Impact of Gödel's incompleteness theorems on a TOE

[D H]

People will happily (and fairly) declare, for example, string theory a TOE simply because it includes a graviton and and standard-model-like constructions at once, even though we don't know how to calculate many things in it or whether important series produce finite results. Physicists can and do get by without being able to actually calculate basic things, even in sub-TOE theories like QFT or Newtonian gravity (n-body problem anyone?).

Exactly. This thread is really just a quibble over what exactly constitutes a TOE. Would a theory that unites the electroweak interaction, strong interaction, and gravitation qualify as a TOE? Would it be a TOE even if it leaves the particle zoo in more or less the same shape as in the standard model? Would it be a TOE if it cannot definitively answer when the last atom in a pile of ¹⁴C decays to nitrogen? Would it still be a TOE if it cannot definitively answer if, when, and where, a perfectly round ball perfectly balanced on Norton's dome reaches the bottom of the dome?

 

[bcrowell]

edit: I could be wrong, but I'm pretty confident that arithmetic can be embedded in first order geometry

I'm pretty sure you're wrong about that. See #38.

Technically speaking, Turing machines do have an infinitely-long tape.

Sire, no sire.

D H is correct. http://en.wikipedia.org/wiki/Turing_machine

 

[yossell]

Hurkyl, I'm not following you. I'm not interested in convenience. I've already said there are limitations in what can be done in Tarksi's system. Your posts seemed to be arguing the contrary, but now it seems they are not.

I pointed out that there are first order systems of geometry, which allow quantification over first order definable regions, in which arithmetic can be embedded. Is this what you disagree with?

 

[yossell]

I'm pretty sure you're wrong about that. See #38.

I thought we'd discussed this, and I explained that I thought that Tarski's system did not capture everything that is meant by first order geometry. This could come down to semantics - if by first order geometry you just mean Tarksi's theory, fine. But if you modify Hilbert's original theory, replacing the second order axioms with first order schemas, then I believe you have a system in which you can get arithmetic.

I'm willing to admit this could be wrong, but post 38 alone doesn't show it.

 

[Lievo]

Technically speaking, Turing machines do have an infinitely-long tape.

D H is correct.

No sire. Technically speaking, a Turing memory do not have an infinite-long tape. Please give wikipedia a closer look.

Additional details required to visualize or implement Turing machines (...)
The tape can be finite, and automatically extended with blanks as needed (which is closest to the mathematical definition)

 

[yossell]

If you're interested, a first order versions of Hilbert's `Foundations of Geometry' were developed in a book by Hartry Field called `Science Without Numbers.'

Stuart Shapiro (the same Shapiro who wrote `Foundations without Foundationalism' - quite a nice reference on second order logic), in a paper called `Conservativeness and Incompleteness' (Journal of Philosophy 1983) argued that arithmetic was embeddable in Field's geometry, and so the system contained a Godel sentence. Since the consistency of arithmetic was provable in ZFC, Shapiro took this to undermine Field's claim that mathematics was conservative. Field responded in a paper called `On Conservativeness and Incompleteness' (Journal of Philosophy 1983), effectively admitting the technical point, but arguing that it did not have quite the conceptual significance.

Unfortunately, these papers don't seem to be freely available.

 

[D H]

No sire. Technically speaking, a Turing memory do not have an infinite-long tape. Please give wikipedia a closer look.

Additional details required to visualize or implement Turing machines (...)
The tape can be finite, and automatically extended with blanks as needed (which is closest to the mathematical definition)

Nice job of quote mining, Lievo. You omitted the very next sentence.

The tape cannot be fixed in length, since that would not correspond to the given definition and would seriously limit the range of computations the machine can perform to those of a linear bounded automaton.

 

[Coin]

No sire. Technically speaking, a Turing memory do not have an infinite-long tape. Please give wikipedia a closer look.

Additional details required to visualize or implement Turing machines (...)
The tape can be finite, and automatically extended with blanks as needed (which is closest to the mathematical definition)

This dodge is common, and it's why I used the word "unbounded" rather than "infinite" in my post on the last page. The thing is though that even if you are using this dodge, the TM tape is still infinite from the perspective of the mathematical formalism. Within the formalism, we have an infinite series of blank boxes, any of which we may move to and write over. Whether this formal property of the tape is implemented in the real world by somehow acquiring an infinitely long strip of paper, or implemented using a finite tape which additional sections are periodically spliced onto by an army of human servants, these different implementation details are equivalent and unimportant from the formalism's perspective. (Especially since in the real world, people generally don't build literal turing machines at all...)

