LHC - the last chance for all theories of everything?

[tom.stoer]

ConradDJ,

thanks for mentioning Heidegger. I studied Sein und Zeit (I am German therefore I need no translation :-) - but I think I only scratched the surface (which means that even in German it's not easy to understand him :-(.

A final remark before turning back to physics: I admire the fathers of QM not only because they were able to figure out how to calculate atomic spectra etc., but because they were able to initiate a paradigm shift. Perhaps something like this is required today as well (we were happy with the standard model for some decades, therefore we still try to solve physical problems with the toolbox of the standard model ...). Reading Heisenberg or Schrödinger I heave the impression that they were educated and able to understand both - science and philosophy; the latter ability is missing today, at least partially (I do not know were it comes from; I have some ideas but it does not make sense to discuss in this thread).

Unfortunately I did not state my reasoning regarding Goedel carefully enough. Consistency is not the (core) issue, you are right. What I wanted to say is that during this discussion we try to develop a better understanding of what a ToE is and means and what the requirements, restriction etc. should be. I have the feeling that this is like a video game were you can enter the next stage but are immediately confronted with similar tasks, alien space ships etc. It's not really a new quality, is the same task with nastier enemies only. But you are never forced to leave the entire video game and cope with something totally new. So looking for physical theories is quite similar. You start with Newtonian mechanics, then Maxwell theory, then relativity, then quantum mechanics, etc. Even strings, holography etc. are no paradigm shift. If you are happy with this next stage everything is fine. This is like science works, it is successfull (for centuries!) - but we will never be able to leave the video game.

So this is my alternative 2) - we will never manage it; we will enter the next stage and after some decades ask ourselves "why green spacecraft s?" we will complete shooting all green alien spacecraft s, enter the next stage and find - red spacecraft s - ****!

Alternative 1) is to leave the entire game and find something totally different (a new game, all games at once, understand how video games are programmed, programm a video game generator, program a winning strategy generator, ...) Once we are able to specify what this means and how we can escape from the endless "next stage dilemma" the hole platonic world becomes directly visible to us, not only indirectly as in Platon's famous allegory of the cave. But this hole platonic world is rather closed to the idea of the MUH (mathematical universe hypothesis) discussed a couple of days or weeks ago.

99.9% percent of all scientists are working according to the "next stage model" - something we want to overcome - at least here in this thread :-). I do not have any idea how this could work (!) but alternative 1) seems to be as far-reaching as possible - even if I reasoned some time ago that I don't believe in it.

Don't get me wrong - a successful description of evolving physical laws, inference etc. is certainly more than just red spacecraft s. It's comparable to the revolution of quantum physics! So even if we see no hope in succeeding with 1) there is much sense in working on 2)

 

[tom.stoer]

that's funny; I tried to post "s..." and it get's replaced ... works with "f..." as well :-)

 

[RUTA]

Alternative 1) is to leave the entire game and find something totally different (a new game, all games at once, understand how video games are programmed, programm a video game generator, program a winning strategy generator, ...) Once we are able to specify what this means and how we can escape from the endless "next stage dilemma" the hole platonic world becomes directly visible to us, not only indirectly as in Platon's famous allegory of the cave. But this hole platonic world is rather closed to the idea of the MUH (mathematical universe hypothesis) discussed a couple of days or weeks ago.

99.9% percent of all scientists are working according to the "next stage model" - something we want to overcome - at least here in this thread :-). I do not have any idea how this could work (!) but alternative 1) seems to be as far-reaching as possible - even if I reasoned some time ago that I don't believe in it.

We avoid the dilemma of "the next stage model" because we admit a priori that the entire enterprise is one of self-consistency rather than fundamental laws governing the motion of fundamental entities in space as a function of time. Thus, the ultimate expression is not something “at the bottom,” begging for justification from something yet “deeper,” but a mathematical articulation of the self-consistency criterion for the process as a whole. It's a very different way of viewing the game itself, even though it's not a departure from the formalism per se (discrete path integrals a la quantum Regge calculus). Once you view the game differently, you begin to ask different questions of your formalism. Is that what you're talking about? If not, just ignore this post

 

[friend]

We avoid the dilemma of "the next stage model" because we admit a priori that the entire enterprise is one of self-consistency rather than fundamental laws governing the motion of fundamental entities in space as a function of time.
...
but a mathematical articulation of the self-consistency criterion for the process as a whole...

