Well, Roger Penrose said in his Emperor's New Mind that the mind was not reducible to algorithms — Wayfarer

This may be true but I do not agree with Penrose's core argument:

https://en.m.wikipedia.org/wiki/Penrose%E2%80%93Lucas_argument

Due to human ability to see the truth of formal system's Gödel sentences, it is argued that the human mind cannot be computed on a Turing machine that works on Peano arithmetic because the latter cannot see the truth value of its Gödel sentence, while human minds can.

How do humans know that a mathematical sentence is true? There is only one way: by proving it. Otherwise, it will be deemed a hypothesis and not a (true) theorem. But then again, we are still able to correctly detect some Gödel sentences, i.e. sentences that are true but not provable, but that requires a rather special situation, such as for example, in the case of the Goodstein's theorem.

The language in which Goodstein's theorem is phrased, is Peano Arithmetic theory (PA). However, the language in which its proof is phrased, is Zermelo-Fränckel set theory (ZF). Its proof uses infinite ordinals, which are defined in ZF but not in PA. Hence, Goodstein's theorem belongs to PA but its proof does not belong to PA. Its proof belongs to ZF. That is why we know that Goodstein's theorem is true in ZF and therefore also in PA. Hence, from the standpoint of PA, Goodstein's theorem is indeed true but not provable, i.e. a Gödel sentence.

A Turing machine could also use ZF to prove an otherwise unprovable theorem in PA. Therefore, it is not something that only human minds can do. What if there is no alternative theory available to prove the Gödel sentence from? In that case, both humans and the machine will not be able to know that the Gödel sentence is true. They will both consider it to be just a hypothesis.

Conclusion. The ability to see the truth of Gödel sentences is not different between human minds and Turing machines. In the general case, they will both fail to do it. That is where I fundamentally disagree with Penrose. The human mind may still be superior to Turing machines but not for its ability to see the truth of Gödelian sentences.

The Nostr protocol claims to be censorship resistant:

https://www.voltage.cloud/blog/nostr-the-decentralized-censorship-resistant-messaging-protocol

Nostr: The Decentralized, Censorship-Resistant Messaging Protocol

In a world where centralized entities increasingly control digital communication, the demand for more transparent, private, and censorship-resistant systems is growing. One of the technologies that promise to deliver this independence is Nostr, a decentralized messaging protocol.

The author claims that it is enough to "distribute data across a network of peers" in order to achieve censorship resistance:

What sets Nostr apart from other messaging services is its decentralized nature. Unlike centralized platforms—where data is stored and managed by a single organization, like a corporation or a government—Nostr doesn't rely on a single server or entity. Instead, it distributes data across a network of peers (nodes known as relays) that participate in storing and transmitting messages.

I have my doubts about that.

In that case, it would also be enough to just distribute money-moving transactions across a network of peers in order to implement something like Bitcoin. There would be no need to burn lots of electricity as proof of work.

I personally do not believe that the principle of "distribution" alone would be enough to fend off the censorship attacks of a determined state actor. In my opinion, it takes a lot more effort to put up a credible anti-Statist defense. Everything else is just an exercise in wishful thinking.

Anti-Statist measures are possible but they are invariably costly.

How’s that?
So they are not well-formed? Something is amiss. — Banno

Doesn’t two plus two equals four qualify? It’s a true statement about natural numbers isn’t it? — Wayfarer

You gave an example of a statement about the natural numbers that can be expressed in language. That is an exception and not the rule.

Take for example {5, 10, 71}. This is a subset of the natural numbers. So, the following statement is a truth about the natural numbers:

The set {5, 10, 71} is a subset of the natural numbers.

This statement cannot be expressed in language about the overwhelmingly vast majority of the subsets of the natural numbers, because there are uncountably infinite many such subsets while language itself is countably infinite.

Hence, the overwhelmingly vast majority of the truths about the natural numbers are ineffable.

