Yeah, if you say determinism means completeness, then "incomplete deterministic" just sounds like "incomplete completeness". Seems like a nosnense term to me.

Yeah, if you say determinism means completeness, then "incomplete deterministic" just sounds like "incomplete completeness". Seems like a nosnense term to me. — flannel jesus

You got caught up in the vocabulary misalignment. The phrase "incomplete deterministic system" is perfectly fine in computer science or mathematics. It means that there is nothing random in the system ("deterministic system"). However, most facts can not be predicted from its theory either ("incomplete"). This is the essence of Gödel's incompleteness theorem.

seems like you're mixing vocabularies a lot here and generating a lot of unnecessary ambiguity.

seems like you're mixing vocabularies a lot here and generating a lot of unnecessary ambiguity. — flannel jesus

I am trying to point out the metaphysical implications of the foundational crisis in mathematics. That is necessarily multidisciplinary, meaning that you end up with two vocabularies that are not necessarily compatible.

Gödel proves the lack of determinism (as in metaphysics) in particular deterministic systems (as in mathematics).

This sounds confusing.

This misalignment in vocabulary is, however, inevitable because people from either field rarely talk to each other or read each other's publications.

and you don't seem to be trying much to disambiguate your incompatible vocabularies, making the arguments seem very non compelling as a whole.

When one definition of determinism is equivalent to "completeness", but then another definition allows you to say "incomplete determinism", and you put pretty close to 0 effort into explaining how that's supposed to make sense, I can't imagine I'm alone in just thinking it's all nonsense from that point on.

When one definition of determinism is equivalent to "completeness", but then another definition allows you to say "incomplete determinism", and you put pretty close to 0 effort into explaining how that's supposed to make sense, I can't imagine I'm alone in just thinking it's all nonsense from that point on. — flannel jesus

The misalignment in vocabulary is something akin to a landmine. You become aware of the problem only after the facts. But then again, I don't think that "determinism" is a much used term in mathematics. You will mostly find the term "deterministic system".

If you Google for "mathematics determinism", the first search result is "deterministic system":

https://www.google.com/search?q=mathematics+determinism

So, even Google is confused here, because "determinism" does not mean "deterministic system" in mathematics. It means "completeness".

So, if even Google puts "pretty close to zero effort" into getting the facts straight, then it means that their 182,000 members of staff are possibly just spouting nonsense instead of properly maintaining their search engine.

Well, the real conclusion is that playing the blame game is pointless. Looking for whom to blame is unproductive. Furthermore, it never fixes the real underlying problem. Two different backgrounds means two different vocabularies. Sometimes it still works flawlessly. Sometimes, it doesn't.

You don't seem interested in trying to make yourself clear, in trying to develop a self-consistent vocabulary for your ideas. You end your post with "Sometimes it still works flawlessly. Sometimes, it doesn't." as if there's nothing at all you could do to clarify your ideas.

Maybe there's not, maybe you can't clarify your ideas.

The task given to the oracle doesn't make sense. The task given to the oracle is "predict the output of this Thw program, after you feed into the Thw program your prediction for the output of the Thw program."

It's recursive in a way that means the oracle can't even begin.

It's like me telling you, Tarski, I have a math problem for you: your job is to give me a number that's 2 more than the answer to this math problem.

Does that even make sense as a task?

There's no problem with this oracle being impossible in the first place, because of course it's impossible, the task itself is inherently recursively impossible.

You don't seem interested in trying to make yourself clear, in trying to develop a self-consistent vocabulary for your ideas. You end your post with "Sometimes it still works flawlessly. Sometimes, it doesn't." as if there's nothing at all you could do to clarify your ideas.

Maybe there's not, maybe you can't clarify your ideas. — flannel jesus

There are landmines when in the combined vocabulary of both metaphysics and mathematics. It is overly optimistic to believe that you can always detect them beforehand. The sentence, Gödel proves the lack of determinism of deterministic systems, even sounds contradictory. If you are lucky, you become aware of the problem after the facts. I can certainly imagine situations in which you actually don't.