The trouble we encounter is which set of consistent math describes physical properties. It seems we need to start with some physical properties and then try to discover a consistent math that describes it. This is the normal curve fitting techiques that science is acustomed to.

The only alternative is to trust with blind faith in consistency itself and see where it leads us. Then the question is how to translate this logical consistency between all facts into mathematical expressions. That seems like a tall order. I don't think we'd have much faith in such an effort until we could derive some familiar physical principles, like Fyenman's path integral or something. But once we started to get something that looks like physics, it would be hard to accept that it would not derive all of physics. Maybe it's worth a Google search to see if anyone has ever found "Physics derived from logic alone". I wonder if it's on the arXiv yet?

 

[RUTA]

The trouble we encounter is which set of consistent math describes physical properties. It seems we need to start with some physical properties and then try to discover a consistent math that describe it. This is the normal curve fitting techiques that science is acustomed to.

The only alternative is to trust with blind faith in consistency itself and see where it leads us. Then the question is how to translate this logical consistency between all facts into mathematical expressions. That seems like a tall order. I don't think we'd have much faith in such an effort until we could derive some familiar physical principles, like Fyenman's path integral or something. But once we started to get something that looks like physics, it would hard to accept that it would not derive all of physics. Maybe it's worth a Google search to see if anyone has ever found "Physics derived from logic alone". I wonder if it's on the arXiv yet?

The self-consistency criterion (SCC) we propose is nothing so grand. Our SCC is just the discrete counterpart to the boundary of a boundary principle as used to construct the action for the transition amplitude. The form of the SCC survives the statistical limit and is responsible for classical field theory by construction. Roughly, we stick "boundary of a boundary is zero" in the bottom (so that it rules fundamental physics) in such a way that it is guaranteed to come out on top (survive the statistical limit and rule classical physics). Thus, on our view, the entire process of physics is one of self-consistency in this very specific (mathematically articulated) sense. See arXiv 0908.4348 for details. We shouldn't discuss this further here, but I would appreciate any comments, just send them to me directly via my PF profile.

 

[tom.stoer]

I still do not understand the role of (self) consistency.

Let's assume a ToE is a mathematical framework, that means it consists of
1) basic rules (alphabet, ...) to define expressions
2) definitions, axioms
3) (derived) theorems = proofs according to 1+2)

This framework will (I guess) rich enough to contain at least logic and natural numbers, that means it will certainly have the Peano axoims as a subset. But then we know by Goedels theorem that it is
- either incomplete = not all true theorems are provable (*)
- or inconsistent = all theorems (true and false ones) are provable;
(*) one central theorem which is not provable is the consistency of the framework;
this last sentence has been proved by Goedel.

What does that mean when it is applied to a physical theory?

 

[friend]

This framework will (I guess) rich enough to contain at least logic and natural numbers, that means it will certainly have the Peano axoims as a subset. But then we know by Goedels theorem that it is
- either incomplete = not all true theorems are provable (*)
- or inconsistent = all theorems (true and false ones) are provable;

What does that mean when it is applied to a physical theory?

Physics is not the attempt to prove the completeness of math. No one expects to need every possible math statement to describe physics. In fact we are looking for a minimum set of equations that describe physics. We are looking for the overall, underlying principle from which all of physics is implied. This principle, by definitinon, would have to include all possible facts in the universe so that it would be impossible to find exceptions. This suggests one underlying principle that all facts exist in conjunction, which means every fact is consistent with every other fact in the universe, so that every fact proves every other. The question is can you derive physical law from this principle?

 

[RUTA]

I still do not understand the role of (self) consistency.

Let's assume a ToE is a mathematical framework, that means it consists of
1) basic rules (alphabet, ...) to define expressions
2) definitions, axioms
3) (derived) theorems = proofs according to 1+2)

This framework will (I guess) rich enough to contain at least logic and natural numbers, that means it will certainly have the Peano axoims as a subset. But then we know by Goedels theorem that it is
- either incomplete = not all true theorems are provable (*)
- or inconsistent = all theorems (true and false ones) are provable;
(*) one central theorem which is not provable is the consistency of the framework;
this last sentence has been proved by Goedel.

