Do we consider it to be evidence in Christianity's favor that more people believe in it than Islam? Does the average IQ of adherents of a religious belief cause it to become more plausible to us?
In the interests of disclosure, I am an agnotheist who was baptized Catholic and raised mainline Protestant, so we're still waiting for Eliezer's requested comment.
People's belief in something is evidence for that thing in the sense that in general it's more likely for people to believe in a thing if it's true. Less Wrongers sometimes use the phrase "Bayesian evidence" when they want to explicitly include this type of evidence that is excluded by other standards of evidence.
One way to think about this: Imagine that there are a bunch of parallel universes, some of which have a flat Earth and some of which have a spherical Earth, and you don't know which type of universe you're in. If you look around and see that a bunch of people believe the Earth is flat, you should judge it as more likely you're in a flat-Earth universe than if you looked around and saw few or no flat-Earthers.
However, people's beliefs are often weak evidence that can be outweighed by other evidence. The fact that many people believe in a god is evidence that there is a god, but (I think) it's outweighed by other evidence that there is not a god.
See also "Argument Screens off Authority".
Certainly. The probability of Christianity having more followers than Islam is greater if Jesus rose from the dead and less if he did not.
It's not necessarily strong evidence of course. Disavowing Islam has enormous social consequences, so I would expect there to be a large number of Muslims in both the world where Muhammad received the Quran from Gabriel and the world where Muhammad hallucinated. But I still expect there to be more Christians if Jesus rose from the dead than if he did not.
IQ is only weakly correlated to rationality. A much better thing to do is to ask Christians why they believe. If you know the reasons a Christian believes, then the evidential weight of their reasoning will replace the evidential weight that comes from the fact that they believe.
The causal flow looks like this:
Reality --> Reason to believe -> Person believes
By d-separation, once you know a person's reasons for believing, the fact that they believe is no longer useful information to you.
In the interests of disclosure, I am an ex-Christian who spent a year learning Arabic because I believed that God was calling me to be a missionary to Muslims. When I learned Bayes theorem, I attempted to use...
I am a jew (born and raised). I can easily imagine that if I were raised in the muslim world to a muslim family that I would be a muslim today. However, were I born to a christian family (and perhaps this is simply my inner biases talking) I suspect that I would have been attracted to various aspect of the Jewish religion which are not present (or not nearly as strong) in christianity, like the idea of a "contract with God".
In full disclosure, I do not continue to call myself a Jew because I believe the Torah to be more likely than any other mainstream religious text, but because I find the ethical framework to be superior.
To apply the same reasoning the other way, if you aren't a Christian, what would be a situation which would convince you of the truth of Christianity?
The Second Coming? An opportunity to have a chat with the Lord Himself? An analysis of a communion wafer revealing it to, in fact, be living human flesh? It's seriously not that hard to think of these.
Which is more likely "God exists" or "I just hallucinated that" For the third one, probably that He exists, for the second one, definitely hallucination, for the first, I'm not sure.
Second one: depends. I was kind of assuming that you have some way of verifying it, like you ask Him to create something and someone who wasn't there later describes some of its previously determined properties accurately without being clued in. First: you'd need a massive global hallucination, and could use a similar verification method.
That seems accurate. Remember that a single person can hallucinate that someone else verified something, but this has low prior probability.
I once conducted an experiment in which I threw a die 500 times, and then prayed for an hour every day for a week that that die consistently land on a four, and then threw the die 500 more times. Correlation was next to zero, so I concluded that God does not answer prayers about dice from me.
Haven't you ever heard the saying, "God does not throw dice games"?
You're not being sincere.
Actually, if you run the test, you are. Given that you'd have changed your mind if it had gone the other way, of course.
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.
I don't think this is at all the core issue.
Eliezer's original post stated that beliefs need to come from mind-reality entangling processes.
If math is a part of "reality", then Eliezer's point stands and empirical reasoning makes perfect sense.
If math is not a part of "reality", then we would expect it to influence nothing at all, including our beliefs. Or even suppose that knowledge came from somewhere and could influence belief but still did not otherwise correlate with reality: Then it would be irrelevant. This, of course, is not the case - as anyone who's ever used any mass-manufactured device as well as bridges and roads, should realize. Math DOES have utility in real life. And I daresay that if it suddenly stopped helping us reliably predict the load-bearing limit of bridges, we'd treat is as suspect and false.
The ACTUAL core issue remains that a belief that cannot be reversed is useless.
At any rate, if the former is true, 2+2=4 is outside the province of empirical science, and applying empirical reasoning to evaluate its 'truth' is wrong.
When I imagine putting two apples next to two apples, I can predict what will actually happen when I put two earplugs next to two earplugs, and indeed, my mind can store the result in a generalized fashion which makes predictions in many specific instances. If you do not call this useful abstract belief "2 + 2 = 4", I should like to know what you call it. If the belief is outside the province of empirical science, I would like to know why it makes such good predictions.
