Even programs have free will

Quoting Tarskian

Conversely, you can prove the existence of free will by proving that it is impossible to construct such app.

So by your argument, you've used Turing's argument to prove free will. Somehow that doesn't follow from the impossibility of such an app since the app is impossible even in a pure deterministic universe.

Quoting noAxioms

Somehow that doesn't follow from the impossibility of such an app since the app is impossible even in a pure deterministic universe.

The natural numbers are also a pure deterministic universe. Most of its truth, however, cannot be predicted by arithmetic theory. A pure deterministic universe can still have free will as long as its theory is incomplete.

The real requirement here, is incompleteness of the theory.

Quoting Tarskian

The narrative above is pretty much the gist of Alan Turing's proof for the halting problem.

Tarskian, 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.

https://arxiv.org/pdf/2407.00680

Quoting Tarskian

In fact, there is no app that can tell minute by minute what even any other app will be doing.

That's too strong a statement. If an app is halted, I can write a program that, given any time t, says, "The program is halted at time t."

Likewise if I'm dead, a program can exactly predict what I'm doing. But of course in that case I wouldn't have much in the way of free will.

Quoting Tarskian

The real requirement here, is incompleteness of the theory.

Penrose thinks free will might be a quantum effect in the microtubules of the brain.

https://philosophy.stackexchange.com/questions/3322/the-emperors-new-mind-and-free-will

By the way, humans may or may not have free will.

Programs, by their very nature, do not have free will.

Quoting Tarskian

In fact, there is no app that can tell minute by minute what even any other app will be doing.

Time is of the essence.

The Thwarter app is not aware (figuratively speaking) of the existence of the Oracle app. All the Thwarter app is aware of is input.

Therefore, we only need to consider the Thwarter app.

Feedback occurs when the output of the Thwarter app then becomes new input. This is a temporal process, in that its output happens at a later time than its input.

The source code of the Thwarter app determines the output from the input using the function F, where output = F (input).

For example, if the input is a set of numbers, such as 3, 5 and 7, the output could be the addition of this set of numbers, such as 15.

At time zero, let there be an input I (1). This input cannot include any subsequent output, as any output happens at a later time.

At time t + 1, the output O (1) can be predicted from the function F operating on input I (1).

At time t + 1, the new input I (2) includes output O (1).

At time t + 2, the new output O (2) can be predicted from the function F operating on input I (2).

At time t + 2, the new input I (3) includes output O (2).

Etc.

At each subsequent time, the output can be predicted from the input. The output is pre-determined by the input.

At any time t + x, the output has been pre-determined by the situation at time zero.

Quoting fishfry

Terrific, readable paper. Hamkins rocks. https://arxiv.org/pdf/2407.00680

Just finished reading it. It is very informative. I must say, though, that it is heavily vested in logic connected to the arithmetical hierarchy. It is still doable but admittedly an obstacle of sorts if you do not use that framework particularly often.

Hamkins acknowledges that the contemporary version of the proof is arguably preferable to Turing's original "detour":

Turing thus showed that the symbol-printing problem is undecidable by mounting a reduction to and through the undecidability of the circle-free problem. But let us illustrate how one may improve upon Turing with a simpler self-referential proof of the undecidability of the symbol-printing problem in the style of the standard contemporary proof of the undecidability of the halting problem. There was actually no need for Turing’s detour through the circle-free problem.

I have tried to turn Hamkins' phrasing of the standard contemporary proof into a narrative:

= Original ==

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.

== Narrative ==

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.

For the original circle-free problem, the proof is actually trivially easy.

Say that we call programs with infinite output "infinitist" ("circle-free") and programs with finite output "finitist" ("circular"). Can we list all possible infinitist programs? No, because if we list their infinite output in a table, then we can create a brand new infinite output by flipping the bit on the diagonal, i.e. by diagonalization.

The real difficulty is related to how Turing uses the circle-free problem to prove that the symbol-printing problem is undecidable:

He mounts an unusual kind of reduction, showing that if symbol-printing were decidable, then also the circle-free problem would be decidable, which he had already proved is not the case.

This is not a straightforward reduction of one problem to another, but rather an argument that if one problem were actually computably decidable, then so would be the other.

I agree with Hamkins' take on the matter. I also find the contemporary standard version of the proof much simpler than Turing's original approach. Turing's "unusual kind of reduction" feels like an exercise in painful shoehorning.

Quoting fishfry

By the way, humans may or may not have free will.
Programs, by their very nature, do not have free will.

I think that humans have a soul while programs do not. However, since programs also make choices, they can just as humans appear to be "free" in making them or not. That is why I think that it is perfectly possible to analyze free will as a computability problem.

