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…

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, during 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.

This essay provides a narration of…

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…

Arnold Newman’s iconic 1956 portraits of Kurt Gödel

The Gödel Essays

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“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…

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…

An Illustrated Five Step Guide

Earlier this week Reuters broke the news that the U.S. House of Representatives Judiciary Committee has called on founder Jeff Bezos to testify on

“Allegations that the online retailer uses data from its own third-party sellers to create competing products”

The allegations are, and have been for some time, that Amazon engages in uncompetitive practices when it comes to how they compete with third-party sellers on the Amazon Marketplace.

Amazon’s platform for buying and selling consumer goods has since 2018 has enjoyed a 50% market share of all e-commerce in the United States. Starting in 2009, Amazon entered this…

The von Neumann Essays

“What, I asked, are the duties of an assistant to Professor von Neumann?”

The year is 1933. A 26 year-old former graduate student of Columbia University, Edgar R. Lorch (1907–1990) has just completed his Ph.D. in mathematics and is, in his own words “by some miracle” awarded a National Research Council Fellowship for a year of postdoctoral studies at Harvard University.

Worried that the ongoing Great Depression will leave him unemployed, as the fellowship nears its end, Lorch applies for a one-year extension to visit Princeton and study under Professor John von Neumann (1903–1957) at the Institute for Advanced Study.

Becoming von Neumann’s Assistant

John von Neumann (1903–1957) and Oswald Veblen (1880–1960)

The von Neumann Essays

The book John von Neumann: Selected Letters* by Rédei (2005) is a fascinating collection of 150 of von Neumann’s correspondences starting in the 1920s until his death in 1957. Among von Neumann’s extensive communications included in the book are:

Jørgen Veisdal

Assistant Professor at NTNU. Author at Privatdozent. Editor-in-Chief at Cantor’s Paradise.

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