"Entropy, heat, and Gödel incompleteness", 2014, by Karl-Georg Schlesinger, — Tarskian
...is there a logically sound argument claiming there is a causal relationship between entropy and incompleteness? — ucarr
No. — 180 Proof
...succinctly express your disagreement with something I have written that you wish for me to further elaborate on... — 180 Proof
Interesting observation. I'm not sure it takes G-incompleteness to reach this point. — jgill
If the forward direction of a phenomenon incorporates information that cannot be decompressed from its theory, then it will also be impossible to decompress the information needed to reverse it, rendering the phenomenon irreversible.
The only problem I have, is that this view makes the details of the compression algorithm (the underlying theory) a bit too fundamental to my taste. — Tarskian
Is there any literature that examines questions about the relationship between Heisenberg Uncertainty and Gödel Incompleteness? — ucarr
Calude & Stay, 2004, "From Heisenberg to Gödel via Chaitin."
https://link.springer.com/article/10.1007/s10773-006-9296-8#preview
In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.” Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the converse implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a “formal uncertainty principle” which implies Chaitin’s information-theoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. In fact, the formal uncertainty principle applies to all systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies algorithmic randomness not only in mathematics, but also in physics.
The accidental can’t be in fact removed from the world, even if that is not what axiomatic determinism wants us to believe. — apokrisis
Similarly, there is absolutely no need for the physical universe to be random, for it to be largely unpredictable. It could be, but it does not have to be. — Tarskian
But then again, it also does not mean that the information forgotten in the compression is "accidental" or "random". It does not even need to be. There is nothing random about arithmetical reality, while it is still full of unpredictable facts. — Tarskian
But It only takes an infinitesimal grain of chance to complete things. — apokrisis
Now you are making points about variety in types of compression algorithms, not about general principles. — apokrisis
https://arxiv.org/pdf/0906.3507
In particular, I use Jaynes’ maximum entropy approach to unify the relations between aggregation and pattern (Jaynes, 2003). Information plays the key role. In each problem, ultimate pattern arises from the particular information preserved in the face of the combined fluctuations in aggregates that decay all non-preserved aspects of pattern toward maximum entropy or maximum randomness.
I am starting to believe that what you are really getting at behind the curtains here is that science and art share common features. — I like sushi
You have not given me any reason to read someone else's thoughts on the matter. Make your philosoophical case, ucarr, and I will respond. — 180 Proof
...is there a logically sound argument claiming there is a causal relationship between entropy and incompleteness? — ucarr
No. — 180 Proof
If the forward direction of a phenomenon incorporates information that cannot be decompressed from its theory, then it will also be impossible to decompress the information needed to reverse it, rendering the phenomenon irreversible. — Tarskian
You understand it perfectly clear. The basic issue here is the negative self-reference. And that issue is similar in Gödel's incompleteness theorem and Turings result (on the Entscheidungsproblem).Let me attempt to clarity: I'm attempting say I can't enact the negation of what I'm doing. →
→
Anything I write will not be something I do not write. — ucarr
Stop. This confuses empiricism with formalism – nonsense (i.e. logic is not "doing work").... for any system that does work, as it goes forward in the systematic process of doing work, the work builds up complexity of detail. This building up of complexity can be observed in two modes: phenomenal (entropy) and epistemic (logic). — ucarr
More nonsense. Formalisms (axiomatic or otherwise) are abstract and therefore do not refer beyond themselves to concrete matters of fact (e.g. entropy), rather they are used as syntax for methods of precisely measuring / describing the regularities of nature. That syntax is fundamentally incomplete / undecidable (re: Gödel / Chaitin) says nothing about nature, only about the (apparently) limits of (our) rationality. In other words, that physical laws are computable does not entail that the physical universe is a computer.This leads to the conclusionthat axiomatic systems are a form of compression of complexity and that the increase of complexity is an irreversible process.
The few rules in the axiomatic theory will not succeed in decompressing themselves back into the full reality. What facts from the full reality that they fail to incorporate does not say particularly much about these facts (deemed "chance", "random", ...). They rather say something about the compression technique being used, which is the principle that chooses what facts will be deemed predictable and what facts will be deemed mere "chance". — Tarskian
Is it true that the extrapolation from an axiomatic system to complexity irreversible to the axiomatic system cannot be certified, and thus axiomatic systems are both incomplete and uncertain? — ucarr
It is simply not possible to decompress and reconstruct the totality of all the information about reality out of an axiomatic system that describes it (if this axiomatic system is capable of arithmetic). — Tarskian
But then again, it also does not mean that the information forgotten in the compression is "accidental" or "random". — Tarskian
Randomness is not a necessary requirement for unpredictability. Incompleteness alone is already sufficient. A completely deterministic system can still be mostly unpredictable. — Tarskian
In each problem, ultimate pattern arises from the particular information preserved in the face of the combined fluctuations in aggregates that decay all non-preserved aspects of pattern toward maximum entropy or maximum randomness — Tarskian
Axiomatic theories do something similar. — Tarskian
The few rules in the axiomatic theory will not succeed in decompressing themselves back into the full reality. What facts from the full reality that they fail to incorporate does not say particularly much about these facts (deemed "chance", "random", ...). They rather say something about the compression technique being used, which is the principle that chooses what facts will be deemed predictable and what facts will be deemed mere "chance". — Tarskian
As I understand it, an axiomatic system is a compressor. — ucarr
The algorithm that generates the axiomatic system has a focal point that excludes info inconsequential to the outcome the axiomatic system tries to predict. — ucarr
Does a lossy axiomatic system also necessarily omit consequential facts because of measurement limitations described by Heisenberg Uncertainty? — ucarr
In fact, the formal uncertainty principle applies to all systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies algorithmic randomness not only in mathematics, but also in physics.
