I saw a post of Dan Piponi on Mastodon where he mentions theorem
shadowshttps://mathstodon.xyz/@dpiponi/117327643845116373We can define a
"shadow of a theorem" as an informal metaphor for a real-world, practical
behavior or rule of thumb that reflects a deeper, idealized mathematical limit.
Historically, people often observe shadows - like Kepler's laws of planetary
motion or floating-point comparison bugs - decades or centuries before
mathematicians prove the abstract "theorems" that actually explain them.Today,
fields like deep learning, medicine, cryptography, and quantum physics still
contain empirical shadows that engineers and scientists rely on, even though
the master theorems behind them remain undiscovered, for example+ how deep
learning model scaling works+ how general anesthesia actually produces
unconsciousness+ the Yang–Mills mass gap problem in Quantum Field Theoriesand
of course+ the hard problem of consciousnessThe FLP impossibility theorem in
distributed computing proves that no deterministic consensus algorithm can
guarantee both safety and liveness in a fully asynchronous distributed system
if even a single process experiences a crash failure. The hard problem of
consciousness is a shadow of such an impossibility theorem, isn't it? I believe
the hard problem is related to path dependence in a complex adaptive system.
Differences in Qualia is related to the topological and structural difference
between two path-dependent graphs. Path dependence can be precisely defined
mathematically. It should be possible to prove such an impossibility theorem
for the hard problem of consciousness by a indirect proof. If we assume it is
solvable then the path-dependent graphs (or emotional valuation matrices) must
be identical, which is not possible in real systems. Exact subjective
equivalence between two conscious agents is impossible.We can consider the hard
problem of consciousness as a shadow of this impossibility theorem. I think the
impossibility theorem solves it. Finally. And we know the solution for an
asymptotic approximation as well: the what-it-is-like-to-be machines of film
industry, cinemas and show business.-J.
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