Preno wrote:No, not really. Decidability has to do with quantification, whereas the question of whether the universe is deterministic is simply the question of whether we can determine its future behaviour. You're mixing different things together.

Not at all. Predictability by us differs from determinism.

Preno wrote:Of course there are propositions that cannot be decided by physical law: for example, "the Earth has a radius of 6378 km". This is a statement that is independent of any of our physical laws.

I disagree. In a deterministic universe this proposition could be derived from the innitial state (or any state for that matter) of the universe and all the governing laws.

Preno wrote:So what you actually mean is whether we can, given sufficiently precise initial conditions, determine the future behaviour of the universe in principle with arbitrary precision. Which is a matter of computation/simulation, not of deciding some statement in some formal system.

Nope. Because the issue is not whether we can do it, but whether the Laplacian demon could do it. Given the state of the universe at a point in time t and the laws governing it, the state of the universe at time t' is decidable within the system of axioms containing the laws and the state (which pretty much defines a boundary condition) precisely if the laws are deterministic. In that case there is precisely one state at time t' which is consistent with the axioms. If the laws are stochastic there are more states and thus which state appears at time t' is not decidable. Fundamentally, the universe is governed by laws, which can be expressed logically. Turing showed that if you have a set of axioms that is of cardinality c, you can have a logical system not restricted to FOL that is both consistent and complete (Gödels proof rests on the diagonal argument, and just as R has a complete list of numbers, an uncountably infinite set of axioms can escape Gödel).

Preno wrote:Computability doesn't deal with sentences like for all x there is an y such that for all z ..., which are precisely the root cause of undecidability, so Godel's theorems have no direct bearing on determinism.

You are ignoring that you can have a function from the space of logical statements to N. Which is where the equivalence stems from. For any such function from a logical statement A to a number n, it´s equivalent to state that A is undecidable and that n is not computable.

Paul Almond wrote:Suppose a zardonk can go left or right when it reaches a zoltok.

Is this any different to Roger's zig-zag structure, printed in The Road to Reality, of the electron?

The rule for working out whether a zardonk goes left or right when it reaches a zoltok is based on some specified set of Penrose tiles. The exact set of Penrose tiles is based on the type of zardonk and the type of zoltok involved.
If the specified set of Penrose tiles can be used to tile the infinite plan, it goes left.
If the specified set of Penrose tiles can be used to tile the infinite plan, it goes right.

susu.exp wrote:

Preno wrote:No, not really. Decidability has to do with quantification, whereas the question of whether the universe is deterministic is simply the question of whether we can determine its future behaviour. You're mixing different things together.

Not at all. Predictability by us differs from determinism.

I suppose, depending on how broadly you construe "predictability by us, but that's besides the point I was making, which had to do with your misconstrual of decidability (see below).

Preno wrote:Of course there are propositions that cannot be decided by physical law: for example, "the Earth has a radius of 6378 km". This is a statement that is independent of any of our physical laws.

I disagree. In a deterministic universe this proposition could be derived from the innitial state (or any state for that matter) of the universe and all the governing laws.

Uh, yeah, that the physical laws by themselves aren't sufficient and need to be supplemented by an account of the initial state was precisely my point, so I'm not sure why you're disagreeing.

Preno wrote:So what you actually mean is whether we can, given sufficiently precise initial conditions, determine the future behaviour of the universe in principle with arbitrary precision. Which is a matter of computation/simulation, not of deciding some statement in some formal system.

Nope. Because the issue is not whether we can do it, but whether the Laplacian demon could do it. Given the state of the universe at a point in time t and the laws governing it, the state of the universe at time t' is decidable within the system of axioms containing the laws and the state (which pretty much defines a boundary condition) precisely if the laws are deterministic.

I don't disagree, but my point was that you're substantially reducing the definition of decidability, so much that Godel's theorems (whose relevance to this topic you claimed earlier) become irrelevant.

Decidability means that every sentence in some formal language is decidable in some theory. Here, however, you're only talking about sentences of the form "at time t, the state of our universe will be such and such". These are much simpler than the sentences to which undecidability might apply, which would look like, for example "no matter what the initial state is, there is a time t in the evolution of that state such that ...". Such sentences might be undecidable on Godelian grounds, but not sentences of the former kind. So computability (which we haven't even defined yet, btw) is not equivalent to decidability. The decidability of the fragment of our language consisting of simple, unquantified sentences like "at time t, ... will hold" doesn't entail the decidability of the whole language (which includes arbitrarily long sequences of quantifiers), and computability only requires this weaker kind of decidability.