Quoting RussellA

At each subsequent time, the output can be predicted from the input. The output is pre-determined by the input. At any time t + x, the output has been pre-determined by the situation at time zero.

Yes, the oracle may perfectly well know that thwarter will do the opposite of what he predicts, but he has committed to his prediction already. It will be too late already.

This mirrors the liar paradox:
"The following sentence is true" (the oracle predicting the impending outcome). "The previous sentence is false" (the thwarter doing the opposite to what was predicted to thwart the oracle's veracity/render it false.

It is a dichotomy/pair of mutually cancelling phenomena. The result: lack of utility of either. They're entangled and self defeating.

Quoting Tarskian

Yes, the oracle may perfectly well know that thwarter will do the opposite of what he predicts, but he has committed to his prediction already. It will be too late already.

The Thwarter app has a source code which specifies how the Thwarter app performs a calculation when input information

The Thwarter app is given an input and performs a calculation to arrive at an answer.

It may be that the Oracle app knows that the answer is contained within the input information.

However, the Thwarter app would only know that the answer was contained in the input information after it had completed its calculation, and then it would be too late to change what type of calculation it had used.

IE, the calculation that the Thwarter app uses cannot be determined by an answer that is only known by the Thwarter app after it has completed its calculation.

Quoting Tarskian

Just finished reading it. It is very informative. I must say, though, that it is heavily vested in logic connected to the arithmetical hierarchy. It is still doable but admittedly an obstacle of sorts if you do not use that framework particularly often.

I enjoyed it.

Quoting Tarskian

Hamkins acknowledges that the contemporary version of the proof is arguably preferable to Turing's original "detour":

Turing thus showed that the symbol-printing problem is undecidable by mounting a reduction to and through the undecidability of the circle-free problem. But let us illustrate how one may improve upon Turing with a simpler self-referential proof of the undecidability of the symbol-printing problem in the style of the standard contemporary proof of the undecidability of the halting problem. There was actually no need for Turing’s detour through the circle-free problem.

I have tried to turn Hamkins' phrasing of the standard contemporary proof into a narrative:

[... details omitted]

You have made an impressively detailed reading of the article, way more than I did.

Quoting Tarskian

I think that humans have a soul while programs do not.

I am in complete agreement. But just try to explain that to the simulation theorists, the mind-uploading freaks, the singularitarians, the AGI proponents, etc. They have the mindshare these days.

Quoting Tarskian

However, since programs also make choices, they can just as humans appear to be "free" in making them or not. That is why I think that it is perfectly possible to analyze free will as a computability problem.

Hmmm. Let me mull that over. I don't agree. Computability, by its nature, is deterministic. Whatever free will is, it is not computable.

Quoting fishfry

Hmmm. Let me mull that over. I don't agree. Computability, by its nature, is deterministic. Whatever free will is, it is not computable.

Computability may be deterministic but is fundamentally still unpredictable too. It is generally not possible to predict what a program will be doing at runtime:

https://en.wikipedia.org/wiki/Rice%27s_theorem

In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable.

A semantic property is one about the program's behavior (for instance, "does the program terminate for all inputs?"), unlike a syntactic property (for instance, "does the program contain an if-then-else statement?"). A non-trivial property is one which is neither true for every program, nor false for every program.

The theorem generalizes the undecidability of the halting problem. It has far-reaching implications on the feasibility of static analysis of programs. It implies that it is impossible, for example, to implement a tool that checks whether a given program is correct, or even executes without error.

The theorem is named after Henry Gordon Rice, who proved it in his doctoral dissertation of 1951 at Syracuse University.

A deterministic system is unpredictable when its theory is incomplete. There is no need for randomness for a system to be unpredictable. Free will is essentially the same as unpredictability.

Quoting Tarskian

Computability may be deterministic but is fundamentally still unpredictable too. It is generally not possible to predict what a program will be doing at runtime:

I am not sure if Rice's theorem means what you say it does.

If you give me a program, say its listing printed out on paper; and you give me its inputs; and you give me a lot of pencils, paper, and time; I can deterministically and with no ambiguity determine exactly what it's going to do. I can not imagine this being false, and therefore Rice must be full of beans! :-)

Quoting Tarskian

A deterministic system is unpredictable when its theory is incomplete. There is no need for randomness for a system to be unpredictable. Free will is essentially the same as unpredictability.

Not how I understand this. A chaotic system is deterministic yet unpredictable. Nothing to do with incompleteness. There's no free will, none whatsoever, in a chaotic system.

Quoting fishfry

If you give me a program, say its listing printed out on paper; and you give me its inputs; and you give me a lot of pencils, paper, and time; I can deterministically and with no ambiguity determine exactly what it's going to do. I can not imagine this being false, and therefore Rice must be full of beans!

