The latent simplicity posits that our limited perception results in complicated models of a simple underlying reality. Many phenomena we encounter are simplified manifestations of general rules presented in specific contexts. By recognizing this principle and striving for the simplest possible explanation, we can avoid overfitting and develop more accurate models of the world around us.
Example 1: The Two-Dimensional Plane Dwellers
Imagine beings living in a two-dimensional plane, unaware of the three-dimensional world. When a three-dimensional sphere intersects their plane, they perceive it as a two-dimensional circle. As the sphere moves through the plane, they observe the circle appearing to grow and shrink, leading them to create complex models to explain the phenomenon. In reality, the circle’s size changes due to the motion of the sphere in the third dimension, which the two-dimensional beings cannot perceive.
Example 2: Plato’s Allegory of the Cave
Plato’s Allegory of the Cave is another illustration. In the allegory, a group of people have been chained inside a cave since birth, facing a blank wall. They cannot see the actual objects outside the cave, nor can they turn their heads to view the world behind them. A fire behind the prisoners casts shadows of various objects, such as animals and people, onto the wall, and these shadows become their only perception of reality.
The prisoners are unaware of the true nature of the objects and the world outside the cave. They observe the shadows and their movements, noticing patterns and creating meanings for them. For example, they may see a shadow of a tree swaying in the wind and believe that the shadow itself possesses some inherent power causing the movement, without realizing it’s merely a representation of a real tree outside the cave. They develop complex theories and beliefs to explain the behavior of the shadows, creating an elaborate understanding of a reality that is, in fact, much simpler than they perceive.
Example 3: Euler’s Formula and its Special Cases
Euler’s Formula can be expressed as follows:
$$e^{ix} = \cos(x) + i \sin(x)$$This formula has several interesting special cases that showcase the Latent Simplicity in the context of mathematical relationships.
Euler’s Formula simplifies trigonometric identities. For instance, the addition formulas for sine and cosine can be derived from Euler’s Formula:
$$\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)$$$$\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)$$One can start assigning intuitions to all derived equations out of Euler’s formula, but there is one equation that drives them all.
On Overfitting
Occam’s Razor suggests that out of all explanations, the simplest one should have the highest prior probability, i.e., it is the most likely one. In the context of the two-dimensional plane dwellers, if they were guided by Occam’s Razor, they might be more inclined to search for a simpler explanation for the circle’s behavior, rather than relying on elaborate theories about mysterious forces or expanding materials.