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byrnema comments on The mathematical universe: the map that is the territory - Less Wrong
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The simulation wouldn't know what 'cousin_it' is or what a pheasant is. These are higher level things that evolve from the lower level rules. (They don't exist independently, and don't really exist except as categories in our mind.) All of the rules in the universe have to be in terms of the lowest level things because reductively, these are the things that are real. So the universe wouldn't be able to say "cousin_it morphs into a pheasant on turn N", it would only be able to say that "things like cousin_it morph into pheasants under these conditions", in which case we would discover that the rule happens reliably and predictably, not arbitrarily.
Another way of expressing this is that a universe which has our laws of physics is a shorter encoding than a universe that has our laws of physics, plus a detailed exception for "cousin_it turns into a pheasant." This makes it more likely, but, the set of all universes with rules much more complicated than ours which would still allow conscious observers makes it unlikely that we made it into such a simple universe. This has been covered on lesswrong and overcomingbias before.
I disagree that you're responding to my argument. I'm not making an argument about whether the universe is simple or not: I'm making the argument that if the universe has an encoding, "cousin_it turns into a pheasant", it's not going to be an exception. If the universe has that encoding, we would find that cousin_it turns into a pheasant, and upon further study, would find this was predicted all along by the lower level rules. Simply because nothing exists beyond the lower level rules. We can't expect inconsistencies at higher levels because the higher levels are just derivative of the lower ones.
What do you mean? There's no law of nature saying laws of nature must be low-level in all possible worlds - that's just an observation about our world. I can write a simulator that runs the low-level rules as you'd expect, but also it's preprogrammed to search for cousin_it in the simulation at a certain moment, turn him into a pheasant, then resume business as usual. If all simulated worlds "exist", this one "exists" too.
On the one hand, it a matter of the definition of 'low-level': all laws of nature must be low-level or derived from a low-level law because if you had a law that wasn't derived from a lower-level law, that would make it low-level.
Yes, I agree and I'm generally interested in this case. But as I explained, this is only possible in a subset of a universe. The whole universe could not be such a simulation, because there would be nothing 'outside it' to swoop down and make the arbitrary change.
The hypothesis says that all universes that can be simulated by computer programs exist. It doesn't restrict those computer programs by saying they must use "only local laws", or "can't swoop down", or whatever. What does this even mean? Moving the universe one step forward in time according to the Schroedinger equation qualifies as "swooping down" just as much as turning me into a pheasant, they're both things that the program just does.
I assume you're thinking of some other hypothesis, like "all universes that exist must start from a simple core and proceed logically from there", but unfortunately no one has a "logicalness predicate" that would say whether a given program simulates a "logical" universe without "swooping down". In fact, looking at a program you may not even tell if it's "simulating" any universe at all, as opposed to moving some bytes around in weird patterns.
Perhaps it is a matter of relying on different analogies for our intuitions. I was thinking of each possible universe as being identified with a single self-consistent mathematical structure. In which case, I expect the universe to be organized and coherent because any set of mutually consistent initial facts would generate a universe with structure rather than one with haphazard and inexplicable events.
I hadn't thought of all possible computer programs each mapping to a universe ... what is the truth value of a random string of instructions? But since I don't know much about computer science, I don't expect that to work as an intuition pump for me.
Instead I was thinking along the lines of there being a universe -- say -- for each exotic algebra there might be and all the facts that are derived from it. An "algebra-generated-universe" wouldn't say something random or arbitrary; instead everything about the universe would be derivable from a few facts. (Note you couldn't have too many facts, or they'd self-contradict.)
I can remove contradictions by making up a new rule that specifies which rules take precedence over other ones.
Let's build a simpler intuition pump: instead of universes, we'll talk about infinite sequences of integers that can be specified by finite sets of rules. For example, (1, 1, 1, ...) is such a sequence. (1, 2, 3, 4, 5, ...) is another. Those look regular all right. But this sequence: (1, 2, 3, 4, ..., 999999, 1000000, 12345, 1000001, 1000002, ...) can also be specified by a finite and self-consistent set of rules, even though something seems to have "swooped down" and changed it in one place. There's no hard difference between "regular-looking" and "irregular-looking" sequences. All finite sets of rules have equal footing.