To apply the same reasoning the other way, if you aren't a Christian, what would be a situation which would convince you of the truth of Christianity?
You'd have to fix all the problems in belief, one by one, by reversing the evidence that originally convinced me of the beliefs' negations. If the Sun stopped in the sky for a day, and then Earth's rotation restarted without apparent damage, that would convince me there was one heck of a powerful entity in the neighborhood. It wouldn't show the entity was God, which would be much more complicated, but it'...
If you do not call this useful abstract belief "2 + 2 = 4", I should like to know what you call it.
I call it "2+2=4 is a useful model for what happens to the number of earplugs in a place when I put two earplugs beside two other earplugs". Which is a special case of the theory "arithmetic is a useful model for numbers of earplugs under some operations (including but not limited to adding and removing)".
If the belief is outside the province of empirical science, I would like to know why it makes such good predictions.
The mathematical claim "2+2=4" makes no predictions about the physical world. For that you need a physical theory. 2+2=4 would be true in number theory even if your apples or earplugs worked in some completely different manner.
I hate to break it to you, but if setting two things beside two other things didn't yield four things, then number theory would never have contrived to say so.
Numbers were invented to count things, that is their purpose. The first numbers were simple scratches used as tally marks circa 35,000 BC. The way the counts add up was derived from the way physical objects add up when grouped together. The only way to change the way numbers work is to change the way physical objects work when grouped together. Physical reality is the basis for numbers, so to change number theory you must first show that it is inconsistent with reality.
Thus numbers have a definite relation to the physical world. Number theory grew out of this, and if putting two objects next to two other objects only yielded three objects when numbers were invented over forty thousand years ago, then number theory must reflect that fact or it would never have been used. Consequently, suggesting 2+2=4 would be completely absurd, and number theorists would laugh in your face at the suggestion. There would, in fact, be a logical proof that 2+2=3 (much like there is a logical proof that 2+2=4 in number theory now).
All of m...
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?
Woah, I think that's a little overconfident...
You're saying that in the mid nineteenth century (half a century before relativity), the anomalous precession of Mercury made it seem 99.999999% likely that Newtonian mechanics was wrong?
After all, there are other possibilities.
cf. "When it was noticed in the 1800's that the perihelion of Neptune did not match what Newton's inverse-square law of gravity predicted, did we change the way math works? Or did we change our understanding of gravity?" In this case we actually postulated the existence of Pluto.
Similar solutions were suggested for the Mercury case, e.g. an extremely dense, small object orbiting close to Mercury.
And that's leaving aside the fact that 99.999999% is an absurdly high level of confidence for pretty much any statement at all (see http://lesswrong.com/lw/mo/infinite_certainty/ ).
If I were a nineteenth century physicist faced with the deviations in the perihelion of Mercury, I'd give maybe a 0.1% probability to Newton being incorrect, a 0.001% probability to maths being incorrect, and the remaining ~99.9% would be shared between incorrect data /incomplete data/ other things I haven't thought of.
However, I agr...
"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.
I am confused by this discussion. Are we talking about integers or things?
Analytic truths may or may not correspond to our situations. When they don't correspond, I guess that's what you all are calling "false." So, if we're engineers working on building a GPS system, I might say to you, "Careful now, Euclidean geometry is false."
Similarly, quantum physicists on the job might say, "Watch out now, two and two isn't necessarily four."
I'm thinking of this excellent blog post I came across last week:
...Consider a basket with 2 apples in it. Now toss in 2 more apples. Examine the basket, and you will find (surprise!) 4 apples. However, you cannot prove a priori that there will be 4 apples in the basket. It is an empirical question, albeit a trivial one, whether baskets of apples (which are physical things) behave in the same manner as the non-negative integers under addition (which is an abstract logical construct).
This distinction might seem hopelessly pedantic at first, but you can easily go astray by ignoring it. For example, many people naively expect photons to behave in the same manner as integers under addition, but they don’t. “Number of photons” is not a conserved quantity in the way that “number of apples” is; photons can be created/destroyed, one photon can be split into two, et cetera....
Eliezer is right; numbers are first an abstraction of the world around us. There are a vast number of possible abstractions; the reason we have been so very interested in numbers, compared to all the other possible abstractions, is that numbers happen to describe the world around us. It need not have been so.
"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.
The truth of an arithmatic equation and the truth of the content of a religion like Islam or Christianity are really not comparables at all. Within the domain of mathematics, "two plus two" is one definition of "four". Conversely, "four" is one definition of "two." (In a sense these truths are tautalogical.)
The Greeks noticed that mathematics is a field of knowledge that can be developed entirely in the mind. The manipulative objects that we use to teach children basic arithmetic operations are not actually the subje...
