How to Convince Me That 2 + 2 = 3

by Eliezer Yudkowsky3 min read27th Sep 2007403 comments

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EpistemologyLogic & Mathematics Rationality
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In “What is Evidence?” I wrote:1

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

Cihan Baran replied:2

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

I admit, I cannot conceive of a “situation” that would make 2 + 2 = 4 false. (There are redefinitions, but those are not “situations,” and then you’re no longer talking about 2, 4, =, or +.) But that doesn’t make my belief unconditional. I find it quite easy to imagine a situation which would convince me that 2 + 2 = 3.

Suppose I got up one morning, and took out two earplugs, and set them down next to two other earplugs on my nighttable, and noticed that there were now three earplugs, without any earplugs having appeared or disappeared—in contrast to my stored memory that 2 + 2 was supposed to equal 4. Moreover, when I visualized the process in my own mind, it seemed that making xx and xx come out to xxxx required an extra x to appear from nowhere, and was, moreover, inconsistent with other arithmetic I visualized, since subtracting xx from xxx left xx, but subtracting xx from xxxx left xxx. This would conflict with my stored memory that 3 - 2 = 1, but memory would be absurd in the face of physical and mental confirmation that xxx - xx = xx.

I would also check a pocket calculator, Google, and perhaps my copy of 1984 where Winston writes that “Freedom is the freedom to say two plus two equals three.” All of these would naturally show that the rest of the world agreed with my current visualization, and disagreed with my memory, that 2 + 2 = 3.

How could I possibly have ever been so deluded as to believe that 2 + 2 = 4? Two explanations would come to mind: First, a neurological fault (possibly caused by a sneeze) had made all the additive sums in my stored memory go up by one. Second, someone was messing with me, by hypnosis or by my being a computer simulation. In the second case, I would think it more likely that they had messed with my arithmetic recall than that 2 + 2 actually equalled 4. Neither of these plausible-sounding explanations would prevent me from noticing that I was very, very, very confused.3

What would convince me that 2 + 2 = 3, in other words, is exactly the same kind of evidence that currently convinces me that 2 + 2 = 4: The evidential crossfire of physical observation, mental visualization, and social agreement.

There was a time when I had no idea that 2 + 2 = 4. I did not arrive at this new belief by random processes—then there would have been no particular reason for my brain to end up storing “2 + 2 = 4” instead of “2 + 2 = 7.” The fact that my brain stores an answer surprisingly similar to what happens when I lay down two earplugs alongside two earplugs, calls forth an explanation of what entanglement produces this strange mirroring of mind and reality.

There’s really only two possibilities, for a belief of fact—either the belief got there via a mind-reality entangling process, or not. If not, the belief can’t be correct except by coincidence. For beliefs with the slightest shred of internal complexity (requiring a computer program of more than 10 bits to simulate), the space of possibilities is large enough that coincidence vanishes.4

Unconditional facts are not the same as unconditional beliefs. If entangled evidence convinces me that a fact is unconditional, this doesn’t mean I always believed in the fact without need of entangled evidence.

I believe that 2 + 2 = 4, and I find it quite easy to conceive of a situation which would convince me that 2 + 2 = 3. Namely, the same sort of situation that currently convinces me that 2 + 2 = 4. Thus I do not fear that I am a victim of blind faith.5

1See Map and Territory.

2Comment: http://lesswrong.com/lw/jl/what_is_evidence/f7h.

3See “Your Strength as a Rationalist” in Map and Territory.

4For more on belief formation and beliefs of fact, see “Feeling Rational” and “What Is Evidence?” in Map and Territory. For more on belief complexity, see “Occam’s Razor” in the same volume.

5If there are any Christians reading this who know Bayes’s Theorem, might I inquire of you what situation would convince you of the truth of Islam? Presumably it would be the same sort of situation causally responsible for producing your current belief in Christianity: We would push you screaming out of the uterus of a Muslim woman, and have you raised by Muslim parents who continually told you that it is good to believe unconditionally in Islam.

Or is there more to it than that? If so, what situation would convince you of Islam, or at least, non-Christianity? And how confident are you that the general kinds of evidence and reasoning you appeal to would have been enough to dissuade you of your religion if you had been raised a Muslim?

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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.

0Sengachi8yPeople can believe wrong things by the millions (Yay! the Earth is flat), that does not make it right. Stupid people can believe correct things and intelligent people can believe incorrect things. And if you were basing the belief system you believe in on average IQ, you'd go with atheism anyway. But none of these things are evidence. Something would be evidence even if everybody in the world disagreed. Relativity was true even when every single physicist (a group of rather educated people) just knew that Newtonian physic was truth.

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

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

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

See also "Argument Screens off Authority".

2[anonymous]6y'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?
2Andreas Källberg2yThe 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.
00ctoDragon5ySomething 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)

1wizzwizz49moBut 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

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.

-3yaro8yChristians substitute the heavy yoke of the law for a lighter burden (faith and hope) in Christ. So yes, you can appreciate a more rigorous ethical code, but Christians wish not to live under the law. You can call it weakness but I think it is humility grounded in the realization that we all know the good we ought to do but fail in doing, and the bad we hate we find ourselves doing. If a superior ethical framework makes it easy to do that good and shun that evil, we're back to keeping contracts with God and herein lies the real challenge - contracts are hard to keep and easy to break. What you have is a student with a textbook who has to get a perfect score on a test. I'd prefer a test where I don't need a perfect score to pass, and even better if the professor (Jesus) takes it for me. This my concept of Christianity and I hope other Christians will agree.
0Alicorn8yThis probably isn't the website for you.

Are you sure Alicorn? As a former christian now atheist thanks to LW(especially EY's writings) I've learned a lot here.

4Alicorn8yI wouldn't turn away arbitrary Christians, but this one in particular at least needs to lurk more and would probably be better served by a different website with more basic material and a different mix in the audience.
7roland8yWell, he just wrote one comment so I'm wondering how you arrived at that conclusion. Wouldn't it be better to give some positive reinforcement [http://lesswrong.com/lw/cu2/the_power_of_reinforcement/] to newcomers?
9Alicorn8yYaro's comment is entirely about theology. It's actually an on-topic response to its immediate parent, but it appears completely isolated from any other context or relevance. It makes assertions and states opinions without demonstrating the underlying comprehension of productive argument that is necessary for a theist, or anyone, to make worthwhile comments (at least within a topic which is this kind of substantive matter of fact [http://lesswrong.com/lw/2zl/meta_tiered_discussions/2ve3].) Positive reinforcement is to reward behaviors that I would like to see continue. I think the gap between yaro's level and the level I want to see on this site is big enough that it is not worth a reinforcement project.

Alicorn,

Good effort, you are almost there, if you used the explanation you gave above to Yaro he probably would be more enlightened than by "this is not the right website for you."

Quoting from The Power of Reinforcement:

The reason you should ignore poor performance if you say "No, you're doing it wrong!" you are inadvertently punishing the effort. A better response to a mistake would be to reinforce the effort: "Good effort! You're almost there! Try once more."

In spite of his comment being not in line, why not praise the effort, especially considering that this was his first comment on LW. I'm assuming he did an honest effort to present his point of view, why not praise that and point out the way this site works?

4Alicorn8yI do not wish to reward yaro's effort, and thereby get more of it. I believe that more of yaro's effort here would result in more comments that are of comparable quality and topicality at least for a prolonged learning curve, and do not believe that the project of nudging them towards improvement is worth community time. Please don't use an article about the use of reward to encourage good behavior as a way to say that all behavior is worth rewarding. ... Dude. Subtlety?
1roland8yStraw man, I'm not talking about rewarding his comment, but his participation. Isn't one of the goals of LW to raise the sanity waterline? I agree here but I see you are wasting precious community time(actually it's your time) by engaging in a discussion with me. You would have wasted less of our time if you were to give a more constructive reply in the first place. Another straw man, I didn't say that. Please don't use straw men when arguing with me. I won't waste any more time with this, so I'll leave you the last answer.
0RichardKennaway8yYou quoted this: and previously said [http://lesswrong.com/lw/cu2/the_power_of_reinforcement/6w33] of the larger passage: But when you tried it out: this was her response: Didn't work. Shall I praise your effort and urge you to do better next time? Would that work for you any better than it did for Alicorn?
3keen8yWe hardly have enough evidence after just one attempt. Additionally, subtlety is a form of deceit, and not generally encouraged in rational discussion. Anyway, my complaint would have to be toward the out-of-hand dismissal of yaro's post, rather than offering a substantive disagreement or at least a link regarding the perceived flaws in yaro's argument. That's proper rationalist encouragement. No subtlety required.
2TimS8yReinforcement is for things one wants more off. I can't say I want more posts without knowing something about the content. In this case, I think Alicorn is right that this poster is not ready to contribute things this community would find valuable.
-2roland8yDoesn't LW want more participation? I agree that the comment is not in line with the thinking of LW, but why not point it out to the commenter in a polite manner if you want to say anything at all?
2wedrifid8yNot especially. I want better participation. Not absolutely ridiculous religious tripe. The person Yaro is welcome to stay, so long as he leaves his bullshit [http://en.wikipedia.org/wiki/On_Bullshit] at the door. Making that kind of comment will, and should, prompt a negative response.
5Personable1yI would disagree with you completely on this. You are outright dismissing a point of view with no supporting evidence other than a vague, insulting comment. It appears you haven't actually learned what this site is promoting. It's one thing to disagree with Yaro here, it's another to outright insult his beliefs and simply raise his barriers in agreeing with anything you believe. I've seen similar instances occur enough times to know (within a reasonable margin of doubt) that if you want to actually make a person agree with you, you start by making yourself agreeable.

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.

1MTGandP8yGiven 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.
0MaxNanasy6yBut what if you're hallucinating the increase in mental capacity and resulting discernment?
2MTGandP6yIt 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.
9mszegedy8yI 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"?

0mszegedy8yWasn't that what Einstein said about QM?
4JoshuaZ8yAlmost. Eliezer is making a bad wordplay with what Einstein said.
5MixedNuts8yI 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.
2Lethalmud7yDo those studies have a placebo group?
5Rixie7yI 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.
2Tintinnabulation6yI recall the same study - and I seem to remember that the patients who knew they were being prayed for did a bit worse.
6notsonewuser7yActually, if you run the test, you are. Given that you'd have changed your mind if it had gone the other way, of course.
3Document7y(Related: Religion's Claim to be Non-Disprovable [http://lesswrong.com/lw/i8/religions_claim_to_be_nondisprovable/])
0Jake Heiser1moconversely, 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.
0AlexanderRM6yThere'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.
2jeronimo1969moGod 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.

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.

0rkr141010yThere are some points of view that sometimes do require mathematical statements to be dependent on reality (i.e. constructivism, actual versus potential infinity debate, etc). Sometimes it is intuitive to require mathematics to behave this way, i.e. 'natural' numbers are called that for a reason, and they better behave like the apples or I'm postulating a change in nomenclature. P.S. Ii seems to me the OP's wording wasn't precise enough. I can very well imagine a situation in which some basic addition would yield non obvious results (like addition inside modulo N number space).
3Will_Sawin10yWhen 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"?
0rkr141010yI guess when one reasons inside a fully axiomatized formal system, this something the rules for symbol manipulation depend on is the set of axioms. Now I'm putting on my uneducated hat, so excuse me if this is heresy: Starting with the axioms you apply logic to formulate more specific rules (in this case the abstract is empirically falsifiable, since we're working on natural numbers). So, to arrive at SS0+SS0=SSS0, you'd have to venture outside the realm of reason I'm afraid.Tthat would maybe manifest itself as magic - getting 4 apples on the table during night, but 3 during day when you put 2 and 2 apples side by side. And could mean ability to produce something from nothing by clever arrangement of apples. and waste disposal would become easy :) In other words my opinion is it's not possible even as thought experiment unless you introduce some random factor from beyond the scope of axioms.
4Will_Sawin10ywell there's the special other thing, the reason you can't explain Peano Arithmetic to a rock, which is that axioms are static sequences of signals, but in addition you have these dynamics. Best source on this is Lewis Carroll http://www.ditext.com/carroll/tortoise.html [http://www.ditext.com/carroll/tortoise.html] These dynamics are contained within the structure of our thoughts, which is why they're preserved in a thought experiment. But we still have to actually check our thoughts, which are part of reality. Sorry if this wasn't very coherent.
1MarkusRamikin9yHm... 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)

7wedrifid10yVerbal expressions almost certainly predate physical notations. Unfortunately the echos don't last quite that long.
2randallsquared10yIn 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?

4randallsquared10yYup. I think we agree. My disagreeing post was a mere misunderstanding of what you were saying.
4bigjeff510yAfter 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 http://lesswrong.com/lw/mo/infinite_certainty/ ).

