All of AnotherIdiot's Comments + Replies

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.

1mwengler12y

Or you can leave it, take your karma lumps which seem to be somewhat finite, and expect that to the extent there is something useful here, people stumbling across it will be influenced. Missing the nuance is the right thing to do with most nuances, IF you are interested in making technical progress. And to the extent that there is a bias towards things that would help you build an AI, that is a valid purpose of this board.

0TimS12y

Don't worry about it - I may be missing some nuance. I would recommend reading some more advanced math before trying to make a philosophy of math. Integers make intuitive sense in the physical world in a way that more advanced math tends not to. Godel, Escher, Bach gets high marks around these parts, and rightfully so.

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.

AnotherIdiot12y10

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 d... (read more)

2TimS12y

Sorry, unintended inferential distance. In a previous post, Eliezer distinguishes between "true" and "valid" because only empirical things can be true, and he doesn't think mathematics is empirical. Thus, propositions that follow from proposed axioms are "valid" - what a mathematician would call true - to avoid confusing vocabulary. You avoid the confusion by asserting that mathematical assertions really do correspond to some physical state (i.e. are empirical). Under the correspondence theory of truth, that allows some mathematical statements to be true, not simply valid. Nonetheless, I assume you don't think all mathematical statements are true (2 + 2 != 3, etc). The problem with asserting that mathematical statements are empirical is that there are certain mathematical assertions that are valid but do not have any physical basis. Consider the proposition, "The Pythagorean theorem follows from Euclid's axioms." The statement is valid, but cannot meaningfully be called true because there is no physical fact that corresponds to the assertion by virtue of the fact that the physical universe is not a Euclidean space. But the statement is not false because there is no physical fact that corresponds to "The Pythagorean theorem is not deducible from Euclidean axioms." In other words, your theory of mathematics has no room for "validity", only "truth." The Pythagorean theorem is interesting to mathematicians, but adopting your philosophy of mathematics would hold that generations of mathematicians have been interested in a theorem that we now know can never be true or false. That's just too weird for most people to accept.

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 p... (read more)

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, excep... (read more)

1Decius12y

Quantities can be converted to and from numbers: (32 ft lbf / (lbm sec^2))=1 (64 ft lbf / (lbm sec^2))=2 It is true, but not useful, to say that the area of a circle with radius R is equal to or equivalently, Taking the sec^2lbf root does not have an analogue in reality, but the units output from taking the time^2*Force root of a distance raised to the power of scalar*distance*mass is area in this specific case.

AnotherIdiot12y10

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 rule... (read more)

3TimS12y

Ok, if you want a serious response instead of a snarky one, here goes: You may have learned about Euclidean geometry in school. Two points define a line. A line and a point not along the line define a plane. As Euclid defined the geometry, parallel lines never intersect. However, we don't live in a Euclidean geometry. To a first order of approximation, we live on a sphere. If Line A is perpendicular to Line X and Line B is also perpendicular to Line X, they are parallel in Euclidean geometry. Nonetheless, on a sphere, Line A and Line B will eventually intersect. So we've got all this neat mathematics deriving interesting results from Euclidean axioms, but nothing in the real world is Euclidean. If we take your thesis (that math can be reduced to physics) seriously, that means that Euclidean geometry is not simply invalid - it is incoherent (i.e. wronger than wrong). You might be willing to bite the bullet and throw Euclidean geometry in the trash, but no one who takes math seriously is willing to do so. ---------------------------------------- For further reading, you might consider following the links from this post by Wei_Dai. In short, the issue here - how to talk about the "truth" of mathematics - is a basic problem with the correspondence theory of truth. Eliezer is making an attempt to bridge the gap in the post you highlighted, but he is deliberately avoiding the philosophical choice you made - I suspect because he is unwilling to throw out non-physical mathematics, which I've argued above is a requirement for your theory of mathematical truth.

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 ... (read more)

1Decius12y

I put two apples in my one bag, then walk two miles and add two more apples to the bag. There is one hole in the bag, and one of the apples falls out. 2 (apples) + 2 (miles) + 1 (bag) - 1 (hole) - 1 (apple) + 2 (apples) = 5 I end up five miles from where I started, because I dropped units from my quantities and did operations on the numbers. 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.

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 apple... (read more)

3Decius12y

You can't get a number out of a quantity- it's simply that just about all of the rules that apply to numbers also apply to quantities. That's because numbers were created to be useful in manipulating quantities. Mathematics interacts very well with physics from a combination of factors: Physics appears to obey similar underlying laws as mathematics, and mathematics is created by people who want to explain physical phenomena. When you forget that it's apples and miles that you are talking about, 2 apples and 2 miles never yields 4 anything, even though 2 apples plus 2 apples is 4 apples, and 2 miles plus 2 miles is 4 miles. There's even a term for what you describe in the academic world- Magic Units. It is used to provide partial credit for showing mathematical calculations on numbers rather than showing mathematical calculations on quantities.

AnotherIdiot12y-30

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

... (read more)

0mwengler12y

There seems to be no problem representing numbers using machines that have many fewer pieces than the number represented. With only 10 bits I can represent more than 1000 numbers as a trivial example. WIth only 10 billion neurons I can represent infinity (in the human mind) although it might be difficult to prove my representation was perfectly accurate.

8tim12y

Is your claim that because the mind is itself physical, any idea stored in a mind is necessarily reducible to something physical? Because this seems like a map-territory confusion. ETA: minds can contain gods, magic and any number of wonders that are fundamentally irreconcilable with physical reality.

Logical Pinpointing

Open Thread, October 16-31, 2012

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?

Harry Potter and the Methods of Rationality discussion thread, part 16, chapter 85

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 fro... (read more)

0wedrifid12y

It's more fun to think of the reverse relationship!

Causal Reference

AnotherIdiot12y100

Because epiphenomenalist theories are common but incorrect, and the goal of LessWrong is at least partially what its name implies.

Causal Reference

AnotherIdiot12y-20

'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 t... (read more)

AI cooperation in practice

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?

Magical Healing Powers

AnotherIdiot13y-20

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?

-2Shmi13y

I am saying nothing of the sort. My point is that I distrust anyone who tells me that I am obligated to do stuff that they think is "right".

-1saturn13y

http://en.wikipedia.org/wiki/Ethical_egoism

AI cooperation in practice

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 tru... (read more)

2cousin_it13y

Your proof of eval(box) relied on the fact that eval(box) can't be false and also provably true. But that fact is equivalent to consistency of the formal theory, so it can't be proven within the formal theory itself, by Godel's theorem. Of course, since eval(box) is a terminating computation that returns true (your proof of that is correct), the formal theory can eventually prove that by step-by-step simulation. But that proof will be much longer than your proof above, and much longer than n1. In fact, proves(box,n1) will return false, and your comment serves as a nice proof of that by reductio. Don't get discouraged, it's a very common mistake =)

Thoughts on moral intuitions

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 D... (read more)

0cousin_it13y

I don't understand how the first statement can be used to prove anything... The second statement might be true for every statement, but it's not necessarily provable for every statement, which is required in the proof. In fact, the second statement is provable for "outputs(1)" by inspection of the program (because the program searches for proofs of "outputs(1)"), but not provable for "2==3" (because then "2==3" would be true, by Lob's theorem).