Computing

The life of John von Neumann

Timeline 1903Born on 28 December in Budapest, the son of a banker. 1913His father receives a title of nobility; the son becomes “von Neumann”. 1921Finishes the Lutheran gymnasium in Budapest, where he studied alongside Eugene Wigner. 1923Publishes, at 19, his definition of ordinal numbers. 1926Diploma in chemical engineering in Zurich and doctorate in mathematics in Budapest. 1927Becomes the youngest Privatdozent at the University of Berlin. 1928Publishes the minimax theorem, the basis of game theory. 1930Moves to Princeton. In 1933 he is one of the first professors of the Institute for Advanced Study. 1932Publishes the Mathematical Foundations of Quantum Mechanics. 1943Begins working at Los Alamos, on the atomic bomb project. 1944Publishes, with Oskar Morgenstern, the Theory of Games and Economic Behavior. 1945Writes the EDVAC report, which describes the stored-program computer. 1946With Stanisław Ulam, develops the Monte Carlo method. 1952The IAS computer comes into operation. 1955Appointed to the Atomic Energy Commission. Discovers he has cancer. 1957Dies on 8 February, in Washington, aged 53. ◀ Previous Follow the journey Next ▶ 1903–1921 BudapestHe is born on 28 December, the son of a banker. He studies at the Lutheran gymnasium, where Eugene Wigner is a schoolmate. 1921–1923 BerlinHe studies chemistry, at his father’s insistence, while enrolled for a doctorate in mathematics in Budapest. 1923–1926 ZurichHe graduates in chemical engineering at the Polytechnic. He attends classes by George Pólya, who would say he was the only student he was ever afraid of. 1926–1927 GöttingenHe works with David Hilbert on the foundations of mathematics and of the new quantum mechanics. 1927–1930 BerlinThe university’s youngest Privatdozent. In 1928 he publishes the minimax theorem. 1930–1957 PrincetonProfessor at the university and, from 1933, at the Institute for Advanced Study, alongside Einstein and Gödel. There he builds the IAS computer. 1943–1955 Los AlamosConsultant to the atomic bomb project. With Ulam, he creates the Monte Carlo method. 1955–1957 WashingtonMember of the Atomic Energy Commission. He dies of cancer on 8 February 1957, at Walter Reed Hospital. Enable JavaScript to see the map. The stages are listed below. From Budapest to Washington, by way of Zurich, Göttingen, Berlin, Princeton and Los Alamos. Map: © OpenStreetMap contributors

October 26, 2026 Read

The life of Alan Turing

Timeline 1912Born on 23 June in London. His parents live in India. 1926Starts at Sherborne boarding school. 1930Christopher Morcom, his best friend, dies. 1931Enters King’s College, Cambridge. 1935Elected a fellow of King’s, aged 22. 1936Writes “On computable numbers”. Goes to Princeton. 1938Doctorate at Princeton. Returns to Cambridge. 1939Starts work at Bletchley Park, on breaking the Enigma. 1940The first electromechanical bombe comes into operation. 1945Moves to the National Physical Laboratory, where he designs the ACE. 1948Moves to the University of Manchester. 1950Publishes the paper on the imitation game. 1952Publishes the theory of morphogenesis. Convicted for homosexuality. 1954Dies on 7 June, in Wilmslow, aged 41. ◀ Previous Follow the journey Next ▶ 1912 LondonHe is born on 23 June in Maida Vale. His parents live in India, and he is brought up by a foster family in the south of England. 1926–1931 SherborneAt boarding school he meets Christopher Morcom, the friend whose death in 1930 leads him to think about the relation between mind and matter. 1931–1936 CambridgeHe studies at King’s College, becomes a fellow at 22 and writes “On computable numbers”. 1936–1938 PrincetonHe does his doctorate with Alonzo Church, who had reached the same result by another route. 1939–1945 Bletchley ParkDuring the war he heads the section that breaks the German naval Enigma and designs the electromechanical bombe. 1945–1947 TeddingtonAt the National Physical Laboratory he designs the ACE, a stored-program computer. 1948–1954 ManchesterHe programs one of the first computers, proposes the imitation game and studies morphogenesis. He dies in Wilmslow, near the city, on 7 June 1954. Enable JavaScript to see the map. The stages are listed below. From London to Manchester, by way of Princeton and Bletchley Park. Map: © OpenStreetMap contributors

October 17, 2026 Read

The Turing machine: calculating with pencil, tape and rules

In 1936 there was no computer. The English word computer meant a person who did calculations for a living, with pencil and paper, in insurance offices, observatories and government departments. Turing started from that person. He asked what they really do when they calculate and stripped away everything that is not essential. What is left of someone calculating Turing’s reasoning, in the paper “On computable numbers”, is almost a list of simplifications:

October 18, 2026 Read

The halting problem: the question no machine can answer

In 1928 the German David Hilbert set a challenge he called the Entscheidungsproblem, the “decision problem”: find a mechanical procedure that, given any mathematical statement written in the language of logic, would answer in finite time whether it is true or false. If it existed, mathematics would become an assembly line: just write down the question and turn the handle. Hilbert believed the answer was yes. Eight years later, Turing and the American Alonzo Church showed, each by his own route, that it is no.