In that case there is precisely one state at time t' which is consistent with the axioms.

Essentially, yes, although strictly speaking, there is no such thing as a state when we're talking about decidability, what you have is an infinite set of sentence in some language describing that state.

If the laws are stochastic there are more states and thus which state appears at time t' is not decidable. Fundamentally, the universe is governed by laws, which can be expressed logically.

I assume you mean that they can be expressed formally. Being expressed logically presumably means being expressed by means of purely logical vocabulary (logical connectives, quantifiers, identity), which they obviously cannot, being laws of physics.

Turing showed that if you have a set of axioms that is of cardinality c, you can have a logical system not restricted to FOL that is both consistent and complete (Gödels proof rests on the diagonal argument, and just as R has a complete list of numbers, an uncountably infinite set of axioms can escape Gödel).

I'm not sure which results you're alluding to here, but at any rate assuming we're not talking about some outlandish language with an uncountable number of symbols (which, if you are, you should declare and justify), there is only a countable number of formulas, so there can hardly be an uncountable number of axioms.

Preno wrote:Computability doesn't deal with sentences like for all x there is an y such that for all z ..., which are precisely the root cause of undecidability, so Godel's theorems have no direct bearing on determinism.

You are ignoring that you can have a function from the space of logical statements to N. Which is where the equivalence stems from. For any such function from a logical statement A to a number n, it´s equivalent to state that A is undecidable and that n is not computable.

I don't understand what it means to say that a natural number is not computable. (Also, I can't help making the same terminological nitpick again - you mean formal sentences, not "logical statements".)

byofrcs wrote:

UndercoverElephant wrote:

jamest wrote:Does this thread boil down to indeterminism and quantum mechanics?

No. It is about the nature of physical laws. From my point of view physical laws are necessarily mathematical. I don't understand what a "non-computable physical law" could be.

Random can't be computed and the universe has a lots of random (sources of entropy).

Random is also not bound by laws. You can't have random laws, can you?

Preno wrote:Uh, yeah, that the physical laws by themselves aren't sufficient and need to be supplemented by an account of the initial state was precisely my point, so I'm not sure why you're disagreeing.

I think we´re disagreeing on whether the initial state does fall under physical law. I would hold that it does. Let´s assume a deterministic dynamical system dx/dt(t)=1. It is compatible with various initial states x(t=0)=x0, x0 in R. With a given innitial state, we can state for any state (t,x(t)), whether the system will reach it or not. In a stochastic system, say dx/dt=Y(t), with Y(t) normally distributed with mean 1 and SD 0.5, we can not decide whether the system will reach any particular state (T,x(tT)), even if x0 is known. To make that question decidable, we need to know Y(t) for 0<t<T, giving us uncountably many statements of the type Y(t=a)=b.

Preno wrote:Decidability means that every sentence in some formal language is decidable in some theory. Here, however, you're only talking about sentences of the form "at time t, the state of our universe will be such and such". These are much simpler than the sentences to which undecidability might apply, which would look like, for example "no matter what the initial state is, there is a time t in the evolution of that state such that ...". Such sentences might be undecidable on Godelian grounds, but not sentences of the former kind.

I think you´re right there and one has to be careful in constructing this, but one can note that there are states in the universe in which a formal argument is presented. It´s back to the drawing board for me, but I suspect a Gödel type argument can be brought forward about this.

Preno wrote:I'm not sure which results you're alluding to here, but at any rate assuming we're not talking about some outlandish language with an uncountable number of symbols (which, if you are, you should declare and justify), there is only a countable number of formulas, so there can hardly be an uncountable number of axioms.

I have to look this up, but yea, it basically requires an uncountable number of symbols. It´s doubtful such an axiomatic system is constructable, but it does escape the problem.

Preno wrote:I don't understand what it means to say that a natural number is not computable.

You are right, I don´t know what (or for that matter if) I was thinking there.

UndercoverElephant wrote:Random is also not bound by laws. You can't have random laws, can you?

Why do you say that?

Because the whole point of physical laws is to predict what is going to happen, isn't it? They are for calculating what we are likely to observe in the future. And randomness is, by definition, unpredictable.