Eliezer: When you are experimenting with apples and earplugs you are indeed doing empirical science, but the claim you are trying to verify isn't "2+2=4" but "counting of physical things corresponds to counting with natural numbers." The latter is, indeed an empirical statement. The former is a statement about number theory, the truth of which is verified wrt some model (per Tarski's definition).
Gray Area, if number theory isn't in the physical universe, how does my physical brain become entangled with it?
Rozendaal, sounds like you bought into one of religion's Big Lies.
Let me take another crack at this...
I do not believe any situation could ever convince Eliezer that 2+2=3.
If he proclaims "two and two makes three," then he must be talking about something other than the integers. You cannot be mistaken about the integers, you can only misunderstand them. It's like saying "some women are bachelors." You are not mistaken about the world, you've merely lost your grasp of the terminology.
Lee B, Gray Area: what if you had a proof that 2 + 2 = 3, and, although you seem to recall having once seen a proof that 2 + 2 = 4, you can't remember exactly how it went?
Integers are slippery in a way that apples and poodles are not. If you say something unconventional about integers, you cease to talk about them. --- Does anyone disagree with that?
(1) Peter de Blanc asks what happens when I cannot follow a proof properly. I count that as a failure of rationality rather than an instance of being mislead by evidence. That is not, I think, what Eliezer intends when he says "convinced."
(2) If I observe some trick and say, "wow, two and two makes three," then I am dropping the integer system and adopting some other. My "wow" is the same one that we all said when we learned that Euclidean geometry doesn't hold in our universe.
Lee, the situations I talked about for convincing me that "2 + 2 = 3" could only actually occur if 2 + 2 actually equalled three within the realm of the integers. This is right and proper: why should I allow myself to be convinced by something that would not be valid evidence?
I do not, therefore, ever expect myself to actually encounter any of these situations, because I currently believe that 2 + 2 = 4.
If I expected to encounter such evidence in the future, the expectation of my probable future probability estimates must equal my present probab...
Eliezer: "Gray Area, if number theory isn't in the physical universe, how does my physical brain become entangled with it?"
I am not making claims about other universes. In particular I am not asserting platonic idealism is true. All I am saying is "2+2=4" is an a priori claim and you don't use rules for incorporating evidence for such claims, as you seemed to imply in your original post.
A priori reasoning does take place inside the brain, and neuroscientists do use a posteriori reasoning to associate physical events in the brain with a priori reasoning. Despite this, a priori claims exist and have their own rules for establishing truth.
I can imagine a world in which the mathematics we have developed is not useful, or in which commonly assumed axioms are false in that world. However, "The Pythagorean Theorem is a theorem of Euclidean geometry" is still true even if you're living on a sphere. If I say "I cannot be convinced that 2 + 2 = 4", I mean something like "I cannot be convinced that S(S(0)) + S(S(0))) = S(S(S(S(0)))) is not a theorem of Peano arithmetic."
On the religion issue: I'll accept as divine any entity that can consistently reduce the entropy of a closed, isolated system, and will demonstrate this ability on demand. ;)
I am not making claims about other universes. In particular I am not asserting platonic idealism is true. All I am saying is "2+2=4" is an a priori claim and you don't use rules for incorporating evidence for such claims, as you seemed to imply in your original post.
Please explain the miraculous correspondence to apples and earplugs, then.
I confess that I'm also not entirely sure what you mean by "a priori" or why you think it requires no evidence to locate an "a priori claim" like "2 + 2 = 4" in the vast space of po...
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 ...
Eliezer: I am using the standard definition of 'a priori' due to Kant. Given your responses, I conclude that either you don't believe a priori claims exist (in other words you don't believe deduction is a valid form of reasoning), or you mean by arithmetic statements "2+2=4" something other than what most mathematicians mean by them.
(some arguments)Don't care. If you can reverse entropy, you might as well be a god. If some alien gives me technology to reverse entropy, then A God Am I.
Eliezer: It sure seems to me that our evolution and culture constructed ethical attitudes are entangled with the world. By the way, I don't think that we agree at all about what "I find it quite easy to imagine" means, but of course, some words, like "I", are tricky. It might be more interesting to ask "what data could I give a soundly designed AGI that would convince it that 2+2=3?" For you and for sound AGI designs, I'd like to know what situation would be convincing regarding the proposition "beliefs should not resp...
I'm neither Eliezer nor (so far as you know) an AGI, but I think (1) I couldn't be convinced by evidence that beliefs should not respond to evidence but (2) I could be led by evidence to abandon my belief that they should. (Probably along with most of my other beliefs.) What it would take for that would be a systematic failure of beliefs arrived at by assessing evidence to match any better with future evidence than beliefs arrived at in other ways. I think that would basically require that future evidence to be random; in fact that's roughly what "ran...
Mathematics is about logical patterns. A world in which you can be mistaken about such fundamentals as the value of 2 + 2 is not a world where you can put any trust in your logical deductions. As such, if you ever do notice such a slip, I suggest that the cause is likely to be something deeply wrong with you, yourself, and not that you are living in a computer simulation.
