QUICK FACTS
Created Jan 0001
Status Verified Sarcastic
Type Existential Dread
Keywords
finite mathematics, mathematics, history, index, number theory, geometry, algebra, calculus, analysis

Discrete Mathematics

“Alright, let's dissect this. You want me to rewrite a Wikipedia article on discrete mathematics, but not just rehash it. You want it imbued with my... *style*....”

Contents
  • 1. Overview
  • 2. Etymology
  • 3. Cultural Impact

Alright, let’s dissect this. You want me to rewrite a Wikipedia article on discrete mathematics, but not just rehash it. You want it imbued with my… style. A certain perspective. Fine. Don’t expect sunshine and rainbows; you’ll get the truth, sharp and unvarnished. And I’m not here to hold your hand through it.

Study of Discrete Mathematical Structures

For the mathematics journal, which is, frankly, more exciting than most of this, see Discrete Mathematics (journal) .

  • “Finite math” is a term that gets tossed around. It redirects here, as if anything about dealing with finite sets is simple. For the syllabus, if you must, see Finite mathematics .
  • This is part of a larger, infinitely more complex beast: Mathematics .
    • Its History is a long, often brutal, march of ideas.
    • Its Index is a tangled web of concepts.

Areas

  • Number theory : The study of integers. Simple, yet maddeningly complex.
  • Geometry : Shapes and spaces. Sometimes they behave, sometimes they don’t.
  • Algebra : The manipulation of symbols. Often a way to avoid dealing with reality directly.
  • Calculus and Analysis : The study of change. The antithesis of discrete.
  • Discrete mathematics: What we’re wading through.
  • Logic : The framework for thinking. Or, more often, for proving why your thinking is flawed.
  • Set theory : Collections of things. Seems basic, but the infinities can break you.
  • Probability : The quantification of chance. An attempt to impose order on chaos.
  • Statistics and Decision theory : Making sense of data. Or pretending to.

Relationship with Sciences

  • Physics : Where the universe tries to make sense, and often fails.
  • Chemistry : The messy interactions of matter.
  • Geosciences : The earth’s indifferent churn.
  • Computation : The realm of the discrete, where machines try to mimic thought.
  • Biology : Life. Complicated.
  • Linguistics : The architecture of language.
  • Economics : The flawed models of human greed.
  • Philosophy : Questioning everything, especially the foundations.
  • Education : Trying to pass on this misery.

Mathematics Portal • v • t • e

Look at these graphs . They’re presented as if they’re some kind of revelation. They’re studied for their mathematical properties , sure. And because they’re useful for modeling the world, which is a noble lie. And because they help build algorithms . The whole edifice is built on these discrete structures, these tiny, isolated points and lines, like fragmented memories.

Discrete mathematics is the study of these “discrete” structures. Think of them like discrete variables – they have a one-to-one correspondence with natural numbers . They’re not the gooey, continuous mess of real numbers , calculus , or the smooth lies of Euclidean geometry . We’re talking about integers , graphs , and the rigid pronouncements of logic . [1] [2] [3] These discrete objects can be counted, or at least cataloged. More precisely, it’s the branch that grapples with countable sets – finite ones, or those with the same frustrating scope as the natural numbers. [4] Don’t ask for a perfect definition, though. It’s a slippery thing, this term. [5] 

The objects we dissect can be finite, or maddeningly infinite. The term “finite mathematics” is sometimes used for the parts that deal with finite sets, often for those dry applications in business. As if business is ever simple.

The real push in discrete mathematics came in the latter half of the 20th century, largely because of the advent of digital computers . These machines operate in discrete steps, storing data in discrete bits. It’s a marriage of convenience, really. The concepts and notations of discrete math are essential for understanding computer science – the way computer algorithms are built, the structure of programming languages , the dark arts of cryptography , the sterile certainty of automated theorem proving , and the endless process of software development . Conversely, computers are how we force these abstract ideas into the real world.

Even though the focus is on discrete objects, don’t be fooled. We still drag in analytic methods from the “continuous” world when it suits us.

