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

Yes, it is.

I need your help in understanding how I'm being unfair. — ucarr

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

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.

"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

It's all cleared up. I like Art Garfunkel's rendering of "Always Look On The Bright Side of Life (Monty Python)." Have you heard it?

I know the song well - have done for years. It always gets a smile. Don't know if it's Art Garfunkel's rendering. How many are there?

"Always Look On The Bright Side Of Life"

Links to renderings:

Mozart Group

Eric Idle

North Korean Version

Art Garfunkel

Life of Brian

Links to renderings: — ucarr

Thanks. It would be quite a festival to play all of those at the same time.

NORTH KOREAN VERSION!!!

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

Yes, but if you have a copy of the theory of the physical universe, you could conceivably ask a computing device to check if a given fact is provable from this theory, or not.

In absence of this theory, we indeed don't know if a fact is provable or unprovable from it.

In fact, even when we have the theory of a particular reality, such as for arithmetical reality, we still cannot figure out if something like the Riemann hypothesis is provable from it or not.

Independent from whether the Riemann hypothesis is true or not, "Is the Riemann hypothesis independent from Peano arithmetic theory?" remains an unanswered question.

In fact, a computing device is unlikely to produce a new original proof anyway. All non-trivial proofs have historically been produced manually from a particular theory.

Therefore, having a sound theory to prove a given fact from is a necessary condition to assess its provability but not a sufficient one.

Therefore, having a sound theory to prove a given fact from is a necessary condition to assess its provability but not a sufficient one. — Tarskian

What if a given fact is unprovable within a given theory, but provable within another one. Would that be consistent with Godel?

What if a given fact is unprovable within a given theory, but provable within another one. Would that be consistent with Godel? — Ludwig V

Yes, that is exactly the case for Goodstein's theorem.

The theorem itself can be expressed in the language of Peano arithmetic but the proof cannot.

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.

The proof that there cannot be a proof for Goodstein's theorem in Peano arithmetic is deemed much harder than the proof itself:

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.

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.

Examples for Godel's theorem are in fact always such contorted corner cases. Otherwise, they can generally not even be detected with arithmetical vision. Unlike in physical reality, in arithmetical reality we typically know that a theorem is true because we can prove it. No need for proof in physical reality to perceive its facts. 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.

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

OK. I think understand what is going on, even though I cannot understand the proofs. Thanks.

Examples for Godel's theorem are in fact always such contorted corner cases. — Tarskian

I'm not surprised.

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

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.

No need for proof in physical reality to perceive its facts. — Tarskian

Why is the Copernicus_Galileo debate with the Catholic Church (re: the earth orbiting the sun) not a counter-narrative to this claim?

Is there any literature that examines questions about the relationship between Heisenberg Uncertainty and Gödel Incompleteness?

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

Is this an example of uncertainty rooted within incompleteness?

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

Here's the really big question: Is there a foundational relationship between uncertainty, incompleteness and entropy?

Consider: The second law of thermodynamics is a physical law based on universal empirical observation concerning heat and energy interconversions. A simple statement of the law is that heat always flows spontaneously from hotter to colder regions of matter (or 'downhill' in terms of the temperature gradient). Another statement is: "Not all heat can be converted into work in a cyclic process."[1][2][3]

So, heat (unfocused energy) without constraints, always spontaneously flows out of an energetic system, such that necessarily only a fraction of the contained energy of a thermo-dynamical system can be converted (brought into focus) into work.

The second law of thermodynamics establishes the concept of entropy as a physical property of a thermodynamic system. It predicts whether processes are forbidden despite obeying the requirement of conservation of energy as expressed in the first law of thermodynamics and provides necessary criteria for spontaneous processes. For example, the first law allows the process of a cup falling off a table and breaking on the floor, as well as allowing the reverse process of the cup fragments coming back together and 'jumping' back onto the table, while the second law allows the former and denies the latter.