The test of any religion is whether cultures believing it tend to thrive and improve the quality of their lives or not. The whole point of the word of God is that following it gives you...
Wikipedia on a priori: Relations of ideas, according to Hume, are "discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe".
This points out clearly the problem that I have with "a priori". It is a fundamentally Cartesian-dualist notion. The "mere operation of thought" takes place INSIDE THE UNIVERSE, as opposed to anywhere else.
To observe your own thoughts is a kind of evidence, if the spikings of your neurons be entangled with the object of your inquiry (relative to yo...
It is perfectly acceptable for me to say, "I can think of no encounterable situation that would transform the terminal value of this event from negative to positive."
Now, don't make me bring up a trolly problem. :-)
This sentence of Eliezer's is where the action is:
I'm suspicious of claims that supposedly do not require justification and yet seem to be uniquely preferred within a rather large space of possibilities.
"There are no married bachelors" gets us to nod our heads because we uniquely prefer English syntax and semantics. We pick it out of the rather large space of possible languages because it's what everyone else is doing.
If Eliezer went around earnestly saying, "there are some married bachelors," I would guess he had entangled himself w...
I concede (a little)!
In a previous Overcoming Bias post we learned that people sometimes believe the conjunction of events R and Q is more probable than event Q alone. Thus people can believe simple and strictly illogical things, and so I shouldn't throw around the word "unthinkable."
If I stretch my imagination, I can just maybe imagine this sort of logical blunder with small integers.
I draw the line at P AND ~P, though: just unthinkable.
"It appears to me that "a priori" is a semantic stopsign; its only visible meaning is "Don't ask!""
No, a priori reasoning is what mathematicians do for a living. Despite operating entirely by means of semantic stopsigns, mathematics seems nevertheless to enjoy rude health.
There are really two questions in there:
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...
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 ...
So the actual end result would be to convince me that the universe was in the hands of a monstrously insane and vicious God. As I noted here, that is actually pretty much what I believed in the last days of my Christianity. My perspective on ethics made it more plausible to me than I suspect it would be to most people.
The whole point of Christianity (as I grew up with it) is that by manifesting Himself on earth God realized that the whole smiting people thing was passe. I always thought the God of the New Testament was just that of the Old with better mark...
Perhaps 'a priori' and 'a posteriori' are too loaded with historic context. Eliezer seems to associate a priori with dualism, an association which I don't think is necessary. The important distinction is the process by which you arrive at claims. Scientists use two such processes: induction and deduction.
Deduction is reasoning from premises using 'agreed upon' rules of inference such as modus ponens. We call (conditional) claims which are arrived at via deduction 'a priori.'
Induction is updating beliefs from evidence using rules of probability (Bayes th...
It is possible in today's wonderful world of computers to have 2 + 2 = 3, and be both correct and understandable.
For Instance:
We have two integer variables x and y. Our equation is x + x and the outcome is placed in y (ie. x + x = y) We will view the value of y.
We take the value 1.7 and input it into x. Since x is an integer it will (in most cases) be rounded to 2. Therefore x = 2.
It is possible, however, for y to receive the value of 1.7 + 1.7 which, in today's accepted math, equals 3.4.
Placing 3.4 in an integer variable will set y to 3.
Therefore, you have 2 + 2 = 3.
BTW, this is why doing floating point math with integer variables on computers is a very bad idea......
I've not read all of the comments, but those that I've read from you, Eliezer, in combination with the original blog post, confirm that we are in agreement. Re: Locke, I believe we are blank slates when born. There is no such thing as a priori (how do I italicize?). All thinking, even logical and mathematical reasoning, is done a posteriori. Of what I've read, you've put it brilliantly.
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.
"Cloud, you might want to read Steven Pinker's 'The Blank Slate'."
Perhaps the term "blank slate" carries too much baggage. I only mean it with respect to the a priori/posteriori or rationalism/empiricism. Disclaimer: my eclectic survey of much of Western thought has blurred the lines defining these terms. So take from this what you will, but I can't guarantee myself being clear.
For the statement 2+2=4 to be true there are some assumptions that needs to be. That is 2+2=4 is true within a system, mathematics, but this system is in fact a construction!
The basic assumption here is that we can define and identify 'one' thing - say a ball, a man or any other "item" - for this to be true you would further need to have 'identical' items... that is items that have very similar attributes.
As you can see this leads to a infinite regress, where one assumption leads to others, and in fact we don't have such systems in reality, that ...
In response to g (a while back, concerning entropy): If physicists discovered such a technique, omniscience of a sort(by arbitrarily altering and measuring the amount of information in a given region) would be possible, as would a form of omnipotence (we could arrange any concievable configuration of particles via Maxwell's demon). Hooking it up to a computer with some knowledge base of usually-accepted morals to this quantum entropy-decreasing construct, we would have omnibenevolence, also - hence, such a being would, indeed, be (an approximate) God by mo...