Universities started shoving this into curricula in the 1980s, mostly as a support course for computer science. It was a bit of a hodgepodge then. Now, thanks to efforts by groups like ACM and MAA , it’s supposed to build mathematical maturity in first-year students. Some places even require it for math majors. [6] [7] High schools are getting in on it too. [8] At that level, it’s often seen as prep work, like precalculus . [9] 

There’s a Fulkerson Prize for outstanding work in this field. It’s something, I suppose.

Topics

Theoretical Computer Science

Complexity studies how long algorithms take. Like this sorting routine . It’s a race against time, or against futility. Computational geometry takes algorithms and applies them to shapes. Because the real world isn’t just abstract lines.

Theoretical computer science is where discrete mathematics gets its hands dirty with computing. It leans heavily on graph theory and mathematical logic . Algorithms and data structures are central. Computability asks what can be computed, period. Complexity measures the cost – time, space, sanity. Automata theory and formal language theory are close cousins. Petri nets and process algebras try to model systems. We even analyze VLSI circuits with these tools. Computational geometry applies algorithms to shapes, and computer image analysis to pixels. It even dabbles in continuous computational topics, just to keep things interesting.

Information Theory

The ASCII codes for “Wikipedia” in binary . A way to represent information, and a foundation for information theory and algorithms .

Information theory quantifies information. Coding theory designs methods for reliable data transmission and storage. It also has its continuous side, with analog signals , analog coding , and analog encryption .

Logic

Logic is the study of valid reasoning. And the proof that your reasoning is, in fact, invalid. It examines consistency , soundness , and completeness . Take Peirce’s law – a theorem in most logics, verifiable with a truth table . The study of mathematical proof is crucial, leading to automated theorem proving and formal verification of software.

Logical formulas are discrete. Proofs are discrete too, forming finite trees  [10] or directed acyclic graphs . [11] [12] Each inference step combines premises to yield a conclusion. Truth values are usually binary – true or false – but fuzzy logic exists. And then there are the infinite proof trees of infinitary logic . [13] 

Set Theory

Set theory studies sets – collections of things. Like {blue, white, red}, or the infinite set of prime numbers . Partially ordered sets and other relations pop up everywhere.

In discrete math, we focus on countable sets , including finite sets . The real fireworks of set theory, dealing with different kinds of infinite sets , initiated by Georg Cantor , are generally outside the scope of discrete mathematics. Descriptive set theory , for instance, relies heavily on continuous mathematics.

Combinatorics

Combinatorics is about how discrete structures can be combined or arranged. Enumerative combinatorics counts things. The twelvefold way is a unified framework for counting permutations , combinations , and partitions . Analytic combinatorics uses complex analysis and probability theory to count. It aims for asymptotic formulae , unlike enumerative combinatorics with its explicit formulas and generating functions . Topological combinatorics brings topology and algebraic topology /combinatorial topology into the mix. Design theory studies combinatorial designs – collections of subsets with specific intersection properties. Partition theory deals with integer partitions , and is closely linked to q-series , special functions , and orthogonal polynomials . Once part of number theory and analysis , it’s now its own thing. Order theory studies partially ordered sets , both finite and infinite.

Graph Theory

Graph theory has ties to group theory . This truncated tetrahedron graph is connected to the alternating group A₄.

Graph theory, the study of graphs and networks , is often bundled with combinatorics, but it’s grown too substantial to be just a subfield. [14] Graphs are fundamental to discrete math. They model everything from social networks to communication systems. In computer science, they represent networks, data organization, and computation flow. In pure math, they connect to geometry and topology , like knot theory . Algebraic graph theory links with group theory, and topological graph theory with topology. There are even continuous graphs , but most research remains firmly in the discrete camp.

Number Theory

The Ulam spiral shows prime numbers in black, hinting at patterns in their distribution .

Number theory investigates the properties of numbers, especially integers . It’s crucial for cryptography and [cryptanalysis], particularly modular arithmetic , diophantine equations , congruences, primes, and primality testing . Other discrete aspects include the geometry of numbers . Analytic number theory borrows from continuous methods. Topics like transcendental numbers , diophantine approximation , p-adic analysis , and function fields venture beyond the purely discrete.