 

[PhysDrew]

One more thing I would like to propse which was brought up earlier, is what the everything in the TOE actually means? I've heard it discussed that a TOE may have predictive power, but however I have also heard by other physicists that the TOE will be nothing more than a joining of mathematical realtionships to give one unique 'formula' with little to no predictive power, rendering it to low importance in the grand scheme of things. I think Godels theorem/s may or may not apply depending on which scheme it is applied to?
By the way has anyone noticed the self referential hint I put into the title?!

 

[Fra]

I wasn't sure wether to bother entering the discussion, because maybe it's not possible to be brief so maybe I should keep quiet, but I'll just throw in my wet socks and be done with it.

Well, this is just the good ol' problem of inference. In general, we can only prove a theory to be false. There is no observational way to say that a theory is genuinely true.

So while we cannot prove everything in a TOE that falls under Goedel's incompleteness theorem, we can prove some things. And if we prove some things that then turn out to be contrary to observation, the theory is falsified. In the way we typically deal with inference, then, we would progressively gain confidence that the TOE is the correct TOE as repeated attempts to falsify the theory fail, and no alternative TOE that also fits those observations is produced.

Let me state, however, that it may be exceedingly difficult, perhaps even impossible in practice, to falsify a TOE.

If you believe in that information is bounded and that any observer process and hold only finite info, there is a limit of the amount of confidence any information processing agent/observer can accumulate. At some point there is a saturation and you can't inflate the theory by adding the gödel scentences as another axiom, where the progression ends, and all the TOE means is an effective basis for further actions. Also there is not guarantee that there is any convergence to a unique TOE, it may be that it just keeps evolving by destruction of axioms and generation of new ones.

I'll just make the story short and att my OPINION that I think an analysis of all this, decidability, inference, incompletness theorems support by plausability (but not imply in any logical sense) the idea of evolving law and that it's impossible to distinguish cleanly between law and state, simply because they are both merely results of inference, and the limit of infinite confidence in a unique limit seems seems highly unphysical.

If you think about the limits of a TOE, it seems to be at least that is subject to constraints similar to the information state. The TOE represents the observers "state of information" of the expected laws of evolution of the information state. So it's another layer of information.

The more common idea that laws of physics are eternal, correspond as I see it to the "infinite confidence" limits in the progression mentioned. But this limit may not exists, for two reasons (unknow non-uniqueness/convergence) and the lack of information capacity to encode this limit/confidence (if at hand).

This can be expanded a lot, but I agree it does enter philosophy, and it's also not very conclusive therefor I think it's value is mainly suggestive and inspirational. Therefor there is a limit to how much it makes sense to elaborate. This is already a long thread but I've added my short input.

/Fredrik

 

[Hurkyl]

Hurkyl, I'm not following you. I'm not interested in convenience. I've already said there are limitations in what can be done in Tarksi's system. Your posts seemed to be arguing the contrary, but now it seems they are not.

I'm stating that Tarski's system is not as limited as you make it out to be. There are clear limitations -- e.g. that the theory of integer arithmetic cannot be expressed in it -- but it is not as limited as you claim -- e.g. it is able to discuss lines.

The objection you had against lines was that you were rejecting the syntactic tools of first-order logic. Constructing the type of Euclidean Lines from the type of Euclidean Points, for example, is pure first-order logic -- you are not creating a new theory by doing so.

 

[Chalnoth]

If you believe in that information is bounded and that any observer process and hold only finite info, there is a limit of the amount of confidence any information processing agent/observer can accumulate. At some point there is a saturation and you can't inflate the theory by adding the gödel scentences as another axiom, where the progression ends, and all the TOE means is an effective basis for further actions. Also there is not guarantee that there is any convergence to a unique TOE, it may be that it just keeps evolving by destruction of axioms and generation of new ones.

Yes, this is pretty obvious.

Basically our only hope of finding the TOE is if it turns out that mathematical consistency severely limits the possible TOE's so that the correct one can be experimentally distinguished from the others. There is obviously no guarantee that this is the case. And there is certainly no guarantee that we will be able to genuinely demonstrate that there aren't any TOE's that are also consistent but experimentally indistinguishable.

The more common idea that laws of physics are eternal, correspond as I see it to the "infinite confidence" limits in the progression mentioned. But this limit may not exists, for two reasons (unknow non-uniqueness/convergence) and the lack of information capacity to encode this limit/confidence (if at hand).

