Discrete and Continuous

Another aspect of the two paintings posted in Why Art is that they illustrate the difference between discrete and continuous, at least in terms of color.

In Keith Haring’s painting, every point is either yellow, red, green, or purple: four discrete possibilities. In Wayne Thiebaud’s painting, there are many continuous transitions from one color to another. Notably, for me, a field that changes from gray to brown to indigo. If you look at the body of work of these artists, you would see that they each stay pretty much in one color camp. Haring in the discrete camp, Thiebaud in the continuous.

In mathematics the discrete and continuous camps exist too. People typically pick one or the other. What camp they choose is an individual matter probably much as it was for Haring and Thiebaud. Is one camp “easier” or “simpler” than the other? Not necessarily. It’s tempting to think that the discrete camp is easier – all whole numbers, no messy extended decimals. No Zeno’s Paradox.

However, the continuum can be comforting because it is filled with solutions. We know, for example, that two non-parallel lines in the plane definitely intersect, which means that the system of equations defining two such lines always has a solution. Ask a similar question in the discrete world – does a given line intersect any lattice points in the plane? And that question is harder.

(By lattice points in the plane, I mean points with whole-number coordinates in the X-Y plane, such as (2,1) and (3, 56). These points form a regular grid of dots.)

Then there is the famous Diophantine equation (love that name, Diophantine, makes me think of an equation in a flowing white dress with a long trunk):


In the continuum are many solutions. But if you ask about discrete solutions, you get Fermat’s Last Theorem, which stumped mathematicians for centuries.

On the other hand, the continuum is a very mysterious place. That’ll have to be another post.