- #176
tom.stoer
Science Advisor
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MTd2 said:Hey Tom, I didn't you also liked exotic smoothness. Right?
What do you mean exactly?
MTd2 said:Hey Tom, I didn't you also liked exotic smoothness. Right?
tom.stoer said:What do you mean exactly?
tom.stoer said: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.
tom.stoer said: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.
MTd2 said: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.
Fra said:I guess it is possible to try to map this into a kind of sujbective axiomatic approach, ... ... 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. ...
tom.stoer said:No, it's not just putting it in different words.
1) If you succeed with this reformulation you will have an axiomatic approach which describes how axiomatic systems can evolve. It describes (sets the rules) how axioms can be lost and how one can go from one axiomatic system to another. It will be very interesting to learn if this meta-level is again an axiomatic system, if it is subject to evolution, if it can be constructed explicitly or if you can only prove existence, what about my "next stage ..." reasoning etc.
2) If you fail you will hopefully learn from failing if your approach is richer than the purely axiomatic one; perhaps failing with the construction will be a success for your approach ...
So you will learn if your approach is immune to my criticism - and if this is a feature or a weakness (weakness in the sense that self-immunization may be non-scientific; refer to arguments against the anthropic principle).
tom.stoer said:It will be very interesting to learn if this meta-level is again an axiomatic system, if it is subject to evolution, if it can be constructed explicitly or if you can only prove existence, what about my "next stage ..." reasoning etc.
Yes. I am not convinced that your approach is right, but IF it is right, this must be true, of course.Fra said:Do you mean wether it's possible to describe what I propose - the evolving axiom system - again as a *fixed* objective larger axiom system?
I think that is not possible - it would in fact not be very conceptually consistent with the spirit of the idea.
Again my question is how these laws of evolution become subject to evolution.Fra said:What I think is possible OTOH, which also part of the idea here, is that it IS possible to describe the evolution of axiom systems, RELATIVE to another axiomsystem. But there is no objective universal axiomsystem which we can see in a realist view.
ConradDJ said: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 said:Can you give me examples?
tom.stoer said: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 ...)
MTd2 said:Exotic smoothness exists for an infinite type of different topologies!
Fra said:I see NO better option. ... If you have a better one and are able to convince me, I am open.
tom.stoer said:Look at strings: there you do not have several differential structures but its the basic topology of the world sheet that matters.
tom.stoer said:Wouldn't that mean that your approach became a somehow fixed system of rules and statements? Wouldn't that mean that it is no longer subject to evolution? Wouldn't that mean that your meta-program became an "axiomatic system"?
...
If you say that the answer to the above raised questions is "no" there is only one conclusion, namely that the approach is somehow inconsistent or incomplete as we agreed to accept it,but at the same time we concluded (right now!) that it is not fixed but subject to change and therefore we have to withdraw our acceptance unless you have finished your work.
tom.stoer said:Mathematics cannot answer questions regarding existence. ...Therefore there is a missing link between physical existence and mathematical truth.
Fra said:My answer to the above would be the no option.
At no point in this process, must the fact that we are still on a journey, or the fact that our inference system can improve, inhibit us from actually doing so.
Fra said:I have a feeling thta your quest for "perfection" or ultimate consistency, ambitious as it may be, is actually inhibiting. You almost run into a halt. Nature don't do that. ... My attempt of TRYING, lead to my my current strategy. So I did consider the rigid inference system, but I can't make sense out of it. Thereof my position.
friend said:The only representation we have of the states of existence or non-existence is binary logic with its algebric manipulation of true and false. Here true represents what exists, and false represents what does not exist. We always consider true those statements that describe what really is, and visa versa.
friend said:So any ToE that proports to determine what is, as opposed to what is not, would have to be based in logic. Can you imagine a theory of physics which does not comply with logic? I think logic is a ToE since we assume it applies to everything physical. The question is how do we extract laws of physics from this.[/QUOTE
Very good starting point.
friend said:... every proposition would have to be assigned numbers, or coordinates to keep track of them.[/QUOTE
Then you face the usual trouble with natural numbers, Gödel etc. Binary logic alone seems to not powerfull enough to do the job; using natural numbers may lead to inconsistent or incomplete systems w/o any chance to prove or disprove it.
Let me ask some questions to get a better understanding of your second statement:
Do you believe that natural numbers exist?
Do you believe that this is not only a platonic existence, but that there is a level of mathematics (say the natural numbers) that IS physically EXISTING = fundamental physical entities (which we haven't discovered so far).
Do you believe that you can somehow construct physical laws (which rely on more than logic) from logic itself? That means do you think that natural numbers, real numbers etc. emergy somehow from logic alone?
Or do you think that if we uncover the ultimate laws of science those concepts fade away as purely approximate descriptions and we are faced with a system of logical automata interacting with each other?
friend said:The only representation we have of the states of existence or non-existence is binary logic with its algebric manipulation of true and false. Here true represents what exists, and false represents what does not exist... So any ToE that proports to determine what is, as opposed to what is not, would have to be based in logic... I think logic is a ToE since we assume it applies to everything physical. The question is how do we extract laws of physics from this.
It may not be that hard. Whatever the tiniest constituents of reality are, it's fair to assume they can be describe with statements or propositions...
tom.stoer said:Let me ask some questions to get a better understanding of your second statement:
Do you believe that natural numbers exist?
