Ever since 2005, I have been pondering the subject of dreaming greatly, and have finally came to some useful philosophical conclusions (but I still haven’t found sure proof that this world is real), but one particular milestone is when I watched Inception, the second greatest movie I’ve seen. Out of many interesting things I noted while watching the movie, by far the greatest idea was a model for dreams that I came up with while watching it. It is the representation of dream levels as functions, and the representation of dream transactions as calculus.

Let’s, for purposes of introduction, call real life 1, and by “1” I mean the function f(x)=1, not the number 1. This is a sensible assignment, initially because the simplicity of “1” represents how reality makes sense straightforwardly, but also because of reasons to be explained later. What happens after you die? Well, nonexistence quickly leads to thinking of “0” (or d(x)=0) to represent what occurs when life ends.

What would be a good expression to represent your dream? (I’ll just call it ‘dream’ for now and ‘Dream Level 1’ later.) A dream is mysterious and doesn’t always make sense—much of it is a representation of unknown, so a sensible expression for a dream would be “x” (as a function, g(x)=x).

Of course, some things about “x” don’t make sense, like what the slope represents in terms of properties of the perceived mental world; for example, what does the slope represent? Insanity?

In any case, let’s go back on track. What if we had a dream within a dream? Would that just be x²? That would make an intuitive guess, but these new dreams are very weird…they are more unstable and have greater time dilation. Perhaps h(x)=x²/2 is a better representation for Dream Level 2.

At this point, something just has to be noticed. h'(x)=g(x), g'(x)=f(x), and f'(x)=d(x). That’s right, the relation between worlds of existence is analogous to the relation between dream levels. Philosophically, it makes sense, in terms of the derivations of spontaneous actions from the unconscious. The factor that the power rule tells us to bring around represents closely the shift in apparent time speed, although in this case it’s by factorials rather than powers of 2, but both of these functions show the correct exponential growth of time dilation in deeper dream levels. Knowing this, it can be seen that entering a dream is not just inception, but also integration—integration of our secret emotional reserves into our interaction with the world around. The world of “1” is thus distinctly different from dream worlds, because attempting to derive the source of entering the f(x)=1 world leads to death (or perhaps limbo, but see later), where more attempts of deriving are futile for change and integration from 0 (reincarnation?) leads to too much uncertainty for returning to where one was. The integral of d(x)=0 is just “C,” and who knows which world will be picked for you: 2? 9001? π?

Once a full connection is made though, the next thing to do is extrapolation. What exactly is limbo? Think about the delicate nature of how the process of transferring between levels uses the discrete package of a difference of “1” between dream levels. When the framework of meta-dream mechanics is sabotaged or vandalized, perhaps the error of inputting a fractional difference could occur. What does this imply? One could end in a world with a fractional exponent of x, and unless or until another such mistake occurs of coincidental fractional fix, one must wander this stack of functions related by integrals and derivatives. This is like limbo. Why? There is only mindless wandering here. Unlike the case with dream worlds of positive integral degree, looking down the dream stack reveals a well with no bottom. Differentiating again and again leads to worlds that are indeed one exponential degree apart, but there is no stopping at zero, no getting rid of the x in representation, and an infinite descent down the negative degrees. There isn’t really any dream level more dreamy than another now, as they are all infinitely far from any world that can be distinguished as reality. All the worlds are equally dreamy: it is dreams all the way up and dreams all the way down.

But what if this, instead of representing limbo, represented an alternative structure for a universe over the parameter of dreams? It shows that perhaps there isn’t really any reality (which I could talk about more but would be tangential here), thus in some way nulling the meaningfulness of the distinction of a world as “dreaming.” That would be a pretty weird alternative world, but one worth thinking about.

Yet we could extrapolate further. Consider each function as possible of representing a world in reference to dreaminess. What happens in an f(x)=sin x world? In that interesting hypothetical world, entering a dream consecutively four times leads you back to where one was—a cyclical dream stack. That is a possibility that deserves some thinking as well. What about an f(x)=e^x world? A dreamless world? What does integrating in f’(x)=1/x to f(x)=ln|x| represent? What would doing these things in polar coordinates represent?

And most importantly, what do all of these imply about the juices that mold life and its meanings and properties—the elusive top that has been spinning in the minds of humans for millennia?