ec429

Karma: 133

ec429 Sep 24, 2011, 4:07 AM
1 point
in reply to: [deleted]’s comment on: Syntacticism

I am still arguing with you because I think your misstep poisons more than you have yet realized, not to get on your nerves.

I wasn’t suggesting you were trying to get on my nerves. I just think we’re talking past each other.

“A proof exists” is a much murkier statement and it is much more difficult to discuss its probability.

As a first approximation, what’s wrong with “\lim_{t → \infty} P(I can find a proof in time t)”?

Also, I don’t see why the prior has to be oracular; what’s wrong with, say, P(the 3^^^3th decimal digit of pi is even)=½? But then if the digit is X, then surely a proof exists that it is X (because, in principle, the digit can be computed in finitely many steps); it must be some X in [[:digit:]], so if it is even a proof exists that it is even; otherwise (sharp swerve) one does not, and P=½. Not sure about that sharp swerve; if I condition all my probabilities on |arithmetic is consistent) then it’s ok. But then, assuming I actually need to do so, the probabilities would be different if conditioned on |arithmetic is inconsistent), and thus by finding a proof, you find evidence for or against the assertion that arithmetic is consistent. But things you can find evidence on, exist! (They are the sheep that determine your pebbles.) So where did I go wrong? (Did I slip a meta-level somewhere? It’s possible; I was juggling them a bit.)

ec429 Sep 24, 2011, 3:57 AM
−1 points
in reply to: [deleted]’s comment on: Syntacticism

lengthiness is not expected to be the only obstacle to finding a proof

True; stick a ceteris paribus in there somewhere.

You are trying to reason about reality from the point of view of a hypothetical entity that has infinite resources.

Not so; I am reasoning about reality in terms of what it is theoretically possible we might conclude with finite resources. It is just that enumerating the collection of things it is theoretically possible we might conclude with finite resources requires infinite resources (and may not be possible even then). Fortunately I do not require an enumeration of this collection.

I am certainly not saying that feasible proofs cause things to be true. Our previous slow computer and our new fast computer cause exactly the same number of important things to be true: none at all. That is the formalist position, anyway.

So either things that are unfeasible to prove can nonetheless be true, or nothing is true. So why does feasibility matter again?

P(I will prove the negation of your theorem in fewer than m+1 minutes) = p

No, it is > p. P(I will prove 1=0 in fewer than m+1 minutes) = p + epsilon. P(I will prove 1+1=2 in fewer than m+1 minues) = nearly 1. This is because you don’t know whether my proof was correct.

ec429 Sep 24, 2011, 3:41 AM
1 point
in reply to: loqi’s comment on: The Apparent Reality of Physics

Paul Almond

To Minds, Substrate, Measure and Value Part 2: Extra Information About Substrate Dependence I make his Objection 9 and am not satisfied with his answer to it. I believe there is a directed graph (possibly cyclic) of mathematical structures containing simulations of other mathematical structures (where the causal relation proceeds from the simulated to the simulator), and I suspect that if we treat this graph as a Markov chain and find its invariant distribution, that this might then give us a statistical measure of the probability of being in each structure, without having to have a concept of a physical substrate which all other substrates eventually reduce to.

However, I’m not sure that any of this is essential to my OP claims; the measure I assign to structures for purposes of forecasting the future is a property of my map, not of the territory, and there needn’t be a territorial measure of ‘realness’ attached to each structure, any more than there need be a boolean property of ‘realness’ attached to each structure. I note, though, that, being unable to explain why I find myself in an Everett branch in which experiments have confirmed the Born rule (even though in many worlds (without mangling) there should be a ‘me’ in a branch in which experiments have consistently confirmed the Equal Probabilities rule), I clearly do not have an intuitive grasp of probabilities in a possible-worlds or modal-realistic universe, so I may well be barking up the wrong giraffe.

EDIT: In part 3, Almond characterises the Strong AI Hypothesis thus:

A mind exists when the appropriate algorithm is being run on a physical system.

I characterise my own position on minds thus:

A mind exists when there is an appropriate algorithm, whether that algorithm is being run on a physical system or not. If the existence-of-mind inheres in the interpretative algorithm rather than the algorithm-that-might-be-run, then the interpretative algorithm is the appropriate one; but the mind still exists, whether the interpretative algorithm is being run on a physical system or not.