Note: you do not need set-theoretical language to express the notion of "finite subset" in arithmetic, given the bi-interpretability of arithmetic theory (PA) with bounded set theory (ZF-inf). You can express it entirely in the language of arithmetic itself: "On Interpretations of Arithmetic and Set Theory" by Richard Kaye, Tin Lok Wong, https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-48/issue-4/On-Interpretations-of-Arithmetic-and-Set-Theory/10.1305/ndjfl/1193667707.full

All of this leads me to conclude that the hubbub over misinformation is a campaign for more power rather than a legitimate plight for public safety. — NOS4A2

It is actually possible to set up decentralized censorship-resistant information publication networks.

For example, Bitcoin is one. The powers-that-be cannot prevent the publication of money-moving transactions on the Bitcoin blockchain. It is beyond the capacity of any government on earth to do that.

Unfortunately, it would be prohibitively expensive to cover ordinary non-financial speech with electricity-based proof-of-work protection.

Conclusion. We would have censorship-resistant social media, if we agreed to pay for that.

This would be impractical in its application in the real world and would serve no use apart from counting, though the points in a line are infinite naming/identifying each point in a line would be an unnecessary exercise. — kindred

This is just one example of the problem.

The overwhelmingly vast majority of true statements about the natural numbers cannot be expressed in language. The arithmetical universe is actually replete with ineffable truth.

If the physical universe is sufficiently structurally similar to the arithmetical universe, then this is also the case for physical truth.

While pure reason, and therefore arithmetic itself, are blind, physical truth is much more observable. Unlike arithmetical truth, we do not even need to understand it, in order to see it. However, even though we can see it, we cannot express it in language. There is fundamental mathematical support for this case.

Furthermore, not all of these ineffable truths will turn out to be irrelevant.

That’s because the points on a line are infinite, why can’t there be infinite words ? — kindred

There are infinite identifiers possible, but still a countable number of them. There are uncountably infinite points in a line.

Therefore, it is not possible to create a one-to-one mapping between them.

If aleph0 is the number of countably infinite elements, then the number of uncountably infinite elements is 2^aleph0.

There is not just one measure for an infinite number of elements. Sets can have different possible infinite sizes ("cardinalities").

This is the essence of Cantor's 1891 diagonal argument:

https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Cantor's diagonal argument (among various similar names[note 1]) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which in some sense contain more elements than there are positive integers. Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers, which Cantor began.

No you won’t, you can just create new identifiers (words). — kindred

You can count identifiers. You cannot count points in a line. So, you will never be able to create an identifier for each point.

But, if you could have infinite planets, you could name each one as a number. I don't follow how there can be an infinite set of planets but not an infinite set of names. — Hanover

Planets are indeed countable. There are uncountable sets too. For example, the individual points in a line.

You can just assign a newly discovered planet for which we don’t have word for, a made up word or letter-number designation. That word then would be its identifier… — kindred

If you know that there are uncountably many of them, you cannot keep assigning countably many identifiers. You will run out of those.

https://en.m.wikipedia.org/wiki/Cardinality_of_the_continuum

Cardinality of the continuum

The real numbers R are more numerous than the natural numbers N.

Planets are actually a bad example because they are obviously countable.

That I've not seen all the planets doesn't mean those I've not seen are ineffable. — Hanover

If there are more planets than possible words then you can't give each of them a different name. It doesn't matter if you have seen all of them.

Is there anything that language can’t express ? — kindred

Language expressions are countable.

(You could conceivably create a list of them)

Even though there may be an infinite number of language expressions, their countability severely restricts what can be expressed.

For example, there are uncountably infinite real numbers. Therefore, it is impossible to express all of them in language.

For example, "true arithmetic" is the set of all true statements about the natural numbers. This set is uncountable. Therefore, it cannot be expressed in language. Hence, most of the truth about the natural numbers is ineffable.

If the physical universe is structurally sufficiently similar to the natural numbers, then most of the truth about the physical universe is also ineffable, meaning that only a very small fraction can be expressed in language.