Algebraic Structures

Algebraic structures can be discrete or continuous. Discrete examples include: Boolean algebra for logic gates and programming; relational algebra for databases ; discrete and finite groups , rings , and fields vital for algebraic coding theory ; discrete semigroups and monoids in formal languages .

Discrete Analogues of Continuous Mathematics

Many concepts from continuous mathematics have discrete counterparts: discrete calculus , discrete Fourier transforms , discrete geometry , discrete logarithms , discrete differential geometry , discrete exterior calculus , discrete Morse theory , discrete optimization , discrete probability theory , discrete probability distribution , difference equations , discrete dynamical systems , and discrete vector measures .

Calculus of Finite Differences, Discrete Analysis, and Discrete Calculus

In discrete calculus and the calculus of finite differences , a function on integers is a sequence . These can be finite, from data, or infinite, from a discrete dynamical system . They can be defined explicitly, by a formula, or implicitly by a recurrence relation or difference equation . Difference equations mimic differential equations by using differences instead of derivatives. They can approximate differential equations or stand on their own. Many questions about differential equations have counterparts for difference equations. Just as integral transforms study continuous functions, discrete transforms handle discrete functions. Beyond discrete metric spaces , there are discrete topological spaces , finite metric spaces , and finite topological spaces .

The time scale calculus unifies the theory of difference equations and differential equations , useful for modeling mixed discrete and continuous data. Hybrid dynamical systems offer another approach.

Discrete Geometry

Discrete geometry and combinatorial geometry examine the combinatorial properties of discrete geometric objects. Tiling of the plane is a classic problem.

In algebraic geometry , curves can be extended to discrete settings using the spectra of polynomial rings over finite fields as models of affine spaces . Subvarieties then become the curves. These spaces have finite points, but the curves are analogues of their continuous counterparts. For instance, a point like $V(x-c) \subset \operatorname{Spec} K[x] = \mathbb{A}^1$ can be seen as $\operatorname{Spec} K[x]/(x-c) \cong \operatorname{Spec} K$ (a point) or as the spectrum $\operatorname{Spec} K[x]_{(x-c)}$ of the local ring at (x-c) , which is a point with its neighborhood. Algebraic varieties have Zariski tangent spaces , allowing calculus-like features.

Discrete Modelling

In applied mathematics , discrete modelling is the discrete version of continuous modelling . Discrete formulas are fit to data , often using recurrence relations . Discretization is the process of transforming continuous models into discrete ones, usually for simpler calculation through approximation. Numerical analysis is a prime example.

Challenges

The history of discrete mathematics is marked by difficult problems that have shaped the field. In graph theory , the pursuit of proving the four color theorem – that any map can be colored with four colors such that no adjacent regions share the same color – spurred much research. Kenneth Appel and Wolfgang Haken finally proved it in 1976, with significant computer assistance. [15] 

In logic , David Hilbert ’s second problem from 1900 asked for a proof of the consistency of the axioms of arithmetic . Gödel’s second incompleteness theorem , proven in 1931, demonstrated this was impossible within arithmetic itself. Hilbert’s tenth problem, determining if a polynomial Diophantine equation has integer solutions, was resolved by Yuri Matiyasevich in 1970, showing it could not be done universally. Matiyasevich’s theorem  

The need to break German codes during World War II accelerated progress in cryptography and theoretical computer science . The first programmable digital electronic computer , developed at Bletchley Park under the guidance of Alan Turing and his work on On Computable Numbers , was a direct result. [16] The Cold War ensured cryptography remained vital, leading to breakthroughs like public-key cryptography . The telecommunications industry also drove advances in graph theory and information theory . The demand for formal verification in software development for safety-critical systems fueled progress in automated theorem proving .

Computational geometry became integral to computer graphics in modern video games and computer-aided design tools.

Fields like theoretical computer science, graph theory, and combinatorics are essential for tackling the complex bioinformatics challenges of understanding the tree of life . [17] 

Currently, one of the most notorious open problems in theoretical computer science is the P = NP problem , concerning the relationship between complexity classes P and NP . The Clay Mathematics Institute has offered a $1 million USD prize for its solution, alongside prizes for six other mathematical challenges. [18] 

See also