This is because the idea of a ‘physical system’ is an attachment to physical realism which I reject in the OP.

ec429 Sep 24, 2011, 2:39 AM
0 points
in reply to: lavalamp’s comment on: The Apparent Reality of Physics
But then, how do you determine whether information exists-in-the-universe at all? Does the number 2 exist-in-the-universe? (I can pick up 2 pebbles, so I’m guessing ‘yes’.) Does the number 3^^^3 exist-in-the-universe? Does the number N = total count of particles in the universe exist-in-the-universe? (I’m guessing ‘yes’, because it’s represented by the universe.) Does N+1 exist-in-the-universe? (After all, I can consider {particles in the universe} union {{particles in the universe}}, with cardinality N+1) If you allow encodings other than unary, let N = largest number which can be represented using all the particles in the universe. But I can coherently talk about N+1, because I don’t need to know the value of a number to do arithmetic on it (if N is even, then N+1 is odd, even though I can’t represent the value of N+1). Does the set of natural numbers exist-in-the-universe? If so, I can induct—and therefore, by induction on induction itself, I claim I can perform transfinite induction (aka ‘scary dots’) in which case the first uncountable ordinal exists-in-the-universe, which is something I’d quite like to conclude.

So where does it stop being a heap?

ec429 Sep 24, 2011, 2:31 AM
−2 points
in reply to: [deleted]’s comment on: Syntacticism
I am aware it can be very small. The only sense in which I claimed otherwise was by a poor choice of wording. The use I made of the claim that “Agents implementing the same deduction rules and starting from the same axioms tend to converge on the same set of theorems” was to argue for the proposition that there is a fact-of-the-matter about which theorems are provable in a given system. You accept that my finding a proof causes you to update P(you can find a proof) upwards by a strictly positive amount—from which I infer that you accept that there is a fact-of-the-matter as to whether a proof exists. In which case, you are not arguing with my conclusion, merely with a step I used in deriving it—a step I have replaced—so does that not screen off my conclusion from that step—so why are you still arguing with me?

ec429 Sep 24, 2011, 2:10 AM
2 points
in reply to: David_Allen’s comment on: Syntacticism

Your conclusion on truth is a physical state in your mind, generated by physical processes. The existence of a metaphysical truth is not required for you to come to that conclusion.

I think a meta- has gone missing here: I can’t be certain that others tend to reach the same truth (rather than funny hats), and I can’t be certain that 2+2=4. I can’t even be certain that there is a fact-of-the-matter about whether 2+2=4. But it seems damned likely, given Occamian priors, that there is a fact-of-the-matter about whether 2+2=4 (and, inasmuch as a reflective mind can have evidence for anything, which has to be justified through a strange loop on the bedrock, I have strong evidence that 2+2 does indeed equal 4).

That “truth” in the map doesn’t imply truth in the territory, I accept. That there is no truth in the territory, I vehemently reject. If two minds implement the same computation, and reach different answers, then I simply do not believe that they were really implementing the same computation. If you compute 2+2 but get struck by a cosmic ray that flips a bit and makes you conclude “5!”, then you actually implemented the computation “2+2 with such-and-such a cosmic ray bitflip”.

I am not able to comprehend the workings of a mind which believes arithmetic truth to be a property only of minds, any more than I am able to comprehend a mind which believes sheep to be a property only of buckets. Your conclusion on sheep is a physical state in your mind, generated by physical processes. But the sheep still exist outside of your mind.

ec429 Sep 24, 2011, 1:55 AM
0 points
in reply to: [deleted]’s comment on: Syntacticism

A positive but minuscule amount.

Right—but if there were no ‘fact-of-the-matter’ as to whether a proof exists, why should it be non-zero at all?

ec429 Sep 24, 2011, 12:53 AM
0 points
in reply to: ArisKatsaris’s comment on: The Apparent Reality of Physics

we find it hard to taboo words that are truly about the fundamentals of our universe, such as ‘causality’ or ‘reality’ or ‘existence’ or ‘subjective experience’.

I tabooed “exist”, above, by what I think it means. You think ‘existence’ is fundamental, but you’ve not given me enough of a definition for me to understand your arguments that use it as an untabooable word.

words like ‘mathematical equations’

I’d say that (or rather ‘mathematics’) is just ‘the orderly manipulations of symbols’. Or, as I prefer to phrase it, ‘symbol games’.

