ec429

Karma: 133

ec429 Sep 18, 2011, 5:16 AM
2 points
in reply to: g’s comment on: Relative Configuration Space
I agree that taking quotients of the configuration space is a more natural way of doing things. But, when you say

quotienting R^6 by this action of E, you’re left with a 2-dimensional quotient space

don’t you mean you’re left with a 3-dimensional quotient space? Counting degrees of freedom: wherever we put A, that eats the translation. Wherever we put B, that eats the rotation and we’re left with the distance |AB| (one dimension). Wherever we put C, that eats reflection and we’re left with the position of C up to reflection. So, the space of triangles ends up as R x R x (R / ~), where a~b iff |a|=|b|.

But then, this space should be homeomorphic to the one Eliezer gives, with the relative distances. We’ll take a point (x,y,z) in R x R x (R / ~). Then |AB|=x, |AC|=hypot(y, z), |BC|=hypot(y-x, z), clearly this is continuous and nice, and also clearly the image doesn’t change if we replace z by -z (so the function is well-defined despite the domain being a quotient space, which generally needs to be checked). Showing that the mapping is invertible, with continuous inverse, is left as exercise for the reader.

Consider now the apparent boundary when we embed this in R³; it’s z=0, which corresponds to “A, B and C form a straight line”, which (triangle inequality) corresponds to the boundary of the subset of distance-space. But if you imagine the particles moving, it’s a lot more obvious that you should bounce off the ”/ ~” surface than that you should bounce off the “if you cross this surface you get a distance-tuple that’s un-geometric” surface. Similarly, straight lines in R x R x (R / ~) correspond to fixing any two particles and moving the third in a straight line.

I would conclude from this that the equations of physics in the quotient space are likely to be much nicer than the equivalent equations in distance-tuple space.

So why bother formulating the relational configuration space in distance-tuples? After all, with the distance-tuples, you still end up having to quotient afterwards on particle-swapping to get the quantum-mechanical picture. Isn’t it easier to just use quotients, rather than an odd mix of quotients, new bases, and subsets?

ec429 Sep 17, 2011, 8:37 PM
0 points
in reply to: Liron’s comment on: No Individual Particles
I was also having a problem with this, but I think it can be resolved by saying that every operator is symmetric in A and B. Therefore, whenever we take a measurement, we will fail to distinguish between a blob at (1,0) and a blob at (0,1). More fundamentally, any interaction with a third particle will be the same whether the blob is at (1,0) or (0,1).

Even more important is that a blob at (0,1) plus a blob with the opposite phase at (1,0) can have a subtractive effect, even cancelling out entirely if both have the same amplitude.

In a sense, then, factoring the topology of configuration-space by the identity A==B causes the symmetry of operators to be an unavoidable consequence of the topology, rather than some freakish coincidence.

ec429 Sep 17, 2011, 4:59 PM
0 points
in reply to: [deleted]’s comment on: How to Convince Me That 2 + 2 = 3
Well, of course I don’t win automatically. It’s just that there’s a kind of Godwin’s Law of philosophy, whereby the first to invoke Gödel loses by default.

ec429 Sep 16, 2011, 4:13 PM
15 points
0
in reply to: james_gnz’s comment on: Disguised Queries
That doesn’t answer the question “Is a fœtus a person”, it just supplies a definition of “person”, which may or may not be relevant to any given query.

Suppose my real query is “Can a fœtus talk?” Now, just because I choose to define “person” in such a way that most “person”s can talk, and in such a way that a fœtus classes as a “person”, that doesn’t make the probability that a fœtus can talk any different to if I’d defined “person” differently.

The whole point of these examples of disguised queries is that if you find yourself trying to answer them, you’re doing it wrong.

Suppose we call the horse’s tail a leg.

ec429 Sep 16, 2011, 2:03 PM
1 point
in reply to: Vladimir_Nesov’s comment on: The Crackpot Offer
Well, I’m studying for an undergraduate degree in mathematics at a good university; the “trying to construct...” is just one of several things I do in my copious free time. Also, I’m spending a much smaller proportion of my time on this project than I was spending on trying to disprove the ITs. So it looks to me as though I’m actually behaving rationally, but maybe that’s just how the algorithm looks from the inside.