Language expressions and digital models are both countable. Therefore, a digital model will be able to capture only a very small fraction of the truth about the physical universe.

Conclusion. The majority of truths cannot be expressed in language. Most truth is ineffable.

But all those theorems rely on axioms which have not been proven, so they rest on a foundation that isn't objectively sound, which is why I question if 3+5 equaling 8 being an objective truth. — noAxioms

You can reject arithmetical truth, until you make use of it, or of a statement that happens to be equi-consistent with it.

actual truth seems not to depend on proof or even anything being aware of it. — noAxioms

For physical truth, you can observe it. No need for ulterior justification. Arithmetical truth, however, cannot be physically observed. Pure reason is essentially blind.

In almost all cases, we make use of arithmetical soundness theorem to ascertain the truth of a statement: The statement is true because it is provable.

For example, is the sum of 3 and 5 equal to 8, or is that just a property of our universe? Mathemaical 'truths' are often held as objective, but proving that is another thing. — noAxioms

Yes, if there is no objective justification ("proof") then there won't be (an objective) consensus on whether it is true or not, rendering such truth subjective. Even in arithmetic, most truth is unprovable and therefore lacking objectivity.

Not all true statements have an objective justification, but some do.

This is a fundamental tenet in mathematics.

According to Godel's incompleteness theorem, some true statements in arithmetic theory are provable but the vast majority is not.

("Arithmetic" being standard Peano arithmetic or similar)

In the set of true arithmetical statements, the unprovably true statements vastly outnumber the provable ones.

For example, the laws of physics or mathematical truths are often cited as examples of objectivism in action. — Cadet John Kervensley

If a statement is provable in arithmetic, then it is true in all "models" ("interpretations") of arithmetic.

The reverse is not true.

It is not because a statement in arithmetic is true that there would be an objective justification for it ("proof").

In that case, the statement is true in one interpretation ("model") of the theory but false in other interpretation(s).

If there is no objective justification for an allegedly true statement, there does not need to be a consensus on its truth.

How can you even be sure that it is true, since there is not even a common understanding on why that would be the case?

Some small proportion of arithmetical reality is provably true. Most of it, however, is unprovably true.

Therefore, arithmetical reality is generally not objectivist. Only a very small part is.

I’m wondering if conventional wisdom thinks causation a part of physics, and if it’s thought causation directly the report of empirical experience. — ucarr

Pinpointing a previous event that would be the cause of a next one, the effect, is often too restrictive.

The next state in a system may be predictable from the previous state without resorting to such precise pinpointing.

I think that the notion of causality fails to allow for complex system-wide inputs leading to a particular output.

I suppose the Muslim version of this claim might be "if anyone slaps on the right cheek, slap them back so hard that they don't dare ever slap you again." Now that would be more in line with human nature. — BitconnectCarlos

Yes.

However, the general biological rule that governs all sovereign primate groups remains applicable.

In-group violence between individuals or subgroups is considered a breakdown in law and order, to be adjudicated by the ruler, who judges which of both sides is at fault.

Violence is legitimate only between sovereign groups ("war"). We share this biological rule with chimps, baboons, and gorillas.

An in-group cycle of violence is preferably cut short by means of victim compensation:

Quran 2:178. O ye who believe! the law of equality is prescribed to you in cases of murder: the free for the free, the slave for the slave, the woman for the woman. But if any remission is made by the brother of the slain, then grant any reasonable demand, and compensate him with handsome gratitude, this is a concession and a Mercy from your Lord. After this whoever exceeds the limits shall be in grave penalty.

Proportional retaliation is to be deemed a natural reaction and cannot be held against the parties in the conflict. Furthermore, no party in the conflict is expected to offer the other cheek.

The ruler must intervene, however, and the judge will attempt to solve the conflict by means of financial compensation. Such conflict-resolution process at the societal level of the sovereign group is simply a biological necessity.

Our laws must be compatible with our fundamental biological nature. Otherwise, the alternative is mayhem.

