# How to Convince Me That 2 + 2 = 3

53 27 September 2007 11:00PM

In "What is Evidence?", I wrote:

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

Cihan Baran replied:

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

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

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

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

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

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

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

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

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

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

If there are any Christians in the audience who know Bayes's Theorem (no numerophobes, please) might I inquire of you what situation would convince you of the truth of Islam?  Presumably it would be the same sort of situation causally responsible for producing your current belief in Christianity:  We would push you screaming out of the uterus of a Muslim woman, and have you raised by Muslim parents who continually told you that it is good to believe unconditionally in Islam.  Or is there more to it than that?  If so, what situation would convince you of Islam, or at least, non-Christianity?

Part of the Overly Convenient Excuses subsequence of How To Actually Change Your Mind

Next post: "Infinite Certainty"

Previous post: "Absolute Authority"

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Comment author: 28 September 2007 12:07:27AM 1 point [-]

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.

Comment author: 28 September 2007 12:15:33AM 2 points [-]

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.

Comment author: 28 September 2007 12:16:44AM 8 points [-]

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?

Comment author: 08 January 2011 02:34:12AM 39 points [-]

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.

Comment author: 10 January 2011 01:49:06AM 10 points [-]

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.

Comment author: 29 April 2011 07:49:50AM 9 points [-]

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.

Comment author: 30 April 2011 12:54:21AM 9 points [-]

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

Comment author: 28 September 2007 12:21:09AM 3 points [-]

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.

Comment author: 03 January 2011 04:55:36PM 0 points [-]

There are some points of view that sometimes do require mathematical statements to be dependent on reality (i.e. constructivism, actual versus potential infinity debate, etc).

Sometimes it is intuitive to require mathematics to behave this way, i.e. 'natural' numbers are called that for a reason, and they better behave like the apples or I'm postulating a change in nomenclature.

P.S. Ii seems to me the OP's wording wasn't precise enough. I can very well imagine a situation in which some basic addition would yield non obvious results (like addition inside modulo N number space).

Comment author: 03 January 2011 08:44:15PM 3 points [-]

When I reason inside a fully axiomatized formal system, the axioms don't depend on reality, but the rules for manipulating symbols depend on ... something. You could define it as "if I perform these manipulations in reality, I will get this result" but what if performing the manipulations in different places gets different results?

What if, when you applied the rule "(x+Sy) => S(x+y)" twice and the rule "(x+0)=>x" once, to "(SS0+SS0)", you got "SSS0" instead of "SSSS0"?

Comment author: 05 January 2011 11:14:32PM 0 points [-]

I guess when one reasons inside a fully axiomatized formal system, this something the rules for symbol manipulation depend on is the set of axioms.

Now I'm putting on my uneducated hat, so excuse me if this is heresy: Starting with the axioms you apply logic to formulate more specific rules (in this case the abstract is empirically falsifiable, since we're working on natural numbers).

So, to arrive at SS0+SS0=SSS0, you'd have to venture outside the realm of reason I'm afraid.Tthat would maybe manifest itself as magic - getting 4 apples on the table during night, but 3 during day when you put 2 and 2 apples side by side. And could mean ability to produce something from nothing by clever arrangement of apples. and waste disposal would become easy :)

In other words my opinion is it's not possible even as thought experiment unless you introduce some random factor from beyond the scope of axioms.

Comment author: 06 January 2011 02:10:29AM *  3 points [-]

well there's the special other thing, the reason you can't explain Peano Arithmetic to a rock, which is that axioms are static sequences of signals, but in addition you have these dynamics.

Best source on this is Lewis Carroll http://www.ditext.com/carroll/tortoise.html

These dynamics are contained within the structure of our thoughts, which is why they're preserved in a thought experiment. But we still have to actually check our thoughts, which are part of reality.

Sorry if this wasn't very coherent.

Comment author: 29 August 2011 04:21:11PM 1 point [-]

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).

Hm... not precise enough for what? I think we all know what was meant... unless Eliezer did a ninja edit after you posted ;) this seems to cover it:

There are redefinitions, but those are not "situations", and then you're no longer talking about 2, 4, =, or +.

What you suggested, that's not a "basic addition" any more, is it?

Comment author: 03 March 2011 07:53:05AM 11 points [-]

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.

Comment author: 28 September 2007 12:38:57AM 29 points [-]

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's an example of how one small piece of my model could be flipped from the negation of Christianity (in that facet) to the non-negation.

Getting all the pieces of the factual model (including the parts I was previously convinced were logically self-contradictory) to align with Christianity's factual model, would still leave all the ethical problems. So the actual end result would be to convince me that the universe was in the hands of a monstrously insane and vicious God. But then there does not need to be any observable situation which convinces me that it is morally acceptable to murder the first-born children of Egyptians - morality does not come from environmental entanglement.

Comment author: 15 July 2009 01:10:15PM 14 points [-]

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.

Comment author: 03 March 2011 06:53:16PM 19 points [-]

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 mathematics are, in reality, nothing more than extremely advanced counting. If it is not related to the physical world, then there is no reason for it to exist. It follows rules first derived from the physical world, even if the current principles of mathematics have been extrapolated far beyond the bounds of the strictly physical. I think people lose sight of this far too easily (or worse, never recognize it in the first place).

Mathematics are so firmly grounded in the physical reality that when observations don't line up with what our math tells us, we must change our understanding of reality, not of math. This is because math is inextricably tied to reality, not because it is separate from it.

Comment author: 04 March 2011 01:09:43AM *  7 points [-]

Numbers were invented to count things, that is their purpose. The first numbers were simple scratches used as tally marks circa 35,000 BC.

Verbal expressions almost certainly predate physical notations. Unfortunately the echos don't last quite that long.

Comment author: 04 March 2011 02:07:40AM 2 points [-]

In your last paragraph you turn everything around and inexplicably claim that math is more primary than observation of reality, though you did a good job -- and one I agree with -- of pointing out the opposite in the previous part of the comment.

Comment author: 04 March 2011 04:12:24AM 9 points [-]

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?

Comment author: 04 March 2011 04:23:02AM 4 points [-]

Yup. I think we agree. My disagreeing post was a mere misunderstanding of what you were saying.

Comment author: 04 March 2011 04:53:16PM 4 points [-]

After a few recent posts of mine it looks like I need to work on my phrasing in order to make my points clear.

No harm no foul.

Comment author: 21 March 2011 03:58:34AM 12 points [-]

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 agree that we can probably be more confident of results in maths than results in experimental science. (I was going to distinguish between mathematical/empirical results, but given that the OP was to do with the empirical confirmation of maths, I thought "mathematical/experimental" would be a safer distinction)

Comment author: 28 September 2007 12:48:19AM 35 points [-]

"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.

Comment author: 28 September 2007 01:01:54AM 20 points [-]

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....

Comment author: 28 September 2007 01:10:44AM 6 points [-]

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.

Comment author: 29 May 2011 03:50:36PM 3 points [-]

What's an example of another possible abstraction?

Comment author: 28 September 2007 01:17:42AM 10 points [-]

"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.

Comment author: 28 September 2007 01:32:22AM 2 points [-]

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 subjects of arithmetic, but crude illustrations of ideas (ideals) that are universal in the most absolute sense of the word - they are part of the universe.

Religions seek knowledge by entirely different methods, methods that are not subject to any kind of proofs ore verifications. (I think it weird that religious people consider it a virtue to cling to ideas for which no data of any kind can be summoned for support.)

Comment author: 28 September 2007 01:45:57AM 8 points [-]

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).

Comment author: 28 September 2007 01:54:56AM 2 points [-]

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.

Comment author: 28 September 2007 02:03:57AM 5 points [-]

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.

Comment author: 28 September 2007 02:19:30AM 1 point [-]

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?

Comment author: 28 September 2007 02:46:56AM 2 points [-]

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.

Comment author: 28 September 2007 02:55:27AM 4 points [-]

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 probability estimate, so I would have to not really believe that 2 + 2 = 4 in order to expect to encounter evidence that 2 + 2 = 3.

Comment author: 28 September 2007 03:41:55AM 3 points [-]

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.

Comment author: 28 September 2007 03:43:32AM 2 points [-]

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.

Comment author: 28 September 2007 04:01:32AM 3 points [-]

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. ;)

Comment author: 28 September 2007 06:38:53AM 9 points [-]

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 possible a priori claims that includes "2 + 2 = 498034". 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. Are you sure "a priori" isn't just functioning as a semantic stopsign?

I'll accept as divine any entity that can consistently reduce the entropy of a closed, isolated system

This could just be a manifestation of an entity running our world as a computer simulation. Or even simpler, it could be an alien that knows an important fact you don't know about the real laws of physics. Even if the entity is running our world as a computer simulation, it could itself be made of atoms, go to the bathroom, have a boss screaming at it, etc.

As Damien Broderick observed: "If you build a snazzy alife sim ... you'd be a kind of bridging `first cause', and might even have the power to intervene in their lives - even obliterate their entire experienced cosmos - but that wouldn't make you a god in any interesting sense. Gods are ontologically distinct from creatures, or they're not worth the paper they're written on."

Comment author: 15 July 2009 01:41:11PM *  4 points [-]

I'm suspicious of claims that supposedly do not require justification

Mathematical claims do require justification. They even require stronger justification than empirical claims: mathematical proof. As Doug S explained, the proof that 2+2=4 is

2+2 = 2+(1+1) = (2+1)+1 = 3+1 = 4 QED.

(Using the definition of 2, the associativity of +, the definition of 3 and the definition of 4 in that order).

Empirical claims, such as "2+2=4 is related to earplugs or apples" do not require proofs, but they do require evidence.

Comment author: 27 July 2011 09:13:05PM 0 points [-]

"Gods are ontologically distinct from creatures, or they're not worth the paper they're written on."

What an interesting argument... but I know of at least one religion that would tend to disagree with this anti-definition of God.

Comment author: 28 September 2007 07:11:19AM 2 points [-]

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 way, and Eliezer is most likely not one of them. One could probably convince someone to believe a false mathematical formula if it were sufficiently complicated for the individual to have trouble understanding it, and it had a sufficiently crafty explanation.

Basically, believing 2+2=3 to be true would require the evidence necessary to believe in married bachelors: evidence that confuses the hell out of you effectively.

Some people are arguing that mathematics is not a priori. If so, then the situation with putting two pairs of apples together and getting 3 apples would be the appropriate type of evidence. If mathematics is a posteriori, the answer is thus quite simple.

Sorry if this is overly redundant with previous posts.

Comment author: 19 February 2011 11:21:55AM 0 points [-]

There is an example that often floats around, where it is 'proven' mathematically that 1=2 (or some other such equality, by the principle of explosion it doesn't really matter). The trick is that at some step in the proof, a non obvious division by zero occurred.

Comment author: 03 March 2011 07:14:13PM 0 points [-]

I imagine it's the same proof that makes 2+2=5. There is a point in the proof where the correct result is obviously 0=0 (though never explicitly written), yet it continues as though it didn't happen.

It's an example of making the problems so complex that you make a mistake, it's not a valid proof.

The proof for 2+2=4, incidentally, is almost 400 pages long. The simplistic versions most use take for granted many things for granted (like + and = and 2) that the actual proof does not.

Comment author: 28 September 2007 07:42:06AM 2 points [-]

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.

Comment author: 28 September 2007 10:49:54AM 1 point [-]

(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.

Comment author: 28 September 2007 12:08:42PM 0 points [-]

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 respond to evidence or to reasoning".

Doug: we can all reduce entropy in physical systems. This does, however, require that we act upon those systems, but the god you are discussing would also be acting upon the systems where it reversed entropy, right?

Ben: I’m pretty sure that “Christianity is true” is not a hypothesis. Believing something more specific, like Jesus was born to a Virgin and returned 3 days after being buried doesn't present any serious problems.

Comment author: 28 September 2007 01:31:36PM 0 points [-]

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 "random" *means*. I'm not sure that I can actually imagine a world like that, though.