I think that by “make the ITs become irrelevant” I mean that I’m trying to find a philosophy in which the things that make me want the ITs to be false are no longer represented, because if I have any assumption that implies “And therefore the ITs are false” then that assumption is wrong. But again, is that just me rationalising?

ec429 Sep 16, 2011, 12:26 PM
3 points
on: The Crackpot Offer
I fear that I might be currently trapped in this error: I’ve always resented Gödel’s Incompleteness Theorems. When I was about 17 I thought I’d disproved 1IT (turned out I’d just reconstructed the proof of 2IT and missed the detail that Con(T)≠ProvT(Con(T))). It took me about a year after that to realise that, no, I wasn’t going to disprove the ITs no matter how much I wanted to, and I accepted that trying to disprove them anyway would be a crackpot thing to do. Since then I’ve been trying to construct a philosophical framework of mathematics in which the ITs become irrelevant. Have I, in fact, taken the Crackpot Offer?

ec429 Sep 16, 2011, 8:45 AM
1 point
in reply to: [deleted]’s comment on: A Case Study of Motivated Continuation
Of course. Except that I think you mean trillion^trillion, not trillion*trillion.

ec429 Sep 16, 2011, 3:58 AM
1 point
in reply to: Will_Sawin’s comment on: What is Evidence?
Um, but isn’t that just a convention? Why should we treat the “amplitude” of a classical probability as being the probability?

Does the problem have something to do with the extra directionality quantum probabilities have by virtue of the amplitude being in C? (so that |0> and (-1*|0>) can cancel each other out)

ec429 Sep 16, 2011, 3:29 AM
2 points
on: A Case Study of Motivated Continuation
Hmm. I seem to be flinching away from both answers, and I think I know why. It’s because I’m unable to decide whether utility really does multiply (after all, one could advocate the utility function “The minimum happiness within the population”, instead of the sum).

So I’m happy to make the factual claims that “Sum-utility ⇒ pick ‘torture’” and “Min-utility ⇒ pick ‘specks’”; I just can’t see any procedure for choosing between sum and min. So I’ll formulate a test to see which I believe: I’ll gradually reduce the severity of the non-speck option. So, specks versus someone getting tortured for 25 years, I’m still unsure. Specks versus someone getting slapped in the face, I choose slap over specks. Therefore I’m not following min-utility, so I’m willing to accept that really I’m following sum-utility, so in the original problem I pick Torture. I don’t like this, because my brain wants to be scope-insensitive and refuses to understand 3^^^3, but when I made one of the outcomes not flinch-worthy that outcome got picked, and I’m pretty sure that my reasons for picking Slap ought to scale up to Torture, so there it is.

I started this post not knowing what answer I would reach despite having spent several minutes on the question. I think I’ve now been trying to resolve this for over half an hour, and I still feel uncomfortable. My mind has just now come up with a third alternative, which is that utilities should perhaps be rated with hyperreals, so that 3^^^3 1 is still less than 1 H (for an infinite hyperinteger H), in which case we could pick Specks without discarding sum-utility. But I probably wouldn’t have thought of that while I was locked up and couldn’t choose an answer. I am now feeling comfortable, which suggests that this is what I actually believe about utilities. Of course, now there is an experiment I could do to try and falsify this: try to construct a chain of things starting at a dust speck and ending at torture, where each link in the chain is only a finite amount worse than the one before. I know I can get from speck to slap, because I chose Slap over Specks. I also think a hefty kick up the arse is only finitely worse than a slap. I next try to get to a broken arm, but I’m unwilling to do that (at least, in a single step), so I need to find something intermediate. In fact I think I should try and find something I can jump down to from a broken arm, because a broken anything seems scary in a new way. A deep-bruised hand? Yes, I think that relates finitely to a broken arm. I also think that finitely many kicks up the arse are worse than a deep-bruised hand. Given that my instinctive feeling about the relation of kick to arm was very similar to my feeling about the relation of speck to torture, I conclude that in fact my scale of utilities is constrained to the finite.

The point to this post (if there is one) is that a useful method seems to be to vary the parameters of the problem until you can get an answer, and then look to see whether that illuminates the original problem. (Come to think of it, ISTR that’s one of Polyà′s How To Solve It tips.) But to evaluate this method, I need to see whether it can also produce the opposite result. So, I need to vary the parameters in ways that favour Specks. If I reduce the 3^^^3 to something smaller, I eventually pick Specks because I get a number I think I can comprehend—but that number has to be so much smaller than 3^^^3 that I don’t think it’s relevant to the original problem. If I make the ‘torture’ option involve something worse than torture, I still pick it—I can’t think of anything that’s sufficiently worse than torture that doing that to someone could make me pick Specks when I didn’t pick Specks against torture.