But even any 'theological' defence of 'self-defence' in Christianity is IMO questionable — boundless

It is obvious that there are situations in which fighting is simply necessary. That is indeed difficult to reconcile with the ambiguous, nebulous and misleading notion of fake pacifism typically advocated by Christians, which I consider to be in violation of the most fundamental laws of nature.

This problem does not exist in Islam. The following is a typical jurisprudential ruling in Islam on the matter:

https://islamqa.info/en/answers/21932/islamic-ruling-on-self-defence

Protecting oneself and one’s honour, mind, wealth and religion is a well-established basic principle in Islam. These are the five essentials which are well known to Muslims. A person has to defend himself; it is not permissible for him to consume that which will harm him, and it is not permissible for him to allow anyone to harm him. If a person or a vicious animal etc attacks him, he has to defend himself, or his family or his property, and if he is killed he is counted as a shaheed (martyr), and the killer will be in Hell.

This ruling is completely in line with human nature, with biology, and with the laws of nature. On the other hand, I reject the following statement:

Matthew 5:39. But I tell you, do not resist an evil person. If anyone slaps you on the right cheek, turn to them the other cheek also.

I will never endorse this view. In general, I have converted from Christianity to Islam for various reasons but mostly because I consider a large number of Christian teachings to be in violation of the laws of nature and to be contrary to very basic tenets of fundamental biology.

Not being a brute' is hardly the same as 'being a coward'. If 'not being a brute' means to be 'non violent', I hardly see how being 'non violent' is being a coward. — boundless

Refusing to "go over the top" or to open fire when instructed, is an act of cowardice.

http://www.worcestershireregiment.com/shot_at_dawn.php

Shot at Dawn

Offences under the British Army Act, which resulting in a court martial with a sentence to be shot at dawn included alleged acts of cowardice, desertion, sleeping at post, casting away arms and disobedience.

Christianity is deemed to have some responsibility for the fact that Germany lost both world wars:

https://www.wsj.com/articles/book-review-ataturk-in-the-nazi-imagination-by-stefan-ihrig-and-islam-and-nazi-germanys-war-by-david-motadel-1421441724

‘It’s been our misfortune to have the wrong religion,” Hitler complained to his pet architect Albert Speer. “Why did it have to be Christianity, with its meekness and flabbiness?”

Islam was a Männerreligion—a “religion of men”—and hygienic too. The “soldiers of Islam” received a warrior’s heaven, “a real earthly paradise” with “houris” and “wine flowing.”

This, Hitler argued, was much more suited to the “Germanic temperament” than the “Jewish filth and priestly twaddle” of Christianity.

Except during the Battle of the Warsaw Ghetto in 1944, Judaism did not encourage the Jews either to put up a fight. It was all too easy to mass transport them to the extermination camps.

There are moments in the life of a nation in which the day is carried away by the courage of their men, effectively turning them into murderous brutes. Judaism and Christianity are deemed to be liabilities and not assets, when the going gets tough.

Does a number have any type of connection to matter? — ucarr

Information is connected to matter.

For example, a pendulum is able to swing back because it physically stores the information necessary to do so. An explosion does not store the information necessary to implode again. Storing the information to do so, is a necessary condition for reversibility. That is why an explosion is irreversible.

https://arxiv.org/pdf/1404.7433
Schlesinger (2014)

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.

Numbers can represent some but not necessarily all of the information in the physical world required to reverse physical processes.

If some claim is information-theoretically impossible then it is also physically impossible.

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

Interesting.

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.

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

Yes, informational incompleteness (Chaitin) and uncertainty (Heisenberg) are deemed.directly related (Calude & Stay, 2004).

The irreversibility of particular physical processes (entropy) is also deemed directly related (Schlesinger 2014) to informational incompleteness (Chaitin).

Furthermore, arithmetical incompleteness (Godel) is provable (Zisselman 2023) from informational uncertainty (Chaitin).

However, only some part of all the above is effectively provable.

