Things a Perfect logician would Discover

[krogoth]

I don't know if this should be it's own thread. I was going to comment in the serious business topic about rational atheism but I feel this it too far off-topic.
I'm also not sure If this should just be in the science or maybe even the serious business forum.

But starting with basic's, A perfect logician born with no memories of any history but of the basic's of math, and he is of a nature that he wants to find and list/describe all physical laws mathematically, or such that he can develop math, and create any devices needed given to measure his world however he wish's. Also given an arbitrarily long time.
(I hope I've worded this accurately)

Would is be rational he would discover and investigate gravity? friction? speed of light?

I came to this from the "is atheism is rational?" and it's side "is religion rational?" and I would have answered no to the second because I don't see any reason a perfect logician should come up with religion.

What things might or might not a perfect logician or group of perfect logicians discover?

[Proginoskes]

I would hope they'd discover how to use apostrophes correctly ...

[WarDaft]

Given an unbounded amount of time to interact with the universe, one would expect that they would tend towards (as the time increased) the exact definition of that universe. Note that even an utterly chaotic universe still has an exact definition, it's just more akin to an NDTM than a TM.

[mfb]

My ideads for this topic: He will try to find out how he was created, as he will consider his as part of the universe to explore. If there is a reasonable explanation (like: "he had humans as parents", "humans evolved on this planet"), that is fine. But if he is the only human on a planet, this might be tricky. He might come to the conclusion that he has no good explanation (read: much better than anything else).
But I am certain that he will not develop some creation myth including turtles, snakes, or gods. In addition, his analysis of the human psychology might reveal why so many humans believe in one ore more gods.

He would come up with concepts similar to gravity, friction, speed of light, ....
Maybe with a framework which is a bit different, but I think it would be close enough to recognize it. If he finds a theory which is more fundamental than our Standard Model, it should be possible to derive that as low-energy approximation.

[Charlie!]

Facts. They would discover facts. It might also be helpful to think of the things they would not discover, which would fall under the category not facts.

[Quizatzhaderac]

1) This rationalist is human (I assume)
2) They have "no memories of any history"
3) They have "the basic's of math"
4) They have "a nature that he wants to find and list/describe all physical laws mathematically..."

Starting with 1, since a human being isn't naturally a rationalist the rules for rationality have to be expressed in terms of the basic mental processed humans actually have.
Number 4 is a complex aesthetic even taking curiosity as a given.
Now, number 2: This person may have no explicit memory of history, but this person has significant history and culture already by virtue and 1,3 and 4.

As for your actual questions:
"Would is be rational he would discover and investigate gravity? friction? speed of light?" By giving this person number 4 and implicitly concepts of "physical laws" and mathematics they're pretty much born running in that direction. So yes, it seems obvious they'd do it.

"is atheism is rational?" Given that this is a platonic rationalist created as an adult with a complete culture and mind? No, they'd conclude they were created by a god in order to produce an interesting xkcd form conversation. So much of our culture and rationality comes from religion it would be difficult to describe a condition where rationality emerged before religion.

Assuming we did find a situation resembling reality that would produce a rationalist first: The question "is religion rational?" is a rather vague since the word "religion" is rather vague. So I'm going to split it into three questions.

Anthropologically: Would a group worthy of a census category under "religion" be created? With enough, such similar rationalists, I'd say yes. They'd have similar word views, priors, metaphysics, and (potentially) ethics to justify a social and ideological grouping. More so than say, Hinduism.

Theologically: Would they be theists? depends how strict a definition you use. Omniscient, omnipotent, omnibenevolent, white guy with a beard? No, people who believe that scare even priests.Omniscient, omnipotent and omnibenevolent of any shape? That's a specific (if popular) interpretation of Abrahamic religions, so I'd say no. It also either assumes God and humanity share the same sense of justice/good, or placing their ethics below God's without changing theirs to match, something I'm assume a perfect rationalist would either succeeded in or not try at all. The full range of what people call "God" is so broad I won't guess what they would believe beyond that it'd probably be original, nonanthropomorphic, and get put in the "God" bin by regular people.

