Shannon's information theory and foundations of mathematics

X.Y. Newberry newberryxy at
Mon Jul 25 11:24:05 EDT 2022

"All things physical are information-theoretic in origin." This statement
by J. Wheeler is false. What is this mysterious substance called
"information"? It arises because the value of a quantity in point A cannot
be replicated with infinite precision in point B. Part of the problem is
the ubiquity of noise. And finite precision implies quantization. Shannon
has a formula for this:

m = sqrt(P/N)

(N = signal power, N = noise power.) Log of m is information. It is
measured with respect to the "noise floor"; it is dimensionless. Any medium
can be used for transmission.

Furthermore a transition from one level to another such as from 0 to 1
cannot occur infinitely fast. This is limited by the channel bandwidth i.e.
the maximum frequency the channel can transmit. The measurement and
transmission of a quantity (at least in general) is thus effectively
quantized and time sampled. This is a law of physics, but it acquires
particular salience in telecommunication engineering.

In the Newtonian/Laplacian deterministic world view it is implicitly
assumed that the value of a quantity CAN be transmitted with infinite
precision. In such a universe the concept of information would not arise.
We could transmit an equivalent of an entire DVD in 1 ns waveform not
exceeding 1 mV.

Shannon's paper caused a sensation ... and confusion. People thought that
he discovered some substance that perhaps predated the Universe itself.
People confused Shannon information with the meaning of the transmitted
narrative. It IS possible to correlate the meaning of messages with their
probabilities. When monitoring weather reports from Sahara you know that
"it is sunny" is more probable than "it rains." Nevertheless Shannon
information and the meaning of the transmitted sentences are two different
things, and it may not always be possible to correlate them.

X.Y. Newberry

*There are two ways to be fooled. One is to believe what isn't true; the
other is to refuse to believe what is true.*
― Søren Kierkegaard
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