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Shannon 1948 — A Mathematical Theory of Communication

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Shannon 1948 — A Mathematical Theory of Communication

System notes

Information is physical. Uncertainty can be measured in bits, and there is a hard limit to how much a signal can be compressed before information is lost.

The Source Coding Theorem states that the minimum average number of bits needed to encode a message equals its Shannon entropy, and no code can beat this limit without losing information.

The Noisy Channel Coding Theorem states that information can be transmitted through a noisy channel with arbitrarily low error if and only if the transmission rate stays below the channel capacity C = max_{p(x)} I(X;Y).

The statistical-mechanical formula for entropy H = -Σ p log p is an identity that measures both disorder in a gas and uncertainty in a communication channel, not merely an analogy.

Shannon's theory did not prove that information erasure costs energy; Landauer proved the physical cost of erasing one bit (at least kT ln 2 of heat) thirteen years later in 1961.

Shannon entropy is observer-relative in a way that Boltzmann's physical entropy is not: the same signal has different Shannon entropy under different codebooks.

Shannon entropy measures average compressibility across an ensemble, whereas Kolmogorov complexity measures compressibility of a single object; the gap between ensemble and individual compressibility matters for some incompressible strings drawn from compressible distributions.

shannon-1948 · condition map

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system
Information is physical. Uncertainty can be measured in bits, and there is a hard limit to how much a signal can be compressed before information is lost.
sources: S1
system
The Source Coding Theorem states that the minimum average number of bits needed to encode a message equals its Shannon entropy, and no code can beat this limit without losing information.
sources: S1
system
The Noisy Channel Coding Theorem states that information can be transmitted through a noisy channel with arbitrarily low error if and only if the transmission rate stays below the channel capacity C = max_{p(x)} I(X;Y).
sources: S1
system
The statistical-mechanical formula for entropy H = -Σ p log p is an identity that measures both disorder in a gas and uncertainty in a communication channel, not merely an analogy.
sources: S1, S4
system
Shannon's theory did not prove that information erasure costs energy; Landauer proved the physical cost of erasing one bit (at least kT ln 2 of heat) thirteen years later in 1961.
sources: S1, S2
3 more ranked claims
system0.90
Shannon entropy is observer-relative in a way that Boltzmann's physical entropy is not: the same signal has different Shannon entropy under different codebooks.
sources: S1
system0.85
Shannon entropy measures average compressibility across an ensemble, whereas Kolmogorov complexity measures compressibility of a single object; the gap between ensemble and individual compressibility matters for some incompressible strings drawn from compressible distributions.
sources: S1, S5
speculative0.60
Jaynes and the objective Bayesian camp argue that entropy is a measure of our ignorance, not a property of the world, and that the thermodynamic convergence is formal analogy rather than physical identity.
sources: S3
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