Every model of a complex situation is an analogy, highlighting one set of features while suppressing others with the goal of gaining understanding. Mathematicians have developed a precise language, or category theory, for systematizing analogies. For example, because volume is related to containment, it is often more significant than length: larger balls cannot be put in smaller ones, while the small intestine is longer than the body that contains it. This perspective focuses on extracting meaning from relations among measurements. It has become a powerful technical tool in the application of mathematics across many scientific areas. The panel will present examples from networks, evolution, cancer research, and dynamical systems where the exploitation of analogy has led to important new discoveries and insights. Whereas traditional approaches to modeling nature have relied on classical techniques of continuous mathematics and linear approximation, the modern scientist is faced with massive amounts of discrete data, usually obtained from highly nonlinear dynamics where the “rules of nature” are opaque at best. This session will demonstrate how mathematical notions of persistence and naturality can be used to understand statistical properties, patterns, and features hidden in the massive experimental data being collected at an ever-increasing rate across science. The speakers have been selected for their ability to communicate to audiences with diverse backgrounds and scientific interests.