October 18, 2026 Read

Computable numbers: almost every number is impossible to calculate

The title of the 1936 paper does not talk about machines. It talks about numbers: “On computable numbers”. Turing defined them like this: a real number is computable when there is a machine that writes out, one after another, all of its digits, without ever getting any wrong. Who is computable Almost all the numbers that appear at school are computable. Fractions: $\frac17=0.142857142857\ldots$ The machine does the long division and repeats the cycle forever. Roots: $\sqrt2=1.41421356\ldots$ To find the next digit, the machine tests from 0 to 9 which is the largest that, when appended, still gives a number whose square does not exceed 2. $\pi$ and $e$: there are series that approximate them as closely as we like, such as $e=1+\frac1{1!}+\frac1{2!}+\frac1{3!}+\cdots$, and with them a machine produces any digit, sooner or later. It looks as if every “real” number is computable. And here comes the surprise: almost none is.

October 18, 2026 Read

The von Neumann architecture: the EDVAC report

The ENIAC, finished in 1945 at the University of Pennsylvania, was the fastest electronic computer in the world: it did five thousand additions a second with almost eighteen thousand valves. But to change calculation one had to rewire cables and switches on a panel, a job of days done by a team of programmers. Two programmers operating the ENIAC’s control panel, at the Moore School of the University of Pennsylvania, around 1946.United States Army · Wikimedia Commons · public domain The ENIAC’s engineers, J. Presper Eckert and John Mauchly, were already planning the next machine, the EDVAC, in which the program would be kept in memory. Von Neumann began attending the meetings and, in June 1945, wrote a text that organised the group’s ideas in an abstract, almost biological language, speaking of “organs” and “neurons”.

October 26, 2026 Read

The universal machine: the program is data too

Each Turing machine does one thing only. The one that adds 1 adds 1; the one that writes $\frac17$ writes $\frac17$. To do a different calculation, you would need a different machine. In 1936 that was normal: mechanical calculators, punched-card tabulators and tide-predicting machines were built for one task, and changing task meant changing machine or rewiring. In the same paper, Turing showed that this is not necessary. A machine that reads machines The table of rules of any machine can be written as text on a tape. Turing then described a special machine, $U$, the universal machine, which does the following:

October 18, 2026 Read

Binary arithmetic: writing numbers with 0 and 1

Why do we count in tens? Probably because we have ten fingers. There is nothing special about the number 10: any number can be written in any base. Leibniz took this idea to the extreme and asked: what is the smallest number of digits that can write everything? The answer is two: 0 and 1. Each place is worth double The numbers 0 to 32 written in binary, in the table from the paper Explanation of Binary Arithmetic, which Leibniz published in 1703 in the memoirs of the Paris Academy of Sciences.G. W. Leibniz (1703) · Wikimedia Commons · public domain In the decimal system the places are worth 1, 10, 100, 1000… and each digit says how many times we use that value. In the binary system the places are worth 1, 2, 4, 8, 16, 32… and each digit can only say yes (1) or no (0).

October 6, 2026 Read

Rotating in space with quaternions

Complex numbers of modulus 1 rotate the plane. Hamilton hoped his numbers would do the same in space, and they do, but in a subtler way than he first imagined. The recipe Write a point of space $(x, y, z)$ as the pure quaternion $v = xi + yj + zk$. To rotate by an angle $\theta$ about an axis given by a unit vector $u = u_1 i + u_2 j + u_3 k$, take

October 27, 2026 Read

Merge sort: sorting by joining

Sorting a list is one of the tasks computers do most: names in a register, prices in a search, election results. In 1945, when the EDVAC was still on paper, von Neumann wanted to show that the machine in his report was good for work that was not just arithmetic, and wrote by hand a program to sort and merge lists. It is one of the first programs written for a stored-program computer, and the method he used, now called merge sort, is still one of the most widely used.

October 26, 2026 Read

The Enigma: 159 quintillion keys and one flaw

A three-rotor Enigma machine on display at the National Cryptologic Museum of the American security agency. At the top, the rotors; in the middle, the lamps; below, the keyboard; at the front, the plugboard.National Security Agency · Wikimedia Commons · public domain The Enigma looked like a typewriter inside a wooden box. The operator pressed a letter on the keyboard and another letter lit up on a panel of lamps. A second operator wrote down the letters that lit up, and that sequence was sent by radio, in Morse code. At the other end, an operator with an identical machine, set up the same way, typed in the enciphered message and read the original text on the lamps.

October 18, 2026 Read

Leibniz's machine: turning sums into motion

In 1642 the young Blaise Pascal built a machine to help his father, a tax collector. The Pascaline added and subtracted by turning gears. To multiply, however, you had to repeat the addition by hand, many times. Leibniz saw a Pascaline in Paris and wanted to go further: a machine that would multiply and divide on its own. Multiplying is adding and shifting The trick is the same as in long multiplication at school. To multiply $23\times14$, we split 14 into $4+10$:

October 6, 2026 Read