I think Doug should amend his criterion to say "... with no sign of any increase in entropy elsewhere". But it seems to me that a being with no power other than (say) being able to induce modestly sized temperature gradients would not thereby qualify to be called a god. (If physicists announce tomorrow that the second law of thermodynamics can be cheated by some cunning technique with the word "quantum" in it, are we suddenly all gods?) And if the power is sufficiently limited (it takes time, and only operates on a small region of space, and the temperature gradient induced is very small) then it doesn't even qualify as god-like power in my book. But I expect Doug wasn't being perfectly serious.

What do you mean by "doesn't present any serious problems"? That you have no trouble thinking of evidence that would suffice to convince you of those things? (If so, I agree.) (It's not fair to blame Ben for picking "Christianity is true" instead of something more specific; he was just copying Eliezer.)

Comment author: 28 September 2007 03:33:15PM 0 points [-]

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 your life "eudaimonia", as Aristotle put it. The Communist religion, for example, failed miserably, and the current secular liberal religion seems to be failing at the "thrive" part. Western flavors of the Christian religion seem to have done pretty well over the last millennium or so, so the move away from it over the last century seems strange. Islam is good at thriving, but seems poor at improving quality of life.

Incidentally, the most fundamental test of Christianity is meant to be belief in the Nicene creed, which is perhaps the best test of whether you believe if "Christianity is true" or not.

Comment author: 28 September 2007 04:16:45PM 8 points [-]

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 your current state of uncertainty about both). If, for example, I do not know what will happen with two earplugs and two earplugs on the nightstand, I can visualize two apples plus two apples to find out. All of this takes place in the same, unified, physical universe, with no ontological border between the atoms in my skull and the atoms outside my skull. That's why the trick works. It would work just as well if I used a pocket calculator. Is the output of a pocket calculator an a priori truth? Why not call the earplugs themselves a priori truths, then? But if neither of these are a priori, why should I treat the outputs of my neurons as "a priori"? It's all the same universe.

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

Vassar: It sure seems to me that our evolution and culture constructed ethical attitudes are entangled with the world.

They're causal products of the world, and yes, if I was ignorant about some evolution-related factual question, I might be able to use my ethical attitudes as evidence about conditions obtaining in my ancestral environment. That's not the same as my stating an external truth-condition for it being wrong to slaughter the first-born male children of the subjects of an unelected Pharaoh. 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."

Spear: The test of any religion is whether cultures believing it tend to thrive and improve the quality of their lives or not.

Ah, yes, the old theory that there are reasons to believe2 in an assertion-of-fact besides its being true.

Lee: 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.

Just to be clear, when I say "be convinced that 2 + 2 = 3", I mean being convinced that the system of Peano axioms with standard deductive logic and:

\a.(a + 0 = a) \ab.(a + Sb = S(a + b))

does not have as a theorem

SS0 + SS0 = SSSS0

but does have as a theorem

SS0 + SS0 = SSS0

and is consistent. Just as I currently believe that PA is consistent and has a theorem SS0+SS0=SSSS0 but not SS0+SS0=SSS0. So yes, this blog post is about what it would take to convince me that 2 + 2 actually equalled 3. I am not supposed to be convinced of this, if I am sane, and if it is not true. But at the same time, my belief in it should not be unconditional or nonevidential, because there are particular evidences which convinced me that 2 + 2 = 4 in the first place.

I also note that if you do not believe that there is a finite positive integer which encodes a proof of Godel's Statement, then you clearly are not using Peano Arithmetic to define what you mean by the word "integer".

Comment author: 29 March 2011 07:08:47PM 0 points [-]

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.

What has this to do with Peano Arithmetic and a mathematical proof "PA proofs 2+2=4" which is merely a string of symbols? On the other hand, what has PA to do with reality of earplugs except the evidence that PA is a good model for them?

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

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

Comment author: 22 June 2011 03:33:46PM *  2 points [-]

I think you've pointed out an issue of semantics, not falsified 2 + 2 = 4. If you pour one glass of water into another glass of water, you have one glass of water—but " one glass", in that case, is qualitative and not quantitative; it's not math.

Comment author: 28 September 2007 05:45:42PM 0 points [-]

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. :-)

Comment author: 28 September 2007 07:38:37PM 0 points [-]

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 with an environment where people go around saying such things, with a different possible language.

Eliezer insists that he could be entangled with evidence such that he believes "there are some married bachelors" is true in English as we know it. I don't think he could; that proposition is unthinkable in good English.

Comment author: 28 September 2007 08:17:13PM 0 points [-]

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.

Comment author: 28 September 2007 08:56:56PM 0 points [-]

"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.

Comment author: 28 September 2007 09:41:47PM 1 point [-]

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 David Deutsch warns, we shouldn't mistake the study of absolute truths for the possession of absolute truths. We can ask ourselves how we computed whether 2+2=4, conscious that our means of computing it may be flawed. We could in principle try many means of computing whether 2+2=4 that seem to obey the Peano axioms: fingers, abacus, other physical counters, etc. Then we could call into question our means of aggregating the computations into a single very confident answer and then our means of retaining the answer in memory.

Seems a pointless exercise to me, though. Evolution either has endowed us with mental tools that correspond to some basic neccessary truths or it hasn't. If it hadn't, we would have no good means of exploring the question.

Comment author: 28 September 2007 10:05:28PM 0 points [-]

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 logically possible worlds with different physics, but I can't imagine how I could model (as opposed to loosely imagine) a universe where 2+2=3. Eliezer, would you hold that such a world is actually constructable and modelable in addition to imaginable? Do you think "necessary" and "contingent" or "logically possible"/"impossible" are semantic stopsigns too?

Comment author: 29 September 2007 07:19:34AM 0 points [-]

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 marketing, though of course I found the latter more interesting.

Eliezer, the things you are saying here about math are just the type of things I was attempting to here, but you're much better at it.

Comment author: 29 September 2007 09:25:35AM 7 points [-]

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 theorem, etc). We call (conditional) claims which are arrived at via induction 'a posteriori.'

Note: both the rules of inference used in deduction and rules of evidence aggregation used in induction are agreed upon as an empirical matter because it has been observed that we get useful results using these particular rules and not others.

Furthermore: both deduction and induction happen only (as far as we know) in the physical world.

Furthermore: deductive claims by themselves are 'sterile,' and making them useful immediately entails coating them with a posteriori claims.

Nevertheless, there is a clear algorithmic distinction between deduction and induction, a distinction which is mirrored in the claims obtained from these two processes.

Comment author: 01 October 2007 03:38:07PM 0 points [-]

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......

Comment author: 03 December 2007 04:44:00AM 0 points [-]

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.

Comment author: 03 December 2007 05:51:06AM 1 point [-]

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

Comment author: 03 December 2007 07:04:12AM 0 points [-]

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.

Comment author: 03 March 2011 07:54:41PM 0 points [-]

Yet if you counted the grains of sand, you would have as much sand as is contained in four piles of sand - 2+2=4.

This is the same as saying when I add 2+2 and get 4, I start with two numbers and only get one number. It's true, but you've fooled yourself into believing this is some profound mathematical truth (and in a sense it is, but not the way you originally thought), when in fact it was so obvious to anybody who wasn't trying to fool themselves that it did not need pointing out.

This is also the same as starting with two groups of two apples, adding them together, and getting one group of four apples. I'm not disappointed by this result. In fact, the very reason I have four apples is because I have merged two groups of two apples into a single group. The result of this merger is four apples.

Comment author: 27 July 2011 09:15:53PM 0 points [-]

Yet if you counted the grains of sand, you would have as much sand as is contained in four piles of sand - 2+2=4.

Could you please define a "pile"? :3

Comment author: 04 December 2007 05:43:15AM 0 points [-]

"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.

Comment author: 22 July 2008 08:01:49PM 1 point [-]

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 is such systems are only 'real' in our minds.

Thus we can construct system where one object can be exchanged 1:1 directly, say money in a computer system,, but there are few if any such systems in the real world. In our constructions we can agree that one ant is equal to an other, but in the real world they may be very different.

So the idea of the piles of sand are very much to the point. There is never such a thing as one man or one cow - and 2 + 2 cows is 4 cows only if we agree to what one cow can be.

Regards Per

Comment author: 23 July 2008 10:29:23PM 0 points [-]

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 most standards. Except for having created the universe (could it possibly be used with some as-yet-unknown theory of quantum gravity to create some multiverses, or a loope in space-time back to the Big Bang?), such a being is about as close to a God as is logically possible.

Comment author: 27 September 2008 12:30:04AM 0 points [-]

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.

Comment author: 28 July 2009 07:57:08AM -2 points [-]

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's axioms (even if their representation is physical, inside the brain), and this experiment doesn't prove that.

If the 2+2=3 gets usual in every empirical test I do, as suggested in this article (no matter how absurd it can seem to be), I wouldn't stop believing in the integer numbers: I would have a NEW number system (axioms/definitions) with this characteristic (2+2=3). That's a new model, and what was empirically falsified before, was the link between the old model and the physical reality I could notice, but not the old model itself.

I've got curious about paraconsistent logics in this case...

Comment author: 31 July 2009 01:21:12AM 8 points [-]

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 groups of two, their minds mistakenly "see" the pattern 0 0 0 as composed of two distinct 0 0 groups. Although such a mind would be unlikely to arise within natural selection, there's nothing impossible about engineering a mind with this error, or rewiring a mind within a simulation to have this error.

These minds, of course, would notice empirical contradictions everywhere: they would put two objects together with two more, count them, and then count four instead of three, when it's obvious by visualizing in their heads that two and two ought to make three instead. They would even encounter proofs that 2 + 2 =4, and be unable to find an error, although it's patently absurd to write SSSS0 = SS0 + SS0. Eventually, a sufficiently reflective and rational mind of this type might entertain the possibility that maybe two and two do actually make four, and that its system of visualization and mental arithmetic are in fact wrong, as obvious as they seem from the inside. We would consider such a mind to be more rational than one that decided that, no matter what it encountered, it could never be convinced that 2 and 2 made 4 rather than 3.

Now, given all that, why exactly should I refuse to ever update my arithmetical beliefs if given the sort of experiences in Eliezer's thought experiment? Wouldn't the hypothesis that I am such an agent get a lot of confirmation? (Of course, I very strongly don't expect to encounter such experiences, because of all the continuing evidence before me that 2 + 2 = 4; but if I did wake up in that situation, I'd have to accept that some part of my mind is probably broken, and the part that tells me 2 + 2 = 4 is as likely a candidate as any.)

Comment author: 31 July 2009 03:13:18AM *  1 point [-]

I had parallel thoughts at one time, and discovered with some effort that I could train myself to believe that 1+1=3. It took about five minutes of mental practice. What eventually happened was that every time I combined two objects together mentally (abstractly), I simultaneously imagined a third which had the bizarre property that it only existed when the two objects were considered simultaneously. If I thought of just one object, the third disappeared, if I thought of the other object, it again disappeared -- it only appeared as an emergent property of the pair. Thus imagining 1+1=3 was discovering the following "operation":

{E} + {F} = { {E} , {F} , { {E},{F} } }

Looking at the cardinality of the sets, we have: 1 + 1 = 3

Could such an operation be 'logical' and yield a consistent number theory? (I don't know. I think it's a question in abstract algebra. (Rings, fields, groups, etc.) Are there any algebraists here that can comment?)

Yet orthonormal is suggesting the case that 2+2=3 doesn't result in a logical, consistent theory -- the possible minds just believe it due to an internal error, and they can use the inconsistency of their theory to deduce the internal error. However, I find it really difficult to think of 2+2=3 happening as a mistaken Peano arithmetic instead of the assertion of another type of arithmetic. The possible logical self-consistency of this arithmetic further confounds: if it's self-consistent, they may never deduce that they got Peano arithmetic wrong. If its not self-consistent, they can prove all propositions and how will they know where the error lies? Or even understand what error means? If there is an error in our reasoning, it cannot be so fundamentally embedded in our understanding of logic.

Comment author: 02 August 2009 07:48:30AM *  -1 points [-]

Nice, but the difference with this "belief" is that you're talking about sensory "counting" (visual grouping), and I was talking about the numbers themselves, as models for games, other phenomena, etc., and not just as a "counting" tool.

In the 1+1=3 example, to define the cardinality, he/she used the Peano's axioms, didn't he/she?

