Thanks, that was an awesome read!

I'll attempt a translation. If I'm engaging with the world, then I notice new things about it, or I see things in new ways. For example: once, looking at the sky, I noticed that it was brightest near the horizon and darkest at the zenith. Suddenly I realized the reason: there was more air between me and the horizon than there was between me and the space directly above me. The scene snapped into focus, and I found I could distinctly see the atmosphere as a three-dimensional mass. If I turn my attention inward, on the other hand, I tend to draw connections between pieces of information I already have - suddenly intuiting the behaviour of quantum wave packets, for example, or drawing an analogy between my social networking behaviour and annealing. These two cases are the "dots" and the connections between the dots, respectively.

Information theory generally defines information to be about some event with an uncertain outcome: if you know a coin has been flipped, you will need additional information to determine whether it came up heads or tails. By contrast, if you already know that five coins were flipped and three came up heads, you don't need any additional information to deduce that the number of heads was prime. Anything that you could in principle figure out from the information you've already got isn't treated as new information; In this sense, mathematical truths (connections) are separated from information proper ("dots").

While this separation may be useful for theory, it doesn't capture all aspects of the way we learn and process information. For starters, we're often rather surprised to learn mathematical facts; this is because we need to use physical hardware (our brains) to compute proofs, and we don't know what the outcome of the computation will be. Also, our brains seem to treat things, states, patterns, and pieces of information all in the same way - hence, for example, we can refer to "the economy" as if it were a single thing rather than a complex system of interrelationships; or, going in the opposite direction, we can break down a tree into its component cells and, moreover, recognize each cell as a fantastically complicated system, and so on.

Meanwhile, our ability to draw links between different levels of organization allows us, in particular, to see that certain mathematical patterns are reflected the world around us. Once we've found a model that fits the system, we can make predictions we couldn't make before: the more confident I am, say, that every fifth coin flip will come up as heads, the less information will be conveyed when this does indeed happen. It goes in the other direction too: sometimes we see patterns in nature which point us toward new mathematical understanding. The example I gave was of soliton waves, which you can read about here - even if you have no technical background, I think you'll find the History section enlightening.

For all these reasons, I suspect that a better model of information might loosen the hard distinction that's made between new information and new deductions.