So the method does constrain, and I pick Torture. There, finally finished this post.

ec429 Sep 16, 2011, 1:25 AM
3 points
on: Positive Bias: Look Into the Dark
Thought experiment. Suppose you have two oracles, and your task is to find out whether or not they have the same rule. If each oracle is considered as “A lookup table produced by a coin flip for each possible input, except that there’s a 50% chance that the second is just a copy of the first” then of course any input is as likely as any other to exhibit a difference, and you can easily compute the probability of no difference after n tests fail to exhibit one. But if you have an assumption that simpler rules are more likely (eg. your prior is 2^-complexity) then what’s your optimal strategy?

A plausible strategy is to follow the same strategy as you would if you had to find the rule of a single oracle; you always send the input that gives you the most bits about Oracle A’s rule. That way, you maximise the probability of exhibiting a difference given that one exists. So if you can generate an input which, under your current model of the space of A’s possible rules (and the probability of each), has exactly a 50% chance of matching A, then it also has a 50% chance of matching B; moreover these probabilities are independent, so you have 25%+25%=50% chance of exhibiting a difference. If instead you picked an input with a 30% chance of matching A, your chance of exhibiting a difference is 21%+21%=42%.

ec429 Sep 15, 2011, 8:27 PM
4 points
on: Human Evil and Muddled Thinking
Clearly, Winston was just an actor, and O’Brien was being tested by a researcher at Miniluv, whose name was Stanley Milgram.

ec429 Sep 15, 2011, 7:04 PM
0 points
in reply to: moshez’s comment on: How to Convince Me That 2 + 2 = 3
You’re missing the essential point about deductives, which is this:

Changing the substrate used for the calculations does not change the experiment.

With a normal experiment, if you repeat my experiment it’s possible that your apparatus differs from mine in a way which (unbeknownst to either of us) is salient and affects the outcome.

With mathematical deduction, if our results disagree, (at least) one of us is simply wrong, it’s not “this datum is also valid but it’s data about a different set of conditions”, it’s “this datum contains an error in its derivation”. It is the same experiment, and the same computation, whether it is carried out on my brain, your brain, your brain using pen and paper as an external single-write store, theorem-prover software running on a Pentium, the same software running on an Athlon, different software in a different language running on a Babbage Analytical Engine… it’s still the same experiment. And a mistake in your proof really is a mistake, rather than the laws of mathematics having been momentarily false leading you to a false conclusion. To quote the article, “Unconditional facts are not the same as unconditional beliefs.” Contrapositive: conditional beliefs are not the same as conditional facts.

The only way in which your calculation entangled with the world is in terms of the reliability of pen-and-paper single-write storage; that reliability is not contingent on what the true laws of mathematics are, so the bits that come from that are not bits you can usefully entangle with. The bits that you can obtain about the true laws of mathematics are bits produced by computation.

ec429 Sep 15, 2011, 12:14 AM
1 point
in reply to: moshez’s comment on: How to Convince Me That 2 + 2 = 3
I think the point is that mathematical reasoning is inherently self-correcting in this sense, and that this corrective force is intentionistic and Lamarckian—it is being corrected toward a mathematical argument which one thinks of as a timeless perfect Form (because come on, are there really any mathematicians who don’t, secretly, believe in the Platonic realism of mathematics?), and not just away from an argument that’s flawed.

An incorrect theory can appear to be supported by experimental results (with probability going to 0 as the sample size goes to \infty), and if you have the finite set of experimental results pointing to the wrong conclusion, then no amount of mind-internal examination of those results can correct the error (if it could, your theory would not be predictive; conservation of probability, you all know that). But mind-internal examination of a mathematical argument, without any further entangling (so no new information, in the Bayesian sense, about the outside world; only new information about the world inside your head), can discover the error, and once it has done so, it is typically a mechanical process to verify that the error is indeed an error and that the correction has indeed corrected that error.

This remains true if the error is an error of omission (We haven’t found the proof that T, so we don’t know that T, but in fact there is a proof of T).

