Thinking like a Mathematician

Implications of Condorcet’s jury theorem

There once lived a man named Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (1743–1794). A mathematician and philosopher, his work was mainly focused on advancing social progress towards a more egalitarian society. For instance, he strongly advocated for gender equality as early as in 1787, when he wrote:

“The rights of men stem exclusively from the fact that they are sentient beings, capable of acquiring moral ideas and of reasoning upon them. Since women have the same qualities, they necessarily also have the same rights.”

Despite his best efforts, the majority of French society and indeed all the “enlightened” societies did not agree with Condorcet’s view about women’s suffrage in the 1700s. I was surprise to learn that it took more than 100 years before New Zealand became the first country to recognize women’s right to vote in parliamentary elections in 1893. …

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Left: A doodle from one of Abel’s notebooks. Right: The only known portrait of Abel, by Norwegian painter Johan Gørbitz (1826, edited and colorized)

“Although Abel shared with many mathematicians a complete lack of musical talent, I will not sound absurd if I compare his kind of productivity and his personality with Mozart’s.” — Felix Klein

Niels Henrik Abel (1802–1829) died at age 26. Largely self-taught, in his short life the young Abel made pioneering contributions to variety of subjects in pure mathematics, including: algebraic equations, elliptic functions, elliptic integrals, functional equations, integral transforms and series representations. …

The Ramanujan Essays

“I had never seen anything in the least like [it] before” — G.H. Hardy

On or about the 31st of January 1913, mathematician G.H. Hardy of Trinity College at Cambridge University received a parcel of papers from Madras, India. The package included a cover letter where a young clerk by the name of Srinivasa Ramanujan (1887–1920) provided an introduction of himself and his precarious situation, as well as various mathematical claims about the domain of the gamma function and the distribution of prime numbers. The content of the letter is discussed in detail here.

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This essay provides a narration of Ramanujan and Hardy’s correspondence regarding the former’s early work on continued fractions, in addition to an overview of the fractions and their history. The essay is based in large part on the wonderful book Ramanujan: Letters and Commentary* by Berndt & Rankin (1991). Estimated reading time is 12 minutes. …

Photos of nine Martians of Budapest: von Neumann, Erdos, Wigner, Teller, Szilard, Karman, Halmos, Polya and Kemeny
Photos of nine Martians of Budapest: von Neumann, Erdos, Wigner, Teller, Szilard, Karman, Halmos, Polya and Kemeny
John von Neumann, Paul Erdős, Eugene Wigner and Edward Teller, Leó Szilárd, Theodore von Kármán, Paul Halmos, George Polya and John G. Kemeny

“There is a rumor in America that there are two intelligent races on Earth: Humans and Hungarians” — Isaac Asimov

“The Martians of Budapest”, sometimes simply “The Martians” is a colloquial term used to describe a group of prominent Hungarian physicists and mathematicians who emigrated to the United States following the Great Purge of 1933. The term refers to — what appeared, from the perspective of Americans —to be a group of men with superhuman intellects, arriving from an obscure country speaking an incomprehensible foreign language and English with strong, characteristic accents (later popularized by Bela Lugosi in Dracula). …

Left: Colorized photograph of Georg Cantor. Right: Cantor’s publication
Left: Colorized photograph of Georg Cantor. Right: Cantor’s publication

Proof Theory

“Diagonalization seems to show that there is an inexhaustibility phenomenon for definability similar to that for provability” — Franzén (2004)

In addition to his inventions of set theory and transfinite numbers, Georg Cantor (1845–1918) is remembered as the brilliant inventor of the popular diagonalization argument later employed by both Kurt Gödel (1906–1978) and Alan Turing (1912–1954) in their most famous papers.

The Diagonal Argument

In set theory, the diagonal argument is a mathematical argument originally employed by Cantor to show that

“There are infinite sets which cannot be put into one-to-one correspondence with the infinite set of the natural numbers” — Georg Cantor…

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Left: Feynman on the day of his Graduation from Princeton University (1942)

The Feynman Essays

“I went through fire on my first.”

While still a graduate student at Princeton University in 1940, Richard P. Feynman (1918–1988) gave his first lecture in a seminar on electrodynamics, the topic that would eventually earn him the 1965 Nobel Prize in physics. In front of a prestigious audience consisting of Nobel laureates Albert Einstein (1879–1955), Wolfgang Pauli (1900–1958), and Eugene Wigner (1902–1995) as well as the Hungarian polymath John von Neumann (1903–1957), Feynman lectured on the current state of what is now known as the Wheeler- Feynman absorber theory.