Behaviorally: Would they worship? Ceremonies, prayer, unneededly complicated customs? Yes, that's not a cultural thing, that's human nature. The cultural aspect is how we do so. Though I'd assume they do it without expecting sub-vocalizations to heal strangers after carefully removing all other forms of casual interaction.

[flicky1991]

He would have to assume that something put him in this world, even if it's not a god. Or maybe whatever he comes up with would be equivalent to a god, from his point of view. It's hard for anyone to tell because we've been raised in... well, in this world, which is full of history and culture, and it's hard to imagine living in any other kind of place.

[Charlie!]

He would have to assume that something put him in this world, even if it's not a god. Or maybe whatever he comes up with would be equivalent to a god, from his point of view. It's hard for anyone to tell because we've been raised in... well, in this world, which is full of history and culture, and it's hard to imagine living in any other kind of place.

Are you familiar with Boltzmann's multiverse?

[flicky1991]

He would have to assume that something put him in this world, even if it's not a god. Or maybe whatever he comes up with would be equivalent to a god, from his point of view. It's hard for anyone to tell because we've been raised in... well, in this world, which is full of history and culture, and it's hard to imagine living in any other kind of place.

Are you familiar with Boltzmann's multiverse?

No. What exactly is it about?

[PM 2Ring]

From Boltzmann’s Anthropic Brain, by Sean Carroll.

Faced with the deep puzzle of why the early universe had a low entropy, Boltzmann hit on the bright idea of taking advantage of the statistical nature of the Second Law. Instead of a box of gas, think of the whole universe. Imagine that it is in thermal equilibrium, the state in which the entropy is as large as possible. By construction the entropy can’t possibly increase, but it will tend to fluctuate, every so often diminishing just a bit and then returning to its maximum. We can even calculate how likely the fluctuations are; larger downward fluctuations of the entropy are much (exponentially) less likely than smaller ones. But eventually every kind of fluctuation will happen.



You can see where this is going: maybe our universe is in the midst of a fluctuation away from its typical state of equilibrium. The low entropy of the early universe, in other words, might just be a statistical accident, the kind of thing that happens every now and then. On the diagram, we are imagining that we live either at point A or point B, in the midst of the entropy evolving between a small value and its maximum. It’s worth emphasizing that A and B are utterly indistinguishable. People living in A would call the direction to the left on the diagram “the past,” since that’s the region of lower entropy; people living at B, meanwhile, would call the direction to the right “the past.”

During the overwhelming majority of such a universe’s history, there is no entropy gradient at all — everything just sits there in a tranquil equilibrium. So why should we find ourselves living in those extremely rare bits where things are evolving through a fluctuation? The same reason why we find ourselves living in a relatively pleasant planetary atmosphere, rather than the forbiddingly dilute cold of intergalactic space, even though there’s much more of the latter than the former — because that’s where we can live. Here Boltzmann makes an unambiguously anthropic move. There exists, he posits, a much bigger universe than we can see; a multiverse, if you will, although it extends through time rather than in pockets scattered through space. Much of that universe is inhospitable to life, in a very basic way that doesn’t depend on the neutron-proton mass difference or other minutiae of particle physics. Nothing worthy of being called “life” can possibly exist in thermal equilibrium, where conditions are thoroughly static and boring. Life requires motion and evolution, riding the wave of increasing entropy. But, Boltzmann reasons, because of occasional fluctuations there will always be some points in time where the entropy is temporarily evolving (there is an entropy gradient), allowing for the existence of life — we can live there, and that’s what matters.

Here is where, like it or not, we have to think carefully about what anthropic reasoning can and cannot buy us. On the one hand, Boltzmann’s fluctuations of entropy around equilibrium allow for the existence of dynamical regions, where the entropy is (just by chance) in the midst of evolving to or from a low-entropy minimum. And we could certainly live in one of those regions — nothing problematic about that.

