It means that a sufficiently small change in the initial value will cause an arbitrarily small change in the final value. This is true for real life systems, but if you had a truly binary system, it wouldn’t be. For example, if you have a switch that conducts electricity if it’s flipped more than half way, and you flip it exactly half way, then if you flip it any more, no matter how little you move it, it will instantly go from being an insulator to being a conductor.
This is true of a particular model of reality, namely classical mechanics, but it seems to be a contentious claim that it’s true of reality itself. AFAIK physicists take the possibility that reality might be discrete fairly seriously.
To my knowledge, it’s true of every model of reality.
There have already been stable time loops in both canon Harry Potter and in Methods. If the universe is discrete, there won’t generally be a stable time loop. It’s not going to take Harry trying to break time to cause a paradox.
The universe is chaotic. Not every aspect is, but that’s not necessary. If you go back in time one way and look at the Brownian motion, and go back with the initial conditions changed by one, the Brownian motion will be completely different. It will basically be random. If you make a random bijection, the probability of there being a fixed point approaches roughly 63% as the domain increases. Thus, if the universe was discrete, and obeyed the law of increasing entropy, then there would only be a working time loop about two thirds of the time even if you didn’t try to prevent one.
What does “behave continuously” mean?
It means that a sufficiently small change in the initial value will cause an arbitrarily small change in the final value. This is true for real life systems, but if you had a truly binary system, it wouldn’t be. For example, if you have a switch that conducts electricity if it’s flipped more than half way, and you flip it exactly half way, then if you flip it any more, no matter how little you move it, it will instantly go from being an insulator to being a conductor.
This is true of a particular model of reality, namely classical mechanics, but it seems to be a contentious claim that it’s true of reality itself. AFAIK physicists take the possibility that reality might be discrete fairly seriously.
To my knowledge, it’s true of every model of reality.
There have already been stable time loops in both canon Harry Potter and in Methods. If the universe is discrete, there won’t generally be a stable time loop. It’s not going to take Harry trying to break time to cause a paradox.
The universe is chaotic. Not every aspect is, but that’s not necessary. If you go back in time one way and look at the Brownian motion, and go back with the initial conditions changed by one, the Brownian motion will be completely different. It will basically be random. If you make a random bijection, the probability of there being a fixed point approaches roughly 63% as the domain increases. Thus, if the universe was discrete, and obeyed the law of increasing entropy, then there would only be a working time loop about two thirds of the time even if you didn’t try to prevent one.