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)

2elharo8yFor 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 [http://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-wrong] .
1matteyas6yAt 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?
0Peterdjones6yOn the other hand... http://en.m.wikipedia.org/wiki/Is_logic_empirical%3F [http://en.m.wikipedia.org/wiki/Is_logic_empirical%3F]
0Lethalmud7yElezier, 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?
3khafra7yIf 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.
2christopherj7yNo, 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.

1tlhonmey1moHaving 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.

3[anonymous]9yWhat's an example of another possible abstraction?
8Kenny9yOther possible abstractions: * groups * rings * modular arithmetic * sets * simple computer programs (like these cellular automata [http://www.wolframscience.com/nksonline/page-52?firstview=1]) These abstractions describe aspects of the world around us too, just not counting macroscopic objects like apples and earplugs.
0SebastianGarren8yYes 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.

0[anonymous]9yAs 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.

0SebastianGarren8yYou 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)

I think the key is to look at religion in economic terms. Rational beings will maximize utility. To a scientist, it may seem irrational not to question every aspect about life and our existence. To a religious person, eternal questioning may seem like a waste of time. What is the difference to me, from a purely selfish point of view, whether or not God created everything and sent down a son/prophet/what have you? I can accept it as truth with no harm to myself, leaving my time free to pursue more productive activity - so why shouldn't I?

There are many peop... (read more)

0Arandur9yI'm going to go ahead and offer a differing opinion regarding Christianity. o_o; First of all, much of the "God is vile and insane" protest can be mitigated by remembering that, if the Christian God exists, then this life is a tiny piece of our entire existence, and that death is far from the worst possible outcome. If you try to justify the Christian God's ethics based on conclusions you've reached by disbelieving in Him, your premises are contradictory. And second, God has indicated [http://lds.org/scriptures/ot/ps/102.27?lang=eng#26] many [http://lds.org/scriptures/ot/mal/3.6?lang=eng#5] times [http://lds.org/scriptures/nt/heb/13.8?lang=eng#7] that He does [http://lds.org/scriptures/bofm/morm/9.9?lang=eng#8] not [http://lds.org/scriptures/dc-testament/dc/20.11-12?lang=eng#10] change [http://lds.org/scriptures/dc-testament/dc/76.4?lang=eng#3]. (Yes, a few of those were not from God directly, but it doesn't matter [http://lds.org/scriptures/dc-testament/dc/1.38?lang=eng#37]). So I don't know why a brief stint on Earth would change His mind, particularly about morality, since He's supposed to be infinitely just... I just... no, I really am curious now. This doctrine seems to be all sorts of self-contradictory. In which church did you grow up?
8Desrtopa9yIf you try to justify the Christian God's ethics based on the bible's own assertions of his justice and infallibility, your argument is circular. Conclusions based on belief or disbelief in God are irrelevant to judging his morality according to the portrait painted in holy texts, we already have our own moral toolkit to work with [http://lesswrong.com/lw/ky/fake_morality/]. I don't see how it remotely mitigates any charges of vileness or insanity to posit that God hands out rewards of eternal bliss or punishments of eternal suffering for beliefs and/or actions without making his expectations clear. That's negligence beyond the scale of criminality. No matter how sure you are that you understand what God wants, if he had actually made his message clear and well evidenced, your beliefs would not be a minority worldwide. Any intelligent and competent human who wanted to get a coherent message across could do better. Hell, scholars have better agreement on what Nietzsche meant, and he was being deliberately obscure.
4lessdazed9yAs morality is entangled with reality, (incorrect) conclusions about anything are potentially relevant to judging morality. I wonder how readily believers would, in a sort of plea bargain, accept and embrace the charge of insanity, with all its connotations, if it were thought the obviously attractive horn of the dilemma, with no other alternatives. I'm imagining what religious institutions and literature would look like if, say, one out of every ten adjectives referring to a deity in hymns, psalms, etc. were replaced with "insane" or one of its synonyms.
1Arandur9ysnerk I'm sad to say that the answer is "rather readily, for some". I've gotten into heated debates - heated because they ended with me throwing my hands up in the air, appalled at the willing ignorance of my opponent - about whether God follows the same basic rules of logic that we do. It's astonishing. I'm sure if you were to start a Protestant movement saying that "God is insane, but we follow Him anyway", you'd get followers by the truckload. By Occam's razor, we must conclude that the basic tenants of logic we have in this life will not cease to be true in the next. Then again, Occam's razor depends on Occam's razor for its veracity... :3 If you wanna talk about circular arguments. And I agree with your statement re: incorrect conclusions; thank you. EDIT: Man, this 10-minute lag is killing me. O_o;
0christopherj7yJust out of curiosity, if you for some reason valued above all else, faith (of the sort where the more evidence the less faith), how would you go about maximizing it? If you made the faith too universal, that itself might be evidence of its truth (and thereby reduce the value of the faith). Can you prove that there is an absolute morality, such that valuing faith above rationality is necessarily immoral (as opposed to really inconvenient, or seems immoral to most people)? PS: Many Christians get annoyed if you suggest that you can just believe in Jesus after you die, like people who lived before Jesus. They'll get even more annoyed if you ask them where in the Bible it says otherwise.
0Rob Bensinger7yTo optimize that kind of faith, just put every person in a separate universe tailored to contain maximally weak evidence for a proposition P, then make them insane (e.g., with torture, brainwashing, or innate cognitive biases) until they delusively believe in P despite the lack of evidence. Since each person gets their own pocket universe, they needn't be exposed to any other person capable of communicating to them supporting evidence for P. Define "absolute morality" and "prove". If nothing can even be evidenced about morality, then I can safely ignore the question and just ask about what actually benefits people. Believing true things generally helps people have happier, more peaceful lives filled with more love and beauty; is the lunatic in the asylum a paradigm of human flourishing? If you think that's not universally the case, then OK. But you must think so because you have actual evidence for the value of faith. If I have no evidence that Zeus is Lord but think that's OK because faith is a virtue, I still need to provide some evidence that faith in the proposition 'Zeus is Lord' is a virtue. If I appeal to faith in faith, as opposed to just faith in Zeus, then I've made the content of faith arbitrary, which means that any religious claim can be justified, not just my own religious claim.
0christopherj7yIn other words, little to no practical difference from what we live in. There's widespread belief in God (very weak evidence but excellent privileging of the hypothesis), the threat of future torture (and current social pressures), indoctrination from an early age, and we certainly have plenty of cognitive biases (many of which are geared to preferring things with positive consequences). The main difference with your scenario is the interaction with other people -- but it can be argued that other people are very poor evidence for the existence of God, compared to eg providing a non-theistic explanation to where you came from. I'm pointing out that P(the world we live in | a God that values faith) is very high, and essentially indistinguishable from P(the world we live in | there is no God). This is no coincidence, as otherwise people who believed in a God that values faith would be constantly surprised by the world we live in. Note that if someone who believes in a God that values faith, claims to have found strong objective evidence for the existence of their God, they're coincidentally also claiming to have found strong evidence that said god doesn't value faith (ie, evidence for some other god). (I'm not trying to misuse the representativeness heuristic, just pointing out that if our world wasn't representative of some popular god, belief in that god would die out, like with Zeus.) If nothing can be evidenced about morality, than what you just did is called "begging the question". I wrote my above post because I was intrigued by Desrtopa's claim that it would be immoral and/or insane for God to hand out epic punishments and rewards based on [the person's unevidenced faith] without making it clear to everyone that He wanted [unevidenced faith].
0Rob Bensinger7y.. No. If our universe were optimized for maximal faith in Christianity, there would be no Christians other than you, much more conclusive empirical evidence (e.g., impossible-to-fake video footage) showing that Jesus did not rise from the dead, etc., and your brain would just come pre-wired to believe in Jesus with equal and maximal confidence no matter what evidence you encountered. Basically your brain would just be a circuit that chants 'JESUS IS LORD', associated with a strong conviction-feeling, and seeks out and absorbs as much Christianity-refuting evidence as possible, without having any mechanism for modifying the 'JESUS IS LORD' conviction pump in your brain. If built-in beliefs aren't allowed, then the second best step would be to literally torture you -- like, you've never lived a day of your life without undergoing something roughly as unpleasant as scaphism [http://en.wikipedia.org/wiki/Scaphism] -- until you're driven literally insane and can do nothing but grind your teeth and gibber endlessly about Jesus being lord. That's much, much closer to what a universe optimized for faith (and faith alone) would look like. We aren't literally in such a universe; any comparison of your own day-to-day existence to actual torture must suggest either a loose grasp on the English language, or a fundamental misunderstanding of just what real torture is like. Certainly vague, hand-wavey threats are not optimizing for unjustified-belief. Threats can't drive most people past the brink of madness, arguably can't inspire belief at all. Try waving a gun around and threatening to kill someone, or kill eir family, if ey doesn't spontaneously believe that the Moon is made of green cheese. The person might claim to believe the Moon is made of green cheese, but they wouldn't actually believe. That's not how human belief works. Humans do not automatically notice their confusion in all instances, nor do they automatically form internally consistent belief structures withou
1christopherj7yYes, but I didn't claim it as evidence of anything other than that such belief isn't particularly maladaptive (consequenceless beliefs won't interfere with your predictive power). If it helps you understand my point, P(the world we live in | invisible unicorns in a far away galaxy fart rainbows) is also very high, which is not at all an argument in favor of invisible unicorns (only that there's no evidence against unicorns nor will believing in unicorns therefore result in poor predictions). To the extent that belief in a God that values faith is consequenceless, it can't be maladaptive. To the extent that such belief has consequences, it will make predictions about the real world. Making predictions about the real world means they can test the belief, which regardless of the result will be suspect (one way, it is evidence against their belief in God, the other way evidence against their belief that God values faith). My point is that most Christians don't think through the consequences of claiming that something is evidence for God. Why have faith if they're arguing that their belief is a scientifically verifiable one -- in such a case, faith would be more likely to mislead them than science. Long story short, if you find yourself arguing with a Christian about whether something is or isn't evidence for God, you could have cut the whole thing short by asking them "Will your suggestion let me test for the existence of God?" Not only is there only one reasonable way they can answer, there's specifically a command against testing God.

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. ;)

0AlexanderRM6yIf 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)

5tut11yMathematical 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.
0Arandur9yWhat an interesting argument... but I know of at least one religion that would tend to disagree with this anti-definition of God.
0AlexanderRM6yActually, 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?
0Jayson_Virissimo6ySome scholars of religion have claimed that a straight-forward reading of the early Old Testament suggests it is better described as henotheistic [http://en.wikipedia.org/wiki/Henotheism] 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)

0Skarey10yThere 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.
0bigjeff510yI 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)

0AlexanderRM6y"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)

2AlexanderRM6y"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)

0shrimp10yWhat 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? ...now 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
2AmagicalFishy9yI 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.
0mytyde8yIn regards to Hume's interesting contributions to the question, I stumbled across this video a while back which I think will be interesting: http://www.youtube.com/watch?v=BVZG0G-jnAM [http://www.youtube.com/watch?v=BVZG0G-jnAM] (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.

0bigjeff510yYet 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.
0Arandur9yCould you please define a "pile"? :3
4DSimon8yA 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)

1byrnema11yI 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.
-1danilobellini11yNice, 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"
0[anonymous]11yNice, 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"
0Kingreaper10yUpon 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
0orthonormal10yYou'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".
0Kingreaper10yIf 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".
0orthonormal10yThis 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.
2wedrifid10yWhy 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.
0Kingreaper10yGood 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.
0wedrifid10yExactly. 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.

1tlhonmey1mohttps://www.smbc-comics.com/comic/technically-beautiful [https://www.smbc-comics.com/comic/technically-beautiful] 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.