What does that mean when it is applied to a physical theory?

As friend said, physics isn't responsible for the self consistency of logic and math. The kind of self consistency we habor in physics is much less complicated. Einstein’s equations of general relativity (GR) provide an example. Momentum, force and energy all depend on spatiotemporal measurements (tacit or explicit), so the stress-energy tensor cannot be constructed without tacit or explicit knowledge of the spacetime metric (technically, the stress-energy tensor can be written as the functional derivative of the matter-energy Lagrangian with respect to the metric). But, if one wants a ‘dynamic’ spacetime in the parlance of GR, the spacetime metric must depend on the matter-energy distribution in spacetime. GR solves this dilemma by demanding the stress-energy tensor be ‘consistent’ with the spacetime metric per Einstein’s equations.

 

[tom.stoer]

We avoid the dilemma of "the next stage model" because we admit a priori that the entire enterprise is one of self-consistency rather than ...

The trouble we encounter is which set of consistent math describes physical properties. ...

The only alternative is to trust with blind faith in consistency itself and see where it leads us. ...

Physics is not the attempt to prove the completeness of math.

I only wanted to stress that we cannot trust in consistency of theorems or math systems. Constructing a math system as a candidate-ToE automatically excludes the proof of its consistency.

That's why I am a bit more willing now to think about the idea that the ToE is not one single math system but the collection of all consistent systems.

 

[RUTA]

I only wanted to stress that we cannot trust in consistency of theorems or math systems. Constructing a math system as a candidate-ToE automatically excludes the proof of its consistency.

That's why I am a bit more willing now to think about the idea that the ToE is not one single math system but the collection of all consistent systems.

That may be true for a Theory of Everything. Do you think physics alone can produce such a theory? It seems to be ruled out by virture of the fact that physics is concerned with the objective, not the subjective, so by its very nature any theory of physics cannot explain "everything."

 

[friend]

That's why I am a bit more willing now to think about the idea that the ToE is not one single math system but the collection of all consistent systems.

A collection of consistent systems that are each consistent with every other should be regarded as one consistent system.

so by its very nature any theory of physics cannot explain "everything."

Are you suggesting that there are somethings that cannot be explained, that are illogical and deny all reason itself?

 

[RUTA]

Are you suggesting that there are somethings that cannot be explained, that are illogical and deny all reason itself?

It may be true that some phenomena defy logic, but even if all phenomena are logical, I don't think physics can explain everything. As I argued previously, physics, as a system to explain the objective, cannot explain the subjective. Perhaps, you can appreciate my claim per the hard problem of consciousness.

Suppose we were able to explain all brain activity via physics (neurology and neurochemisty explained via physics). Using this you construct an explanation of what happens in the brain when a person sees the color red. You give this explanation to Alice who has never seen the color red, but she is very bright so she understands the explanation. You then show her something red. Does she have new knowledge after actually seeing red that she didn't possesses prior thereto, even though she had and totally comprehended a complete physical explanation of seeing red? If you answer affirmatively, then you share my belief that physics cannot explain the subjective.

Similarly, Penrose claimed we cannot create a mathematical description of everything because part of everything is the recognition of the truth of a mathematical proof, which is itself beyond mathematics.

 

[tom.stoer]

That may be true for a Theory of Everything. Do you think physics alone can produce such a theory? It seems to be ruled out by virture of the fact that physics is concerned with the objective, not the subjective, so by its very nature any theory of physics cannot explain "everything."

I don't think that physics alone (as it is understood today) is able. The question is what a ToE really IS. If you give me a set of equations and claim that this is the ToE, then what are good reasons for me to believe you?
That the theory makes physically correct predictions? The standard model would do the job!
That it explains some (all) free parameters? OK, let's assume string theory eventually provides a selection principle which essentially predicts U(1)*SU(2)*SU(3) + three generations; Of course my next question would be: why strings?
That it is consistent (whatever that means)? We believe that the standard model plus gravity is somehow inconsistent, but as I said we will never be able (for no complex mathematical framework) toprove its consistency.

