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This article seems aimed to resolve the "Interesting Number Paradox"[0], but does so in a pretty weak way. The author concludes: > So the split into interestin

This article seems aimed to resolve the "Interesting Number Paradox"[0], but does so in a pretty weak way. The author concludes: > So the split into interesting and boring numbers seems to stem from the judgments we make, such as attaching importance to prime numbers. Given the context of the rest of the article, about what numbers appear in OEIS[1] sequences, I'd argue that the author has assumed the conclusion from the start. If you look only at numbers that people have deemed interesting, you....only find numbers that people have deemed interesting. There's a much deeper (maybe-philosophical) question left unasked about the nature and structure of numbers. I was hoping the author would at least wave to that. [0] https://en.wikipedia.org/wiki/Interesting_number_paradox https://en.wikipedia.org/wiki/Interesting_number_paradox [1] https://oeis.org/ https://oeis.org/

TeMPOraL 4y ago
I think there's one more deep, interesting (!) question hidden, adjacent to the post and your reply: what exactly makes something interesting? I feel it's a real question, because the things we find interesting aren't all that arbitrary. I have only a handwavy notion of it, based on some on-line articles (IIRC one of them might have been a blogpost or a paper by Scott Aaronson), but the general idea is that the things we find interesting exist in the middle between low and high-entropy states. For example, take matches. A box of matches is relatively low-entropy: they all sit there neatly arranged in parallel rows, pointing the same direction. Matches in a box are boring. Now, dump the box onto the table. The matches are now arranged in a random pattern. This is a high-entropy state, and it's even more boring than the low-entropy one. Now, start putting those matches into some pattern - suddenly, things get interesting. One of the particularly interesting configurations would be a self-supporting 3D structure made of those matches - and that's, in terms of entropy, about half-way between matches in a box, and matches on a pile. Put in information-theoretic terms, it seems that we find both highly predictable and highly unpredictable configurations boring: the former likely because there is just not much new information to pay attention to - but the latter, seemingly because it's too unpredictable, and we round it up to "pure noise". So my question is, is there any formal description of this concept? A mathematical answer to why we find both order and disorder boring, and are most interested in things in-between? Some kind of natural quantity that tracks -E² (with E being entropy), and which would be fundamentally useful to maximize?
- mankutimma 4y ago
  Thank you for your great comment, I made an account just to reply. Please see Robert Sapolsky, Stanford Class Day 2009 lecture on "Uniqueness of Humans". He shares experiments on primates involving dopamine, which closely mirror your comment on entropy buy without using the term. If you can find it, please share the blog or paper where you found the idea.
- Doxin 4y ago
  I figure both order and disorder compress easily, whereas stuff in the middle doesn't. A box of matches describes the situation pretty succinctly. A pile of matches is also pretty succinct. A number of matches arranged in the shape of a dragon fractal is getting harder to denote succinctly.
joe_the_user 4y ago
I think this more or less has to do with formal languages for specifying numbers. If you have a formal language in which any string might (or might not) be interpreted to mean a single integer, then by the pigeonhole principle, half the integers of a given string-length will require strings of greater information length expression to be referred to. So relative to this given formal language, you can have a smaller integer that doesn't have a compact representation, doesn't have representation "smaller" than itself (and so might be labeled "boring"). And you can refer to the smallest of these integers in some meta-language but you can't refer to this smallest integer within the given formal language (and still have the language be consistent). Related to: Kolmogorov complexity, Godel's Second Incompleteness Theorem, Chaitin's Number Omega

TeMPOraL 4y ago

I think there's one more deep, interesting (!) question hidden, adjacent to the post and your reply: what exactly makes something interesting? I feel it's a real question, because the things we find interesting aren't all that arbitrary. I have only a handwavy notion of it, based on some on-line articles (IIRC one of them might have been a blogpost or a paper by Scott Aaronson), but the general idea is that the things we find interesting exist in the middle between low and high-entropy states. For example, take matches. A box of matches is relatively low-entropy: they all sit there neatly arranged in parallel rows, pointing the same direction. Matches in a box are boring. Now, dump the box onto the table. The matches are now arranged in a random pattern. This is a high-entropy state, and it's even more boring than the low-entropy one. Now, start putting those matches into some pattern - suddenly, things get interesting. One of the particularly interesting configurations would be a self-supporting 3D structure made of those matches - and that's, in terms of entropy, about half-way between matches in a box, and matches on a pile. Put in information-theoretic terms, it seems that we find both highly predictable and highly unpredictable configurations boring: the former likely because there is just not much new information to pay attention to - but the latter, seemingly because it's too unpredictable, and we round it up to "pure noise". So my question is, is there any formal description of this concept? A mathematical answer to why we find both order and disorder boring, and are most interested in things in-between? Some kind of natural quantity that tracks -E² (with E being entropy), and which would be fundamentally useful to maximize?

mankutimma 4y ago
Thank you for your great comment, I made an account just to reply. Please see Robert Sapolsky, Stanford Class Day 2009 lecture on "Uniqueness of Humans". He shares experiments on primates involving dopamine, which closely mirror your comment on entropy buy without using the term. If you can find it, please share the blog or paper where you found the idea.
Doxin 4y ago
I figure both order and disorder compress easily, whereas stuff in the middle doesn't. A box of matches describes the situation pretty succinctly. A pile of matches is also pretty succinct. A number of matches arranged in the shape of a dragon fractal is getting harder to denote succinctly.

joe_the_user 4y ago

I think this more or less has to do with formal languages for specifying numbers. If you have a formal language in which any string might (or might not) be interpreted to mean a single integer, then by the pigeonhole principle, half the integers of a given string-length will require strings of greater information length expression to be referred to. So relative to this given formal language, you can have a smaller integer that doesn't have a compact representation, doesn't have representation "smaller" than itself (and so might be labeled "boring"). And you can refer to the smallest of these integers in some meta-language but you can't refer to this smallest integer within the given formal language (and still have the language be consistent). Related to: Kolmogorov complexity, Godel's Second Incompleteness Theorem, Chaitin's Number Omega