As I said, it’s an interesting question! I think I’ve found a paper describing something like the scenario you mentioned (Dhar, A. (1993). Nonuniqueness in the solutions of Newton’s equation of motion. American Journal of Physics, 61(1), 58–61. doi:10.1119/1.17411). It’s a apparently shows that for certain conditions (such as the balanced knife you mentioned, or a particle in a field that would accelerate it away from the origin proportionally to it’s distance) Newton’s equations of motion have non-unique solutions, although I confess that the author rather lost me during some of his leaps in mathematics. The discussion section is interesting, a couple of key conclusions stood out to me: ‘In this sense we may say that Newton’s equation has a unique solution even for singular forces like x1/3 but x(0)=0 and derivative(x(0))=0 in such cases do not uniquely specify the initial state.’ and ‘Infinitesimal disturbance in position or velocity will change the state and one of the other solutions will become effective.’
From what I have understood from the paper, the author seems to be mostly pointing out that there are certain conditions under which Newton’s equations do not have a unique solution, but that in reality a deterministic, but chaotic, outcome will occur due to infinitesimal disturbances. Ultimately, no matter how carefully you balance the knife, it’s going to fall over, and the direction it falls will be determined by a multitude of forces rather than pure chance.
@[email protected] has also made a thoughtful reply regarding quantum field theory and it’s implications on determinism, and I need to respond to that too as it’s a fascinating, if baffling, topic.
Your question about predicting your own future is interesting; you’re making the assumption that a prediction must continue to be true after the point at which it is made, but I would suggest that you can resolve the apparent contradiction by considering that any prediction of the future is only true at the instant it is made. After all, if someone else predicted your future, wrote it down, but did not tell you, you would eat the avocado, however seen as you changed the conditions of your future by gaining additional information the result changed. If you predicted your future a second time, directly after having resolved to not eat the avocado, the prediction would have you not eating it.
If we assume the universe is deterministic, and that we have the ability to perfectly replicate it and run that replica forward in time without time passing in our universe it would seem that we could accurately predict the future of our universe just be seeing what happened in the replica. However, that would involve the replica creating it’s own replica as it would evolve in exactly the same way as our universe. That replica would create it’s own replica, and so on. I’m not quite sure of what the implications of that are, and it’s late here, so I’m going to have to call it a night, but if if could be done it would be a clear way to distinguish between a random or non-deterministic universe and a chaotic one. If the predictions sometimes proved incorrect it would suggest true randomness rather than just a chaotic system.
yeah you’re right, i’ve also been thinking about developments in a parallel universe a lot.
i cannot convince myself that the universe is purely deterministic because the universe is huge (it literally includes all things in existence); surely some of all those things must have non-deterministic behavior!
I have started to think of it as deterministic behavior being actually pretty rare, with just a few things which we can actually predict reliably about the future and a huge sea of chance! It fits my intuition nicely. I must and will investigate these things into a lot of detail, for sure. I’m especially intrigued by seeing the world as a network of events, where deterministic connections run like trails through the network. It’s interesting to see the structure of it all.
As I said, it’s an interesting question! I think I’ve found a paper describing something like the scenario you mentioned (Dhar, A. (1993). Nonuniqueness in the solutions of Newton’s equation of motion. American Journal of Physics, 61(1), 58–61. doi:10.1119/1.17411). It’s a apparently shows that for certain conditions (such as the balanced knife you mentioned, or a particle in a field that would accelerate it away from the origin proportionally to it’s distance) Newton’s equations of motion have non-unique solutions, although I confess that the author rather lost me during some of his leaps in mathematics. The discussion section is interesting, a couple of key conclusions stood out to me: ‘In this sense we may say that Newton’s equation has a unique solution even for singular forces like x1/3 but x(0)=0 and derivative(x(0))=0 in such cases do not uniquely specify the initial state.’ and ‘Infinitesimal disturbance in position or velocity will change the state and one of the other solutions will become effective.’
From what I have understood from the paper, the author seems to be mostly pointing out that there are certain conditions under which Newton’s equations do not have a unique solution, but that in reality a deterministic, but chaotic, outcome will occur due to infinitesimal disturbances. Ultimately, no matter how carefully you balance the knife, it’s going to fall over, and the direction it falls will be determined by a multitude of forces rather than pure chance.
@[email protected] has also made a thoughtful reply regarding quantum field theory and it’s implications on determinism, and I need to respond to that too as it’s a fascinating, if baffling, topic.
Your question about predicting your own future is interesting; you’re making the assumption that a prediction must continue to be true after the point at which it is made, but I would suggest that you can resolve the apparent contradiction by considering that any prediction of the future is only true at the instant it is made. After all, if someone else predicted your future, wrote it down, but did not tell you, you would eat the avocado, however seen as you changed the conditions of your future by gaining additional information the result changed. If you predicted your future a second time, directly after having resolved to not eat the avocado, the prediction would have you not eating it.
If we assume the universe is deterministic, and that we have the ability to perfectly replicate it and run that replica forward in time without time passing in our universe it would seem that we could accurately predict the future of our universe just be seeing what happened in the replica. However, that would involve the replica creating it’s own replica as it would evolve in exactly the same way as our universe. That replica would create it’s own replica, and so on. I’m not quite sure of what the implications of that are, and it’s late here, so I’m going to have to call it a night, but if if could be done it would be a clear way to distinguish between a random or non-deterministic universe and a chaotic one. If the predictions sometimes proved incorrect it would suggest true randomness rather than just a chaotic system.
yeah you’re right, i’ve also been thinking about developments in a parallel universe a lot.
i cannot convince myself that the universe is purely deterministic because the universe is huge (it literally includes all things in existence); surely some of all those things must have non-deterministic behavior!
I have started to think of it as deterministic behavior being actually pretty rare, with just a few things which we can actually predict reliably about the future and a huge sea of chance! It fits my intuition nicely. I must and will investigate these things into a lot of detail, for sure. I’m especially intrigued by seeing the world as a network of events, where deterministic connections run like trails through the network. It’s interesting to see the structure of it all.