What we are talking about is more than a ToE with a set of equations, rules etc. which is (as Marcus said) predictive up to arbitrary high energies. It is a kind of meta-theory that provides in some sense a good reasoning why it is the way it is. And I think it's exactly this question we are not able to answer.

 

[tom.stoer]

A collection of consistent systems that are each consistent with every other should be regarded as one consistent system.

Of course it only makes sense to distinguish between two systems if they are either independet from each other or inconsistent with one another

Is [set theory with continuum hypothesis] and [set theory with the negation of the continuum hypothesis] one system? I do not know if a ToE requires a specific choice for the continuum hypothesis; but if it does, we have to ask what about a different ToE were "somebody" made a different choice.

The continuum hypothesis may be artifical, but homeomorphic, non-diffeomorphic manifolds may very well be physically interesting!

 

[tom.stoer]

... I don't think physics can explain everything.

Similarly, Penrose claimed we cannot create a mathematical description of everything because part of everything is the recognition of the truth of a mathematical proof, which is itself beyond mathematics.

This seems to be very reasonable and indicates that physics is limited in its explanatory ability.

But - isn't it possible to transcend this reasoning? Let's assume that the only requirement for a ToE in a mathematical multiverse is that the ToE must be consistent. Of course one cannot proof consistency of the theory, but one can claim (in a platonic sense) that one specific theory either IS consistent or it is NOT (just like any natural number either IS prime or it is NOT prime, regardless if we already checked it). If we do that we reduce all theories to a subset, namely the consistent theories.

 

[RUTA]

This seems to be very reasonable and indicates that physics is limited in its explanatory ability.

But - isn't it possible to transcend this reasoning? Let's assume that the only requirement for a ToE in a mathematical multiverse is that the ToE must be consistent. Of course one cannot proof consistency of the theory, but one can claim (in a platonic sense) that one specific theory either IS consistent or it is NOT (just like any natural number either IS prime or it is NOT prime, regardless if we already checked it). If we do that we reduce all theories to a subset, namely the consistent theories.

Of course, if what you mean by "everything" is restricted to a small enough domain of discourse, you can get a ToE in any context! I'm a physicist working on the unification of physics, so mine is purely self criticism here. It's ridiculous to think physics is the be-all end-all of knowledge and reason.

 

[BigFairy]

Of course, if what you mean by "everything" is restricted to a small enough domain of discourse, you can get a ToE in any context! I'm a physicist working on the unification of physics, so mine is purely self criticism here. It's ridiculous to think physics is the be-all end-all of knowledge and reason.

Oh? i would beg otherwise..

 

[moniker2]

Of course it only makes sense to distinguish between two systems if they are either independet from each other or inconsistent with one another

Is [set theory with continuum hypothesis] and [set theory with the negation of the continuum hypothesis] one system? I do not know if a ToE requires a specific choice for the continuum hypothesis; but if it does, we have to ask what about a different ToE were "somebody" made a different choice.

The continuum hypothesis may be artifical, but homeomorphic, non-diffeomorphic manifolds may very well be physically interesting!

I have seen something like that before, around two years ago. I posted here and was called a crackpot and banned.

 

[Fra]

FWIW - here is some more fuel on the fire.

About this discussions about deductive systems of inference and consistency and how it relates to physics, I think there are different ways to view the relation between physics and logic, but I'll just trow in these reflections.

From my point of view of physical law, and physical action as pretty much one to one with an inference system, I see some analogies that I can related to, but I think in terms of inductive inference and actions beein constrained by the inference system, and feedback from the environment to change the inference system.

This Gödel stuff is quite different in that it applies to deductive systems of inference, which from my point of view makes no sense in physics for the very reason that I think that if you take the inference seriously, then even the inference system itself must be inferred, which effectively measn that even for the special case of deductive systems, the axioms are not chosen at will, they are the result of an inference process. But not a deductive one.

The consistency I would personally usually make sense of in physics, is just that if we see the laws of physics as a system of inference (whereby initial conditions implies the future) then consistency would normally mean that different observers should arrive at the "same future", or alternatively that different observers making an inference/abduction of laws from experience they would end up with the same laws of physics.

