of this event from negative to positive."

    Now, don't make me bring up a trolly problem. :-)

    This sentence of Eliezer's is where the action is:

    I'm suspicious of claims that supposedly do not require justification and yet seem to be uniquely preferred within a rather large space of possibilities.

    "There are no married bachelors" gets us to nod our heads because we uniquely prefer English syntax and semantics. We pick it out of the rather large space of possible languages because it's what everyone else is doing.

    If Eliezer went around earnestly saying, "there are some married bachelors," I would guess he had entangled himself w... (read more)

    I concede (a little)!

    In a previous Overcoming Bias post we learned that people sometimes believe the conjunction of events R and Q is more probable than event Q alone. Thus people can believe simple and strictly illogical things, and so I shouldn't throw around the word "unthinkable."

    If I stretch my imagination, I can just maybe imagine this sort of logical blunder with small integers.

    I draw the line at P AND ~P, though: just unthinkable.

    "It appears to me that "a priori" is a semantic stopsign; its only visible meaning is "Don't ask!""

    No, a priori reasoning is what mathematicians do for a living. Despite operating entirely by means of semantic stopsigns, mathematics seems nevertheless to enjoy rude health.

    There are really two questions in there:

    • Whether the Peano arithmetic axioms correctly describe the physical world.
    • Whether, given those axioms and appropriate definitions of 2 and 4 (perhaps as Church numerals), 2 + 2 = 4.

    One is a question about the world, the other about a neccessary truth.

    The first is about what aspect of the world we are looking at, under what definitions. 2 rabbits plus 2 rabbits may not result in 4 rabbits. So I have to assume Eliezer refers to the second question.

    Can we even meaningfully ask the second question? Kind of. As... (read more)

    I draw the line at P AND ~P, though: just unthinkable.

    I've heard religious people profess beliefs of this nature. I don't think they actually believe it, but I don't think it's pure belief-in-belief either; I see it as an attempt to explain a deeply unusual subjective experience in poorly suited language. (Which is not to say I think any statements like that are metaphysically true or anything.)

    I do think there's something to "a priori" besides a mere semantic stopsign, though. I could model physically possible worlds with different contents, or ... (read more)

    So the actual end result would be to convince me that the universe was in the hands of a monstrously insane and vicious God. As I noted here, that is actually pretty much what I believed in the last days of my Christianity. My perspective on ethics made it more plausible to me than I suspect it would be to most people.

    The whole point of Christianity (as I grew up with it) is that by manifesting Himself on earth God realized that the whole smiting people thing was passe. I always thought the God of the New Testament was just that of the Old with better mark... (read more)

    Perhaps 'a priori' and 'a posteriori' are too loaded with historic context. Eliezer seems to associate a priori with dualism, an association which I don't think is necessary. The important distinction is the process by which you arrive at claims. Scientists use two such processes: induction and deduction.

    Deduction is reasoning from premises using 'agreed upon' rules of inference such as modus ponens. We call (conditional) claims which are arrived at via deduction 'a priori.'

    Induction is updating beliefs from evidence using rules of probability (Bayes th... (read more)


    It is possible in today's wonderful world of computers to have 2 + 2 = 3, and be both correct and understandable.

    For Instance:

    We have two integer variables x and y. Our equation is x + x and the outcome is placed in y (ie. x + x = y) We will view the value of y.

    We take the value 1.7 and input it into x. Since x is an integer it will (in most cases) be rounded to 2. Therefore x = 2.

    It is possible, however, for y to receive the value of 1.7 + 1.7 which, in today's accepted math, equals 3.4.

    Placing 3.4 in an integer variable will set y to 3.

    Therefore, you have 2 + 2 = 3.

    BTW, this is why doing floating point math with integer variables on computers is a very bad idea......

    I've not read all of the comments, but those that I've read from you, Eliezer, in combination with the original blog post, confirm that we are in agreement. Re: Locke, I believe we are blank slates when born. There is no such thing as a priori (how do I italicize?). All thinking, even logical and mathematical reasoning, is done a posteriori. Of what I've read, you've put it brilliantly.

    Cloud, you might want to read Steven Pinker's "The Blank Slate".


    I recall my music teacher once put a quote on the board which I shall now adjust to the problem: Take 2 piles of sand and 2 more piles of sand and add them together. What do you get? 1 or more piles of sand.

    Not directly applicable to the general understanding of integers, but amusing to me. You could also do similar quibbles with musical tones or beats.

    Then again it could all be rubbish...for I don't think I could argue any of the points argued so far, though I do find my attempt at understanding it enjoyable if not complete.

    Yet if you counted the grains of sand, you would have as much sand as is contained in four piles of sand - 2+2=4. This is the same as saying when I add 2+2 and get 4, I start with two numbers and only get one number. It's true, but you've fooled yourself into believing this is some profound mathematical truth (and in a sense it is, but not the way you originally thought), when in fact it was so obvious to anybody who wasn't trying to fool themselves that it did not need pointing out. This is also the same as starting with two groups of two apples, adding them together, and getting one group of four apples. I'm not disappointed by this result. In fact, the very reason I have four apples is because I have merged two groups of two apples into a single group. The result of this merger is four apples.
    Could you please define a "pile"? :3
    A pile is a collection of 0 or more grains of sand.

    "Cloud, you might want to read Steven Pinker's 'The Blank Slate'."

    Perhaps the term "blank slate" carries too much baggage. I only mean it with respect to the a priori/posteriori or rationalism/empiricism. Disclaimer: my eclectic survey of much of Western thought has blurred the lines defining these terms. So take from this what you will, but I can't guarantee myself being clear.

    For the statement 2+2=4 to be true there are some assumptions that needs to be. That is 2+2=4 is true within a system, mathematics, but this system is in fact a construction!

    The basic assumption here is that we can define and identify 'one' thing - say a ball, a man or any other "item" - for this to be true you would further need to have 'identical' items... that is items that have very similar attributes.

    As you can see this leads to a infinite regress, where one assumption leads to others, and in fact we don't have such systems in reality, that ... (read more)

    In response to g (a while back, concerning entropy): If physicists discovered such a technique, omniscience of a sort(by arbitrarily altering and measuring the amount of information in a given region) would be possible, as would a form of omnipotence (we could arrange any concievable configuration of particles via Maxwell's demon). Hooking it up to a computer with some knowledge base of usually-accepted morals to this quantum entropy-decreasing construct, we would have omnibenevolence, also - hence, such a being would, indeed, be (an approximate) God by mo... (read more)

    Thanks for an excellent post. I think you have summed up the distinction between beliefs arising out of blind faith and those that are observation based.


    This time I disagree with Eliezer...this experiment won't convince me that 2+2=3...wouldn't even convince me that physical maxim "everything goes somewhere" is wrong...I would find where the phones are (even if they sublimated). That still don't make that an "imutable belief".

    There's nothing wrong in switching lexically 3 and 4 ( S(2) = 4; S(4) = 3; S(3) = 5 )...sounds unuseful, and don't attack Peano's axioms. That would make me believe in 2+2=3.