Well, you can make any laws of physics eternal simply by having the laws of physics describe any changes that occur. But in any event, obviously infinite confidence isn't possible, and I wasn't attempting to imply it was.

 

[bcrowell]

I thought we'd discussed this, and I explained that I thought that Tarski's system did not capture everything that is meant by first order geometry. This could come down to semantics - if by first order geometry you just mean Tarksi's theory, fine. But if you modify Hilbert's original theory, replacing the second order axioms with first order schemas, then I believe you have a system in which you can get arithmetic.

I'm willing to admit this could be wrong, but post 38 alone doesn't show it.

Like Hurkyl, I'm not convinced by your assertion, and I haven't seen you offer any evidence for it. My #38 does offer evidence for my assertion, in the form of a reference to Tarski's paper "A decision method for elementary algebra and geometry." Anyway, I don't know if any of this even matters for the purposes of this thread. It seems that we all agree that there are weaker and stronger formulations of geometry, that the weaker ones don't give enough arithmetic for Godel to apply, and that the stronger ones do.

 

[bcrowell]

I don't think decidability is a necessary or even particularly desirable property for a TOE.

I agree disagree only because IMO this is putting it too mildly. I don't think there is even any clearly defined notion of what it would mean for a physical theory to be decidable.

This thread is really just a quibble over what exactly constitutes a TOE.

And again I disagree only because IMO this is putting it too mildly. I think it's even less than a quibble over what constitutes a TOE. If we get a TOE, we'll know it's a TOE because it will unite the four forces, reconcile GR with quantum mechanics, and make testable predictions that are verified by experiment. That's how we'd know what constituted a TOE. We can't use decidability to define what a TOE would be, because the notion of decidability is fundamentally inappropriate for talking about physical theories.

 

[Lievo]

Nice job of quote mining, Lievo. You omitted the very next sentence.

Which states that the tape is unbounded, and that's not the same, technically speaking, as infinite.

the TM tape is still infinite from the perspective of the mathematical formalism.

This is just making no sense! Ok two ways to see it:
-the tape of any TM that halts is bound by the busy beaver function. It's large, it's unbounded, and it's finite. If the TM doesn't halt, you don't need any tape at all, because you'll never get the result of the TM -which is define as what remains when the TM stops. Of course you can't, in general, know if your TM belong to one or the other class. Doesn't change the tape you need is finite.
-if you think it's infinite, then precise what infinite you're talking about. Is it aleph 0? No, or you'd enter hypercomputation. Is it more? No, that's worse! Is it less than aleph 0? Well, if you believe there is such a thing as an infinite less than aleph 0, go and publish!

Sorry, I'm beginning to be fed up of stating the obvious. Last time I comment this.

 

[D H]

Which states that the tape is unbounded, and that's not the same, technically speaking, as infinite.

Technically speaking, saying that the length is unbounded exactly the same as saying it is infinite.

 

[Chalnoth]

-the tape of any TM that halts is bound by the busy beaver function. It's large, it's unbounded, and it's finite. If the TM doesn't halt, you don't need any tape at all, because you'll never get the result of the TM -which is define as what remains when the TM stops. Of course you can't, in general, know if your TM belong to one or the other class. Doesn't change the tape you need is finite.

If you don't know whether or not the program halts, then you don't know beforehand how much tape is required, which means you need infinite-length tape or risk your computer crashing.

 

[Lievo]

Chalnoth, as you previously teach me some stuff I should not refuse to answer you.

If you don't know whether or not the program halts, then you don't know beforehand how much tape is required, which means you need infinite-length tape or risk your computer crashing.

What you need, mathematically speaking, is at most the busy beaver corresponding to the number of non blank in the initial state. Nothing more, and that's finite. Pratically speaking, be sure your computer will crash before reaching it.

 

[D H]

Baloney. Busy beavers halt. Halting is not a requirement of a Turing machine program. An algorithm to compute the decimal representation of an irrational computable numbers such as the square root of 2 will not stop. The decimal representation of course requires an infinitely-long tape.

 

[Coin]

We are getting a little off topic. But.

What you need, mathematically speaking, is at most the busy beaver corresponding to the number of non blank in the initial state. Nothing more, and that's finite. Pratically speaking, be sure your computer will crash before reaching it.

I think some of this disagreement might be avoidable if we speak precisely.

1. A turing machine must have access to an unlimited amount of tape in order to capture the full power of the turing machine formalism.

