tag:blogger.com,1999:blog-5801278565856116215.post3845206886383994159..comments2017-06-09T20:32:56.717+01:00Comments on Disagreeable Me: Rescuing Computationalism with PlatonismDisagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comBlogger105125tag:blogger.com,1999:blog-5801278565856116215.post-86626542445973257282016-11-10T13:57:43.522+00:002016-11-10T13:57:43.522+00:00> I'm just pointing out that we don't k...> I'm just pointing out that we don't know the conditions under which something is possible in terms of physical stuff. You're assuming that mathematical consistency suffices;<br /><br />It's more that I'm not assuming there is anything else to consider. We already know that if something is mathematically inconsistent (i.e. impossible) then it cannot exist. You're postulating some other unknown set of conditions that need to hold. Not only do I not see any reason to assume this, I don't think it gets us anywhere, because it is logically possible that that set of conditions could be otherwise, and then we can still consider the class of possible worlds that would be possible if the set of conditions were otherwise.<br /><br />> But if math is all there is, then what makes them true? What's the truthmaker of these propositions?<br /><br />That it is not mathematically possible to prove them false. For instance, if the Goldbach conjecture is true but unprovable, then it is not mathematically possible to construct a counter-example. This is the truth-maker of the Goldbach conjecture, and it is far from independent of mathematics.<br /><br />> The natural numbers are a model of the Peano axioms, but if you add something to them, then you need a new model<br /><br />I don't think this is true as long as the natural numbers are consistent with the new axioms.<br /><br />For example, if I adopted a new axiom that 1 < 2, that would not mean the natural numbers are no longer a model of the new system.<br /><br />> But that would be wrong (provided that the Peano axioms are indeed consistent). <br /><br />First, a minor correction, I should have said "they cannot prove their own consistency" rather than "they can prove their own inconsistency".<br /><br />Provided that they are consistent, then they cannot be complete. If we are assuming that they are consistent, that means we can add the statement "The Peano axioms are not complete".<br /><br />So far we have.<br /><br />The Peano axioms are consistent (A)<br />The Peano axioms are not complete (~B).<br />The Peano axioms are complete => The Peano axioms can prove their own consistency. (B=>C)<br /><br />I am claiming that "The Peano axioms are complete => the peano axioms cannot prove their own consistency" (B=>~C) is derivable, and you disagree.<br /><br />That ought to be simple enough to work out.<br /><br />Working backwards, B=>~C is equivalent to ~B v C<br />But we already have ~B, therefore ~B v C is true.<br /><br />QED.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-16122670058582199702016-11-10T13:57:34.647+00:002016-11-10T13:57:34.647+00:00> Because there's nothing (apart from, mayb...> Because there's nothing (apart from, maybe, aesthetics) to motivate saying something else.<br /><br />OK, so you are advocating the rationality of adopting as many groundless metaphysical assumptions as you like, simply on the basis that it pleases you. So, for instance, it is rational to believe in invisible pink unicorns and a rich pantheon of non-intervening deities. On this I will just have to flatly disagree. If we can't find common ground here, then the conversation is futile.<br /><br />> and there are propositions true in every world without the mathematics deciding their truth;<br /><br />I think here we're just talking at cross purposes. For me, even if the Goldbach conjecture is intrinsically unprovable, its truth is logically and mathematically necessary, and this truth is determined by mathematics. For you, it is true for no reason and it is not logically or mathematically unnecessary. This is a trivial distinction of terminology. For me, logical necessity is just what has to be true in all possible worlds, even if we could not derive a contradiction from assuming the opposite. You may therefore have a problem with the way I express my view, but this is not in fact a criticism of the view itself.<br /><br />> This is a mixing of syntactic levels with semantic ones that, I think, really only generates trouble.<br /><br />Well, fair enough, but I did say "and with the evidence" a number of times previously. I thought a little shorthand would be excused in this context.<br /><br />> This is disanalogous: since for a battery to perform work actually requires energy<br /><br />That depends on what you mean by "energy". Again, you're assuming that we're talking the same language. In this case, we normally would be, but on existence we are not. So if to me, "energy" means being blue, then you don't require DM-energy to perform work. The truth of any statement depends not only on the world but on the way that statement is interpreted. If you and I interpret existence differently, then statements that are true for you may be false for me and vice versa.<br /><br />> No; there are some properties of QFTs that aren't decidable, some which are.<br /><br />I dealt with that in my next paragraph. It doesn't bother me if there are some convergent properties and some divergent properties of infinite lattices, as long as the divergent properties of infinite lattices never manifest in the real world. And it seems they don't.<br /><br />> -both quantum theory and general relativity depend on the continuum of real numbers, almost all of which aren't computable.<br /><br />True. I take the computability of physics to mean that outcomes/predictions can be computed to an arbitrary precision. I think this is what is generally meant when the computability of physics is debated. Pi is one of the computable reals, for instance, even though we can't compute it to infinite precision.<br /><br />> Well, for one, the laws themselves don't specify the behavior of a given system---you also need the initial conditions for that.<br /><br />Well, OK. The laws specify behaviour given initial conditions. Behaviour given initial conditions is laws, laws are behaviour given initial conditions.<br />Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-69647258212182342092016-11-10T13:57:19.367+00:002016-11-10T13:57:19.367+00:00> No! I could prove it if and only if it is cor...> No! I could prove it if and only if it is correct:<br /><br />I can see where you are coming from but explosion means you are wrong. The proposition that you can solve the halting problem is a contradiction. Anything follows from a contradiction. Again, I know you are not actually assuming that you can solve the halting problem, but you are using the proposition that you can solve the halting problem as the antecedent in an implication, and anything follows from using a contradiction as an antecedent in an implication.<br /><br />> Take the violation of the second law (X): it suffices to build a perpetuum mobile (Y), but it isn't necessary for it, which is shown by the fact that the violation of the first law (Z) also suffices. <br /><br />This is where the last part of my sentence comes in, although I realise I need to modify it a little to be strictly correct.<br /><br />I said "unless you show that it is possible for Z to hold without X holding". I should have said "unless you show that Z suffices without X holding".<br /><br />A violation of the first law is enough to build a perpetuum mobile if and only if the second law is also violated. A violation of the first law permits the construction of a perpetuum mobile just because it permits a violation of the second law. So just because you can build a perpetuum mobile by violating the first law does not mean that you can build a perpetuum mobile without violating the second law.<br /><br />> How could Jochen-existence depend on DM-existence, when both are simply mutually exclusive concepts?<br /><br />If they are mutually exclusive, and X is true for B, then I must take Z to be false for B. Z must therefore be the proposition that B Jochen-exists (as opposed to the proposition that B does not Jochen-exist). If this is what you're saying, then Z cannot depend on X. Since they are mutually exclusive, your argument simply doesn't work. This is where I complain that "in your example, Z (Jochen-existence) doesn't even hold for B". Your argument says both that Z suffices for B to be implementable, and that Z is not true for B. This is confused. Your argument would only show that Z suffices for B to be implementable if Z were true for B.