As you may have read, the name Google originated from the misspelling by Larry Page and Sergey Brin of the word googol, a number whose value is 10^1001. This word, in turn, was coined by a child, the nephew of mathematician Edward Kasner (1878-1955) in 1940. As the story goes:

“Dr. Kasner's nine-year-old nephew [Milton] was asked to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "googolplex". A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would happen if one tried to write a googolplex, but different people get ti…

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Alan Turing (1912-54) is a well-known name for a variety of reasons. To the general public, he is probably best known for his work on the cryptanalysis of the Enigma ciphering system during World War II. To mathematicians, however, he is perhaps best known for being one of the founders of computer science. He was the person who formalized the core concepts of computation and algorithms via his invention of the Turing machine. He also solved Hilbert’s Entscheidungsproblem (see essay below).

Few, however, are aware of his significant contributions to analytic number theory via his work on the non-trivial zeros of the Riemann zeta function and, by extension, the Riemann hypothesis.

The Problem

The first essay I ever wrote about mathematics was on the Riemann zeta function and its association with the distribution of prime number, the so-called Riemann hypothesis (see technical ess…

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Mathematician John F. Nash Jr (1928-2015) was anything but inhibited, at least in his willingness to reach out and enter into debates about big issues with prestigious scientists. Famously, while still a 20-year old a graduate student, he went to see both Albert Einstein (1879-1955) and John von Neumann (1903-1957) to discuss, respectively, the “germ of an idea” related to relativity theory and “his rival idea” regarding equilibrium points in noncooperative games (Nasar, 1998) . In her biography of the man, Sylvia Nasar describes the first of these meetings, which took place in the fall of 1948, as follows:

Just a few weeks into his first term at Princeton, Nash made an appointment to see Einstein in his office in Fuld Hall. Einstein’s office, a large airy room with a bay window that let in plenty of light, was messy. Einstein’s twenty-two-year-old Hungarian assistant — an intense, chain-smoking logician named John Kemeny, who would later invent the computer language BASIC, become president of Dartmouth College, and head a commission to investigate Three Mile Island — ushered Nash in. Einstein’s handshake, which ended with a twist, was remarkably firm, and he showed Nash to a large wooden meeting table on the far side of the office.

The late-morning light streaming through the bay window produced a sort of aura around Einstein. Nash, however, quickly got into the substance of his idea while Einstein listened politely, twirled the curls on the back of his head with his finger, sucked on his tobaccoless pipe, and occasionally muttered a remark or asked a question. As he spoke, Nash became aware of a mild form of echolalia: “deep, deep, interesting, interesting”.

Nash had an idea about “gravity, friction, and radiation,” as he later recalled. The friction he was thinking of was the friction that a particle, say a photon, might encounter as it moved through space due to its fluctuating gravitational field interacting with other gravitational fields. Nash had given his hunch enough thought to spend much of the meeting at the blackboard scribbling equations. Soon, Einstein and Kemeny were standing at the blackboard as well. The discussion lasted the better part of an hour. But in the end all that Einstein said, with a kindly smile, was “You had better study some more physics, young man.”

The extent of Nash’s schooling in theoretical physics (as economics) was limited to some undergraduate courses at Carnegie Tech (now Carnegie Mellon University). His later foray into differential geometry (which would result in the publication of the so-called Nash embedding theorem1 in 1954) suggests that he either had or later obtained advanced knowledge of Riemannian geometry (which Einstein’s general theory of relativity is formulated in). Clearly, the “cocky youngster”, as Nasar describes him, didn’t shy away from either venturing into new fields or conjecturing about difficult problems (see essay below).

His visit to von Neumann’s office the following autumn (in October of 1949) appears no less audacious (Nasar, 1998):

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A movement which began as a response to German soldiers’ acts during the first world war spread to academia and began gaining support among prominent colleagues of Einstein, including Arnold Sommerfeld (1868-1951) and Johannes Stark (1874-1957). At its core, the academic dimension of the debate regarded the increasing use of the English language in scientific writing, including by German authors, editors and translators. Following the burning of the library at Louvain/Leuven (see essay below) a front developed between German and British intellectuals, spearheaded by a handful of prominent physicists.

“Manifestos rained down. Under the heading of Louvain, English universities and institutions of higher learning g…

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I’ve written before about the book burning ceremonies (“bücherverbrennung”) that the Nazi Party arranged in 1933 following their seizure of power and passing of the Berufsbeamtengesetz (“Law for the Restoration of the Professsional Civil Service”) dismissing public servants from German Universities (see essay below).

In that context, book burnings functioned as an overt, public and aggressive rejection of ideas considered sympathetic with socio-democratic, intellectual, left-leaning, and/or “Jewish values”. In addition, of course, the campaigns also served as mechanisms for the promulgation of censorship and suppression, including of free speech and the dissemination of knowledge. The latter, partly in order to promote a nationalistic alternative to theoretical physics later known as “German” physics (see essay below).