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

Quoting fishfry

A chaotic system is deterministic yet unpredictable. Nothing to do with incompleteness. There's no free will, none whatsoever, in a chaotic system.

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.

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.

Quoting Tarskian

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

What makes you convinced thwarter is a genuinely possible program? Has anyone programmed one?

Reply to fishfry 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.

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?

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

Quoting flannel jesus

What makes you convinced thwarter is a genuinely possible program? Has anyone programmed one?

Thwarter is trivially easy to implement. On input of string "halt" it prints "loop forever" and on input "loop forever" it prints "halt".

So, if the prediction (which is the input string) is that Thwarter will print "halt" or "loop forever", it won't.

The problem is rather to implement oracle. Example:

https://github.com/Solidsoft-Reply/Halting-Problem

ah my bad, didn't read the op carefully enough.

I really don't see that as free will in any meaningful sense.

Quoting flannel jesus

I really don't see that as free will in any meaningful sense.

It is a contorted example.

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

Even in a perfectly deterministic environment, free will can still exist, as long as its theory is incomplete.

Therefore, we don't even need the physical universe to be nondeterministic for free will to be possible. It just needs to be incomplete.

Reply to Tarskian I don't agree that free will has anything to do with oracle-like programs at all

Reply to flannel jesus incompatibilism implies that if the oracle exists, free will doesn't.

Reply to Tarskian no, incompatibilism implies that if determinism is true, free will doesn't exist - but you explained yourself that this oracle can be impossible even if determinism is true, no?

So one can imagine a world where determinism is true, this oracle is impossible, and free will doesn't exist because determinism is true, regardless of this oracle.

Quoting flannel jesus

no, incompatibilism implies that if determinism is true, free will doesn't exist

incompatibilism implies that if predeterminism is true, free will doesn't exist

There is a massive difference between predeterminism and deterministic systems. If a deterministic system is incomplete, its future is not predetermined.

Quoting flannel jesus

So one can imagine a world where determinism is true, this oracle is impossible.

It is exactly in a predetermined world, that the oracle can make flawless predictions.

The universe consisting of just the oracle app and the thwarter app, however, is not predetermined because of its incompleteness. The construction theory of this world is capable of arithmetic. That is enough to make it incomplete and therefore not possibly predetermined.

It is perfectly possible to deterministically build machines that are not predetermined.

Reply to Tarskian what is this incomplete determinism? I googled it but none of the results seemed like what you're talking about.

Quoting flannel jesus

what is this incomplete determinism?

Predeterminism implies that the system's theory is complete. In that case, every true fact about the system can be derived from its theory. If this is not possible, then the system's theory is incomplete.

For example, the arithmetic theory about the natural numbers is incomplete.

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of natural numbers.

The arithmetic of the natural numbers is obviously a deterministic system. There is nothing random about it. Still, its truth is mostly unpredictable.

Reply to Tarskian I think you've invented your own special kind of incompatibilism here. If you ask an incompatibilist if they think free will is possible in a universe that is deterministic, but in which an oracle is impossible, I guarantee you 99%+ of incompatibilists will say "screw oracles, free will is incompatible with determinism period".

I don't think anything about the oracle or the thwarter says anything interesting about free will at all, personally, i think it's a red herring.

Quoting flannel jesus

I guarantee you 95%+ of incompatibilists will say "screw oracles, free will is incompatible with determinism period".

I use the term "predeterminism" instead of "determinism" because of the possible confusion with the term "deterministic system".

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

Predeterminism is the philosophy that all events of history, past, present and future, have been already decided or are already known (by God, fate, or some other force), including human actions. Predeterminism is closely related to determinism.

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

In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system.

A deterministic system that is incomplete is not predetermined. Further confusion is also caused by tying the term "determinism" to "causality":

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

Determinism is the philosophical view that all events in the universe, including human decisions and actions, are causally inevitable.

Causality is not a usable notion in mathematics. It is replaced by "provable from its theory". We don't need to know what the individual causes are for a particular fact. We don't care about that. We just need to know that the system can correctly predict the fact. Hence, the idea that all facts are "causally inevitable" translates into all facts being "provable from theory". So, the term "determinism" in mathematical terms means "completeness". It does not mean "deterministic system".

There are two fields involved here: metaphysics and mathematics. The vocabulary is not completely aligned.

Reply to Tarskian I don't think any of that goes any distance towards demonstrating what I said was incorrect. Incompatibilists say free will is incompatible with determinism, not oracles

Quoting flannel jesus

I don't think any of that goes any distance towards demonstrating what I said was incorrect. Incompatibilists say free will is incompatible with determinism, not oracles

The existence of a functioning oracle is equivalent to determinism (with the notion of determinism equivalent to the notion of completeness). The oracle fails. It doesn't function. Therefore, there is no determinism.