    To stop believing in the integer numbers, it's needed to prove an inconsistency in Peano's ... (read more)

    This time I disagree with Eliezer...this experiment won't convince me that 2+2=3...wouldn't even convince me that physical maxim "everything goes somewhere" is wrong...I would find where the earplugs are (even if they sublimated). That still don't make that an "imutable belief".

    There's nothing wrong in switching lexically 3 and 4 ( S(2) = 4; S(4) = 3; S(3) = 5 )...sounds unuseful, and don't attack Peano's axioms. That would make me believe in 2+2=3.

    To stop believing in the integer numbers, it's needed to prove an inconsistency in Peano'... (read more)

    It's often poor form to quote oneself, but since this post (deservedly) continues to get visits, it might be good to bring up the line of thought that convinced me that this post made perfect sense:

    The space of all possible minds includes some (aliens/mental patients/AIs) which have a notion of number and counting and an intuitive mental arithmetic, but where the last of these is skewed so that 2 and 2 really do seem to make 3 rather than 4. Not just lexically, but actually; the way that our brains can instantly subitize four objects as two distinct group... (read more)

    I had parallel thoughts at one time, and discovered with some effort that I could train myself to believe that 1+1=3. It took about five minutes of mental practice. What eventually happened was that every time I combined two objects together mentally (abstractly), I simultaneously imagined a third which had the bizarre property that it only existed when the two objects were considered simultaneously. If I thought of just one object, the third disappeared, if I thought of the other object, it again disappeared -- it only appeared as an emergent property of the pair. Thus imagining 1+1=3 was discovering the following "operation": {E} + {F} = { {E} , {F} , { {E},{F} } } Looking at the cardinality of the sets, we have: 1 + 1 = 3 Could such an operation be 'logical' and yield a consistent number theory? (I don't know. I think it's a question in abstract algebra. (Rings, fields, groups, etc.) Are there any algebraists here that can comment?) Yet orthonormal is suggesting the case that 2+2=3 doesn't result in a logical, consistent theory -- the possible minds just believe it due to an internal error, and they can use the inconsistency of their theory to deduce the internal error. However, I find it really difficult to think of 2+2=3 happening as a mistaken Peano arithmetic instead of the assertion of another type of arithmetic. The possible logical self-consistency of this arithmetic further confounds: if it's self-consistent, they may never deduce that they got Peano arithmetic wrong. If its not self-consistent, they can prove all propositions and how will they know where the error lies? Or even understand what error means? If there is an error in our reasoning, it cannot be so fundamentally embedded in our understanding of logic.
    Nice, but the difference with this "belief" is that you're talking about sensory "counting" (visual grouping), and I was talking about the numbers themselves, as models for games, other phenomena, etc., and not just as a "counting" tool. In the 1+1=3 example, to define the cardinality, he/she used the Peano's axioms, didn't he/she? I don't see the "visual sensory counting" as the only use for "2+2=4", that's why I don't think this experiment would refute such a priori content. Another idea: let Ann be a girl with hemispatial neglect in a extinction condition. Ann has problems detecting anything on the left, and she can possibly see 2+2=3 as idealized above, due her brain damage. Will she think that 2+2=3? I don't think so...but if she does...will that be a model for all "integer numbers" aplications? I think in "integer" as a framework for several phenomena, other models, other knowledge, not only the counting one. For the minds that see 2+2=4 as something patently absurd, because 2+2=3 is part of their intuitive arithmetic, these minds probably won't see the 2+2=4 even when brought to a world like ours. After a time in the 2+2=4 world, they probably won't forget that 2+2=3, unless the 2+2=3 wasn't modeling anything else. But the 2+2=3 was modeling something in their past history, at least the counting principle of their world. So they still have the 2+2=3 belief in their lives while they remember their past. If they forget their past, the 2+2=3 belief might became unuseful, but that still don't make the 2+2=3 an absurd or replaced by the 2+2=4: there are 2 number systems here. For me, 2+2=3 isn't an absurd. That might be seem as a "common sum with a 3/4 multiplier" or a "X + Y = X p Y/X" where "p" is our common sum and "/" is our division, etc.. This way, like the 1+1=3 example above, only overloads the "+" operator. But, again, this "+" isn't the same from the "2+2=4"
    Nice, but the difference with this "belief" is that you're talking about sensory "counting" (visual grouping), and I was talking about the numbers themselves, as models for games, other phenomena, etc., and not just as a "counting" tool. In the 1+1=3 example (byrnema answer, just below), to define the cardinality, he/she used the Peano's axioms, didn't he/she? I don't see the "visual sensory counting" as the only use for "2+2=4", that's why I don't think this experiment would refute such a priori content. Another idea: let Ann be a girl with hemispatial neglect in a extinction condition. Ann has problems detecting anything on the left, and she can possibly see 2+2=3 as idealized above, due her brain damage. Will she think that 2+2=3? I don't think so...but if she does...will that be a model for all "integer numbers" aplications? I think in "integer" as a framework for several phenomena, other models, other knowledge, not only the counting one. For the minds that see 2+2=4 as something patently absurd, because 2+2=3 is part of their intuitive arithmetic, these minds probably won't see the 2+2=4 even when brought to a world like ours. After a time in the 2+2=4 world, they probably won't forget that 2+2=3, unless the 2+2=3 wasn't modeling anything else. But the 2+2=3 was modeling something in their past history, at least the counting principle of their world. So they still have the 2+2=3 belief in their lives while they remember their past. If they forget their past, the 2+2=3 belief might became unuseful, but that still don't make the 2+2=3 an absurd or replaced by the 2+2=4: there are 2 number systems here. The "2" in the "2+2=3" is different from the "2" in the "2+2=4". For me, 2+2=3 isn't an absurd. That might be seem as a "common sum with a 3/4 multiplier" or a "X + Y = X p Y/X" where "p" is our common sum and "/" is our division, etc.. This way, like the 1+1=3 example, only overloads the "+" operator. But, again, this "+" isn't the same from the "2+2=4"
    Upon suddenly discovering that the whole world looks different this morning than it did last night is the rational belief "I guess I was deluded for my whole life up to this point" or "I guess I'm deluded now"? Considering the fact that you're not waking up in a mental institution, but the world still seems to contain them (and if you get 2 sets of 2 of them, you have 3); the latter is a much more likely situation
    You're neglecting the hypothesis "my memories of the past are being distorted to convince me that 2 and 2 make 4 instead of 3". Given how easily we distort our memories under conditions of sanity, this is as likely as "I'm deluded now".
    If you suddenly gain a set of memories indicating that the raptor conspiracy is taking over the world, you would be considered deluded. If you suddenly gain a set of memories indicating that 2+2 equals something other than what it DOES in fact equal, you are likewise deluded. So your suggestion is in fact a subset of being deluded*. At which point you should voluntarily seek out psychological/psychiatric help. * (which I assign a low probability, as I have never heard of such a type of delusion existing) If you believe (as you seem to suggest by use of the aactive rather than the passive voice) that this delusion is being deliberately induced, it is important to remember that anyone with the power to induce that delusion could also reduce you to a gibbering wreck; and hence that going to get help is highly unlikely to be "part of their plan".
    This is a distraction from the actual point; of course if this happened to me, then my first priority would be getting help (I might be having a stroke, for instance). But once I'm at the hospital and they tell me that I'm all right, but something strange happened to my brain so that it falsely remembers 2 and 2 having made 4, instead of the obviously correct 3... If you don't agree that some set of circumstances like this should conspire to make me rationally accept 2+2=3, then if the scenario happened to you (with 3 and 4 reversed), you're asserting that you could never rationally recover from that metal event. Since I'd prefer, should I go through a hallucination that 2 and 2 always made 3, to be able to recover given enough evidence, I have to take the "risk" of being convinced of something false, in a world where events conspired against me just so.
    Why completely leave out the possibility that you aren't deluded at all? Depending on just what kind of 'different' you wake up in that is a distinct possibility. I would, by the way, start with a high prior for 'deluded now' which would be altered one way or the other by extensive reality testing. I experience that in dreams all the time. I know from personal experience it is easier for me to be confused about the transient sensory experience of the present than the broad structure of all my memories. Results may vary somewhat.
    Good point, in the case of waking up in a logically possible world, remembering a previous logically possible world, there is a non-zero possibility that you've actually gone from one to the other somehow. How low the probability is depends on the nature of the differences I was too caught up in the case of waking up in a world where the world you remember is logically impossible.
    Exactly. And with slightly different wording a world in which it seems like you have changed from one logical world to another is itself a just a logically possible world. That would be awkward! It would require an awful lot of reality testing on the question of just how logically impossible things were. Even after that your confidence in just about anything would be fubared.