I don't see the "visual sensory counting" as the only use for "2+2=4", that's why I don't think this experiment would refute such a priori content.

Another idea: let Ann be a girl with hemispatial neglect in a extinction condition. Ann has problems detecting anything on the left, and she can possibly see 2+2=3 as idealized above, due her brain damage. Will she think that 2+2=3? I don't think so...but if she does...will that be a model for all "integer numbers" aplications? I think in "integer" as a framework for several phenomena, other models, other knowledge, not only the counting one.

For the minds that see 2+2=4 as something patently absurd, because 2+2=3 is part of their intuitive arithmetic, these minds probably won't see the 2+2=4 even when brought to a world like ours. After a time in the 2+2=4 world, they probably won't forget that 2+2=3, unless the 2+2=3 wasn't modeling anything else. But the 2+2=3 was modeling something in their past history, at least the counting principle of their world. So they still have the 2+2=3 belief in their lives while they remember their past. If they forget their past, the 2+2=3 belief might became unuseful, but that still don't make the 2+2=3 an absurd or replaced by the 2+2=4: there are 2 number systems here.

For me, 2+2=3 isn't an absurd. That might be seem as a "common sum with a 3/4 multiplier" or a "X + Y = X p Y/X" where "p" is our common sum and "/" is our division, etc.. This way, like the 1+1=3 example above, only overloads the "+" operator. But, again, this "+" isn't the same from the "2+2=4"

Comment author: 26 November 2010 01:39:33AM 0 points [-]

Upon suddenly discovering that the whole world looks different this morning than it did last night is the rational belief "I guess I was deluded for my whole life up to this point" or "I guess I'm deluded now"?

Considering the fact that you're not waking up in a mental institution, but the world still seems to contain them (and if you get 2 sets of 2 of them, you have 3); the latter is a much more likely situation

Comment author: 27 November 2010 10:57:43PM 0 points [-]

You're neglecting the hypothesis "my memories of the past are being distorted to convince me that 2 and 2 make 4 instead of 3". Given how easily we distort our memories under conditions of sanity, this is as likely as "I'm deluded now".

Comment author: 28 November 2010 12:08:32PM *  0 points [-]

If you suddenly gain a set of memories indicating that the raptor conspiracy is taking over the world, you would be considered deluded.

If you suddenly gain a set of memories indicating that 2+2 equals something other than what it DOES in fact equal, you are likewise deluded.

So your suggestion is in fact a subset of being deluded*. At which point you should voluntarily seek out psychological/psychiatric help.

• (which I assign a low probability, as I have never heard of such a type of delusion existing)

If you believe (as you seem to suggest by use of the aactive rather than the passive voice) that this delusion is being deliberately induced, it is important to remember that anyone with the power to induce that delusion could also reduce you to a gibbering wreck; and hence that going to get help is highly unlikely to be "part of their plan".

Comment author: 28 November 2010 08:58:11PM 0 points [-]

This is a distraction from the actual point; of course if this happened to me, then my first priority would be getting help (I might be having a stroke, for instance). But once I'm at the hospital and they tell me that I'm all right, but something strange happened to my brain so that it falsely remembers 2 and 2 having made 4, instead of the obviously correct 3...

If you don't agree that some set of circumstances like this should conspire to make me rationally accept 2+2=3, then if the scenario happened to you (with 3 and 4 reversed), you're asserting that you could never rationally recover from that metal event. Since I'd prefer, should I go through a hallucination that 2 and 2 always made 3, to be able to recover given enough evidence, I have to take the "risk" of being convinced of something false, in a world where events conspired against me just so.

Comment author: 28 November 2010 12:21:16PM *  2 points [-]

Upon suddenly discovering that the whole world looks different this morning than it did last night is the rational belief "I guess I was deluded for my whole life up to this point" or "I guess I'm deluded now"?

Why completely leave out the possibility that you aren't deluded at all? Depending on just what kind of 'different' you wake up in that is a distinct possibility.

I would, by the way, start with a high prior for 'deluded now' which would be altered one way or the other by extensive reality testing. I experience that in dreams all the time. I know from personal experience it is easier for me to be confused about the transient sensory experience of the present than the broad structure of all my memories. Results may vary somewhat.

Comment author: 28 November 2010 01:55:03PM *  0 points [-]

Good point, in the case of waking up in a logically possible world, remembering a previous logically possible world, there is a non-zero possibility that you've actually gone from one to the other somehow. How low the probability is depends on the nature of the differences

I was too caught up in the case of waking up in a world where the world you remember is logically impossible.

Comment author: 29 November 2010 12:27:23AM *  0 points [-]

Good point, in the case of waking up in a logically possible world, remembering a previous logically possible world, there is a non-zero possibility that you've actually gone from one to the other somehow. How low the probability is depends on the nature of the differences

Exactly. And with slightly different wording a world in which it seems like you have changed from one logical world to another is itself a just a logically possible world.

I was too caught up in the case of waking up in a world where the world you remember is logically impossible.

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.

Comment author: 11 March 2010 09:01:59PM *  1 point [-]

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.

Comment author: 13 April 2010 11:04:20PM 2 points [-]
Comment author: 11 May 2010 01:20:14AM *  0 points [-]

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?

Comment author: 13 May 2010 09:10:39AM *  13 points [-]

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.

Comment author: 29 July 2010 06:57:21PM *  1 point [-]

I wish I could vote you up so much more! The distinction between a-convincing-argument and what-it-would-take-to-convince-me is very real and overlooked by almost everyone posting here.

To take my own experience in becoming convinced of atheism, I sometimes like to think I accept atheism for the same reason that I accept evolution--because of the evidence/lack thereof/etcetera. But that is simply not the case. I accept atheism because of a highly personal history of what it took to get me, personally, to stop believing in Christianity, and start believing in something else that, as much as I would like to pat my rational self on the back, has fairly little to do with the arguments and evidence I heard on the matter.

When asking someone why they believe something or are convinced of it, "what is the reason?" and "what is your reason?" are two totally different questions.

Comment author: 30 July 2010 01:52:11AM 4 points [-]

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.

It's possible that the reason you accepted atheism is different to the reason you currently accept atheism. To make an analogy, I use the QWERTY keyboard now because it's the industry standard and therefore the most likely keyboard layout for me to encounter on an unfamiliar machine, regardless of the fact that I learned the QWERTY keyboard because that's what was the setting on my computer when I started posting obsessively in the Dominic Deegan forums.

Comment author: 03 August 2010 02:37:14PM 28 points [-]

In fact I once had this sort of mathematical experience.

Somehow, while memorizing tables of arithmetic in the first grade, I learned that 11 - 6 = 7. This equation didn't come up very often in elementary school arithmetic, and even more seldom in high school algebra, and so I seldom got any math questions marked wrong. Then one day at university, I received back a Math 300 homework assignment on which I'd casually asserted that 11 - 6 - 7. My TA had drawn a red circle around the statement, punctuating it with three question marks and the loss of a single point.

I was confused. There was nothing wrong with 11 - 6 = 7. Why would my TA have deducted a point? Everyone knew that 11 - 6 = 7, because it was just the reverse of 7 + 6 = wait-a-minute-here.

Pen. Paper. I grabbed eleven coins and carefully counted six of them away. There were not seven of them left. I started writing down remembered subtraction problems. 11 - 4 = 7. 11 - 5 = 6. 11 - 6 = 7. 11 - 7 = 4. One of these sums was clearly not like the others. I tried addition, and found that both 7 + 6 = 13 and 6 + 7 = 13.

The evidence was overwhelming. I was convinced. Confused, yes—fascinated by where my error could have come from, and how I could have held onto it so long—but convinced. I set to work memorizing 11 - 6 = 5 instead.

It didn't entirely take. Twenty years later, the equation 11 - 6 = 7 still feels so right and familiar and uncontroversial that I've had to memorize 11 - 6 = stop. I know the answer is probably either 5 or 7, but I work it out manually every time.

Comment author: 03 August 2010 03:40:47PM 5 points [-]

I don't know if the American elementary curriculum is better than it was (I hope so) but this mistake is less likely to happen now. My niece in 2nd grade is learning different methods of 'knowing' arithmetic. They still memorize tables, but they also spend a lot of time practicing what they call 'strategies for learning the addition facts'.

For example..

11-6 = (10-6)+1 = 5 is the compensation approach.

and 11-6 = 10-5 is the equal additions approach.

They also spend a lot of time doing mental math. I'm impressed with how different things are, and hope that students are doing better with this more empirical, constructivist approach. (My niece is good at math anyway, so I don't know if she's getting more out of it than average.)

Comment author: 03 August 2010 04:22:23PM 3 points [-]

I don't know very much about the American curriculum, having grown up with the Canadian one. But I also didn't pay very much attention in math class. I preferred to read the textbook myself, early in the year, and then play around with as many derivations and theorems as I could figure out, occasionally popping my head above water long enough for a test.

I wrote and memorized my own subtraction tables, and invented a baroque and complicated system for writing negative numbers -- for example, 1 - 2 = 9-with-a-circle-around-it, and 5 - 17 = 8-with-two-circles-around-it. Really this is the sort of mistake which could only have happened to me. :)

I'm glad that they're teaching these sort of strategies in US schools. My experience tutoring elementary school math (my son attended an alternative school in which parents all volunteered their own skills & experience) is that every kid has a slightly different conception of how numbers interact. The most important thing I could teach them was that every consistent way of approaching math is correct; if you don't understand the textbook's prescription for subtracting, there are dozens of other right ways to think about the problem; it doesn't matter how you get to the answer as long as you follow the axioms.

Comment author: 03 August 2010 04:31:06PM 2 points [-]

I never bothered to memorize trig equivalences. Instead, I just reduced sine, cosine, and tangent (and their inverses) to ratios of the sides of a triangle, and then used the Pythagorean theorem.

Comment author: 03 August 2010 04:39:54PM 2 points [-]

Well, it's so much easier and more robust that way! Instead of a long list of confoundingly similar equations, you're left with a single clear understanding of why trigonometry works. After that you can memorize a few formulas as shortcuts if it helps.

Of course this principle completely breaks down when you start working with a child who's already convinced that they can't do math—or with a group of 30 kids at once, a third of whose mathematical intuitions will be far enough from the textbook norm that no one teacher has enough time to guide them through to that first epiphany.

Comment author: 03 August 2010 04:52:11PM 1 point [-]

it doesn't matter how you get to the answer as long as you follow the axioms.

Well, it does also matter in practice that you can communicate effectively (a lesson I had to learn myself at that age). But learning how to translate from an idiosyncratic system into a standard one can be a source of even better learning, so I agree that kids should not be discouraged from inventing nonstandard but valid systems.

Comment author: 03 August 2010 05:01:57PM 1 point [-]

Your method of subtraction is similar to being the p-adic numbers, you might want to look them up!

Comment author: 22 October 2010 04:53:55PM -1 points [-]

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 3 – 2 = 2 so than if 2+1 = 3 we can substitute it for 3 (because it ‘is the very same thing as 3’) and get 2+1-2=2 and since addition is commutative (its just putting 2 things together and I can’t conceive of the order mattering) we would again have 1 = 0. Now, you can write a post about an imaginary world where addition is not commutative or where things have different properties than themselves (so I can’t substitute 2+1 for 3) or where the set of integers is not closed under addition but they wouldn’t be conceivable. Yes I can conceive of putting 2 ear plugs next to 2 ear plugs and being left with 3 but even if that happened I would still believe that 2+2 = 4

Comment author: 20 November 2010 11:13:24AM 0 points [-]

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 power or similiar disposition, does this make them muslim, christian, neither or both? Furthermore, if somebody really wants to believe that sound travels in space, then their belief may well prevail despite all evidence against it as the person creates 'exceptions' to their beliefs that they believe do not disprove their theory. Hope this made sense...

Comment author: 20 November 2010 01:14:42PM 2 points [-]

One might indeed "believe" all that. But a belief has no use if it isn't true.

Comment author: 03 March 2011 09:18:08PM 1 point [-]

Apples and oranges have more ways they are alike than not alike.