UndercoverElephant wrote:Because the whole point of physical laws is to predict what is going to happen, isn't it? They are for calculating what we are likely to observe in the future. And randomness is, by definition, unpredictable.

Shouldn't we include things which only predict probabistic outcomes as physical laws ? Like Schrodinger's equation.

UndercoverElephant wrote:Because the whole point of physical laws is to predict what is going to happen, isn't it? They are for calculating what we are likely to observe in the future. And randomness is, by definition, unpredictable.

Random is not that unpredictable. It follows very clear laws based on the distribution which are related to well known mathematical functions and constants e.g.a Poisson distribution uses 'e'.

An individual trial may be random but in the long run it follows a rule and more so in that when a random system deviates from being random that it becomes more predictable in the short term though we may not know what the long term distribution will look like.

With a fair roulette wheel I predict that the distribution of numbers will be equal in the long term, though I may not know what the next number will be. With an unfair wheel I may better guess the short term outcome but I would not have predicted that before the game had started. Once I realise that the wheel is untrue then I can better predict the results.

Random therefore follows a rule and we know when it breaks this rule.

The thing is that I cannot calculate a random number with a TM without using an external source of entropy.

twistor59 wrote:

UndercoverElephant wrote:Because the whole point of physical laws is to predict what is going to happen, isn't it? They are for calculating what we are likely to observe in the future. And randomness is, by definition, unpredictable.

Shouldn't we include things which only predict probabistic outcomes as physical laws ? Like Schrodinger's equation.

Yes, we can include them as probabilistic laws...but we also have to acknowledge that they have a non-lawful component to them. No law can predict when a specific nucleus decays or explain why a specific nucleus decayed when it did. It just happens. I think we have to say that there are (at least) two fundamentally different sorts of physical law - those which are fully deterministic and those which are probabilistic. Probabilistic laws offer philosophical problems/opportunities which determinstic laws do not.

byofrcs wrote:

UndercoverElephant wrote:Because the whole point of physical laws is to predict what is going to happen, isn't it? They are for calculating what we are likely to observe in the future. And randomness is, by definition, unpredictable.

Random is not that unpredictable. It follows very clear laws based on the distribution which are related to well known mathematical functions and constants e.g.a Poisson distribution uses 'e'.

An individual trial may be random but in the long run it follows a rule and more so in that when a random system deviates from being random that it becomes more predictable in the short term though we may not know what the long term distribution will look like.

With a fair roulette wheel I predict that the distribution of numbers will be equal in the long term, though I may not know what the next number will be. With an unfair wheel I may better guess the short term outcome but I would not have predicted that before the game had started. Once I realise that the wheel is untrue then I can better predict the results.

The roulette wheel is a bad comparison. It is only random in the sense that we don't have the measuring/computing power to make an accurate prediction. By contrast, all the measuring and computing power concievable won't help to tell us when a specific uranium nucleus will decay. This is the difference between subjective and objective randomness. The nuclear decay is said to be "objectively random" because it is necessarily random at a metaphysical level. The roulette wheel is only random from our point of view as humans. It's not really random at all, just complex to predict.

My point was that the roulette wheel has a predictable distribution function and a block of Uranium has a similarly predictable (though different) distribution function.

Say I used an alpha particle counter pointed at a block of uranium and use an alpha particle hit to time when the mechanism fires the ball into the roulette wheel. Do you think the roulette wheel then suddenly fails to follow an even distribution ?.

It won't but what I have now done is made it impossible for you to predict the roulette wheel outcome no matter how much computing power you apply to the problem because you claim it is impossible to compute the decay time of the uranium particle.

Therefore a roulette wheel is subjectively random because of the timing of the ball release and how this synchronises to the wheel. That doesn't sound like subjectively random unless a human is able to manipulate this in some way with a resolution that is probably far smaller than a human can function at.

The roulette results will still trend to an even distribution (assuming a fair wheel).

I therefore question how a roulette wheel appears only random from a human point of view (though it has a well defined distribution) and yet the decay of uranium is somehow hidden (though it has a well defined distribution).

I say that both systems are objectively random though the roulette ball and wheel appears more predictable because that system has been quantised many trillions of atoms that make up the ball and wheel whereas you brought in us looking at just one uranium atom decaying to thorium.

susu.exp wrote:

Preno wrote:Uh, yeah, that the physical laws by themselves aren't sufficient and need to be supplemented by an account of the initial state was precisely my point, so I'm not sure why you're disagreeing.