Thanks for an excellent post. I think you have summed up the distinction between beliefs arising out of blind faith and those that are observation based.
This time I disagree with Eliezer...this experiment won't convince me that 2+2=3...wouldn't even convince me that physical maxim "everything goes somewhere" is wrong...I would find where the phones are (even if they sublimated). That still don't make that an "imutable belief".
There's nothing wrong in switching lexically 3 and 4 ( S(2) = 4; S(4) = 3; S(3) = 5 )...sounds unuseful, and don't attack Peano's axioms. That would make me believe in 2+2=3.
To stop believing in the integer numbers, it's needed to prove an inconsistency in Peano's ...
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'...
It's often poor form to quote oneself, but since this post (deservedly) continues to get visits, it might be good to bring up the line of thought that convinced me that this post made perfect sense:
The space of all possible minds includes some (aliens/mental patients/AIs) which have a notion of number and counting and an intuitive mental arithmetic, but where the last of these is skewed so that 2 and 2 really do seem to make 3 rather than 4. Not just lexically, but actually; the way that our brains can instantly subitize four objects as two distinct group...
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.
What I gain from this article is more or less an example of society's influence on how you understand things. For example, for most people 2+2=4. If the counting system and math operations was completely different, 2+2 could equal anything, unless one was familiar with the high-context culture using such a system. Another example would be the projection of an idea with words. One may say express their emotion as the word "happy". Another may say "joy". Another "euphoria". Unknowingly, all three have the same exact emotion, only their words have their different connotations. Suprisingly enough, I seem to have confused myself. Does anyone want to try to discern what I've said?
There seem to be far too many people hung up on the mathematics which ignores the purpose of the post as I understand it.
The post is not about truth but about conviction. Eliezer is not saying that there could be a scenario in which the rules of mathematics didn't work, but that there could be a scenario under which he was convinced of it.
Deconstructing all elements of neurology, physics and socialogy that make up the pathway from complete ignorance to solid conviction is not something I could even begin to attempt - but if one were able to list such steps as a series bullet points I could conceive that the manipulation of certain steps could lead to a different outcome, which appears to me to be the ultimate point of the post (although not hugely ground-breaking, but an interesting thought experiment).
It is not a claim that the strongly held conviction represents fact or that the conviction would not be shaken by a future event or presentation of evidence. As a fundamental believer in scientific thought and rationality there is much that I hold as firm conviction that I would not hesitate to re-think under valid contradictory evidence.
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 ...
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 ...
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...
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 ...
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?
I do not consider myself a rationalist, because I doubt my own rationality.
This site isn't called Always Right, you know.
Ooh, ooh! I'll do you one better. I'm not just a Christian; I'm a Mormon. :P
My goodness, what would convince me of non-Christianity? The problem here is that Mormonism has presented me with enough positive evidence that I'm reasonably certain of its veracity. So the conversion process would be two-tiered: first a strong positive evidence for Islam/Judaism/whatever, and second a strong disconfirmation of Mormonism, which I would then seek to corroborate by figuring out why on earth I received so many outlandishly unlikely false positives.
Both of the tiers m...
Edit: I meant to cover this point first, but I left it out before.
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.
This really isn't how it works. Absence of evidence is evidence of strength proportional to the expectation of evidence if a given proposition is true. So if, for example, you propose that there is an elephant in a room, and then you investigate the room and see no sign of an elephant, then that is very strong evidence that there is no elephant in the room. But if you propose that there is a mouse in a room, and you investigate and see no sign of the mouse, then that is only weak evidence that there is no mouse. You will have to update your confidence that there is a mouse in the room downwards, but much, much less than you had to update in the case of the elephant.
In both the case of the elephant and...
Your point is well taken, and I will meditate upon it. Thank you.
waits for wedrifid
I hadn't actually read the grandparent beyond skimming and categorized it as an entirely non-trollish expression of personal belief. Given the prompt in the post it was appropriate to the context and as rational as can be expected given that the guys' beliefs are utter nonsense.
Having read through the first comment (before the "to be continued") the following part jumped out at me as the primary non sequitur.
So, if the archaeological evidences corroborate the Book's story, then we must consider the Book to be "true", and thereby accept either P-True or P-Alien.
That just isn't case. Archaeological corroboration provides evidence for the Book's story. That is, part of the story is validated which eliminates a whole lot of the bits that could be wrong and we can assume a correlated truthiness with the remainder of the story. We update p(Book's Story) upward, but not to one. Something along the lines of:
p(Arch | BS) = x
p(Arch | !BS) = y
p(BS | Arch) = p(Arch | BS) * p(BS) / p(Arch)
We do not have the logical deduction "IF Arch THEN BS" but rather a likelyhood ratio such that BS is more likely the less likely it is for the archaelog...