<br /><br />Whether Z is true for B is a predicate. Whether Z is the definition of existence is not a predicate. It doesn't hold or not hold. It's just a convention or a definition. I think this is where you are going wrong.<br /><br />> That's not my argument. I'm saying that B's DM-existence isn't necessary, because its Jochen-existence (i.e. non-existence) is sufficient.<br /><br />Now, here, you seem to be saying the opposite. You seem to be interpreting Z as the proposition that B does not Jochen-exist. In this case, X and Z can both hold at the same time, so the fact that Z is sufficient for B to be implementable does not mean that X is not necessary for B to be implementable. To show that X is not necessary for B to be implementable, you need to show a case where X does not hold for B but B is implementable. That is not logically possible, because what DM-exists is defined (more or less) as what it is possible to implement in principle.<br />Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-51468587067535983672016-11-10T13:56:59.192+00:002016-11-10T13:56:59.192+00:00> Arbitrary values of all (infinitely many) add...> Arbitrary values of all (infinitely many) additional bits can be added as axioms to S without producing an inconsistency<br /><br />So this is not quite true. As n grows, then arbitrary values for bits can yield inconsistencies.<br /><br />> And no 'potential for inconsistency'<br /><br />So there is a potential for inconsistency, because we don't know how far we can go with n.<br /><br />> Soundness means that every theorem of the system holds true in every model;<br /><br />I'm not sure this is a sensible definition. If a theorem doesn't hold true in some putative model, that just means that it is not a model, surely. Soundness, in my view, assumes a model. A system is sound for a putative model if every theorem of the system holds true for that model. Indeed, it is a model only if the system is sound for that model. This is what soundness means in logic and I presume it is similar in mathematics.<br /><br />The abstract syllogism<br /><br />For All x: A(x) => B(x)<br />A(y)<br />Therefore B(y)<br /><br />Is valid. Whether it is sound depends on what model we adopt. If A is modelled as the predicate "x is a man", and B is modelled as "x is mortal", and y is modelled as Socrates, then this is a sound syllogism.<br /><br />If B is instead modelled as "x is immortal", then this is not a sound syllogism. Equivalently, I can say that this interpretation is not a model of the abstract syllogism. Soundness is therefore a property of formal system + model taken together, not a property of a formal system alone. Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-12881426748165499792016-11-10T13:56:44.165+00:002016-11-10T13:56:44.165+00:00Hi Jochen,
> What good is an algorithm that ma...Hi Jochen,<br /><br />> What good is an algorithm that may or may not solve the halting problem?<br /><br />No good at all. I'm not saying it's any good, I'm saying that I believe you to be using the term "undecidable" incorrectly. I don't think that it is meaningful to say that the halting of a specific algorithm A is undecidable. It is, on the other hand, meaningful to hypothesize that A does not halt and that this cannot ever be proven by any means. I take it this is what you mean to say.<br /><br />I'm not convinced that it has been proven that such an algorithm exists. The proofs we have are subtly different, that there is no universal effective procedure for solving halting problems (or, equivalently, in Gödel terms, that there are true statements for which there are no proofs in any given axiomatic system). It seems to me that this doesn't quite rule out the possibility of there being a proof of non-halting for every algorithm that doesn't halt, although personally I doubt it. In Gödelian terms, this would be the unlikely possibility that for every true statement in an axiomatic system, there exists a proof of truth if we can bring in arguments from without the axiomatic system (as Gödel himself does to show that his unprovable statements are in fact true).<br /><br />> But if *every* specific instance of the halting problem were decidable, then the halting problem would be decidable in general.<br /><br />Not so, since a general solution which merely delegated to specific solutions would need to specify which specific solution to give for every possible algorithm, of which there are infinitely many. A general solution would therefore need to be infinitely long, and so is not an effective procedure and not an algorithm.<br /><br />> But then, there must exist some T for which you can't solve the halting problem for A.<br /><br />Of course. For any A, there always exists a T which A cannot decide whether it halts or not. But that doesn't mean T is undecidable. It only means that T is undecidable by A. There may be another algorithm B which can decide it. So undecidability is not a property of a Turing machine T alone but a property of a two-place relation (A, T). The Halting Problem shows that there exists no A for which (A,T) is always decidable, given any T. It doesn't show that there exists no T for which (A,T) is always undecidable, given any A. As I understand decidability, this is certainly false. However I am prepared for the sake of argument (though I am not sure if it is true or not) that there exists a T for which there is no way to prove that it doesn't halt.<br /><br />> S can only determine C scattered bits of its binary expansion.<br /><br />This is neither here nor there, but I don't think this is technically true. The bits we can determine are not scattered as far as I understand -- we ought to be able to determine the first n bits. The value of n is not known -- it depends on how clever and inventive we get with our proofs. There is no upper bound on n, but however clever we get, n will always be finite.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-35764472830151850372016-11-10T10:46:19.803+00:002016-11-10T10:46:19.803+00:00Hi Jochen,
> sorry, I'm losing track of wh...Hi Jochen,<br /><br />> sorry, I'm losing track of who replied where to what... But anyway:<br /><br />My fault, didn't mean to start a new top-level comment. Also, I don't really have the bandwidth to keep up, so I'm posting comments piecemeal and replying to things you said a couple of days ago. Perhaps email would be a better venue to continue the conversation. You can contact me via the contact form on the right sidebar if that interests you.<br /><br />Anwyay, I'm over to the other thread to continue for now...Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-90483348054311271062016-11-09T22:05:43.391+00:002016-11-09T22:05:43.391+00:00Hi DM,
sorry, I'm losing track of who replied...Hi DM,<br /><br />sorry, I'm losing track of who replied where to what... But anyway:<br /><br />>his view would be incompatible with any result which proved that the physics of this world are not computable<br /><br />Well, even without this result, physics isn't computable in its current form---both quantum theory and general relativity depend on the continuum of real numbers, almost all of which aren't computable. So if, say, the position of a particle can take on arbitrary real values, then almost none of these are computable.<br /><br />>but an identity relation (laws are behaviour, behaviour is laws).<br /><br />Well, for one, the laws themselves don't specify the behavior of a given system---you also need the initial conditions for that. Consequently, systems obeying the same laws may behave quite differently. So I don't think I see this identity.<br /><br />>Anything at all can happen, and whatever happens we would describe with a law.<br /><br />I'm not sure about that---there may be systems that don't behave according to some finitely specifiable set of laws.<br /><br />>logically possible because this entails no contradiction that I can see.<br /><br />I'm just pointing out that we don't know the conditions under which something is possible in terms of physical stuff. You're assuming that mathematical consistency suffices; I simply see no reason for making this assumption. <br /><br />>It's an ungrounded assumption which I guess may be true<br /><br />As is the assumption that anything that's mathematically consistent may possibly exist as a physical world. This is a metaphysical hypothesis that may be, in fact, wrong.