Was this correct:
I am starting to believe that what you are really getting at behind the curtains here is that science and art share common features. — I like sushi
Followed by the possibility of uniting/transcending the differences held by many? — I like sushi
I simple yes/no or suffice. If it is a bit more than this then a sketchy - yet straight forward - outline would be all I need. — I like sushi
We don't live within a universe; instead, we live within a vital approach to a universe strategically forestalled by entropy_uncertainty_incompleteness. Science and Humanities are the two great modes of experiencing the uncontainable vitality. — ucarr
Can we generalize to the following claim: our material creation, as we currently understand it, supports: the determinism of axiomatic systems, the incompleteness of irreversible complexity and the uncertainty of evolving dynamical systems, and, moreover, this triad of attributes is fundamental, not conditional? — ucarr
... for any system that does work, as it goes forward in the systematic process of doing work, the work builds up complexity of detail. This building up of complexity can be observed in two modes: phenomenal (entropy) and epistemic (logic). — ucarr
...logic is not "doing work" — 180 Proof
This leads to the conclusionthat axiomatic systems are a form of compression of complexity and that the increase of complexity is an irreversible process.
More nonsense. Formalisms (axiomatic or otherwise) are abstract and therefore do not refer beyond themselves to concrete matters of fact (e.g. entropy), rather they are used as syntax for methods of precisely measuring / describing the regularities of nature. — 180 Proof
We will show that algorithmic randomness is equivalent to a “formal uncertainty principle” which implies Chaitin’s information-theoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. In fact, the formal uncertainty principle applies to all systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies algorithmic randomness not only in mathematics, but also in physics. — Tarskian quoting Calude and Stay
Does a lossy axiomatic system also necessarily omit consequential facts because of measurement limitations described by Heisenberg Uncertainty? — ucarr
Yes. Technically, the resulting imprecision is the due to the fundamental properties of wave functions. — Tarskian
Do you have any interest in the Beckenstein bound, from the Holographic Principle (Gerard t'Hooft)? It describes a limit to the amount of information that can be stored within an area of spacetime at the Planck scale. Among other things, this limit establishes the physical nature of information. There's an algorithm for measuring the Beckenstein bound: it's a fraction of the area of the event horizon of a black hole. — ucarr
https://en.wikipedia.org/wiki/Bekenstein_bound
In physics, the Bekenstein bound (named after Jacob Bekenstein) is an upper limit on the thermodynamic entropy S, or Shannon entropy H, that can be contained within a given finite region of space which has a finite amount of energy—or conversely, the maximum amount of information required to perfectly describe a given physical system down to the quantum level.
Formalisms (axiomatic or otherwise) are abstract and therefore do not refer beyond themselves to concrete matters of fact (e.g. entropy)... — 180 Proof
...rather they are...measuring / describing the regularities of nature. — 180 Proof
Uncertainty is a precision problem.
More precision means more information.
According to Chaitin's incompleteness, sufficiently higher precision will indeed at some point exceed the amount of information that the system can decompress.
According to the literature on the subject, both incompleteness and imprecision ("uncertainty") can be explained by the principle of lossy compression that results in a particular maximum amount of information that could ever be decompressed out of the system. — Tarskian
Does a lossy axiomatic system also necessarily omit consequential facts because of measurement limitations described by Heisenberg Uncertainty? — ucarr
Yes. Technically, the resulting imprecision is the due to the fundamental properties of wave functions.
However, the paper mentioned , Calude & Stay, 2004, "From Heisenberg to Gödel via Chaitin.", connects uncertainty to Chaitin's incompleteness:
In fact, the formal uncertainty principle applies to all systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies algorithmic randomness not only in mathematics, but also in physics.
They conclude that it is not possible to decompress more precise information out of an axiomatic system than the maximum precision imposed by the fundamental properties of wave functions. — Tarskian
...that physical laws are computable does not entail that the physical universe is a computer. — 180 Proof
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