[...]

But, having taken a bite of the apple, we have no choice but to swallow. If the only thing that one’s multiverse does is to allow for regions that resemble our observed universe, we haven’t accomplished anything; it would have been just as sensible to simply posit that our universe looks the way it does, and that’s the end of it. We haven’t truly explained any of the features we observed, simply provided a context in which they can exist; but it would have been just as acceptable to say “that’s the way it is” and stop there. If the anthropic move is to be meaningful, we have to go further, and explain why within this ensemble it makes sense to observe the conditions we do. In other words, we have to make some conditional predictions: given that our observable universe exhibits property X (like “substantial entropy gradient”), what other properties Y should we expect to measure, given the characteristics of the ensemble as a whole?

And this is where Boltzmann’s program crashes and burns. (In a way that is ominous for similar attempts to understand the cosmological constant, but that’s for another day.) Let’s posit that the universe is typically in thermal equilibrium, with occasional fluctuations down to low-entropy states, and that we live in the midst of one of those fluctuations because that’s the only place hospitable to life. What follows?

The most basic problem has been colorfully labeled “Boltzmann’s Brain” by Albrecht and Sorbo. Remember that the low-entropy fluctuations we are talking about are incredibly rare, and the lower the entropy goes, the rarer they are. If it almost never happens that the air molecules in a room all randomly zip to one half, it is just as unlikely (although still inevitable, given enough time) that, given that they did end up in half, they will continue on to collect in one quarter of the room. On the diagram above, points like C are overwhelmingly more common than points like A or B. So if we are explaining our low-entropy universe by appealing to the anthropic criterion that it must be possible for intelligent life to exist, quite a strong prediction follows: we should find ourselves in the minimum possible entropy fluctuation consistent with life’s existence.

And that minimum fluctuation would be “Boltzmann’s Brain.” Out of the background thermal equilibrium, a fluctuation randomly appears that collects some degrees of freedom into the form of a conscious brain, with just enough sensory apparatus to look around and say “Hey! I exist!”, before dissolving back into the equilibrated ooze.

You might object that such a fluctuation is very rare, and indeed it is. But so would be a fluctuation into our whole universe — in fact, quite a bit more rare. The momentary decrease in entropy required to produce such a brain is fantastically less than that required to make our whole universe. Within the infinite ensemble envisioned by Boltzmann, the overwhelming majority of brains will find themselves disembodied and alone, not happily ensconsed in a warm and welcoming universe filled with other souls. (You know, like ours.)

[Xanthir]

I think the last paragraph doesn't make it very clear - it makes it seem like most brains will know that they're all alone, and so we can tell that we're obviously not one of them.

Properly, though, a brain pre-loaded with patterns that give it memories of a life in an orderly universe is also ridiculously more likely than actually having an orderly universe. This is Last Thursdayism writ large. (By the way, welcome to the universe, created just this morning with all the appearances of having existed for 15 billion years!)

[eSOANEM]

Properly, though, a brain pre-loaded with patterns that give it memories of a life in an orderly universe is also ridiculously more likely than actually having an orderly universe. This is Last Thursdayism writ large. (By the way, welcome to the universe, created just this morning with all the appearances of having existed for 15 billion years!)

The thing is, last thursdayism is a prediction of statistical mechanics and therefore classical thermodynamics. All the laws of physics leading to statistical mechanics are reversible which means that, when you run the laws of physics for a large body of particles backwards, you find that, in almost all cases, the entropy increases. Statistical mechanics says that any observation made is almost certainly made at the bottom of a fluctuation away from thermal equilibrium with the entropy increasing as you move away from it into the past or future.

Of course this runs into problems because Boltzmann's multiverse also suggests that I am almost certainly a lone Boltzmann brain and that the history and observations I make of a low entropy universe cannot be trusted which would render science (and hence the arguments in favour of Boltzmann's multiverse) a futile exercise.