Interestingly, Deutsche Physik — which by the 1930s had grown to be overtly anti-semitic — had its origins not

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As Nasar writes, “when the book appeared in 1944, von Neumann’s reputation was at its peak. It got the kind of public attention — including a breathless front-page story in the New York Times — that no other densely mathematical work had ever received” (Nasar, 1998). The timing, as Morgenstern had predicted, was perfect. World War II…

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In the early 1930s, the provincial town of Princeton, New Jersey was already well on its way to becoming a hotbed for mathematical exploration. Although plans for an Institute for Advanced Study (IAS) had been in the making for a while, 1930 would be the year of its founding and 1933 the year of its opening. Until 1938, “the Institute” was located in the same building as the math department at the nearby Princeton University, then called Fine Hall (now Jones Hall). On its permanent faculty “of unrivalled prestige” was “the last great universal mathematician” Hermann Weyl (1885-1955) and an elite group of topologists, including Oswald Veblen (1880-1960), James Alexander (1888-1971) and Marston Morse (1892-1977). Topologist Solomon Lefschetz (1884-1972) was there as well, although formally affiliated with Princeton, as was Veblen’s former Ph.D. student Alonzo Church (1903-95) who had…

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The book John von Neumann: Selected Letters* by Miklós Rédei (2005) is a fascinating collection of 150 of John von Neumann (1903-57)’s correspondences from the period 1920 until his death in 1957. Among his extensive communications included in the book are:

• Four letters to Kurt Gödel (1906–1978), from 1930-33

• Two letters to Norbert Wiener (1964–1909), from 1945–46

• One letter to J. Robert Oppenheimer (1903–1957), from 1948

• Four letters to Stanislaw Ulam (1909–1984), from 1936–43

• One letter to Paul Dirac (1902–1984), from 1934

• One letter to Edwin Schrödinger (1887–1961), from 1936

• One letter to Edward Teller (1908–2003), from 1947

• One letter to Hans Bethe (1906–2005), from 1941

• Two letter to Pascual Jordan (1902–1980), from 1949–50

Also included are seven letters to von Neumann’s friend and colleague at the Institute for Advanced Study, mathematician Oswald Veblen (1880–1960) from the period 1930 (when the IAS was founded, se…

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By 1926, mathematician John von Neumann (1903-57) was a professional researcher in two disparate fields: set theory and quantum mechanics. That year, he took it upon himself to write a paper on a completely different topic entirely: mathematical modelling of strategic interactions. von Neumann’s result, what is now referred to as the von Neumann’s minimax theorem, was published two years later under the title Zur Theorie der Gesellschaftsspiele, “On the Theory of Games of Strategy” (see essay below).

Despite the (later) significance of this work, following its publication the man himself returned to his research on set theory and quantum mechanics, as well as his developing interests in ergodic theory, rings of operators, the Haar measure and theoretical computer science. Indeed, von Neumann did not publish on the topic of ‘strategic interactions’ again for another ten years, until 19371. Then, seven years after that, seemingly out of nowhere von Neumann (with Oskar Morgenstern) produces what has since been called the “groundbreaking text that created the interdisciplinary research field of game theory”, Theory of Games and Economic Behavior* (TGEB).

The book has been cited more than 50,000 times as of 2024. This despite the fact that it (at least for economists) is a famously overwhelmingly complex mathematical text. Since its release in 1944, the work of 18 Nobel laureates has been in game theory, including John F. Nash Jr. (1928-2015), Kenneth Arrow (1921-2017), Gérard Debreu (1921-2004) and Jean Tirole (1953-).

This is the fifth in my series of essays on the von Neumann-Morgenstern collaboration (see the 1st, 2nd, 3rd and 4th here), and their work on TGEB in the period 1938 to 1943.

Their First Meeting (1938)

Although they would later be close friends, neither Morgenstern or von Neumann remembered the first time they met. This is odd, because Morgenstern would later write that von Neumann’s presence at the Institute for Advanced Study was one of the mai reasons why he pursued a position at the nearby Princeton University when he emigrated from Austria:

“The principal reason for my wanting to go to Princeton was the possibility that I might become acquainted with von Neumann and the hope that this would be a great stimulus for my future work.”

According to Morgenstern’s later recollections, the two mens’ first meeting must have been “soon after the university opened” in the fall of 1938 (Morgenstern, 1976). The circumstances of their second meeting are known in more detail. It happened after a lecture Morgenstern gave during a luncheon at the Nassau Club in Princeton on February 1st, 1939. As he later wrote, von Neumann and Niels Bohr (1885-1962) invited him for tea in Fine Hall. There, the three conversed for several hours about games and experiments:

“This was the first time that we had a talk on games. The occasion was heightened by Bohr's presence.”

