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

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

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

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?

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? — Tarskian

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.

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

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.

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

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?

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

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.

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. — fishfry

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.

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. — Tarskian

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.

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

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

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

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.

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.

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

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.

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. — Janus

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.

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

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.

I was referring to real physical systems which are not conceptual — Janus

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.

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 — ssu

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 α : Y → Y without a fixed point (for all y ∈ Y , α(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 α (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.

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. — Tarskian

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.

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

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.

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

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".

Deep down humans could also be deterministic. — Tarskian

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.

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

We all have moral choice.

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

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?

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

a bit eccentric — fishfry

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

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

Yanofsky phrases a generalized Cantor theorem in terms of the sets Y, T and the functions α(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.

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. — fishfry

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.

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? — fishfry

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?

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. :)

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

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.

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. — Tarskian

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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."

"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." — fishfry

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.

I believe that the soul is non-algorithmic. — Tarskian

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.

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. — Tarskian

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.

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

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

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

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

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

I predict that conversation will never end. :grin: