> Now let’s step back and think about what’s going on.
> Lately I’ve been trying to unify a bunch of ‘extremal principles’, including:
> 1) the principle of least action
> 2) the principle of least energy
> 3) the principle of maximum entropy
> 4) the principle of maximum simplicity, or Occam’s razor
> In my post on quantropy I explained how the first three principles fit into a single framework if we treat Planck’s constant as an imaginary temperature. The guiding principle of this framework is
> maximize entropy
> subject to the constraints imposed by what you believe
> And that’s nice, because E. T. Jaynes has made a powerful case for this principle.
> However, when the temperature is imaginary, entropy is so different that it may deserves a new name: say, ‘quantropy’. In particular, it’s complex-valued, so instead of maximizing it we have to look for stationary points: places where its first derivative is zero. But this isn’t so bad. Indeed, a lot of minimum and maximum principles are really ‘stationary principles’ if you examine them carefully.
> What about the fourth principle: Occam’s razor? We can formalize this using algorithmic probability theory. Occam’s razor then becomes yet another special case of
> maximize entropy
> subject the constraints imposed by what you believe
> once we realize that algorithmic entropy is a special case of ordinary entropy.
https://johncarlosbaez.wordpress.com/2012/01/19/classical-me...
https://johncarlosbaez.wordpress.com/2012/01/23/classical-me...
https://johncarlosbaez.wordpress.com/2021/09/23/classical-me...
https://johncarlosbaez.wordpress.com/2021/09/26/classical-me...
An excerpt from the first article:
> The big picture
> Now let’s step back and think about what’s going on.
> Lately I’ve been trying to unify a bunch of ‘extremal principles’, including:
> 1) the principle of least action
> 2) the principle of least energy
> 3) the principle of maximum entropy
> 4) the principle of maximum simplicity, or Occam’s razor
> In my post on quantropy I explained how the first three principles fit into a single framework if we treat Planck’s constant as an imaginary temperature. The guiding principle of this framework is
> maximize entropy
> subject to the constraints imposed by what you believe
> And that’s nice, because E. T. Jaynes has made a powerful case for this principle.
> However, when the temperature is imaginary, entropy is so different that it may deserves a new name: say, ‘quantropy’. In particular, it’s complex-valued, so instead of maximizing it we have to look for stationary points: places where its first derivative is zero. But this isn’t so bad. Indeed, a lot of minimum and maximum principles are really ‘stationary principles’ if you examine them carefully.
> What about the fourth principle: Occam’s razor? We can formalize this using algorithmic probability theory. Occam’s razor then becomes yet another special case of
> maximize entropy
> subject the constraints imposed by what you believe
> once we realize that algorithmic entropy is a special case of ordinary entropy.