    For a 5-year-old, saying "You're not not not not fat" is just another way of saying "You're fat."

    For an adult, saying "(the sum of) 2 + 2" is just another way of saying "4."

    For an entity far more intelligent than humans, stating the appropriate set of axioms is just another way of stating Euler's identity, Cauchy's integral theorem, and all sorts of other things.

    1tlhonmey This one also seem apropos.

    What I gain from this article is more or less an example of society's influence on how you understand things. For example, for most people 2+2=4. If the counting system and math operations was completely different, 2+2 could equal anything, unless one was familiar with the high-context culture using such a system. Another example would be the projection of an idea with words. One may say express their emotion as the word "happy". Another may say "joy". Another "euphoria". Unknowingly, all three have the same exact emotion, only their words have their different connotations. Suprisingly enough, I seem to have confused myself. Does anyone want to try to discern what I've said?


    There seem to be far too many people hung up on the mathematics which ignores the purpose of the post as I understand it.

    The post is not about truth but about conviction. Eliezer is not saying that there could be a scenario in which the rules of mathematics didn't work, but that there could be a scenario under which he was convinced of it.

    Deconstructing all elements of neurology, physics and socialogy that make up the pathway from complete ignorance to solid conviction is not something I could even begin to attempt - but if one were able to list such steps as a series bullet points I could conceive that the manipulation of certain steps could lead to a different outcome, which appears to me to be the ultimate point of the post (although not hugely ground-breaking, but an interesting thought experiment).

    It is not a claim that the strongly held conviction represents fact or that the conviction would not be shaken by a future event or presentation of evidence. As a fundamental believer in scientific thought and rationality there is much that I hold as firm conviction that I would not hesitate to re-think under valid contradictory evidence.

    I wish I could vote you up so much more! The distinction between a-convincing-argument and what-it-would-take-to-convince-me is very real and overlooked by almost everyone posting here. To take my own experience in becoming convinced of atheism, I sometimes like to think I accept atheism for the same reason that I accept evolution--because of the evidence/lack thereof/etcetera. But that is simply not the case. I accept atheism because of a highly personal history of what it took to get me, personally, to stop believing in Christianity, and start believing in something else that, as much as I would like to pat my rational self on the back, has fairly little to do with the arguments and evidence I heard on the matter. When asking someone why they believe something or are convinced of it, "what is the reason?" and "what is your reason?" are two totally different questions.
    It's possible that the reason you accepted atheism is different to the reason you currently accept atheism. To make an analogy, I use the QWERTY keyboard now because it's the industry standard and therefore the most likely keyboard layout for me to encounter on an unfamiliar machine, regardless of the fact that I learned the QWERTY keyboard because that's what was the setting on my computer when I started posting obsessively in the Dominic Deegan forums.

    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 ... (read more)

    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 students are doing better with this more empirical, constructivist approach. (My niece is good at math anyway, so I don't know if she's getting more out of it than average.)
    I don't know very much about the American curriculum, having grown up with the Canadian one. But I also didn't pay very much attention in math class. I preferred to read the textbook myself, early in the year, and then play around with as many derivations and theorems as I could figure out, occasionally popping my head above water long enough for a test. I wrote and memorized my own subtraction tables, and invented a baroque and complicated system for writing negative numbers -- for example, 1 - 2 = 9-with-a-circle-around-it, and 5 - 17 = 8-with-two-circles-around-it. Really this is the sort of mistake which could only have happened to me. :) I'm glad that they're teaching these sort of strategies in US schools. My experience tutoring elementary school math (my son attended an alternative school in which parents all volunteered their own skills & experience) is that every kid has a slightly different conception of how numbers interact. The most important thing I could teach them was that every consistent way of approaching math is correct; if you don't understand the textbook's prescription for subtracting, there are dozens of other right ways to think about the problem; it doesn't matter how you get to the answer as long as you follow the axioms.
    I never bothered to memorize trig equivalences. Instead, I just reduced sine, cosine, and tangent (and their inverses) to ratios of the sides of a triangle, and then used the Pythagorean theorem.
    Well, it's so much easier and more robust that way! Instead of a long list of confoundingly similar equations, you're left with a single clear understanding of why trigonometry works. After that you can memorize a few formulas as shortcuts if it helps. Of course this principle completely breaks down when you start working with a child who's already convinced that they can't do math—or with a group of 30 kids at once, a third of whose mathematical intuitions will be far enough from the textbook norm that no one teacher has enough time to guide them through to that first epiphany.
    Well, it does also matter in practice that you can communicate effectively (a lesson I had to learn myself at that age). But learning how to translate from an idiosyncratic system into a standard one can be a source of even better learning, so I agree that kids should not be discouraged from inventing nonstandard but valid systems.
    Your method of subtraction is similar to being the p-adic numbers, you might want to look them up!

    I cannot conceive of a possible world where “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.” Unless, in that possible world I did not know how to reason. If 2 + 2 really was 3, what would 1 + 2 be? Not 4, since then 2+2 = 2+1 and since subtraction is defined as the inverse of addition (if its not, its not subtraction) we would have 0 = 1. Not 3, since in the world you’re imagining ... (read more)

    I tend to think that a physical system of numbering and an entirely nonphysical system of belief as apples and oranges- entirely incomparable. Specifically, adding or subtracting earplugs is an entanglement of reality and belief whereas choosing eg. christianity or islam is simply something of belief- yes, that spiritual belief is affected by your reality (environmental factors like schooling, parents and location, obviously) but in the end, it is still a belief- for example, if a person never heard of Jesus or Muhammad but nonetheless believed in a higher... (read more)

    One might indeed "believe" all that. But a belief has no use if it isn't true.
    Apples and oranges have more ways they are alike than not alike. I always have to bring this up when someone makes the "apples to oranges" statement. It's only true so long as you are purposefully ignoring all the ways they are alike. In other words, it is just as valid to compare apples to oranges as it is to compare fuji apples to granny smith apples. That's just me being pedantic, but it really seems to apply in this particular case.
    Apples and oranges are alike in more ways than they are not alike. I always have to bring this up when someone makes the "you can't compare apples to oranges" statement. In fact, it is quite reasonable to compare apples to oranges. It's also reasonable to compare apples to eighteen-wheelers. It's only unreasonable when you are explicitly ignoring all the ways they are alike. Even then, it isn't particularly unreasonable to compare two things that are completely dissimilar. I'm being a little pedantic, but it really seems to apply in this particular case.

