Shannon Information Paradox
Previously, I wrote about how information presents its own form of virtual gravity, which can shape the space-time of the Infosphere, and thus our experience of reality, in ways that parallel the shaping of space-time in the Biosphere.
This Infosphere black hole emerges when information convergence occurs on a massive scale, requiring (as Hawking observed) the massive amounts of energy. Energy has mass, and with massive mass comes massive gravity. In this way, the “informational gravity” can be understood as bending, or even at its outermost limits breaking, the virtual space-time fabric of the Infosphere.
What is missing from that is a sense of location. After all, the universe of full of mass. It is only when that mass is “sufficiently compact” that a black hole can take space-time to the limits of comprehension. In the Infosphere, however, location doesn’t quite work the same way. There is no there there, as Gertude Stein might put it. So, how to proceed with issues compactness, or density, of information, if information isn’t bound by form like matter?
Good question.
I think there’s a way that I haven’t come across elsewhere to quantify the density effect upon the space-time of the Infosphere. What this offers, or proposes, is an approach for understanding how a observer-participant in any informational event perceives (and therefore comes to know) the information and how the information’s reality itself (or existence or ontology) is in turn altered by the presence of observer-participants.
Here it is…after some context.
In a Biosphere black hole (the kind out there in outer space), at its center is a gravitational singularity. There the mass to volume ratio approaches a point of infinite density. In the Infosphere, however, information doesn’t have mass or volume in the Biosphere sense, but, because information requires energy, and energy has mass, we can nevertheless conceive of the idea of information density. Information Density then frames the conversation around gravity.
Mathematically, density is defined as mass divided by volume:

For mass: Shannon Information Content, because the mass describes the quantity of matter and Shannon Information describes the quantity of information. For example, in a coin toss, the quantity of information produced by the outcome of each coin flip is one bit of information. That’s because the odds in a fair coin toss are 50/50, or .5 probability, and the result of putting that probability (p) into Shannon’s equation:
log2 (1/p)
is 1.
For volume: Metcalfe’s Law, because network connection capacity may be seen as a de facto measure of space in an otherwise dimensionless Infosphere. For quantifying this volume, rather than the more traditional n², I’m adopting Odlyzko and Tilly’s equation:
nlog2(n)
In both formulas above, the logarithm is to base two (anyone know how to do subscript in Medium?).
Information Density is therefore:
ρ = mass/volume = log2(1/p)/nlog2(n) = logn(1/p)/n
which we can graph for probability values of p between 0 and 1:

Here y is graphing (something like) the measure of informational density, rho (ρ).
For probability values of p = .5 (like in our coin toss), this produces the following density values:
when n=2, ρ=.5
when n=4, ρ=.125
when n=8, ρ=.0417
(Perhaps this gets us a specific kind of density, however, because both Shannon and Metcalfe are something of density calculations themselves — how dense with surprise and how dense with nodes/users, respectively. So, we may even consider this as a ratio of densities, akin to informational specific gravity relative to the reference gravity of n=2. In this dimensionless conceit, N=2 may work as the I and Thou baseline for network ontology.)
So…what?
Good question.
So, as the ‘volume’ of an information network (n) grows, and approaches infinity, ρ approaches zero for the same information content (p).
What is zero ρ? It is a constant, regardless of network size, occuring when p=1. When p=1, there is no information generated by an event. For example, in a double headed coin flip, the coin is coming up heads every time, so there is no information coming out of the coin flip. There is no ‘surprise’ at the end, and therefore, according to Shannon, there is no information communicated by the event. The outcome of the flip isn’t really an outcome, because it was never in question. No bits of information or ‘mass’ is produced by the event. It’s an empty event.
Here’s how ρ looks at different, increasing network values:




As these table above show, zero information always carries zero ρ.
As bits approach infinity, p approaches zero, aka the probability of some event is so infinitesimal, that the surprise generated by the outcome — its information content — is correspondingly great…like a double-headed coin coming up tails, perhaps. Here ρ remains undefined and also approaches infinity.
(And, furthermore, at n=1, all values of p result in ρ undefined, because you can’t have a connection (network) without at least two things to connect. I think this also ties in with the Bishop Berkeley precept to be is to be perceived due to the nature of one observer (of a coin flip, for instance) being unable to validate the information in that outcome. Basically, how do you know you are seeing what you’re seeing — in that real philosophical sense of Knowing? If you see a fair coin come up heads 100 times in a row, even though there’s nothing wrong with that, you would surely feel better about your state of mind if there were an outside observer (outside of you) to agree with you by saying, “I saw it too.”)
But…so what?
So, as the participant-observational energy increases (n approaches infinity), our 1-bit of information (p=.5) results in a ρ value that increasingly resembles the ρ of a 0-bit event. The ρ curve flattens for all probabilities, in fact, as network size increases. Any information begins, in this manner, to resemble no information.
Something that turns outcomes into non-outcomes deontologizes information. It makes information un-become itself, leeching it of its mass, emptying it. And, as I said before, deontology is the basis of information black holes.
So, in this way, increasing numbers of connected informational agents engaging with any amount of information decreases the Being-ness of the information to the point that the information ceases to be real.
This information deontologization effect may warp some experiences of reality by warping our perception of the information it takes to know that reality. Is it possible that we’ve never really had to deal with this before? It is, or possibly not at the level now or eventually made possible by global ICT. It is also possible that we already have some sense of how networks of information warp information, like in ‘echo chambers’ or with ‘group think’ or mobs or conspiracies or any number of ways. Is there a break-even? And how does the information entropy of events more complex than a coin toss change the resulting information ‘density’? And how does one measure the true network size of informational observer-participants? All good questions, which may merit further thought.
What I have thought about, and hoped to answer with this mathematical approach, is how information, apparently losing mass via network effects, can cease to matter because, in a way, it ceases to be. Or it becomes more like information that isn’t, like isn’tformation, to use a term introduced previously.
This information-isn’tformation relationship is, in the end, why this equation matters. Information content quantifies the information in an event, but, to paraphrase Dretske, information doesn’t need us to exist. How do we then, as dependent informational organisms, alter the nature of information’s independent existence?
Great question, maybe the very one behind this story and the last one.
To return to where we started, at black holes: instead of density apparently becoming infinite, the gravitational singularity in an Infosphere black hole approaches zero density, despite the presence of information (mass), which, when you think about it, is improbable to the point that the information content of that singularity event is, itself, undefined.
This is all in keeping with black holes, of course. A black hole could emit anything. What are the odds of that?
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