The task given to the oracle doesn't make sense. The task given to the oracle is "predict the output of this Thw program, after you feed into the Thw program your prediction for the output of the Thw program."

It's recursive in a way that means the oracle can't even begin. — flannel jesus

It is actually a description of the standard contemporary proof for Alan Turing's halting problem. The oracle must predict if thwarter will print a zero or not.

Hamkins describes it as following:

https://arxiv.org/pdf/2407.00680

Namely, assume toward contradiction that the symbol-printing problem were computably decidable, and fix a method of solving this problem. Using this as a subroutine, consider the algorithm q which on input p, a program, asks whether p on input p would ever print 0 as output. If so, then q will halt immediately without printing 0; but if not, then q prints 0 immediately as output. So q has the opposite behavior on input p with respect to printing 0 as output than p has on input p. Running q on input q will therefore print 0 as output if and only if it will not, a contradiction.

Just for the hell of it, I rewrote Hamkins wording in terms of the oracle and the thwarter:

Namely, assume toward contradiction that the symbol-printing problem were computably decidable, and fix a method of solving this problem. Using the oracle as a subroutine, consider the thwarter program which asks to the oracle whether any program p on input p would ever print 0 as output. If the oracle answers that it will print 0, then thwarter itself will not print 0; but if the oracle says that thwarter doesn't print 0, then thwarter does print 0. Running thwarter on itself as input will therefore print 0 as output if and only if the oracle says that thwarter will not, a contradiction.

The sentence, Gödel proves the lack of determinism of deterministic systems, even sounds contradictory. — Tarskian

And who came up with that sentence? Typed that into google, no hits. Is that one of yours?

And who came up with that sentence? — flannel jesus

The real question is, who confused the vocabulary? Well, the pretty much complete absence of communication between both fields.

Looks like the answer to both is, you.

The sentence, Gödel proves the lack of determinism of deterministic systems, even sounds contradictory. — Tarskian

It's kind of hilarious, it seems like you're using this as an example of some unavaoidable language landmine just about anybody could walk into, but... it's not, it's just another landmine YOU personally chose to walk into.

Like, we're in a sitcom and you see a landmine on the ground and you just actively, knowingly step right on it, and your leg blows off a hundred yards away, and you look right in the camera and the Curb Your Enthusiasm music plays and you say "Damn, these landmines are so hard to avoid."

They... aren't that hard to avoid. You're literally not trying.

They... aren't that hard to avoid. You're literally not trying. — flannel jesus

The problem is that this is not the only problem. It is just one of the problems. The language in which the foundational crisis of mathematics is worded, is usually "impenetrable". So, I first need to translate it into a narrative with an oracle and a thwarter, because otherwise, it is absolutely not suitable for interdisciplinary use.

For example, Hamkins paper:

Tarskian, You may be interested in a recent paper by Joel David Hamkins. [...] Terrific, readable paper. Hamkins rocks.
https://arxiv.org/pdf/2407.00680

Indeed, it is actually surprisingly readable for a paper on this subject. The following paragraph, however, is unsuitable for interdisciplinary use:

At bottom, the logic of the argument is like this: if we had a computable way of finding whether existential statements are true, then we could iterate this with negation to also compute ∀∃ assertions, since ∀k∃n ϕ fails just in case there is some k for which the existential statement about it fails. In short, if in general existential statements are decidable, then the whole arithmetic hierarchy collapses.

If I cannot not find an alternative way of phrasing this differently, it will be pointless to use this particular argument. Fortunately, I don't need this argument for anything.

The environment of the oracle and the thwarter is perfectly deterministic. There is nothing random going on. Still, the oracle cannot ever predict correctly what is going to happen next. The oracle is therefore forced to conclude that the thwarter has free will. — Tarskian

The effects of diagonalization are important and should be discussed here in PF. It's great that this pops up in several threads and people obviously are understanding it!

Basically the oracle is similar to the Laplace's demon, that we have been talked about, for example here (real world example) in the "The Argument There Is Determinism And Free Will"-thread. One simply cannot say what one doesn't say or predict what one doesn't predict. Yet in some occasions this obviously can be the correct prediction. In your example, you make the diagonalization with the "Thwarter app".