[idobox]

Even if Boltzmann brains are much more frequent than large low-entropy universes, in a infinite multiverse, there must an infinity of large universes with low entropy.

And of course, the argument is built on the assumption a large low-entropy universe is much less likely than a Boltzmann brain. Large scale variations of entropy in the multiverse might be explained by some yet undiscovered theory.
As a parallel, we can look at waves in the sea. If you take a very tank of water in the world of spherical cows in vacuum, and do some statistical thermodynamics, you will find it absurdly improbable that waves spontaneously appear. But in the real world, a lot of things disturb the water, and create waves. I'm not suggesting a low-entropy higher level multiverse, just saying that thermodynamics can be insufficient to explain the phenomenon.

[eSOANEM]

Even if Boltzmann brains are much more frequent than large low-entropy universes, in a infinite multiverse, there must an infinity of large universes with low entropy.

And of course, the argument is built on the assumption a large low-entropy universe is much less likely than a Boltzmann brain. Large scale variations of entropy in the multiverse might be explained by some yet undiscovered theory.
As a parallel, we can look at waves in the sea. If you take a very tank of water in the world of spherical cows in vacuum, and do some statistical thermodynamics, you will find it absurdly improbable that waves spontaneously appear. But in the real world, a lot of things disturb the water, and create waves. I'm not suggesting a low-entropy higher level multiverse, just saying that thermodynamics can be insufficient to explain the phenomenon.

The reason large scale fluctuations are less likely than small scale ones is simply a matter of probability. If the pdf of entropy fluctuations was not vastly higher at low fluctuations than at high ones, the entropy of any system you gave me would not be almost constant as we observe, but would instead dance around so wildly it would be difficult if not impossible to determine a value.

So clearly large fluctuations are less likely than smaller ones and so the vast majority of beings capable of observing their own existence should be Boltzmann brains.

This is a problem because it means that the anthropic advantages of Boltzmann's multiverse (which isn't so much a multiverse per se as a single infinite universe with various regions of lower entropy which could be termed a "universe") in explaining why the universe seems to have had a low entropy beginning are lost because it suggests that it is far more likely that we simply find ourselves as a cloud of almost-thermal gas thinking "I exist" than that we find ourselves living alongside 7 billion other sentient beings on a planet orbitting a star in a spiral galaxy part of a 13.2 billion year old universe and arguing about boltzmann brains.

Yet we are clearly in the latter case. So how do we explain that?

Saying, we're in one of the vanishingly improbable fluctuations is not a satisfying or useful answer because it provides no better explanation than a shrug and worse explanation than the usual anthropic argument that life complex enough to do science and cosmology requires that the universe have certain properties.


...


The waves analogy is a complete red-herring as well. We get waves because of the wind. We get wind because the sun does not heat the whole surface of the earth evenly (and topographical effects on how the wind can blow). The sun does not heat the whole surface of the earth evenly because the earth is not a sphere (and the earth is not tidally locked). The earth is not a sphere because ...

If I take a sufficiently large tank of water and leave it in a controlled environment (with the whole thing at a constant temperature and pressure), I won't see any waves, even in the real world.

Waves are not an example of an unlikely thing happening, they are an example of something behaving exactly as you'd expect.

[Xanthir]

This is a problem because it means that the anthropic advantages of Boltzmann's multiverse (which isn't so much a multiverse per se as a single infinite universe with various regions of lower entropy which could be termed a "universe") in explaining why the universe seems to have had a low entropy beginning are lost because it suggests that it is far more likely that we simply find ourselves as a cloud of almost-thermal gas thinking "I exist" than that we find ourselves living alongside 7 billion other sentient beings on a planet orbitting a star in a spiral galaxy part of a 13.2 billion year old universe and arguing about boltzmann brains.

Yet we are clearly in the latter case. So how do we explain that?

Two things.