‘correspond to concepts in the material universe in order to predict happenings in said material universe’

That’s applied mathematics (or, perhaps, physics), an entirely different beast with an entirely different epistemic status.

Why don’t you taboo the words “mathematics” and “equations” first, and see if your argument still makes any sense

Manipulations of symbols according to formal rules are the ontological basis, and our perception of “physical reality” results merely from our status as collections of symbols in the abstract Platonic realm that defines the convergent results of those manipulations, “existence” being merely how the algorithm feels from inside.

Yup, still makes sense to me!

ec429 Sep 24, 2011, 12:26 AM
0 points
in reply to: [deleted]’s comment on: Syntacticism
But why should feasibility matter? Sure, the more steps it takes to prove a proposition, the less likely you are to be able to find a proof. But saying that things are true only by virtue of their proof being feasible… is disturbing, to say the least. If we build a faster computer, do some propositions suddenly become true, because we now have the computing power to prove them?

Me saying I have a proof of a theorem should cause you to update P(you can find a proof) upwards. (If it doesn’t, I’d be very surprised.) Consequently, there is something common.

Similarly, no matter how low your prior probability for “PA is consistent”, so long as that probability is not 0, learning that I have proved a theorem should cause you to decrease your estimate of the probability that you will prove its negation.

ec429 Sep 24, 2011, 12:16 AM
0 points
in reply to: ArisKatsaris’s comment on: The Apparent Reality of Physics
Ok, now taboo your uses of “reality” and “preexisted” in the above comment, because I can’t conceive of meanings of those words in which your comment makes sense.

ec429 Sep 24, 2011, 12:14 AM
0 points
in reply to: DuncanS’s comment on: The Apparent Reality of Physics

Surely can’t be exactly what you mean, as exists(our Univese) and ¬exists(everything else) seems coherent if rather unlikely

I would dispute this, on the grounds that my deductions in formal systems come from somewhere that has a causal relation to my brain—the formal system causes me to be more likely to deduce the things which are valid deductions than the things that aren’t. So, if I ‘exist’, I maintain that the formal systems have to ‘exist’ too, unless you’re happy with ‘existing’ things being causally influenced by ‘non-existing’ things—in which case there’s not a lot of point in asserting that ¬exists(infinite sets). A definition of ‘exists’ which doesn’t satisfy my coherence requirements is, I am attempting to argue, simply a means of sneaking in connotations.

ec429 Sep 24, 2011, 12:08 AM
1 point
in reply to: Manfred’s comment on: The Apparent Reality of Physics

So if you’ve ever read Probability Theory, by E.T. Jaynes

I haven’t; I probably should.

the position that, in order to make sense when applied to the real world, infinite things have to behave like limits of finite things.

Is this “limits” in the sense of analysis (epsi-delta limits), or is it “limit points” (like ω)? If the former, then that position involves not believing that arithmetic makes sense when applied to the real world. If the latter, then the position doesn’t seem different from what most mathematicians believe, because allowing limit points gets you transfinite induction… but then, as I intend to show, that gets you the first uncountable ordinal, by the power of ‘scary dots’. So… either EY doesn’t believe arithmetic applies to the real world, or EY doesn’t know the logical consequences of his beliefs, or EY doesn’t believe the above position, or I’ve made an error. Of course, at this point the last of those is rather likely, which is exactly why I want to formalise my argument and lay it out as coherently as possible.

Also, if I can’t say “exists”, how come you can talk about “the real world”? Double standards if you ask me ;)

ec429 Sep 23, 2011, 11:40 PM
1 point
in reply to: DuncanS’s comment on: The Apparent Reality of Physics
Getting rid of the ‘exists’ concept is more or less what I’m trying to do—or rather, show that if you have an ‘exists’ concept such that ¬exists(infinite sets) then your ‘exists’ concept is incoherent; moreover, that an ‘exists’ defined by exists(our Universe) and ¬exists(everything else) is not an important concept and should be detached from the connotations the savannah brain likes to associate with things that ‘exist’.