Asserting incompatibilism, as a notion in metaphysics, translates into proving the impossibility of constructing an oracle, as a notion in computer science. It is effectively equivalent. The difficulty here is that we are mapping concepts from one field to another.

Quoting Tarskian

with the notion of determinism equivalent to the notion of completeness

So then when you were talking about incomplete determinism, you were... what? What is that? An oxymoron? Nonsense? A contradiction? What is that?

Quoting flannel jesus

So then when you were talking about incomplete determinism

incomplete deterministic system.

Quoting Tarskian

Asserting incompatibilism, as a notion in metaphysics, translates into proving the impossibility of constructing an oracle, as a notion in computer science. It is effectively equivalent.

This is just factually untrue. You've got chaos theory which makes future-predicting oracles impossible, to start with.

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

Quoting flannel jesus

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

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.

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

Quoting flannel jesus

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

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.

Reply to Tarskian 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.

Quoting flannel jesus

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.

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.

Reply to Tarskian 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.

Quoting flannel jesus

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.

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.

Quoting flannel jesus

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 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.

Quoting Tarskian

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

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

Quoting flannel jesus

And who came up with that sentence?

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

Reply to Tarskian Looks like the answer to both is, you.

Quoting Tarskian

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

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.

Quoting flannel jesus

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

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 ? 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.

Quoting Tarskian

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.

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.

Quoting fishfry

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.

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.

Quoting Tarskian

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

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.

Reply to Tarskian 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.

Quoting Tarskian

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.

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?

Quoting Hanover

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?

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.

Quoting Hanover

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

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?

Quoting Tarskian

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

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

Quoting Tarskian

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.

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.

Quoting Tarskian

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.

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.

Quoting flannel jesus

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.

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

Quoting flannel jesus

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?

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

Quoting flannel jesus

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

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.

Quoting fishfry

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

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.

Quoting ssu

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

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

Quoting ssu

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

He's awesome.

Quoting flannel jesus

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.

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.

Quoting fishfry

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

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

Quoting Tarskian

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.

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 [I]that[/I], 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."

Quoting Patterner

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."

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

Quoting Tarskian

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

Ok. Oracle gives a final spoken prediction, but secretly writes down what it knows thwarter will do at that point.

Quoting Patterner

Ok. Oracle gives a final spoken prediction, but secretly writes down what it knows thwarter will do at that point.

Yes, of course, Oracle can perfectly know what is truly going to happen. However, his knowledge of the truth is not actionable. What else is he going to do with it?

Quoting Tarskian

Yes, of course, Oracle can perfectly know what is truly going to happen. However, his knowledge of the truth is not actionable. What else is he going to do with it?

This was your idea. I didn't know you were looking for a purpose for Oracle. What did your have in mind? Off the top of my head, I'd say there's money to be made at the roulette wheel.

Quoting Tarskian

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

Yes, But notice that the Oracle staying silent can be also viewed as an input. So when the Oracle is silent and doesn't make a prediction, the Thwarter can do something (perhaps mock the Oracle's limited abilities to make predictions), which should be easily predictable.

Quoting Tarskian

Yes, of course, Oracle can perfectly know what is truly going to happen. However, his knowledge of the truth is not actionable.

Oracle can know perfectly what is going to happen if your Thwarter app is a Turing Machine that runs on a program that tells exactly how Thwarter will act on the Oracle's prediction.

And this is why you have to go a step forward from just declaring what that the Thwarter has free will. After all, what's the "free will" in the following?

Oracle predicts A -> Thwarter does B
Oracle predicts B -> Thwarter does A
Oracle predicts something else or is silent -> Thwarter does B

Notice the simple diagonalization. Now, here really both the Oracle and the Thwarter can be basically Turing Machines. Turing Machines don't have free will.

However, you do get to the really interesting point of free will when from this (which is basically a result from the Church-Turing thesis) when you make the following question: If the Oracle knows it's limitations in predicting the Thwarter, but can write Thwarter's actions down on a paper, when does the Oracle have problems even with writing the actions of the Thwarter on a paper?

The Thwarter cannot be a simple predictable program that simple reacts to the Oracle's prediction. The Oracle can easily write this down as it knows Thwarter's program.

The Thwarter app basically has to be an Oracle itself with an ability that no Turing Machine has: it has to understand it's programs it itself is running on and then change it's behaviour/action in a way that it hasn't changed ever before.

