Paul Erdős

Paul Erdős was a Hungarian mathematician renowned for his extraordinary contributions to number theory, combinatorics, graph theory, and probability. One of the most prolific mathematicians in history, Erdős authored or co-authored more than 1,500 research papers with hundreds of collaborators across the world. His distinctive lifestyle, collaborative ethos, and deep influence on modern mathematics made him a singular figure in twentieth-century science.
Erdős is remembered not only for his technical achievements but also for reshaping how mathematics is practised, particularly through large-scale collaboration and the open exchange of ideas. His work laid foundational results across multiple branches of discrete mathematics that remain central to contemporary research.

Early Life and Education

Paul Erdős was born in Budapest in 1913 to a family of mathematicians. Both of his parents were secondary-school mathematics teachers, and he was introduced to mathematical thinking at an exceptionally young age. Childhood tragedy, including the loss of two sisters, contributed to a close family environment and an intense focus on intellectual pursuits.
Erdős displayed prodigious talent early on, publishing his first mathematical paper at the age of nineteen. He studied mathematics at the University of Budapest, earning his doctorate in 1934. His early work already demonstrated the originality and depth that would characterise his lifelong research.

Mathematical Itinerancy and Lifestyle

Unlike most academics, Erdős did not settle at a single institution. Instead, he lived a nomadic life, travelling constantly between universities and research centres across Europe, North America, and later Israel. He carried few personal possessions and relied on the hospitality of colleagues, often staying with fellow mathematicians while working intensely on problems.
This itinerant lifestyle was closely tied to his philosophy of mathematics as a communal endeavour. Erdős believed that mathematical ideas flourished through collaboration, discussion, and shared problem-solving rather than solitary work.

Contributions to Number Theory

Number theory was one of Erdős’s primary areas of research. He made significant contributions to both elementary and analytic number theory, often favouring ingenious and elementary methods over heavy formal machinery. His work addressed problems concerning prime numbers, additive number theory, and arithmetic functions.
Erdős was particularly known for posing deep and accessible questions, many of which stimulated decades of further research. Some of these problems remain unsolved, reflecting their depth and enduring significance within mathematics.

Combinatorics and Graph Theory

Erdős played a central role in establishing combinatorics as a major branch of modern mathematics. At a time when the field was often viewed as peripheral, he demonstrated its depth, power, and broad applicability.
In graph theory, Erdős introduced fundamental ideas and results concerning connectivity, colouring, and extremal properties. His work helped formalise the field and inspired systematic study of networks, structures, and discrete systems that later found applications in computer science and information theory.

Probabilistic Method

One of Erdős’s most influential innovations was the probabilistic method, a powerful technique for proving the existence of mathematical objects by showing that a randomly chosen example has a desired property with positive probability.
This approach was revolutionary in that it established existence without explicitly constructing examples. The probabilistic method became a standard tool in combinatorics, theoretical computer science, and discrete geometry, profoundly influencing modern mathematical practice.

Collaboration and the Erdős Number

Erdős collaborated with an unprecedented number of mathematicians, publishing joint papers with over 500 co-authors. This extensive network led to the informal concept of the Erdős number, which measures collaborative distance from Erdős based on co-authorship of mathematical papers.
While partly humorous, the Erdős number reflects the real impact of his collaborative culture. It symbolises the interconnectedness of the mathematical community and highlights Erdős’s role as a catalyst for research across generations.

Mathematical Philosophy and Style

Erdős believed that mathematics was a universal and timeless pursuit, often referring to an imaginary “Book” in which the most elegant proofs were written. He valued simplicity, clarity, and beauty in mathematical arguments, preferring short and insightful proofs over technically complex ones.
His approach encouraged mathematicians to seek deeper understanding rather than merely technical solutions. This emphasis on elegance strongly influenced the culture of problem-solving in modern mathematics.

Personal Traits and Intellectual Character

Erdős was known for his eccentric personality and distinctive vocabulary. He referred to children as “epsilons”, caffeine as a primary working aid, and mathematics as the ultimate purpose of life. Despite his unconventional lifestyle, he was deeply generous with his time, ideas, and financial resources, often donating prize money to students and causes he supported.
His personal identity was inseparable from his intellectual life, and his singular devotion to mathematics became legendary within the academic community.

Influence on Computer Science and Discrete Mathematics

Many of Erdős’s ideas found lasting application in theoretical computer science, particularly in algorithms, complexity theory, and network analysis. His work on random graphs and combinatorial structures anticipated developments in data science and modern network theory.
As computing grew in importance, Erdős’s emphasis on discrete structures proved increasingly relevant, further amplifying his long-term influence beyond pure mathematics.