I always have to bring this up when someone makes the "apples to oranges" statement. It's only true so long as you are purposefully ignoring all the ways they are alike.

In other words, it is just as valid to compare apples to oranges as it is to compare fuji apples to granny smith apples.

That's just me being pedantic, but it really seems to apply in this particular case.

Comment author: 03 March 2011 09:22:00PM 0 points [-]

Apples and oranges are alike in more ways than they are not alike.

I always have to bring this up when someone makes the "you can't compare apples to oranges" statement. In fact, it is quite reasonable to compare apples to oranges. It's also reasonable to compare apples to eighteen-wheelers. It's only unreasonable when you are explicitly ignoring all the ways they are alike. Even then, it isn't particularly unreasonable to compare two things that are completely dissimilar.

I'm being a little pedantic, but it really seems to apply in this particular case.

Comment author: 17 December 2010 02:31:31AM *  0 points [-]

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 Christian because of direct revelation by God himself, and I was not high.

A similar or better revelation could convince me of a different religion, though Islam would require a much bigger revelation. (Extraordinary claims require extraordinary evidence, and I consider the claim that Islam is true much less likely than the claim that Judaism, Christianity or Buddhism is true, which in turn I consider less likely than the claim that Deism is true.)

Edit: Thanks all for the lovely discussion, I think I've presented my theory of Life, the Universe and Everything in enough detail to get a fair understanding of it.

I hope you like the data point and a lotta luck synthesizing your own theory of L,U and E.

Comment author: 17 December 2010 02:43:27AM *  6 points [-]

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

Comment author: 17 December 2010 03:06:07AM 3 points [-]

To the extent that their experiences do not contradict mine, I see no reason to doubt. There is nothing in Christianity that prevents non-Christians from having religious experiences.

But when the experiences of others do contradict mine, such as the revelations Joseph Smith or Mohammed received, I have to doubt their sincerity or their sanity (I don't know which) for the same reason you doubt mine: Because I can't see in their mind and I wasn't in their body when it happened. And if I have to choose between my own experiences and another persons experiences, I choose my own.

But I should mention that of all the people I trust and who have told me their religious experiences (mostly hindu family members) to date none of them has proven a challenge to my Christianity.

Comment author: 17 December 2010 03:20:39AM *  2 points [-]

And if I have to choose between my own experiences and another persons experiences, I choose my own.

Luckily, we need not be limited to those hypotheses. Neither you nor many of the others with similar experiences need be lying or insane. And the existence of an omnipotent, omniscient and omni-beneficent deity need not enter into it either. You just have to have brains.

Welcome, by the way.

Comment author: 17 December 2010 03:36:50AM 0 points [-]

I've considered those kind of explanations, but the nature of the particular experiences which caused me to convert does not lend itself to that kind of explanation.

My policy is to never discuss the details with someone I do not personally know and trust, but I will say this much: the evidence was external and observed and confirmed by trusted others.

In fact if you are familiar with Zero Knowledge Proofs (I'm a crypto geek) the evidence was a type of ZKP that allows me to know with certainty (to the extent that I can trust my own rationality and senses) without enabling me to duplicate the proof.

Comment author: 17 December 2010 05:51:28AM *  3 points [-]

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.

It is important to understand that if no religious experiences were mutually exclusive with Christianity (nobody ever saw Ganesh or Mohammed), then they would count a lot more strongly as evidence for Christianity. But many are mutually exclusive, and doubting the sincerity of every Sufi mystic who saw God is a move that requires strong evidence.

As to another person's experiences vs your own: I sympathize, I really do. But you need to have some epistemic humility here, and realize that "you" are encoded in about half a kilo of mushy grey stuff that is often very untrustworthy. I for one do not doubt your sincerity (or the Sufis') but I do doubt that you correctly interpreted your experience.

Comment author: 17 December 2010 03:05:30AM 6 points [-]

Hello! As you're no doubt aware, the general tenor of Less Wrong tends toward non-belief in religion. However, in contrast to many religious believers, you have expressed a willingness to alter your views in the face of evidence. Watch out! Even your tentative suggestion that you might "drift towards atheism" might cause you to be regarded as a heretic or at least untrustworthy in some churches. But if you're willing to commit yourself to pursuing the truth wheresoever it may lead, then congratulations!

As has been mentioned already in this thread, Judaism and Christianity historically do not claim to be non-disprovable. Elijah bet his God against Baal and (in the Biblical narrative) won. Do you think this experiment can be replicated? Alternatively, is there something equivalent to a "similar or better revelation" that could convince you that no organized religion is correct at all?

Comment author: 17 December 2010 03:26:42AM 1 point [-]

My parents don't consider me a real Christian, somehow I cope. ;-)

Not only do I believe the Elijah experiment can be replicated, I believe it is being replicated today along with many other miracles. Just hidden for most people, because in Christianity, God reveals the truth to those who he chooses (poor/humble/righteous people) and keeps other people (rich/wicked/prideful) blind. So God might raise someone from the dead but in a way that could not be publicly verified, lest the rich proud people who think they're so smart find out the truth.

I fail to see how a supernatural revelation could prove no (organized) religion is correct, short of God saying "no religion is correct", which would then cause me to create my own organized religion...

But Christianity could surely be disproved in many different ways. For one, aliens or real sentient AI would disprove Christianity AFA I'm concerned. I'm not yet 30, so maybe I'll discover it in my lifetime.

If Christianity were disproved, that would leave Buddhism and Deism as the only viable religions left IMH(current)O. And Deism is only necessary in so far as I find the evidence for abiogenesis and humans-created-by-evolution lacking.

So except miracles and creation, I could be an atheist.

Comment author: 17 December 2010 03:32:30AM 3 points [-]

and keeps other people (rich/wicked/prideful) blind

Now I'm wondering which of those categories I fit in to. They all sound a tad appealing. :)

Comment author: 17 December 2010 03:39:59AM *  0 points [-]

If you are familiar with Christianity, all humans fall into the wicked and prideful categories.

The fact that you are on the internet suggests you additionally fall into the rich one too.

Now whether God sovereignly chooses his people (calvinism), or humans can also choose e.g. by humbling themselves (arianism) is an open question.

Edit to add: Just because God hasn't revealed the truth to someone today, doesn't mean he won't do it tomorrow or even (though this is heresy) after death.

So I certainly don't consider all non-Christians to be hopeless, after all I was a non-Christian too, once. And I also don't consider all who call themselves Chrstian to be chosen.

Comment author: 17 December 2010 03:51:10AM 1 point [-]

If you are familiar with Christianity,

I was sincere Christian right up until I realised the religion could be better explained by tribal signalling than magic.

all humans fall into the wicked and prideful categories.

You just finished saying:

God reveals the truth to those who he chooses (poor/humble/righteous people) and keeps other people (rich/wicked/prideful) blind.

Comment author: 17 December 2010 04:04:34AM *  0 points [-]

A basic doctrine of Christianity is that poor, humble and righteous people are wicked and prideful too.

Only Jesus is perfect.

Some strains believe God choses for reasons we can't grasp and then those people become more humble and less prideful, etc.

Others believe that if you do your best to be humble and righteous eventually God will reveal himself (though no guarantee that it will happen before your last minute on earth).

I don't know which of the two it is, or perhaps it is something else entirely.

Comment author: 17 December 2010 03:40:19AM *  6 points [-]

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.

You can see how non-Christians might find that to be a suspiciously convenient excuse, right?

Comment author: 17 December 2010 03:49:01AM 0 points [-]

So because it makes sense it's suspiciously convenient?

Obviously if there was a God (e.g. the Christian one) and he wanted the whole world to be nominal Christians he would do another Elijah like demonstration of his power, recorded on camera.

This is obviously not the case. So either the Christian god does not exist (suspiciously convenient for the non-Christian?) or he does not actually want all those non-Christians to self-identify as Christians (suspiciously convenient for the Christian god?)

Comment author: 17 December 2010 04:01:56AM *  5 points [-]

So because it makes sense it's suspiciously convenient?

It's suspiciously convenient because your claim implies that that evidence of Christianity's truth is only available to people who already believe in it (or who are already much closer to believing it than their epistemic state actually warrants).

Comment author: 17 December 2010 04:11:51AM 0 points [-]

Obviously, if the evidence of Christianity's truth was available to all then all would be Christians. Assuming the Christian god does not want all to be Christians the evidence should not be available to all.

Anyway, when I received my experience I certainly did not want to believe in it. And even now many years later, I would prefer to abandon Christianity and its morality but find myself unable because of my experiences.

I also know of a few other stories similar to mine, enough to convince myself I'm not delusional.

Comment author: 17 December 2010 05:54:59AM 2 points [-]

I also know of a few other stories similar to mine, enough to convince myself I'm not delusional.

I assume you mean stories of religious experiences similar to your own. This should not be evidence that you are not delusional, since many people throughout history have claimed to have had such experiences, with reference to different, mutually exclusive religions. On average, therefore, most (if not all) people having such experiences must have been delusional. You should have a probability that you are mistaken at least as high as this proportion.

Comment author: 17 December 2010 06:18:15AM *  3 points [-]

Minor quibble, but "delusional" would seem overly inflammatory as it implies the delusionality is a persistent property of Xaway's person, rather than the one-off occurrence it more likely was.

Comment author: [deleted] 17 December 2010 02:53:44PM 4 points [-]

As some people have pointed out, it's not a binary choice between you being crazy or delusional, and Christianity being right. Human brains complete patterns, in predictable ways. I don't know what your experience was (since you're keeping that private) but there are probably multiple possible worlds that are consistent with your experience: not just "Xaway is nuts" or "Jesus is the Savior." Think about what might have actually happened and what it might actually mean, and resist pattern-matching for a while.

Just a word of info on this site: this is not a place where people generally debate religion. You sound like you have your doubts; I recommend you read the best atheist arguments (Bertrand Russell comes to mind), and read about the history of the Bible and early Church from a secular academic writer. Let it marinate for a while. Read widely and see what happens to your views. Sometimes debating on the internet isn't the best way to learn; it crystallizes whatever ideas you started off with and makes it hard to change your mind.

If you would "prefer to abandon Christianity" but your experiences won't let you, you should really take some time to think about whether your experiences have religious implications. There are naturalistic explanations for religious visions, and no, they don't all mean you're crazy. (Check out Oliver Sacks on Hildegard of Bingen, and Robert Sapolsky on St. Paul.)

Comment author: 17 December 2010 03:07:02PM 0 points [-]

There are naturalistic explanations for religious visions, and no, they don't all mean you're crazy. (Check out Oliver Sacks on Hildegard of Bingen, and Robert Sapolsky on St. Paul.)

I'd also recommend Why God Won't Go Away: Brain Science and the Biology of Belief by Gene D'Aquili and Vince Rauss.

Comment author: 17 December 2010 03:50:01AM 2 points [-]

There were some things I thought of saying, but I think I'll hold my tongue for now. In short, I think your assertions have some logical errors. This is not a put-down or a personal comment -- I'm certainly no more than an aspiring rationalist, at best, myself. I hope you stick around this forum. In the spirit of Tarski I would ask you to join me in saying:

If Christianity is true I desire to believe that Christianity is true. If Christianity is not true, I desire to believe that Christianity is not true.

I would say this, and do!

Comment author: 17 December 2010 03:59:26AM 0 points [-]

If you spot a logical error, bring it on.

Obviously I don't want to believe untrue things.

But if there is two things I am sure about, it's (1) that humans are not rational, especially not me and (2) there are things that are true which can not be proven to be true (the real world analogue to Godels theorems).

I frequent this site, but I generally do not participate in internet discussions. I only registered this account and gave my two cents because Eliezer asked for a Christian who speaks Bayes to chime in.

I'm afraid that once I log off, I will probably forget the password to this account.

Comment author: 17 December 2010 04:23:49AM *  2 points [-]

Again, I hope you stick around. No need to burn yourself out as the lone voice of Christianity -- pacing yourself is fine.

Also, this truly is a rationalist site. If you can present well-thought out arguments, people here will listen to you. If you can make a rational argument demonstrating the truth of Christianity, then (according to some denominations) you could save some souls. (I understand the Calvinists would not necessarily agree.) But according to some traditions, good works (not just fide sola) have merit, and evangelizing is one of the greatest of all good works. Is it not?