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Left: Richard Feynman in 1943 (Photo: unknown) Right: Wheeler and Feynman’s 1945 paper on the Wheeler-Feynman absorber theory

Richard Feynman (1918–1988) attended college at the Massachusetts Institute of Technology, graduating in 1939 with a B.Sc. Although he had originally studied mathematics, he eventually switched to electrical engineering because he considered mathematics to be “too abstract”. Noticing later that he had “gone too far” in the direction of the practical, he again switched to physics. For his Ph.D, he applied and was accepted to Princeton University after achieving an — unprecedented — perfect score on the prestigious graduate school’s entrance exam in physics as well as an outstanding score in mathematics. His Ph.D. thesis advisor Professor John A. Wheeler (1911–2008), who was only seven years older, described his enthusiasm of Feynman as…

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Arnold Newman’s iconic 1956 portraits of Kurt Gödel

The Gödel Essays

“Modern math’s absolute Prince of Darkness” — David Foster Wallace

Hungarian polymath John von Neumann (1903–1957) once wrote that Kurt Gödel was “absolutely irreplaceable” and “in a class by himself”. Describing his 1931 proof of Gödel’s incompleteness theorem, von Neumann called the achievement

“Singular and monumental — indeed it is more than a monument, it is a landmark which will remain visible far in space and time. The subject of logic will never again be the same.”

von Neumann was not alone in his admiration of Gödel. A young Alan Turing (1912–1954) sought out Gödel in 1936 to inquire about his own monumental reformulation of Gödel’s incompleteness result which showed the limits of proof and computation. …

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El Farol on Canyon Road in Santa Fe, New Mexico

Game Theory

“Oh that place. It’s so crowded nobody goes there anymore.” — Yogi Berra

The El Farol Bar Problem, sometimes known as the “Santa Fe Bar problem” (SFBP) is a constrained resource allocation problem for non-cooperating agents defined by economist William Brian Arthur (1945-) in 1994. The problem deals with ways of achieving an optimal collective resource allocation in situations where, if everyone uses the same pure strategy that strategy is guaranteed to fail no matter what it is.


The problem begins with explaining that on Thursday night every week 100 people decide independently whether or not to go to a bar in El Farol, Santa Fe that offers live Irish music. Each person knows that they will only have fun if a certain number of people show up. …

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Three frames of Albert Einstein and Leo Szilárd from the documentary film Atomic Power (1946) for TIME’s The March of Time series

The Einstein Essays

and the birth of the Manhattan Project

The now famous Einstein-Szilárd letter was written at the initiative of Hungarian nuclear physicist Leó Szilárd with help from Edward Teller and Eugene Wigner in 1939. It was signed by Albert Einstein and sent to the President of the United States, Franklin D. Roosevelt in October 1939. The letter argued that the United States should engage in uranium research. Its writing was motivated by the news of the discovery of uranium fission by Otto Hahn and Fritz Strassmann nine months prior.

The letter prompted Roosevelt to propose the undertaking which would later become the Manhattan Project, producing the first nuclear weapons and — following the atomic bombings of Hiroshima and Nagasaki — leading to the unconditional surrender of Imperial Japan and the conclusion of World War II. …

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Richard Feynman and Stephen Wolfram (Photo: Wikimedia Commons and Stephen Wolfram’s PR team)

The Feynman Essays

«You don’t understand “ordinary people”. To you they are “stupid fools”. »

Entrepreneur Stephen Wolfram is a unique egg. By age 14, he had written three books on particle physics. He earned his Ph.D. at age 20 and began publishing research papers at the age of 18, some of which have been cited thousands of times. His software package Mathematica is in its 12th edition. His 1197 page book “A New Kind of Science” was a best-seller, reaching #1 on Amazon when it was published in 2002.

On the 5th of February 2020, the once-boy-genius-turned-software-mogul took to Twitter to announce that he had been “working more intensely than ever … as a result of an unexpected scientific breakthrough I’m hoping to share soon”. Two months later, on April 14th, he posted a blog entry entitled Finally We May Have a Path to the Fundamental Theory of Physics…and It’s Beautiful”, which starts off on a similarly swaggering…


Jørgen Veisdal

Editor-in-Chief at Cantor’s Paradise. Research Fellow at the Norwegian University of Science and Technology.

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