Interestingly, Erwin Schrödinger's seminal paper "What is Life?" came out in 1944, almost contemporaneously with the creation of information theory in 1948. Unbeknownst to most Claude Shannon's (the researcher who created the field of information theory and whose Masters thesis literally launched the digital revolution) Ph.D. thesis (1940) was entitled: "An Algebra for Theoretical Genetics," so he could certainly be said to be the god-father of the entire area.

Since then, certainly dozens of researchers have looked at the potential of using information theory to create a better hard core definition of what life really is (certainly the place where any self-respecting Platonist would begin.)

One of the best definitions and frameworks I've seen thus far has to be that of Christoph Adami. To start, and depending on your level of sophistication, take a look at his recent arXiv paper (Information-theoretic considerations concerning the origin of life - http://arxiv.org/abs/1409.0590) and then take a crack at this popular press article about it in Medium (https://medium.com/the-physics-arxiv-blog/information-theory-and-the-origin-of-life-4cf6b93d156c). If it's something that blows your skirt up, then you can certainly begin to delve more deeply into some of his journal articles over the past decade or so.

For further references, I maintain a nice list of resources at Information Theory and Biology Resources (http://boffosocko.com/itbio/), as well as a "journal club" of sorts at Mendeley: ITBio: Information Theory, Microbiology, Evolution, and Complexity (https://www.mendeley.com/groups/2545131/itbio-information-theory-microbiology-evolution-and-complexity/).

If you really want to blow the top off of your definitions, you might also consider taking a much broader look at the universe (a distant reading, so-to-speak) and read through David Christian's conceptualization (Maps of Time: An Introduction to Big History: David Christian, William H. McNeill http://www.amazon.com/gp/product/0520271440/ref=as_li_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0520271440&linkCode=as2&tag=itbio-20&linkId=R3CFNWGXPDTSA5GG) of the relatively new area known as "Big History" (see http://www.bighistoryproject.org) . There, he posits the universe, the stars, and other structures as living things. Admittedly their structures are much simpler than what we might "traditionally" think of as life, but which when considered deeply, certainly are living by the broadest definition - they have less "complexity" but are far longer lived than human beings.

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.)