However, for various reasons I do not think that there is any meaning in that because such consistency is not inferrable from the point of view of an inside observer. To me this is largely intuitive, but there are also some other arguments. This is maybe vaguely similar to gödels incompleteness theorem, but still not quite of course.

Anway, Instead, my view is that inconsistencies in the above sense might actually occur, but observation of those inconsistencies is exactly what causes evolution or defomration of the inference system as represented by a physical observer.

So instead of thinking that physics is a system of consistent "inference systems", I personally think of it as systems of *interacting* inference systems, and interaction terms can sort of be traced to partial inconsistencies between the actions/inference systems of the interacting observers.

Because I do not think in terms of deductive inference, but rather inductive inference, the "inconsistencies" are not truly the kind of thing that crashes the logical system, instead the inconsistency produces a "negative" feedback that will decrease the confidence in the inconsisntent inference system, which will instead evolving to something MORE consisntent, but not necessarily perfectly so.

/Fredrik

 

[Fra]

then consistency would normally mean that different observers should arrive at the "same future", or alternatively that different observers making an inference/abduction of laws from experience they would end up with the same laws of physics.

the trick I envison to make sense out of this: Almost needless to say, the futures we are talking about here are not actual futures, it's only the INFERRED future, based on whatever incomplete premises and imperfect inference system at hand, and the only FUNCTION of this inferred future is that it impacts the ACTION of the inference system itsef.

Just like the behaviour of the human, depends on what expectations of the future the brain has; the action is clearly invariant with regards to the ACTUAL future. In reality, the inferece systems itself is update during the evolution; not just jumping from initial to final state.

Thus, in this view I think it's crystal clear that this is not a true contradiction - it is rather the basis for an interaction.

One might ask why can't this defined by the class of "disagreements", is well defined and can be treated like we usually do on physics with gauge symmetries and gauge interactions and thus this transformation IS then the REAL law that is invariant, and thus restores consistency.

I think the problem with this "resolution" is that it isn't realisable, when you acknolwedge that the information required to actuall infere the symmetry, is not localised to ONE observer, it's an entire local population of observers. But this is also why this scheme DOES make sense when you study a confined subsystem; like is the case in particle experiments. but this is not equally obvious when you consider how a particle experiences it's own environment, or how things behave very very far from equilibrium. Such as the big band or other hypotetical scenarios. This scheme is also (IMHO at least) quote doubtful for cosmological thinking.

/Fredrik

 

[ConradDJ]

The question is what a ToE really IS. If you give me a set of equations and claim that this is the ToE, then what are good reasons for me to believe you?
That the theory makes physically correct predictions? The standard model would do the job!

What we are talking about is more than a ToE with a set of equations, rules etc. which is (as Marcus said) predictive up to arbitrary high energies. It is a kind of meta-theory that provides in some sense a good reasoning why it is the way it is. And I think it's exactly this question we are not able to answer.

Physics has never been concerned to answer "why it is the way it is." But I agree it's become important to ask this question now. It's certainly possible that the LHC will uncover phenomena that once again point theorists in a new direction. And it's possible that technical developments in quantum gravity will come up with something that seems compelling. But to me it's amazing how much is known about fundamental physics, and how little we've been able to learn from that vast body of knowledge, about what's going on here.

In biology, the question "why things are the way they are" has a remarkably clear answer, that provides a foundation for the entire field. It's not unreasonable to expect something similar in physics.

My own sense is that whatever's going on in physics is not just mathematical. If we just regard the world as a pattern of given facts, we can try to model it mathematically -- and this has essentially been accomplished, with the Standard Model. We can push on in the direction of Unification... but is there really any deep basis for our belief that the world must be built on a single structure? Once we get beneath the atomic level, is there really any empirical support for this? For every step toward unification, we've opened up new kinds of differences between fundamental structures.

But the thing is, our world is much more than a body of fact. Among other things, it's a system that makes its facts physically meaningful -- i.e. observable and definable in terms of other facts, that are also physically observable and definable.

