The Gödel Essays

“The examiner was intelligent enough to quickly quieten Gödel and say ‘Oh god, let’s not go into this’ and broke off the examination at this point, greatly to our relief” — Oskar Morgenstern

Kurt Gödel (1906–1978) was the greatest logician who ever lived. At the age of 24, he revolutionized our understanding of the limits of epistemology — the theory of knowledge—by proving mathematically that all formal systems of logic are inherently incomplete. In a highly technical paper of 26 pages, Gödel showed how, no matter how comprehensive a system of rules, laws or axioms we devise, there will always…

Thinking like a Mathematician

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…

“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…

“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). …

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…

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.

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…