<br /><br />>which I would be inclined to discard as unparsimonious<br /><br />Parsimony is no indicator for truth, or even likelihood. All it gets you is falsifiability in empirical contexts---which we don't have in a metaphysical setting. Aside from that, I don't see why 'mathematical consistency dictates what is possible' is any more parsimonious than 'physical realizability dictates what is possible'. Sure, we might not know what exactly is meant by physical realizability; but it's not like consistency is an easily settled question, either.<br /><br />>I would call them necessarily true but unprovable.<br /><br />But if math is all there is, then what makes them true? What's the truthmaker of these propositions?<br /><br />>I don't see why they wouldn't be a model.<br /><br />Because you've changed the axioms. The natural numbers are a model of the Peano axioms, but if you add something to them, then you need a new model, i.e. a mathematical structure that satisfies the properties as laid out by the axioms. <br /><br />>but I also accept "If the Peano axioms were complete, they could prove their own inconsistency"<br /><br />But that would be wrong (provided that the Peano axioms are indeed consistent). You're simply misapplying explosion here: nowhere is a contradiction assumed to be true.Jochenhttps://www.blogger.com/profile/07418841955052661428noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-7987380891073250162016-11-09T15:35:49.970+00:002016-11-09T15:35:49.970+00:00>we can get the predictions of QFT by postulati...>we can get the predictions of QFT by postulating an infinite lattice, which would mean that these properties must be decidable.<br /><br />No; there are some properties of QFTs that aren't decidable, some which are. This really isn't different from, e.g., the situation within the GoL: whether a certain pattern ever develops is undecidable; yet, for any given initial configuration, this pattern either develops, or fails to.<br /><br />>I don't think that taking the limit as a approaches 0 is the same as postulating an infinite lattice in any case.<br /><br />Well, be that as it may, the authors of the paper certainly do believe their result to be (potentially) applicable to real-world quantum field theories. Jochenhttps://www.blogger.com/profile/07418841955052661428noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-47428608035031541742016-11-09T15:32:13.573+00:002016-11-09T15:32:13.573+00:00>Indeed you could both prove and disprove it
N...>Indeed you could both prove and disprove it<br /><br />No! I could prove it if and only if it is correct: I can set up a computer that runs through all numbers, trying to find a counterexample, and that halts if it does. So if I can decide this computer's halting problem, then I know, if it halts, that a counterexample exists, and, if it fails to halt, that Goldbach's conjecture is true.<br /><br />>As I see it, we were disagreeing about whether a perpetuum mobile was logically possible or not.<br /><br />I was arguing that the violation of the second law implies the possibility of a perpetuum mobile; that there are other ways of building one doesn't make that false. (Indeed, this would be denying the antecedent.)<br /><br />>You can't show that X is not necessary for Y by showing that Z suffices for Y unless you show that it is possible for Z to hold without X holding. <br /><br />Huh? If I show that Z suffices for Y, then what I've shown is exactly that X isn't necessary for Y. Take the violation of the second law (X): it suffices to build a perpetuum mobile (Y), but it isn't necessary for it, which is shown by the fact that the violation of the first law (Z) also suffices. <br /><br />I don't understand the last part of your sentence, though, so maybe you meant something different. How could Jochen-existence depend on DM-existence, when both are simply mutually exclusive concepts?<br /><br />>What I'm saying is that B must DM-exist for B to be implementable. That B is implementable without Jochen-existing has no bearing on what I am saying.<br /><br />That's not my argument. I'm saying that B's DM-existence isn't necessary, because its Jochen-existence (i.e. non-existence) is sufficient.<br /><br />>Why ever would you say that?<br /><br />Because there's nothing (apart from, maybe, aesthetics) to motivate saying something else. My aesthetic prejudices simply might differ from yours!<br /><br />>OK, so show the MUH to be false.<br /><br />Well, to me, implying that structure is all there is to the world already does that. But you're unlikely to accept this, I'd wager.<br /><br />However, we've already seen two ways for how the MUH could turn out to be false: the Gödelian worry could be justified, and there are propositions true in every world without the mathematics deciding their truth; or, the world could not correspond to a single, consistent, mathematical structure. Neither of which would render the MUH inconsistent---merely wrong.<br /><br />>I'm taking consistency here to mean consistency with the evidence as well as with itself.<br /><br />This is a mixing of syntactic levels with semantic ones that, I think, really only generates trouble. There's good reasons these levels are kept well separate in mathematical logic.<br /><br />>so since there is a concept of energy on which a red battery doesn't have energy, and can neverthleless do work, then energy isn't necessary to do work.<br /><br />This is disanalogous: since for a battery to perform work actually requires energy, you of course descend into absurdity. However, to instantiate a computation, its existence isn't required.Jochenhttps://www.blogger.com/profile/07418841955052661428noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-43623548025570227452016-11-09T15:12:32.576+00:002016-11-09T15:12:32.576+00:00Hi DM,
>This conversation is a welcome distrac...Hi DM,<br /><br />>This conversation is a welcome distraction from what's going on in the world!<br /><br />Yes, it is. Thinking about possible worlds is infinitely more pleasant than thinking about the real one right now.<br /><br />>One algorithm which decides that it halts is either the algorithm which always returns true, or alternatively the algorithm that always returns false<br /><br />Typically, one would require of an algorithm that it's possible to know if it implements the function one is interested in. What good is an algorithm that may or may not solve the halting problem? What good is an algorithm that may or may not compute the digits of pi, if one wants to know the digits of pi?<br /><br />>Whether any specific algorithm halts is not, nor is it sensible to suppose there is a fact of the matter on whether it undecidable.<br /><br />But if *every* specific instance of the halting problem were decidable, then the halting problem would be decidable in general. Since it's not decidable in general, it's not decidable in some specific case.<br /><br />Take an algorithm A which takes as input the description of some Turing machine T, and then simulates the performance of this machine on an initially blank tape. It's clear that you can't solve the halting problem for A for every input t: if you could, the halting problem would be decidable. But then, there must exist some T for which you can't solve the halting problem for A.<br /><br />>If the program halts, you can prove that it halts, just by running through the steps until you hit the end.<br /><br />OK, yes: there is an algorithm which eventually spits out the description of every Turing machine that halts. <br /><br />But my original assertion still works for the digits of Chaitin's constant: for every axiomatic system S, there exists a constant C such that for some given Chaitin constant, S can only determine C scattered bits of its binary expansion. Arbitrary values of all (infinitely many) additional bits can be added as axioms to S without producing an inconsistency. (And no 'potential for inconsistency', either: we know that there is no proof/disproof of the proposition 'the value of the nth bit is 1' within the system if the nth bit is not within the C bits that can be proven to be a certain value.)<br /><br />>but that none of the axioms are inconsistent with facts which are necessary given those axioms.<br /><br />Well, but the values of the bits aren't really necessary given the axioms: by Gödel's completeness theorem, since both S+v(n)=1 and S+v(n)=0 are consistent (where v(n)=x is the axiom asserting that the value of the nth bit is x, where the value of the nth bit is not derivable from S), both have a model; consequently, neither v(n)=1 nor v(n)=0 is necessary given the axioms.