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Niels Henrik Abel (1802–1829) died at the modest age of 26 years old. 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.

Born on the small island of Finnøy in Rogaland, Norway, the early years of Abel’s short life was dominated by the instability of an alcoholic father who died when he was 16 years old. More or less self-taught, at 21, Abel provided the first complete proof demonstrating that there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four. At that point, the problem had been unresolved for over 250 years. In the process of writing the proof, he laid the foundation — independently of Galois — for the branch of mathematics now known as group theory. At 22, he also wrote a fundament…

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Sir Arthur Eddington (1882-1944) was quite a lucky man. He was a Quaker, which helped keep him out of World War I. He also had a talent for being at the right place at the right time. A mathematically inclined English astronomer, he happened to be working as a secretary at the Royal Astronomical Society when Dutch astronomer Willem de Sitter (1872-1934)’s letters and papers on Einstein’s new theory of general relativity began arriving in England. Quick to comprehend the implications of Einstein’s ideas, already in 1916 Eddington became a supporter of Einstein. This despite the fact that the theory had been formulated by a German at a time when anti-German sentiments were at an all-time high in England due to the war. Eddington, a pacifist, was among the first to teach the theory in the West (both at Cambridge University and at a meeting of the British Association). In 1918, he also produced a report on the theory for the Physical Society1. One issue however remained. The theory had no…

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The Dartmouth Summer Research Project on Artificial Intelligence was an academic workshop sometimes considered to be the founding moment for artificial intelligence as a field of research. Held for eight weeks in Hanover, New Hampshire in 1956, the conference gathered twenty or so of the brightest minds in computer- and cognitive science for a workshop committed to the idea that:

"..every aspect of learning or any other feature of intelligence can in principle be so precisely described that a machine can be made to simulate it. An attempt will be made to find how to make machines use language, form abstractions and concepts, solve kinds of problems now reserved for humans, and improve themselves. We think that a significant advance can be made in one or more of these problems if a carefully selected group of scientists work on it together for a summer."

— A Proposal for the Dartmouth Summer Research Project on Artificial Intelligence (McCarthy et al, 1955)

Now seventy or so years later, …

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Famously, after obtaining his Ph.D. from Princeton University in June of 1938, Alan Turing (1912-54) turned down an offer from John von Neumann (1902-55) to come work with him as a postdoctoral assistant at the Institute for Advanced Study (Macrae, 1999). Rather than stay in America (see essay below), Turing instead returned to Cambridge, where he had been a fellow at King’s College since 1936.

For about a year, starting in September 1938, Turing worked part-time at the Government Code and Cypher School (GC&CS), engaging in research on the cryptanalysis of the German Enigma cipher machine. In his spare time, he attended various lectures at Cambridge, including a graduate seminar held by philosopher Ludwig Wittgenstein (1889-1951) on the foundations of mathematics.

Wittgenstein’s seminar, as he himself described it, regarded investigations into “the role that logic plays in mathematics, or the relation supposed to hold between logic and mathematics”. In particular, Wittgenstein was interested in interrogating mathematicians’ attitudes towards contradictions, an attitude of “avoiding them at all costs”, or as he put it, of “a germ which shows general illness” (Vanrie, 2024; Berg, 2021). Finding a contradiction in a system, Wittgenstein argued, is to mathematicians akin to “finding a germ in an otherwise healthy body” and arguing that this “shows that the whole system or body is diseased”. This view ran counter to his own views, which—in his later work—maintained that mathematics is a human invention and that contradictions more-so indicate flaws in this invention than truths about the absurdity of reality. Wittgenstein in his seminar, thus, sought to discredit ‘contradictory calculus’, a contraption he regarded as inherently defective.

Turing disagreed. His most celebrated result, entitled ‘On Computable Numbers, with an Application to the Entscheidungsproblem’ employed proof by contradiction in showing that Hilbert’s “decision problem” did not have a solution—that some problems are indeed undecidable. The result had concrete implications for the development of the computer. Turing—in other words—was building a career on a result obtained using the very contraption that Wittgenstein was looking to suppress.

This essay is about the colliding of two worlds in one Cambridge classroom in 1939—the anti-Platonist late Wittgenstein and the realist views of Turing and, in turn, his hero Kurt Gödel (1906-1978).

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Prior to the 1930s, Germany had dominated the natural sciences for more than one hundred and fifty years. Its reputation for excellence in chemistry, physics, biology, medicine and mathematics was rivaled, if at all, only by Britan (Medawar & Pyke, 2000). Of the 100 Nobel Prizes awarded between the years of 1901 and 1932 (the year before Hitler came to power), 33 had been awarded to Germans and/or scientists working in Germany. For comparison, Brits and scientists in British universities had won 18 and the United States a mere six.