How does the Oracle now write down what is going to happen, as in this case there is not historical example of what the Thwarter will do? Well, it cannot use past information and extrapolate from it.

It should be understood here that computers cannot follow an order of "do something else". They can follow it only if in their program there's instructions what to do when asked to "do something else". And now what the "Twarter app" has to do is even more. And something doing the above, basically a "double diagonalization", if one can coin a new term.

But of course it should be evident that nobody here will crack philosophical question of free will, because the counterargument to this is that even we cannot know our own "metaprogram". Well, I would argue that as we can understand our behaviour at least partly and can learn from the past, this "double diagonalization" is at least partly something that we can do. Yet this deep philosophical question of free will won't go away.

In my view, this is an extremely important discussion, because it just shows how profound philosophical impact the findings of Turing and the Church-Turing thesis have. Just what lies beyond computability is a very important question. It's not just a limitation in mathematics for computability, it's also a deep philosophical limitation.

Comments?

Quoting ssu

It should be understood here that computers cannot follow an order of "do something else".

If a program knows a list of things it can do [ A1, A2, A3, ..., An], and it receives the instruction "do something else but not Ak", then it can randomly pick any action from [A1, A2, ..., A(k-1),A(k+1) .... An] as long as it is not Ak.

Quoting fishfry

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.

Chaos theory has already been brought up twice, which he ignored, like he does everytime his incorrigible nonsense is challenged. Prediction of choices has nothing to do with free will — and this is nonsensical woo disguised in logical language. If you know your friend likes cake over pie, it is possible to predict he will choose cake, it doesn't mean he has no free will.

Quoting Tarskian

If a program knows a list of things it can do [ A1, A2, A3, ..., An], and it receives the instruction "do something else but not Ak", then it can randomly pick any action from [A1, A2, ..., A(k-1),A(k+1) .... An] as long as it is not Ak.

Randomly picking some action from [A1, A2, ..., A(k-1),A(k+1) .... An] as long as it is not Ak is surely not "do something else". It is an exact order that is in the program that the Oracle can surely know. Just like "If Ak" then take "Ak+1". A computer or Turing Machine cannot do something not described in it's program.

Quoting Tarskian

If a deterministic system is incomplete, its future is not predetermined.

It's not systems that are "incomplete": the idea makes no sense at all, but our understanding of systems.

Quoting Janus

It's systems that are "incomplete": the idea makes no sense at all, but our understanding of systems.

Understanding of a system amounts to having perfect knowledge of its construction logic, i.e. its theory.

For example, the axioms of arithmetic theory are perfectly well known. Every claim that we can prove from it, is also true in the universe of the natural numbers. However, most of the truth about the natural numbers is still unpredictable.

So, it is not because you build a system -- and therefore know how you have built it -- that you will be able to predict its entire truth. Only some of its truth will be predictable.

Reply to Tarskian I was referring to real physical systems which are not conceptual, I was not referring to mathematical systems, which are conceptual. It makes no sense to say that the Universe, a real physical system, is incomplete, but of course our understanding of the universe is incomplete, and always will be. So, the future is not comprehensively predictable, but it does not follow that it is incomplete or in possession of free will.

Quoting Janus

I was referring to real physical systems which are not conceptual, not I was not referring to mathematical systems, which are conceptual.

The idea is that every physical system has a sound theory, albeit possibly unknown. Every collection of truth has a sound theory.

Every claim that necessarily follows from this sound theory will be true about the physical system.

This is even true about the entire physical universe. It is not because we do not know this theory that it does not exist.

Quoting Janus

It makes no sense to say that the Universe, a real physical system, is incomplete, but of course our understanding of the universe is incomplete, and always will be.

The universe is not a theory. It is a collection of truth, i.e. a "model" or "interpretation" of its unknown theory.

If its unknown theory is complete, it can predict its entire history, akin to Laplace's demon. No free will could possibly exist in it.

If we knew its incomplete theory, we would still not be able to predict most of its truth or future. We know the theory of the natural numbers. However, because it is incomplete, we cannot explain most of its truth.

Quoting Janus

So, the future is not comprehensively predictable, but it does not follow that it is incomplete or in possession of free will.

If some of its truth is unpredictable, its theory must be incomplete.

The alternative would be in violation of Godel's completeness theorem. If a theory is complete, every fact in its universe is provable and therefore predictable.

Without unpredictability, free will is not possible. Therefore, incompleteness is a firm requirement for free will.

Free will necessarily implies incompleteness, according to the impossibilist assessment.

You only need to discover one true sentence that is not provable from the system's theory to conclude that most of the system's truth isn't predictable.

Quoting Janus

I was referring to real physical systems which are not conceptual

Every collection of truth has a sound theory. However, only some part of its truth may follow from it.

