Wouldn't also any finite complexity class only have finitely many hypotheses in it
Think of it as like the set of all positive integers of finite size. As it turns out, every single integer has finite size! You show me an integer, and I'll show you its size :P But even though each individual element is less than infinity, the size of the set is infinite.
Why would the mapping between the language the hypotheses are framed in have impact on which statements are most likely to be true?
Choosing which language to use is ultimately arbitrary. But because there's no way to assign the same probability to infinitely many discrete things and have the probabilities still add up to one, we're forced into a choice of some "natural ordering of hypotheses" in which the probability is monotonically decreasing. This does not happen because of any specific fact about the external world - this is a property of what it looks to have hypotheses about something that might be arbitrarily complicated.
The article mentions that in domains where the correct hypotheses are complex in the proof language the principle tends to be anti-productive.
Well... it's anti-productive until you eliminate the simple-but-wrong alternatives, and then suddenly it's the only thing allowing you to choose the right hypothesis out of the list that contains many more complex-and-still-accurate hypotheses.
If you want a much better explanation of these topics than I can give, and you like math, I recommend the textbook by Li and Vitanyi.
9 has 4 digits as "1001" in binary and 1 in decimal, so no function from integers to their size. There is no such thing as the size of a integer independent of any digit system used (well you could refer to some set constructions but then the size would be the integer itself).
As surreals we could have ω pieces of equal probability ɛ that sum to 1 exactly (althought ordinal numbers are only applicaple to orders which can be different than cardinal numbers. While for finites there is no big distinciton from ordinal and cardinal, "infinitely ma...
This essay claims to refute a popularized understanding of Occam's Razor that I myself adhere to. It is confusing me, since I hold this belief at a very deep level that it's difficult for me to examine. Does anyone see any problems in its argument, or does it seem compelling? I specifically feel as though it might be summarizing the relevant Machine Learning research badly, but I'm not very familiar with the field. It also might be failing to give any credit to simplicity as a general heuristic when simplicity succeeds in a specific field, and it's unclear whether such credit would be justified. Finally, my intuition is that situations in nature where there is a steady bias towards growing complexity are more common than the author claims, and that such tendencies are stronger for longer. However, for all of this, I have no clear evidence to back up the ideas in my head, just vague notions that are difficult to examine. I'd appreciate someone else's perspective on this, as mine seems to be distorted.
Essay: http://bruce.edmonds.name/sinti/