    OK, I'm a Christian. Bit of history: -raised christian -As a teen became agnostic/deist, atheist at 17 -Converted to Christianity at 18

    Based on rational thinking I drift towards deism/agnosticism. I'm skeptical of microbes-to-man evolution and abiogenesis. But if abiogenesis could be demonstrated, or if evolutionary processes could be demonstrated to be capable of producing the kind of complexity we see in biology (e.g. evolutionary algorithms run on supercomputer clusters producing real AI) then I'd probably drift towards atheism.

    Anyway, at 18 I became a ... (read more)

    Many other people have such experiences, high or no. Some Hindu, some Muslim, some Pagan, some even atheists. To be blunt, do you doubt their sincerity, or their sanity? Why are you epistemically privileged?

    To the extent that their experiences do not contradict mine, I see no reason to doubt. There is nothing in Christianity that prevents non-Christians from having religious experiences. But when the experiences of others do contradict mine, such as the revelations Joseph Smith or Mohammed received, I have to doubt their sincerity or their sanity (I don't know which) for the same reason you doubt mine: Because I can't see in their mind and I wasn't in their body when it happened. And if I have to choose between my own experiences and another persons experiences, I choose my own. But I should mention that of all the people I trust and who have told me their religious experiences (mostly hindu family members) to date none of them has proven a challenge to my Christianity.
    Luckily, we need not be limited to those hypotheses. Neither you nor many of the others with similar experiences need be lying or insane. And the existence of an omnipotent, omniscient and omni-beneficent deity need not enter into it either. You just have to have brains. Welcome, by the way.
    I've considered those kind of explanations, but the nature of the particular experiences which caused me to convert does not lend itself to that kind of explanation. My policy is to never discuss the details with someone I do not personally know and trust, but I will say this much: the evidence was external and observed and confirmed by trusted others. In fact if you are familiar with Zero Knowledge Proofs (I'm a crypto geek) the evidence was a type of ZKP that allows me to know with certainty (to the extent that I can trust my own rationality and senses) without enabling me to duplicate the proof.
    You're being a very good sport about this; and seconding Jack, welcome! It is important to understand that if no religious experiences were mutually exclusive with Christianity (nobody ever saw Ganesh or Mohammed), then they would count a lot more strongly as evidence for Christianity. But many are mutually exclusive, and doubting the sincerity of every Sufi mystic who saw God is a move that requires strong evidence. As to another person's experiences vs your own: I sympathize, I really do. But you need to have some epistemic humility here, and realize that "you" are encoded in about half a kilo of mushy grey stuff that is often very untrustworthy. I for one do not doubt your sincerity (or the Sufis') but I do doubt that you correctly interpreted your experience.
    Hello! As you're no doubt aware, the general tenor of Less Wrong tends toward non-belief in religion. However, in contrast to many religious believers, you have expressed a willingness to alter your views in the face of evidence. Watch out! Even your tentative suggestion that you might "drift towards atheism" might cause you to be regarded as a heretic or at least untrustworthy in some churches. But if you're willing to commit yourself to pursuing the truth wheresoever it may lead, then congratulations! As has been mentioned already in this thread, Judaism and Christianity historically do not claim to be non-disprovable. Elijah bet his God against Baal and (in the Biblical narrative) won. Do you think this experiment can be replicated? Alternatively, is there something equivalent to a "similar or better revelation" that could convince you that no organized religion is correct at all?
    My parents don't consider me a real Christian, somehow I cope. ;-) Not only do I believe the Elijah experiment can be replicated, I believe it is being replicated today along with many other miracles. Just hidden for most people, because in Christianity, God reveals the truth to those who he chooses (poor/humble/righteous people) and keeps other people (rich/wicked/prideful) blind. So God might raise someone from the dead but in a way that could not be publicly verified, lest the rich proud people who think they're so smart find out the truth. I fail to see how a supernatural revelation could prove no (organized) religion is correct, short of God saying "no religion is correct", which would then cause me to create my own organized religion... But Christianity could surely be disproved in many different ways. For one, aliens or real sentient AI would disprove Christianity AFA I'm concerned. I'm not yet 30, so maybe I'll discover it in my lifetime. If Christianity were disproved, that would leave Buddhism and Deism as the only viable religions left IMH(current)O. And Deism is only necessary in so far as I find the evidence for abiogenesis and humans-created-by-evolution lacking. So except miracles and creation, I could be an atheist.
    Now I'm wondering which of those categories I fit in to. They all sound a tad appealing. :)
    If you are familiar with Christianity, all humans fall into the wicked and prideful categories. The fact that you are on the internet suggests you additionally fall into the rich one too. Now whether God sovereignly chooses his people (calvinism), or humans can also choose e.g. by humbling themselves (arianism) is an open question. Edit to add: Just because God hasn't revealed the truth to someone today, doesn't mean he won't do it tomorrow or even (though this is heresy) after death. So I certainly don't consider all non-Christians to be hopeless, after all I was a non-Christian too, once. And I also don't consider all who call themselves Chrstian to be chosen.
    I was sincere Christian right up until I realised the religion could be better explained by tribal signalling than magic. You just finished saying:
    A basic doctrine of Christianity is that poor, humble and righteous people are wicked and prideful too. Only Jesus is perfect. Some strains believe God choses for reasons we can't grasp and then those people become more humble and less prideful, etc. Others believe that if you do your best to be humble and righteous eventually God will reveal himself (though no guarantee that it will happen before your last minute on earth). I don't know which of the two it is, or perhaps it is something else entirely.
    You can see how non-Christians might find that to be a suspiciously convenient excuse, right?
    So because it makes sense it's suspiciously convenient? Obviously if there was a God (e.g. the Christian one) and he wanted the whole world to be nominal Christians he would do another Elijah like demonstration of his power, recorded on camera. This is obviously not the case. So either the Christian god does not exist (suspiciously convenient for the non-Christian?) or he does not actually want all those non-Christians to self-identify as Christians (suspiciously convenient for the Christian god?)
    It's suspiciously convenient because your claim implies that that evidence of Christianity's truth is only available to people who already believe in it (or who are already much closer to believing it than their epistemic state actually warrants).
    Obviously, if the evidence of Christianity's truth was available to all then all would be Christians. Assuming the Christian god does not want all to be Christians the evidence should not be available to all. Anyway, when I received my experience I certainly did not want to believe in it. And even now many years later, I would prefer to abandon Christianity and its morality but find myself unable because of my experiences. I also know of a few other stories similar to mine, enough to convince myself I'm not delusional.
    I assume you mean stories of religious experiences similar to your own. This should not be evidence that you are not delusional, since many people throughout history have claimed to have had such experiences, with reference to different, mutually exclusive religions. On average, therefore, most (if not all) people having such experiences must have been delusional. You should have a probability that you are mistaken at least as high as this proportion.
    Minor quibble, but "delusional" would seem overly inflammatory as it implies the delusionality is a persistent property of Xaway's person, rather than the one-off occurrence it more likely was.
