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Biomedical and Electrical Engineer with interests in information theory, evolution, genetics, abstract mathematics, microbiology, big history, Indieweb, and the entertainment industry including: finance, distribution, representation

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This sounds a tad like the debate in early Greek philosophy between the relativists (pre-Socratics) and the Platonists.

I'm curious what your calculation would look like to prove that they either contain the same information or that Mount Fuji contains more?

One of the subtleties of Shannon's original paper on information theory is in the second paragraph where he explicitly states that he's leaving out the semantic processing of the message by the receiver and just concentrating on the sending and receipt of the signal. (Put another way we can play the game telephone and you can hear and repeat the exact words I say, but concepts like <i>double entendre</i> and nuance may prevent you from understanding exactly what I meant to say. Shannon is leaving the second problem understanding out of the picture and solely working on the question of did you hear the words I actually said.) The way you're framing your question, it would appear that you're only focusing on the interpretation of the information after it's been received, in which case Shannon's information doesn't really have anything to say.

Now, to take a look at it from a purely mathematical standpoint within a Shannon framework, once could count the totality of the number of atoms and their states in Fuji and compare it to that of Rushmore and the one with more information is simply going to be the larger one.

If we're looking at photos of the two, then the one with more information is going to be the one that doesn't compress down as much under whatever compression algorithm we might choose. (ie, it won't have anything to do with the "information" contained in the faces and whose faces they are, which again is a semantic issue and not a mathematical one.)