Meta Math!

What Hacker News says

1. bonoboTPSep 2025
   Yes, I mean exactly this type of insight. Basically taking a digital photo with a camera technically also just picks out the "address" of your current environment within the space of all images. Any 4K 2-hour-length feature film in a digital format is also just an address in the space of all possible videos. The director, the actors, the whole crew did all that work in order to select that point from the space of possibilities, they didn't "create" anything. That movie already existed.

   Of course this is silly, but interesting nonetheless. And we routinely speak about such high-dimensional spaces in research and engineering. Or we can imagine optimization as traversing a pre-existing search space. It may be structured as a graph or perhaps a Euclidean space. And in that space we can imagine a loss surface, that sits there in peace all along, with its global minimum somewhere. And instead of "constructing" a solution, we are simply hiking in this space and trying to spot that valley. But this is a bit fictional. We never physically "instantiate" this surface. It's an imagined abstraction. In reality we just have a vector and some rules as to how we change that vector. But we can imagine those changes to be movements in an imagined space.

   It's like the idea that the sculptor doesn't create the sculpture, the sculpture was there all along, he just had to remove the superfluous matter to reveal what was already there (i.e. the atoms belonging to the final sculpture).

   The most interesting thing is kind of on the border, between these absurdly large spaces and the more manageable ones that are feasible to enumerate.

   Another similar mindblow thing was when I forgot the password to a file that I encrypted. It's a fascinating thing that the bit pattern on the disk is functionally random now, and cracking it would take longer than the age of the universe. But if only I knew the password, it would only take just a second. There is a definite sequence of keystrokes I can execute to bring the universe in a state where the content will appear on my screen, it's so close, yet it's so-so far if you don't remember the password. Just a little difference in your brain state and it flips from trivial to hopeless.

   PS, if you like thinking about such things, I recommend Meta-Math by Gregory Chaitin, it's very fun (providing an address VS constructing the thing is basically the gist of algorithmic information theory).

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2. bentonaMay 2020
   This is by Gregory Chaitin, who also published a pretty interesting & related book called "META MATH!" [1]

   I especially like the Turing / Godel equivalency stuff he discusses.

   [1] https://arxiv.org/pdf/math/0404335.pdf

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3. jaymzcampbellJan 2020
   If you want to read a bit more from the mathematician himself on this very topic he wrote an accessible "pop-math" book about it, "Meta Math!: The Quest for Omega" though you'll need to look beyond the author's rather strange choices of metaphor (https://www.goodreads.com/book/show/249849.Meta_Math_).
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4. mathgeniusDec 2019
   Chaitin's "meta math" is also a good read. Less arduous than G.E.B.
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5. GregBuchholzJun 2018
   A good book to read to get aquainted with these notions is "Meta Math" by Gregory Chaitin. The impatient can jump to chapter 5.

   https://arxiv.org/abs/math/0404335

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6. cghMar 2017
   I try to read Godel's Proof once a year or so. You might also be interested in Gregory Chaitin's Meta Math! The Quest For Omega, which relates the halting problem (among other things) to Godel. A word of warning though, the book is rather idiosyncratic, as you might have seen from the title.
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