    As some people have pointed out, it's not a binary choice between you being crazy or delusional, and Christianity being right. Human brains complete patterns, in predictable ways. I don't know what your experience was (since you're keeping that private) but there are probably multiple possible worlds that are consistent with your experience: not just "Xaway is nuts" or "Jesus is the Savior." Think about what might have actually happened and what it might actually mean, and resist pattern-matching for a while. Just a word of info on this site: this is not a place where people generally debate religion. You sound like you have your doubts; I recommend you read the best atheist arguments (Bertrand Russell comes to mind), and read about the history of the Bible and early Church from a secular academic writer. Let it marinate for a while. Read widely and see what happens to your views. Sometimes debating on the internet isn't the best way to learn; it crystallizes whatever ideas you started off with and makes it hard to change your mind. If you would "prefer to abandon Christianity" but your experiences won't let you, you should really take some time to think about whether your experiences have religious implications. There are naturalistic explanations for religious visions, and no, they don't all mean you're crazy. (Check out Oliver Sacks on Hildegard of Bingen, and Robert Sapolsky on St. Paul.)
    I'd also recommend Why God Won't Go Away: Brain Science and the Biology of Belief by Gene D'Aquili and Vince Rauss.
    There were some things I thought of saying, but I think I'll hold my tongue for now. In short, I think your assertions have some logical errors. This is not a put-down or a personal comment -- I'm certainly no more than an aspiring rationalist, at best, myself. I hope you stick around this forum. In the spirit of Tarski I would ask you to join me in saying: If Christianity is true I desire to believe that Christianity is true. If Christianity is not true, I desire to believe that Christianity is not true. I would say this, and do!
    If you spot a logical error, bring it on. Obviously I don't want to believe untrue things. But if there is two things I am sure about, it's (1) that humans are not rational, especially not me and (2) there are things that are true which can not be proven to be true (the real world analogue to Godels theorems). I frequent this site, but I generally do not participate in internet discussions. I only registered this account and gave my two cents because Eliezer asked for a Christian who speaks Bayes to chime in. I'm afraid that once I log off, I will probably forget the password to this account.
    Again, I hope you stick around. No need to burn yourself out as the lone voice of Christianity -- pacing yourself is fine. Also, this truly is a rationalist site. If you can present well-thought out arguments, people here will listen to you. If you can make a rational argument demonstrating the truth of Christianity, then (according to some denominations) you could save some souls. (I understand the Calvinists would not necessarily agree.) But according to some traditions, good works (not just fide sola) have merit, and evangelizing is one of the greatest of all good works. Is it not? My ulterior motive in making that argument is that I also think this forum could benefit from the perspective of a Christian who speaks Bayes.
    I think I'd rather have a better calibrated Frequentist.
    I'd rather have a rock. Or a Christian who doesn't speak Bayes. At least that implies less doublethink.
    Christianity here is actually a memetic hazard. It's a set of beliefs that has so many things wrong with it all of us feel compelled to address all of the bad thinking and wrong evidence all at once. It immediately draws everyone away from whatever productive comments they were making and into an attempt to deconvert the interlocutor. The interlocutor then responds to these attempts with more nonsense in different places which draws still more people in to the battle. Better to just keep the Hydra's out than try and chop off all those heads. No one here is actually at risk but we don't get anything to justify the strain on the immune system.
    I can think of some counterexamples. We "got" SarahC, for instance (according to her own words), and that was an unadulterated boon. Also, the claims of religion are varied enough that they provide a range of topics, many trivial but some interesting. E.g., if we were in a sim and somebody changed it from outside in violation of the sim's internal physical law, that would constitute a "miracle" at this level of reality. How would we recognize such an event from inside?
    A lot of Sarah's comments were made this summer when I wasn't around, so I may have missed something but I quick glance confirms that she is not a believing Christian. She certainly hasn't argued for the truth Christianity, which is really my concern. Which we can discuss successfully without real Christians.
    Sorry, I was unclear in speaking. I meant she acknowledged LW's influence in her deconversion, and is no longer religious. I think she started out Jewish actually. I can't seem to find the relevant comment/post.
    I was never Christian, I was raised Jewish, and now I don't believe in God. And, yes, LessWrong contributed. (I think, IIRC, we also have a member who was raised Muslim and recently became an atheist since he found LW.) I don't think you can randomly deconvert someone who isn't already seeking a change. Like most major changes in belief or lifestyle, deconversion has to be self-motivated. But if a Christian (or other religious person) is hanging around LW and not trolling, then he's probably looking for some alternatives, and there's no harm pointing him in that direction.
    My reason to abjure God was mainly due to ethical reasons. I didn't want to follow something anymore that had deliberately designed such an hellhole of a universe. Later I became an atheist mainly for noticing that nothing natural really appeared to be intelligently designed. Just look at the moon, the shape of the continents etc., or that we live on the surface of a sphere rather than inside a Dyson sphere. The next big step came via science fiction, when I noticed how easy it would have been to design a universe where nothing could suffer horribly. What Less Wrong added on top of all else I learnt is that Occam's razor has been formalized. I didn't know about that before LW. I just don't see that anyone would need Less Wrong to stop beliving into one of the Abrahamic religions. It should be obvious to anyone who isn't morally bankrupt or a psychopath that God is not your friend, rather it is your worst enemy. If that doesn't convince you, why not just read the Bible:
    And yet simple observation confirms that it is not obvious to many people who are clearly not sociopaths or more morally bankrupt than usual. It's completely ordinary for people to rationalize away inconsistencies or flaws in their beliefs with as little revision as possible. Making large alterations to account for large errors is a rare and difficult to learn skill.
    Yes, obviously, as I am used to from my parents. Sadly none of them would read LW or not rationalize away what is being said here like so much else. I believe that those who abjured religion because of reading something like LW are rather an exception. I was really addressing religious people, with what I call my shock and awe approach to crack their stronghold of subjective moral superiority. To paraphrase what I said, you are dumb, ignorant and morally abhorrent if you do not abjure your God. Yep, that might not work, but it does reflect my weariness. So never mind my little tirade, I lost my sense of location awareness for a moment there ;-)
    I was aware of the moral aspects; but I was confused by the notion that I seemed to disagree with God and I thought this was my fault. I had a problem with the story of Pinchas, but I thought that was me just being "soft" or "secularized" and I was really unsure whether to trust my own sense of morality. (One thing we should all understand here is that "conscience" is very far from infallible.) What changed my mind is a sense that my brain is all I've got. I may be wrong about many things, but I'm not going to become less wrong by throwing out the majority of what I know in favor of one ancient and rather bloody book; if "conscience" isn't trustworthy, it's still probably more trustworthy than simple conformism.
    If you replace God with Yudkowsky, story of Pinchas with AI going FOOM and soft, respectively secularized, with irrational and sense of morality with education (or worse, intelligence), then you got how I feel about another topic. I've always felt that conscience was just a matter of taste. So it was never really a question about how trustworthy my moral judgement is but that I care about it. I abjured God when I still believed that it exists. Only later I became an atheist. I suppose that is the difference between you and me here. You wanted to do the right thing (in an objective sense) and for me the right thing has always been that what I want.
    Although I appreciate some of the articles on this site, I don't think I'll participate much in the discussion. Although I speak Bayes and know more logic than a human should know, I do not consider myself a rationalist, because I doubt my own rationality. It wouldn't make sense for an inherently irrational person to spend his time trying to talk rationally when he could be dancing or programming. Also, I firmly believe that Christianity can not be proven by argument, only by evidence (miracles). And only God himself, not the Christian, can provide the evidence, which he does on his own terms.