Then, starting in 1933 with the Nazi party’s seizure of control in Germany and their passing of the Berufsbeamtengesetz (“Law for the Restoration of the Professsional Civil Service”), certain groups of publ…

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In 1932, physicist Albert Einstein (1879-1955) took it upon himself to write an open letter to neurologist Sigmund Freud (1856-1939) to initiate a public conversation on the origins of war and peace, human nature and the role of the sciences in these matters.

At the time of the writing of the letter, public scrutiny of Jewish intellectuals in Germany and Austria was mounting. The following May, a viciously anti-Semetic illustrated brochure entitled Juden Sehen Dich An (“Jews are Watching You”) would feature Einstein in its prologue alongside some sixty other prominent (alleged and actual) Jewish intellectuals (Robinson, 2019a p. 225). Written by a close collaborator of Hitler’s propaganda minister Joseph Goebbels, the author of the brochure would include a photograph of Einstein alongside the caption:

“Discovered a…

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The Institute for Advanced Study is one of the few institutions in the world where the pursuit of knowledge for its own sake is the ultimate raison d'être. Speculative research, the kind that is fundamental to the advancement of human understanding of the world of nature and of humanity, is not a product that can be made to order. Rather, like artistic creativity, it benefits from a special environment. — IAS Bluebook

I’ve written in the past about the University of Göttingen and its historical status as a ‘center of the mathematical universe’ in the period from Gauss’ appointment as the Director of the Göttingen University Observatory in 1807 to the 1933 fleeing of much of its senior faculty, including Born, Franck, Wigner, Szilárd, Teller, Landau and Courant. This essay is the second part to that story—about communities of mathematicians that form in small, seemingly arbitrary geographical locations such as G…

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“He was so incredibly himself” — Herta Newman

Mathematician John Forbes Nash, Jr (1928-2015). was born in Bluefield, West Virginia in 1928. He died in a car crash in New Jersey on the 23rd of May, 2015, on his way back home after receiving the renowned Abel Prize in Oslo a few days prior.

Popularized in the 2001 Academy Award winning movie A Beautiful Mind, Nash’s life story has since captured the fascination and imagination of the world. The goal of this article has been to celebrate that life. Here’s to John Forbes Nash, Jr., surely one of the most fascinating and brilliant people who has ever lived.

Early years (1928–45)

John Forbes Nash, Jr. was born on the 13th of June, 1928, in Bluefield, West Virginia. His father John Sr. was an electrical engineer for the Appalachian Electric Power Company. His mother, Margaret Virginia, had been a schoolteacher of Latin before she marr…

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“The baseball announcer has it, of course, conveniently all wrong. Ted Williams is due to hit because he hasn’t hit for seven days. That’s red noise. Ted Williams is hot, he’s sure to hit tomorrow because he’s been hitting for seven days. That’s blue noise. The better description of an efficient market with random movements, which is white noise, is in between.”

The above anecdote was made by economist Paul Samuelson (1915-2009) in a 2004 interview conducted by fellow Nobel Laureate Robert C. Merton (1944-) for the American Finance Association (AFA). Lauded for his many contributions and often hailed as the ‘last of the great general economists’, the interview regarded one of Samuelson’s main contributions to finance, work on the so-called efficient-market hypothesis.

The efficient-market hypothesis (EMH) states that asset prices fully reflect all available information in a market. That is, beating an efficient market consistently on a risk-adjusted basis should in theory be impossible…

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Historians of science sometimes lament the fact certain historical figures are famous for the wrong reasons. Alan Turing (1912-54) is perhaps the best example of this phenomenon. Most people associate Turing with the breaking of an encryption device called the German Navy Enigma, rather than his—arguably—much more important contributions to theoretical computer science. Another example is Albert Einstein (1879-1955), who is typically recognized for his mass-energy equivalence formula E = mc² rather than for general relativity. J. Robert Oppenheimer (1904-67) is another one, more frequently associated with being the director of the Manhattan Project than with the Born-Oppenheimer approximation, arguably a much more important scientific contribution.

In this essay I will argue that Abraham Wald (1902-50) is another such figure. His name, for…

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

Georg Cantor (1845-1918)’s correspondence with mathematician Richard Dedekind (1831-1916) in the years 1873-74 has been narrated in a previous newsletter. Their letters detail the process by which Cantor arrived at the discovery that the set of the real numbers is uncountably infinite. That is, Cantor showed that although both the natural numbers and the real numbers are infinite in number and so go on forever, there “aren’t enough” natural numbers to create a one-to-one correspondence between them and the real numbers. Cantor’s brilliant discovery, in other words, showed rigorously and undeniably that infinity comes in different sizes, some of which are larger than others. One of his methods of proving this assertion, Cantor’s diagonal ar…

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