For proving a proposition P I have to show for all x P(x), but for contradiction I only have to show that there exists an x such that not P(x).

While I agree there could be a lot of theory crafting to reduce the search space of possible x's to find not P(x), but with for all x P(x) you have to be able to produce a larger framework that explains why no counter example exists.

Assuming humans are more powerful than regular languages I could maybe agree that these methods may not eventually yield entirely human like intelligence, but just better and better approximations.

The vibe I get though is that we aren't more powerful than regular languages, cause human beings feel computationally bounded. So I could see given enough "human signal" these things could learn to imitate us precisely.

I agree with one of the mathematician's responses in the linked PDF that this is somewhat less interesting than proving the actual conjecture was true.

In my eyes proving the conjecture true requires a bit more theory crafting. You have to explain why the conjecture is correct by grounding it in a larger theory while with the counterexample the model has to just perform a more advanced form of search to find the correct construction.

Obviously this search is impressive not naive and requires many steps along the way to prove connections to the counterexample, but instead of developing new deep mathematics the model is still just connecting existing ideas.

Not to discount this monumental achievement. I think we're really getting somewhere! To me, and this is just vibes based, I think the models aren't far from being able to theory craft in such a way that they could prove more complicated conjectures that require developing new mathematics. I think that's just a matter of having them able to work on longer and longer time horizons.

  Unless someone eventually finds the consciousness center in the brain I will continue to hold the position that it is just another property of "things". I know consciousness must be real because it's the first thing I have access to without any sort of reasoning attached on top of it. Its realness is more visceral than atoms or any other physical theory because it is the way in which the world is conveyed to me, but I don't think I'm unique in any way for having it.

  I feel like all systems, in a panpsychist sense, participate in consciousness, so in some way it's a property of matter or systems in our universe that we have somehow failed to account for in physics. We miss it because systems only exhibit consciousness internally like on top of having all the physical properties of rocks, rocks also have an internal state of being. That internal state of being for the most part is uninteresting cause it doesn't dictate the rocks actual form or function in the universe.

  I'd argue human consciousness is the same. My conscious experience has nothing to do with the thoughts that are actually being produced. By this I mean there is no authorship of the thoughts and actions I perform by my consciousness. To me it seems more like a stage in which elements of my experience appear for brief moments before fading away, so much like the rock's internal experience my internal experience does not have any affect on the physical world.

  Part of me then starts to worry why worry about consciousness at all if it's something that doesn't participate in the physical world because then what's the point of it all? Also, if all systems get to participate, then what stops things like basic logic gates on a PC from having consciousness as well. I tend to lean towards thinking that those feelings are similar to the same kinds of feelings humans used to have about thinking they were the center of the universe, but I'm not sure.

Austin