Yes, it is.In the grapevine mesh of existing things, for each thing, there's always one observer who sees that thing as it is in truth. Is this not a charming article of faith warding off depression? — ucarr
I need your help in understanding how I'm being unfair. — ucarr
You seemed a bit depressed when you said this. I was trying to be encouraging. "Be fair" is an expression I use - perhaps it is not as widely used as I thought - to signal that there is a brighter side to what seems so depressing. It's not an accusation or criticism.Yes, our experience is rooted within interrelationships. There seems not to be any existing thing utterly isolated and alone. There's always the hope of being understood. — ucarr
"Be fair" is an expression I use - perhaps it is not as widely used as I thought - to signal that there is a brighter side to what seems so depressing. It's not an accusation or criticism. — Ludwig V
But not knowing why my observation is true is not the same as its being unprovable. Surely that will only work if what I observe is incapable of being proved, as opposed to my not knowing how to prove it. If I knew that it was unprovable, I think I would either not believe my eyes or at least suspend judgement. — Ludwig V
What if a given fact is unprovable within a given theory, but provable within another one. Would that be consistent with Godel? — Ludwig V
https://en.m.wikipedia.org/wiki/Goodstein%27s_theorem
Goodstein's theorem
Laurence Kirby and Jeff Paris[1] showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic or Zermelo-Fraenkel set theory). This was the third example of a true statement about natural numbers that is unprovable in Peano arithmetic.
While this proof of Goodstein's theorem is fairly easy, the Kirby–Paris theorem,[1] which shows that Goodstein's theorem is not a theorem of Peano arithmetic, is technical and considerably more difficult.
OK. I think understand what is going on, even though I cannot understand the proofs. Thanks.The proof makes use of infinite ordinals. Transfinite numbers are not defined in Peano arithmetic, pushing the proof outside the capabilities of this theory. The difficulty is to prove that the proof must make use of them. — Tarskian
I'm not surprised.Examples for Godel's theorem are in fact always such contorted corner cases. — Tarskian
I've been changing my view of mathematics for a couple of years now - since I came back to it, in fact. I no longer think of it as an eternally peaceful, ordered world, as in Plato's heaven. (Although they did already know about the irrationality of sqrt2). As you say, it's coming to look much more like physical reality.That is why arithmetical reality appears so orderly to us, while in reality, it is highly chaotic, just like physical reality. We just cannot see the chaos. — Tarskian
No need for proof in physical reality to perceive its facts. — Tarskian
The Fourier transforms won't allow us to accurately measure both position and momentum of an elementary particle; it's one measurement at a time being accurate, with the other measurement being far less accurate. — Tarskian
Unlike in physical reality, in arithmetical reality we typically know that a theorem is true because we can prove it... That is why arithmetical reality appears so orderly to us, while in reality, it is highly chaotic, just like physical reality. We just cannot see the chaos. — Tarskian
"Uncertainty" is epistemic, "incompleteness" is mathematical and "entropy" is physical. I don't think they are related at a deeper – "foundational" – level unless Max Tegmark's MUH is the case. :chin:Is there a foundational relationship between uncertainty, incompleteness and entropy? — ucarr
"Uncertainty" is epistemic, "incompleteness" is mathematical and "entropy" is physical. I don't think they are related at a deeper – "foundational" – level unless Max Tegmark's MUH is the case. :chin: — 180 Proof
If there is any logical equivalence, mathematical incompleteness (or undecidability) would be something equivalent to the problem in physics of the measurement of an object effecting itself what is to be measured and hence ruining what was supposed to be an objective measurement in the first place. The undecidability results simply show that not all is computable (or in the case of Gödel's theorems, provable), even if there is a correct model for the true mathematical object (namely itself).Even though the parallel breaks down at b) incompleteness because, in the example, it's due to the entropy of electromagnetic transduction (albeit mathematically describable), nonetheless the physics of entropy causes the incompleteness and, in turn, the incompleteness causes the uncertainty. — ucarr
The undecidability results simply show that not all is computable (or in the case of Gödel's theorems, provable), even if there is a correct model for the true mathematical object (namely itself). — ssu
There is a lot of text which you won't ever write, but anything you write will automatically be something you do write (and hence not in the category of all the texts you will never write). So is this a limitation on what you can write? Of course not. You can still write anything you want. It's a bit similar with the undecidability results. — ssu
This is not an example of a "fundamental relationship of uncertainty, incompleteness & entropy". Not even close. — 180 Proof
Are you proceeding from the premise causal relationships are not fundamental in nature? — ucarr
Nope. — 180 Proof
causal relationship between entropy and incompleteness? — ucarr
is there a logically sound argument claiming there is a causal relationship between entropy and incompleteness? — ucarr
https://arxiv.org/pdf/1404.7433
Irreversible phenomena – such as the production of entropy and heat – arise from fundamental reversible dynamics because the forward dynamics is too complex, in the sense that it becomes impossible to provide the necessary information to keep track of the dynamics.