    I do not consider myself a rationalist, because I doubt my own rationality.

    This site isn't called Always Right, you know.

    The truth status of Christianity is something that Less Wrong should be able to consider a settled question. We can debate about things like the Simulation Argument, etc. and other reductionist non-supernaturalist claims that look sorta like deism if you squint, but Jehovah did not create the universe, and Jesus is not Lord, and I don't think there's any point in humouring someone who disagrees, or encouraging them to come up with smarter-sounding rationalizations. Let's not push Less Wrong in the direction of becoming the sort of place where these old debates are rehashed; there are more interesting things to think and talk about. Although it seems that Xaway in particular may not have come here with the intention of actually convincing anyone to believe in Christianity, I would propose in the general case that anyone who does want to should be referred to some place like /r/atheism instead.
    Well, I certainly don't think Jehovah created the universe. On the other hand, this thread is devoted to the consideration of the proposition that 2 + 2 = 3 -- arguably a settled question -- with the understanding that "a belief is only really worthwhile if you could, in principle, be persuaded to believe otherwise." I don't know if Xaway is going to be participating any more (hi, Xaway, if you're reading this!), but I was hoping that this might be a good exercise in practicing rational discussion. In part, I thought we could win him over to the dark side (joking about it being the dark side.)
    That is not 'truly rationalist'. Well thought out arguments for a preselected bottom line are bullshit.
    Perhaps you could go to 'Preferences' on the right and change your password to something easier to remember. Regarding your revelation and direct experience with God, I am very curious as to whether the revelation specified in any way which religion God would prefer you to participate in. (You wrote above that you think the Judeo-Christian religions seem more likely, only, so this leads me to believe the revelation wasn't that specific.) (Echoing Costanza's questions) How much error do you allow for knowing about God, but following the wrong religion? Even if Christianity seems most likely to you, what probability do you assign to any current organized religion being correct? I suppose the reason why I'm asking is that something like Christianity seems unnecessarily specific if you are potentially deist or atheist. Probabilistically, God could exist in a lot of different ways, and provide true revelations, long before all the specific things are true about Christianity.
    Did. Didn't work. Wrote you off. :)
    Arguably this is the case for everything (until we solve the problem of induction). In the meantime, I don't know of anything you can't assign a probability to or collect evidence about. As for whether this is an analogue to Godel's theorem (or, in times gone by, Russel's paradoxical catalogues - or in times yet to come, the halting problem) - no. Mathematical systems are useful ways to carve reality at its joints. So are categories, and so is computation. They can't answer questions about themselves. But reality quite clearly can answer questions about itself.
    How about the question of whether there is anything you can't assign a probability to or collect evidence about?
    You can assign a probability to that. I hadn't considered the question strongly enough to have a mathematical number for you, but I would estimate there is a 10% chance that there are things which I cannot assign a probability to or collect evidence about. (Note that I assign a much lower probability to the claim "you can't assign a probability to or collect evidence about x"; empirically those statements have been made probably millions of times in history and as far as I know not a single one has been correct) That said, "I don't know of anything you can't assign a probability to or collect evidence about" is true with a probability of 1 - 4x10^-8 (the chance I am hallucinating, or made a gross error given that I double-checked).
    Christianity isn't true.
    You're just mad they refused to canonize Samuel B. Fay.
    A God is a very complex entity. Positing one does not, therefore, help to explain biological complexity (unless you have an explanation for God). Even though we don't know how abiogenesis happened it is still orders of magnitude more likely than God existing given the relative complexities involved. That Christianity is true is also orders of magnitude more unlikely than you and your companions hallucinating your direct revelation-- the former being an extraordinarily complex hypothesis and hallucinations and general irrationality being quite common.
    Well that is just your biases... Because a God is supernatural any probability assigned to it existing is as arbitrary as any other. Obviously, if the P=1/3^^^^^3 then it would be absurd to see biogenesis or biological complexity as evidence for God. But if the P =0.01 then I, for one, see it as very strong evidence. I see no reason to prefer theism vs. atheism and I consider an extraordinarily low P to be biased towards atheism, but if that rocks your boat, have fun. That I am irriational and delusional is highly probable, in fact I am sure of it. But I have no choice but to trust my own faulty brain. I would certainly not consider you rational if you were to convert to Christianity solely based on reading my story on the internetz.
    This is really wrong, obviously, but my hopes that the inferential distance was manageable have been dashed. You might start here. I'm done though.
    What was that like? In particular, how could you tell that it was really a revelation and not any kind of temporary brain malfunction?
    Ha! Creepy. :3

    Ooh, ooh! I'll do you one better. I'm not just a Christian; I'm a Mormon. :P

    My goodness, what would convince me of non-Christianity? The problem here is that Mormonism has presented me with enough positive evidence that I'm reasonably certain of its veracity. So the conversion process would be two-tiered: first a strong positive evidence for Islam/Judaism/whatever, and second a strong disconfirmation of Mormonism, which I would then seek to corroborate by figuring out why on earth I received so many outlandishly unlikely false positives.

    Both of the tiers m... (read more)

