Mathematical, a Rational World
In this short note, I want to make one point: all mathematical models exist only in our rational world and are abstractions of the real world. No mathematical model can perfectly replicate reality, but their advantage lies in helping us focus on the core issues when solving specific problems in real life by discarding irrelevant details. As the statistician George Box famously said, “All models are wrong, but some are useful.” This understanding is very beneficial; if you grasp it, it will help you elevate your thinking and eliminate confusion.
Let’s start our discussion with a familiar mathematical concept. Circle is a fundamental concept in geometry, but where did it come from? The process of delineating the circle likely unfolds as follows: within the tangible world, a multitude of objects exhibit analogous characteristics in their forms. Take, for instance, the full moon, ripples in a lake, a wheel, and so forth. All of these things have a “full” shape, or their shapes appear to be very consistent from all directions. Consequently, individuals distilled this observation into a succinct principle, thereby defining the circle as follows:
Circle is a collection of points that are equal distances to a fixed point.
This definition aptly encompasses the majority of objects with a circular shape. Nevertheless, it’s worth noting that once mathematicians have established the circle’s definition, the notion of a perfect circle, I mean mathematical circle, no longer finds a direct counterpart in the tangible (real) world. This is due to the challenge of locating a precise circle-like form within the physical realm. Even if one were to stumble upon an exceptionally round iron ring, ensuring that the distances from the ring’s center remain consistently uniform presents a challenge. In essence, the concept of a perfect circle, as defined in mathematics, resides solely within the confines of our abstract realm of reason, rather than within the tangible world.
While this might evoke a sense of melancholy, it underscores a fundamental connection between mathematical models and the tangible world. Mathematical models are, in essence, abstractions that represent real-world phenomena. Conversely, the real world can be approximated and understood through the lens of mathematical models.
Probability theory and real-world data illustrate this relationship well. A probability model is essentially a mathematical construct that exists solely in our rational world, while data represent observations of the real world, vibrant and alive. The bridge between the two is precisely statistics. Clearly, statistics cannot exist without data, but it also relies on probability models to “tame” data, making it useful to us. Of course, data will never be fully tamed, just as if we were trying to grasp the mind of a higher power. As the saying goes, all models are wrong, but some are truly useful.