My ulterior motive in making that argument is that I also think this forum could benefit from the perspective of a Christian who speaks Bayes.

Comment author: 17 December 2010 04:34:58AM 2 points [-]

I also think this forum could benefit from the perspective of a Christian who speaks Bayes.

I think I'd rather have a better calibrated Frequentist.

Comment author: 17 December 2010 05:10:24AM 1 point [-]

I'd rather have a rock. Or a Christian who doesn't speak Bayes. At least that implies less doublethink.

Comment author: 17 December 2010 05:53:52AM *  1 point [-]

Christianity here is actually a memetic hazard. It's a set of beliefs that has so many things wrong with it all of us feel compelled to address all of the bad thinking and wrong evidence all at once. It immediately draws everyone away from whatever productive comments they were making and into an attempt to deconvert the interlocutor. The interlocutor then responds to these attempts with more nonsense in different places which draws still more people in to the battle. Better to just keep the Hydra's out than try and chop off all those heads.

No one here is actually at risk but we don't get anything to justify the strain on the immune system.

Comment author: 17 December 2010 06:07:29AM 1 point [-]

I can think of some counterexamples. We "got" SarahC, for instance (according to her own words), and that was an unadulterated boon.

Also, the claims of religion are varied enough that they provide a range of topics, many trivial but some interesting. E.g., if we were in a sim and somebody changed it from outside in violation of the sim's internal physical law, that would constitute a "miracle" at this level of reality. How would we recognize such an event from inside?

Comment author: 17 December 2010 04:36:09AM 0 points [-]

Although I appreciate some of the articles on this site, I don't think I'll participate much in the discussion.

Although I speak Bayes and know more logic than a human should know, I do not consider myself a rationalist, because I doubt my own rationality. It wouldn't make sense for an inherently irrational person to spend his time trying to talk rationally when he could be dancing or programming.

Also, I firmly believe that Christianity can not be proven by argument, only by evidence (miracles). And only God himself, not the Christian, can provide the evidence, which he does on his own terms.

Comment author: 17 December 2010 04:39:01AM 10 points [-]

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

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

Comment author: 17 December 2010 04:46:22AM *  4 points [-]

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

The truth status of Christianity is something that Less Wrong should be able to consider a settled question. We can debate about things like the Simulation Argument, etc. and other reductionist non-supernaturalist claims that look sorta like deism if you squint, but Jehovah did not create the universe, and Jesus is not Lord, and I don't think there's any point in humouring someone who disagrees, or encouraging them to come up with smarter-sounding rationalizations. Let's not push Less Wrong in the direction of becoming the sort of place where these old debates are rehashed; there are more interesting things to think and talk about. Although it seems that Xaway in particular may not have come here with the intention of actually convincing anyone to believe in Christianity, I would propose in the general case that anyone who does want to should be referred to some place like /r/atheism instead.

Comment author: 17 December 2010 04:58:24AM *  5 points [-]

Well, I certainly don't think Jehovah created the universe. On the other hand, this thread is devoted to the consideration of the proposition that 2 + 2 = 3 -- arguably a settled question -- with the understanding that "a belief is only really worthwhile if you could, in principle, be persuaded to believe otherwise." I don't know if Xaway is going to be participating any more (hi, Xaway, if you're reading this!), but I was hoping that this might be a good exercise in practicing rational discussion. In part, I thought we could win him over to the dark side (joking about it being the dark side.)

Comment author: 17 December 2010 05:17:19AM 7 points [-]

Also, this truly is a rationalist site. If you can present well-thought out arguments, people here will listen to you.

That is not 'truly rationalist'. Well thought out arguments for a preselected bottom line are bullshit.

Comment author: 17 December 2010 05:04:40AM *  0 points [-]

I'm afraid that once I log off, I will probably forget the password to this account.

Perhaps you could go to 'Preferences' on the right and change your password to something easier to remember.

Anyway, at 18 I became a Christian because of direct revelation by God himself, and I was not high.

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.

Comment author: 17 December 2010 05:19:03AM 0 points [-]

If you spot a logical error, bring it on.

Did. Didn't work. Wrote you off. :)

Comment author: 17 December 2010 03:21:28PM 0 points [-]

(2) there are things that are true which can not be proven to be true (the real world analogue to Godels theorems).

Arguably this is the case for everything (until we solve the problem of induction). In the meantime, I don't know of anything you can't assign a probability to or collect evidence about.

As for whether this is an analogue to Godel's theorem (or, in times gone by, Russel's paradoxical catalogues - or in times yet to come, the halting problem) - no. Mathematical systems are useful ways to carve reality at its joints. So are categories, and so is computation. They can't answer questions about themselves. But reality quite clearly can answer questions about itself.

Comment author: 17 December 2010 03:46:20PM 0 points [-]

there are things that are true which can not be proven to be true (the real world analogue to Godels theorems).

Arguably this is the case for everything (until we solve the problem of induction). In the meantime, I don't know of anything you can't assign a probability to or collect evidence about.

How about the question of whether there is anything you can't assign a probability to or collect evidence about?

Comment author: 18 December 2010 05:07:04AM 1 point [-]

You can assign a probability to that. I hadn't considered the question strongly enough to have a mathematical number for you, but I would estimate there is a 10% chance that there are things which I cannot assign a probability to or collect evidence about. (Note that I assign a much lower probability to the claim "you can't assign a probability to or collect evidence about x"; empirically those statements have been made probably millions of times in history and as far as I know not a single one has been correct)

That said, "I don't know of anything you can't assign a probability to or collect evidence about" is true with a probability of 1 - 4x10^-8 (the chance I am hallucinating, or made a gross error given that I double-checked).

Comment author: 17 December 2010 04:10:12AM 0 points [-]

Christianity isn't true.

Comment author: 17 December 2010 04:24:17AM 5 points [-]

You're just mad they refused to canonize Samuel B. Fay.

Comment author: 17 December 2010 04:33:28AM *  1 point [-]

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.

A God is a very complex entity. Positing one does not, therefore, help to explain biological complexity (unless you have an explanation for God). Even though we don't know how abiogenesis happened it is still orders of magnitude more likely than God existing given the relative complexities involved. That Christianity is true is also orders of magnitude more unlikely than you and your companions hallucinating your direct revelation-- the former being an extraordinarily complex hypothesis and hallucinations and general irrationality being quite common.

Comment author: 17 December 2010 05:15:38AM *  0 points [-]

Well that is just your biases...

Because a God is supernatural any probability assigned to it existing is as arbitrary as any other.

Obviously, if the P=1/3^^^^^3 then it would be absurd to see biogenesis or biological complexity as evidence for God. But if the P =0.01 then I, for one, see it as very strong evidence.

I see no reason to prefer theism vs. atheism and I consider an extraordinarily low P to be biased towards atheism, but if that rocks your boat, have fun.

That I am irriational and delusional is highly probable, in fact I am sure of it. But I have no choice but to trust my own faulty brain.

I would certainly not consider you rational if you were to convert to Christianity solely based on reading my story on the internetz.

Comment author: 17 December 2010 05:37:01AM 1 point [-]

This is really wrong, obviously, but my hopes that the inferential distance was manageable have been dashed. You might start here. I'm done though.

Comment author: 17 December 2010 05:53:17AM 1 point [-]

Anyway, at 18 I became a Christian because of direct revelation by God himself, and I was not high.

What was that like? In particular, how could you tell that it was really a revelation and not any kind of temporary brain malfunction?

Comment author: 16 May 2011 04:29:31PM *  6 points [-]
Comment author: 28 July 2011 06:14:07PM 0 points [-]

Ha! Creepy. :3

Comment author: 27 July 2011 10:49:37PM 4 points [-]

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 may be explained by telling you about the positive evidences I've received for the religion I currently espouse.

Many of you may be aware that Mormons (obligatory semantic note: we are technically called the Church of Jesus Christ of Latter-Day Saints; the 'Mormon' appellation is non-official and in fact was originally a term of offense, but it sure makes for a good abbreviation!) have a bit more scripture than most Christian denominations: in addition to the Bible (we use the KJV, thanks!) we have the Book of Mormon (hence our nickname) and a few other bits and pieces. But it is the Book of Mormon that's the most well-known book of our scripture, and rightly so: it is often referred to as the "keystone of our religion", in that, were it to be disproved, our church would crumble like an arch with its keystone displaced.

Those of you who are not familiar with the circumstances of the translation of the Book of Mormon may seek to enlighten yourselves; I will proceed from this point under the assumption that you are familiar with the actual story, and not with the, erm, South Park dramatization.

(In addition, you may wish you educate yourselves regarding the differences between Mormonism and mainstream Christianity; they are many. Here is a primer on what we believe.)

Here we have an example of God talking to Joseph Smith, giving him the power to speak in His name, and giving him physical evidence of such: the Book of Mormon. Therefore, the veracity of the Church of Jesus Christ of Latter-Day Saints depends wholly on the veracity of the Book of Mormon: if it is false, then Joseph made everything up and the Church is false. If the book is true, then the Church upon which it is founded must also be true.

I will here make an aside to state that yes, I am treating the truth of the Book of Mormon as a binary value. There are three possibilities: Either the vision(s) were true, the translation was by God, and the Book really is what it claims to be; the vision(s) never happened, the "translation" was a lie, and the Book is fiction; or the vision(s) were false visions, the translation occurred by a supernatural power (Occam suggests the same power behind the visions), and the Book is still fiction (which, being backed by a supernatural power, may have false evidence to back it up), and also "God" is a perverse and conniving creature acting with motives beyond our ken. These possibilities will forward be referred to, respectively, as P-True, P-False, and P-Alien.

Therefore we have reduced the problem of whether this religion is true to a much simpler one: If the Book is true, then either the Church is true (P-True) or P-Alien. If the Book is false, then the Church is false (P-False).

Now we will dwell for a moment on what it means for the Book of Mormon to be true. In this case, we have a very precise working definition: The Book makes several historical claims, which would, if true, leave archaeological evidences. 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. Keep in mind that, in order to disprove P-False, we must use archaeological evidence that would not have been available to Joseph Smith; that is, evidence lacking from what we must assume would be the knowledge available to a 21-year old unschooled farmhand in upstate New York in 1827.

So does such archaeological evidence exist? Why yes, it does.

(comment split for length)

Comment author: 27 July 2011 10:49:57PM *  0 points [-]
• The entirety of Lehi's journey from Jerusalem to the sea has been found to match up with actual geographical and cultural sites in the Arabian Peninsula. Lehi's journey follows what we now know to be ancient trading routes. Lehi's family buried Ishmael at a place which Nephi called Nahom, which has been found by name, exactly where it should be. More or less directly east of that site, on the coast of Oman, have been found two candidate sites for Bountiful, Wadi Sayq and Salalah, right where they ought to be, with every feature described in the book including: reasonable access from Nahom (i.e. no mountains in the way!), an inlet for launching a ship, fertile ground with "much fruit and... honey", timber to build a durable ship, year-round access to fresh water, a nearby mountain upon which Nephi could offer his prayer, available ore and flint, and wind and ocean currents favorable for launching a ship out to sea. Candidate sites for the Valley of Lemuel and the River of Laman have been found, right where they ought to be.
• The practice of writing on metal plates, laughable in 1830, has now been well established as a legitimate ancient practice; one that Joseph Smith could never have known. He was mocked for his claim of receiving a "book written on gold plates" more than any other he made... even to this day! The practice of burying said records in stone boxes in the earth has been similarly credited.
• The practice of writing in what Mormon called "reformed Egyptian" (Mormon 9:32-34) has recently been shown to be rather more accurate than might be expected, as demonstrated by Daniel C. Peterson in Review of Books on the Book of Mormon, Vol. 5, 1993:

The statement "When modern Jews copy their scripture, they use Hebrew. They do not use Egyptian or Arabic, the language of their historic enemies" is quite an astonishing display of ignorance. Since the Egyptian language has been dead for centuries, it is hardly remarkable that modern Jews do not read the Bible in Egyptian. On the other hand, "the first and most important rendering [of the Old Testament] from Hebrew [into Arabic] was made by Sa'adya the Ga'on, a learned Jew who was head of the rabbinic school at Sura in Babylon (died 942)" (George A. Buttrick, ed., The Interpreter's Dictionary of the Bible [hereafter IDB], 4 vols. and supplement [Nashville: Abingdon, 1962-1976], 4:758b). Thus, Jews have indeed translated the Bible into "Arabic, the language of their historic enemies." They also have translated it into the language of their "historic enemies" the Greeks (IDB 4:750b on the Septuagint) and Aramaeans (IDB 1:185-93; 4:749-50, on the Aramaic Targums).