    * The entirety of Lehi's journey from Jerusalem to the sea has been found to match up with actual geographical and cultural sites in the Arabian Peninsula. Lehi's journey follows what we now know to be ancient trading routes. Lehi's family buried Ishmael at a place which Nephi called Nahom, which has been found by name, exactly where it should be. More or less directly east of that site, on the coast of Oman, have been found two candidate sites for Bountiful, Wadi Sayq and Salalah, right where they ought to be, with every feature described in the book including: reasonable access from Nahom (i.e. no mountains in the way!), an inlet for launching a ship, fertile ground with "much fruit and... honey", timber to build a durable ship, year-round access to fresh water, a nearby mountain upon which Nephi could offer his prayer, available ore and flint, and wind and ocean currents favorable for launching a ship out to sea. Candidate sites for the Valley of Lemuel and the River of Laman have been found, right where they ought to be. * The practice of writing on metal plates, laughable in 1830, has now been well established as a legitimate ancient practice; one that Joseph Smith could never have known. He was mocked for his claim of receiving a "book written on gold plates" more than any other he made... even to this day! The practice of burying said records in stone boxes in the earth has been similarly credited. * The practice of writing in what Mormon called "reformed Egyptian" (Mormon 9:32-34) has recently been shown to be rather more accurate than might be expected, as demonstrated by Daniel C. Peterson in Review of Books on the Book of Mormon, Vol. 5, 1993: * Many tourists to Mexico will be well familiar with the use of cement in ancient America, for example in Teotihuacan; this information is had in the Book of Mormon, but was unknown in Joseph Smith's time. * Jacob chapter 5 offers many, many details regarding olive cultivation which match precisely with what i
    The places exist, but is there evidence of the actual journey? If I adopt this theory of evidence, I accept American Gods as non-fiction, because most of the places in that book exist. What evidence is there that Smith knew nothing of the practice of writing on metal plates? Who says it was laughable in 1830? It was known in Europe -- used almost everywhere in Rome. Are there specific architectural details that were unprecedented? Jacob 5 agrees with what, as Darwin would say, "every animal husbander knows." What exactly are the details that match? Are they unexpected? What proportion of random 3-character Hebrew strings do not correspond to personal names? I have read the Book of Mormon in the past, but I hereby precommit to reading it again and "searching in my heart" (I have a copy on my bookshelf) if you can demonstrate that my skepticism regarding your evidence is unwarranted.
    I will answer your points in the order received. First: your analogy is flawed, and, I'm sorry to say, rather obviously so. Neil Gaiman knew of the places where he set the events of American Gods, having either traveled there himself or else at least seen them on a map. (I can't name any specifically, never having read the book, but I can surmise as much from the context of your objection, I should think! x3) Smith, on the other hand, could not have credibly known anything about the location or name of an ancient burial site in the Arabian Peninsula, or of the location of such a place as "Bountiful" in the same part of the world... particularly since "common knowledge" of the Arabian Peninsula makes the notion of finding anything that could be described as "bountiful" there subject to skepticism. Second: Here are various sources deriding Joseph's claim of metal plates. John Hyde, Jr., Mormonism: Its Leaders and Designs (New York: Fetridge, 1857), 217-18; M.T. Lamb, The Golden Bible (New York: Ward and Drummond, 1887), 11; Stuart Martin, The Mystery of Mormonism (London: Odhams Press, 1920), 27. A quote by Hugh Nibley in 1957 seems amusingly prescient: "it will not be long before men forget that in Joseph Smith's day the prophet was mocked and derided for his description of the plates more than anything else." (Hugh Nibley, Lehi in the Desert, CWHN 5:107). A quote I have on hand: "No such records were ever engraved upon golden plates, or any other plates, in the early ages" /[M.T. Lamb, The Golden Bible, or, the Book of Mormon: Is It from God? (New York: Ward & Drummond, 1887), p. 11/]. More information can be had here, thanks to Jeff Lindsay, who is my primary (though not my sole!) source for Book of Mormon evidences. He has done a wonderful job compiling them. I must, for the sake of completeness, humbly admit fault: To say that the practice was one "that Joseph Smith could never have known" is in fact false; it is within the realm of possibility that Joseph mig
    First: Very well, the analogy was flawed. I'm unclear as to what the name "Bountiful" is supposed to refer to. Do either of the places mentioned as candidates translate to "Bountiful"? Further, I want to point out that "Critics doubt the link between Nahom and NHM, as well as having other criticisms." This will dovetail with our forthcoming conversation on Hebrew/English transliteration. Second: While such things were unknown archaeologically, the practicing of inscription on gold is referenced in the Bible; some googling uncovers Ex 39:30; see also the references here. Whoever the author of the BoM was, they were very well versed in the Bible. Third: The quote demonstrates that the actual existence of pre-Columbian American cement houses is irrelelvant. If they had not been found in our time, surely you also would maintain that they would be found... eventually. As you do elsewhere.
    First: The name "Bountiful" has no significance other than indicating a place of bounty. The candidate sites are those which match the description I noted above: The only reason I am able to use the Nahom - NHM theory as evidence is because the language Nephi uses indicates that the name of the place was given by someone prior to Lehi's travel. Speaking of which, yes, Critics do doubt the link, but if you read on, those criticisms are somewhat less than moving... The first is not really comprehensible as a counter-argument; contact with outsiders is not requisite for Lehi to know the name. The second is a mere lack of evidence. The third is merely a complaint of ambiguity inherent in Hebrew, and is answered elsewhere in the article. The fourth is simply an alternate theory, and a right flimsy one at that, if it's meant to explain away the consonance between Joseph's "guess" and the actual place. Second: I'll just note that the practice of engraving on metal jewelry and plaques is something much different than the practice of writing sacred records on books of precious metal. Third: The story of Einstein's Arrogance is relevant. :3 But at this point, I have enough positive evidence behind the Book of Mormon to start taking some of its as-of-yet unverified claims on faith. And what about you? What of the evidences that do stand? What is the chance that these could have come about by pure luck? Certainly we've acquired enough bits of evidence to raise the Book of Mormon to the level where it merits our attention, at least.
    Reasonably high. We have many examples of charismatic people constructing obviously fictive religions whose followers then retroactively find evidence, exploiting hindsight/confirmation bias. Scientology, Baha'i, Theosophy, and the various tibetan tulkus are examples.
    In each of these cases, the amount of retroactive evidence is far outweighed by the number of evidences against the religion's teachings. The opposite is true of Mormonism. None of its claims are disproven; we are only lacking evidence to support them. And the number of claims unsubstantiated by physical evidence is shrinking. Every time a discovery has been made that relates to the Book of Mormon, it supports the text. I will admit that there have been discoveries that have challenged popular understandings of the Book of Mormon. Once upon a time, it was in vogue to suppose that the narrative spanned the entire American continent (that is, both of them). This has been shown to be probably false, and in fact the text of the Book of Mormon itself seems to contradict that notion. However, the difference between, say, Scientology and the Book of Mormon is that we have in the latter a document that is not changing, but is still matching up to the evidence thrown at it. This document has been around for some 200 years in its present form, and the only alterations that have been made to it have been to repair grammatical errors - errors that, in fact, speak more strongly for the Book of Mormon than against it, since the first printing had "errors" that, while atrocious English, actually made very good Hebrew. I will supply you with references to this claim if you wish, but I thought it behooved me to stick to physical evidence first, as those are, in my opinion, the strongest claims. But you say "reasonably high". I'm afraid I'll have to hand you the burden of proof. With this counter, you chose to comment on an afterthought of a question and dismiss it out of hand, instead of talking about my arguments. We started this conversation - at least I did - under the premise that the physical evidences I supplied were worth discussing. I thought that you were under the same premise, but now with this post you attempt to dismiss any physical evidences as "hindsight/confirmatio
