Posts
Comments
Taking your definition of an abstract model (so we don't squabble over mere definitions), I don't think that just by removing information you'll go from an actual baseball to the 'abstract concept' of a sphere. You'll also be adding information. For example, for your model you can provide the formula that will yield the exact volume of the sphere - you can't do that as precisely for your baseball. Will your abstract models typically be more compact / contain less information than your baseball, sure. However, the information may be partially different, not just a subset, which it would be if you were just ignoring information.
That's true. Balls are very complex, so there isn't actually much you can ignore about them without invalidating your results. But you can ignore a lot of things and get approximately correct results, which is usually good enough when talking about balls.
Numbers, however, tend to be a little more convenient. If there's a hole in the bag of apples which you don't take into account, you'll get bad results, because that's a detail which impacts the numeric aspect of the apples. But we don't really care that it's a hole when talking about the number of apples. All we need to keep in mind is that the number decreased. If 1 apple fell through the hole, you can abstract that to a simple -1.
Anyway, this post has gotten out of hand, mostly because I was unclear, so I'll retract it and use these comments to write a hopefully clearer version. Thanks for the feedback.
You seem to misinterpret what I mean, but that's my fault for explaining poorly. This post has been getting out of hand with all the clarifications, so I will retract it and post a hopefully clearer version later on. Maybe as I write it, I'll notice a problem with my view which I hadn't seen before, and I'll never actually post it.
Out there in reality, there are just atoms.
I know. But it's easier to talk about apples than atoms. And the apples are just another level of abstractions. From atoms emerge apples, and from apples emerge [natural] numbers.
Take a look at my response to tim. Replace god with Euclidean Geometry, and forget the fluff about god being inconsistent, and you can see that Euclidean Geometry is still coherent, because our minds can represent it with consistent rules, so these rules exist as an abstraction in the universe. So my view doesn't make Euclidean Geometry incoherent. I'm not sure what exactly you mean by validity, but the only thing that my view says is "invalid" about Euclidean Geometry is that it is not the same as the geometry of our universe.
Now it gets a bit difficult to write about clearly, I'm sorry if it's not clear enough to be understandable. Things we figure out about numbers using Euclidean Geometry can still be valid, simply because when we abstract the details about Euclidean Geometry to be left with only numbers, we get the same thing as when we abstract apples to numbers, and the same thing is true about our mental representation of PA. So proofs from one can be "transferred" over to another. But "transfer" doesn't really describe it well. What's really happening is that from the abstract numbers, you can un-abstract them by filling them in with some details. So you can remember that the apples were in a bag, and that gravity was acting on them. If, when you add in the details, the abstract number behavior still holds, then the object follows the rules of numbers. So if the added details about apples don't affect the conclusions you make using PA, by abstracting PA into numbers, and then filling in the details about apples, you have shown that things that are true about PA are true about apples too. And all this is done using physical processes.
So my view doesn't entail anything about accepting or rejecting mathematical statements. What it says is that mathematical concepts are abstract concepts, which we obtain by ignoring details in things in this world, and thanks to our awesome simple and universal laws of physics, the same abstract concepts emerge again and again.
Is your claim that because the mind is itself physical, any idea stored in a mind is necessarily reducible to something physical?
...
ETA: minds can contain gods, ...
No, I'm claiming that the idea of god exists physically.
In our universe, the map is part of the territory. So the concept of god which a human stores in his mind is something physical. God himself might not exist, but the idea of god, and the rules this idea follows, exist, despite being inconsistent. And these rules which the idea of god follows can be represented in many ways, all of them physical.
For example, in the human mind, in computers, in mathematical logic (despite inconsistencies), etc. All these ways of representing god are done using completely different configurations of molecules. What is the common ground between them? Certainly not that the idea of god and it's rules are some special thing with special properties. So what do the hard drive and the human mind have in common when representing the idea of god?
By my theory, what they have in common is abstraction. Ignore all the specific details about how hard drives and human minds work, and just look at the specific abstract rules which we define as "god". These are complex, so we can't easily visualize this removal of details. It's much easier when talking about apples and numbers. You can see that when you have 2 apples, you can get the idea of 2 by ignoring the fact that it's apples, and that they're in a bag, and that gravity is affecting them. It's also easy to see when talking about balls. You get the idea of a ball by taking a sphere of matter, forgetting what it's composed of and forgetting it's radius. This abstract idea of a "ball" fits many things, because it's just ignoring details which vary from ball to ball.
So my claim is that the idea of axiomatic systems exists in the physical universe. In fact, all the ideas we ever have, and there rules, exist in the physical universe. But if we take PA as an example, the idea of PA exists in a mathematician's mind, and numbers emerge inside this idea of PA, because numbers do emerge inside PA. So by removing the details of how PA is stored in the mathematician's mind, we obtain numbers, which is just like getting numbers by removing the details about apples.