• Many tourists to Mexico will be well familiar with the use of cement in ancient America, for example in Teotihuacan; this information is had in the Book of Mormon, but was unknown in Joseph Smith's time.
• Jacob chapter 5 offers many, many details regarding olive cultivation which match precisely with what is now known about ancient practices in Israel.
• Many seemingly non-Hebraic personal names in the Book of Mormon (Alma, Sariah, Lehi, Mosiah, Aha) have been independently attested. These names are not just "Hebrew-ish", mind. They have been shown to be actual attested Hebrew names.

There are more evidences, but these are the strongest, in my opinion.

Now, are there negative archaeological evidences for the Book of Mormon? Unfortunately, there is rarely such a thing as a "negative archaeological evidence"; there are certainly none that disprove anything the Book of Mormon says. All that can be said is that the Book of Mormon makes claims that do not match up with our current archaeological knowledge... but the same was said, at various points in the past, for all the above claims. It is true that the statement "lack of evidence is not evidence of lack" is blatantly false... but "lack of evidence" certainly has a lot less weight than the positive evidence above.

Eliezer, you said that it is more rational to believe that Occam's Razor will always yield useful results than to believe that, although it has yielded useful results up to the present, it will cease to at some future date. Forgive me if I make an error here, but by application of the same argument, I should think that it is more likely that the Book of Mormon will corroborate with all future archaeological evidence than that the Book of Mormon will fail to match up, having so far met all of the above and more.

It is true that I do not have a 100% certainty that the Book of Mormon is true. But having seen all the evidence for its veracity, I am convinced enough of it to base my life and worldview on the religion predicated upon it.

So in order to convert me to Islam? First, you'd have to convince me that the Book of Mormon is not true, in order to get me back to a baseline. Then you'd have to convince me that Islam is true... and you've seen above the weight of evidence that will convince me.

DISCLAIMER: The above is not an attempt to convert anyone. It is an honest response to the question (challenge?) that Eliezer posed. I do not believe that anyone can or ought to be converted to a religion solely based upon logical evidence... though logical evidence can certainly be a gateway drug! :3 If you're intrigued, I would urge you to read the Book of Mormon for yourself. If you have questions, I urge you to comment, or email me at vl (period) arandur (at) gmail (dot) com.

Comment author: [deleted] 27 July 2011 11:10:36PM *  4 points [-]

The entirety of Lehi's journey...

The places exist, but is there evidence of the actual journey? If I adopt this theory of evidence, I accept American Gods as non-fiction, because most of the places in that book exist.

The practice of writing on metal plates, laughable in 1830, has now been well established as a legitimate ancient practice; one that Joseph Smith could never have known.

What evidence is there that Smith knew nothing of the practice of writing on metal plates? Who says it was laughable in 1830?

Many tourists to Mexico will be well familiar with the use of cement in ancient America, for example in Teotihuacan; this information is had in the Book of Mormon, but was unknown in Joseph Smith's time.

It was known in Europe -- used almost everywhere in Rome. Are there specific architectural details that were unprecedented?

Jacob chapter 5 offers many, many details regarding olive cultivation which match precisely with what is now known about ancient practices in Israel.

Jacob 5 agrees with what, as Darwin would say, "every animal husbander knows." What exactly are the details that match? Are they unexpected?

Many seemingly non-Hebraic personal names...

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.

Comment author: 28 July 2011 01:44:30AM *  0 points [-]

First: your analogy is flawed, and, I'm sorry to say, rather obviously so. Neil Gaiman knew of the places where he set the events of American Gods, having either traveled there himself or else at least seen them on a map. (I can't name any specifically, never having read the book, but I can surmise as much from the context of your objection, I should think! x3) Smith, on the other hand, could not have credibly known anything about the location or name of an ancient burial site in the Arabian Peninsula, or of the location of such a place as "Bountiful" in the same part of the world... particularly since "common knowledge" of the Arabian Peninsula makes the notion of finding anything that could be described as "bountiful" there subject to skepticism.

Second: Here are various sources deriding Joseph's claim of metal plates. John Hyde, Jr., Mormonism: Its Leaders and Designs (New York: Fetridge, 1857), 217-18; M.T. Lamb, The Golden Bible (New York: Ward and Drummond, 1887), 11; Stuart Martin, The Mystery of Mormonism (London: Odhams Press, 1920), 27. A quote by Hugh Nibley in 1957 seems amusingly prescient: "it will not be long before men forget that in Joseph Smith's day the prophet was mocked and derided for his description of the plates more than anything else." (Hugh Nibley, Lehi in the Desert, CWHN 5:107). A quote I have on hand: "No such records were ever engraved upon golden plates, or any other plates, in the early ages" /M.T. Lamb, The Golden Bible, or, the Book of Mormon: Is It from God? (New York: Ward & Drummond, 1887), p. 11/]. More information can be had [here, thanks to Jeff Lindsay, who is my primary (though not my sole!) source for Book of Mormon evidences. He has done a wonderful job compiling them.

I must, for the sake of completeness, humbly admit fault: To say that the practice was one "that Joseph Smith could never have known" is in fact false; it is within the realm of possibility that Joseph might have heard of such a thing. In my opinion, the likelihood that he could have known anything about the practice is so small as makes no odds, but I must concede that the probability is not 0. But the ridicule that he received for his claim is well-documented.

Third: I did not mean to imply that the use of cement to make dwellings was unheard of. However, the use of cement by pre-Columbian Americans was unknown to the hoi polloi in Palmyra, late 1820's. Even as late as 1929, ridicule abounded for the sake of this idea:

In 1929, Heber J. Grant (former President of the Church) told the story of a man with a doctorate who had ridiculed him for believing in the Book of Mormon. That learned man cited the mention of cement work as an obvious lie "because the people in that early age knew nothing about cement." President Grant, who was a young man at the time of that conversation, said:

"That does not affect my faith one particle. I read the Book of Mormon prayerfully and supplicated God for a testimony in my heart and soul of the divinity of it, and I have accepted it and believe it with all my heart." I also said to him, "If my children do not find cement houses, I expect that my grandchildren will." He said, "Well, what is the good of talking with a fool like that?" (April 1929 Conference Report, p. 128 ff.)

For more on this, please see Matthew G. Wells and John W. Welch, "Concrete Evidence for the Book of Mormon," Insights (May 1991): 2.

Comment author: [deleted] 28 July 2011 02:27:04AM 0 points [-]

First: Very well, the analogy was flawed. I'm unclear as to what the name "Bountiful" is supposed to refer to. Do either of the places mentioned as candidates translate to "Bountiful"? Further, I want to point out that "Critics doubt the link between Nahom and NHM, as well as having other criticisms." This will dovetail with our forthcoming conversation on Hebrew/English transliteration.

Second: While such things were unknown archaeologically, the practicing of inscription on gold is referenced in the Bible; some googling uncovers Ex 39:30; see also the references here. Whoever the author of the BoM was, they were very well versed in the Bible.

Third: The quote demonstrates that the actual existence of pre-Columbian American cement houses is irrelelvant. If they had not been found in our time, surely you also would maintain that they would be found... eventually. As you do elsewhere.

Comment author: 28 July 2011 02:54:10AM -1 points [-]

First: The name "Bountiful" has no significance other than indicating a place of bounty. The candidate sites are those which match the description I noted above:

...reasonable access from Nahom (i.e. no mountains in the way!), an inlet for launching a ship, fertile ground with "much fruit and... honey", timber to build a durable ship, year-round access to fresh water, a nearby mountain upon which Nephi could offer his prayer, available ore and flint, and wind and ocean currents favorable for launching a ship out to sea.

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 fact that the Book of Mormon does not explicitly mention contact with outsiders during Lehi's journey.
• It is suggested that there is no evidence dating NHM before A.D. 600.
• It is suggested that the pronunciation of NHM is unknown and may not relate to Nahom at all.
• It has been suggested that Joseph Smith simply created the name Nahom as a variant of the Biblical names Naham (1 Chron. 4:19), Nehum (Ne. 7:7) and Nahum (Na. 1:1).

The first is not really comprehensible as a counter-argument; contact with outsiders is not requisite for Lehi to know the name. The second is a mere lack of evidence. The third is merely a complaint of ambiguity inherent in Hebrew, and is answered elsewhere in the article. The fourth is simply an alternate theory, and a right flimsy one at that, if it's meant to explain away the consonance between Joseph's "guess" and the actual place.

Second: I'll just note that the practice of engraving on metal jewelry and plaques is something much different than the practice of writing sacred records on books of precious metal.

Third: The story of Einstein's Arrogance is relevant. :3 But at this point, I have enough positive evidence behind the Book of Mormon to start taking some of its as-of-yet unverified claims on faith.

And what about you? What of the evidences that do stand? What is the chance that these could have come about by pure luck? Certainly we've acquired enough bits of evidence to raise the Book of Mormon to the level where it merits our attention, at least.

Comment author: [deleted] 28 July 2011 02:59:48AM *  5 points [-]

What is the chance that these could have come about by pure luck?

Reasonably high. We have many examples of charismatic people constructing obviously fictive religions whose followers then retroactively find evidence, exploiting hindsight/confirmation bias. Scientology, Baha'i, Theosophy, and the various tibetan tulkus are examples.

Comment author: 28 July 2011 06:02:08PM 0 points [-]

In each of these cases, the amount of retroactive evidence is far outweighed by the number of evidences against the religion's teachings. The opposite is true of Mormonism. None of its claims are disproven; we are only lacking evidence to support them. And the number of claims unsubstantiated by physical evidence is shrinking. Every time a discovery has been made that relates to the Book of Mormon, it supports the text.

I will admit that there have been discoveries that have challenged popular understandings of the Book of Mormon. Once upon a time, it was in vogue to suppose that the narrative spanned the entire American continent (that is, both of them). This has been shown to be probably false, and in fact the text of the Book of Mormon itself seems to contradict that notion. However, the difference between, say, Scientology and the Book of Mormon is that we have in the latter a document that is not changing, but is still matching up to the evidence thrown at it. This document has been around for some 200 years in its present form, and the only alterations that have been made to it have been to repair grammatical errors - errors that, in fact, speak more strongly for the Book of Mormon than against it, since the first printing had "errors" that, while atrocious English, actually made very good Hebrew. I will supply you with references to this claim if you wish, but I thought it behooved me to stick to physical evidence first, as those are, in my opinion, the strongest claims.

But you say "reasonably high". I'm afraid I'll have to hand you the burden of proof. With this counter, you chose to comment on an afterthought of a question and dismiss it out of hand, instead of talking about my arguments. We started this conversation - at least I did - under the premise that the physical evidences I supplied were worth discussing. I thought that you were under the same premise, but now with this post you attempt to dismiss any physical evidences as "hindsight/confirmation bias". I call foul.

Comment author: [deleted] 28 July 2011 06:56:14PM *  3 points [-]

In each of these cases, the amount of retroactive evidence is far outweighed by the number of evidences against the religion's teachings.

Really? I can't think of any evidence contradicting the belief that His Holiness the Dalai Lama is the reincarnation of the previous Dalai Lama. Yet the evidence in favor is much of the same kind of evidence presented here, namely, "How could the young Dalai Lama have known which of many objects were the personal possessions of the previous Dalai Lama, were he not the reincarnation thereof?" In the same vein, "How could Joseph Smith have known X?", asked rhetorically, doesn't provide evidence in itself.

In any case, this was never meant to be an argument about me converting to Mormonism. I wanted to know why you thought a non-Mormon shouldn't be skeptical of these evidences. I'll leave others to judge whether or not you've satisfied the condition of the precommitment in a parallel discussion thread.