Say that a collection of truth has 5 sentences: A, B, C, D, E. From its incomplete theory only B and E necessarily follow. Therefore, A,C, and D are its unpredictable truths.

One major problem in trying to discover this system's theory, is that some of its truth must be ignored. You cannot possibly discover its theory if you take A,C, and D into account. You must ignore it.

The general idea in physics is that we cannot discover a theory because we can see too little. According to mathematics, they are actually wrong. It is exactly the other way around. We cannot discover a theory, because we can see too much.

One reason why mathematics works, is because we cannot easily see its unpredictable truth. It takes a series of rather difficult hacks to even detect that it is there.

Quoting ssu

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

Yanofsky seems to say that all paradoxes listed in his paper are somehow the consequence of Cantor's theorem. Even though I understand Cantor's theorem as described in wikipedia:

https://en.wikipedia.org/wiki/Cantor%27s_theorem

Theorem (Cantor) — Let f be a map from set A to its power set P ( A ). Then f : A ? P ( A ) is not surjective. As a consequence, card ? ( A ) < card ? ( P ( A ) ) holds for any set A.

I cannot fully grasp why it is supposedly the same as how Yanofsky phrases it:

https://arxiv.org/pdf/math/0305282

Theorem 1 (Cantor’s Theorem) If Y is a set and there exists a function ? : Y ? Y without a fixed point (for all y ? Y , ?(y) != y), then for all sets T and for all functions f : T × T ? Y there exists a function g : T ? Y that is not representable by f i.e. such that for all t ? T: g(?) != f (?, t).

In my impression, f(x,y) is Cantor's table while g(r) is the value in the diagonal that is not in the table, or something like that. Concerning Y, a derangement ? (permutation without fixed points) must exist. I can't connect it, though. He does not mention Cantor's power set. His wording for the theorem seems to condense Cantor's diagonalization proof right into the statement of the theorem itself. My intuition says that Yanofsky's version is undoubtedly correct, but I don't fully master its construction.

While Cantor says something simple, i.e. any onto mapping of a set onto its power set will fail, Yanofsky says something much more general that I do not fully grasp.

Quoting Tarskian

While Cantor says something simple, i.e. any onto mapping of a set onto its power set will fail, Yanofsky says something much more general that I do not fully grasp.

Ok, this is very important and seemingly easy, but a really difficult issue altogether. So I'll give my 5 cents, but if anyone finds a mistake, please correct me.

Let's first think about how truly important in mathematics is making a bijection, which is both an injection and a surjection. We can call it a 1 to 1 correspondence or a 1-to-1 mapping. And basically bijections are equations like y=f(x) or 1+1=2. And of course Cantor found the way to measure infinite sets by making bijections between them, like there's a bijection between the natural numbers N and the rational numbers Q.

With the diagonal argument or diagonalization, by negative self-reference we show that a bijection is impossible to make as the relation is not surjective. This is the proof for Cantor's theorem. Yet this is also the general issue that Yanofsky is talking about as this is found on all of these theorems.

Even in the case of your example in the OP (if I have understand correctly, that is) first it is assumed that the Oracle can make a bijection from the past to the future and hence can make correct predictions about everything. Then with the Thwarter app, because of the negative self-reference, means that the situation for the Oracle is that it cannot make a bijection as the new situation with the Thwarter app is not surjective anymore.

And as @noAxioms immediately pointed out, you are basically using Turing's proof in your model. Which itself uses also diagonalization.

Hopefully this was useful for you.

Quoting ssu

And basically bijections are equations like y=f(x)

Just a nitpick. Not every f(x) function is bijective. I don't think there is a general form of a bijective function. 1+1=2 is not bijective either because it is not a function.

Quoting Lionino

Just a nitpick. Not every f(x) function is bijective.

Not too nitpicky, I think it's an important distinction to make. If you don't make this distinction, then... there's no point to the word "bijection", as "function" already exists. This distinction is what makes bijection meaningful over just "function".

Quoting Tarskian

Deep down humans could also be deterministic.

I stipulate that:

1. This is a very hip and TED-talky idea going around; and

2. I personally disagree strenuously; but I concede that I can't prove it.

But given that, my original point stands. That programs can't have free will. And I hope you agree that humans being deterministic would not contradict that point.

Quoting Tarskian

As long as the theory of humans is incomplete, humans would still have free will.

We all have moral choice.

Quoting Lionino

Chaos theory has already been brought up twice, which he ignored, like he does everytime his incorrigible nonsense is challenged.

I try to keep an open mind and take the good with the bad of all, say, a bit eccentric posters. I hope that is not too uncharitable to @Tarskian. Am I being fair?