Mathematical systems seem to work very differently -- they're logical structures built on undefined basic elements / operations. These basic elements are meaningful to us, because we live in a physical world that has analogues to them. But there seem to be very basic features of the physical world that we haven't yet tried to account for in our theoretical models.

 

[tom.stoer]

I have seen something like that before, around two years ago. I posted here and was called a crackpot and banned.

Dear friend,

let's wait and see what will happen to me :-)

btw - a few days ago a got a hint to read the following paper:

http://arxiv.org/abs/gr-qc/0404088
Quantum general relativity and the classification of smooth manifolds
Hendryk Pfeiffer
(Submitted on 21 Apr 2004 (v1), last revised 17 May 2004 (this version, v2))
Abstract: The gauge symmetry of classical general relativity under space-time diffeomorphisms implies that any path integral quantization which can be interpreted as a sum over space-time geometries, gives rise to a formal invariant of smooth manifolds. This is an opportunity to review results on the classification of smooth, piecewise-linear and topological manifolds. It turns out that differential topology distinguishes the space-time dimension d=3+1 from any other lower or higher dimension and relates the sought-after path integral quantization of general relativity in d=3+1 with an open problem in topology, namely to construct non-trivial invariants of smooth manifolds using their piecewise-linear structure. In any dimension d<=5+1, the classification results provide us with triangulations of space-time which are not merely approximations nor introduce any physical cut-off, but which rather capture the full information about smooth manifolds up to diffeomorphism. Conditions on refinements of these triangulations reveal what replaces block-spin renormalization group transformations in theories with dynamical geometry. The classification results finally suggest that it is space-time dimension rather than absence of gravitons that renders pure gravity in d=2+1 a `topological' theory.

Tom
(on behalf of all crackpots)

 

[MTd2]

Hey Tom, I didn't you also liked exotic smoothness. Right?

 

[tom.stoer]

FWIW - here is some more fuel on the fire.

... the "inconsistencies" are not truly the kind of thing that crashes the logical system, instead the inconsistency produces a "negative" feedback that will decrease the confidence in the inconsisntent inference system, which will instead evolving to something MORE consisntent, but not necessarily perfectly so.

Honestly speaking I expected something like that!

I came to the conclusion that insisting on inference and at the same time relying on axiomatic methods would cause big trouble for you approach. It would rely on consistency w/o being able to prove or disprove it. Every question for consistency or completeness would force you to enter the next stage where you eventually face the same kind of problems.

So it seems like a clearance to change the perspective and let the hole thing become subject to evolution w/o ever referring to "something" fixed. I was thinking about consistency being the only selection criterion, but I couldn't make sense out of it because in any axiomatic approach inconsistency is not something you can deal with - you must simply throw it away.

I still see problems with that approach:
1) I still have the feeling that there is "something" in your approach that is NOT subject to evolution and that is NOT immune regarding the following argument:
2) I think that you can change perspective again and map your approach into a kind of axiomatic system or a member of a family of axiomatic systems.
3) Then you are trapped again because now you have to deal with inconsistent axiomatic systems - not a very nice idea.

But I should stress that this is perhaps not the problem of your approach but only my problem in understanding it (or being able to attack my own prejudices).

 

[tom.stoer]

Physics has never been concerned to answer "why it is the way it is."

I disagree. I think these kind of questions are the very driving force for the progress of science over centuries! I think latest since the early days of quantum mechanics every day physics is concerned with questions regarding measurements and predictions, but every revolution in science has its roots in questions like "why is ...". Nobody was able to present a full answer, only approximations to some deeper truth, so perhaps that's why we are not used to ask and analyze these questions.
In addition certain philosophers claimed that those questions are meaningless and must be avoided in all sound systems - which is a reasoning we know from "The Fox and the Grapes".

But to me it's amazing how much is known about fundamental physics, and how little we've been able to learn from that vast body of knowledge, about what's going on here.

Good point! I think Smolin was pointing out that for a further development of quantum gravity, unification etc. we do not necessarily have to wait for new experimental results. Let me explain why: Progress in physics was always a step-wise approach. First there were experimental results w/o a theoretical explanation. Then there was a theory able to post-dict these phenomena - and to predict new ones. Today we are confronted with a couple of phenomena (4-dim. spacetime, dark energy, U(1)*SU(2)*SU(3), three generations, ...) we do not understand and we cannot explain (why spacetime is four-dimensional is a very good question even w/o referring to string theory). So the very first step for a candidate ToE need not be to predict new phenomena but to post-dict the known ones!