It should be noticed that this doesn't refute determinism, it just is that any program itself or predictor himself or herself is part of the universe and once there's interaction with reality to be predicted, situations like where it cannot predict the future will happen. The pathological "Thwarter app" is similar what is describe in Turing's paper about the Entscheidungsproblem. But notice you don't have to have this app and problems will arise. (Btw, have you read Yanofsky's A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points that we discussed on another thread, should be important to this too)

Yet what should be noticed is that this is a limitation that we have or any machine has in the ability to forecast everything. There's much that indeed can be accurately predicted.

And free will?

Well, this doesn't refute determinism, it's only a limitation of basically our computational abilities and logic. So the philosophical question of free will won't go anywhere.

And does the Thwarter app have free will?

Well, the thwarter app does exaclty what the original app doesn't do. Is that free will? The thwarter app still can be a program (Turing Machine) that itself cannot do something else than what is written in it's own program.

You may be interested in a recent paper by Joel David Hamkins. Turing never proved the impossibility of the Halting problem! He actually proved something stronger than the Halting problem; and something else equivalent to it. But he never actually gave this commonly known proof that everyone thinks he did. Terrific, readable paper. Hamkins rocks. — fishfry

Thanks! Again a fine article, @fishfry, that I have to read. I've been listening to Youtube lectures that Joel David Hamkins gives. They are informative and understandable.

It is accepted as proof, however, that no oracle can exist that can predict what choices programs will make. — Tarskian

Couldn't oracle simply lie to the thwarter. It knows what the thwarter will do. It tells it something else.
O: You will produce the number 2.
T: [produces number 7]
Which is exactly what oracle had predicted and [whispered] to the experimenters.

It seems like the scenario is conflated a specific chain of events with an inability to accurately predict the future. Yes, that app if it is forced to say it's conclusion to thwarter might not be able to predict that one part of the future. But that doesn't mean an app couldn't predict the future - though I think there are computing power issues in making such a deity level app.

A deity level app given a self-undermining task has a problem.

I can't see where one can conclude there is free will from the odd restrictions and fantasies in this scenario.

A few issues with this.
One, if an app such as the oracle were to exist, it would only show possibilities in which the thwarted cannot thwart the future, so it may actually work in that case. Obviously, this is impossible if the thwarted is effective, so the thwarter would just not do anything because the oracle could never show a possibility.

While both working is a paradox and a result of impossible apps, I would still ask, how does the oracle conclude the thwarter has free will? There is a distinction between sentient individuals and programmed applications, even if both are able to respond and make decisions. The thwarter does not have free will because its choices are limited. It can only choose to respond in a set of ways, there are some things it cannot choose to do, unlike a sentient entity, which can essentially choose to do anything.

Imagine that you install an app on your phone that can tell you minute by minute what you will be doing at any point in the future along with all possible details?

The existence of this app would prove that you are just an automaton, i.e. a robot. In that case, it would be ridiculous to claim that you have free will. — Tarskian

If I predict you will go to the store and you do, that would not be sufficient for me say you didn't have free when you went to the store.

At what point do you declare my predictive powers eliminate your free will? How many trials must there be and would a single variance re-establish my free will?

If I accurately predict the outcome of 50 coin tosses, does that necessarilymake the coin toss outcomes not random?

At what point do you declare my predictive powers eliminate your free will? How many trials must there be and would a single variance re-establish my free will? — Hanover

This is a good point. An infinite amount would not limit free will. Free will is only limited if the person does not have complete control over the choices they can make.

If I accurately predict the outcome of 50 coin tosses, does that necessarilymake the coin toss outcomes not random? — Hanover

And to this point, they would still be random. What does finding out the outcome earlier have to do with the randomness of the trial?

You will never predict correctly what thwarter is going to do. — Tarskian

I'll concede you the Halting problem, but certainly not that programs have free will, if that was the claim.

When you put thwarter in that chaotic system, you suddenly have something freely making decisions while you can impossibly predict what decisions it will make. — Tarskian

Nothing is "freely making decisions." That's a complete misunderstanding of what programs are. I know you know that, so you must be using free will in a different sense than I understand.

