The mathematics of countable things grew up alongside the digital computer
Calculus deals in smooth, unbroken change. Discrete mathematics deals in separate, countable things: whole numbers, networks, logical statements. It barely existed as a university course before the 1980s, when it arrived as a haphazard support class for computer science students. Today some universities require it of every mathematics major.
Nobody has pinned down an exact definition. The usual rule of thumb is that discrete mathematics handles sets you can count, whether finite or matched one-for-one with the natural numbers, and leaves real numbers, calculus and Euclidean geometry to continuous mathematics. When the focus is strictly on finite sets, especially for business applications, the label finite mathematics is sometimes used.
Research surged in the second half of the twentieth century largely because digital computers work in distinct steps and store information in separate bits. The traffic runs both ways. Discrete ideas describe algorithms, programming languages, cryptography and software, while computers make it practical to apply those ideas to real problems. Two professional bodies, the Mathematical Association of America and the Association for Computing Machinery, helped reshape the early course into one aimed at building mathematical maturity in first-year students, and high school versions now serve as a kind of alternative to precalculus.
Much of everyday technology rests on discrete structures. Boolean algebra drives logic gates, relational algebra underlies databases, and finite versions of groups and fields power error-correcting codes. Graphs, dots joined by lines, model communication networks, social ties and the flow of a computation. Even proofs are discrete objects, finite trees in which each step merges premises into a single conclusion. Peirce's law, a formula about implication, holds in classical logic and can be checked with a truth table, yet fails in intuitionistic logic.
The borders are porous. Many continuous theories have discrete twins, with difference equations standing in for differential equations by comparing neighbouring terms instead of taking derivatives, and time scale calculus merges the two. Tiling the plane is a long-standing problem in discrete geometry. Outstanding papers in the field can win the Fulkerson Prize.
Source: Discrete mathematics