As I understand it, an axiomatic system is a compressor. — ucarr

The axiomatic system is indeed a compression (Chaitin) but we just axiomitize it without using any known compression algorithm.

The axiomatization is actually discovered by human ingenuity without any further justification.

By proving from it, however, we decompress information out of the axiomatic system . So, it is the proving from it that constitutes the decompression algorithm.

According to the Curry-Howard correspondence, a proof is indeed a program, and therefore, an algorithm. Because every proof is potentially different, the decompression algorithm is actually a collection of algorithms, usually, each painstakingly discovered.

So, the compression algorithm is unknown but some part of the decompression algorithm is discovered each time we successfully prove from the system.

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

Yes, the compression result excludes information. This excluded information may be inconsequential but it may also lead to a substantial reduction in desired predictive power. It forgets facts in the reality that it describes. This may or may not be a problem for the application at hand.

The axiomatic system is the result of a compression but we don't know what algorithm led to this result. It is discovered simply by human ingenuity.

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.

Why Russian milbloggers and propagandists are freaking out about Telegram's CEO arrest (— Chris York · Kyiv Independent · Aug 31, 2024) — jorndoe

Forget about Pavel Durov. Elon Musk is next. That is who they really want to bring into line:

https://www.theguardian.com/commentisfree/article/2024/aug/30/elon-musk-wealth-power

Elon Musk is out of control.
Here is how to rein him in.
Robert Reich

Here are six ways to rein in Musk:

1. Boycott Tesla.
2. Advertisers should boycott X.
3. Regulators around the world should threaten Musk with arrest if he doesn’t stop disseminating lies and hate on X.
4. In the United States, the Federal Trade Commission should demand that Musk take down lies that are likely to endanger individuals – and if he does not, sue him under Section Five of the FTC Act.
5. The US government – and we taxpayers – have additional power over Musk, if we’re willing to use it. The US should terminate its contracts with him, starting with Musk’s SpaceX.
6. Make sure Musk’s favorite candidate for president is not elected.

Elon Musk's Gleichschaltung is firmly under way. Elon Musk will find himself mercilessly "synchronized".

The modern communists do not need to turn Tesla, X, and SpaceX into government departments under direct state control. Nazi-style Gleichschaltung works a lot better.

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

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.

This problem becomes very apparent when trying to reverse a particular physical process. The amount of information required to do that may simply not be available, leading to the process becoming irreversible.

For example, an explosion. Can it be reversed? The problem is that the process would need to store an inordinate amount of information to even attempt that. This is a necessary (but not a sufficient) condition for the process to be reversible. Since even just the information to reverse the process is not remembered anywhere by the process, any attempt at reversing it would simply fail.

Prophecy? Mark Zuckerberg, Elon Musk, Zhang Yiming, ... are still "at large" in the open. — jorndoe

They are still at large and in the open ... for now.

It will work until it doesn't anymore.

The French judicial wanted Pavel Durov for ignoring authorities looking into some possible crimes on Telegram. Do you think those laws (are meant to) implement Gleichschaltung? — jorndoe

They could have arrested him for some other vague crime such as "money laundering". That is another favorite one of the modern ruling mafia.

Wasn't his girlfriend sitting in the plane? That's obviously "human trafficking" !!!

From there on, they can demand whatever they want from him, legal or illegal. Anything really. Either he toes the party line or else he will be made to toe the party line. The choice is his.

He apparently refused to build in a backdoor into his platform.

In the USA, they solve that by issuing a secret FISA court order (United States Foreign Intelligence Surveillance Court) along with a gag order. So, he builds in the backdoor as instructed and then shuts up about it, or else !!!

So, yeah, I would say that it is better to have a reputation of being a 'coward' than act as a 'brute'. — boundless

That would be the first-order assessment.

Then, there is the second-order one: Regardless of whether you are yourself a coward or a brute, do you prefer to be surrounded by cowards or by brutes?

You see, I could myself be a coward but if I am surrounded by brutes, I can always count on someone else to do the dirty work for me.

