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September 27, 2007

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

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

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 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.

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.

"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 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.)

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 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.

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

There are many people who convert from one religion to another, thus providing demonstrable evidence that the beliefs can be swayed given an appropriate change in utility function. Perhaps it for marriage or other social interactions, perhaps because a different set of beliefs carries more resonance ethically or logically.

As to the "monstrously insane and vicious God" you associate with Christianity, you (along with a good many Christians) are confusing Him with the Judaic God of the Old Testament. 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.

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 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."

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.

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 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.

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.)

One should be careful with context here...

I'm entirely justified in claiming that 2 + 2 = 1, or 2 + 2 = 0 in some situations, and such a claim is quite mathematically sound.

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.

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".

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 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.

I concede (a little)!

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

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

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

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

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

There are really two questions in there:


  • Whether the Peano arithmetic axioms correctly describe the physical world.

  • Whether, given those axioms and appropriate definitions of 2 and 4 (perhaps as Church numerals), 2 + 2 = 4.

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

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

Can we even meaningfully ask the second question? Kind of. As 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.

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?

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.

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.

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

For Instance:

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

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

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

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

Therefore, you have 2 + 2 = 3.

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

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

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

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

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

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

"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 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

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.

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.

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