Comment author: 28 July 2011 07:14:48PM *  1 point [-]

If you look at the votes for our posts, I think you'll find that they've already been judging. :3 Yes, I'm sorry if you felt I was jumping onto the "Hey, I've convinced you, now you should convert!" bandwagon; that was far from my intent. But I have offered my arguments about why a non-Mormon shouldn't be skeptical - rather, ought to be skeptical, but should be swayed anyway by the weight of evidence - but if it is not enough to convince you, then so be it. It is said that two Bayesians, working from the same set of priors, cannot agree to disagree... but I think we have different priors, which disturbs me to an extent. I will go meditate on this; I hope you will, too.

EDIT: As to the Dalai Lama example, whose word do we have that these objects did in fact belong to the previous Dalai Lama? If the honesty of the ceremony is well-documented, then I would be interested to learn more.

Comment author: 28 July 2011 08:12:13PM 6 points [-]

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.

Absence of evidence is evidence of absence. The book of Mormon makes many claims for which, if they were true, we would expect to find evidence, but we do not. If you only look at the writings of Mormon apologists, you're going to get an extremely slanted picture of how well the Book of Mormon agrees with existing archaeological evidence, but if you look elsewhere, it's not hard to find strong evidence against it. The fact that the Book of Mormon references as being present animals that did not exist in Mesoamerica, or anywhere in the New World at the time, while not mentioning any of numerous common animals that were, is, as I see it, a knockdown argument all by itself. If these animals existed at that time and place, we have an extremely strong expectation of evidence for it given the archaeological and paleontological research we've done, but instead there is none. And the chance that legitimate writings from that time and place would reference as present animals which were not approximates to zero. This is extremely strong evidence against the Book of Mormon being true, and it's only one among its evidential failings.

Comment author: 28 July 2011 10:22:17PM *  0 points [-]

I am well acquainted with the notion of absence of evidence, thank you; I touched on this point above, stating that, although absence of evidence does count as points against the case I make, positive evidence makes stronger points. Were this not the case, then physicists wouldn't be searching for the Higgs Boson; they'd be restricted to theories which are readily explained by only the particles we have evidence of.

A disproof of the Book of Mormon, then, must rest upon just that: disproof. With that in mind, let us examine further those points raised in the link you provided.

Archaeological Fallacies
First, four technologies are mentioned which were "unknown to Mesoamerica": chariots, steel swords, bellows, and silk.

An explanation of the word 'chariot' can be found here.

Many explanations have been made re: steel swords; the reference made in this case comes from the book of Ether, speaking of the Jaredites. I offer the below quote as a counter:

In light of contemporary conditions in Mesoamerica, one can understand this passage a number of ways. Although the blades of most macuahuitls in Mesoamerica were made from obsidian, the Aztecs are known to have had war clubs studded with iron instead of the usual obsidian. There are even examples in Mesoamerica of ceremonial macuahuitls with feathers replacing the obsidian blades.

Various types of material, including iron, replaced the usual obsidian of the macuahuitl, and such a weapon could thus be described as a sword with a metal "blade." Another possibility is to equate this Jaredite steel with the "steel" of the King James translation of the Old Testament, which actually refers to the Hebrew word for "bronze."

Finally, we need to understand that Mosiah translated Ether's plates into social and linguistic concepts with which he was familiar. Mosiah, as king, possessed Laban's sword, a steel weapon that was passed down as one of the insignia of royalty. In translating Ether's record, Mosiah might thus have given the Jaredite kings steel swords, like the one he himself possessed, because in Mosiah's society a king was expected to have a steel sword as his royal weapon.

Bellows are only mentioned in the locale of the old world, not in America, making this a non-point.

Regarding silk: An LDS publication, 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:

As for the account dealing with peoples in the New World, Book of Mormon geography best fits the Isthmus of Tehuantepec (southern Mexico, Guatemala), where a number of sites, cities, etc., have been tentatively correlated with Book of Mormon locations. The best treatment of this is in John Sorenson's An Ancient American Setting for the Book of Mormon. He offers fascinating correlations, very strong (in my opinion), though only a small fraction of the archaeological work has been done that is needed to confirm most of the specific proposals. The Lamanites in the Book of Mormon may correlate well with a part of the early Mayan civilization or one of the other cultural groups in ancient Mesoamerica. The peoples described in the Book of Ether could very well be part of the Olmec peoples from the same area. A number of Mayan legends and the few surviving writings provide interesting parallels with Book of Mormon concepts. Could write much more on this if you're interested. Bottom line: yes, there are real places and there were real people described in what is truly an authentic ancient record. But we are in our infancy when it comes to understanding Mesoamerica. Until scholars are able to do more work there, the argument from silence should be applied with caution.

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

Most stunning of all, the BoM never once indicates that the American continent was anything but uninhabited when the refugees from Jerusalem arrived.

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 intriguing question this is. I'll defer to Jeff Lindsay, who has done much work on this subject, and who has cited many good primary sources, lest there be a complaint against my using his work too many times.

Linguistic fallacies
I once again defer to Jeff Lindsay:

One of the most interesting evidences of transoceanic contact between the Old and New Worlds is the Bat Creek Hebrew inscription found by a Smithsonian expedition in Tennessee in 1889. (The Bat Creek Stone and other interesting oddities of archaeology, including pre-Columbian maize in India, can be seen at the Archaeological Outliers site.) Anti-Mormon writers such as the Tanners have spent much effort trying to argue that the writing on the Bat Creek Stone is not Hebrew. However, non-LDS scholar J. Huston McCulloch has now shown that the Bat Creek inscription, once thought to be Cherokee, "fits significantly better as Paleo-Hebrew" (J. Huston McCulloch, "The Bat Creek Inscription: Cherokee or Hebrew?" Tennessee Anthropologist, Vol. 13, Fall 1988, p. 116, as cited by Matthew Roper, Review of Books on the Book of Mormon, Vol. 4, 1992, p. 212). McCulloch's recent work confirms Cyrus Gordon's original hypothesis about the inscription, namely, that it was from between 70 A.D. and 135 A.D. and represented Old World writing (Science Vol. 2, May 1971, pp. 14-16, as cited by Paul R. Cheesman, BYU Studies, Vol. 13, No. 1, p. 85). Carbon-14 dated wood and brass bracelets associated with the inscription date to between A.D. 32 and A.D. 769 (Ibid., pp.107-12, 116) - definitely before Columbus. Cyrus Gordon, a respected non-LDS scholar, wrote:

The Bat Creek Inscription is important because it is the first scientifically authenticated pre-Columbian text in an Old World script or language found in America; and, at that, in a flawless archaeological context. It proves that some Old World people not only could, but actually did, cross the Atlantic to America before the Vikings and Columbus....The discredited pre-Columbian inscriptions in Old World scripts or languages will have to be reexamined and reevaluated, each on the merits of the evidence, case by case. (Cyrus Gordon, "A Hebrew Inscription Authenticated," in Lundquist and Ricks, eds., By Study and Also by Faith, 1:71,80, as cited by Roper, op. cit.; for more on this controversial issue, see also J. Huston McCulloch, "The Bat Creek Inscription: Did Judean Refugees Escape to Tennessee?" Biblical Archaeology Review, July/August 1993, pp. 46-53, 82, and the differing view of P. Kyle McCarter, Jr., "Let's Be Serious about the Bat Creek Stone," Biblical Archaeology Review, July/August 1993, pp. 54-55, 83.)

While critics will repeat old arguments that the Bat Creek Stone is a forgery, it is important to recognize that "there is absolutely no indication that the inscription is a forgery, in the first place, other than the circular, and therefore unscientific, argument that being Hebrew, it must surely be a fake" (J. Huston McCulloch, "The Bat Creek Stone: A Reply to Mainfort and Kwas," Tennessee Anthropologist, Vol. 18, No. 1, Spring 1993, p. 16, emphasis added, as cited by Matthew Roper, FARMS Review of Books, Vol. 9, No. 1, 1997, p. 142). David H. Kelly has also found serious evidence of several pre-Columbian inscriptions of European origin: "We need to ask . . . where we have gone wrong as archaeologists in not recognizing such an extensive European presence in the New World" (David H. Kelly, "Proto-Tifnagh and Proto-Ogham in the Americas," Review of Archaeology, Vol. 2, Spring 1990, p. 10, as cited by Roper, op. cit.). More evidence for scholarly acceptance of Old World scripts in the ancient Americas can be found in W.R. McGlone et al., Ancient American Inscriptions: Plow Marks or History? (Long Hill, Mass.: Early Sites Research Society, 1993, as cited by Sorenson, 1993, p. 21) and Jacques de Mahieu, "Corpus des inscriptions ruiniques d'Amerique du Sud," Kadath 68, Brussels, 1988, pp. 11-42 (cited by Sorenson, 1993, p. 21). More relevant research has tentatively identified hundreds of possible links between Uto-Aztecan languages (in Book of Mormon territory) with the ancient Hebrew language (work by Brian D. Stubbs, including "A Curious Element in Uto-Aztecan," The Epigraphic Society Occasional Papers, Vol. 23, 1998 [according to second-hand sources - I have not yet read this article]; "Elements of Hebrew in Uto-Aztecan: A Summary of the Data," F.A.R.M.S. paper, 1988; "Looking Over vs. Overlooking Native American Languages: Let's Void the Void," Journal of Book of Mormon Studies, Vol. 5, No. 1, Spring 1996, pp. 1-49).

It Takes a Thief...
I will not deign to justify this. Any examination of the story, from either side, will show that this is neither a rigorous disproof or in fact unreasonable.

Lost in Translation
B. F. Sperry writes a response here to the question of Harris vs. Anthon. As for the Book of Abraham, I would be remiss not to refer you to Jeff Lindsay's excellent three-part piece.

Comment author: 28 July 2011 01:54:50AM *  -1 points [-]

Fourth: In this case, I defer entirely to the experts.

Below is an excerpt from John Gee and Daniel C. Peterson, "Graft and Corruption: On Olives and Olive Culture in the Pre-Modern Mediterranean," in The Allegory of the Olive Tree, pp. 186-247, taken from pages 223-224:

And here is an excerpt from "Botanical Aspects of Olive Culture Relevant to The Allegory of the Olive Tree" by Wilford M. Hess, Daniel J. Fairbanks, John W. Welch, and Jonathan K. Driggs in The Allegory of the Oliver Tree (pp. 552-554):

Based on the botanical and horticultural information present in the archaeological and historical record, and reflected in Jacob 5, we can conclude that the ancients were superb horticulturists and had a profound understanding of vital biological and plant cultural principles. Most of the botanical and horticultural principles in Jacob 5 are sound and are very important for olive culture. In addition, the one or two points, according to our interpretation, that represent unusual or anomalous circumstances are necessary enhancements to the message of the allegory. In this single chapter of the Book of Mormon there are many detailed horticultural practices and procedures that were not likely known by an untrained person, and may not have been fully appreciated by professional botanists or horticulturalists at the time the Book of Mormon was translated. Even today, outside of olive-growing areas, professional horticulturalists may not fully appreciate some of the unique aspects of olive culture. Given the extensive detail about olive culture present in Jacob 5, we must give Zenos much credit for a high degree of horticultural knowledge, which many take for granted. Examples of what the ancients and Zenos evidently knew were how to prune, dig about, dung, and nourish; how to graft tame to wild and wild to tame, and how to graft tame back into tame; how to balance tops and roots by pruning, and the reasons for doing this; how to save the roots of trees whose branches had decayed, and how to transplant branches to preserve the desired traits of good plants; how to preserve and store fruit and how to distinguish between good and bad fruit; how well plants grow on good and bad soil; how to care for trees to cause young and tender branches to shoot forth; that they could graft wild to tame to rejuvenate tame; that specific cultivars produced well in certain areas; . . . that they could burn an orchard to reestablish a new one; that plants grown from seeds would not have desirable characteristics; the importance of elimination of old wood and debris by burning, and how to deal with pests and pathogens; how to prevent heavy bearing one year and no bearing the next by proper pruning; the necessity to plant more than one cultivar for pollination; and how to propagate scions with the desirable genetic material. Interestingly, much of this sophisticated technology was probably lost in the Nephite civilization, for the olive is not mentioned again in the Book of Mormon after Jacob 5, an indication that the lands of the Book of Mormon may not have been suitable for growing olives ... The only regions on the American continents with Mediterranean climates where olive culture is economically feasible are the regions of California, Chile, and Argentina. Joseph Smith probably knew how to prune, dig about, dung, and nourish local fruit trees; he probably knew a little about grafting, and he may have been familiar with some other horticultural principles, but not likely those peculiarly related to olive culture.