One, the point expressed here is very important, and deserves an even more direct restarting:

The entire reason for using Boltzmann's universe is because it seems to give us a way to make an anthropic argument for our observed surroundings, while removing the question of why our entropy is so low in the first place. Anthropic arguments let us handwave away the chance of living in a low-probability universe by arguing that our universe is actually *high* probability within the set of universes that we can potentially live in. However, when you follow Boltzmann's arguments to their conclusion, you find that the set of universes that match our experiences is actually *much* larger than you initially think. The vast, vast majority of them are Last Thursdayism-style, where the entire history of the universe is "fake", and everything just happens to be set up such that it looks really old but isn't. This applies both to time and space - nearly all of the universes that match our observations contain a single brain, which came into being whole and complete, with brain structures already set up to contain memories of living in an orderly universe like what we observe.

Thus, the anthropic argument falls apart. The universe we want to see is no longer high-probability in the set of universes that can support us. In fact, it's still staggeringly unlikely, such that the chance of hitting it, or one like it, randomly is effectively nil.

Two, you got the argument wrong in precisely the way I just said is likely. We are not "clearly in the latter case". We have memories of living in a universe that is filled with space, stars, earth, and 7 billion other human beings. The chance that we actually live in such a universe, rather than one of the vastly more likely Boltzmann universes where we're a brain momentarily appearing from the ether with those same memories, is basically zero.

That is why we have to reject Boltzmann's universe. Not because we can tell that we're obviously not in one of the brain universes, but because we can't, and it's much more likely that we're actually in one of them rather than in the big old universe that our senses tell us we're in.

[eSOANEM]

Two, you got the argument wrong in precisely the way I just said is likely. We are not "clearly in the latter case". We have memories of living in a universe that is filled with space, stars, earth, and 7 billion other human beings. The chance that we actually live in such a universe, rather than one of the vastly more likely Boltzmann universes where we're a brain momentarily appearing from the ether with those same memories, is basically zero.

That is why we have to reject Boltzmann's universe. Not because we can tell that we're obviously not in one of the brain universes, but because we can't, and it's much more likely that we're actually in one of them rather than in the big old universe that our senses tell us we're in.

Ah, when I said we weren't in a Boltzmann brain's universe, I meant that we weren't in the simplest Boltzmann brain's universe, one where it has no memories and exists just long enough to observe its existence.

Such a Boltzmann brain would require far less organisation (and so be far higher entropy) than any other Boltzmann brain and so, we'd expect almost all observers to be this brain with then some very small percentage being more complex Boltzmann brains (such as the one you describe) and a vanishingly small percentage genuinely being what we appear to experience.

When I said we were clearly in the latter case, I communicated very poorly. I meant to say that we clearly are not in the simplest Boltzmann brain's universe despite the fact that it should be the most common. This is far more of a problem than Boltzmann predicting a last-Thursdayist false-reality would be because the latter would at least, in some sense, explain our observations; the fact that Boltzmann predicts that our only observation should be of our existence (for the majority (I have no idea how large) of observers) and this is contrary to our experience.

Boltzmann's multiverse fails the primary scientific test in that it's predictions do not match our observations.

Last Thursdayism on the other hand is only un-falsifiable.

[idobox]

The waves analogy is a complete red-herring as well. We get waves because of the wind. We get wind because the sun does not heat the whole surface of the earth evenly (and topographical effects on how the wind can blow). The sun does not heat the whole surface of the earth evenly because the earth is not a sphere (and the earth is not tidally locked). The earth is not a sphere because ...

If I take a sufficiently large tank of water and leave it in a controlled environment (with the whole thing at a constant temperature and pressure), I won't see any waves, even in the real world.

Waves are not an example of an unlikely thing happening, they are an example of something behaving exactly as you'd expect.

My point was that if we didn't know about the wind, the moon, the uneven heating and all that, thermodynamics couldn't explain the waves, the existence of waves through statistical variations of entropy would appear absurdly improbable.
Boltzmann's brain show us that having a very large very low entropy universe cannot be explained by simple statistical variations of entropy. If universes like ours are more frequent than Boltzmann brains, then some other phenomenon must cause large variations of entropy.
We're trying to explain the existence of waves, and brownian motion can't explain it, so we think we found a paradox. But we just haven't discovered the existence of wind.