So you’re not getting new bits from observed reality, yet you’re making new discoveries and overthrowing past mistakes. The bits are coming from the processing; your ignorance has decreased by computation without the acquisition of bits by entangling with the world. That’s why deductive knowledge is categorically different, and why errors in logical reasoning are not a problem with the idea of logical reasoning itself, nor do they exclude a mathematical statement from being unconditionally true. They just exclude the possibility of unconditional knowledge.

Can you conceive of a world in which, say, ⋀∅ is false? It’s certainly a lot harder than conceiving of a world in which earplugs obey “2+2=3”-arithmetic, but is your belief that ⋀∅ unconditional? What is the absolutely most fundamentally obvious tautology you can think of, and is your belief in it unconditional? If not, what kind of evidence could there be against it? It seems to me that ¬⋀∅ would require “there exists a false proposition which is an element of the empty set”; in order to make an error there I’d have to have made an error in looking up a definition, in which case I’m not really talking about ⋀∅ when I assert its truth; nonetheless the thing I am talking about is a tautological truth and so one still exists (I may have gained or lost a ‘box’, here, in which case things don’t work).

My mind is beginning to melt and I think I’ve drifted off topic a little. I should go to bed. (Sorry for rambling)

ec429 Sep 14, 2011, 6:09 PM
1 point
in reply to: Will_Sawin’s comment on: What is Evidence?
Nonetheless, it is instructive (imho) to consider how (assigned) probability is a property of the observer, and not an inherent property of the system. If a qubit is (|0> + |1>)/sqrt(2), and I measure it and observe 0, then I’m entangled with it so relative to me it’s now |0>. But what’s really happened is that I became (|observed 0> + |observed 1>)/sqrt(2), or rather, that the whole system became (|0,observed 0> + |1,observed 1>)/sqrt(2). This is closely analogous to the Law of Conservation of Probability; if you take Expectations conditional on the observation, then take Expectation of the whole thing, you get the original expectation back. This is because observing the system doesn’t change the system, it just changes you. This is obvious in Bayesian probability in the classical-mechanics world; the only reason it doesn’t seem obvious in the quantum realm is that we’ve been told over and over that “observing a quantum system changes it”.

Quite honestly, I don’t see how a Bayesian can possibly be a Copenhagenist. Quantum probability is Bayesian probability, because quantum entanglement is just the territory updating itself on an observation, in the same way that Bayesian ‘evidence entanglement’ is updating one’s map on an observation.

ec429 Sep 14, 2011, 4:41 PM
6 points
in reply to: thomblake’s comment on: Hand vs. Fingers
It is a lesson hard-learned over many programmer-years of coding that portability should be acquired as an innate reflex; that whenever you are about to sacrifice portability, you ask yourself “Why am I doing so, are there alternatives, and have I done at least a rudimentary cost/benefit analysis of this decision?”

What you don’t do is throw away portability just because you happen to be using a particular machine right now. This is something quickly learned in comp.lang.c.

Incidentally, a better model than PK’s might be:

typedef struct foo {int i} s;

s f;

Now what is ‘f’? Is it an ‘s’, is it a struct foo, or is it an int? And what about f.i? Both have the same address, casting either’s address to any of the pointer types ‘s *’, ‘struct foo *’ or ‘int *’ is valid; writing

*(int *)&f=1;

does just the same as f.i=1; quite possibly the compiler will generate the same code, because after all, the territory is the same. It’s just that “f.i=1” is a higher-level map, which conveniently abstracts things out.

This is made more explicit by considering struct bar {int a; long b;} t; *(long *)(((char *)&bar)+offsetof(bar, b))=1; // same as bar.b=1

Of course, you could go to all the trouble of computing the offset pointer every time, since that’s what happens on the “quark” (assembly language) level of the territory, but the higher-level ‘struct’ map is cognitively useful.

ec429 Sep 14, 2011, 1:20 PM
12 points
in reply to: PK’s comment on: Hand vs. Fingers
As a C programmer who hangs out in comp.lang.c, I’m strongly tempted to get out a copy of C99 so that I can tell you precisely where you’re wrong there. But I’ll content myself with pointing out that there is no guarantee that sizeof(long)==2*sizeof(short)==4*sizeof(char), and moreover that even if that did hold, there is still no guarantee that sizeof(struct {short hi; short lo;})==2*sizeof(short) because the struct might have padding—what if ‘short’ were a 16 bit quantity but stored in 32 bit words (perhaps because the arch can only do 32 bit writes, and has decided that short should be an int_fast16_t rather than an int_least16_t), resulting in alignment requirements?