<br /><br />>This is different from soundness, which is just that none of the axioms are inconsistent with (possibly contingent) facts about something the system is supposed to model.<br /><br />Soundness means that every theorem of the system holds true in every model; that is, the derivability of a formula within S ensures that the formula is true in every model of S. Jochenhttps://www.blogger.com/profile/07418841955052661428noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-13002076936870540432016-11-09T15:11:37.548+00:002016-11-09T15:11:37.548+00:00> If our world turns out not to be described b...> If our world turns out not to be described by a consistent mathematical structure, then the MUH is wrong,<br /><br />Agreed. And it is germane. I'm just trying to keep a handle on the explosion of digressions. It's getting a little unwieldy. I value this conversation greatly but it's getting out of control.<br /><br />> Well, in his paper, he motivates the CUH explicitly with Gödelian worries:<br /><br />I know, but in proposing the CUH he is assuming that the physics of this world are computable, which means that his view would be incompatible with any result which proved that the physics of this world are not computable. Now, he said that before this result, so I guess it's possible he has changed his mind but I haven't heard about it if so.<br /><br />> To my view, the laws of physics are descriptive rather than prescriptive:<br /><br />My view is that the laws of physics just are the behaviour of things. It's neither descriptive (behaviour before laws) nor prescriptive (laws before behaviour), but an identity relation (laws are behaviour, behaviour is laws).<br /><br />But if you have a purely descriptive interpretation (behaviour before laws), I don't think the concept of physical impossibility makes sense. Anything at all can happen, and whatever happens we would describe with a law. So I don't see how some mathematical object would be physically impossible in some other logically possible universe where things might behave differently.<br /><br />> I mean that there may not be possible physical stuff, the right kind of relata, to instantiate the GoL (as a world on its own). <br /><br />Possible in what sense? It is logically possible that in some other world, stuff might behave in such a way as to instantiate the GoL -- logically possible because this entails no contradiction that I can see. If it is logically possible for stuff to behave this way, then it must be logically possible for such a world to be physically possible, because on your account what is physically possible is just how stuff behaves.<br /><br />> and indeed, that it's logically necessitated what that nature is.<br /><br />OK, but you don't seem to have much of an argument to back that up. It's an ungrounded assumption which I guess may be true (although it's hard for me to even accept that since it's so vague -- I really have trouble seeing it as something other than a set of metaphysical laws), but which I would be inclined to discard as unparsimonious. So we're back to judging whether your view or mine is more reasonable if both are consistent. Perhaps better to focus on points where my view may be inconsistent, since we disagree on how to choose between consistent views.<br /><br />> Chaitin then calls these propositions 'true for no reason'---at least none within mathematics.<br /><br />I'm familiar with Chaitin's constant but I don't think it is a problem for my view. I wouldn't call them true for no reason. I would call them necessarily true but unprovable. "for no reason" has the wrong connotations for me -- connotations of contingency.<br /><br />> Sure, but if you do, then the natural numbers will no longer be a model of the axiom system you're proposing;<br /><br />I don't see why they wouldn't be a model. There is nothing in these axioms which are inconsistent with the natural numbers.<br /><br />> how about 'if the Peano axioms were complete, they could prove their own consistency'---does that work for you?<br /><br />I accept that statement as a good analogy, but I also accept "If the Peano axioms were complete, they could prove their own inconsistency", in general because of explosion, but more intuitively simply because no system can be both complete and consistent. Being complete means being inconsistent.<br /><br />So your examples are valid but trivial and useless.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-57910945996807983592016-11-09T14:46:34.841+00:002016-11-09T14:46:34.841+00:00> The problem is applicable to real-world quant...> The problem is applicable to real-world quantum field theories:<br /><br />I think that I'm basically not going to buy this, Jochen, even if I'm not competent to judge it. You can take this as wilful arrogance or ignorance on my part if you like, but as far as I can see it is not the case that this result has persuaded everyone in the physics community that there is no mathematical description of the world. I don't think I am rejecting a consensus, I think I am rejecting your particular interpretation.<br /><br />But, to give a sense of my problem with your interpretation (apart from that it would potentially shatter my world view), you're saying both that the properties of infinite spin lattices are undecidable and that we can get the predictions of QFT by postulating an infinite lattice, which would mean that these properties must be decidable.<br /><br />I don't think that taking the limit as a approaches 0 is the same as postulating an infinite lattice in any case. Taking a limit means that we can see that our predictions are converging even before we reach infinity. The undecidable properties of the spin lattice presumably aren't convergent and this is why we require actual infinities to have a problem. It doesn't bother me if we can compute some properties of some infinite structures, but equally it doesn't bother me if we cannot compute other properties of other infinite structures. It's like the difference between a convergent series and a divergent series. I only have a problem if there is some behaviour we can observe in the real world that cannot be described mathematically, and this does not appear to be the case.<br /><br />Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-20047543592993119752016-11-09T14:46:24.880+00:002016-11-09T14:46:24.880+00:00> 'If I could solve the halting problem, I ...> 'If I could solve the halting problem, I could prove (or disprove) Goldbach's conjecture.'<br /><br />Yes, that's a better example. Indeed you could both prove and disprove it, irrespective of whether it is true or not, because a contradiction (such as solving the halting problem) implies anything.<br /><br />> Perhaps, but that has no bearing on what I said:<br /><br />As I see it, we were disagreeing about whether a perpetuum mobile was logically possible or not. I said it was, you said it wasn't. If the first law is violated, it is logically possible. On the other point relating to statistical flukes, I guess we'll just have to disagree on whether something that is infinitesimally probable is logically possible or not. I accept all you say, but yet I think it is.<br /><br />> your argument needs that DM-existence is necessary for the implementability of B; which is shown not to be right, since Jochen-existence also suffices.<br /><br />That argument doesn't follow. You can't show that X is not necessary for Y by showing that Z suffices for Y unless you show that it is possible for Z to hold without X holding. In any case, in your example, Z (Jochen-existence) doesn't even hold for B, so I don't know why you are saying that you have shown that Jochen-existence suffices for B to be implementable.<br /><br />Perhaps you interpret me as saying that one must define existence as DM-existence for B to be implementable. Of course that's not what I'm saying. What I'm saying is that B must DM-exist for B to be implementable. That B is implementable without Jochen-existing has no bearing on what I am saying.<br /><br />> But since metaphysical hypotheses aren't empirical, this motivation for parsimony vanishes. <br /><br />Parsimony is just making as few ungrounded assumptions as possible. If that isn't self-evidently a sensible heuristic to you then I don't know what to tell you. To reject parsimony is just to assert that it is reasonable to accept ungrounded assumptions. Why ever would you say that?