Mathematical Model at Five Levels
Child: A mathematical model is like a recipe in cooking. You know how in a recipe we use different ingredients in different amounts to make something delicious? In a mathematical model, we use numbers and math operations instead of ingredients to describe something. For example, if you are saving money to buy a toy, we can use a mathematical model to figure out how much you need to save each day!
Teenager: Imagine you are playing a video game, and you want to predict how many points you’ll score in the next hour based on your current gaming skills. A mathematical model would be like a set of rules that take into account your current score, the difficulty of the game, and how fast you are improving to predict your future score. It uses equations to represent a situation and can be used to predict outcomes or understand relationships.
Undergrad majoring in the same subject: Mathematical models are representations of systems or processes using mathematical concepts and language. They can describe a variety of phenomena, from the growth of a population to the motion of a pendulum. By formulating a system mathematically, we can use analytical or numerical methods to solve the equations and predict future behavior or understand underlying mechanisms.
Grad student: A mathematical model provides a quantitative framework for interpreting and predicting behavior of a system. They are established using principles from fields such as calculus, statistics, or linear algebra and are often designed to simplify complex systems. These models can be deterministic or probabilistic, and they can represent static or dynamic systems. The validity of these models often depends on how well they can be calibrated to empirical data and their assumptions.
Colleague (Fellow Researcher/Engineer): Mathematical models are integral tools in our scientific toolbox, aiding us in the exploration, interpretation, and prediction of complex systems. They can range from simple linear regressions to complex nonlinear dynamic systems. Despite their abstraction from reality, these models are critical for hypothesis testing, inference, and simulation in our research. Nevertheless, the assumptions and constraints of our models must be critically evaluated, and the models themselves should be iteratively refined to ensure their robustness and accuracy in the face of empirical evidence.
Richard Feynman Explanation
Alright, so let’s think about a mathematical model as if it’s a map. Now, when you have a map of a city, does it show you every single building, tree, and person in that city? No, of course not! But it does show you the major roads, landmarks, and districts.
Why is that? Well, if a map tried to show you every single little detail, it would be as big as the city itself! That wouldn’t be very useful, would it? Instead, a map simplifies the city down to the most important parts, so you can understand the city and navigate through it more easily.
A mathematical model does something similar, but for a piece of the world or a system we’re interested in understanding or predicting. Let’s say we’re interested in how a disease spreads through a population. Now, we could try to track every single person, who they interact with, their health, their behavior, etc. But that would be incredibly complex and impractical.
Instead, we use a mathematical model. We simplify the problem down to the key parts: the number of people who are susceptible to the disease, the number who are infected, and the number who have recovered. We use equations to represent how people move between these categories over time.
Now, just like the map, our model isn’t a perfect representation of reality. It doesn’t capture every single detail. But it does give us a way to understand the key dynamics of the disease and make predictions about how it will spread. And that’s incredibly useful.
So, a mathematical model is like a map. It’s a simplified representation of a system that helps us understand it and navigate our way to solutions.