The fact that the proof uses only very weak properties of the notion of “complexity” does not show that the proof is invalid. In fact, it suggests (not with certainty) the opposite: that an even stronger result could be proven.
Thanks and apologies, I was being careless when I said the proof “couldn’t be right”. It may be formally right, but fail to explain why we should use Occam’s Razor instead of the Flubby Razor.
Yes a statement like “on average, X holds” needs to be quantitative in order to be useful in practice. Still, this is a valuable argument, especially as far as it clarifies what needs to be done to reach a quantitative refinement.
The proof does allow for different Razors depending on your specific definition of complexity. It’s true that this makes the statement fairly weak. But it is also in a way a virtue, because it corresponds with our actual use of the Razor. There are in fact different definitions of complexity and we use different ones in different contexts; we learn from experience which contexts require which Razors. For example, if you want to describe the path of a ball moving through the air, you expect to be able to describe it with some fairly simple mathematics, and you think this more likely than complicated mathematical descriptions.
On the other hand, if you see a precise human-shaped footprint in the mud in your yard, saying “a human did it” is very complicated mathematically, since the mathematical description would include the description of a human being. So in this way, “Wind and weather did it” should be simpler, in the mathematical way. Nonetheless, you use a different Razor and say that with this Razor, it is simpler to say that a human being did it.
The last paragraph feels wrong to me. This is the explanation for that feeling that came to mind first, but I’m not positive it’s my true objection:
If your only observation were the footprint, sure—but your observations also include that you live on a planet with 7 billion people who leave footprints like that—so despite the comparitive algorithmic simplicity of the weather model, it doesn’t match the observations as well.
Yes, it is likely that a stronger result is possible. However, it is difficult to make it stronger while satisfying my conditions (true in all possible worlds; true according to every logically consistent assignment of priors).
The fact that the proof uses only very weak properties of the notion of “complexity” does not show that the proof is invalid. In fact, it suggests (not with certainty) the opposite: that an even stronger result could be proven.
Thanks and apologies, I was being careless when I said the proof “couldn’t be right”. It may be formally right, but fail to explain why we should use Occam’s Razor instead of the Flubby Razor.
Yes a statement like “on average, X holds” needs to be quantitative in order to be useful in practice. Still, this is a valuable argument, especially as far as it clarifies what needs to be done to reach a quantitative refinement.
The proof does allow for different Razors depending on your specific definition of complexity. It’s true that this makes the statement fairly weak. But it is also in a way a virtue, because it corresponds with our actual use of the Razor. There are in fact different definitions of complexity and we use different ones in different contexts; we learn from experience which contexts require which Razors. For example, if you want to describe the path of a ball moving through the air, you expect to be able to describe it with some fairly simple mathematics, and you think this more likely than complicated mathematical descriptions.
On the other hand, if you see a precise human-shaped footprint in the mud in your yard, saying “a human did it” is very complicated mathematically, since the mathematical description would include the description of a human being. So in this way, “Wind and weather did it” should be simpler, in the mathematical way. Nonetheless, you use a different Razor and say that with this Razor, it is simpler to say that a human being did it.
The last paragraph feels wrong to me. This is the explanation for that feeling that came to mind first, but I’m not positive it’s my true objection:
If your only observation were the footprint, sure—but your observations also include that you live on a planet with 7 billion people who leave footprints like that—so despite the comparitive algorithmic simplicity of the weather model, it doesn’t match the observations as well.
Yes, it is likely that a stronger result is possible. However, it is difficult to make it stronger while satisfying my conditions (true in all possible worlds; true according to every logically consistent assignment of priors).