Free will is a property of a process making choices. If it impossible to predict what choices this process will make, then it has free will. — Tarskian

Oh for gosh sake. That's not true. A coin doesn't have free will when you flip it. And if you say that deep down coin flips are deterministic, so are programs.

you kind of contradict the first half of your post here with the second half. In the first half, you speak as if something being deterministic is basically synonyms with it being predictable, but in the second half you acknowledge that a chaotic system could be deterministic but unpredictable. — flannel jesus

I believe I'm losing this point. I do know about chaos.

If a chaotic system can be deterministic but unpredictable, then you should be able to imagine software that is chaotic, and thus deterministic and unpredictable, no? — flannel jesus

Yes, I think I have lost this debate to @Tarskian. Except that he thinks programs have "free will," and of course they don't.

I think there's a subtly shifting meaning for the word "unpredictable" that's at play there. — flannel jesus

Agreed. But also, chaotic deterministic unpredictability is not the same as Halting problem deterministic unpredictability, and Tarskian is trying to make some kind of connection.

But I concede the point that programs are inherently unpredictable in the sense of Turing. Not in the sense of chaos, and they certainly don't have free will, except for alternative definitions of the phrase.

Yes, I think I have lost this debate to Tarskian. — fishfry

You haven't lost any debate, you just made a post with some mistakes. You seem ready to acknowledge them, which is winning in my book.

Thanks! Again a fine article, fishfry, that I have to read. — ssu

It's relatively short. You can skip most of the technical bits. I did.

I've been listening to Youtube lectures that Joel David Hamkins gives. They are informative and understandable. — ssu

He's awesome.

You haven't lost any debate, you just made a post with some mistakes. You seem ready to acknowledge them, which is winning in my book. — flannel jesus

I make many misteaks :-)

But consider the fact that the halting behaviour of two identical algorithms stands and falls together. So although there does not exist an infallible universal halting tester, there exists an infallible special-case halting tester for any given algorithm, namely a copy of that very algorithm.

Although an epistemic limitation falls short of a metaphysical proof, I am sympathetic to the idea of free will, because in my opinion the conceptual distinction between free will and determinism rests upon a belief in absolute infinity, which i reject.

In my view, to say that "A => B is necessary true" in the sense of material causation, is to say that there exists a Z such that "A => Z is necessarily true" and "Z => B is necessarily true". If we reject the idea that this definition can appeal to actually infinite recursion, then the use-meaning of " A => B is necessarily true" in any given context must eventually bottom out to a finite chain of implicative reasoning, in which the meaning of "necessarily true" is left undefined.

A simpler way of putting it, is to say that we make up the meaning of " A => B is necessarily true" as we go along. This proposition doesn't have precise a priori meaning, and so isn't contradicted by a future discovery that A => B fails to hold, rather the proposition meant by the sentence "A=> B is necessary true" changes on discovery that A => B fails to hold.

And if you say that deep down coin flips are deterministic, so are programs. — fishfry

Deep down humans could also be deterministic. As long as the theory of humans is incomplete, humans would still have free will.

The thwarter first asks the oracle what it predicts that it will be doing. The oracle then looks at the source code of the thwarter and at the inputs that it would be getting from the environment, and then predicts what the thwarter will be doing. Upon receiving the answer from the oracle, the thwarter does something else instead, because that is exactly how it was programmed. — Tarskian

The oracle would know how the thwarter would react to its prediction. It could say, "Now that I've told you you will do X, you will do Y, just to thwart me." Which would make the thwarter do X, or Z, or whatever. And the oracle would know every step of the dance. A dance that might go on forever, thwarter never actually doing anything, as oracle endlessly says, "But now that I've said that, you will..." Which oracle would know ahead of time.

Or, at any point, oracle might say, "I'll (app equivalent of) write it down, and, after you act, you can read it. And you'll see I predicted accurately."

Or, at any point, oracle might say, "I'll (app equivalent of) write it down, and, after you act, you can read it. And you'll see I predicted accurately." — Patterner

Thwarter needs a prediction as input. Otherwise it does not run.