    Really? I can't think of any evidence contradicting the belief that His Holiness the Dalai Lama is the reincarnation of the previous Dalai Lama. Yet the evidence in favor is much of the same kind of evidence presented here, namely, "How could the young Dalai Lama have known which of many objects were the personal possessions of the previous Dalai Lama, were he not the reincarnation thereof?" In the same vein, "How could Joseph Smith have known X?", asked rhetorically, doesn't provide evidence in itself. In any case, this was never meant to be an argument about me converting to Mormonism. I wanted to know why you thought a non-Mormon shouldn't be skeptical of these evidences. I'll leave others to judge whether or not you've satisfied the condition of the precommitment in a parallel discussion thread.
    If you look at the votes for our posts, I think you'll find that they've already been judging. :3 Yes, I'm sorry if you felt I was jumping onto the "Hey, I've convinced you, now you should convert!" bandwagon; that was far from my intent. But I have offered my arguments about why a non-Mormon shouldn't be skeptical - rather, ought to be skeptical, but should be swayed anyway by the weight of evidence - but if it is not enough to convince you, then so be it. It is said that two Bayesians, working from the same set of priors, cannot agree to disagree... but I think we have different priors, which disturbs me to an extent. I will go meditate on this; I hope you will, too. EDIT: As to the Dalai Lama example, whose word do we have that these objects did in fact belong to the previous Dalai Lama? If the honesty of the ceremony is well-documented, then I would be interested to learn more.
    Beh, half of LW downvotes everything remotely theist on sight. It wasn't a judgment of the evidence. I do worry that I have been insufficiently diligent in evaluating the many religions. Hopefully any extant gods will turn out to be understanding.
    I reckoned that was the case, but I wanted to verify my unease. :3 And don't worry! If we Mormons turn out to be right, then the salvation/damnation schema isn't binary. ^_~ We believe that if you're a good person who didn't complete all the mystical rituals you need in order to be "saved", then you'll go to the next-lower degree of heaven, which is still a fair sight better than this place. Also that you'll probably get ample evidence to peruse during the millennium, so you'll be able to make an informed decision. (My own understanding; may be disproven upon further perusal of Church doctrine, but I think I've got it pretty right.)
    " Rationality can't be used to argue for a fixed side, its only possible use is deciding which side to argue." People arguing for their own religion automatically fail this rather basic premise of rationality, so what's the point getting into a discussion with them on finer points of religious doctrine, given that they have no clue about rationality to begin with, regardless of what they say? My question would instead be "Is it important to you for your religion to be right? If so, how does this mesh with rationality, if not, what are the odds that all the available evidence you evaluated pointed you in this convenient direction without any bias involved?".
    If a religion were correct, what would you expect debates with followers of that religion to look like?
    First, it would be interesting to know how one can convince a neutral and mildly rational observer what it means for a given religion to be correct and explain how this correctness can be tested experimentally. I don't have Yudkowsky's imagination, so it's not something I can easily conceive.
    If I interpret JGW's comment correctly, the rhetorical question wouldn't suffer much were it phrased "If X were true, what would you expect debates with people who believed X was true to look like?" The answer is that as humans speaking colloquially, they would first say "X is true" and then rattle off reasons, in the same format apologists use. This pattern of speaking does not strongly imply that the pattern of speaking was the pattern of thinking, it's just how people speak. Some people do think in this pattern, including many theists, so one can lose sight of the fact that the mode of speaking and mode of thinking are not perfectly correlated. As hard as they try, I don't think religions can avoid making testable claims. The untestable claim X is implicitly paired with the testable claim that one should believe X. Even probabilistic beliefs are held because belief systems lead people to expect things. when confronted with inputs. If a Unitarian Universalist says "(One ought to believe that) there is a 99.9% chance Jesus existed," and the scientific consensus is "(One ought to believe that) there is a 99.5% chance Jesus existed," and we fire up the ol' AIXI and it outputs the latter, the UU is wrong even if Jesus existed as one historical character. The UU might as well claim that the Noah's ark tale literally happened, if he isn't his belief system is in one way worse than the fundamentalist's, as his contains the proposition "To hell with reality when it contradicts my religion, if I can defy it without doing so in a flagrant enough way that people notice, including myself", whereas the latter's contains the proposition "To hell with reality when it contradicts my religion." Much simpler.
    Depends on what you mean by "correct". For example, if $religion's teachings correctly constrain expectations in verifiable ways, I expect such debates to look something like this: Skeptic: "Why do you follow the teachings of $religion?" Believer: "Because its teachings correctly constrain expectations. Here, I'll show you: here's a real-world situation. What do you expect to happen next?" Skeptic: "I expect $A." Believer: "Well, applying $religion's teachings I conclude that $B is more likely." Skeptic: "Excellent! Let's see what happens." (lather, rinse, repeat) Eventually one of them says to the other: "Huh. Yeah, it seems you were right!"
    "$religion's teachings correctly constrain expectations in verifiable ways" -- that's where it fails every time. That the universe was created 6000 years ago "should not be taken literally" now, though it was back when it was not testable. There is some nice stuff about it in HPMoR, Ch 22, Belief in Belief. There is no rational argument you can make that would change someone's belief if they are determined to keep it. Our Mormon friend here is a typical example. An honestly religious person would say that "this is what I choose to believe, leave logic out of it."
    I'm not sure if you think I disagree with you, or if you're just echoing me for emphasis. Just to be clear, I was answering JGWeissman's question: if a religion were correct, for at least that understanding of "correct" (which I endorse), that's what I would expect debates with its followers to look like. I've never encountered a religious tradition for which debates with its followers actually looked like that, which I take as evidence that no religious tradition I'm familiar with correctly constrains expectations in verifiable ways.
    My point was that the reason you never encountered it is because it would imply rationality, which is incompatible with faith. Not to say that a religious person cannot be rational about other things, just not about their own beliefs. Thus "if a religion were correct" is not a meaningful statement.
    Ah. Thanks for clarifying. I think your proposed explanation for the observed event is underdetermined by the evidence we're discussing, but it could certainly be true. Nevertheless, I'm still inclined to attend more to how well a practice is observed to constrain anticipated experience than to how rational its practitioners can be inferred to be on general principles... though I'll grant you that inferrable level of rationality correlates pretty well to how much energy I'll devote to making the observations in the first place.
    Absence of evidence is evidence of absence. The book of Mormon makes many claims for which, if they were true, we would expect to find evidence, but we do not. If you only look at the writings of Mormon apologists, you're going to get an extremely slanted picture of how well the Book of Mormon agrees with existing archaeological evidence, but if you look elsewhere, it's not hard to find strong evidence against it. The fact that the Book of Mormon references as being present animals that did not exist in Mesoamerica, or anywhere in the New World at the time, while not mentioning any of numerous common animals that were, is, as I see it, a knockdown argument all by itself. If these animals existed at that time and place, we have an extremely strong expectation of evidence for it given the archaeological and paleontological research we've done, but instead there is none. And the chance that legitimate writings from that time and place would reference as present animals which were not approximates to zero. This is extremely strong evidence against the Book of Mormon being true, and it's only one among its evidential failings.
    I am well acquainted with the notion of absence of evidence, thank you; I touched on this point above, stating that, although absence of evidence does count as points against the case I make, positive evidence makes stronger points. Were this not the case, then physicists wouldn't be searching for the Higgs Boson; they'd be restricted to theories which are readily explained by only the particles we have evidence of. A disproof of the Book of Mormon, then, must rest upon just that: disproof. With that in mind, let us examine further those points raised in the link you provided. Archaeological Fallacies First, four technologies are mentioned which were "unknown to Mesoamerica": chariots, steel swords, bellows, and silk. An explanation of the word 'chariot' can be found here. Many explanations have been made re: steel swords; the reference made in this case comes from the boo