<br /><br />> By showing one to be false, of course.<br /><br />OK, so show the MUH to be false. Accusing it of circularlity is not showing it to be false.<br /><br />> A theory's consistency does not imply its truthfulness!<br /><br />I'm taking consistency here to mean consistency with the evidence as well as with itself. As far as I'm aware, the MUH is consistent in this sense. That still doesn't imply its truthfulness, but we have no better way of choosing between consistent views than parsimony. Of course agnosticism is perfectly acceptable.<br /><br />> Our concepts only differ in what we consider to exist; so since there is a concept of existence on which B doesn't exist, and is nevertheless instantiable, then existence isn't necessary to instantiability.<br /><br />To me, this argument is blatantly absurd. I think energy is the property of being blue in appearance. Our concepts only differ in what we consider to have energy; so since there is a concept of energy on which a red battery doesn't have energy, and can neverthleless do work, then energy isn't necessary to do work. Nonsense! DM-energy isn't necessary to do work. Jochen-energy is.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-26503268705060234112016-11-09T13:48:26.815+00:002016-11-09T13:48:26.815+00:00Hi Jochen,
This conversation is a welcome distrac...Hi Jochen,<br /><br />This conversation is a welcome distraction from what's going on in the world!<br /><br />> While it's true that the halting problem is decidable for certain TMs, it's indeed the case that there exist Turing machines for which it is undecidable. <br /><br />Your argument here strikes me as being a bit like Cantor's diagonal argument, so it is certainly a good effort. I'm not convinced that this kind of argument works in this case, though. <br /><br />I'm going to assume that a question is decidable if it is possible to write an algorithm that gets the right answer to that question 100% of the time. When you say that there are undecidable TM's, you are saying that there exists an algorithm X for which there does not exist an algorithm Y which can decide if it halts.<br /><br />As I have framed the question, it is obvious that you are wrong. There is a (possibly unknown) fact of the matter on whether X halts or not. One algorithm which decides that it halts is either the algorithm which always returns true, or alternatively the algorithm that always returns false, depending on what the fact of the matter is. For instance, for algorithm X1, which does not halt, then the algorithm which returns FALSE decides its halting problem. For the algorithm X2, which does halt, then the algorithm which returns TRUE decides its halting problem. There always exists an algorithm Y to decide whether X halts, but we don't know what that algorithm is and we won't know that it gets the right answer unless we figure out whether X halts.<br /><br />The halting problem only makes sense if you take it to be the generic problem of deciding whether any algorithm halts. That problem genuinely is provably undecidable. Whether any specific algorithm halts is not, nor is it sensible to suppose there is a fact of the matter on whether it undecidable. All that it makes sense to say is that we know that it halts, we know that it does not, or we do not know which.<br /><br />What you may want to say instead is that there exist TMs which do not halt and for which it is impossible to prove that they do not halt. I think this is probably true but I'm not sure.<br /><br />> No, I have merely created one which isn't sound.<br /><br />This is certainly not true if you take the former case (and not my vice versa). If the program halts, you can prove that it halts, just by running through the steps until you hit the end. If you take as an axiom that it does not halt, then you can prove that it does not halt. If you can prove both that it halts and that it does not halt, you have an inconsistent system on your hands.<br /><br />It's harder to demonstrate that a system is inconsistent if we have it the other way around -- if it doesn't halt, but we take it as an axiom that it does halt. I guess the best I can do is say that if it doesn't halt, there is always the potential that we could one day find a proof that it doesn't halt, and then we would have an inconsistent system on our hands. So I guess this situation is more of a system with a strong potential for inconsistency rather than one that I can reliably show to be inconsistent. It might not meet the formal definition of inconsistency but I take it to be inconsistent for my purposes -- I wouldn't consider it to be a viable basis on which to define a mathematical object that platonically exists. We can say it is pseudo-consistent. For my purposes, full consistency (call it ultra-consistency if you like) requires not only that a contradiction cannot be derived, but that none of the axioms are inconsistent with facts which are necessary given those axioms.<br /><br />This is different from soundness, which is just that none of the axioms are inconsistent with (possibly contingent) facts about something the system is supposed to model. For instance, non-Euclidean geometry may not be sound if taken as a model of the geometry of our spacetime at a macroscopic level, but it is (presumably) ultra-consistent.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-57506499246998257052016-11-09T10:06:43.435+00:002016-11-09T10:06:43.435+00:00>Do you realise you're drawing a distinctio...>Do you realise you're drawing a distinction between a logically possible world and a possible world?<br /><br />So no, I really don't think I do. What worlds are physically possible may still be logically determined, but that determination may not be the same as the one the MUH presumes.<br /><br />>If there is an a priori fact of the matter, then what happens is determined by mathematics.<br /><br />Well, many mathematicians would disagree with you on that point. Take <a href="https://en.wikipedia.org/wiki/Chaitin%27s_constant" rel="nofollow">Chaitin's constant</a>: no axiomatic system can determine more than finitely many bits of its binary expansion. But still, there is a definite such expansion: that 'bit n is 1' is either true or false for all n (which follows from the fact that every Turing machine either halts, or fails to halt). Chaitin then calls these propositions 'true for no reason'---at least none within mathematics.<br /><br />>the undecidable scenario never exists in reality.<br /><br />According to our best physical theories, it does, so this carries a commitment to most of modern physics being wrong---to me, this would be far too strong a conclusion to not worry about whether the metaphysical assumptions going into it actually hold up.<br /><br />>I am free to add as an additional axiom either that 5 is "foo" or 5 is not "foo".<br /><br />Sure, but if you do, then the natural numbers will no longer be a model of the axiom system you're proposing; rather, you'll end up with a new mathematical structure in which every element either has the 'foo' property, or not.<br /><br />Oh, and regarding the discussion on coming up with something logically false that leads to something sensible: how about 'if the Peano axioms were complete, they could prove their own consistency'---does that work for you?Jochenhttps://www.blogger.com/profile/07418841955052661428noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-61780256372380318362016-11-09T10:02:34.307+00:002016-11-09T10:02:34.307+00:00Hi DM,