[eSOANEM]

Boltzmann's brain show us that having a very large very low entropy universe cannot be explained by simple statistical variations of entropy. If universes like ours are more frequent than Boltzmann brains, then some other phenomenon must cause large variations of entropy.

This only follows in a Boltzmann-like cosmology.

In other cosmologies, the low entropy of the early universe was not a fluctuation away from a high entropy state and so there is no large variation to be explained. Instead, it must explain why the entropy started so low.

Trying to find phenomena allowing for large variations of entropy is not a useful endeavour. As I said, if large variations of entropy (sufficient to create our universe) are more common than those to create a Boltzmann brain then we would not expect to be able to measure stable values for the entropy of any system and yet we can.

This tells us that the flaw is not in the exact probabilities of various fluctuations, but the fundamentals of the cosmology and so the solution lies not in a phenomenon explaining large variations in entropy (which are completely unobserved), but rather in a phenomenon explaining low initial entropy.

[WarDaft]

This is rather getting extremely close to another thread. I noticed this when I realized the post I was typing up seemed strangely familiar.

The jist of it is, low entropy actually makes sense if you run a UTM on all random infinite input strings. The likelihood of execution is equivalent to 1/2ⁿ, where n is the length of the specification of the universe. A large initially high entropy universe necessarily has an absurdly large description compared to a tiny low entropy universe that expands according to its own rules and introduces entropy by its own internal laws as it ages. In such a model, the probability of an infinitely large universe with non-zero average entropy density is in fact, zero, despite the fact that there are uncountably many of them. Meanwhile, merely unboundedly large universes with tiny initial conditions and simple rules that permit the development of observers will absolutely dominate anthropically.

[eSOANEM]

Meanwhile, merely unboundedly large universes with tiny initial conditions and simple rules that permit the development of observers will absolutely dominate anthropically.

Surely a lone Boltzmann brain will still have a shorter definition than the universe we perceive and so, should lone Boltzmann brains not still dominate?

[Xanthir]

That's actually an interesting question, which probably requires actual math to answer. The complexity of physics + a singularity is very low compared to the universe that emerges from it. I find it plausible that specifying a brain with memories of the universe might actually be more complex in the TM sense.

[idobox]

Trying to find phenomena allowing for large variations of entropy is not a useful endeavour. As I said, if large variations of entropy (sufficient to create our universe) are more common than those to create a Boltzmann brain then we would not expect to be able to measure stable values for the entropy of any system and yet we can.

I don't understand the theory behind it, but I have read that one of the possible consequences of string theory is the existence of branes, and that the big bang was caused by two branes touching. I don't if this is considered a sensible theory, and if what I describe actually corresponds to that theory, but it helps explaining my point of view.
Let say the branes "float" in a universe with lots of dimension, and the system is close to maximal entropy. The collision of two branes cause a local decrease of entropy, on the scale of the whole universe, it is a small variation of entropy, but it more than enough to explain the big bang. And because collisions have a very high energy, they create big bangs and not Boltzmann brains.
This kind of mechanism could explain that large variations of entropy are more probable than low variations, the way some systems do not follow the Boltzmann energy distribution.

I am not able to describe, or even imagine, an actually scientifically sound example of such a system, but it doesn't mean it doesn't exist.

[eSOANEM]

That's actually an interesting question, which probably requires actual math to answer. The complexity of physics + a singularity is very low compared to the universe that emerges from it. I find it plausible that specifying a brain with memories of the universe might actually be more complex in the TM sense.

This is why I'm not dealing with that case. The case I'm dealing with is a universe consisting solely of the minimal Boltzmann brain (so with no memories, only the capacity to observe its own existence) vs the universe as we perceive it.