Do you believe that this is not only a platonic existence, but that there is a level of mathematics (say the natural numbers) that IS physically EXISTING = fundamental physical entities (which we haven't discovered so far).
Do you believe that you can somehow construct physical laws (which rely on more than logic) from logic itself? That means do you think that natural numbers, real numbers etc. emergy somehow from logic alone?
Or do you think that if we uncover the ultimate laws of science those concepts fade away as purely approximate descriptions and we are faced with a system of logical automata interacting with each other?
ConradDJ said:Friend -- The point I was trying to make above (#149, 186) is that this binary representation of physical existence isn’t adequate. For a proposition “A exists” to have any meaning, it must be possible for something to determine something about A through interaction, within some particular frame of reference.
Yes. I also understand why you think my position is a subjective - it is. The question where we differ, if it's avoidable or not - does the real world fit into a static description?tom.stoer said:OK, now you position is clear. You include in the process not only nature's evolution of laws but also our quest for these laws. My feeling as that it becomes al little too subjective, but that's my position - and I don't want to repeat it here.
Yes something like that is what I mean.tom.stoer said:Yes, you are right. As I just said I can only tell you how it does NOT work (what a pitty). If it is this what you mean by "halting" then I agree.
That analogy seems a bit odd :) but it's true that I think the point is the quest for best performance. The perfect performance might be impossible.tom.stoer said:You have chosen to search for evolving laws - and you somehow adapt your search strategy accordingly. You are like Achilles running after the turtle. You never overtake the turtle, but you are coming closer and closer - and perhaps this is the best you can do.
tom.stoer said:If we focus on the nature and existence (or emerence) of laws then we disagree - but that's the interesting point here :-)
I think set theory was required as well.friend said:I don't know how numbers can be derived from logic alone. I understand Whitehead and Russel wrote a book doing just that. They also attempted to proved the completeness of math, but failed.
tom.stoer said:Assume that for some reason nature decided that its existence is based on manifolds...
This is still my belief: Physical existence is based on mathematical principles. These principles allow us not only to describe what exists but to some extend to explain why it exists and why it is the way it is.
Fra said:In a certain sense, I think any starting point must be valid as long as it, itself, is subject to selection and critics...
However, there is one important thing here which is that the initial inference system and hypothesis, must be able to be encodable with respect to the observers...
So I try to ask myself like, what questions would I ask, if I were an electron? :) Quickly one can first post another question, what questions COULD I even FORMULATE if I were an electron?
It's hard to distinguish between mathematical and physical principles. I tried to explain my position a couple of days ago: usually you start with a physical principle and develop a theory out of it = develop a math framework. During the development the physical principle may be transformed into a mathematical one. Look for example at the principles in special and general relativity.ConradDJ said:I understand why mathematical principles can seem more fundamental than physical ones ... Why manifolds?
You are right; it wouldn't explain "why QG at all - and not game of life". So you would have to step back and think about your unique theory again. Hopefully you will find a hidden and more fundamental (physical) principle from which this uniqueness follows. At some point it may be appropriate to think about an evolution process, but as I said: I do not (yet) see a good starting point for an evolution process - and I still like the axiomatic approach more ;-)ConradDJ said:Or suppose it turned out there was only one mathematically viable formulation of quantum gravity. Something like that would at least give us a reliable starting-point for trying to understand why the world is built the way it is – but it would hardly be an answer in itself.
Not really. The standard model plus GR seem to explain the different structures on different scales (different ages) quite well. It's the same with evolution in biology: The fundamental laws (selection due to environmental pressure, survival of the fittest) and the building blocks (DNA, RNA) are static but nevertheless allow for a dynamic evolution process. That's for me a very good reason not to let everything become dynamic, emerging, relative and - hopefully not - arbitrary.ConradDJ said:... that the universe has changed very radically over 13+ billion years, and all or nearly all the physical structure of our world did not exist at the beginning. This in itself seems to point to something very different from mathematical principles, as a starting-point, and raises the possibility of an evolutionary explanation.
I agree that the ontological interpretation of qm is by no means clear; but to take the wave function as a physically real entity is - if you ask me - not a very good choice.ConradDJ said:Whatever happens in the “collapse of the wave-function” seems to be basic to the emergence of definite information in the physical world, but it’s not clear what if anything it has anything to do with mathematical principles, since it happens at random.
Yes, but not less, either!ConradDJ said:... what’s going on in the physical world – even at the foundations – involves more than mathematical consistency.
We should distinguish between the view OF the outside (from the inside), and the view FROM the outside, I'm not sure what you mean here?ConradDJ said:I see the point of taking an “internal” view of the world... but it’s still a view of the world “out there” in which each system participates with others.
ConradDJ said:So I’m trying to see what this informational interaction-context looks like. What would the electron’s “inference” be? What would the input be, and what would be its “action”?
tom.stoer said:It's the same with evolution in biology: The fundamental laws (selection due to environmental pressure, survival of the fittest) and the building blocks (DNA, RNA) are static but nevertheless allow for a dynamic evolution process. That's for me a very good reason not to let everything become dynamic, emerging, relative and - hopefully not - arbitrary.
Fra said:I have to object here. To call the DNA/RNA and the the relevant replication and synthesis machinery in a cell a "static" doesn't seem to nearly appreciate the depth of life. There was certainly a time where there was no earth, no DNA molecules, not even proteins, and not even atoms.
So I think it's clear that the static abstraction there fails.