We suggest that on a fundamental level the impossibility to provide the necessary information might be related to the incompleteness results of Gödel.
If the dynamics of a system becomes so complex that Gödel incompleteness prohibits a complete description of its dynamics, the necessary information – to determine the dynamics – is fundamentally lost on a universal Turing machine. This should – from the results on universal Turing machines, mentioned above – imply a production of entropy and heat.
So, the results of [Moo] offer the possibility for a new Ansatz which could lead to a fundamental understanding of irreversibility and the production of entropy and heat from Gödel incompleteness for dynamical systems of sufficient complexity.
[Moo] C. Moore, Unpredictability and undecidability in dynamical systems, Phys. Rev. Lett. 64, n. 20, 2354-2357 (1990).
"A fifty-pound logical system cannot prove a seventy-five-pound theorem.”
https://arxiv.org/pdf/1404.7433
Gödel incompleteness has a very clear description in terms of complexity. We can attach a degree of complexity to any choice of axiom system A. If the dynamics of a system of differential equations becomes non-predictable, we can understand this as the dynamics of the system becoming too complex, relative to the complexity of A (see [CC], [Cha]).
The entropy should then be a quantitative measure of how much the complexity of the dynamics exceeds that of A, i.e. it should relate to the complexity of the dynamics relative to the complexity of A.
[Cha] G. J. Chaitin, Algorithmic information theory, Cambridge University Press, Cambridge 1992.
https://arxiv.org/pdf/1404.7433
Gödel’s incompleteness results imply that there are problems which are fundamentally
beyond the reach of universal Turing machines (and, therefore, beyond mathematical acessability since mathematical axiom systems are nothing but programs – or program languages – running on a universal Turing machine).
https://arxiv.org/pdf/1404.7433
So, we would need a slightly stronger form of Gödel incompleteness which would make the dynamics non-predictable for any choice of axiom system A.
Possibly between entropy and incompleteness in its more traditional meaning. Not with the math variety. — jgill
For me uncertainty refers to a situation where you don't have all the information, for example. This isn't the case. You can have all the information, yet there's no way out of this. The reason is negative self-reference. And in the case of Gödel's theorems, it's not even a direct self-reference (the statement s is not provable). What should be noted that Gödel's incompleteness theorems are sound theorems, not paradoxes. Even if many relate it to being close to the Liar paradox.If a thing is not computable, thus causing attempted measurements to terminate in undecidability, is it sound reasoning to characterize this undecidability as uncertainty — ucarr
Why would it be not logical? The undecidability results are totally logical. Not all statements are provable and not everything is computable by a Turing Machine. It is totally logical. You can call them preemptive limitations, that's fine. So a Turing Machine has this "preemptive limitation" and hence it cannot compute everything.If so, then why is it not a logically preemptive limitation on what I can write? — ucarr
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
..the beating heart of physics is entropy — I like sushi
Are you proceeding from the premise causal relationships are not fundamental in nature? — ucarr
Nope. — 180 Proof
...is there a logically sound argument claiming there is a causal relationship between entropy and incompleteness? — ucarr
No. — 180 Proof
The implicit but really strong assumption in Schlesinger's paper is that there exists exactly one lossy compression algorithm, i.e. axiom system A, for the information contained in the physical universe.
Schlesinger actually admits this problem:
https://arxiv.org/pdf/1404.7433
So, we would need a slightly stronger form of Gödel incompleteness which would make the dynamics non-predictable for any choice of axiom system A. — Tarskian
According to Schlesinger, a physical phenomenon becomes irreversible and entropy will grow, if reversing the phenomenon would require using more information than allowed by Godel's incompleteness. — Tarskian
If all these alternative compression algorithms always lead to the same output in terms of predicting entropy, then for all practical purposes, they are one and the same, aren't they? — Tarskian
If a thing is not computable, thus causing attempted measurements to terminate in undecidability, is it sound reasoning to characterize this undecidability as uncertainty — ucarr
For me uncertainty refers to a situation where you don't have all the information.. — ssu
You can have all the information, yet there's no way out of this. — ssu
There is a lot of text which you won't ever write, but anything you write will automatically be something you do write (and hence not in the category of all the texts you will never write). So is this a limitation on what you can write? Of course not. — ssu
Is this a logical statement: ¬x ≠ x? If so, then why is it not a logically preemptive limitation on what I can write? — ucarr
Regarding what exactly? — 180 Proof
What do you think of the connection that Schlesinger makes between entropy and Godelian incompleteness in "Entropy, heat, and Gödel incompleteness" (2014)? — Tarskian
If the dynamics of a system becomes so complex that G¨odel incompleteness
prohibits a complete description of its dynamics, the necessary information
– to determine the dynamics – is fundamentally lost on a universal Turing
machine.
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.