>let's just assume I'm right on...Hi DM,<br /><br />>let's just assume I'm right on all the points that we're not actively discussing.<br /><br />I think whether there are alternatives to 'mathematical possibility' as a criterion for existence is very germane to the points we're discussing. If our world turns out not to be described by a consistent mathematical structure, then the MUH is wrong, and is in particular no help with the pixie problem---so if that's sufficient to conclude the failure of computationalism, then the question of whether our world is described by a consistent mathematical structure is essential to the plausibility of computationalism.<br /><br />>when he proposes the CUH he is endorsing the idea that there is no such thing as uncomputable physics<br /><br />Well, in his paper, he motivates the CUH explicitly with Gödelian worries: "I have long wondered whether Gödel's incompleteness theorem in some sense torpedos the MUH".<br /><br />>But what is physically possible is determined by laws of physics of a particular world. <br /><br />To my view, the laws of physics are descriptive rather than prescriptive: a system's time evolution and properties are a certain way, and we formulate concise descriptions thereof; not the other way around. An action produces an equal and opposite reaction not because of Newton's third law, but rather, because an action always produces an equal and opposite reaction, Newton's third law holds.<br /><br />If we want to employ some old-fashioned terminology, I think there is some irreducible 'primitive thisness' or haecceity to physical stuff (Peter, over at Conscious Entities, simply calls that 'reality'). This isn't fixed by the math; rather, the math merely furnishes a description of physical stuff, which only attends to its structural properties. But those properties don't exhaust physical stuff: each relation needs relata it supervenes on.<br /><br />So when I say that the worlds that works according to the GoL may not be physically possible, I mean that there may not be possible physical stuff, the right kind of relata, to instantiate the GoL (as a world on its own). <br /><br />You can view this as espousing some additional metaphysical laws if you like, but I don't: first of all, I don't think the nature of stuff is optional, and indeed, that it's logically necessitated what that nature is.<br /><br />The MUH is one proposal for this necessary determination: on it, physical stuff has no further properties other than the ones that can be captured mathematically, and thus, the determination relation simply becomes logical consistency. But that's not to say that no other options exist.<br /><br />>But to suggest that something is a possible world is not (usually) to say that there is there is actually such a physical world<br /><br />Right, I expressed myself badly there: I should have said that there's some possible physical stuff instantiating the right relations. I don't require the actual existence of possible worlds.Jochenhttps://www.blogger.com/profile/07418841955052661428noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-28820871067964854362016-11-08T17:18:07.089+00:002016-11-08T17:18:07.089+00:00> No: each statement expressible in the languag...<br />> No: each statement expressible in the language of a theory is either true or false given a model of the axioms. <br /><br />I think that is straightforwardly false. I could add an axiom that some integers have a property "foo" and others don't. If that's all I say on the matter, then whether 5 has property "foo" is undecidable. I am free to add as an additional axiom either that 5 is "foo" or 5 is not "foo".<br /><br />I think the axiom of choice is similar. Unless you take as an axiom that it is true or false, whether it is true or false is undecidable.<br /><br />More later or tomorrow!Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-87791351905906740042016-11-08T17:17:55.395+00:002016-11-08T17:17:55.395+00:00Hi Jochen,
There are many points at which my worl...Hi Jochen,<br /><br />There are many points at which my world view departs from yours, and I'd prefer to stay focused on a few at a time. So, since we're exploring my view rather than yours (although I'd be happy to discuss yours also), let's just assume I'm right on all the points that we're not actively discussing.<br /><br />So let's assume that Hawking and Smolin are wrong let's assume that the Many Worlds Interpretation is right, and therefore let us assume that there is a mathematical description of the world.<br /><br />Tegmark is concerned about computability, but when he proposes the CUH he is endorsing the idea that there is no such thing as uncomputable physics in this world. He is also quite skeptical of infinities so the idea of an infinite lattice wouldn't concern him much. I'm relatively agnostic on the idea of the CUH versus the MUH and on the question of infinity -- so far I haven't seen much reason to think that an uncomputable or infinite world could not exist but I acknowledge that there may yet be such a reason I haven't grasped yet.<br /><br />> Or, in other words, the set of physically possible worlds might not include one whose laws are given by GoL<br /><br />But what is physically possible is determined by laws of physics of a particular world. What laws could determine what worlds (and so what sets of laws) are physically possible?<br /><br />Earlier, in answer to a similar point, you said<br /><br />> whether a piece of mathematics 'exists' depends on whether there is something physical that possesses/instantiates that structure.<br /><br />But to suggest that something is a possible world is not (usually) to say that there is there is actually such a physical world, unless like me you think all possible worlds exist. Say I'm wrong about that, and only some possible worlds exist. So what would make GoL not a possible world? It can't be simply that it is not physically instantiated as a world, because not all possible worlds exist.<br /><br />Do you realise you're drawing a distinction between a logically possible world and a possible world? What determines that distinction? I can only make sense of what you are saying by interpreting you as consciously or unconsciously presupposing a set of metaphysical laws that govern all of existence, laws which determine what possible worlds can or cannot exist. But then we can just ask instead "What if the metaphysical laws were otherwise?" and start talking about possible meta-worlds with different metaphysical laws instead. Ultimately you hid bedrock -- logical possibility.<br /><br />So, when I say "possible world", I mean "logically possible world", and in my view this is the only sensible definition of possible world.<br /><br />> You might (and indeed, will) end up with axioms claiming that some system has a spectral gap, while it in fact doesn't<br /><br />Either there is an a priori fact of the matter or there isn't. If there is an a priori fact of the matter, then what happens is determined by mathematics. If there isn't, you can't get it wrong. Whatever you simulate is another possible world. It may not be this one, but it is similar in overall character to this one and this one could be a world described by an analogous mathematical object.<br /><br />But anyway, on reading a bit more, it seems there is an a priori fact of the matter, so my position is actually that this is not really an undecidable problem at all -- the undecidable scenario never exists in reality.<br />Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-46917546785925867712016-11-08T16:27:33.215+00:002016-11-08T16:27:33.215+00:00>You're asserting that it doesn't, and ...>You're asserting that it doesn't, and I'm asserting that it does.<br /><br />As I've pointed out in the other thread, parsimony is needed for falsifiability: without parsimony, all predictions can be derived from a set of observed data on equal grounds. Parsimony picks out one particular theory consistent with the data, thus yielding one particular set of predictions, which we can then test, in order to re-start the cycle upon eventual failure of such a prediction. But since metaphysical hypotheses aren't empirical, this motivation for parsimony vanishes. <br /><br />>I mean, what else would you use to choose between two consistent metaphysical world views?<br /><br />By showing one to be false, of course. (Note that I'm not necessarily convinced there is more than one consistent metaphysical worldview.) A theory's consistency does not imply its truthfulness!<br /><br />>Someone who was identical with you up to the point where your paths diverged and one of you ate pizza and the other didn't.<br /><br />But this wouldn't make 'I didn't eat pizza yesterday' any more possible; it would merely mean that of two entities identical up to yesterday, one (I) had pizza, and the other didn't.<br /><br />>You show no more than that B's Jochen-existence isn't a prerequisite to its instantiability.<br /><br />No; on DM-existence, B exists and is instantiable; on Jochen-existence, B doesn't exist and is instantiable. Our concepts only differ in what we consider to exist; so since there is a concept of existence on which B doesn't exist, and is nevertheless instantiable, then existence isn't necessary to instantiability.<br /><br />>Since the real world does not contain an infinite lattice, this is not a real-world undecidable problem.