Does this make sense to you? Now imagine those integers encoding the time evolution of your toy universe...
It seems to me we're conflating 'possible as an output' and 'true' but since I don't really know what 'true' means in this context, let's conflate them.
The fact that you've written the sequence (1, 2, 3, 4, ..., 999999, 1000000, 12345, 1000001, 1000002, ...) is evidence that it's the possible output of an algorithm. (Indeed, it was the output of an algorithm you ran.) However, this means that the output was possible (and true) for a subset of the universe. How do you know this rule could be universally true?
I say that it could not be universally true, because it has this property of arbitrariness. I think to answer, 'what could possibly be universally true?', you would have to answer the question, 'what can be deduced as true from nothing?' or at best, 'what can be deduced as possibly true from nothing?' From nothing, the universe might deduce the natural numbers. By definition of what the natural numbers are they have an ordering 1, 2, 3, ..., n, n+1, ... This ordering really could not be different.
Suppose that the universe had a way of "knowing about" a single element '12345' that it places in a new position between 1000000 and 1000001. Simultaneously, it could have placed this element in any position, so universally, you would get a much, much larger structure in which (1, 2, 3, 4, ..., 999999, 1000000, 12345, 1000001, 1000002, ...) was only a tiny subset.
My point, whether I can figure out how to make it or not, is that the universe doesn't 'know about' 12345, it only knows about all the numbers and evolves this structure universally. You can look at particular components of the structure and observe that individual components seem arbitrary, but the entire structure cannot be.
Imagine a "universe" that consists of all streams of natural numbers that can be specified by algorithms. Is that fundamental and non-arbitrary enough for you? This universe contains many "sub-universes" that cannot communicate, so we can call them "universes" in their own right. One of them is my 12345 sequence, and many others have me spontaneously turn into a pheasant a week from now.
All that means is that 'cousin_it turns into a pheasant' has to be taken as an axiom for the algebra you're using...
Higher-level behavior can be explicitly coded into lower level rules.
The only way "2+2=4" can exist is if there are first two existent objects and then a mind to come up with the construct describing their addition. "2+2=4" doesn't exist on its own.
My own view for why there is "something" rather than "nothing" is:
A. "Something" has always been here. B. "Something" hasn't always been here.
Choice A is possible but doesn't offer much explanatory power so it won't be
pursued here.
Going with choice B, if "something" hasn't always been here, then "nothing" must have beeen here before it. By "nothing", I mean complete non-existence which would be the lack of all volume, matter, energy, ideas/concepts, etc. However, in "nothing", there is no mechanism to change this "nothing" into "something". So, if "something" is here now, the only possible way is if "nothing" and "something" are one and the same thing. I think this is logically required if we go with choice B.
If it's logically required that "nothing" and "something" are the same thing, the next step is to try and figure out how this can be since they seem different. My view on how this can be is that they only seem different because we're looking at them from two different perspectives. In thinking about "nothingness", we use our mind, which exists. Next to something that exists, "nothing" just looks like nothing. But, in true "nothing", there would be no minds there, and only then would "nothing" be completely self-defining (it says exactly what is there) and therefore existent.
An idea that's helpful in thinking about this topic is that the mind's conception of something ("nothing" in this case) and the thing itself are different. Thanks for listening!
Hi! You seem to be asserting anti-realism and trying to arrive at a correct ontology using an Aristotelian application of deductive logic. Would you like some help with that?
Max Tegmark, the physicist who proposed the mathematical multiverse theory, was aware of the anti-realist position. However, there's good evidence that minds are made out of math, instead of the contrary position. It's a fairly mature debate, and it pays to be aware of the strongest arguments both sides.
This awareness also applies to the universe's beginnings, or lack thereof. Historically, deductive logic has had some problems locating true beliefs.
Also, welcome to Lesswrong! Feel free to post on the introduction thread; and start working your way through the sequences so you understand where other people here are coming from.
Wow! Are you a clippy too? Want to reconcile knowledge and mutually satisfice values?
Kharfa,
o You seem to be asserting anti-realism and trying to arrive at a correct ontology using an Aristotelian application of deductive logic. Would you like some help with that?