That is a strategy that allows me to remain myself Socratically kosher but simultaneously still benefit from useful external effort.

What do you mean by 'defending itself'?? How should religious people defend their religion? — boundless

You can either get accused of being a coward or else of being a brute. Feel free to pick your poison.

Now you are making points about variety in types of compression algorithms, not about general principles. — apokrisis

That is actually also what Steven Frank does in his paper "The common patterns of nature". He argues that the algorithm -- in this case, the statistical distribution -- that maximizes entropy will dominate particular situations.

A statistical distribution compresses the information about a sample into just a few parameters. It is lossy. It will generally not succeed in decompressing these few parameters back into the full sample.

About the general principles, he writes:

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.

Axiomatic theories do something similar.

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".

Why or how has communism lost its appeal, if it really has? — Shawn

The first ones to discover that maximizing the ruling elite's power does not require turning every business into a state department, were the Nazis:

https://en.wikipedia.org/wiki/Gleichschaltung

The Nazi term Gleichschaltung (German pronunciation: [ˈɡlaɪçʃaltʊŋ] ⓘ) or "coordination" was the process of Nazification by which Adolf Hitler — leader of the Nazi Party in Germany — successively established a system of totalitarian control and coordination over all aspects of German society "from the economy and trade associations to the media, culture and education".[1]

It has been variously translated as "coordination",[4][5][6] "Nazification of state and society",[7] "synchronization",[3] and "bringing into line".[7]

The ruling mafia of a modern democracy will also use Nazi-style Gleichschaltung instead of direct control. The same holds true for the modern Chinese and Vietnamese communist regimes. They no longer use direct state control. They also use Nazi-style Gleichschaltung instead.

Using Gleichschaltung instead of direct control is a matter of efficiency and not of ideology. Gleichschaltung is simply much better at maximizing the power of the ruling mafia than direct control.

Communism using Gleichschaltung instead of direct control is equivalent to modern democracy.

Sometimes problems occur, however.

For example, modern social media are not yet fully absorbed by Nazi-style Gleichschaltung.

That is why it is necessary for the ruling mafia to arrest and imprison people like Telegram's owner Pavel Durov. He may have successfully escaped Gleichschaltung by the Russian Federation in the past, but Durov is currently being "synchronized" by the French authorities instead. Elon Musk and his Twitter platform (nowadays called "X") will sooner or later also find themselves more thoroughly "synchronized".

Hey, Seven of Nine, resistance is futile. You will be assimilated.

But It only takes an infinitesimal grain of chance to complete things. — apokrisis

When a compression algorithm forgets particular facts, it says much more about this algorithm than about these facts. Another algorithm may even include them. The term "chance" points to facts forgotten by the compression algorithm. I do not see that qualification as particularly fundamental.

The accidental can’t be in fact removed from the world, even if that is not what axiomatic determinism wants us to believe. — apokrisis

Positivism and scientism incorrectly claim this.

Axiomatic determinism does not claim this.

Even not at all.

On the contrary, axiomatic determinism fully acknowledges that an axiomatic system (capable of arithmetic) is at best a lossy compression of the reality that it describes. This lossy compression necessarily forgets most of the information contained in the uncompressed reality.

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). That is exactly what Chaitin (and Gödel) prove about such systems.

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.

Randomness is not a necessary requirement for unpredictability. Incompleteness alone is already sufficient. A completely deterministic system can still be mostly unpredictable.

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

Yes.

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.

Just like in Schlesinger paper, Calude & Stay switched from Gödel's incompleteness to Chaitin's incompleteness. Gödel is provable from Chaitin. However, Chaitin seems to possess better explanatory power when dealing with entropy or fundamental uncertainty.

So, if a theory is the compressed image of a particular (uncompressed) reality, there is potentially a mismatch between the amount of information contained in the theory versus the amount of information contained in its reality. A theory (capable of arithmetic) contains substantially less compressed information than the uncompressed reality that it describes. This theory is necessarily a lossy compression.