1AndyCossyleon10yI 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.
4RobinZ10yIt'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 [http://lesswrong.com/lw/vk/back_up_and_ask_whether_not_why/] 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)

5byrnema10yI 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 [http://olc.spsd.sk.ca/de/math1-3/p-mentalmath.html] '. 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.)
3Osmium_Penguin10yI 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.
2NancyLebovitz10yI 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.
3Osmium_Penguin10yWell, 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.
1orthonormal10yWell, 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.
1Oscar_Cunningham10yYour 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)

2nshepperd10yOne might indeed "believe [http://lesswrong.com/lw/i4/belief_in_belief/]" all that. But a belief has no use [http://lesswrong.com/lw/i3/making_beliefs_pay_rent_in_anticipated_experiences/] if it isn't true [http://yudkowsky.net/rational/the-simple-truth].
1bigjeff510yApples 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.
0bigjeff510yApples 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)

6simplicio10yMany other people have such experiences [http://www.psychwww.com/psyrelig/james/james4.htm#69], 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?
3Xaway10yTo 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.
2Jack10yLuckily, 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. [http://en.wikipedia.org/wiki/Neurotheology] Welcome, by the way.
0Xaway10yI'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.
4simplicio10yYou'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 [http://en.wikipedia.org/wiki/File:Grey_square_optical_illusion.PNG] very [http://en.wikipedia.org/wiki/Anosognosia] untrustworthy [http://en.wikipedia.org/wiki/Confirmation_bias]. I for one do not doubt your sincerity (or the Sufis') but I do doubt that you correctly interpreted your experience.
6Costanza10yHello! 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 [http://lesswrong.com/lw/i8/religions_claim_to_be_nondisprovable/]. 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?
1Xaway10yMy 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.
3wedrifid10yNow I'm wondering which of those categories I fit in to. They all sound a tad appealing. :)
0Xaway10yIf 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.
1wedrifid10yI was sincere Christian right up until I realised the religion could be better explained by tribal signalling than magic. You just finished saying:
0Xaway10yA 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.
6ata10yYou can see how non-Christians might find that to be a suspiciously convenient excuse, right?
0Xaway10ySo 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?)
6ata10yIt'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 [http://lesswrong.com/lw/19m/privileging_the_hypothesis/] than their epistemic state actually warrants).
0Xaway10yObviously, 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.
2nshepperd10yI 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.
4simplicio10yMinor 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.
4[anonymous]10yAs 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.)
0Desrtopa10yI'd also recommend Why God Won't Go Away: Brain Science and the Biology of Belief by Gene D'Aquili and Vince Rauss.
2Costanza10yThere 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 [http://wiki.lesswrong.com/wiki/Litany_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!
0Xaway10yIf 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.
2Costanza10yAgain, 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.
2Jack10yI think I'd rather have a better calibrated Frequentist.
1wedrifid10yI'd rather have a rock. Or a Christian who doesn't speak Bayes. At least that implies less doublethink.
1Jack10yChristianity 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.
1simplicio10yI 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?
0Jack10yA 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.
0simplicio10ySorry, 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.
3[anonymous]10yI 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.
3XiXiDu10yMy reason to abjure God was mainly due to ethical reasons [http://xixidu.net/var/]. 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 [http://en.wikipedia.org/wiki/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 [http://wiki.lesswrong.com/wiki/Occam's_razor]. 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 [http://en.wikipedia.org/wiki/Abrahamic_religions]. It should [http://www.bricktestament.com/] be obvious [http://www.youtube.com/watch?v=wKtuk0ZpnbY] to anyone who isn't morally bankrupt [http://www.youtube.com/watch?v=TOGK0KHttQ0] 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 [http://www.biblegateway.com/passage/?search=Isaiah%2013:15-18&version=NIV]:
5Desrtopa10yAnd 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 [http://lesswrong.com/lw/i9/the_importance_of_saying_oops/].
1XiXiDu10yYes, 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 ;-)
4[anonymous]10yI 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 [http://en.wikipedia.org/wiki/Phinehas], 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.
-1XiXiDu10yIf 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.
0Xaway10yAlthough 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.

4ata10yThe 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 [http://www.reddit.com/r/atheism/] instead.
5Costanza10yWell, 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.)
7wedrifid10yThat is not 'truly rationalist'. Well thought out arguments for a preselected bottom line are bullshit.
0byrnema10yPerhaps 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.
0wedrifid10yDid. Didn't work. Wrote you off. :)
0shokwave10yArguably 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 [http://lesswrong.com/lw/o0/where_to_draw_the_boundary/] at its joints [http://lesswrong.com/lw/o2/mutual_information_and_density_in_thingspace/]. So are categories, and so is computation. They can't answer questions about themselves. But reality quite clearly can answer questions about itself.
0Perplexed10yHow about the question of whether there is anything you can't assign a probability to or collect evidence about?
1shokwave10yYou 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).
0Clippy10yChristianity isn't true.
6Vaniver10yYou're just mad they refused to canonize Samuel B. Fay.
1Jack10yA 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.
0Xaway10yWell 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.
1Jack10yThis is really wrong, obviously, but my hopes that the inferential distance was manageable have been dashed. You might start here [http://www.scholarpedia.org/article/Algorithmic_probability]. I'm done though.
1jimrandomh10yWhat was that like? In particular, how could you tell that it was really a revelation and not any kind of temporary brain malfunction?

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)

0Arandur9y* 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 [http://lds.org/scriptures/bofm/1-ne/16.34?lang=eng#33], 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 [http://lds.org/scriptures/bofm/1-ne/17.5?lang=eng#4], 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 [http://lds.org/scriptures/bofm/1-ne/17.7?lang=eng#6] upon which Nephi could offer his prayer, available ore [http://lds.org/scriptures/bofm/1-ne/17.9-10?lang=eng#8] and flint [http://lds.org/scriptures/bofm/1-ne/17.11?lang=eng#10], 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 [http://lds.org/scriptures/bofm/1-ne/2.5-10?lang=eng#4] have been found, right where they ought to be. * The practice of writing on metal plates, laughable in 1830, has now been well [http://www.jefflindsay.com/LDSMetal.shtml] established [http://www.novinite.com/view_news.php?id=22691] 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 Egyptia
4[anonymous]9yThe 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.
0Arandur9yI 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 [http://mormanity.blogspot.com/2009/08/those-implausible-plates-with-apologies.html] , 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 nev
0[anonymous]9yFirst: 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." [http://en.wikipedia.org/wiki/Nahom] 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 [http://en.wikipedia.org/wiki/Engraving#Biblical_references]. 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.
-1Arandur9yFirst: 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 [http://lesswrong.com/lw/jo/einsteins_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.
6[anonymous]9yReasonably 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.
0Arandur9yIn 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
4[anonymous]9yReally? 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.
1Arandur9yIf 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.
5[anonymous]9yBeh, 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.
0Arandur9yI 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.)
1shminux9y" 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?".
5JGWeissman9yIf a religion were correct, what would you expect debates with followers of that religion to look like?
0shminux9yFirst, 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.
3lessdazed9yIf 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.
3TheOtherDave9yDepends 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 $B is more likely." Skeptic: "Excellent! Let's see what happens." (lather, rinse, repeat) Eventually one of them says to the other: "Huh. Yeah, it seems you were right!"
1shminux9y"$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."
2TheOtherDave9yI'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.
0shminux9yMy 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.
1TheOtherDave9yAh. 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.
6Desrtopa9yAbsence of evidence is evidence of absence [http://lesswrong.com/lw/ih/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 [http://www.dimmeria.com/?q=node/21], 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.
0Arandur9yI 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 [http://www.mormonfortress.com/wheel3.html]. Many explanations have been made re: steel swords; the reference made in this case comes from the book of Ether, speaking of the Jaredites. I offer the below quote as a counter: Bellows are only mentioned in the locale of the old world, not in America, making this a non-point. Regarding silk: An LDS publication [http://www.shields-research.org/Scriptures/BoM/SILK02.html], and a non-LDS publication, "Silkworm of the Aztecs" by Richard S. Peigler, Ph.D., Curator of Entomology, in Museum Quarterly, Vol. 2, No. 1 (Spring, 1993): pp. 10-11 (published by the Denver Museum of Natural History, both show evidence of silk in the Americas. A note on cities in America comes again from Jeff Lindsay: Further, as noted above, the details of Lehi's journey through the Arabian peninsula have been well correlated with actual places, some with names matching those found in archaeological studies. Anthropological fallacies I'm sorry, but this is plainly wrong. We have known for quite some time that the Nephites were not the only inhabitants of ancient America; the Jaredites are an example attested in the Book of Mormon. Biological fallacies My goodness, what an
7Desrtopa9yEdit: I meant to cover this point first, but I left it out before. This really isn't how it works. Absence of evidence is evidence of strength proportional to the expectation of evidence if a given proposition is true. So if, for example, you propose that there is an elephant in a room, and then you investigate the room and see no sign of an elephant, then that is very strong evidence that there is no elephant in the room. But if you propose that there is a mouse in a room, and you investigate and see no sign of the mouse, then that is only weak evidence that there is no mouse. You will have to update your confidence that there is a mouse in the room downwards, but much, much less than you had to update in the case of the elephant. In both the case of the elephant and the mouse, actually observing the elephant or mouse would be extremely strong evidence; you could still be wrong if you were hallucinating or someone had contrived an extremely clever way of creating an illusion of either, but it would still force you to greatly strengthen your probability estimate for an elephant or mouse being in the room. It's psychologically compelling to try to generalize this into an broad principle, that positive evidence is always stronger, but in fact as with the case of the elephant, negative evidence can reach arbitrarily high strengths depending on how strong the expectation of evidence is. Likewise, positive evidence can reach arbitrarily low strengths depending on how likely it is that the observation would be forthcoming without the proposition being true. For instance, if an alleged psychic describes a crime scene, and the police confirm that the description is accurate, this is not strong evidence that the psychic had any sort of vision of the scene if their description is statistically likely to apply to any crime scene of that type. The defenses you've linked are extremely weak. Apologists of any religion can rationalize this degree of agreement with evidence, bu
8Arandur9yYour point is well taken, and I will meditate upon it. Thank you.
3novalis9yRe "Silkworms of the Aztecs", have you read it? Because these people [http://www.theologyweb.com/campus/showthread.php?120965-Questionable-reference-by-Jeff-Lindsey] say that the evidence for it existing is weak. I don't have access to JSTOR and I don't have Aaron Swartz's hard drive, so I can't look it up myself.
1Arandur9yWell, that's disconcerting. Sounds like everyone's copying off everyone else. ;3 Problems in academia, indeed. The final post on that thread does seem to indicate that the article does exist; would you like me to attempt to gain a photocopy, so I can verify your suspicions?
0novalis9yWell, I have to admit that I'm curious, but really only mildly. I mostly gave up trolling Mormon missionaries after high school. I just thought it might be an interesting article, which is why, while skimming this thread, it is one of the two things I googled -- the other being the Bat Creek stone.
0Arandur9y:3 I am glad to hear you gave up on trolling the missionaries. I realize that they can be annoying... and some of them may deserve a bit of trolling, from the stories I've heard... but most of them are hard-working young men who really do believe in what they're saying.
-1Arandur9yFourth: In this case, I defer entirely to the experts. Fifth: That is entirely the wrong question to ask; so wrong that I wonder if you understood my point. Your question should have been, "What proportion of random 3-6 English character strings correspond both to pronounceable words and as-of-that-time undiscovered Hebraic names". Or perhaps you are acting under the assumption that these names are attested only by consonant matches? That's not quite true. For example, the name "Alma" is not simply written as "lm" in hebrew, but is written with four characters, essentially coming out to 'lm'. For scholars of Hebrew, there is good evidence that the name should be "Alma," which is exactly how the non-LDS scholar, Yigael Yadin, transliterated it. As far as the actual proportion, I have no idea, but one must assume that there are more disallowed combinations than allowed ones, or else the language would become incomprehensible. :P
1[anonymous]9yFourth: I'm not an expert, so I too defer. EDIT: Wait, these aren't random experts. They're all Mormon apologists, with obvious incentive to defend their faith. Where are the unaffiliated archaeologists on this? Fifth: I am admittedly an amateur at biblical Hebrew, so I suppose I should have asked for 3-4 character strings. If I were an evil Joseph Smith, I would construct such plausible-sounding Hebrew strings, and then transliterate them into English. Under this procedure, whether I generate aleph-lamed-mem-aleph, aleph-lamed-mem, ayin-lamed-mem, and etc, I still plausibly generate "Alma". After some familiarity with Hebrew, it does not become overly difficult to guess at vowels; hence the legibility of unpointed text.
1Arandur9yFourth: No, of course not. If you were a non-LDS scholar, would you come out and say, "Oh, by the way, according to this evidence we found, the Book of Mormon might well be true after all." First off, it would be career suicide, and second, if you found scientific evidence supporting the Book of Mormon, I imagine you'd be intrigued, start seeking for more information, and eventually become LDS. :P But very well; I can offer what non-LDS scholars have said about olive culture, and you can compare to Jacob 5 and draw your own conclusions. The following quote courtesy of Jeff Lindsay. Fifth: Yes, of course you're correct about the legibility of unpointed text, but again, this does not mean that a majority of viable consonant strings are eligible names. We can roughly do the same thing in English, ndrstndng t wtht hvng vwls, but this wouldn't work if all of the prior consonant strings were viable names. There must be rather large gaps in morpheme-space for any language to be intelligible, otherwise any errors in pronunciation or data lost in transfer would render the communication unintelligible, or worse, change its meaning entirely. I'll claim a minor position of authority on this point; I'm in college, working on a major in Linguistics.
1MatthewBaker9ywaits for wedrifid

waits for wedrifid

I hadn't actually read the grandparent beyond skimming and categorized it as an entirely non-trollish expression of personal belief. Given the prompt in the post it was appropriate to the context and as rational as can be expected given that the guys' beliefs are utter nonsense.

Having read through the first comment (before the "to be continued") the following part jumped out at me as the primary non sequitur.

So, if the archaeological evidences corroborate the Book's story, then we must consider the Book to be "true", and thereby accept either P-True or P-Alien.