But there seem to be very basic features of the physical world that we haven't yet tried to account for in our theoretical models.

Can you give me examples?

 

[tom.stoer]

Hey Tom, I didn't you also liked exotic smoothness. Right?

What do you mean exactly?

 

[MTd2]

What do you mean exactly?

You said "The continuum hypothesis may be artifical, but homeomorphic, non-diffeomorphic manifolds may very well be physically interesting!"

In 4 dimensions, that is exotic smoothness. Google for it! :) This is the topic that I like the most, and I think it is at the very core of every 4 dimensional theory with fractal dimensions, like Horava gravity, LQG, Loll's universe, Asymptotic Safety, and the likes.

This is also closely relateted to the classificantion of smooth 4-dimensional manifolds. The simple connected case has not classification, and it is also related to the question if the generalized poincare conjecture is true or not in 4 dimensions. That is, about the existence of exotic smooth sphere. Hendryk Pfeiffer paper above is related to all this.

 

[RUTA]

To repeat Dicke's famous response, "Well boys, we've been scooped." The search for the unified force is over:

http://www.theonion.com/content/node/39512

Damn, I had my money on LQG.

 

[Fra]

Honestly speaking I expected something like that!

I came to the conclusion that insisting on inference and at the same time relying on axiomatic methods would cause big trouble for you approach. It would rely on consistency w/o being able to prove or disprove it.

I think you are starting to see the logic I tried to convey.

I still see problems with that approach:
1) I still have the feeling that there is "something" in your approach that is NOT subject to evolution and that is NOT immune regarding the following argument:
2) I think that you can change perspective again and map your approach into a kind of axiomatic system or a member of a family of axiomatic systems.
3) Then you are trapped again because now you have to deal with inconsistent axiomatic systems - not a very nice idea.

I guess it is possible to try to map this into a kind of sujbective axiomatic approach, but then to keep the spirit, the set of subjective axioms would sort of constitute the DNA of the inference system, and those actual systems encoding in their actions a particular inference system, would still be subject to challange/selection, and one could say that the inference system implicit in a set of axioms, can have varying degrees of "success" of actually making inferences that MATCH the actual future, and here there is a selection for the axioms, so that axioms can be lost, like genes can be lost, and new axioms can come.

So if we insiste on a axiomatic approach, and still insist on keeping the spirit of reasoning here, then the axioms are still subject to evolution. One one is faced with the problem of understand the logic behind going from axiomatic system to another.

That might make sense to me, but it doesn't make the problem easier, it's just putting it in different words. IT's still very different from the ordinary axiomatic approach, which is more like an accumulative buildup of an axiomatic system. This does not match what I envision because the "choice of axioms actually contains information" about what's fit and what's not. And the axiomatic system are still bounded in complexity in my view, since it's encoded by a finite inside observer.

So the really ordinary axiomatic method (without this evolving stuff I talk about) does not as I see it match this.

/Fredrik

 

[tom.stoer]

In 4 dimensions, that is exotic smoothness. Google for it! :) This is the topic that I like the most, and I think it is at the very core of every 4 dimensional theory with fractal dimensions, like Horava gravity, LQG, Loll's universe, Asymptotic Safety, and the likes.

This is also closely relateted to the classificantion of smooth 4-dimensional manifolds. The simple connected case has not classification, and it is also related to the question if the generalized poincare conjecture is true or not in 4 dimensions. That is, about the existence of exotic smooth sphere. Hendryk Pfeiffer paper above is related to all this.

Yes, you are right, this is exactly what I am talking about. The paper provides a fresh view on PL manifolds which could be used as "diffeomorphims-invariant triangulation". This is a very interesting idea.

I am not so sure if exotic smoothness for R^4 is the very core of 4 dim. physics. Pfeiffer explicitly excludes topologogically inequivalent manifolds, but I would expect that different topologies will matter as well. (I've seen some statements in this regard in the CDT context, but I can't remember the details ...)