This still leaves the question of why numbers emerge in so many places. My best guess is that they do because the universe is built upon simple and universal laws of physics, so it's only natural that the same patterns would be appear everywhere.
Or, I could apply a constant force of 5 newtons to an object massing 25 kg for a duration of 8 seconds. I change it's velocity by 5N8S(1MKG/(NSEC^2))/25KG 1.6 M/S. By conserving units on all quantities, I convert force-time against a mass into acceleration.
Those units can be preserved through all mathematical operations, including exponentiation and definite integration.
Hmm... Another good argument. This one is harder. But that's just making this more fun, and getting me closer to giving in.
If I abstract a whole bunch of details about apples away, except the number and the fact that it's an apple, I get math with units. So if I have a bag of 3 apples, I can abstract this to 3apples. I can add 1apple to this, and get 4apples. Why? Some ancient mathematicians have shown that when you retain this extra bit of information, the abstraction "math with units" is formed, which is pretty much exactly like normal math, except that numbers are multiplied by their units. "math with units" and plain old math are very similar because they are both numbers, but "math with units" happens to retain a few extra details about what the numbers are talking about.
1apple 1apple=1apple^2 is true, but it doesn't make sense. You can't add in details to turn this into apples because of physical limitations, so 1apple 1apple does not emerge from apples. But it is consistent with the rules of mathematics, which you obtain when you remove the complex details of what an apple is, so this is considered true.
1m * 1m=1m^2, however, does make sense, and we can add in details to transform this into an area. m/s can be transformed into the speed of an object by adding in details. m/s^2 can be transformed into an actual acceleration. kgm/s^2 can be transformed into an actual force by adding in details. I think this is true of all useful units, but if you can think of one for which this is not true, please share, because that will be a big blow to my theory. If this theory survives all the other arguments, I will still need to prove that this is true, and until I prove it, my theory is weak. Thank you for showing me this.
Anyway, a summary of this comment: math with units is an abstraction, one level below plain math, and it mostly follows the rules of math because it's just math+(some simple other thing). Units which don't make sense cannot be un-abstracted into things in the world, but because math is consistent, we can still manipulate them like anything else because we have abstracted the details which makes it not make sense away.
So I needed to extend my theory for this, but thankfully not by making it more complex at the base. This is all just emergent stuff in the universe. So if my theory holds up, I expect Occam's Razor will be in favor of it.
Bananas are constrained by the laws of physics, so when you reach the maximum number of bananas possible in our universe, the '+' operation becomes impossible to apply to it. So using physical bananas, it is impossible to talk about infinity.
But even if bananas aren't suited for talking about infinity, where does infinity come from?
Given that we reason about infinity, I infer that infinity can be represented using physical things (unless the mind is not physical). Also, given what I know about mathematics, I expect that infinity is thought about using rules/axioms/(your word of choice).
So to explain the properties of infinity, we simply defined it and some rules it follows, and from there proved other truths about infinity. Infinity may not exist in the real world, but it does exist as our definition, which is just physical stuff. If infinity exists in the real world and I don't know about it, we have probably observed it and created a model of it which follows the same rules as it. And then, by abstracting our model to ignore the fact that it's just a model, and abstracting the real infininty to ignore whatever it's composed of, we get the same thing, and that's what infinity is.
So while bananas are constrained by the amount of matter in our universe and can therefore not represent any number (some of the details about the bananas just can't be safely ignored), PA (or whichever set of axioms we use to think about infinity) can. And from this, we can find rules about infinity, using a purely physical process.
This is sort of guesswork on my part. I am not certain that this is how we have reasoned about infinity, though I would be surprised if it wasn't. So if it isn't, just say so, and I'll probably be convinced that I'm incorrect, and retract my post.
Very nice and convincing argument. There were some moments when thinking about it when I though your argument defeated my view. Sadly, we're not quite there yet.
Trying to add 2 miles to 2 apples does not make sense. There is no physical representation of such an operation. So you can't try to abstract that into numbers. Here's an example, to clarify:
Let's say I've got a bag of 2 apples, I add 2, and one falls through the hole in my bag. The number of apples in my bag is 2+2-1=3. The first 2 is an abstraction of the original number of apples in my bag, the second 2 is the number of apples I added, and -1 is the number of apples that fell out through the hole, and 3 is the total. Everything in this equation can be mapped back to reality by adding in details. But in 2 apples + 2 miles, the + cannot be mapped back to reality. And so it is not representative of apples.
