A remarkable property of the mathematical world is that sometimes there are unexpected connections within different areas of the world of mathematics. Often, a mathematician will study one area of mathematics, realize there are direct correlations with the properties of another area of math, and connect the two to find new properties about each area.
You may have seen this firsthand if you ever studied algebra or geometry.
Do you remember the Cartesian plane, the x- and y-axis graph that allows you to draw a picture representation of an equation?
If so, this – which may seem simple to us – was actual a crucial innovation that allowed the tech revolutions of the Industrial Age to occur. It quite literally played a central role in getting us into the Industrial Age.
Before Descartes, the worlds of algebra and geometry were considered distinct. Algebra – to describe it in a simplified manner – studies the properties of relationships between numbers – and geometry studies the properties of physical shapes.
What Descartes did was discover a natural connection between these two worlds. He found that the world of geometry, the world of shapes, the world of pictures, provides a natural way to express the world of algebra. It allows us to actually visualize relationships between numbers, by plotting them as points on a graph.
So we can use our understanding of geometry – of shapes – to understand the behavior of abstract equations, generalizations of numbers.
The basic connection between the two is the number line – which pictures real numbers, like 1, -1/2, and pi – on a line. The line has a natural direction to it: left and right. In the same way, the real numbers have a natural direction to them: smaller and bigger, or negative and positive.
Unlike the real numbers, though, the number line gives us a concrete visualization, an image to imagine in our minds. And this image serves as a perfect way to picture the real numbers.
This connection seems so simple to us now, but at the time of Descartes, it was revolutionary. Descartes’ connection also lets us picture pairs of real numbers, as the (x,y) axis, in 2 dimensions, and triplets of real numbers, as the (x,y,z) axis, in 3 dimensions. All of these seem like common sense to us now, but at the time of Descartes, were unexpected and revolutionary.
They allowed Newton to formulate his laws of physics, which in turn gave us the understanding necessary to enter the Industrial Age.