Reply to fishfry If these threads https://thephilosophyforum.com/discussion/9705/god https://thephilosophyforum.com/discussion/15172/is-atheism-illogical are something to go by: not uncharitable enough

Quoting fishfry

a bit eccentric

Let's not be so open-minded our brains fall out.

Quoting ssu

Yet this is also the general issue that Yanofsky is talking about as this is found on all of these theorems.

Yanofsky phrases a generalized Cantor theorem in terms of the sets Y, T and the functions ?(x), f(x), g(x,y). I still do not fully grasp the connection between the symbols that he uses. I suspect that it is indeed equivalent to Cantor's theorem but I don't see how exactly.

Quoting fishfry

But given that, my original point stands. That programs can't have free will. And I hope you agree that humans being deterministic would not contradict that point.

I think that having "free will" versus having a "soul" are not the same thing.

As I see it, the soul is an object in religion while free will is an object in mathematics.

I see free will and incompleteness as equivalent. I don't see why they wouldn't be.

Quoting fishfry

I try to keep an open mind and take the good with the bad of all, say, a bit eccentric posters. I hope that is not too uncharitable to Tarskian. Am I being fair?

I guess so.

As you have probably noticed, @Lionino does not talk about metaphysics or about mathematics but about me. That is apparently his obsession. He incessantly talks about me, very much like I incessantly talk about Godel. I don't know if I should feel flattered.

But then again, the metaphysical implications of the foundational crisis in mathematics, are truly fascinating.

Mathematics proper has exactly zero metaphysical implications:

According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.

How can something that "isn't about anything at all" suddenly become about the fundamental nature of everything?

Reply to Lionino Reply to flannel jesus

Thanks to both of you. And no, it isn't nitpicking. Of course we can talk about surjective or injective functions. What for me it's very irritating that there aren't these general definitions. As a layman I would think that something being an equation, a mathematical statement that shows two or more amounts are equal, would also be a (or could be modeled as a) bijection. But, uh, apparently not. :(

And we haven't even discussed isomorphisms and their relation to bijections. Perhaps it's better simply to talk about bijections, injections and surjections. At least that ought to be simple, I hope. Far more easier than these than to talk about Turing Machines, or (yikes), Gödel numbers!

And if that was the only thing correcting, then I'm not totally wrong in the discussion. :)

Quoting Tarskian

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

That sounds rather the opposite of free will.

But again, as I mentioned in my previous post. Oracle could give it a false input. It says you will produce two. Thwarter thwarts and says five, which is what oracle knew and whispered to the judges.
IOW you have conflated the potential for extreme restrictions on the options with oracle - it must be honest with thrwarter and undermine it's predictions, with an inability to predict the future. Ironically seeming to show that we have free will by radically restricting the free will of this tool (oracle) and its tool using owners. IOW the owners of oracle could just tell it to lie to Thwarter.

Quoting Tarskian

I guess so.

As you have probably noticed, Lionino does not talk about metaphysics or about mathematics but about me. That is apparently his obsession. He incessantly talks about me, very much like I incessantly talk about Godel. I don't know if I should feel flattered.

I've noticed that some posters have personal obsessions with others. For me, when I find it unpleasant to interact with someone, I just don't interact. Don't disagree with them, don't bait them, don't troll them, don't interact with them directly or directly.


Quoting Tarskian

But then again, the metaphysical implications of the foundational crisis in mathematics, are truly fascinating.

Well, you are saying that historically contingent opinions about math, have some bearing on the ultimate nature of math. But I imagine that if there is such a thing as an ultimate nature of math, it incorporates and transcends all such opinions. Math is more than the sum of all philosophies about it.

Quoting Tarskian

How can something that "isn't about anything at all" suddenly become about the fundamental nature of everything?

Well now, that is a great question. Wigner asked about the Unreasonable Effectiveness of Mathematics in the Physical Sciences. How can math be so fictional, so idealized, so much about nothing at all; and yet so relevant and useful. I think one answer is that math is useful to humans in the same way that echoes are useful to bats. Our brains are wired to makes sense of the world through math. Our approach isn't better or worse than any other creature's. We flatter ourselves to imagine that the world is "like" math; when in fact, we're just wired that way.

Quoting Bylaw

IOW the owners of oracle could just tell it to lie to Thwarter.

There is an infinite number of ways to write the same thwarter program. In order to know that the source code does indeed represent a thwarter, oracle needs to be able to prove that this alternative is equivalent to thwarter.

That is the same problem as proving that two lambda expressions are equivalent.

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

There is no algorithm that takes as input any two lambda expressions and outputs TRUE or FALSE depending on whether one expression reduces to the other. More precisely, no computable function can decide the question. This was historically the first problem for which undecidability could be proven.