That just isn't case. Archaeological corroboration provides evidence for the Book's story. That is, part of the story is validated which eliminates a whole lot of the bits that could be wrong and we can assume a correlated truthiness with the remainder of the story. We update p(Book's Story) upward, but not to one. Something along the lines of:

p(Arch | BS) = x
p(Arch | !BS) = y

p(BS | Arch) = p(Arch | BS) * p(BS) / p(Arch)

We do not have the logical deduction "IF Arch THEN BS" but rather a likelyhood ratio such that BS is more likely the less likely it is for the archaelog... (read more)

2TheOtherDave9yUpvoted for amusement value.
3Arandur9yI take offense to any implications about my posterior. Heck, even I upvoted this. Your point is well made, and well taken; even if archaeological evidence corroborates parts of the Book of Mormon, that does not update its probability to 1. I should have been more clear... no, rather, I should have thought of it that rationally, but I was blinded by my own certainty. I apologize; thank you for showing me my error. Were I to rewrite the above, it would take the form of something like this: The Book consists of two pieces of information: data that are archaeologically verifiable, and data that are not archaeologically verifiable. If archaeological evidences corroborate the Book's story, then there are two possibilities: either the non-archaeologically-verifiable bits are also true, or they are not. If the former is the case, then Joseph Smith's story is correct, the Church is true, etc. etc. If the non-archaeologically-verifiable bits are not true, given that the a.v. bits are, then we must conclude one of two things: either a coincidence (which probability becomes smaller with each additional corroborating evidence), or something Stranger, e.g. alien teenagers. I am inclined, given the current state of the evidence, to believe the above scenarios in the following order, in descending order of probability: a) The Book Is True, b) Aliens Are Trolling Us, c) Magnificent Coincidence. I also think that these three possibilities, and their subgroups, comprise the entirety of the probability space, but please correct me if there's a possibility I have overlooked. Oh, as an aside: The proposition that "The mainstream LDS church is not true, but the truth is had by one of the handful of splinter groups that split off from the LDS church and still believe in the Book of Mormon" does in fact fall under possibility a, though considering the legal troubles surrounding some of these groups, this seems rather unlikely to me. After all, Joseph Smith published this as one of our t
0MatthewBaker9yOnly solid piece of evidence i found on the DNA route, most of the rest seems to be arguing that there remains a miniscule chance [http://lesswrong.com/lw/ml/but_theres_still_a_chance_right/] despite the current consensus on DNA data.
0Arandur9yI'm not sure why you chose to post this as a response to my rewrite, but that doesn't detract form the validity of the post. I'm well acquainted with the fallacy you linked to; that's actually been one of my favorite of Eliezer's articles. It is unfortunate that this and other fallacies abound in real-world arguments... however, I trust you understand that the existence of fallacies does not equate to a false conclusion. If I base my conclusion X on arguments A, B, C and D, and D is fallacious, X may still happily rest on A, B and C. In particular, there's a difference between the hopeless grasping-at-straws of the "there's a chance it's a coincidence!" argument and the "This does not necessarily contradict what we're saying" argument. In the latter, there are also positive evidences to bolster the conclusion; it is true then that negative evidences (by that I mean, evidences which show no support for the conclusion, but do not disprove it) nudge the probability does, but not as much as the positive evidences nudge it up. In the former, all there is is wishful thinking.
0MatthewBaker9yI just think that if the DNA evidence isn't there then how can i consider the possibility of the book of Mormon having any truth to it. It feels a lot like considering the possibility of Intelligent Design as the origin of humanity. If A, B, and C preclude the existence of D then X is weakened more by the disproof of D then if it is a standalone piece of evidence.
0Arandur9yBut the DNA evidence is there. You pointed to a piece of it, and then said "but the rest is mostly bull". But that doesn't mean that the evidence you found ceases to be valid.
1wedrifid9yWhile I do not accept certain of your premises (surprisingness of corroborated evidence) your reasoning from there is cogent and the update worthy of respect!
1Arandur9yOh! Well, thank you! I will attempt to be cogent the first time in the future. :3
1Bill_McGrath9yI don't understand this - why would legal troubles make their beliefs any more or less likely to be true? Seems like an entirely irrelevant issue.
3gjm9yI think the point is that not getting into legal trouble is an important tenet of Mormonism (since obeying the law was one of those 13 "Articles of Faith"), so that a group that's got into a lot of legal trouble is unlikely to be The One True LDS Church.
1Bill_McGrath9yCheers. I understand what he means now, but it still seems like a particularly peculiar belief.
0byrnema9yI gather the discussion of the whole thread rests on this unexpected [http://lesswrong.com/lw/6vn/am_i_obligated_to_reread_the_book_of_mormon/4kmu] premise: Stories always have a blend of fact and fiction. Accounts of travels, culture and civilizations may have some seeds of truth, but other parts about God's intentions and angels needn't be true. My sense is that you are collectively underestimating how unpredictably information can pass down family lines and through traveling story-tellers, scholars and historians. There's a lot packed into this. To give an analogy for a non-theistic example, if some details prove correct about the collective community's awareness of the lost location of Atlantis, Hans Christian Anderson shouldn't get credit for 'knowing' these details when he included them in The Little Mermaid.
1lessdazed9yThis seems like a subtle attempt to shift the burden of proof. The probability of something being true plus the probability of it not being true is one. Other things being true may entail the first thing's not being true. But it's all related and of the same type, as the probability of "not Mormonism" is aggregated out of an unimaginably large number of possibilities. To have a similarly peremptory (I can't think of a good word for what I mean, but I hope it's clear) belief system as Mormonism, one would only require what would look like the first tier, sufficient strong positive evidence for Islam/Judaism/whatever, and that would itself disconfirm Mormonism. To make Mormonism unreasonable, one would only need what would look like the second tier, though what would look like the second tier of evidence would work too. When I was very young, I thought that the ingredients section of a food label had to list, as the first ingredient, something that comprised over 50% of the product. If I still believed this, it would be easy to prove to me that a five-bean salad was mostly kidney beans. Simply show that none of the other four bean types made up a majority of the salad, and there you'd have it! Likewise, religions illegitimately try to prove themselves true or probable by showing other beliefs unlikely, but not only doesn't this suffice to show them probable, it isn't even the case that the most likely thing is necessarily probable. Improbable things can be coherently amalgamated into sets, so materialist explanation of consciousness1+materialist explanation of consciousness2... > dualist explanation of consciousness1+dualist explanation of consciousness2.
0Arandur9yIt is true that a proof of Islam, or one of any other religion, would necessarily constitute a disproof of Mormonism. But in order for any other theory to gain enough credence for me to pay attention to it, one would first have to lessen my confidence in Mormonism, so that I could, as it were, hear the background noise. The question was not what would convince someone without prior belief; the question was what would convince me as a Christian, and in order to do that, first you would have to convince me to step off my Christianity tower. Is this a bias? I don't believe so. I've tried my hardest to erase my preconceived notions and start from scratch. I've tried it three ways. Starting without privileging any hypothesis led to rather a paralysis of thought; I realized that, without any prior hint as to which direction I should start searching for truth, I could only rely on the evidence of my senses; hence atheism. Starting by privileging Mormonism led to a reaffirmation of the veracity of Mormonism. Starting by privileging Catholicism, for comparison, led to Mormonism. This is either a proof of deep-rooted bias in my own mind, or evidence - that suffices for me, at least - that Mormonism is the most correct religion. :3 But of course, this is an experiment I will rehash over the course of my entire life, working ever to perfect my strength as a rationalist. When I was very young, I thought that the Nutrition Information percentages had to total up to 100%. x3 I just thought I'd share that with you. I like your point that the most likely thing isn't necessarily probable. I apologize if I'm taking this the wrong way, but that seems to actually be a point in my favor (though not mine specifically! Please don't accuse me of arrogance!): Just because Mormonism is improbable doesn't necessarily mean it's not the most probable thing out there. But time will tell, and in the meantime, I will attempt to keep my mind wide open.
1lessdazed9yThis could mean at least two things, one right, one wrong. I do not know what you mean. If I pick up a book, and read page 54, and then 53, and then 55, I will think certain things about the world. If instead I had read 53, then 54, then 55, and if doing so would have led me to think different things about the world upon concluding my reading, there is a problem with me as an information collecting and judging agent.
0Arandur9yIt means that, having been born into the covenant, and not having any of the qualms and confusion that apparently are a common result of being born into religion, I therefore have a bias, which may or may not be irreparable, which, if it is, may or may not be unfortunate. Eliezer said that noticing one's confusion was the first step to changing one's mind. I can boldly state, without qualm: I am not confused. Everything I have learned about Mormonism is internally consistent, and consistent with my own ideas on morality. There is a God, and He is my Father, who loves each of us as a child. Joseph Smith was a true prophet, ordained of said God to restore His church in these, the latter days of the world.
0lessdazed9yLet possible states of the world be represented by A, B, C, etc. Let's say A is true. An agent that decides to believe that the world is represented by the theory that comes earliest alphabetically will be fortunate as it will believe true things, but it isn't discerning at all. An agent that believes the contents of books when it reads the book's chapters in sequential order and disbelieves the contents of books when it chooses to read the chapters in reverse order is not an agent designed to discern truth, however lucky it gets deciding how to read each book it reads. I'm just trying to ask to what extent you don't resemble an optimal thinker in this particular way no human totally succeeds at, one possibility would be for you to deny that this human tendency is a flaw. Some people may disproportionately be influenced by the last book they read, others by the first, others by the one's with nice covers, etc.. All I'm trying to get at is to see if you agree it's bad to be a decider that is influenced by the order it gets information in (except for to the extent the order constitutes information, but this isn't really an exception). Someone could claim that truth of a proposition is commensurate with the age of the oldest book containing it, and such a person would not mean what anyone else means by "truth", and would be wrong to the extent they are trying to communicate. Likewise truth isn't usually bound to the order of evidence. If I read a pamphlet advocating Islam, and then one advocating Mormonism, I ought to reach the same exact conclusions as if I had read them in the other order. If I don't, I may happen to come to believe the correct thing, but this is true of any decision process, even the alphabetical one. In the first two quotes above, you seem to disagree with what I say, in the latter two, you seem to agree.
0Arandur9yThe confusion, I reckon, comes from my inability to step outside myself. I am not a perfect rationalist; I am trapped to an extent by the concepts taught to me since birth, just as I find myself uncomfortable with my gender identity due to growing up in an abusive household. It is difficult to step outside one's own biases. So yes, my bias may be irreparable. As for "unfortunate", the odds of it being an unfortunate bias are exactly the odds of Mormonism being true. If I believe the truth, then I am fortunate. It is the chance that my bias is unfortunate that drives me ever to refine my understanding, and never stop questioning my premises. It's not not a flaw. I'm just struggling to determine to what extent my belief in my religion is due to prior bias, and to what extent it's due to rational thought.
2Alicorn9yThis sounds very convenient for you. Do you consider the church's consistency with your morality to be evidence that your morality is correct, or that the church is? Especially if the latter, what evidential status do you consider people whose morality disagrees at least partly with the church to have?
0Arandur9yOh, yes, it's very convenient. :P Well, not always. A good example is the recent fight over Prop 8, wherein the Church's morality came into sharp contrast with the morality of many outside it. (I will not say "most", because it was in fact the vote of California citizens which decided the matter, and not the Church.) To showcase the inconvenience without revealing overmuch about my personal life, I will simply state that I have many personal friends who were outraged at my decision to stand with my Church on the matter. The church's consistency with my own morality is, I think, evidence that the Church is correct. Without the church, my morality would still exist. As far as others' conflicting moralities... ..... That's an interesting question, actually. What evidential status does my conflicting morality have on yours?
0Alicorn9yWithout your morality, the church would still exist, too, wouldn't it? Some, but not more than the average dissenter - less than a typical clever consequentialist found around these parts, and not even as much as the ideologically similar votes of Mormons I'm friends with and have had a chance to question in more detail. But that's not quite the same question, because I developed the framework of my own morality independently, and am not backed by a large institution. What I want to know is more along the lines of: why is your morality agreeing with the LDS church evidence for the LDS church, which is not overwhelmed by the majority of human beings whose moralities disagree with yours/the church's, or overbalanced by the humans whose moralities agree with those of other religions? (If you were using "evidence" in a sufficiently technical sense that this overwhelmingness/overbalancedness was in fact noted and simply left unmentioned as strictly irrelevant to what I originally asked, I retract the question, but I suspect otherwise.)
0Arandur9yI was in fact using evidence in that technical a sense, but I'll answer your question anyway. Because morality is not a binary attribute. You can't go out on the street and ask them, "Do you agree with the Mormons, yes or no?" Well, you could, but then if they answered no, you'd have to ask them how many people they killed today. It's exactly that fallacy that leads fundamentalist Christians /shudder/ to claim that atheists love to rape and murder and... I dunno, engage in bestiality or something. So no, other peoples' moralities don't sway me particularly much, because a) they don't matter as much to me as my own morality - as I think you'd agree with, saying "not more than the average dissenter"; and b) because the consonance between my morality and Mormonism isn't that much of an evidence in its favor. I was using it mainly as a contrast between myself and all the people who have posted saying that Christianity made them feel "wrong".
0lavalamp9yI saw this on the side while reading an unrelated post... I'm much more inclined to think it's evidence that you were raised in the church, or in a culture influenced by the church, etc... If I rephrase what you said, it's "Party X's agreement with me on subject Y is evidence that Party X can think well and is probably right about other things, too." Please tell me you meant something else... PS: You seem capable of updating, judging from a few of the comments in this thread, and you seem to care about the truth. The next step is to stop holding your own beliefs to a different standard of evidence than you do other beliefs. I hope you find your time in the soon-to-be-formerly-theistic camp more fun than I did.
0Arandur9yYour point is only applicable inasmuch as you took my quote out of context. I was asked to choose one of two options; I chose the one that seemed most right to me. I could be wrong, but your point doesn't answer to the original question.