I think we´re disagreeing on whether the initial state does fall under physical law. I would hold that it does.

No, it doesn't (or rather, what I said was that it isn't a physical law that ...). A physical law is a general statement, not a particular statement about the Earth's diameter or about the position of the Mona Lisa. Their evolution, however, is governed by physical law, which I assumed to be what you meant to say.

Let´s assume a deterministic dynamical system dx/dt(t)=1. It is compatible with various initial states x(t=0)=x0, x0 in R. With a given innitial state, we can state for any state (t,x(t)), whether the system will reach it or not. In a stochastic system, say dx/dt=Y(t), with Y(t) normally distributed with mean 1 and SD 0.5, we can not decide whether the system will reach any particular state (T,x(tT)), even if x0 is known. To make that question decidable, we need to know Y(t) for 0<t<T, giving us uncountably many statements of the type Y(t=a)=b.

Well, sort of, except we need to distinguish between states of some variable and sentences about those states. The variable x goes through an uncountable number of states, but that doesn't mean we have uncountably many sentences describing it. We don't, as our language is necessarily countable (since we're not capable of an infinitely precise discrimination between symbols). Corresponding to each state, however, there is a set of countable sentences describing it, like "3.1 < x < 3.2", "3.14 < x < 3.15", etc. And there are uncountably many sets of sentences, so every pair of states can be distinguished by some such inequality.

Preno wrote:No, it doesn't (or rather, what I said was that it isn't a physical law that ...). A physical law is a general statement, not a particular statement about the Earth's diameter or about the position of the Mona Lisa. Their evolution, however, is governed by physical law, which I assumed to be what you meant to say.

I think the line between physical law and the description of the initial state is somewhat arbitrary. Take a law using a constant, say, Einsteins famous E=mc2. Now c can either be a constant in that law, or c is a variable in our state space, governed by the dynamical equation dc/dt=0 and an initial condition c(t0)=...

byofrcs wrote:My point was that the roulette wheel has a predictable distribution function and a block of Uranium has a similarly predictable (though different) distribution function.

Say I used an alpha particle counter pointed at a block of uranium and use an alpha particle hit to time when the mechanism fires the ball into the roulette wheel. Do you think the roulette wheel then suddenly fails to follow an even distribution ?.

No, but the experiment has changed.

It won't but what I have now done is made it impossible for you to predict the roulette wheel outcome no matter how much computing power you apply to the problem because you claim it is impossible to compute the decay time of the uranium particle.

Yes.

Therefore a roulette wheel is subjectively random because of the timing of the ball release and how this synchronises to the wheel. That doesn't sound like subjectively random unless a human is able to manipulate this in some way with a resolution that is probably far smaller than a human can function at.

The wheel is now objectively random because of the timing of the ball release. It becomes subjectively random as soon as the nucleus decays.

That's an odd question as that's exactly what computing IS- making calculations using laws programmed in (either machine or organism).
If you can define a law you CAN compute with it.

Maybe the question should be rephrased like "Is it possible that not all behavior of the universe can be predicted/modeled/simulated by an algorithm?".

I think this is more than possible. It's very likely. Algorithms are only a human invention. Why should the universe be so well behaved?

I think it comes down to how we should construct the reference class. If it were possible to construct the reference class of possible universes - to write a description of each down in a book - how should we do this? For example, if we say that each possible universe is modeled an algorithm in some language, and yet it is possible to conceive of a universe which is not, then we have clearly not got a complete reference class. So how is this reference class made? When we have an idea of that, we can then think about whether any "non-computable" universes would occupy a significant amount of that reference class.

Laws are things we do understand. There are all kinds of situations where we understand what we start with and what we end up with, but are foggy on the mechanism. Like the statistics examples above. But no worries. Everything has a rational explanation. Even tunneling, I guess.

http://molecularmodelingbasics.blogspot ... d-stm.html

King Hazza wrote:That's an odd question as that's exactly what computing IS- making calculations using laws programmed in (either machine or organism).
If you can define a law you CAN compute with it.

That is pretty much what I think also. I'm sympathetic to the idea that there might be some connection between QM and consciousness, but I really don't know what a non-computable physical law is supposed to be.