In this case, the universe as we perceive it certainly contains far more information (if nothing else because it contains at least one entity capable of observing its existence as well as other less useful information such as the distribution of stars).

As such, it must require a longer set of instructions to generate it/a longer definition simply from an informational point of view and, as such, a UTM-type argument is not sufficient to justify Boltzmann's multiverse.

Trying to find phenomena allowing for large variations of entropy is not a useful endeavour. As I said, if large variations of entropy (sufficient to create our universe) are more common than those to create a Boltzmann brain then we would not expect to be able to measure stable values for the entropy of any system and yet we can.

I don't understand the theory behind it, but I have read that one of the possible consequences of string theory is the existence of branes, and that the big bang was caused by two branes touching. I don't if this is considered a sensible theory, and if what I describe actually corresponds to that theory, but it helps explaining my point of view.
Let say the branes "float" in a universe with lots of dimension, and the system is close to maximal entropy. The collision of two branes cause a local decrease of entropy, on the scale of the whole universe, it is a small variation of entropy, but it more than enough to explain the big bang. And because collisions have a very high energy, they create big bangs and not Boltzmann brains.
This kind of mechanism could explain that large variations of entropy are more probable than low variations, the way some systems do not follow the Boltzmann energy distribution.

Ah, that's fair enough. I forgot to consider a Boltzmann "multi"verse embedded in some sort of multiverse proper. That could explain why large variations in entropy are more common than simple thermodynamics might suggest although I would be astounded if it turned out it suggested they were more common than the small variations sufficient to create a Boltzmann brain.

[gmalivuk]

Getting back to the original question, I think a perfect logician with no memories and no starting assumptions would be incapable of learning anything about the universe other than what sequence of observations it has made thus far. Without any underlying assumptions, it would only be capable of deductive reasoning, rather than inductive or abductive. No matter how many times it observed solar systems, it could not derive laws of gravity and motion because it would have no purely perfectly logical reason to expect those patterns to continue into the future.

[mfb]

The perfect logician could (in addition) be a perfect physicist, and try to predict his future observations. In order to do that, he would have to come up with a theory of gravity (or something similar), as gravity can explain the observations with less input than other theories. There is a PhD thesis with a formal treatment of intelligence as predicting future observations, based on the past. The basic idea is to use the komogorov complexity of the possible worlds (possible = can produce the input) to weight their relative probability. Worlds with gravity would completely dominate the evaluation, and therefore the predictions would be based on worlds with gravity.

[gmalivuk]

Why should the perfect logician care about such aesthetic things as complexity or the ability to make predictions?

[mfb]

As I said, he could be a perfect physicist as well.
And perfectly intelligent, as defined in the reference .

[WarDaft]

Well, if our perfect logician can get hungry, and lives indefinitely, he'll want to understand the universe maximally to ensure he absolutely minimizes his time spent hungry.

[gmalivuk]

So you're adding an elaborate emotional system of values and desires and preferences? I suspect that alone would lessen the perfection of its logic.

But even if it doesn't, even if it is both a perfect logician *and* a being that really strongly desires food, that's not enough to get it to conclusions like "plant A is good to eat", because you still haven't given it any of the assumptions necessary to logically make conclusions about plants of type A in general.

[WarDaft]

It doesn't have to be as complicated as that. We just need something that, for the logician, weights positively the gathering of some resource. They will then, being a perfect logician, gather it ideally. To gather it ideally, they must have the best possible model of the universe.

Unless you're assuming the logician is also incapable of even moving without a priori deducing that they have a musculoskeletal system that permits a variety of actions, in which case, we have a very nearly useless perfect logician. I suppose you could use them as a hat stand maybe, or a conversation starter. "This here's the smartest person in the world. Smarter than anyone who ever has, or ever will, live. Unfortunately they don't do much because they haven't been able to logically deduce the ability to walk yet. But as you can see, very useful for keeping your hat warm."