If 2+2 equals 3, I desire to believe that 2+2 equals 3. I want my conclusion to be controlled by the abstract fact I seek to discern.

       A middle aged woman is charged with the task of counting objects(oranges, apples, earplugs) individually and in pairs as they roll onto a conveyer belt she notices that they come in order of ( OOO,A A, E) and has performed this job for several years. The woman devises a formula for counting these objects she places a [ 1 in column 3 ] for a set of oranges that pass by, then a [ 1 in column 2 ] for the pair of apples that pass by, she then places a [ 1 in column 1 ] for the earplug. So by looking at her count of the line (OOO,AA,E,OOO,AA,E) or  2+2+2
... (read more)
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For a while this confused me, because I incorrectly identified what part of Eliezer's argument I thought was wrong.

Suppose I were to make all those observations suggesting that combining two objects with two objects produced three objects. I would not conclude that 2+2=3, rather I would conclude that objects were not modelled by Peano Arithmetic. (This much has been said by other commenters). But then I only 'know' Peano Arithmetic through the (physical) operation of my own brain.

Here's how to convince me that 2+2=3. Suppose I look at the proof from (peano axioms) to (2+2=4), and suddenly notice that an inference has been made that doesn't follow from the inference rules (say, I notice that the proof says a + (b⁺) = (a+b)⁺ and I know full well that the correct rule is (a⁺)+(b⁺)=(a+b)⁺). I correct this 'error' and follow through to the end of the proof, and conclude the result 2+2=3. What do I do? I consider that this observation is more likely if 2+2=3 than if 2+2=4, and so I update on that. It's still more likely that 2+2=4, because it's more likely that I've made an error this time than that everyone who's analysed that proof before has made an error (or rather, that I h... (read more)

1komponisto9yExactly. This is one of Eliezer's few genuine philosophical mistakes, one which, four years later, he's still making [http://lesswrong.com/lw/797/harry_potter_and_the_methods_of_rationality/4rn3].
1Eliezer Yudkowsky9yI know very well the difference between a collection of axioms and a collection of models of which those axioms are true, thank you. A lot of people seem to have trouble imagining what it means to consider the hypothesis that SS0+SS0 = SSS0 is true in all models of arithmetic, for purposes of deriving predictions which distinguish it from what we should see given the alternative hypothesis that SS0+SS0=SSSS0 is true in all models of arithmetic, thereby allowing internal or external experience to advise you on which of these alternative hypotheses is true.
1[anonymous]9yReading your essay I wondered whether it would have been more effective if you had chosen bigger numbers than 2, 2, and 3. e.g. "How to convince me that 67+41 = 112."
7ec4299yThat would have been a damn nuisance, because throughout the rest of this comment thread we'd have been writing unhelpfully long strings of Ss. ;)
1[anonymous]9yI was proud of this comment [http://lesswrong.com/lw/612/about_addition_and_truth/4app] and I comfort myself with your explanation for why it got the response it did.
2ec4299yI, at least, was not suggesting that you don't know the difference, merely that your article failed to take account of the difference and was therefore confusing and initially unconvincing to me because I was taking account of that difference. However (and it took me too damn long to realise this; I can't wait for Logic and Set Theory this coming year), I wasn't talking about "models" in the sense that pebbles are a Model of the Theory PA. I was talking in the sense that PA is a model of the behaviour observed in pebbles. If PA fails to model pebbles, that doesn't mean PA is wrong, it just means that pebbles don't follow PA. If a Model of PA exists in which SS0+SS0 = SSS0, then the Theory PA materially cannot prove that SS0+SS0 ≠ SSS0, and if such a proof has been constructed from the axiomata of the Theory then either the proof is in error (exists a step not justified by the inference rules), or the combination of axiomata and inference rules contains a contradiction (which can be rephrased as "under these inference rules, the Theory is not consistent"), or the claimed Model is not in fact a Model at all (in which case one of the axiomata does not, in fact, apply to it). I should probably write down what I think I know about the epistemic status of mathematics and why I think I know it, because I'm pretty sure I disagree quite strongly with you (and my prior probability of me being right and you being wrong is rather low).
9Oscar_Cunningham9yScientists and mathematicians use the word "model" in exactly opposite ways. This is occasionally confusing.
1komponisto9yThen why do you persist in saying things like "I don't believe in [Axiom X]/[Mathematical Object Y]"? If this distinction that you are so aptly able to rehearse [http://lesswrong.com/lw/iq/guessing_the_teachers_password/] were truly integrated into your understanding, it wouldn't occur to you to discuss whether you have "seen" a particular cardinal number. I understand the point you wanted to make in this post, and it's a valid one. All the same, it's extremely easy to slip from empiricism to Platonism when discussing mathematics, and parts of this post can indeed be read as betraying that slip (to which you have explicitly fallen victim on other occasions, the most recent being the thread I linked to).

I don't think people really understood what I was talking about in that thread. I would have to write a sequence about

