1. As part of my philosophy series, I did a post on the philosophy of science. That post suggested that science was a language we used to describe and predict how the world works and would behave next time, but that it isn't actually language of how the world actually is. Our formulas, our language describes things that are real, but our language is like finely tuned myths that describe mysteries.
Science also proceeds by way of paradigms that, thus far, change over time. A paradigm is a way of looking at a particular area of knowledge, a particular way of thinking about a certain set of data, a particular set of glasses. So prior to Copernicus, the predominant astronomical paradigm saw the earth as the center of the universe, with sun, moon, and planets circling around it. Prior to Einstein, it was generally thought that space and time had fixed measures.
So paradigm shifts take place when the anomalies of "normal science" prompt new theories that, over time, come to be adopted as the new normal science.
2. The scientific method, therefore, is not exactly about reality itself. It is not exactly about what is absolutely true. It is about either confirming the usefulness of existing paradigms, about refining them, or perhaps about prompting a paradigm shift that will more precisely describe and predict the world.
We look at the data, we form hypotheses, we test them. When our hypotheses repeatedly seem to explain the data and correctly predict the results we will get the next time we test, we eventually call them theories. From the 1600s to the 1800s, discoveries were sometimes called "laws"--Newton's three laws of motion, the law of universal gravitation. We haven't used language of laws for a long time now.
3. So what is math? Some math seems to mirror reality in a much closer way than any scientific theory. I have ten fingers, so base 10 makes sense to me. But is base 10 really the most fundamental base of reality? Other numbers seem more intrinsic to the universe, like the number known as "e," which seems to pop up all over the place. 
Repeatedly, either mathematicians have realized possibilities that were later "discovered," or math was invented to explain what was already "discovered." The square root of negative 1, "i" is a case in point. Recognized in the 1700s, this curiosity would become essential in the quantum mechanics of the twentieth century. Carl Gauss and others explored the possibility of a non-Euclidean geometry in the early 1800s, but Einstein would apply it to general relativity in 1915.
So if the theories of science are far more precise myths to express reality than the stories of the arts and humanities, then numbers are the characters in those myths.
Of course math is much more than quantities. It is more than geometries. It is also a number of tools to help get at quantities and geometries.
5. Over the next twenty weeks, I hope to survey the main fields of math and science. I'm not exactly doing "what a college student should know," but this may not be too far off that idea.
- The Basic Types of Numbers
- The Atom and Quantum Physics
- The Periodic Table
- Molecules and Ions
- Chemical Reactions
- The Basic Tools of Algebra
- Basic Geometry and Trigonometry
- The Physics of Motion
- Basic Calculus
- The Forces of Motion
- Genetics and Evolution
- Cells and Microbiology
- Basic Probability and Statistics