Proving from theory is equivalent to decompressing some information about its reality out of its compressed theory.

Heisenberg discovered that it is not possible to simultaneously "decompress" precise position and precise momentum information for a particle. So, it looks like Chaitin's incompleteness all over again.

I do not see this phenomenon as "randomness", though.

There is absolutely nothing random about arithmetic theory or arithmetical reality. Natural-number arithmetic is a completely deterministic system. Its arithmetical reality is indeed largely unpredictable but it is not random at all.

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.

Interesting observation. I'm not sure it takes G-incompleteness to reach this point. — jgill

I think that the connection Schlesinger sees, is tied more directly to Chaitin's incompleteness theorem than to Godel's theorem. (But then again, Godel is provable from Chaitin.)

We can view Peano arithmetic theory as a compressed image of arithmetical reality. Proving from PA amounts to decompressing some information of PA's reality out of its theory.

But then again, there is a serious mismatch in the amount of information between the compressed and uncompressed states of PA's reality.

The compressed state (its theory) contains only a small fraction of the total amount of information in the uncompressed state (its reality).

Our problem, however, is that we cannot see directly the decompressed state of PA, i.e. its reality. The only way to see some of it is by decompressing it.

This decompression mechanism can easily mislead us.

That is why, until Godel's 1931 publication, the positivists even insisted that we could decompress the totality of arithmetical reality from its compressed state (its theory). They really believed it.

Positivism (and scientism) therefore amount to the misguided belief that PA (or science) is a lossless compression of arithmetical (or physical) reality.

I think that Schlesinger seems to make sense.

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.

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.

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?

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.

Well, not sure of your point. If, say, one has no economical problems, would you still think that 'giving birth' is morally wrong? — boundless

It is not about economic problems but legal ones. You are very likely to end up at some point becoming the noncustodial parent of the child. That is a stupid hobby.

Above a certain national divorce rate, it looks like a bad decision to have children with a person of that particular nationality:

https://divorce.com/blog/divorce-rates-in-the-world/

It is preferable to pick a country with a much lower divorce rate, choose a romantic interest there, and have children there instead.

I ruthlessly discriminate against anybody from a high divorce rate country. So, yes, I consider it to be morally wrong to have children with these people. Better safe than sorry.

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.

Some of these theoreticians (though not their fault) became popular among fascists, nazi, and communists, who took for grounded that societies always are hierarchical/patriarchal/dominated etc., and for this reason the solution is not to go against a natural trend/inclination of humanity (they thought democracy and liberalism were doing that), but to choose the last worst outcome through placing the group who deserves it the most at the head of these ("inevitable") societal hierarchies. — Eros1982

My conclusion is rather that you can choose to go where you are treated best. That is the most efficient way of choosing the least worst outcome.

Ideology does not matter particularly much.

I am personally treated better in communist Vietnam than in supposedly democratic Denmark or Spain.

We have a poor class in this country that may decide who will govern in 2025 and can hope in some money (through tax reforms) to be transferred from the rich to them. — Eros1982

This is exactly what will never happen.

The ruling elite may transfer money from the middle class to the poor, only when it is sufficiently easy to do. However, they will never, ever transfer money from themselves to the poor. This does not happen in any country. The people in power will never use their power to confiscate money from themselves.

The real piggy bank is the excess income and excess wealth of the middle class. These people cannot protect themselves from confiscation. They have some money but do not have the political power to hang on to it. That is why they will gradually find themselves dispossessed of their money.

So, I am more eager to believe that this country tends to be chaotic (like most of the countries in this continent), more than hierarchical. — Eros1982

There are a lots of special interests, the most powerful of which collectively form the ruling elite.

Communist Vietnam has no hope nor reasonable ambition to forcibly extract money out of my pockets. That is why they are nice to me.

So, either you join the political elite of a country and try to extract money from other people, or else, you allow the competition to play between political elites of different countries as to prevent them from extracting funds from you. Otherwise, they will look at you as just another idiot to strip clean.