Hence, oracle won't know that he is looking at the source code of an alternative version of thwarter.

Therefore, the only solution would be for oracle to lie all the time. Consequently, oracle won't be able to correctly predict the output of a program that does the opposite of thwarter and that just prints oracle's prediction as output.

Quoting Tarskian

I think that having "free will" versus having a "soul" are not the same thing.

They're closely related. A self-awareness and the ability to have preferences and desires, and to be able to act to bring them about.

Quoting Tarskian

As I see it, the soul is an object in religion while free will is an object in mathematics.

I'm using soul in a secular sense. And free will does not appear in any math text that I've ever seen. Free will is not an object of study of math at all.

Quoting Tarskian

I see free will and incompleteness as equivalent. I don't see why they wouldn't be.

I believe Penrose makes that argument.

https://en.wikipedia.org/wiki/The_Emperor%27s_New_Mind

"Penrose argues that human consciousness is non-algorithmic, and thus is not capable of being modeled by a conventional Turing machine, which includes a digital computer. Penrose hypothesizes that quantum mechanics plays an essential role in the understanding of human consciousness. The collapse of the quantum wavefunction is seen as playing an important role in brain function."

Quoting fishfry

"Penrose argues that human consciousness is non-algorithmic, and thus is not capable of being modeled by a conventional Turing machine, which includes a digital computer."

I believe that the soul is non-algorithmic.

Concerning "human consciousness", I don't know how much of it is just mechanical. The term is too vague for that purpose. A good part of the brain can only be deemed to be a machine, i.e. a biotechnological device, albeit a complex one, of which we do not understand the technology, if only, because we did not design it by ourselves.

But then again, even if the brain were entirely mechanical, its theory is undoubtedly incomplete, which ensures that most of its truth is unpredictable.

Even things without a soul can have an incomplete theory and therefore be fundamentally unpredictable.

Quoting Tarskian

I believe that the soul is non-algorithmic.

Ok! You're an anti-computationalist like me. I don't believe we're ever going to "upload our minds," I don't think we live in a computer simulation, I don't think our minds or our universe are Turing machines.

Quoting Tarskian

Concerning "human consciousness", I don't know how much of it is just mechanical. The term is too vague for that purpose. A good part of the brain can only be deemed to be a machine, i.e. a biotechnological device, albeit a complex one, of which we do not understand the technology, if only, because we did not design it by ourselves.

What we know of the human brain does not work like a digital computer. Some people say that neural nets work because they mimic the neural structure of brain. I don't believe that, but I have to admit that some of their recent achievements are impressive. Who knows.

Quoting Tarskian

But then again, even if the brain were entirely mechanical, its theory is undoubtedly incomplete, which ensures that most of its truth is unpredictable.

Something deterministic can be unpredictable, so that doesn't solve the problem.

Quoting Tarskian

Even things without a soul can have an incomplete theory and therefore be fundamentally unpredictable.

You're confusing determinism with predictability, but I thought we'd already covered this.

Quoting fishfry

You're confusing determinism with predictability, but I thought we'd already covered this.

I predict that conversation will never end. :grin:

Quoting fishfry

You're confusing determinism with predictability, but I thought we'd already covered this.

According to the page on the subject, determinism and predeterminism are "closely related":

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

Predeterminism is the philosophy that all events of history, past, present and future, have been already decided or are already known (by God, fate, or some other force), including human actions.

Predeterminism is closely related to determinism.[1]

The concept of predeterminism is often argued by invoking causal determinism, implying that there is an unbroken chain of prior occurrences stretching back to infinity. In the case of predeterminism, this chain of events has been pre-established, and human actions cannot interfere with the outcomes of this pre-established chain. Predeterminism can be used to mean such pre-established causal determinism, in which case it is categorised as a specific type of determinism.[2][3] It can also be used interchangeably with causal determinism—in the context of its capacity to determine future events.[2][4] Despite this, predeterminism is often considered as independent of causal determinism.[5][6]

If you believe that everything has a reason, it does not mean that you also know that reason. Predictability requires indeed both.

Quoting Tarskian

According to the page on the subject, determinism and predeterminism are "closely related":

After reading all that i was a little unclear on pre- versus regular old determinism. The text passages are the kind of philosophical writing that always makes my eyes glaze.

Quoting Tarskian

If you believe that everything has a reason, it does not mean that you also know that reason. Predictability requires indeed both.

Hmmm, determinism doesn't mean that everything has a "reason." If you have some Rube Goldberg machine and you start it, it's perfectly deterministic. But it doesn't have any "reason." It's just one thing causing the next thing.