  • the difference between first-order and second-order logic
  • why the Lowenheim-Skolem theorems show that you can talk about integers or reals in higher-order logic but not first-order logic
  • why third-order logic isn't qualitatively different from second-order logic in the same way that second-order logic is qualitatively above first-order logic
  • the generalization of Solomonoff induction to anthropic reasoning about agents resembling yourself who appear embedded in models of second-order theories, with more compact axiom sets being more probable a priori
  • how that addresses some points Wei Dai has made about hypercomputation not being conceivable to agents using Solomonoff induction on computable Cartesian environments, as well as formalizing some of the questions we argue about in anthropic theory
  • why seeing apparently infinite time and apparently continuous space suggests, to an agent using second-order anthropic induction, that we might be living within a model of axioms that imply infinity and continuity
  • why believing that things like a first uncountable ordinal
... (read more)
3[anonymous]9yLowenheim-Skolem, maybe? But that does not imply that you can't talk about integers or reals in first order logic. And indeed you can talk about integers and real numbers using first-order logic, people do so all the time.
1Eliezer Yudkowsky9yOnly in the same sense that you can talk about kittens by saying "Those furry things!" There'll always be some ambiguity over whether you're talking about kittens or lions, even though kittens are in fact furry and have all the properties that you can deduce to hold true of furry things.
3[anonymous]9yNot in the same sense at all. All of the numbers that you have ever physically encountered were nameable, definable, computable. Moreover they came to you with algorithms for verifying that one of them was equal to another.
2Vladimir_Nesov9yYes, and that's OK. I suspect you can't do qualitatively better than that (viz ambient set-theoretic universe for second-order logic), but it's still possible (necessary?) to work under this apparent lack of absolute control over what it is you are dealing with. Even though (first order) PA doesn't know what "integers" are, it's still true that the statements it believes valid are true for "integers", it's useful that way (just as AIs or humans are useful for making the world better). It is a device that perceives some of the properties of the object we study, but not all, not enough to rebuild it completely. (Other devices can form similarly imperfect pictures of the object of study and its relationship with the device perceiving it, or of themselves perceiving this process, or of the object of study being affected by behavior of some of these devices.) Likewise, we may fail to account for all worlds that we might be affecting by our decisions, but we mostly care about (or maybe rather have non-negligible consequentialist control over) "real world" (or worlds), whatever this is, and it's true that our conclusions capture some truth about this "real world", even if it's genuinely impossible for us to ever know completely what it is. (We of course "know" plenty more than was ever understood, and it's a big question how to communicate to a FAI what we do know.)
1CronoDAS9yIn other words, a first uncountable ordinal may be perfectly good math, but it's not physics?
0Eliezer Yudkowsky9yI don't believe it's good math until it becomes possible to talk about the first uncountable ordinal, in the way that you can talk about the integers. Any first-order theory of the integers, like first-order PA, will have some models containing supernatural numbers, but there are many different sorts of models of supernatural numbers, you couldn't talk about the supernaturals the way you can talk about 3 or the natural numbers. My skepticism about "the first uncountable ordinal" is that there would not exist any canonicalizable mathematical object - nothing you could ever pin down uniquely - that would ever contain the first uncountable ordinal inside it, because of the indefinitely extensible character of well-ordering. This is a sort of skepticism of Platonic existence - when that which you thought you wanted to talk about can never be pinned down even in second-order logic, nor in any other language which does not permit of paradox.
6Sniffnoy9yYou seem to keep forgetting that the whole notion of "second-order logic" does not make sense without some ambient set theory. (Unless I am greatly misunderstanding how second-order logic works?) And if you have that, then you can pin down the natural numbers (and the first uncountable ordinal) in first-order terms in this larger theory.
0Eliezer Yudkowsky9yOnly to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense. It's just that the ambient models in the second-order theory include collections of possible predicates of any objects that get predicates attached, or if you prefer, people who speak in second-order logic think that it makes as much sense to say "all possible collections that include some objects and exclude others, but still include and exclude only individual objects" as "all objects".
6[anonymous]9yWell, it makes sense to me without any models. I can compute, prove theorems, verify proofs of theorems and so on happily without ever producing a "model" for the natural numbers in toto, whatever that could mean.
0CronoDAS9yHmmm... ::goes and learns some more math from Wikipedia:: Okay... I now know what an ordinal number actually is. And I'm trying to make more sense out of your comment... So, re-reading this: So if I understand you correctly, you don't trust anything that can't be defined up to isomorphism in second-order logic, and "the set of all countable ordinals" is one of those things? (I never learned second order logic in college...)
6komponisto9yEverything sounded perfectly good until the last bullet: ERROR: CATEGORY. "Wavefunction" is not a mathematical term, it is a physical term. It's a name you give to a mathematical object when it is being used to model the physical world in a particular way, in the specific context of that modeling-task. The actual mathematical object being used as the wavefunction has a mathematical existence totally apart from its physical application, and that mathematical existence is of the exact same nature as that of the first uncountable ordinal; the (mathematical) wavefunction does not gain any "ontological bonus points" for its role in physics. Pinning down a single model up to isomorphism might be a nice property for a set of axioms to have, but it is not "reality-conferring": there are two groups of order 4 up to isomorphism, while there is only one of order 3; yet that does not make "group of order 3" a "more real" mathematical object than "group of order 4".
6ata9yI would like very very much to read that sequence. Might it be written at some point?
4ec4299yHmm, funny you should treat "I don't believe in [Mathematical Object Y]" as Platonism. I generally characterise my 'syntacticism' (wh. I intend to explain more fully when I understand it the hell myself) as a "Platonic Formalism"; it is promiscuously inclusive of Mathematical Objects. If you can formulate a set of behaviours (inference rules) for it, then it has an existing Form - and that Form is the formalism (or... syntax) that encapsulates its behaviour. So in a sense, uncountable cardinals don't exist - but the theory of uncountable cardinals does exist; similarly, the theory of finite cardinals exists but the number '2' doesn't. This is of course bass-ackwards from a map-territory perspective; I am claiming that the map exists and the territory is just something we naïvely suppose ought to exist. After all, a map of non-existent territory is observationally equivalent to a map of manifest reality; unless you can observe the actual territory you can't distinguish the two. Taking as assumption that the observe() function always returns an object Map, the idea that there is a territory gets Occamed out. There is a good reason why I should want to do something so ontologically bizarre: by removing referents, and semantics, and manifest reality; by retaining only syntax, and rejecting the suggestion that one logic really "models" another, we finally solve the problems of Gödel (I'm a mathematician, not a philosopher, so I'm allowed to invoke Gödel without losing automatically) and the infinite descent when we say "first order logic is consistent because second-order logic proves it so, and we can believe second-order logic because third-order logic proves it consistent, and...". When all you are doing is playing symbol games stripped of any semantics, "P ∧ ¬P" is just a string, and who cares if you can derive it from your axiomata? It only stops being a string when you apply your symbol games to what you unknowingly label as "manifest reality", when you (essenti
0[anonymous]9yAs a fellow mathematician, I want to point out that it doesn't mean you win automatically, either. Just look at Voevodsky's [http://en.wikipedia.org/wiki/Vladimir_Voevodsky] recent FOM talk at the IAS.
0ec4299yWell, of course I don't win automatically. It's just that there's a kind of Godwin's Law of philosophy, whereby the first to invoke Gödel loses by default.
0Andreas_Giger9yMaybe I'm misinterpreting you, but could you explain how any non-symmetric equation can possibly be true in all models of arithmetic?
2Eliezer Yudkowsky9ySS0 isn't a free variable like "x", it is, in any given model of arithmetic, the unique object related by the successor relation to the unique object related by the successor relation to the unique object which is not related by the successor relation to any object, which is how mathematicians say "Two".
2ec4299yAlthough as a mathmo myself I should point out that, to save time, we usually pronounce it "Two". :)
1Andreas_Giger9yI am quite familiar with TNT. However either you are talking about models of arithmetic based on peano axioms, in which case e.g. SS0 + SS0 = SSS0 simply cannot be true, for it contradicts these axioms and if both the peano axioms and said equation were true, you wouldn't have a model of arithmetics; or (what I'm assuming) you are actually talking about non-peano arithmetics, in which case there is no compelling reason why any equation of this kind should generally be true anyway. On another note, it seems that bayesianism is heavily based on peano arithmetic, so refuting peano arithmetic by means of bayesianism seems like refuting bayesianism rather than refuting peano arithmetic, at least to me.
7Tyrrell_McAllister9yThe purpose of the article is only to describe some subjective experiences that would cause you to conclude that SS0+SS0 = SSS0 is true in all models of arithmetic. But Eliezer can only describe certain properties that those subjective experiences would have. He can't make you have the experiences themselves. So, for example, he could say that one such experience would conform to the following description: "You count up all the S's on one side of the equation, and you count up all the S's on the other side of the equation, and you find yourself getting the same answer again and again. You show the equation to other people, and they get the same answer again and again. You build a computer from scratch to count the S's on both sides, and it says that there are the same number again and again." Such a description gives some features of an experience. The description provides a test that you could apply to any given experience and answer the question "Does this experience satisfy this description or not?" But the description is not like one in a novel, which, ideally, would induce you to have the experience, at least in your imagination. That is a separate and additional task beyond what this post set out to accomplish.
1Andreas_Giger9yYes, I am aware of that. However, I don't think two pebbles on the table plus another two pebbles on the table resulting in three pebbles on the table could cause anyone sane to conclude that SS0 + SS0 = SSS0 is true in all models of arithmetic. In order to be convinced of that, you would have to assign "PA doesn't apply to pebbles" a lower prior probability than "PA is wrong". The statement "PA applies to pebbles" (or anything else for that matter) doesn't follow of the peano axioms in any way and is therefore not part of peano arithmetic. So what if peano arithmetic doesn't apply to pebbles, there are other arithmetics that don't either, and that doesn't make them any wrong. You're using them everyday in situations where they do apply. A mathematical theory doesn't consist of beliefs that are based on evidence; it's an axiomatic system. There is no way any real-life situation could convince me that PA is false. Saying "SS0 + SS0 = SSS0 is true in all models of arithmetic" sounds like "0 = S0" or "garble asdf qwerty sputz" to me. It just doesn't make any sense. Mathematics has nothing to do with experience, only to what extent mathematics applies to reality does.
1Tyrrell_McAllister9yThat you have certain mathematical beliefs has a lot to do with the experiences that you have had. This applies in particular to your beliefs about what the theorems of PA are.
1Andreas_Giger9ySorry, I edited the statement in question right before you posted that because I anticipated a similar reaction. However, you're still wrong. It has only to do with my beliefs to what extent peano arithmetic applies to reality, which is something completely different. Edit: Ok, you're probably not wrong, but it rather seems we are talking about different things when we say "mathematical beliefs". Whether peano arithmetic applies to reality is not a mathematical belief for me.
0Andreas_Giger9yAnd another thing: It might be possible that if peano arithmetic didn't apply to reality I wouldn't have any beliefs about peano arithmetic because I might not even think of it. However there is no way I could establish the peano axioms and then believe that SS0 + SS0 = SSS0 is true within peano arithmetic. It's just not possible.
1Tyrrell_McAllister9yConsider the experiences that you have had while reading and thinking about proofs within PA. (The experience of devising and confirming a proof is just a particular kind of experience, after all.) Are you saying that the contents of those experiences have had nothing to do with the beliefs that you have formed about what the theorems of PA are? Suppose that those experiences had been systematically different in a certain way. Say that you consistently made a certain kind of mistake while confirming PA proofs, so that certain proofs seemed to be valid to you that don't seem valid to you in reality. Would you not have arrived at different beliefs about what the theorems of PA are? That is the sense in which your beliefs about what the theorems of PA are depend on your experiences.
0Andreas_Giger9yI'm not sure I 100% understand what you're saying, but the question "which beliefs will I end up with if logical reasoning itself is flawed" is of little interest to me.
1Tyrrell_McAllister9yIs the question "Which beliefs will I end up with if my faculty of logical reasoning is flawed" also of little interest to you?
0Andreas_Giger9yYes, because if I assume that my faculty of logical reasoning is flawed, no deductions of logical reasoning I do can be considered certain, in which case everything falls: Mathematics, physics, bayesianism, you name it. It is therefore (haha! but what if my faculty of logical reasoning is flawed?) very irrational to assume this.
1Tyrrell_McAllister9yBut you know that your faculty of logical reasoning is flawed to some extent. Humans are not perfect logicians. We manage to find use in making long chains of logical deductions even though we know that they contain mistakes with some nonzero probability.
0Andreas_Giger9yI don't know that. Can you prove that under the assumption you're making? As I see it, my faculty of logical reasoning is not flawed in any way. The only thing that's flawed is my faculty of doing logical reasoning, i.e. I'm not always doing logical reasoning when I should be. But that's hardly the matter here. I would be very interested in how you can come to any conclusion under the assumption that the logical reasoning you do to come to that conclusion is flawed. If my faculty of logical reasoning is flawed, I can only say one thing with certainty, which is that my faculty of logical reasoning is flawed. Actually, I don't think I could even say that. Edit: I don't consider this to be a problem of actual faculty of logical reasoning because if someone finds a logical mistake I will agree with them.
1Tyrrell_McAllister9ySorry for not being clear. By "faculty of logical reasoning", I mean nothing other than "faculty of doing logical reasoning".
0Andreas_Giger9yIn that case I have probably answered your original question here [http://lesswrong.com/lw/jr/how_to_convince_me_that_2_2_3/4tx7].
2moshez9ySo you don't consider mistakes in logical reasoning a problem because someone might point them out to you? What if it's an easy mistake to make, and a lot of other people make the same mistake? At this point, it seems like you're arguing about the definition of the words "problem with", not about states of the world. Can you clarify what disagreement you have about states of the world?
1Andreas_Giger9yI don't consider these mistakes to be no problem at all. What I meant to say is that the existence of these noise errors doesn't reduce the reasonabliness of me going around and using logical reasoning to draw deductions. Which also means that if reality seems to contradict my deductions, then either there is an error within my deductions that I can theoretically find, or there is an error within the line of thought that made me doubt my deductions, for example eyes being inadequate tools for counting pebbles. To put it more generally: If I don't find errors within my deductions, then my perception of reality is not an appropriate measure for the truth of my deductions, unless said deductions deal in any way with the applicability of other deductions on reality, or reality in general, which mathematics does not. It's not as if errors in perceiving reality weren't much more numerous and harder to detect than errors in anyone's faculty of doing logical reasoning.
3Tyrrell_McAllister9yAnd the probability of an error in a given logical argument gets smaller as the chain of deductions gets shorter and as the number of verifications of the argument gets larger. Nonetheless, the probability of error should never reach zero [http://lesswrong.com/lw/mp/0_and_1_are_not_probabilities/], even if the argument is as short as the proof that SS0 + SS0 = SSSS0 in PA, and even if the proof has been verified by yourself and others billions of times. ETA: Where ever I wrote "proof" in this comment, I meant "alleged proof". (Erm ... except for in this ETA.)
0Andreas_Giger9yThe probability that there is an error within the line of thought that lets me come to the conclusion that there is an error within any theorem of peano arithmetic is always higher than the probability that there actually is an error within any theorem of peano arithmetic, since probability theory is based on peano arithmetic and if SS0 + SS0 = SSSS0 were wrong, probability theory would be at least equally wrong.
4Tyrrell_McAllister9y(Emphasis added.) Where ever I wrote "proof" in the grandparent comment, I should have written "alleged proof". We probably agree that the idea of "an error in a theorem of PA" isn't meaningful. But the idea that everyone was making a mistake the whole time that they thought that SS0 + SS0 = SSSS0 was a theorem of PA, while, all along, SS0 + SS0 = SSS0 was a theorem of PA — that idea is meaningful. After all, people are all the time alleging that some statement is a theorem of PA when it really isn't. That is to say, people make arithmetic mistakes all the time.
0Andreas_Giger9yThat is true. However, if your perception of reality leads you to the thought that there might be an error with SS0 + SS0 = SSSS0, and you can't find that error, then it is irrational to assume that there actually is an error with SS0 + SS0 = SSSS0 rather than with your perception of rationality or the concept of applying SS0 + SS0 = SSSS0 to reality. Can we agree on that?
3Tyrrell_McAllister9yI think so, if I understand you. But I think that you're referring to a more restricted class of "perceptions of reality" than Eliezer is. In the kind of scenario that Eliezer is talking about, your perceptions of reality include seeming to find an error in the alleged proof that SS0 + SS0 = SSSS0 (and confirming your perception of an error sufficiently many times to outweigh all the times when you thought you'd confirmed that the alleged proof was valid). If that is the kind of "perception of reality" that we're talking about, then you should conclude that there was an error in the alleged proof of SS0 + SS0 = SSSS0.
0Andreas_Giger9yThat is all good and valid, and of course I don't believe in any results of deductions with errors in them just based on said deductions. But that has nothing to do with reality. Two pebbles plus two pebbles resulting in three pebbles is not what convinces me that SS0 + SS0 = SSS0; finding the error is, which is nothing that is perceived (i.e. it is purely abstract). If we're defining "situation" in a way similar to how it's used in the top-level post (pebbles and stuff), then there simply can't exist a situation that could convince me that SS0 + SS0 = SSSS0 is wrong in peano arithmetic. It might convince me to check peano arithmetic, of course, but that's all. I try to not argue about definition of words, but it just seems to me that as soon as you define words like "perception", "situation", "believe" etcetera in a way that would result in a situation capable of convincing me that SS0 + SS0 = SSS0 is true in peano arithmetic, we are not talking about reality anymore.
0Andreas_Giger9yOkay, I just thought of a possible situation that would indeed "convince" me of 2 + 2 = 3: Disable the module of my brain responsible for logical reasoning, then show me some stage magic involving pebbles or earplugs, and then my poor rationalization module would probably end up with some explanation along the lines of 2 + 2 = 3. But let's not go there.
1ec4299yI think the point is that mathematical reasoning is inherently self-correcting in this sense, and that this corrective force is intentionistic and Lamarckian - it is being corrected toward a mathematical argument which one thinks of as a timeless perfect Form (because come on, are there really any mathematicians who don't, secretly, believe in the Platonic realism of mathematics?), and not just away from an argument that's flawed. An incorrect theory can appear to be supported by experimental results (with probability going to 0 as the sample size goes to \infty), and if you have the finite set of experimental results pointing to the wrong conclusion, then no amount of mind-internal examination of those results can correct the error (if it could, your theory would not be predictive; conservation of probability, you all know that). But mind-internal examination of a mathematical argument, without any further entangling (so no new information, in the Bayesian sense, about the outside world; only new information about the world inside your head), can discover the error, and once it has done so, it is typically a mechanical process to verify that the error is indeed an error and that the correction has indeed corrected that error. This remains true if the error is an error of omission (We haven't found the proof that T, so we don't know that T, but in fact there is a proof of T). So you're not getting new bits from observed reality, yet you're making new discoveries and overthrowing past mistakes. The bits are coming from the processing; your ignorance has decreased by computation without the acquisition of bits by entangling with the world. That's why deductive knowledge is categorically different, and why errors in logical reasoning are not a problem with the idea of logical reasoning itself, nor do they exclude a mathematical statement from being unconditionally true. They just exclude the possibility of unconditional knowledge. Can you conceive of a world in wh
0moshez9yI guess there are my beliefs-which-predict-my-expectations and my aliefs-which-still-weird-me-out. In the sense of beliefs which predict my expectation, I would say the following about mathematics: as far as logic is concerned, I have seen (with my eyes, connected to neurons, and so on) the proof that from P&-P anything follows, and since I do want to distinguish "truth" from "falsehood", I view it as (unless I made a mistake in the proof of P&-P->Q, which I view as highly unlikely -- an easy million-to-one against) as false. Anything which leads me to P&-P, therefore, I see as false, conditional on the possibility I made a mistake in the proof (or not noticed a mistake someone else made). Since I have a proof from "2+2=3" to "2+2=3 and 2+2!=3" (which is fairly simple, and I checked multiple times) I view 2+2=3 as equally unlikely. That's surely entanglement with the world -- I manipulated symbols written by a physical pen on a physical paper, and at each stage, the line following obeyed a relationship with the line before it. My belief that "there is some truth", I guess, can be called unconditional -- nothing I see will convince me otherwise. But I'm not even certain I can conceive of a world without truth, while I can conceive of a world, sadly, where there are mistakes in my proofs :)
0ec4299yYou're missing the essential point about deductives, which is this: Changing the substrate used for the calculations does not change the experiment. With a normal experiment, if you repeat my experiment it's possible that your apparatus differs from mine in a way which (unbeknownst to either of us) is salient and affects the outcome. With mathematical deduction, if our results disagree, (at least) one of us is simply wrong, it's not "this datum is also valid but it's data about a different set of conditions", it's "this datum contains an error in its derivation". It is the same experiment, and the same computation, whether it is carried out on my brain, your brain, your brain using pen and paper as an external single-write store, theorem-prover software running on a Pentium, the same software running on an Athlon, different software in a different language running on a Babbage Analytical Engine... it's still the same experiment. And a mistake in your proof really is a mistake, rather than the laws of mathematics having been momentarily false leading you to a false conclusion. To quote the article, "Unconditional facts are not the same as unconditional beliefs." Contrapositive: conditional beliefs are not the same as conditional facts. The only way in which your calculation entangled with the world is in terms of the reliability of pen-and-paper single-write storage; that reliability is not contingent on what the true laws of mathematics are, so the bits that come from that are not bits you can usefully entangle with. The bits that you can obtain about the true laws of mathematics are bits produced by computation.
2[anonymous]9yMaybe "earplugs do not model PA," not the other way around? (Edit: just saw this [http://lesswrong.com/lw/jr/how_to_convince_me_that_2_2_3/4tqt] excellent clarification.) Number-handling is an older science than Peano arithmetic, and especially older than model theory. The numbers 2 and 3 would "exist" even if PA were shown to have no models. At least, the notation 2 and 3 would still be relevant to things that really exist. It is very easily verified that 2 + 2 does not equal 3, but not effortlessly verified. It takes a positive amount of effort to verify it, and there is a positive amount risk of having made a mistake while doing so.