I can think of an easy attack on this, by saying that 2 apples + 2 oranges does make sense in our world. But this is a disguised incorrectness. The oranges do not fall into the criterion of "apple", so it does not actually make sense to add them using the '+apples' operator. Of course, we could generalize the '+' operation to mean '+fruit', but then it becomes 2 fruit + 2 fruit, which does make sense, and is true whether you reject my current views or not.
I'm looking forward to your next argument :).
If the physical facts of apples were to change such that 2 apples added to 2 more apples did not give you 4 apples, then removing the detail that it's an apple would not yield numbers. In such a case, you would not be able to abstract apples into numbers. They would abstract away into something else.
Likewise, if you changed the mental processes which makes Peano Arithmetic, you would not change numbers; you would merely have changed what Peano Arithmetic can be abstracted into.
The thing to get from my post is that numbers are an abstraction: they are apples, when you forget that it's apples that you're talking about. They are Peano Arithmetic, when you forget that it's mental processes you're talking about. They are bits, when you forget it's bits you're talking about. But this does not make them special, just like the concept "sphere" is not special. We're just lucky that numbers emerge all over the place in the universe, probably because the laws of physics are the same everywhere so everything is built upon the same base.
Axiomatic Systems ... can all be reduced to physics. I think most LessWrongers, being reductionists, believe this.
I would be suprised if this were true. In fact, I'm not even sure what you mean by it.
Well, given that mathematicians store axiomatic systems in their minds, and use them to prove things, they cannot help but be reducible to physical things, unless the mind itself is not physical.
However, I think you're confusing the finitude of our proofs with some sort of property of the models. I mean, I can easily specify models much bigger than the physical universe.
You can specify models much bigger than the physical universe. But that's just extrapolating the rules by assuming they would keep on working. We do have good reason to believe that they would keep on working, though, because if they stop, then contradictions take place, and from contradictions anything is true, so we would be living in one very strange universe.
Edit:
It has also occurred to me that it doesn't even matter if a number is larger than the total amount of atoms in this universe. Because as I've said in my post, a number is what you get when you abstract away all the little details which aren't shared by all the places where numbers emerge, like the fact that it's atoms you're counting.
So a system for representing a number larger than the total number of atoms represents a number anyway, as long as it follows the rules of numbers. And a model of something much bigger than the universe works simply because the details of how large the universe actually is are ignored in the model.
try pondering this one. Why does 2 + 2 come out the same way each time? Never mind the question of why the laws of physics are stable - why is logic stable? Of course I can't imagine it being any other way, but that's not an explanation.
Do you have an answer which will be revealed in a later post?
My [uninformed] interpretation of mathematics is that it is an abstraction which does exist in this world, which we have observed like we might observe gravity. We then go on to infer things about these abstract concepts using proofs.
So we would observe numbers in many places in nature, from which we would make a model of numbers (which would be an abstract model of all the things which we have observed following the rules of numbers), and from our model of numbers we could infer properties of numbers (much like we can infer things about a falling ball from our model of gravity), and these inferences would be "proofs" (and thankfully, because numbers are so much simpler than most things, we can list all our assumptions and have perfect information about them, so our inferences are indeed proofs in the sense that we can be certain of them).
But it seems like a common view that mathematics has some sort of special place in the universe, above the laws of physics, and I don't really know what arguments people have for believing this. What are the arguments for this belief?
Edit: Reformulated my question to make it more specific.
Because epiphenomenalist theories are common but incorrect, and the goal of LessWrong is at least partially what its name implies.
'2+2=4' can be causally linked to reality. If you take 2 objects, and add 2 others, you've got 4, and this can be mapped back to the concept of '2+2=4'. Computers, and your brain, do it all the time.
This argument falls when we start talking about things which don't seem to actually exist, like fractions when talking about indivisible particles. But numbers can be mapped to many things (that's what abstracting things tends to do), so even though fractions don't exist in that particular case, they do when talking about pies, so fractions can be mapped back to reality.
But this second argument seems to fall when talking about things like infinities, which can't be mapped back to reality, as far as I know (maybe when talking about the number of points in a distance?). But in that case, we are just extrapolating rules which we have already mapped from the universe into our models. We know how the laws of physics work, so when we see the spaceship going of into the distance, where we'll never be able to interact with it again, we know it's still there, because we are extrapolating the laws of physics to outside the observable universe. Likewise, when confronted with infinity, mathematicians extrapolated certain known rules, and from that inferred properties about infinities, and because their rules were correct, whenever computations involving infities were resolved to more manageable numbers, they were consistent with everything else.
So our representations of numbers are a map of the territory (actually, many territories, because numbers are abstract).