I'm not denying the reality of anything that you can show me. Please show me where "2+2=4" is or where it exists. Using that type of argument that things like this exist is like saying Santa Claus exists. That's possible, but we can't prove it or disprove it, and you can't show him to me. There's no point in discussing it. And, by the way, I don't need any help with that. Patronizing attitudes especially when not backed up by sound reasoning are of no interest to me.
o there's good evidence that minds are made out of math, instead of the contrary position.
If I take two rocks, and put them in a cup where you can't see, and then put two more rocks in the cup, and then rattle it around and dump out the contents of the cup on the table, just look at it: two plus two equals four inside the cup.
If I flip this switch on the left, two lights come on, and then if I flip this other switch the other half come on, and then there are four lights.
I could list more examples, and you should have no trouble verifying them experimentally. Were you maybe expecting a stone tablet somewhere, that, if modified, would cause plusOf(2,2) to output five?
That doesn't prove that 2+2=4. That proves that rocks obey a regularity that is concisely describable by reference to an axiom set under which 2+2=4. That's not the same thing. Math still "exists" only as a (human) representation of other real things.
(I will note in passing that Steven Landsburg, who promotes the "Math exists independently" belief -- indeed, pretty much defines his worldview by it -- argues for the position in his book The Big Questions essentially by cheating and slipping in the definition that "Math exists iff math is consistent." Go fig.)
No, but I can think of experiences under which I would keep the belief
a) "rocks obey a regularity that is concisely describable by reference to the standard axiom set for math"
but discard the belief
b) "under that axiom set, 2+2=4"
However, to get better insight into why this would happen, you should replace "2+2=4" with "5896 x 5273 = 31089508".
These discussions about whether 2+2=4 are confusing because 2+2 is 4 by definition of this operation of addition, and then we see in what cases real world phenomena are described by this operation. If you define 2+2 as anything but 4, then you're just describing a different operation. There are many operations where 2 and 2 give 5.
The problem, always, is that there's no causal connection between mathematics and reality. Suppose you try to force one and say that however many rocks you have in a cup when you combine two cups (each with two rocks), that is going to be THE operation of addition. Then asking what 2+2 has to be is asking how many rocks you can have in the final cup. Well, it's not logically impossible for rocks to follow a rule that every four rocks in a certain small area will make a new rock and increase their number to 5. It just happens, in our reality, that rocks satisfy Peano arithmetic (and don't resonate daughter rocks).
Those interested in a rigorous proof are advised to examine Principia Mathematica.
Not disagreeing, but fleshing out part of what it seems you're trying to say:
Numbers don't exist, that much ought to be clear. I think Eliezer says that numbers are in our minds, and our minds exist, but this is not the case: it's not numbers that are in our minds but representations of numbers.
Mathematical Platonism is, to me, religion for intellectuals. Mathematicians as esteemed at Kurt Goedel have even gone so far as to postulate that mathematics exists in an alternate universe. This is a basic error or at least wildly unparsimonious, akin to saying that modus ponens exists in an alternate universe.
To see how silly this is, it helps to realize that a sufficiently intelligent being would find all our mathematical theorems just alternative ways of stating the axioms, and all our mathematics just axioms and definitions with a bunch of obvious rephrases of the same. It would find our most advanced theorems as simple and obvious as modus ponens is to us - as just rewordings of the axioms and definitions.
From the perspective of a sufficiently intelligent being, mathematics is just a set of initial statements (axioms and definitions), along with humans' silly little demonstrations to help each other realize that a bunch rewordings of those statements (theorems) all mean the same thing.
From the perspective of a sufficiently intelligent being, physics is just a set of initial statements, along with a silly demonstration that history is what you get when you apply those statements over and over. How dull!
Amanojack,
One thing I've noticed that is probably covered somewhere in LW archive (I hope!) is that really smart and rational people can sometimes just be really good at hiding the truth from themselves. In other words, the smarter you are, the better you are at Dark Arts, and the easiest person to trick with Dark Arts is sometimes yourself.
[Heads up: your comments are displaying as one long line requiring side-scrolling instead of with natural line breaks.]
Is there? Evidence from simulations running on material comptuers doens't show you can make minds out of immaterial math.