You're over-thinking this. Take a look at this real-world example of a "neurological fault":

Now I knew where I was. Soon I would come to interchange 27 with its two ramps, A and B. B led away from my destination and A directly into it. It had always struck me as strange that one reached 27B before 27A. I recalled drawing that on a map to give to someone who was going to visit me. My breathing has returned to normal and my panic had disappeared. I come up to the first sign for the interchange.

"27A"

I could hardly breathe. That was not possible. 27A was after 27B. I knew that. I considered for a moment the possibility that on the previous night, shortly after I drove on this very highway, construction workers had descended en masse on the interchanges and somehow moved them. That seemed far more possible than that my clear (and detailed) memory could be so wrong. 27A looked exactly as I remembered, except that now I could see 27B clearly in the distance and in the past I had to turn my head to see it.

I exited on the ramp that I knew wasn’t there twenty-four hours previously to find myself on a well-remembered road. And soon I was home.

Now imagine that happenin... (read more)

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?

Why does this need to go out to Christians? I suspect that most, if not all, people reading this are non-Muslims who know Bayes's Theorem. What would convince you of the truth of Islam?

1ArisKatsaris9yIf some fundamental constant, or the ratio between two fundamental constants, encoded in its binary digits the Qu'ran, that would cause me to believe with greater than 50% certainty in the truth of Islam.
1TheOtherDave9yAre you constraining the encoding scheme at all?
2ArisKatsaris9yIn just commonsense ways, e.g. it must not be encoded in a way so arbitrary that you would be able to derive any other similar-length piece of literature by similar methodologies.
3thomblake9yIt's believed by some (but not proven) that you can find any sequence of digits you'd like in Pi. And even though everybody knows that 2Pi is the much more sensible constant, that transformation helpfully changes the sequence of binary digits not at all.
1ArisKatsaris9yBut you generally need a number that has about as many bits as the sequence itself just to pinpoint which digit to start reading from. If that number is represented by some other constant, that counts. If we just have to figure it out ourselves, in the same way that we'd be able to figure out which digit to start if we wanted to find "The Hobbit" encoded in the digits of pi, it doesn't count.
0RichardKennaway9yWhy? It would certainly be an astounding thing (subject to the obvious caveats about the complexity of the encoding scheme), but why would it imply that the encoded material was true?
-2Multiheaded9yYeah. As someone said here, even true "miracles" aren't proof of a theistic God (much less a particular version of one) - it might be e.g. some alien teenagers pulling a prank.
1ArisKatsaris9y"even"? The evidence I mentioned would be much more impressive than a mere miracle. And the Qu'ran is quite clearly indicative of the Islamic version of God. And you may also be an alien teenager pulling a prank, but I'm nonetheless convinced you're human. Demanding an infinite amount of proof before you're convinced of anything isn't actually rationality. Nobody's talking about 100% certainty here.
0Multiheaded9yThat's because I'm interacting with you in an entirely ordinary human way and not displaying any "miracles" or such.
1ArisKatsaris9yI think we must be having different discussions, because I don't understand what your point is, and you don't seem to understand my point either.
2dlthomas9yYour point is that you're forgetting about priors. This should also be Multiheaded's point, however poorly expressed. Our prior for "alien pranksters" is not high - the question is just how low it is compared to alternate explanations. Any reasonable priors assign vastly more probability that Multiheaded is human than... well, anything else, but even if we rejected that it would take a while before we got to aliens. The question of whether aliens or the supernatural is to be assigned higher probability when faced with something as striking as apparent manipulation of the physical constants underlying this universe is a much harder question.
6TheOtherDave9yIf I find the text of Moby Dick suitably encoded (whatever that means) into the foundation of a building, and I don't find other texts encoded into that building, it seems reasonable to take seriously the theory that there exists some process or entity which has a special relationship both with that building, and with the text of Moby Dick, different from the relationship it has with any other text. If I find the text of the Koran suitably encoded into the fundamental constants of the universe, it seems equally reasonable to take seriously the theory that there exists some process or entity which has a special relationship both with that universe, and with the text of the Koran, different from the relationship it has with any other text. You're right, of course, that it doesn't follow from that that either the Koran or Moby Dick is true. Neither does it follow from the truth of the Koran (whatever that means) that Islam is true (whatever that means). OTOH, converting to a belief in Islam on the basis of that evidence seems more justified than remaining indifferent to Islam in the face of that evidence. Of course, those aren't the only options. Granted, it's not really clear to me what is a reasonable response to that evidence. "Investigate the Koran," of course, but I have no sense of what such an investigation might even look like.
3ArisKatsaris9yI don't think it's a question that deserves an answer, if you're truly asking whether finding the Qu'ran embedded into the physical/mathematical structure of the universe should be considered evidence supporting the truth of Islam. It would quite obviously not be evidence AGAINST Islam, nor would it be evidence UNCORRELATED to the reality of Islam. As such it can only be evidence in support of Islam. If you're just arguing that it shouldn't make me raise my confidence level to over 50%, to what percentage do you think one should raise it to, given such a finding?
-3RichardKennaway9yWhere did you pull that out of? I mean, we're talking blue tentacle scenarios [http://lesswrong.com/lw/gu/some_claims_are_just_too_extraordinary] here.
1ArisKatsaris9yDownvoted for lack of clarity. What exactly are you saying here? Are you truly saying that if the Qu'ran is firmly embedded in the physics of the world, that's uncorrelated to Islam having validity? Stop being coy with your responses please, then perhaps I"ll also be able to answer whatever-your-ACTUAL-question-is better.
0RichardKennaway9yI see no reason to presume anything about this correlation. P(the Qu'ran is embedded in pi or similar) is so low as to render the probability of anything conditional on the hypothesis incalculable. That is what I mean by the reference to blue tentacle scenarios. I repeat, why would discovering a large and complex yet parochial assertion, such as the Qu'ran, embedded eternally in the world without human agency, lead you to believe it? You have done no more than assert this as "obvious".
9ArisKatsaris9yOkay, simple logic here: There's a number of given possibilities. A) The Qu'ran is a completely human construction, much like any other work of literature. B) The Qu'ran is a false message by some non-human (but still located within our universe) entity to humanity (e.g. teenage alien pranksters, fairies, Satan, whatever) C) The Qu'ran is a false message by some entity/entities that constructed (or helped construct) the universe (e.g. lying "simulators") D) The Qu'ran is a true message by God to humanity E) ...other possibilities I may be neglecting. These options must sum up to 100%. (A) is what I (and pretty much all other atheists) currently believe. (B) is believed by e.g. those Christians who think that Qu'ran is the work of the devil. (D) is what Muslims believe. If we have enough data to exclude possibilities (A) and (B), don't you see that this must increase the probability of both (C) and (D)? That now the possibilities (C) and (D) must sum up to almost 100%, much like probabilities (A), (B), (C) and (D) previously did? This is obvious to me.
0RichardKennaway9yI can't see why (B) got excluded, but more significantly, what happened to (E)? In blue tentacle scenarios, that's the most important one, the unknown unknowns. You list it explicitly, then completely ignore it. Unknown unknowns are important long before the outer wilds of twenty sigma. It's those that kill you when you thought you had all bases covered. And out in the weird realms where parochial messages can be hidden in eternal mathematics or cosmological engineering, it is impossible to conceive what else may be lurking there.
3MarkusRamikin9yB got excluded because in-universe agents don't write the universe's laws.
4ArisKatsaris9yYou're missing the point. Keep including (B) as a possibility if you like (I could debate with you on the probability of aliens so powerful that they can alter the fundamental constants of the universe they inhabit and be willing to do so just to prank humanity with a religion -- but it's not worth the trouble). Also add as many (E)s, (F)s, (G)s you like. Add a hundred different possibilities if you want. Either way, if (A) gets significantly reduced as a probability, and we have no reason to simultaneously reduce the probability of (D) (or adjust the relative likelihood of the remaining probabilities in a manner that disadvantages (D)), then (D)'s estimated probability must increase by a factor analogous to the weight that (A) previously held. If I currently believe (A) to be 99.99% percent likely, then its exclusion automatically increases the probability of (D) by a factor of 10,000 -- no matter how many alternate possibilities Es, Fs, Gs you also provide. At this point I think you're just being allergic at the idea of anything even hypothetically increasing the probability of Islam being true. What hypothetical amount of evidence would cause you to believe in Islam, if the scenario I provided isn't sufficient?
-6RichardKennaway9y
0thomblake9yI don't think this is obvious. Since (E) is nonzero and unspecified, it might be significant once (A) and (B) are excluded. Probably after excluding (A) and (B), it would be worth spending some time fleshing out (E) a bit better. In my case, the priors on (C) and (D) are so low I'd just as well include them in (E) from the outset.
0[anonymous]9yIf pi is a normal number, which it likely is, then the Qu'ran is embedded in it, and so is any other text.
5Alicorn9yA blue tentacle scenario, however unlikely, is not evidence against the hypothesis that I am a changeling from the octopus people.
-1Multiheaded9yHehe, it shouldn't be long until someone demands rule 34 of this.

I don't quite get what happens. Does imagining two and two together give same mental image as imagining two and one together? Does putting two and two earplugs together give same result as putting two and one earplug together? If it does, then I take 4 earplugs, put two and one together and put other into my ear, then two and one are same as two and two together, so I should be able to separate it into two and two, and and then I have two earplugs on my hands, two in a box, and one in my ear. I do it the second time and I can't hear anything, but I have al... (read more)

I just operate under the assumption that I will never actually encounter a situation where 2+2 does not equal 4, and therefore do not spend time worrying about such a hypothetical situation. This assumption has never failed me before.

I understand the point Eliezer's trying to make here. However, you (whoever's reading this) could not convince me that ss0 + ss0 =sss0 in Peano arithmetic (I define the scenario in which my mind is directly manipulated so that I happen to believe this not to constitute "convincing me"). Here's why I believe this position to be rational:

A)In order for me to make this argument, I have to presume communication of it. It's not that I believe the probability of that communication to be 1. Certainly many people might read this comment and not know ... (read more)

0somejan8yExtrapolating from Eliezers line of reasoning you would probably find that although you remember ss0 + ss0 = ssss0, if you try to derive ss0 + ss0 from the peano axioms, you also discover it ends up as sss0, and starting with ss0 + ss0 = ssss0 quickly leads you to a contradiction.

Wouldn't such an occurrence involve an overhaul of the world on part of some Force/Entity? And why would you, and only you, be able to note that something changed, i.e. that you believed 2+2=4 and now you no longer don't? Much more importantly, since you use it as an example, Winston would not bother to write about 2+2=3, he would probably actually write about 2+2=4, or even 5, thus shaking your world even further...

Hello, I'm a Christian. And, yes, I'm also a rationalist gasp!. I was born and raised a Christian, and I honestly am not sure if I would believe, say, Budhism if I was raised that way- My gut answer is 'No', but I cannot really be sure, as I would be a completely different person. There's no way no one can truthfuly say yes or no for sure to that question.