<br /><br />The problem is applicable to real-world quantum field theories: essentially, a quantum field is equivalent to an infinite spin lattice (for instance, in lattice gauge theory, one generally models a quantum field theory as a lattice with some lattice spacing a; but it's only upon taking the limit a-->0 that one gets the actual QFT predictions). Jochenhttps://www.blogger.com/profile/07418841955052661428noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-4125682629594868692016-11-08T16:25:52.536+00:002016-11-08T16:25:52.536+00:00>But that's not the same as saying that whe...>But that's not the same as saying that whether a given Turing machine halts is undecidable.<br /><br />While it's true that the halting problem is decidable for certain TMs, it's indeed the case that there exist Turing machines for which it is undecidable. Say you have an algorithm that decides the question for some set S1 of TMs, one for some set S2, one for some set S3, and so on. Then the undecidability of the halting problem implies that the conjunction of these sets never exhausts the set of all TMs: because you could just use an algorithm that first uses the algorithm deciding whether it's in S1, then the one for S2, and so on; so if the sequence of sets would exhaust all Turing machines, then we'd have an algorithm deciding the halting problem for all TMs, which is impossible.<br /><br />>If the program in fact halts, and you take an axiom that states that it does not halt, then you have created an inconsistent system, and vice versa.<br /><br />No, I have merely created one which isn't sound. Many formal systems prove theorems which aren't true, but that doesn't make them inconsistent. (I can provide an example if you like.)<br /><br />>Come up with an analogy where you assume that some necessary truth is false and try to conclude something sensible.<br /><br />'If I could solve the halting problem, I could prove (or disprove) Goldbach's conjecture.'<br /><br />>One is that there is a very improbable possible world where entropy always happens to decrease, via statistical fluke. <br /><br />It seems dubious to me that such a world exists---the set of such worlds is of measure 0, so it's impossible to actually point to such a world. For instance, any world you'd find yourself in, if you were to randomly choose a world, would obey the second law.<br /><br />>It seems to me that there is a logically possible world where energy is created ex nihilo and injected into the system<br /><br />Perhaps, but that has no bearing on what I said: that I could build a perpetuum mobile if the second law is violated does not imply that I could only then build a perpetuum mobile. I could perhaps also do it by violating the first law, but that doesn't invalidate my inference.<br /><br />>But that just means you're talking about Jochen-existence rather than DM-existence.<br /><br />Yes, but your argument needs that DM-existence is necessary for the implementability of B; which is shown not to be right, since Jochen-existence also suffices.Jochenhttps://www.blogger.com/profile/07418841955052661428noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-82641724410988698512016-11-08T16:16:59.888+00:002016-11-08T16:16:59.888+00:00Hi DM,
>For instance, the rules of Conway'...Hi DM,<br /><br />>For instance, the rules of Conway's game of life constitute a possible set of physical laws for that world<br /><br />Only if one assumes that 'to be a possible set of physical laws' is the same as 'to be a consistent mathematical object'. But what I'm saying is precisely that there's no reason to make that assumption. Mathematical objects, to me, are just real inasmuch as they are instantiated physically; hence, they can't decide what can be instantiated physically.<br /><br />Or, in other words, the set of physically possible worlds might not include one whose laws are given by GoL. (It might well do, of course---I make no pretenses at knowing what's sufficient to make a world physically possible. I'm merely pointing out that we don't know.)<br /><br />>there are people much smarter than I (Tegmark in particular) who still seem to think that there is a mathematical object which describes this world<br /><br />Even smart people sometimes believe all sorts of strange things. But in this case, I think Tegmark is well aware of the difficulty---isn't that basically why he introduces his computable universe hypothesis (CUH)? (Which, incidentally, would point to a different way in which existence could not be co-extensive with mathematical consistency.)<br /><br />I'm not really sure there's a good popular summary of these results, by the way, but <a href="http://phys.org/news/2015-12-quantum-physics-problem-unsolvable-godel.html" rel="nofollow">this</a> might at least be a start.<br /><br />>simply assume, for the sake of argument, that there is a mathematical description of this world.<br /><br />Even if there are no undecidability problems, I don't think I'd be willing to go along with this. There are still other problems: for instance, Stephen Hawking proposes what he calls 'model-dependent realism', in which there are several distinct, overlapping, not necessarily mutually consistent mathematical structures which are all jointly needed to describe the world. Lee Smolin, on the other hand, proposes that there is no one set of mathematical laws that governs the temporal development of the universe. (I should note I'm not a fan of either proposal, however.)<br /><br />And indeed, the way things appear right now, something like this might well be right: quantum mechanics appears inconsistent with general relativity; more seriously, from my point of view, the description of quantum mechanics follows one set of mathematical laws during ordinary evolution, and another during measurement (which probably doesn't seem terribly serious from the point of view of the many-worlds idea, but I think that fails for other reasons).<br /><br />>But we could have a mathematical object that just decides on the fly to add an axiom (perhaps determining that there is or is not a gap randomly) wherever a novel situation like this crops up.<br /><br />But the problem with this is that there's no way to ensure how to get this correct: if an axiom is added randomly, then we don't know whether it's actually right. You might (and indeed, will) end up with axioms claiming that some system has a spectral gap, while it in fact doesn't, or vice versa.<br /><br />>I think there are statements which have no truth value and we are free to take either as an additional axiom.<br /><br />No: each statement expressible in the language of a theory is either true or false given a model of the axioms. Different models may disagree there, but that's a different thing.Jochenhttps://www.blogger.com/profile/07418841955052661428noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-33749712688183233612016-11-08T13:03:28.160+00:002016-11-08T13:03:28.160+00:00I did a bit more reading about the spectral gap pr...I did a bit more reading about the spectral gap problem and found the following:<br /><br />http://www.nature.com/news/paradox-at-the-heart-of-mathematics-makes-physics-problem-unanswerable-1.18983<br /><br />It seems that this issue is only undecidable if you are dealing with an infinite lattice. It is decidable for any finite lattice. Since the real world does not contain an infinite lattice, this is not a real-world undecidable problem. I feel my view would only run into trouble if there were some actual experiment in the real world that had results that were undecidable (without simply being random).Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-14206334861575168192016-11-08T10:57:32.621+00:002016-11-08T10:57:32.621+00:00> There really isn't: the second law is a l...> There really isn't: the second law is a law of statistics, not of physics<br /><br />I understand this.<br /><br />There are a couple of ways to address this. One is that there is a very improbable possible world where entropy always happens to decrease, via statistical fluke. Another is that the way that the 2nd law prohibits perpetuum mobiles depends on the 1st law, or on the conservation of energy at any rate. It seems to me that there is a logically possible world where energy is created ex nihilo and injected into the system so as to keep a perpetuum mobile running. Again, if I could write a computer program that would simulate a perpetuum mobile (and there's no reason I could not) then a perpetuum mobile is logically possible, which to me means that it could exist in some (logically) possible world.