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

Comment author: [deleted] 28 July 2011 02:35:30AM *  1 point [-]

Fourth: 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.

Comment author: 28 July 2011 05:49:24PM 1 point [-]

Fourth: 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.

For online verification of olive culture principles from non-LDS resources, consider "The Secrets of Olive Trees" from BienManger.com (also LeGourmetMarket.com), from which the following excerpts are taken. That page verifies several concepts in Jacob 5, such as the ability of olive trees to grow in rich and poor soils, the importance of grafting, the ability to regenerate or rejuvenate a decaying olive tree, and the practice of applying dung:

SOILS The olive tree often grows on poor and dry soils, but gives remarkable results on rich soils (California) or by irrigation (Spain and Oranie). . . . GRAFTING : the propagation of a given variety of table olives is done by grafting, except in special cases (cuttings, stump chips of the same variety). Depending on what has to be grafted, the following techniques are being used : For the seedlings and the sprouts coming from stocks of a different variety, you can use cleft grafting or budding. In the case of older trees, be it the grafting of wild olive trees or of olive groves whose production is to be modified, it is advised to use inarching or bark grafting. . . . REGENERATION : It may be necessary to rejuvenate an olive grove if it has not been maintained for a long period or if it has suffered accidents, thus becoming unable to produce a normal crop. It is sufficient to cut away all branches, except the largest ones and then graft the remaining stumps. The grove should then bear a unique variety of table olives and be able of bearing fruit in excellent conditions. A trunk in very bad shape should be cut at the base in order to start with three replacing shoots. . . . MANURE : Although manuring largely pays off, olive trees are still too rarely manured. Manure should be organic, on a basis of dung or cattle cake. When possible, a culture of green fertilizers (vetch, lupin, etc.), mowed at maturity and ploughed in, will complete the dressing of organic matter. . . . Here is some information from a modern olive-tree cultivator, as viewed April 27, 2008: Some ancient olive trees in Jerusalem at the Mount of Olives date back 2000 years. When old large limbs are pruned on large aged olive trees, new branches grow and a new olive crop grows. . . . The leaves of olive trees are gray-green and are replaced at 2-3 year intervals during the spring after new growth appears. Pruning yearly and severely is very important to insure continued production. The trees have the unproductive limbs removed, "so that it will be more fruitful" John 15:2. An olive tree can grow to 50 feet with a limb spread of 30 feet, but most growers will keep the tree pruned to 20 feet to assure maximum production. >>New sprouts and trees will emerge from the olive tree stump roots, even if the trees are cut down. Some olive trees are believed to be over a thousand years old, and most will live to the ripe old age of 500 years. Olives generally are beaten off trees with poles, harvested mechanically or by shaking the fruit from the trees onto canvas. Most ripening olives are removed from the trees after the majority of the fruit begins to change in color. It is important to squeeze out the olive oil within a day after harvesting or else fermentation or decline in flavor and quality will occur. The olive oil can be consumed or used in cooking immediately after its collection from the press. Olive oils are unique and distinct, each brand of olive oil having its own character, as determined by many factors, like those unique flavor differences found in fine wines. Prepared commercial olive oils can vary greatly in aroma, fruit flavor; whether the taste is, flowery, nutty, delicate, or mild, and the coloring of olive oil is quite variable. . . . Olive trees can survive droughts and strong winds, and they grow well on well drained soils up to a pH of 8.5 and the trees can tolerate salt water conditions. In Europe, olive trees are normally fertilized every other year with an organic fertilizer. Alternate bearing can be avoided by heavy pruning and generally the trees respond to this very quickly and favorably. Olive trees should be purchased that have been vegetatively propagated or grafted, because the seed grown trees will revert to a wild type that yields small olives with an insipid taste. Olive trees are more resistant to diseases and insects than any other fruit tree and, therefore, are sprayed less than any other crop. Other olive-related resources are provided by the University of Georgia (note the discussion of soils, indicating that olive trees can grow on soil too poor for ordinary cultivation, consistent with Jacob 5) and the California Rare Fruit Growers.

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.

Comment author: 28 July 2011 11:12:29PM 1 point [-]

waits for wedrifid

Comment author: 29 July 2011 12:40:12AM *  12 points [-]

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 archaelogical evidence is to exist given that the BS is false. Because p(Arch | BS) > p(Arch | !BS), Arch does something to overcome the incredibly low prior probability p(BS).

To put it another way there is a critical observation that deductively "(Book's Story) = (Archeologically verifiable parts) && (parts that are not archeologically verifiable)". This is before we go ahead and calculate p(Mormonism | BS) vs p(Alien | BS).

Another interesting probability calculation to consider is how likely it is that wedrifid will write out a bunch of probability calculations given the clearly false 'truth' proposition is the acronym of "Book's Story" vs wedrifid writing out a bunch of calculations about bullshit given the acronym is not BS. From this we can go ahead and chain our inferences to calculate p(wedrifid is peurile | wedrifid writes out the calculations). For what it is worth I think you should find that the likelyhood ratio is small even if your posterior is enormous.

Comment author: 29 July 2011 01:04:13AM 2 points [-]

Upvoted for amusement value.

Comment author: 29 July 2011 12:33:26PM 3 points [-]

I 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 thirteen Articles of Faith: "We believe in being subject to kings, presidents, rulers, and magistrates; in obeying, honoring, and sustaining the law."

Comment author: 29 July 2011 03:39:23PM *  0 points [-]

Again in 2008 that same team of genetic scientists republished essentially the same article under the same title, making a similar point: "Here we show, by using 86 complete mitochondrial genomes, that all Native American haplogroups, including haplogroup X, were part of a single founding population, thereby refuting multiple-migration models." (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2427228/?tool=pmcentrez)

Only solid piece of evidence i found on the DNA route, most of the rest seems to be arguing that there remains a miniscule chance despite the current consensus on DNA data.

Comment author: 29 July 2011 04:12:49PM 0 points [-]

I'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.

Comment author: 30 July 2011 04:17:11AM 0 points [-]

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.

I 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.

Comment author: 30 July 2011 05:11:04AM 0 points [-]

But 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.

Comment author: 29 July 2011 08:43:39PM 1 point [-]

While I do not accept certain of your premises (surprisingness of corroborated evidence) your reasoning from there is cogent and the update worthy of respect!

Comment author: 29 July 2011 11:33:33PM 1 point [-]

Oh! Well, thank you! I will attempt to be cogent the first time in the future. :3

Comment author: 29 July 2011 12:59:51PM *  0 points [-]

I gather the discussion of the whole thread rests on this unexpected premise:

we can assume a correlated truthiness with the remainder of the story

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.

we must use archaeological evidence that would not have been available to Joseph Smith; that is, evidence lacking from what we must assume would be the knowledge available to a 21-year old unschooled farmhand in upstate New York in 1827

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.

Comment author: 29 July 2011 02:13:51AM 1 point [-]

...what would convince me of non-Christianity? Mormonism has presented me with enough positive evidence...the conversion process would be two-tiered: first a strong positive evidence for Islam/Judaism/whatever, and second a strong disconfirmation of Mormonism

This 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.

Comment author: 29 July 2011 05:02:06AM 0 points [-]

It 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.

Comment author: 29 July 2011 05:09:27AM 1 point [-]

The question was not what would convince someone without prior belief; the question was what would convince me as a Christian,

This 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.

Comment author: 29 July 2011 12:18:18PM 0 points [-]

It 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.

Comment author: 29 July 2011 01:45:22PM 0 points [-]

may or may not be unfortunate

Let 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.

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.

the conversion process would be two-tiered: first a strong positive evidence for Islam/Judaism/whatever, and second a strong disconfirmation of Mormonism

I've tried my hardest to erase my preconceived notions and start from scratch.

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

In the first two quotes above, you seem to disagree with what I say, in the latter two, you seem to agree.

Comment author: 29 July 2011 04:05:58PM 0 points [-]

The 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.

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.

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.

Comment author: 29 July 2011 02:41:12PM 2 points [-]

Everything I have learned about Mormonism is internally consistent, and consistent with my own ideas on morality.

This 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?

Comment author: 29 July 2011 04:23:54PM 0 points [-]

Oh, 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?

Comment author: 29 July 2011 05:41:36PM 0 points [-]

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.

Without your morality, the church would still exist, too, wouldn't it?

What evidential status does my conflicting morality have on yours?

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.)

Comment author: 29 July 2011 11:54:40PM 0 points [-]

I was in fact using evidence in that technical a sense, but I'll answer your question anyway.

...why is your morality agreeing with the LDS church... 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?

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".

Comment author: 29 July 2011 07:33:37PM 0 points [-]

I saw this on the side while reading an unrelated post...

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.

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.

Comment author: 29 July 2011 11:36:44PM 0 points [-]

Your 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.

Comment author: 01 August 2011 11:23:18PM 3 points [-]

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.

Comment author: 14 September 2011 04:48:15AM *  14 points [-]

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 have not heard of anyone else spotting this error). But clearly there is something to update on, so my prior probability that 2+2=3 is not zero. However, I also maintain that if in fact the proof of 2+2=4 is correct, then it remains correct whether or not I am convinced of it, whether or not I exist, and even whether or not physical reality exists. So it is a priori true, but my knowledge of it is not a priori knowledge (because the latter does not exist).

I think this is what Eliezer was trying to say with "Unconditional facts are not the same as unconditional beliefs.", but this seems to be glossed over and almost lost within the confusion about earplugs. The article's failure to distinguish between a mathematical theory and a mathematical model (map and territory, possibly?) came very close to obscuring the actual point. This article does not explain how to convince Eliezer that 2+2=3, it explains how to convince Eliezer that PA does not model earplugs - and since the latter is not an a priori truth, it is much less interesting that knowledge of it is not a priori either.

Comment author: 14 September 2011 05:07:40AM 1 point [-]

The article's failure to distinguish between a mathematical theory and a mathematical model (map and territory, possibly?)

Exactly. This is one of Eliezer's few genuine philosophical mistakes, one which, four years later, he's still making.

Comment author: 14 September 2011 05:41:38AM 1 point [-]

I 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.

Comment author: [deleted] 14 September 2011 05:53:52AM 1 point [-]

Reading 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."

Comment author: 14 September 2011 06:05:19AM 7 points [-]

That would have been a damn nuisance, because throughout the rest of this comment thread we'd have been writing unhelpfully long strings of Ss. ;)

Comment author: [deleted] 14 September 2011 06:13:55AM 1 point [-]

I was proud of this comment and I comfort myself with your explanation for why it got the response it did.

Comment author: 14 September 2011 06:03:20AM 2 points [-]

I, 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).

Comment author: 14 September 2011 06:05:56AM *  1 point [-]

I know very well the difference between a collection of axioms and a collection of models of which those axioms are true, thank you.

Then 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 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).

Comment author: 14 September 2011 06:26:53AM *  10 points [-]

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 can contain reality-fluid in the same way as the wavefunction, or even be uniquely specified by second-order axioms that pin down a single model up to isomorphism the way that second-order axioms can pin down integerness and realness, is something we have rather less evidence for, on the surface of things, than we have evidence favoring the physical existability of models of infinity and continuity, or the mathematical sensibility of talking about the integers or real numbers.
Comment author: 14 September 2011 06:20:58AM *  0 points [-]

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

Maybe I'm misinterpreting you, but could you explain how any non-symmetric equation can possibly be true in all models of arithmetic?

Comment author: [deleted] 14 September 2011 06:08:07AM *  2 points [-]

Maybe "earplugs do not model PA," not the other way around? (Edit: just saw this 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.