# A Brief History of Mathematics: 1950–2000

We explore the field of mathematics from 1950-2000.

The history of 20th Century mathematics can be cleanly divided between the pre-war years, and the half-century between World War II and the Internet Age (which begran roughly around the turn of the millennium). Here we will look at the major developments in the field of Mathematics from 1950-2000. Numerous major advances in math occurred in the early part of the 20th century, such as the emergence of a field called mathematical logic. Indeed, this field would be a leading catalyst in the advent of digital computers, as well as some surprising proofs of limitations, including Austrian mathematical logician Kurt Gödel ’s publication of his famous Incompleteness Theorems in 1931.

Despite these groundbreaking developments, 1950 was a true pivot point for the maturation of mathematics into the modern era. Not only did Claude Shannon ’s information theory and Alan Turing ’s theory of computation find permanent homes in the related discipline of Computer Science (and continued as pure research in Mathematics), but major new disciplines like fractal geometry, chaos theory, and the mathematics of shapes emerged, fundamentally changing our understanding of the universe. Mathematics since 1950 has hardly stalled. It has grown into a multifaceted and exciting discipline. Let’s take a look at some of the major advances.

**The field of Complex Dynamics** was developed early in the 20th century but was largely ignored until the 1970s, when French mathematician Benoit Mandelbrot developed his theory of fractals, which are recursive structures defined by quadratic polynomial equations (having degree 2). The so-called Mandelbrot set generates shapes that have infinite complexity no matter the granularity of observation. These are fractals. The discovery of fractals in nature has led to paradoxes, such as the well-known coastline paradox, where it was found that a coastline (as found in nature) does not have a well-defined length. Because of fractal effects, choosing the scale at which to measure the coast will result in longer or shorter measurements. Fractal geometry has proven useful when investigating nature generally, and has led to a greater understanding of the non-Euclidean world we live in.

**Chaos theory**, a field in mathematics closely connected to fractals, also emerged in the latter part of the 20th century. The initial seminal discovery of chaotic systems resulted from a now-famous accident, when Edward Norton Lorenz was working on weather prediction and accidentally inputted rounded numbers into his computer program for predicting the behavior of weather systems. The rounded input (to three decimals rather than six) resulted in radically different predictions. The effect became known as “sensitive dependence on initial conditions,” where changing the starting points of certain calculations can radically affect outcomes. Lorenz’s discovery led to Chaos Theory in Mathematics, and many people now have heard of its catchphrase, the “butterfly effect.” Chaotic systems cannot be predicted using conventional mathematics like calculus. Another result of chaos theory states that apparently random systems may have underlying order.

**Games theory**, too, received serious attention in the years after 1950, notably with the work of John Forbes Nash Jr. , a mathematician made memorable by the major motion picture A Beautiful Mind. Games theory has been adopted in diverse fields, including economics, accounting, military theory, computer science, and Artificial Intelligence. One offshoot of games theory (broadly speaking) came with British mathematician John Horton Conway ’s development of the computer algorithm for playing the “Game of Life,” where simulated single cell automata (simple programs) interact to produce complex, seemingly intelligent outcomes without following specific recipes. Much work in computer science has flowed from prior work in mathematics this way, and games theory is a great example.

No summary of mathematics in the latter part of the 20th century would be complete without mention of Paul Cohen ’s proof of the **Continuum Hypothesis**. More precisely, Cohen proved that 19th century mathematician G. Cantor’s Continuum Hypothesis about the possible sizes of infinite sets had not one but two solutions: in one solution (perfectly valid) the Continuum Hypothesis is true. In the other (also perfectly valid), it’s false. Cohen’s proof about the Continuum Hypothesis so affected the mathematical world that even today, math proofs must all first specify whether they are taking the Continuum Hypothesis as true or false. It’s part and parcel of mathematics. Cohen proved this pure result in the 1960s, as computers were rapidly drawing attention away from pure math to applied mathematics.

The 1970s saw the formulation of a still unsolved problem that straddles the line between applied and pure math, in the field of computational complexity. The famous “P = NP” problem was posed in 1971 by American-Canadian mathematician Stephen Cook , and remains an important and unsolved problem in mathematics. The problem—really a conjecture—states that computer programs that have simple checks for their solution also have simple solutions. Or, rather, the problem states that there are no programs that are extraordinarily hard to solve but that can be easily checked for correctness. No one knows if this is true in any case, or whether there are indeed fundamental differences between “P,” or simply computable problems, and “NP,” ones with no known simple solutions at all. Much foundational work on algorithms rests on the theory of complexity classes (like P or NP), and so Cook’s problem persists as a core contribution, though one that still evades solution.

By the late 1970s, the role of computing had become clear to a new generation of mathematicians. A famous 19th century problem known as the ”**Four Color Conjecture**,” which stated that no more than four colors are required to paint (cover) any pattern such that no adjacent shapes in the pattern share the same color, was solved in 1976 using (what else?) a computer program, which cranked through over 1,200 cases (different ways of painting the space) to verify that the conjecture was in fact true. The proof evaded the brightest minds in mathematics until finally yielding to brute computational power. Still, while pure mathematicians might balk, the conjecture did receive a perfectly valid proof...with the aid of computers.

Closing out the 20th century, we have diverse but important areas of research emerging. Significantly, the 1970s, 80s and 90s led to interesting developments in geometry-based mathematics like **knot theory** (theory of knot shapes) and even origami. The elegant Japanese art form proved to have interesting mathematical properties, as elucidated by Japanese mathematician Kazuo Haga , who has at least three important mathematical theorems about origami to his name. Knot theory and mathematical theories of shapes have proven helpful in explicating difficult higher-order problems in physics, such as string theory.

Also during this period, British mathematician Andrew Wiles finally realized his lifelong quest to prove **Fermat’s Last Theorem** true in 1995, closing out one of the most famous and longest standing puzzles in number theory (conjectured by French mathematician Pierre de Fermat in 1637). And the search for greater and greater prime numbers made it on the Internet by the 1990s, with the still ongoing Great Internet Mersenne Prime Search (GIMPS).

By 2000, while some great mathematicians remained, many had turned increasingly to computational methods and techniques to tackle some of the deepest and most fundamental questions in the field. This trend would surely come to define the next twenty years of mathematics research, right up to the present day.

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*Find out which influencers have most contributed to advancing the field of mathematics over the last two decades with a look at The Most Influential People in Mathematics, for the years 2000 – 2020.*

*To discover which schools are driving the mathematics field forward today, check out The Most Influential Schools in Mathematics, for the years 2000 – 2020.*

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