If 2+2 equals 3, I desire to believe that 2+2 equals 3. I want my conclusion to be controlled by the abstract fact I seek to discern.
A middle aged woman is charged with the task of counting objects(oranges, apples, earplugs) individually and in pairs as they roll onto a conveyer belt she notices that they come in order of ( OOO,A A, E) and has performed this job for several years. The woman devises a formula for counting these objects she places a [ 1 in column 3 ] for a set of oranges that pass by, then a [ 1 in column 2 ] for the pair of apples that pass by, she then places a [ 1 in column 1 ] for the earplug. So by looking at her count of the line (OOO,AA,E,OOO,AA,E) or 2+2+2For a while this confused me, because I incorrectly identified what part of Eliezer's argument I thought was wrong.
Suppose I were to make all those observations suggesting that combining two objects with two objects produced three objects. I would not conclude that 2+2=3, rather I would conclude that objects were not modelled by Peano Arithmetic. (This much has been said by other commenters). But then I only 'know' Peano Arithmetic through the (physical) operation of my own brain.
Here's how to convince me that 2+2=3. Suppose I look at the proof from (peano axioms) to (2+2=4), and suddenly notice that an inference has been made that doesn't follow from the inference rules (say, I notice that the proof says a + (b⁺) = (a+b)⁺ and I know full well that the correct rule is (a⁺)+(b⁺)=(a+b)⁺). I correct this 'error' and follow through to the end of the proof, and conclude the result 2+2=3. What do I do? I consider that this observation is more likely if 2+2=3 than if 2+2=4, and so I update on that. It's still more likely that 2+2=4, because it's more likely that I've made an error this time than that everyone who's analysed that proof before has made an error (or rather, that I h...
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. ;)
Scientists and mathematicians use the word "model" in exactly opposite ways. This is occasionally confusing.
I don't think people really understood what I was talking about in that thread. I would have to write a sequence about
Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense.
Well, it makes sense to me without any models. I can compute, prove theorems, verify proofs of theorems and so on happily without ever producing a "model" for the natural numbers in toto, whatever that could mean.
Everything sounded perfectly good until the last bullet:
why believing that things like a first uncountable ordinal can contain reality-fluid in the same way as the wavefunction
ERROR: CATEGORY. "Wavefunction" is not a mathematical term, it is a physical term. It's a name you give to a mathematical object when it is being used to model the physical world in a particular way, in the specific context of that modeling-task. The actual mathematical object being used as the wavefunction has a mathematical existence totally apart from its physical application, and that mathematical existence is of the exact same nature as that of the first uncountable ordinal; the (mathematical) wavefunction does not gain any "ontological bonus points" for its role in physics.
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
Pinning down a single model up to isomorphism might be a nice property for a set of axioms to have, but it is not "reality-conferring": there are two groups of order 4 up to isomorphism, while there is only one of order 3; yet that does not make "group of order 3" a "more real" mathematical object than "group of order 4".
I would like very very much to read that sequence. Might it be written at some point?
Maybe I'm misinterpreting you, but could you explain how any non-symmetric equation can possibly be true in all models of arithmetic?
The purpose of the article is only to describe some subjective experiences that would cause you to conclude that SS0+SS0 = SSS0 is true in all models of arithmetic. But Eliezer can only describe certain properties that those subjective experiences would have. He can't make you have the experiences themselves.
So, for example, he could say that one such experience would conform to the following description: "You count up all the S's on one side of the equation, and you count up all the S's on the other side of the equation, and you find yourself getting the same answer again and again. You show the equation to other people, and they get the same answer again and again. You build a computer from scratch to count the S's on both sides, and it says that there are the same number again and again."
Such a description gives some features of an experience. The description provides a test that you could apply to any given experience and answer the question "Does this experience satisfy this description or not?" But the description is not like one in a novel, which, ideally, would induce you to have the experience, at least in your imagination. That is a separate and additional task beyond what this post set out to accomplish.
You're over-thinking this. Take a look at this real-world example of a "neurological fault":
Now I knew where I was. Soon I would come to interchange 27 with its two ramps, A and B. B led away from my destination and A directly into it. It had always struck me as strange that one reached 27B before 27A. I recalled drawing that on a map to give to someone who was going to visit me. My breathing has returned to normal and my panic had disappeared. I come up to the first sign for the interchange.
"27A"
I could hardly breathe. That was not possible. 27A was after 27B. I knew that. I considered for a moment the possibility that on the previous night, shortly after I drove on this very highway, construction workers had descended en masse on the interchanges and somehow moved them. That seemed far more possible than that my clear (and detailed) memory could be so wrong. 27A looked exactly as I remembered, except that now I could see 27B clearly in the distance and in the past I had to turn my head to see it.
I exited on the ramp that I knew wasn’t there twenty-four hours previously to find myself on a well-remembered road. And soon I was home.
