Over the past four thousand years, four waves of mathematical innovation have swept the world, leading to rapid advances and significant changes in society:

**The invention of written number (Fertile Crescent, 3000 BC)**. This allowed civilization to*exist*, because if you want to live with more than your extended family, record keeping is essential…and that means keeping track of numerical amounts.**The invention of geometry (Greece, 300 BC)**. Yes, geometry existed before then; what I’m using is the date of Euclid’s*Elements*, which is the oldest surviving deductive geometry. The idea that you could, from a few simple principles, deduce an entire logical structure has a profound impact on society. How important? Consider a rather famous line: “We hold these truths to be self-evident…”*The Declaration of Independence reads like a mathematical theorem, proving the necessity of revolution from some simple axioms*.**The invention of algebra (Iraq, 900)**. The problem “A number and its seventh make 19; find the number” appears in a 4000-year-old manuscript from ancient Egypt, so finding unknown quantities has a very long history. What algebra adds is an important viewpoint: Any of the infinite variety of possible problems can be*transformed*into one of a small number of types. Thus, “A farmer has 800 feet of fence and wants to enclose the largest area possible” and “Find a number so the sum of the number and its reciprocal is 8” and “The sum of a number is 12 and its product is 20” can all be reduced to and solved using the quadratic formula .**The invention of calculus (Europe, 1600)**. Algebra is the mathematics of*what is*. Calculus is the mathematics of*how things change*. Calculus makes physics possible, and from physics comes chemistry and engineering.**The invention of statistics (Europe, 1900).**Both algebra and calculus deal with single objects: a bridge, a number, a moving planet. But the universe consists of many similar objects: the human population; the planetary climate; the trash generated by a city. Statistics aggregates the data on the individual in a way that can be used to describe a population…then uses the information on a population to predict information about an individual. Everything in modern society, from the pain relievers you use to the road you travel to work, incorporates such a statistical analysis.

Many people, myself included, believe we are on the verge of a sixth wave. That sixth wave will have the transformative power of calculus and statistics, and fundamentally reshape society.

The sixth wave is based around *discrete mathematics*. That’s not mathematics you whisper in dark corners. Rather, it’s the mathematics of things that can be *counted* as opposed to *measured*. For example, length is continuous: a length can have any value, and no one looks at you strangely if you say “I traveled 1.38924 miles today…” (You might get *some* strange looks, but it’s because you specified the distance so precisely and not because of the distance itself) But if you continued “…and met 2.35 people,” you *would* get strange looks, because the number of people you meet is a counted number: it’s a discrete quantity.

How important is discrete mathematics? If calculus is the basis for physics and engineering, then **linear algebra** is the basis for discrete mathematics. But a first-year calculus problem would have a hard time solving even a simple question in statics (the physics of structures). In contrast, * Google’s search algorithm is based on mathematics learned in the first half of a standard college linear algebra course*.

I’ll talk more about this later. But if you’re interested in learning some linear algebra, the video lectures for the course I teach are available on YouTube.