2. Any given turing machine, when run on an input for which it terminates, will only use a finite amount of that tape.

The lack of a limitation on space can't be waved away by saying "but that's only for terminating programs", because a significant portion of the power of turing machines in the first place comes from the fact that turing machines have the option of never terminating. The decidable languages are a smaller set than the recognizable languages. If there is a known bound on either runtime or space used, then all programs are "decidable" and you have something weaker than a turing machine.

You are correct that the busy beaver function does define an upper bound on space usage for halting programs. However this statement is not useful to know, because BB(n) is noncomputable. (In other words, BB(n) is a bound, but it is an unknown and unknowable bound). In fact your statement here is basically tautological, because BB(n) is defined as the maximum number of tape squares used up by an eventually-terminating program of a given complexity. So what you are saying is that the number of squares used by a terminating program will be equal to or less than the number of squares it uses.

Do you disagree?

 

[friend]

I think it's even less than a quibble over what constitutes a TOE. If we get a TOE, we'll know it's a TOE because it will unite the four forces, reconcile GR with quantum mechanics, and make testable predictions that are verified by experiment. That's how we'd know what constituted a TOE. We can't use decidability to define what a TOE would be, because the notion of decidability is fundamentally inappropriate for talking about physical theories.

A Theory of everything would at least explain why there are particles, fields, and even spacetime to begin with. Are we going to be fully satisfied if we predict and/or discover smaller, higher energy particles? No, we'll wonder where those came from, and so on, etc, etc. I think we will not be satisfied until we have explained everything in terms of the principles of reason. Once you derive physics from logic alone, then what is there left to question? There would be nothing left except to maybe question your sanity. But to predict the properties of things does not explain where they came from.

I think a theory of quantum gravity would probably be a TOE, because it would unit QM with GR, particles and fields with spacetime. And to unite spacetime and particles fields would probably require a theory from reason alone. For physically, there is nothing more fundamental than spacetime and particles/fields. And to explain something more fundamental than what is physical would have to rely on complete, abstract, generality about anything true; it would have to rely on principle alone.

So here we are discussing whether physics can be reduced to a complete axiomatic system. The question is not even relevant unless we can derive physics from a system of reasoning. I don't think it will reduce to the axioms of geometry because we would still have to explain why there is geometry, or spacetime, to begin with. It would have to reduce to reason, or we would still have questions and it would not be complete. (am I using "complete" equivocally here?)

 

[Chalnoth]

A Theory of everything would at least explain why there are particles, fields, and even spacetime to begin with. Are we going to be fully satisfied if we predict and/or discover smaller, higher energy particles? No, we'll wonder where those came from, and so on, etc, etc. I think we will not be satisfied until we have explained everything in terms of the principles of reason. Once you derive physics from logic alone, then what is there left to question? There would be nothing left except to maybe question your sanity. But to predict the properties of things does not explain where they came from.

I think a theory of quantum gravity would probably be a TOE, because it would unit QM with GR, particles and fields with spacetime. And to unite spacetime and particles fields would probably require a theory from reason alone. For physically, there is nothing more fundamental than spacetime and particles/fields. And to explain something more fundamental than what is physical would have to rely on complete, abstract, generality about anything true; it would have to rely on principle alone.

So here we are discussing whether physics can be reduced to a complete axiomatic system. The question is not even relevant unless we can derive physics from a system of reasoning. I don't think it will reduce to the axioms of geometry because we would still have to explain why there is geometry, or spacetime, to begin with. It would have to reduce to reason, or we would still have questions and it would not be complete. (am I using "complete" equivocally here?)

The difficulty is that there is no guarantee that there exists only one unique TOE. There may be a great many potential ones. Even after discovering a potential TOE, we would still need to determine whether or not it applies to our reality. Pure rational deduction can never ever tell us whether or not a TOE applies to our reality.

 

[D H]

It would have to reduce to reason, or we would still have questions and it would not be complete. (am I using "complete" equivocally here?)

You are using complete incorrectly here. This is a thread on Godel's incompleteness theorems, and whether they have any relevance to a TOE.

 

[Lievo]

I should not refuse to answer you.

I should not refuse to answer anyone. My apologies for this misbehavior.

An algorithm to compute the decimal representation of an irrational computable numbers such as the square root of 2 will not stop. The decimal representation of course requires an infinitely-long tape.

In case you care, please notice there is no such thing. Precisely because you can't compute something that need an infinitely-long tape. You may wish to read http://en.wikipedia.org/wiki/Computable_number" on this.

the computable numbers (...) are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.

