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.