So, the entropy of a thermo-dynamic system is uni-directional; it always increases; it never decreases.

The second law may be formulated by the observation that the entropy of isolated systems left to spontaneous evolution cannot decrease, as they always tend toward a state of thermodynamic equilibrium where the entropy is highest at the given internal energy.[4] An increase in the combined entropy of system and surroundings accounts for the irreversibility of natural processes, often referred to in the concept of the arrow of time.[5][6]

2nd Law of Thermodynamics

That uncertainty might be rooted in incompleteness suggests a relationship between the two phenomena.

Now we want to see if we can connect uni-directional entropy as the arrow of time to this duad in order to make a very meaningful triad: uncertainty_incompleteness_arrow of time.

Does our triad tell us that entropy insures the material existence is always moving forward into a future that is never a simple cyclic repetition of past phenomena, such as an oscillation of the universe between big bang and big crunch?

Does our triad likewise tell us that our movement towards the future is more than a statistically probable permutation of conserved laws?

Does the discovery of QM tell us that our future is truly unknown and unknowable, albeit partially predictable?

Is entropy the engine driving uncertainty_incompleteness_arrow of time?

Is entropy the mortal enemy of the T.O.E.?

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:

"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

Consider a chain of causation: a) live music is performed in a radio station studio for a live broadcast; b) the live music in the studio as rendered to radio listeners is incomplete because of noise in the transmission; c) the listeners - because of the noisy transmission - are uncertain whether three successive brass instrument solos are a flugelhorn, a trumpet and a cornet, successively, or some other configuration according to the possibilities.

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.

Our triad, in spite of variations, remains intact.

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

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).

I always give the example of trying writing something you will never in your life write.

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.

This is not an example of a "fundamental relationship of uncertainty, incompleteness & entropy". Not even close.

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

If a thing is not computable, thus causing attempted measurements to terminate in undecidability, is it sound reasoning to characterize this undecidability as uncertainty (about a conjectured definitive measurement)?

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

Is this a logical statement: $\neg x$ ≠ $x$? If so, then why is it not a logically preemptive limitation on what I can write?

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?

The $2^{nd}$ law of thermo-dynamics tells us that no isolated system can convert all of its internal energy into work. Why is this statement - among its other related implications - not an implication systematic utilization of available energy is always incomplete?

If it is one of its implications, how is it the case the 2nd law of thermo-dynamics is essential to nature, but one of its implications isn’t?

Are you proceeding from the premise causal relationships are not fundamental in nature? — ucarr

Nope. https://thephilosophyforum.com/discussion/comment/929175

Are you proceeding from the premise causal relationships are not fundamental in nature? — ucarr

Nope. — 180 Proof

So if, for example, the 2$^{nd}$ law of thermo-dynamics establishes that systematic utilization of heat is always incomplete, and that this unconstrained thermal energy always travels to a cooler state toward equilibrium in randomization, is there a logically sound argument claiming there is a causal relationship between entropy and incompleteness?

is there a logically sound argument claiming there is a causal relationship between entropy and incompleteness? — ucarr

Possibly between entropy and incompleteness in its more traditional meaning. Not with the math variety.

causal relationship between entropy and incompleteness? — ucarr

This does not really make sense. entropy is basically the thing revealed empirically when we measure things temporally. Necessarily, not consequentially, the beating heart of physics is entropy (which is a place-holder for "we do not know").

I am starting to believe that what you are really getting at behind the curtains here is that science and art share common features. That the subject and object distinction is merely one of convenience.

To which I can say. Yes. And yes, it can be extremely hard to pull people away from their microscopes to look at the larger picture. To attempt this is often futile so pick your targets well.

No.

is there a logically sound argument claiming there is a causal relationship between entropy and incompleteness? — ucarr

The following paper, "Entropy, heat, and Gödel incompleteness", 2014, by Karl-Georg Schlesinger, suggests that:

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.