“exist” doesn’t have a referent. Any attempt to define it will either be special pleading (my universe is special, it “exists”, because it’s the one I live in!), or will give a definition that applies equally to all mathematical structures.

ec429 Sep 23, 2011, 11:34 PM
3 points
in reply to: [deleted]’s comment on: The Apparent Reality of Physics
Well, I’ve seen him go into quite some detail in comment threads about uncountable ordinals; it looks to me very much as though there is a position there that he’s taking (it’s fairly clear that it’s more than just a joke). OTOH, I haven’t seen an article by him setting out his position in a rigorous or articulated fashion, though if there is one I’d like to know about it.

ec429 Sep 23, 2011, 11:29 PM
4 points
in reply to: [deleted]’s comment on: Syntacticism
To a conscious agent within a mathematical system which models rocks and trees and people and the Moon, it looks very much as though rocks and trees and people and the Moon exist; physics appears to be real. But that appearance is in no way contingent upon anything we could reasonably define as the ‘existence’ of rocks and trees and people and the Moon.

Is that gleanable, or am I still talking gibberish? (I never rule out the latter, particularly on the subject of philosophy.)

ec429 Sep 23, 2011, 11:25 PM
0 points
in reply to: [deleted]’s comment on: Syntacticism
Um, that’s not quite what I meant… if I prove a theorem you’re unlikely to prove its negation, and if I tell you I have a proof of a theorem you’re likely to be able to find one. Moreover, in the latter case, if you’re not able to find a proof, then either I am mistaken in my belief that I have a proof, or you haven’t looked hard enough (and in principle could find a proof if you did). That doesn’t mean that every agent with “I think therefore I am” will deduce the existence of rice pudding and income tax, it only means that if one agent can deduce it, the others in principle can do the same. For any formal system T, the set {X : T proves X} is recursively enumerable, no? (assuming ‘proves’ is defined by ‘a proof of finite length exists’, as happens in Gödel-numbering / PA “box”)

ec429 Sep 23, 2011, 11:17 PM
0 points
in reply to: Manfred’s comment on: The Apparent Reality of Physics
Yes, but I’m actually going somewhere with this: EY doesn’t believe that “infinite sets exist” (loosely put). So I’m trying to deduce constraints on what “exist” can mean and still be coherent; at this point I don’t think we can even claim that lavalamp’s kangaroos “don’t exist”.

ec429 Sep 23, 2011, 11:14 PM
0 points
in reply to: DuncanS’s comment on: The Apparent Reality of Physics

We start off by saying that the universe that I can sense exists, along with me.

I can see no justification whatever for that method. What’s so special about you, that you imbue things with “existing-ness” by sensing them? Surely that’s an egregious Mind Projection Fallacy? “It’s in my map, so it must be in some territory somewhere”?

ec429 Sep 23, 2011, 11:09 PM
0 points
in reply to: lavalamp’s comment on: The Apparent Reality of Physics

I said information, you said theorem—I don’t think it’s the same.

“Theorem Foo is true in Theory T” is information. Though the 3^^^3rd digit of pi is good too; I want to hear what you have to say about it.

I think it’s time to taboo “exist”.

Ok… “exist” doesn’t have a referent. Any attempt to define it will either be special pleading (my universe is special, it “exists”, because it’s the one I live in!), or will give a definition that applies equally to all mathematical structures.
What links here?
- ec429's comment on The Apparent Reality of Physics by ec429 (Sep 23, 2011, 11:40 PM; 1 point)

ec429 Sep 23, 2011, 11:04 PM
1 point
in reply to: lavalamp’s comment on: The Apparent Reality of Physics

Does your model differ from “Everything that could exist, does”?

Not quite. Every mathematical structure (strictly, every syntactic language) that could exist, does (and causes its corresponding apparent physical reality to appear to exist) - so the kangaroos do indeed exist (and did so even before you thought of them). After all, how could you have an effect on the kangaroos merely by simulating them? (I maintain that the causal relation runs from the kangaroos to you, not the other way round).

It may well be impossible to devise an example of something that doesn’t exist (because, if we can think about it, it is modellable), but that doesn’t mean to say that things that don’t exist, um, don’t exist… I mean, {X : not exist(X)} need not be empty, it’s just that we can’t exhibit any of its elements. It might be provably empty, but I have no proof here and can’t imagine what form one would take.

Also, you have to be a bit careful about the causal relations; if you start thinking of them as one formal system modelling another, you get into Gödel-space. Hence the Syntacticism post; I think it’s accurate to say that one formal system models a quotation of another, but I’m not sure of that.