Remark: I thought of this idea with the great Karl Tamraz.
The point I am trying to make is not that there is no useless information. My claim is that the way we relate to the world makes it such that we can never prove that there is a piece of information that is useless. Jane Long talked to me about the subjectivity of the scientist and how the scientist tries to hide it, and I thought about the idea of this blogpost as a way to display the subjectivity of the mathematician with respect to the proof.
Theorem: We can never prove that there exists a useless piece of information
Proof: We prove the theorem by contradiction.
Assume that we were trying to prove that there exists a useless piece of information. We would need to find one piece of information A that is useless. Assume (by contradiction) that we could find such a piece of information. Then this would mean that this piece of information was useful insofar as it was used to prove that there exists a useful piece of information -- which is a contradiction. QED.
PS: [remark for compsci geeks] Maybe someone will find an analogous proof of why we can never prove that P = NP.
PPS: Just to clarify, this is a proof that we cannot prove the theorem, not that the theorem is false.
The point I am trying to make is not that there is no useless information. My claim is that the way we relate to the world makes it such that we can never prove that there is a piece of information that is useless. Jane Long talked to me about the subjectivity of the scientist and how the scientist tries to hide it, and I thought about the idea of this blogpost as a way to display the subjectivity of the mathematician with respect to the proof.
Theorem: We can never prove that there exists a useless piece of information
Proof: We prove the theorem by contradiction.
Assume that we were trying to prove that there exists a useless piece of information. We would need to find one piece of information A that is useless. Assume (by contradiction) that we could find such a piece of information. Then this would mean that this piece of information was useful insofar as it was used to prove that there exists a useful piece of information -- which is a contradiction. QED.
PS: [remark for compsci geeks] Maybe someone will find an analogous proof of why we can never prove that P = NP.
PPS: Just to clarify, this is a proof that we cannot prove the theorem, not that the theorem is false.
