Every great historical mathematician was a person. They lived their lives and grappled with great ideas, all while navigating the human constraints we experience every day.

Names like Emmy Noether and Srinivasa Ramanujan are familiar to undergraduate mathematics majors, but non-mathematicians usually have never heard of them. The upcoming biopic *The Man Who Knew Infinity* (see my review) may change things for Ramanujan, but there are so many other fascinating characters in the recent history of mathematics.

I was inspired to write short blurbs for five giants of modern mathematics. Each has had a profound impact on the subject, and the applications of their ideas are far reaching not just in mathematics, but also across the sciences and engineering.

### David Hilbert

Hilbert was a superstar among 19th century mathematicians. Where Hilbert went, others followed. His work has a major impact in many areas: analysis, number theory, geometry, even applied mathematics and physics. Hilbert spaces, named in his honor, are one of the cornerstones of modern functional analysis. The Hilbert Nullstellensatz, the Hilbert syzygy theorem, Hilbert’s theorem 90, Hilbert cubes, the Hilbert Basis theorem, Hilbert C* modules, … the list of theorems and mathematical constructs in his name seems endless.

Hilbert may be best known for his 23 problems stated at the International Congress of Mathematicians in Paris in 1900. These problems had a profound influence on 20th century mathematics. Several of these problems and their solution led to deep and potent new fields of mathematics. Some of these, like the Riemann hypothesis, remain open today.

We have record of Hilbert speaking from a 1930 radio broadcast.

A transcript with English translation is here. Hilbert ends with his famous pronouncement “Wir müssen wissen, Wir werden wissen,” or “We must know, we will know.”

Hilbert spoke out in the mid-1930s about the Nazi purge of Jewish scholars at the University of Göttingen (see Emmy Noether’s bio below), but he had little influence against that tide of history. But by the time of his death in 1943, the great mathematical center he helped build at Göttingen was only a memory. Only ten people attended the funeral of one the greatest mathematicians of modern times.

### Georg Cantor

There was mathematics before and after Cantor. Georg Cantor may be viewed as one the progenitors of *set theory*, which is the precise language with which we describe modern mathematics. Think of it like the grammar of our work, and it is difficult to imagine how we got by before it. Cantor formalized the notions of ordinals and cardinals, which are the modern tools for counting and measuring sets. Much of set theory was formalized later on by the likes of Bertrand Russell and Ernst Zermelo, but Cantor was one the first to realize the power and place of set theory in mathematics.

Cantor discovered the *transfinite numbers*: he found that there is not just a single notion of infinity, but an infinite hierarchy of infinities. Cantor’s ideas were so disruptive at the time that some prominent mathematicians like Poincaré and Kronecker denounced them. Indeed, Kronecker made personal attacks against him. Cantor suffered from chronic depression, and was hospitalized several times throughout his life.

### Srinivasa Ramanujan

G.H Hardy described his work and interaction with Ramanaujan as the most romantic episode in his life. An accounting clerk in Chennai, India, Ramanujan wrote to Hardy a nine page letter containing mathematics that deeply impressed him. As a result, Hardy arranged for Ramanujan to come to Cambridge. Without formal training, Ramanujan discovered formulas and insights in number theory without proofs. He could just *see* the formulas in his mind. Hardy pressed him to write out his ideas more formally and with rigorous justification.

Together, Ramanujan and Hardy derived the asymptotic number of partitions of an integer. That breakthrough is a centerpiece of the upcoming biopic featuring Dev Patel as Ramanujan, and Jeremy Irons as Hardy. Today, Ramanujan’s ideas continue to inspire new generations of number theorists and combinatorialists, and his work has even found applications to the study of black holes.

Ramanujan died from illness on return to Mumbai at the too young age of 32.

### Emmy Noether

One of the most brilliant stars in abstract algebra of her time, Noether was a giant in the field of ring theory (not the rings you wear on your finger). Many of her ideas still are taught today in every university level course in abstract algebra. Hilbert invited her to the University of Göttingen, despite resistance that she should work as a professor as she was a woman. While at the University of Göttingen in 1920s, she and her students such as Fitting and Krull revolutionized the study of commutative and noncommutative algebra. Hilbert invited her to collaborate on work with Einstein on providing a mathematical framework for general relativity, and she made considerable progress on that theory.

A German Jew, she was expelled from Göttingen in 1933 and fled to the US, where she worked as a professor at Bryn Mawr. Her most famous students there was Olga Tausky Todd, whose work focused on matrix theory. Noether died at the age of 53 in 1935 from cancer.

### Paul Erd**ő**s

My last flash bio is the happier tale of Paul Erdős, who I think of as the pied piper of mathematics. For decades he roamed the world homeless and jobless, but constantly working on his mathematics with a vast network of collaborators. His famous saying, “Is your brain open?” was a calling card to his hosts, who gladly took Erdős into their homes.

The most prolific mathematicians of all time, he had 1,525 published papers, and over 500 collaborators. Erdős studied number theory, graph theory, and combinatorics, and made a deep and lasting impact on each of these fields. For instance, the field of random graphs, whose models have been used to model things like the web and the growth of social networks, would not exist without his work. Everyone has an *Erdős number*, which counts your distance to him in the co-author graph. I have Erdős number 2, as I wrote a paper with Peter Cameron who co-authored with Erdős (but I didn’t write one with Erdős).

Despite Erdős’s unusual lifestyle, he lived a long and engaged life, dying of heart attack at 83 while at a conference in Warsaw. Here is a great excerpt from the documentary *N is a Number: A Portrait of Paul Erdős *where he gives his own charming perspective of aging. I hope to keep my sense of humor if I ever reach his age.

Anthony Bonato, soon to be P.G.O.M., L.D., A.D., and C.D.

Can you suggest some good books by/about Hilbert and Cantor?

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Check out Rudy Rucker’s book Infinity and the Mind. It is older, but a great popular reference which talks about Cantor and Hilbert. Also check out Dauben’s book on Cantor. For Hilbert, I don’t know any books per se, but there quite a few on-line historical references you can find with Google/Bing.

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Thanks!

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Actually, not everybody has an Erdos number. Take, for example, all those who never wrote a single paper in their lives–let alone coauthored one. It is true that most mathematicians who have collaborated on coauthored research have an Erdos number. It is believed that all those with an Erdos number have one that is at most seven.

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Those who’ve never written a paper are assigned an Erdos number of infinity. That would be most people. 🙂

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