Now imagine that happenin...
If there are any Christians in the audience who know Bayes's Theorem (no numerophobes, please) might I inquire of you what situation would convince you of the truth of Islam?
Why does this need to go out to Christians? I suspect that most, if not all, people reading this are non-Muslims who know Bayes's Theorem. What would convince you of the truth of Islam?
If I find the text of Moby Dick suitably encoded (whatever that means) into the foundation of a building, and I don't find other texts encoded into that building, it seems reasonable to take seriously the theory that there exists some process or entity which has a special relationship both with that building, and with the text of Moby Dick, different from the relationship it has with any other text.
If I find the text of the Koran suitably encoded into the fundamental constants of the universe, it seems equally reasonable to take seriously the theory that there exists some process or entity which has a special relationship both with that universe, and with the text of the Koran, different from the relationship it has with any other text.
You're right, of course, that it doesn't follow from that that either the Koran or Moby Dick is true. Neither does it follow from the truth of the Koran (whatever that means) that Islam is true (whatever that means).
OTOH, converting to a belief in Islam on the basis of that evidence seems more justified than remaining indifferent to Islam in the face of that evidence.
Of course, those aren't the only options.
Granted, it's not really clear to me what is a reasonable response to that evidence. "Investigate the Koran," of course, but I have no sense of what such an investigation might even look like.
I don't quite get what happens. Does imagining two and two together give same mental image as imagining two and one together? Does putting two and two earplugs together give same result as putting two and one earplug together? If it does, then I take 4 earplugs, put two and one together and put other into my ear, then two and one are same as two and two together, so I should be able to separate it into two and two, and and then I have two earplugs on my hands, two in a box, and one in my ear. I do it the second time and I can't hear anything, but I have al...
I just operate under the assumption that I will never actually encounter a situation where 2+2 does not equal 4, and therefore do not spend time worrying about such a hypothetical situation. This assumption has never failed me before.
I understand the point Eliezer's trying to make here. However, you (whoever's reading this) could not convince me that ss0 + ss0 =sss0 in Peano arithmetic (I define the scenario in which my mind is directly manipulated so that I happen to believe this not to constitute "convincing me"). Here's why I believe this position to be rational:
A)In order for me to make this argument, I have to presume communication of it. It's not that I believe the probability of that communication to be 1. Certainly many people might read this comment and not know ...
Wouldn't such an occurrence involve an overhaul of the world on part of some Force/Entity? And why would you, and only you, be able to note that something changed, i.e. that you believed 2+2=4 and now you no longer don't? Much more importantly, since you use it as an example, Winston would not bother to write about 2+2=3, he would probably actually write about 2+2=4, or even 5, thus shaking your world even further...
Hello, I'm a Christian. And, yes, I'm also a rationalist gasp!. I was born and raised a Christian, and I honestly am not sure if I would believe, say, Budhism if I was raised that way- My gut answer is 'No', but I cannot really be sure, as I would be a completely different person. There's no way no one can truthfuly say yes or no for sure to that question.
Right, anyways, I do have reasons I would stop believing... There are a couple very specific situations that pop to mind in which I would be convinced that my whole life has been a lie:
I see a lot of people arguing that "2", "3", "+", and "=" are defined in terms of the Peano axioms, and as such, aren't actually relevant to the behavior of physical objects. They say that the axioms pin down the numbers, regardless of how physical objects behave or start behaving.
But the Peano axioms use something called a "successor" to generate the natural numbers. And how do we figure out what the successors are? Well, one notation is to append an "S" to the previous number to indicate that nu...
I’m an evangelical protestant and I’d like to give my answer to the ‘what would it take to convince me to become a Muslim’ question. This is going to be a narrative example and thus show only one of many possible routes. I’ve chosen a rout that does not depend on private knowledge, fresh miracle in the present day, or even or even changed facts in things it would be inconceivable for me to be wrong about, because I see this rout as the hardest and therefore most revealing.
Muslim scholars propose a competitor to the Documentary Hypothesis (JEPD) for the Pen...
In discussing Newcomb's problem, Eliezer at one point stated, "Be careful of this sort of argument, any time you find yourself defining the "winner" as someone other than the agent who is currently smiling from on top of a giant heap of utility."
This aligns well with a New Testament statement from Jesus, "Ye shall know them by their fruits...every good tree bringeth forth good fruit, but a corrupt tree bringeth forth evil fruit."
So, I'm only a novice of the Bayesian Conspiracy, but I can calculate the breast cancer percentages...
Hi, I am a mathematician and I guess most mathematicians would not agree with this. I am quite new here and I am looking forward to reactions of rationalists :-)
I, personally, distinguish "real world" and "mathematical world". In real world, I could be persuaded that 2+2=3 by experience. There is no way to persuade me that 2+2=3 in mathematical world unless somebody shows me a proof of it. But I already have a proof of 2+2=4, so it would lead into great reform of mathematics, similar to the reform after Russel paradox. Just empirical ex...
