The University of Arizona

Puzzles, Proofs and Patterns

Mathematics has been called “The Queen of the Sciences”. It describes how things are structured and ordered, growing from our very earliest impulses, as humans, to count and measure the world around us. Not just a science, mathematics underlies other sciences: It is the basis of our understanding of physics, astronomy, chemistry, engineering, materials sciences, and many other disciplines. It allows us to describe patterns in precise and elegant ways. Mathematics is a body of laws, embodied in axioms, theorems, and proofs. Mathematics gives order to philosophy. It is the language of technology, and of the universe itself.

As you will see in the exhibit that follows, mathematics takes many shapes. It begins with that most concrete of matters: How many of any given thing—say, measures of grain—do we hold in our hands? This is arithmetic, from the Greek word for “number,” the first mathematical subject we learn in school. We remain in the realm of the visible for our earliest lessons in geometry, from the Greek word meaning “measure of the earth”: We learn how to measure length, width, and depth, the three dimensions of our physical world, as well as other wildly more difficult things like angle, volume, and density
But there is a fourth dimension, time, and many permutations of the things we must measure, from speed of curves to curvature of loops and knots to fractal dimensions, and charge and spin of subatomic particles. As our knowledge of the world has grown, so has our mathematics: with arithmetic and geometry, we have algebra, calculus, topology, probability, set theory, statistics, and many, many other branches and sub-branches of the mathematical sciences. In the last 500 years, mathematicians have expanded a basic knowledge that grew only slowly over thousand of years before it, taking us from the visible into the invisible. In the last 100 years or so, our knowledge has leaped with the contributions of mathematical scientists who, as Isaac Newton once said, have been able to see faraway vistas only because they have stood on the shoulders of giants.

We stand on their shoulders, too, and many discoveries await us still. Because we are human, we count and measure. Because we have human minds, we think about mathematical questions all the time, sometimes without really realizing it. We are all mathematicians. Join us in exploring the endlessly fascinating world of numbers, measures, and dimensions—of puzzles and proofs, postulates and paradoxes.