The dynamic system simply "forgets" how to go back. It would have to remember too much information for that purpose:

We suggest that on a fundamental level the impossibility to provide the necessary information might be related to the incompleteness results of Gödel.

The suggested connection would be as following:

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).

The principle (Chaitin's incompleteness theorem) which Yanovsky mentioned in "True But Unprovable":

"A fifty-pound logical system cannot prove a seventy-five-pound theorem.”

would apply as following:

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.

So, entropy would be the result of the gap between the 75 pounds of the system dynamics while having only 50 pounds to explain it on the basis of axiom system A. This 25-pound shortage of explanatory power would fuel the entropy in the dynamics, as it leads to loss of information in the process, rendering the dynamics irreversible.

Schlesinger makes extensive use of the Curry-Howard correspondence ("Every proof is a program") in his paper:

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).

The problem that I have with this view is the very strategic choice of axiom system A.

Nothing guarantees that there exists an axiom system A. Nothing guarantees that there is only one such A. In the meanwhile, entropy still occurs as a physically observable phenomenon, regardless of any choice of A.

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.

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?

Possibly between entropy and incompleteness in its more traditional meaning. Not with the math variety. — jgill

What do you think of the connection that Schlesinger makes between entropy and Godelian incompleteness in "Entropy, heat, and Gödel incompleteness" (2014)?

https://arxiv.org/pdf/1404.7433

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.

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, 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 so, then why is it not a logically preemptive limitation on what I can write? — 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.

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

This is important generally, and here more specifically. If what you say immediately above is true, then the 2$^{nd}$ law of thermo-dynamics, if, as I herein theorize, is directly and deeply tied to the always incomplete systematic utilization of heat energy for work, then there seems to me to be good reason to think incompleteness -- not the existential measurement uncertainty of the Fourier transforms applied to elementary particles, but instead the garden variety of incompleteness: not all of the potential has been utilized -- exhibits a pattern in possession of an underlying logic. If we can learn to read that underlying logic, I conjecture it will tell us a foundational story about the passage of time and events into the future.

Science: The What - the 2$^{nd}$ law of thermo-dynamics; Humanities: The How - clinical depression of the human psyche resembles the conjectured heat death of a material system wherein all is at a lifeless equilibrium, the cosmic tendency of matter energy systems. In human terms of the "how is it experienced": the no-affect grayscale of a flatlined inner emotional life.

Entropy_uncertainty_incompleteness keep our material environment alive. Life will not be understood in the terms of wholeness, completion and closure. Since vitality tends towards these things, it's natural to seek after them. I don't think we'll find them, and that's a good thing because life, the supremely good thing, depends on us not finding them.

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.

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

Please supply: known facts pertinent; corroborating evidence supporting known facts pertinent; logical analysis; valid conclusions drawn from your logical analysis.

Regarding what exactly?

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

I'm glad to see I've joined some estimable thinkers who have preceded me. My gratitude to Tarskian for the citations.

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

:up:

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

This situation - soft uncertainty - doesn't preclude uncertainty from also being applied to:

You can have all the information, yet there's no way out of this. — ssu

which is hard uncertainty.

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

I attempt to refute your above denial (underlined) with:

Is this a logical statement: ¬x ≠ x? If so, then why is it not a logically preemptive limitation on what I can write? — ucarr

Let me attempt to clarity: I'm attempting say I can't enact the negation of what I'm doing. $\rightarrow$ Anything I write will not be something I do not write.

Regarding what exactly? — 180 Proof

Please use the links below to Tarskian's posts. After re-reading the posts, refute the citations referenced by Tarskian by: providing known facts pertinent; corroborating evidence supporting known facts pertinent; logical analysis; valid conclusions drawn from your logical analysis.

Tarskian_Schlesinger 1

Tarskian_Schlesinger 2

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.

Interesting observation. I'm not sure it takes G-incompleteness to reach this point. Dynamical system structures may be just too complex to handle without going into the realm of uncountable math garbage.