<br /><br />> I'm pointing out that one can vary this, and still implement B.<br /><br />Of course you can vary a definition and nothing will change. But that just means you're talking about Jochen-existence rather than DM-existence. The point I'm making pertains to DM-existence, so the fact that B's instantiability is independent of Jochen-existence doesn't show that it is independent of DM-existence. Again, for your point to work, you would have to believe that there is some ideal concept of existence out there independent of how you or I use the term. It's as if existence is a natural phenomenon and we're trying to construct a hypothetical model as a scientist does. But I think that is not the right way to think about it. We only have the definitions we adopt and nothing else.<br /><br />> Actually, parsimony has no force for metaphysical explanations<br /><br />Actually, it does. You're asserting that it doesn't, and I'm asserting that it does. Where does that get us? I mean, what else would you use to choose between two consistent metaphysical world views?<br /><br />> he isn't me: for one, he hasn't eaten pizza yesterday, but I have<br /><br />Fair enough, but you can deal with these objections with sufficient caveats and circumlocution. Someone who was identical with you up to the point where your paths diverged and one of you ate pizza and the other didn't.<br /><br />> then I show that B's existence isn't a prerequisite to its instantiability. <br /><br />You show no more than that B's Jochen-existence isn't a prerequisite to its instantiability. You show nothing about platonic existence as I think of it.<br /><br />> An example may be the solvability of the halting problem: there's no possible way to solve it, but there exists a whole theory detailing the capabilities of systems capable of solving the halting problem.<br /><br />Such a theory depends on those systems simply knowing the answers. They are oracles. For there to be a contradiction, you have to assume that there is an algorithm which solves the halting problem.<br />Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-64799756198731911992016-11-08T10:57:18.034+00:002016-11-08T10:57:18.034+00:00> it's that if there was a way to mathemat...> it's that if there was a way to mathematically figure this out, then we could construct contradictory objects<br /><br />I am familiar with the halting problem and the self-applicability problem but I don't agree with how you interpret the conclusions. The halting problem is just that there is no general algorithm which will determine if any arbitrary input program will halt. I accept that. But that's not the same as saying that whether a given Turing machine halts is undecidable. For a lot of Turing machines, it is trivially decidable. For some, it is difficult to decide, and for some, we don't know the answer. There is no Turing machine, and there cannot be a Turing machine, for which the halting problem is provably undecidable. So what I said stands -- the best you can say is that there are Turing machines for which we don't know how to figure out if they halt or not.<br /><br />> However, any real computation either halts, or fails to; it's just that there isn't sufficient reason within mathematics to decide which it is to be. <br /><br />I do not think of it like this. For me, whether any real computation halts or does not halt is mathematically necessary and all the results you provided show is that there is no algorithm to determine which is true for an arbitrary program. I do not agree that you are free to take an axiom to determine whether a program halts or not. If the program in fact halts, and you take an axiom that states that it does not halt, then you have created an inconsistent system, and vice versa. The only issue is we don't know it is inconsistent until we figure out whether the program halts or not.<br /><br />> Well, first of all, that something is a tautology hardly makes it trivial: all theorems of, e.g., propositional logic are tautologies.<br /><br />Agreed. But something that is an obvious, staring-you-in-the-face tautology doesn't really tell you anything you didn't already know. So observing that FALSE=>X is not informative.<br /><br />> But you're still assuming I assume a contradiction. I'm not, and I think I've run out of ways to try and make this clear.<br /><br />OK, it's not exactly that you're assuming a contradiction, it's that you're using a contradiction (a necessary truth is false) as the antecedent in an implication. What follows is not informative because anything follows from such an antecedent.<br /><br />I think if you abandoned your Superman disanalogy (disanalogy because Superman is not necessary) you would see the problems. Come up with an analogy where you assume that some necessary truth is false and try to conclude something sensible.<br /><br />For instance: if not all bachelors are unmarried, then could I be an married bachelor? Well, I guess so, it seems that way, but on the other hand I couldn't be an married bachelor because that is an oxymoron. Such a hypothetical can only be made sense of if it means changing the definition of bachelor. So I can only make sense of your hypothetical by your changing the definition of existence to one I don't understand, and this means that you are not concluding anything of value about how I think of existence.<br />Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-52698427378401307332016-11-08T10:56:55.596+00:002016-11-08T10:56:55.596+00:00Hi Jochen,
> But there may be structures such ...Hi Jochen,<br /><br />> But there may be structures such that no physical system (under any kind of physical laws) instantiates them.<br /><br />What constitutes a set of physical laws here? I don't have any preconceptions about what a set of physical laws looks like, other than that they form a mathematical structure. For instance, the rules of Conway's game of life constitute a possible set of physical laws for that world. If something is mathematically possible, then the context in which it is defined is a possible set of physical laws in which it could physically exist in some possible world.<br /><br />> Well, the paper is relatively recent---<br />here's the arxiv version. But there are plenty similar results.<br /><br />Thanks, Jochen. I would like to understand this but this is a little beyond me for the moment. I would be looking for something pitched more at the level of a PBS Space Time video. Perhaps, if there are plenty of similar results, there is such a result (or a popular science discussion of such a result) that might be easier to grasp?<br /><br />I guess I am unconvinced by your interpretation because there are people much smarter than I (Tegmark in particular) who still seem to think that there is a mathematical object which describes this world. I recognise that the point you are making here is very important, and potentially fatal to my world view, but since I can't discuss it competently and since we risk opening one too many threads here, I suggest that for now we set it aside and simply assume, for the sake of argument, that there is a mathematical description of this world.<br /><br />> Mathematically speaking, a world in which a spin system has a gap, and one in which it doesn't, are both consistent. But it may be that for every physical instantiation of that system, it does have a gap. <br /><br />I still don't understand this, but I'll have a go at another possible interpretation. If such systems have a gap, might that not be just another axiom we could add? I understand there's probably something Godelian going on, so that for every axiom we add there's always going to be another system that is undecidable. But we could have a mathematical object that just decides on the fly to add an axiom (perhaps determining that there is or is not a gap randomly) wherever a novel situation like this crops up. It seems like such an approach could work in simulating a possible world.<br /><br />> for an undecidable statement S<br /><br />I think there are two kinds of undecidable statements which I feel you may be conflating. I think there are statements which are actually true or false given a set of axioms, but we can't prove it either way, and I think there are statements which have no truth value and we are free to take either as an additional axiom.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.com