The moment a dream ends, and the dreamer awakens, they realize they are awake. Like dreaming, awakening is also a feeling, yet it is perceived as real. The primary characteristic that makes the waking state feel real is its continuity (not in the strict mathematical sense; unless stated otherwise, the broader sense is to be understood) with preceding waking states.
Upon awakening from a dream, I recognize my surroundings and recall that this is the bed in which I slept last night. The feeling of the current waking state seamlessly connects with the preceding waking state. After a few moments, I may start looking for my keys. Memory of a prior waking state reminds me that they are inside the drawer. I follow this suggestion of memory and find them there. This illustrates continuity, consistency, and coherence among waking states. The current waking state is a natural succession of the preceding waking states.
This characteristic is generally absent among dreaming states. Dreams are typically disconnected from one another. A dream begins abruptly, while a waking state always has a definable starting point. Even when a dream incorporates elements from the preceding waking state, it lacks full continuity.
For instance, suppose I am waiting for a friend. While waiting, I nap and dream that my friend arrives, and we share memories from the past. In this case, the dream exhibits a partial succession of events from the prior waking state. However, it remains a dream because the continuity of succession is incomplete. Upon waking, my friend would not recognize or verify the conversation we had in the dream.
Here, an interesting analogy can be drawn between the continuity of waking states and the mathematical concept of removable discontinuity (in its strict sense). In mathematics, a removable discontinuity occurs at an
x-value in a function where the two one-sided limits exist, are finite, and equal, but the function is not defined at that point.
For example, consider the function:
f(x)=(x^2-4)/(x-2)
This function is undefined at
x=2, but both the left-hand limit and the right-hand limit exist and are equal to 4. Thus, this is an example of removable discontinuity.
Similarly, a dreaming state is like a point of discontinuity where the function (representing waking experience) is not defined, as the waking experience is not accessible to the dreamer. The preceding and succeeding waking states resemble the left-hand and right-hand limits, respectively. Both limits approach the same event, ensuring continuity.
It is worth noting that in mathematics, a point is said to be discontinuous, while in the "function" of waking states, the discontinuity spans a duration. This analogy provides an intriguing perspective on the nature of waking states and their relation to dreams.