...I hope this is not quote mining too much.

In fact your statement here is basically tautological (..) Do you disagree?

Well said (...) No, except for some details: decidable and recognizable that'shttp://xw2k.nist.gov/dads/html/decidableLanguage.html" , I think it's usefull to know that BB(n) is the bound because basically, this is what separate computing from hypercomputing, and finally the fact that the bound is uncomputable doesn't make it an infinite. But I begin to desesperate here.

 

[yossell]

I'm stating that Tarski's system is not as limited as you make it out to be. There are clear limitations -- e.g. that the theory of integer arithmetic cannot be expressed in it -- but it is not as limited as you claim -- e.g. it is able to discuss lines.

The objection you had against lines was that you were rejecting the syntactic tools of first-order logic. Constructing the type of Euclidean Lines from the type of Euclidean Points, for example, is pure first-order logic -- you are not creating a new theory by doing so.

Don't know what you've got in mind when you talk about the `construction of types'. The existence of lines is not a first order consequence of the existence of points. The existence of first order definable regions is not a consequence of a theory that only talks about points. I Can see what you might have in mind if you've got first order set-theory in the background. But that is to add something to Tarksi's theory.

 

[friend]

Originally Posted by friend
It would have to reduce to reason, or we would still have questions and it would not be complete. (am I using "complete" equivocally here?)

You are using complete incorrectly here. This is a thread on Godel's incompleteness theorems, and whether they have any relevance to a TOE.

I'm not so sure. "reduce to reason" is a reference to logic which is relevant to Godel's Theorems. "still have questions" asks about how the subject in question is proven, which goes to Godel's incompleteness theorem. So it might be my use of the word "complete" is on topic.

 

[friend]

The difficulty is that there is no guarantee that there exists only one unique TOE. There may be a great many potential ones.

As I understand it, if two systems of complete and consistent sets of axioms agree at any point, then they are equivalent. If reality is completely logical and rational in every detail, then it can be reduced to logic.

Even after discovering a potential TOE, we would still need to determine whether or not it applies to our reality. Pure rational deduction can never ever tell us whether or not a TOE applies to our reality.

Are you suggesting that reality is somewhere, somehow not completely logical? I think you'd have a hard time actually proving that.

 

[Lievo]

Are you suggesting that reality is somewhere, somehow not completely logical?

In my view he's just suggesting that pure logic is not powerfull enough to find the TOE. There have been some https://www.amazon.com/dp/0521657296/?tag=pfamazon01-20about this.

 

[yossell]

Like Hurkyl, I'm not convinced by your assertion, and I haven't seen you offer any evidence for it.

So my citation of papers doesn't count? Ok - what would you like?

My #38 does offer evidence for my assertion, in the form of a reference to Tarski's paper "A decision method for elementary algebra and geometry."

But I've acknowledged the existence of this paper, and everything you say about it. I've not denied this. What I questioned is that every first order theory of geometry reduces to Tarski's formulation. I've even given the theory that I have in mind: the first order weakening of Hilbert's theory in his Foundations of Geometry. #38 just doesn't speak to this.

I'll sketch the idea, but it's going from memory so, as ever, I could be wrong.

If you like, think of Hilbert's second order axioms as being replaced with schemas, plus axioms asserting the existence of first order definable regions to give us the points to give us the lines, planes and regions of the theory (I disagree with Hurkyl that these regions exist as a matter of first order logic from the theory of points). Then one can define a region R which consists of an infinite sequence of points p1, p2, p3..., such that all points lie on the same line, p1p2 Cong p2p3, p2p3Congp3p4..., there is no point p on R such that p1 lies between p2 and p...

Basically, we can define the existence of a region using the points of an arbitrarily, infinite spaced region with one endpoint, and these then behave like the natural numbers. Certain geometric constructions allow us, within this theory, to define addition and multiplication on this points of this region, and then we have enough material to do the Godel construction.

 

[friend]

In my view he's just suggesting that pure logic is not powerfull enough to find the TOE.

The other criticism is that logic is too powerful and applies to more than just physical situations. But the fact that logic can apply to fiction as well as to fact means nothing. For all physical laws apply to fictional situations as well as to factual situations. Every problem in a physics textbook does not refer to any actual facts in nature, but they refer to fictional situations invented in the imagination of the author.

Your response still sounds to me like you're suggesting that reality is not completely logical. I don't think that is the view of science. I think that we go off in search of answers because we think things are rational. I don't think that any credible scientist is trying to prove that reality is illogical.