In conclusion, PK should learn some basic C, and forget about the ++. (Old joke: what does C++ mean? Take C, add to it, then use the old version)

EDIT: thanks, paper-machine, and I approve of Markdown’s choice of escape character. Now, if it’ll just let me use \033[1;35m to change the colour...

ec429 Sep 14, 2011, 6:59 AM
4 points
in reply to: Eliezer Yudkowsky’s comment on: How to Convince Me That 2 + 2 = 3
Although as a mathmo myself I should point out that, to save time, we usually pronounce it “Two”. :)

ec429 Sep 14, 2011, 6:46 AM
2 points
in reply to: [deleted]’s comment on: About addition and truth
I believe that “2+2=4 is either necessarily true or necessarily false”. I believe 2+2=4 is necessarily true (modulo definitions). I don’t believe it’s necessarily true that “2+2=4 is necessarily true”.

There’s some pretty strong evidence that the proof that 2+2=4 doesn’t have a mistake in it (heckuva lot of eyeballs). I have good reasons (well, reasons anyway) to believe that mathematical truths are necessary. Thus most of my mass is on “2+2=4 is necessarily true”. Yet, even if it’s necessarily true that “2+2=4 is either necessarily true or necessarily false”, and 2+2=4 is true, it still needn’t be necessarily true that “2+2=4 is necessarily true”, even though 2+2=4 is necessarily true.

If your eyes have glazed over at this point, I’ll just say that Provable(X) doesn’t imply Provable(Provable(X)), and if you think it does, it’s because your ontology of mathematics is wrong and Gödel will eat you.

ec429 Sep 14, 2011, 6:34 AM
4 points
in reply to: komponisto’s comment on: How to Convince Me That 2 + 2 = 3
Hmm, funny you should treat “I don’t believe in [Mathematical Object Y]” as Platonism. I generally characterise my ‘syntacticism’ (wh. I intend to explain more fully when I understand it the hell myself) as a “Platonic Formalism”; it is promiscuously inclusive of Mathematical Objects. If you can formulate a set of behaviours (inference rules) for it, then it has an existing Form—and that Form is the formalism (or… syntax) that encapsulates its behaviour. So in a sense, uncountable cardinals don’t exist—but the theory of uncountable cardinals does exist; similarly, the theory of finite cardinals exists but the number ‘2’ doesn’t.

This is of course bass-ackwards from a map-territory perspective; I am claiming that the map exists and the territory is just something we naïvely suppose ought to exist. After all, a map of non-existent territory is observationally equivalent to a map of manifest reality; unless you can observe the actual territory you can’t distinguish the two. Taking as assumption that the observe() function always returns an object Map, the idea that there is a territory gets Occamed out.

There is a good reason why I should want to do something so ontologically bizarre: by removing referents, and semantics, and manifest reality; by retaining only syntax, and rejecting the suggestion that one logic really “models” another, we finally solve the problems of Gödel (I’m a mathematician, not a philosopher, so I’m allowed to invoke Gödel without losing automatically) and the infinite descent when we say “first order logic is consistent because second-order logic proves it so, and we can believe second-order logic because third-order logic proves it consistent, and...”. When all you are doing is playing symbol games stripped of any semantics, “P ∧ ¬P” is just a string, and who cares if you can derive it from your axiomata? It only stops being a string when you apply your symbol games to what you unknowingly label as “manifest reality”, when you (essentially) claim that symbol game A (Peano arithmetic) models symbol game B (that part of the Physics game that deals with the objects you’ve identified as pebbles).

Platonism is not a means of excluding a mathematical object because it’s not one of the Forms; it is a means of allowing any mathematical object to have a Form whether you like it or not. I don’t believe in God, but there still exists a Form for a mathematical object that looks a lot like “a universe in which God exists”. It’s just that conceiving of a possible world only makes an arrow from your world to its, not an arrow in the reverse direction, hence why “A perfect God would have the quality of existence” is such a laughable non-starter :) What if someone broke out of a hypothetical situation in your room right now?

ec429 Sep 14, 2011, 6:05 AM
12 points
in reply to: [deleted]’s comment on: How to Convince Me That 2 + 2 = 3
That would have been a damn nuisance, because throughout the rest of this comment thread we’d have been writing unhelpfully long strings of Ss. ;)