It seems to me that the horcrux doesn't need memories. The stored fragment of the soul serves not as a means of resurrection, but to sort of "anchor" the soul to the living world. So the main part of the soul, the part that stays within the living body until death, is left to linger. There is evidence for this: in canon, the first time Voldemort dies, his soul still lives, gathers strength, and then gets a servant to help him, without any contact with the horcruxes.
And I expect that Voldemort actually planned on making Harry a horcrux; what better protection against a prophetic rival than to make him have to suicide to kill you?
Wait, so you're saying that your right to freedom is more important than making this world as good as possible? By all moral systems I know of, that's morally wrong (though I'll admit I don't know many). Do you have a well-defined moral system you could point me to?
I'm sorry, my comment grew into a mess, I should have cleaned it up a bit before posting. Anyway, I agree fully about the second statement only applying to this program, that's what I realized in the edit.
But for the first statement, I'll try to be a bit more clear.
My first claim is that "eval(box) == implies(proves(box, n1), eval('2==3'))" is a true statement, proven by the Diagonal Lemma. I'll refer to it as "statement 1", or "the first statement".
If "eval(box)" returns false, then for the first statement to be true, "implies(proves(box, n1), eval('2==3'))" must return false. "implies" only returns false if "proves(box, n1)" is true and "eval('2==3')" is false. Therefore for statement 1 to be true when "eval(box)" is false, then "proves(box, n1)" has to be true, which is a contradiction: "eval(box)" can't be false and also provably true. Therefore, "eval(box)" must be true.
So let's say "eval(box)" is true, that means that "implies(proves(box, n1), eval('2==3'))" must also return true for statement 1 to be true. One way for the "implies" statement to return true is for "proves(box, n1)" to return false. Since I have proven above that "eval(box)" is true, any good definition of the "proves" function will also return true, because if it finds no other, it will at least find my proof. Therefore, "proves(box, n1)" will return true.
So there is only one other way for the "implies" statement to return true: "eval('2==3')" must return true. Therefore, "eval('2==3')" returns true, and it follows from this that 2=3.
So, where did I go wrong?
Edit: Wow, I really am an idiot. I assumed the second statement was true about every statement, but I just realized (by reading one extra comment after posting) that by Lob's Theorem that's not true. But my original idea, that the first statement is all that's required to prove anything, still seems to hold.
Okay, I can follow the first proof when I assume statement 1, but I don't quite understand how cousin_it got to 1. The Diagonal Lemma requires a free variable inside the formula, but I can't seem to find it.
In fact, I think I totally misunderstand the Diagonal Lemma, because it seems to me that you could use it to prove anything. If you replace "outputs(1)" by "2==3", the proof still seems to hold. Statement 2 would still be true with "2==3" (it is true about any statement, after all), and all the logic that follows from those two statements would be true. In fact, by an unclearly written chain of reasoning which I originally intended to post before realizing that it would be much simpler to just say this, all you seem to require is the first statement to be able to prove anything. If I am mistaken, which is probable, then I expect my error lies in the assumption that "outputs(1)" could be replaced by any string of code.
For my original unclear explanation of why the first statement in the proof seems to allow anything to be proven, in case anyone cares for it:
Also, I must be an idiot, but it seems to me that you can prove pretty much anything using 1. As far as I can tell, the "eval(outputs(1))" could be "2==3" or any other statement, and the only reason "eval(outputs(1))" is used is because it's useful in the proof. Given that statement 1 is proven, "eval(box)" must return true, because if it returns false, then "proves(box,n1)" cannot return true, and therefore "implies(proves(box, n1),eval(outputs(1)))" must return true, and false!=true. Assuming that my reasoning is correct (a very questionable assumption, I know), I have just proven that "eval(box)" is true, therefore "prove(box,n1)" is also true for some arbitrarily large n1. Therefore, the "implies" will return "eval(outputs(1))", which must be true for "eval(box)" to equal it. So given my earlier assumption (which I suspect is my error) that "eval(outputs(1))" can be replaced by anything, then I can replace it by any statement, which must return true to equal "eval(box)". So through this, I seem to have proven that "2==3".
Anyone care to point me to my mistake (or, to satisfy my wildest of dreams, say that I appear to be right ;) )?
To be fair, this post does point out a reason why debating morality is different from debating most other subjects (using different words from mine): people have very different priors on morality, and unlike in, say, physics, these priors can't be rebutted by observing the universe. Reaching an agreement in morality is therefore often much harder than in other subjects, if an agreement even can be reached.
This is sort of avoiding the question. What if you made the choice, but then had your memory erased about the whole dilemma right afterwards? Assuming you knew before making your choice that your memory would be erased, of course.