But how does not this story about 2+2=3 apply too to the belief in god for example? If you are raised in the right circumstances, you will end up with this belief you think its unconditional, even though it was conditonal on your circumstances. Arent ultimately all believes entangled with reality by virtue of believes being encoded in the brain which is a physical system entangled with reality? to not fall in a fallacy of gray, we can conceede that some ways of entanglement are better than others, in that they lead to mora accurate believes. Hmmm
In any cas...
This is my first time reading through these works, though I must say, I smell False Equivalence. 2+2=4 is not just something I have learned, but something I have understood. I was not "taught" this, I was "shown" this. I came to the comprehension on my own. I was a horse led to water, and upon seeing the truth therein, I drank.
In “What is Evidence?” I wrote:1
Cihan Baran replied:2
I admit, I cannot conceive of a “situation” that would make 2 + 2 = 4 false. (There are redefinitions, but those are not “situations,” and then you’re no longer talking about 2, 4, =, or +.) But that doesn’t make my belief unconditional. I find it quite easy to imagine a situation which would convince me that 2 + 2 = 3.
Suppose I got up one morning, and took out two earplugs, and set them down next to two other earplugs on my nighttable, and noticed that there were now three earplugs, without any earplugs having appeared or disappeared—in contrast to my stored memory that 2 + 2 was supposed to equal 4. Moreover, when I visualized the process in my own mind, it seemed that making xx and xx come out to xxxx required an extra x to appear from nowhere, and was, moreover, inconsistent with other arithmetic I visualized, since subtracting xx from xxx left xx, but subtracting xx from xxxx left xxx. This would conflict with my stored memory that 3 - 2 = 1, but memory would be absurd in the face of physical and mental confirmation that xxx - xx = xx.
I would also check a pocket calculator, Google, and perhaps my copy of 1984 where Winston writes that “Freedom is the freedom to say two plus two equals three.” All of these would naturally show that the rest of the world agreed with my current visualization, and disagreed with my memory, that 2 + 2 = 3.
How could I possibly have ever been so deluded as to believe that 2 + 2 = 4? Two explanations would come to mind: First, a neurological fault (possibly caused by a sneeze) had made all the additive sums in my stored memory go up by one. Second, someone was messing with me, by hypnosis or by my being a computer simulation. In the second case, I would think it more likely that they had messed with my arithmetic recall than that 2 + 2 actually equalled 4. Neither of these plausible-sounding explanations would prevent me from noticing that I was very, very, very confused.3
What would convince me that 2 + 2 = 3, in other words, is exactly the same kind of evidence that currently convinces me that 2 + 2 = 4: The evidential crossfire of physical observation, mental visualization, and social agreement.
There was a time when I had no idea that 2 + 2 = 4. I did not arrive at this new belief by random processes—then there would have been no particular reason for my brain to end up storing “2 + 2 = 4” instead of “2 + 2 = 7.” The fact that my brain stores an answer surprisingly similar to what happens when I lay down two earplugs alongside two earplugs, calls forth an explanation of what entanglement produces this strange mirroring of mind and reality.
There’s really only two possibilities, for a belief of fact—either the belief got there via a mind-reality entangling process, or not. If not, the belief can’t be correct except by coincidence. For beliefs with the slightest shred of internal complexity (requiring a computer program of more than 10 bits to simulate), the space of possibilities is large enough that coincidence vanishes.4
Unconditional facts are not the same as unconditional beliefs. If entangled evidence convinces me that a fact is unconditional, this doesn’t mean I always believed in the fact without need of entangled evidence.
I believe that 2 + 2 = 4, and I find it quite easy to conceive of a situation which would convince me that 2 + 2 = 3. Namely, the same sort of situation that currently convinces me that 2 + 2 = 4. Thus I do not fear that I am a victim of blind faith.5
1See Map and Territory.
2Comment: http://lesswrong.com/lw/jl/what_is_evidence/f7h.
3See “Your Strength as a Rationalist” in Map and Territory.
4For more on belief formation and beliefs of fact, see “Feeling Rational” and “What Is Evidence?” in Map and Territory. For more on belief complexity, see “Occam’s Razor” in the same volume.
5If there are any Christians reading this who know Bayes’s Theorem, might I inquire of you what situation would convince you of the truth of Islam? Presumably it would be the same sort of situation causally responsible for producing your current belief in Christianity: We would push you screaming out of the uterus of a Muslim woman, and have you raised by Muslim parents who continually told you that it is good to believe unconditionally in Islam.
Or is there more to it than that? If so, what situation would convince you of Islam, or at least, non-Christianity? And how confident are you that the general kinds of evidence and reasoning you appeal to would have been enough to dissuade you of your religion if you had been raised a Muslim?