The hope is to use the complexity of the statement rather than mathematical taste.
I understand the hope, I just think it’s going to fail (for more or less the same reason it fails with formal proof).
With formal proof, we have Godel’s speedup, which tells us that you can turn a Godel statement in a true statement with a ridiculously long proof.
You attempt to get around this by replacing formal proof with “heuristic”, but whatever your heuristic system, it’s still going to have some power (in the Turing hierarchy sense) and some Godel statement. That Godel statement is in turn going to result in a “seeming coincidence”.
Wolfram’s observation is that this isn’t some crazy exception, this is the rule. Most true statements in math are pretty arbitrary and don’t have shorter explanations than “we checked it and its true”.
The reason why mathematical taste works is that we aren’t dealing with “most true statements”, we’re only dealing with statements that have particular beauty or interest to Mathematicians.
It may seem like cheating to say that human mathematicians can do something that literally no formal mathematical system can do. But if you truly believe that, the correct response would be to respond when asked “is pi normal” with “I don’t know”.
The reason why your intuition is throwing you off is because you keep thinking of coincidences as “pi is normal” and not “we picked an arbitrary CA with 15k bits of complexity and ran it for 15k steps but it didn’t stop. I guess it never terminates.”
‘taste’, ′ mathematical beauty’, ′ interesting to mathematicians’ aren’t arbitrary markers but reflect a deeper underlying structure that is, I believe, ultimately formalizable.
It does not seem unlikely to me at all that it will be possible to mathematically describe those true statements that are moreover of particular beauty or likely interest to mathematicians (human, artificial or alien).
The Godel speedup story is an interesting point. I haven’t thought deeply enough about this but IIRC the original ARC heuristic arguments has several sections on this and related topics. You might want to consult there.
I understand the hope, I just think it’s going to fail (for more or less the same reason it fails with formal proof).
With formal proof, we have Godel’s speedup, which tells us that you can turn a Godel statement in a true statement with a ridiculously long proof.
You attempt to get around this by replacing formal proof with “heuristic”, but whatever your heuristic system, it’s still going to have some power (in the Turing hierarchy sense) and some Godel statement. That Godel statement is in turn going to result in a “seeming coincidence”.
Wolfram’s observation is that this isn’t some crazy exception, this is the rule. Most true statements in math are pretty arbitrary and don’t have shorter explanations than “we checked it and its true”.
The reason why mathematical taste works is that we aren’t dealing with “most true statements”, we’re only dealing with statements that have particular beauty or interest to Mathematicians.
It may seem like cheating to say that human mathematicians can do something that literally no formal mathematical system can do. But if you truly believe that, the correct response would be to respond when asked “is pi normal” with “I don’t know”.
The reason why your intuition is throwing you off is because you keep thinking of coincidences as “pi is normal” and not “we picked an arbitrary CA with 15k bits of complexity and ran it for 15k steps but it didn’t stop. I guess it never terminates.”
‘taste’, ′ mathematical beauty’, ′ interesting to mathematicians’ aren’t arbitrary markers but reflect a deeper underlying structure that is, I believe, ultimately formalizable.
It does not seem unlikely to me at all that it will be possible to mathematically describe those true statements that are moreover of particular beauty or likely interest to mathematicians (human, artificial or alien).
The Godel speedup story is an interesting point. I haven’t thought deeply enough about this but IIRC the original ARC heuristic arguments has several sections on this and related topics. You might want to consult there.