Right, anyways, I do have reasons I would stop believing... There are a couple very specific situations that pop to mind in which I would be convinced that my whole life has been a lie:

  1. The apocalypse h
... (read more)
2notsonewuser8yEdit: Also, welcome to Less Wrong. Sorry, politeness should have come first. Really...that's all you mean when you say I'm a Christian? Do you just mean "The book of Revelation is true, we'll never completely defeat death, and time travel is impossible?" In that case, probably even a lot of people here agree with 2/3 of that. I assume the answer to the first question is "no". In that case, please explain more of what you mean by calling yourself a Christian [http://lesswrong.com/lw/nu/taboo_your_words/]. That will open up more ground for a serious discussion, if you actually do want to change your mind if it is the case that Christianity is false.
0Lauryn8yWell, I should have specified- those are the first examples that jumped to mind. I mean a whole lot more by saying that I am a Christian, I suppose I would define it as I believe that all the Bible says is true- that God created the universe, Jesus is our Saviour, and we exist to glorify God (I know, cached thought, but that one I have thought about.). I really do mean that I do not want to believe in anything that is wrong, but I have yet to see anything that is definitive evidence that my beliefs are incorrect. (And I also admit that I do not want to give up my current beliefs, and I'm going to be heavily biased against any information shown me, but I will try.)
2CCC8yLiterally, to the last decimal point, or do you make some allowances for figurative language, imprecise measurement and/or unmarked parables?
2Lauryn8yThe problem with going there is that it's easy to go to far, to a point where the Bible isn't true anymore and it's just your interpretation of bits and pieces of the Bible. Anyways, I don't really think of figurative language as something you need to make allowances for, it just is how it was written- and most of the time is fairly obvious too. I've never seen one instance of imprecise measurement, but if you know of one, fire away, and unmarked parables are also fairly easy to spot.
1Kingoftheinternet8y1 Kings 7:23
1CCC8yYes, true. But it's possible to go too far the other way, too, which causes a lot of problems (see: the creationist movement in America). 2 Corinthians 4:2: [http://www.biblegateway.com/passage/?search=2%20Chronicles%204:2&version=GNT] I consider it more probable that the measurement is imprecise than that pi is three for that tank. I mean, it's a minor detail, but it's there. That's what I thought, too, but apparently some people take the Garden of Eden literally.
3notsonewuser8yMy first question would be, why do you believe in Christianity, specifically, instead of Hinduism, Islam, or something else (like that there are 4 gods, and they created the Earth with 4 seasons)? Why would you say that the Christian God created the universe, rather than saying that you just don't know?
0Lauryn8yFirst, from a scientific standpoint, there's a good bit of evidence for creation as is told in the Bible- a flood and all. And it really isn't anything I can convince you of from there on- reasons such as that it makes sense that we cannot make ourselves good enough, the Bible makes far more sense than the Quaran (which I have read a good bit of), experiences, so on, so forth. And just pure faith, which of course makes no sense to a good atheist like you. (No offense. No offense.)
1notsonewuser8yI have never heard of such evidence. Could you direct me to where to find it? I think the evidence points in the opposite direction. See the Wikipedia article on flood geology [http://en.wikipedia.org/wiki/Flood_geology], to begin with. If you believe the Earth is under 10000 years old, then you should also read this article, "Is There Really Scientific Evidence for a Young Earth?" [http://www.lpl.arizona.edu/~matthewt/yeclaimsbeta.html]. It was written by a Christian, and I'd like to point out the following quote from the introduction: Explain what you mean when you use the phrase, "pure faith". [http://lesswrong.com/lw/nu/taboo_your_words/]

I see a lot of people arguing that "2", "3", "+", and "=" are defined in terms of the Peano axioms, and as such, aren't actually relevant to the behavior of physical objects. They say that the axioms pin down the numbers, regardless of how physical objects behave or start behaving.

But the Peano axioms use something called a "successor" to generate the natural numbers. And how do we figure out what the successors are? Well, one notation is to append an "S" to the previous number to indicate that nu... (read more)

0nshepperd6yUm, but the (+) operator in peano arithmetic is actually defined in terms of Sx + y = x + Sy. It would be somewhat circular to "suggest counting the S's up" in a method of defining numbers, after all. So the way you calculate 2 + 2 is more like SS0 + SS0 (thing on the left starts with an S) S(S0) + SS0 (move the S to the right) S0 + SSS0 (thing on the left starts with an S) S(0) + SSS0 (move the S to the right) 0 + SSSS0 (eliminate "0 + " with axiom) SSSS0 I imagine you'd need a rather more devious brain modification to prevent one from carrying out these steps correctly, in such a way that the result is SSS0.
0dxu6yAll right. Let me take a stab at it. Okay. Following you so far... Eh? Where'd you get the extra "S" from? (This hack would have the unfortunate side effect of making every addition with at least one term less than or equal to 3 and a result greater than 3 come out to 1 less than it's supposed to, however. If you wanted to only make 2 + 2 = 3, and preserve all other additions as-is, I can't think of any brain hack that could do that. That's not to say no such hack is possible; I'm sure one is, but I just can't think of one.)
0nshepperd6yI think it would even result in any addition with a term ≤ 3 and result > 3 come out to exactly 3, unless you have some sort of rule for S + SSS0 sometimes becoming SSSS0 instead of SSS0. Note also that an enterprising soul can line up the two steps: S0 + SSS0 ⁠ 0 + SSS0 And notice that they are confused, because the SSS0's are identical, even though they shouldn't be, because Sx + y = x + Sy was the rule applied and Sy ≠ y. A brain hack that made all of this work is surely possible, of course, but it seems like it would have to be a bit more systematic.
0dxu6yThat seems fair. Would you agree that my original point (that your grasp of logic stems from a physical brain and can be muddled) stands, though?
0IlyaShpitser6yNot sure who "they" are. May I suggest reading up on model theory? Maybe Enderton's "mathematical logic" (?). -------------------------------------------------------------------------------- It's important to think separately about the "model" (the thing we are studying), the "language" (where we make statements about the model), and the "theory" (a set of statements in a language). The same theory in a particular language may and in fact generally does apply to multiple distinct models. The same model (object) may be described by theories in different languages of different strengths. So in one sense Peano axioms "pin down" the natural numbers, but in another sense they don't because we can invent crazy objects that contain a lot more than just the natural numbers to which Peano axioms also apply (this is the content of the Lowenheim-Skolem theorem). http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem [http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem] We can use a more powerful language than first order logic to describe the natural numbers, and that would rule out some of the crazy models. That will capture more properties of the natural number line, but not everything. -------------------------------------------------------------------------------- Physicists play a similar game to logicians, except perhaps a bit less formally. But their models (in the sense of 'object of study') "bite back." -------------------------------------------------------------------------------- It's confusing that to a model theorist "the model" refers to the territory, while to a statistician "the model" refers to the map.
0dxu6yMy bad; I wasn't clear. "They" refers, not to any person or group of people in academia, but some of the commenters on this LW post. As an example: this comment [http://lesswrong.com/lw/jr/how_to_convince_me_that_2_2_3/faq].

I’m an evangelical protestant and I’d like to give my answer to the ‘what would it take to convince me to become a Muslim’ question. This is going to be a narrative example and thus show only one of many possible routes. I’ve chosen a rout that does not depend on private knowledge, fresh miracle in the present day, or even or even changed facts in things it would be inconceivable for me to be wrong about, because I see this rout as the hardest and therefore most revealing.

Muslim scholars propose a competitor to the Documentary Hypothesis (JEPD) for the Pen... (read more)

0thrawnca5y"if, for example, there was an Islamic theologian who offered to debate the issues with me then I would be inclined to do it and follow where the belief updates lead." Is that an open offer to theologians of all stripes?
0Alia1d5yYes, theologians of all strips, and philosophers and logicians of all perspectives. As long a they are willing to respond to my questions as well as having me respond to their's. (Though if someone is rude, engages in rhetorical hyperbola, etc. I reserve the right to do those things back to them.) I'll try to check back here to see if anyone wants to do that or e-mail me at alia1dx@gmail.com and I'll give you my private e-mail to carry on a dialog.
0thrawnca5yEmail sent about a week ago. Did it get spam-filtered?

In discussing Newcomb's problem, Eliezer at one point stated, "Be careful of this sort of argument, any time you find yourself defining the "winner" as someone other than the agent who is currently smiling from on top of a giant heap of utility."

This aligns well with a New Testament statement from Jesus, "Ye shall know them by their fruits...every good tree bringeth forth good fruit, but a corrupt tree bringeth forth evil fruit."

So, I'm only a novice of the Bayesian Conspiracy, but I can calculate the breast cancer percentages... (read more)

1Personable1ySo to be converted to a religion (Islam only being an example here) it would have to provide a better moral and positive emotional framework than Christianity? Side-note: This is separate to the question above, but on the topic of the post you provided on positive thinking, I think it may have something to do with religion being less common in a trend among those who are both wealthy and in a higher quality of education (i.e. tertiary education has a lower instance of religion than secondary, public has a lower instance of religion than private, CEOs are less religious than white collar workers, etc) along with a number of other factors that I can't recall, and that these factors do increase tendency towards negative thinking. There is a truth in the stated Christian doctrine (as is shared in Buddhism, Taoism, Hinduism and others) that material goods do not bring happiness or 'salvation'. I personally do not believe, however, that this makes those who are atheists less valid in their beliefs (I would hope so, being an atheist myself).

Hi, I am a mathematician and I guess most mathematicians would not agree with this. I am quite new here and I am looking forward to reactions of rationalists :-)

I, personally, distinguish "real world" and "mathematical world". In real world, I could be persuaded that 2+2=3 by experience. There is no way to persuade me that 2+2=3 in mathematical world unless somebody shows me a proof of it. But I already have a proof of 2+2=4, so it would lead into great reform of mathematics, similar to the reform after Russel paradox. Just empirical ex... (read more)

0Raemon4yThe point was less about the physical world applications of 2+2=4, and more about the fact that any belief you have is ultimately based on the evidence you've encountered. In the case of purely theoretical proofs, it's still based on your subjective experience of having read and understood the proofs. Humans are sometimes literally insane (for example, not being able to tell that they're missing an arm). Also, even the best of us sometimes misunderstand or misremember things. So you need to leave probability mass for having misunderstood the proof in the first place. (The followup to this post is this one: http://lesswrong.com/lw/mo/infinite_certainty/ [http://lesswrong.com/lw/mo/infinite_certainty/] which addresses this in some more detail)
1mirefek4yI see. It seemed to me that it was about the experimental method which did not fit to a mathematical statement. I understand the possibility of being mistaken. I was mistaken many times, I am not sure with some proofs and I know some persuasive fake proofs... Despite this, I am not very convinced that I should do such things with my probability estimates. After all, it is just an estimate. Moreover it is a bit self-referencing when the estimate uses a more complicated formula then the statement itself. If I say that I am 1-sure, that 1 is not 1/2, it is safe, isn't it? :-D Well, it does not matter :-) I think that I got the point, "I know that I know nothing" is a well known quote.
0Raemon4yIt's actually a somewhat different point he's trying to make (it's spaced out over several blogposts) - the idea is not to say "all knowledge is fallible." You should be very confident in math proofs that have been well vetted. It's useful to have a sense of how certain your knowledge is. (like, could you make 100 similar statements without being wrong once? 1,000? 10,000?) (i.e. "the sun will rise tomorrow" is a probability, not a certainty, and "Ghosts could be real" is a probability, not a certainty, but they are very different probabilities.) If you're interested, I do recommend the sequences in more detail - a lot of their points build on each other. (For example, there are multiple other posts that argue about what it's useful to think in probabilities, and how to apply that to other things).
0hairyfigment4yAhem. [http://lesswrong.com/lw/jr/how_to_convince_me_that_2_2_3/4u4e] I can think of many ways that some broadly defined "experimental method" could come into play there.
0mirefek4yPlease, be more specific. I am not sure exactly what are you responding to. Do you mean that a math proof (or knowledge of it) can be considered as experimental method in some sense?
0hairyfigment4yI don't think you've responded to my linked comment. But OK, looking up a result in a math book could count as an experiment, as could any method by which you might learn about dyslexia or whatever you suspect might be confusing you. If you don't believe anything like that could happen to you, either you made that judgement based on experience and science or you are very badly misguided.
0mirefek4yTo be honest, your comments confuse me. I knew about the link but I didn't see a connection between the link and experimental method and where the citations in the link came from. I am not sure what you mean by "anything like that" in your last comment and I am not very interested in it. But I prefer to keep the original problem: If looking up a result in a math book could count as an experiment what is the (broader) definition of an experiment, then?
0Raemon4y(I think this may have came across a bit more confrontational than was optimal) ((Also, on that note, mirefek, if I came across as more confrontational than seemed appropriate, apologies.))