[gmalivuk]

It doesn't have to be as complicated as that. We just need something that, for the logician, weights positively the gathering of some resource. They will then, being a perfect logician, gather it ideally. To gather it ideally, they must have the best possible model of the universe.

An efficient model of the universe requires assumptions that are not logically deducible from observation, such as "yes, it does that every time". All the perfect logician could conclude, if it doesn't start with any prior assumptions, is that "yes, it has done that every time so far".

[Quizatzhaderac]

Are you familiar with Boltzmann's multiverse?

Actual physics aside, that doesn't seem all that useful to the current discussion. We don't really know how the most probable "perfectly logician" would work in that situation, or what would be available to be observed.

Also if the logician is in a different enough universe, any conclusions we draw might not be applicable to our universe. For instance if it's actions don't have consequences, there's no reason to make anything but the most aesthetically pleasing narrative for reality, as true beliefs would have no utility.

[douglasm]

An efficient model of the universe requires assumptions that are not logically deducible from observation, such as "yes, it does that every time". All the perfect logician could conclude, if it doesn't start with any prior assumptions, is that "yes, it has done that every time so far".

You are correct about the perfect logician not deducing "it does that every time" facts, but that goes a lot less far than you seem to think. Deducing a high statistical confidence level that it does that every time is well within the domain of perfect logic and has near equal utility in designing efficient models.

It looks like we need to give our perfect logician a goal designed to encourage the creation of universe models. I suggest something along the lines of designing a universe-simulating Turing machine while minimizing its complexity (there are plenty of quantifiable traits of Turing machines we could use to construct a measure of this) and maximizing the statistical confidence of its accuracy.

[gmalivuk]

Deducing a high statistical confidence level that it does that every time is well within the domain of perfect logic

Describe how statistical confidence would be deduced given no prior assumptions about how the world works.

[douglasm]

Statistical confidence is a purely mathematical concept. The OP explicitly states that the perfect logician is created with knowledge of math.

[gmalivuk]

I know confidence intervals are mathematical. It's just that they don't mean what people often seem to think they mean.

Suppose that our logician has pushed a certain button 100 times over the past couple weeks, and every time a food pellet has been dispensed one minute later. The button is set in an apparently infinite wall, so the logician literally has no idea what might be going on behind it.

How confident should the logician be that every button press will lead to a food pellet?

[douglasm]

That every button press will lead to a food pellet? 0% confidence because experimental statistics can never pin down exact probabilities.

That pushing the button leads to a pellet with at least 99% probability? It's been 10 years since I studied confidence intervals in class so I could easily be remembering the math wrong, but I think it's 63.4% confidence. By the same (possibly incorrect) math, he should be 99.6% confident that pushing the button produces a pellet with at least 95% probability.

The only difference in this kind of thing between the perfect logician and real world scientists is that real world scientists tend to drop the "almost" from "almost 100% confidence" and collapse especially narrow value ranges into exact values if one exists in the range that has an inherent unique aspect.

[gmalivuk]

Why would a perfect logician reject the hypothesis that the button only results in food the first 100 times it's pressed?

More generally, why would it presuppose the hypothesis that there is some fixed probability p that the button results in food? And even under that hypothesis, why is it rejecting the hypothesis that, before the first button press, p was chosen randomly to be either 50% or 100%, with for example 2¹⁰⁰:1 odds against its being 100%? (Under that hypothesis, the odds of it being 100%, given that there's been food 100 out of 100 button presses, are 50:50.)

[mfb]

Why would a perfect logician reject the hypothesis that the button only results in food the first 100 times it's pressed?

He would not reject it. But he could assign a low probability to this, using the approach to look for the minimal turing machine (probably with a source of randomness) with this output.

[douglasm]

I think all we've really demonstrated these last few posts is that the concept of a perfect logician is not well defined in the context of problems that are not simple progressions of absolute proofs from known facts. To make a functional definition that serves the intended purpose of the OP, I think we need to include some formulation of Occam's Razor in our perfect logician's reasoning methods.