Logic

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 16, 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 16, 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 16, 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 16, 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 16, 2026 Read

Calculemus: the idea of calculating with reasoning

The only way to rectify our reasonings is to make them as tangible as those of the mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: let us calculate!, without further ado, to see who is right. — Leibniz, The Art of Discovery (1685) Leibniz was a lawyer, diplomat and philosopher. He spent his life watching arguments that never ended, between religions, between countries, between schools of philosophy. A mathematician, on the other hand, does not argue about whether $2+2$ is 4: he does the sum. What if arguments could be calculated like numbers?

October 11, 2026 Read

The legacy of Alan Turing

Turing believes machines think. Turing lies with men. Therefore machines do not think. — Alan Turing, letter to Norman Routledge (1952) The bitter syllogism Turing wrote to a friend, just after his conviction, sums up his life: a man who thought with logic and was judged by people who did not understand him. The work he left behind, however, is everywhere. In computer science The Turing machine is still the official definition of what it means to calculate. When a computer scientist says a problem “has no algorithm”, they mean no Turing machine solves it; when they say a problem is “hard”, they count the steps a Turing machine needs.

October 16, 2026 Read

Image credits

Historical portrait The portrait of Leibniz painted by Christoph Bernhard Francke around 1695 comes from Wikimedia Commons and is in the public domain. We use the 960-pixel thumbnail provided by Commons, without editing. Source page. Public domain mark. Blog figures The cover and the sketch of the multiplication $23\times14$ are original illustrations. The sketch explains the calculation and does not reproduce the mechanism of the historical machine. The interactive figures for the eight bits, the terms of the product rule and the sums of the series for π were created for this trail. The examples in the text let you follow the calculations even without JavaScript.

October 11, 2026 Read

Image credits

Historical images and photographs All of them come from Wikimedia Commons. We use the thumbnails provided by Commons, with no editing beyond compression. Turing at 16, passport photo attributed to Arthur Reginald Chaffin: source page · public domain. Turing in 1951, portrait by the Elliott & Fry studio: source page · public domain. King’s College chapel, photo by Martinvl: source page · CC BY-SA 4.0. The Bletchley Park mansion, photo by DeFacto: source page · CC BY-SA 4.0. A three-rotor Enigma, National Security Agency: source page · public domain. The bombe rebuild, photo by Mike Peel: source page · CC BY-SA 4.0. The Pilot ACE, photo by Philafrenzy: source page · CC BY 2.0. Replica of the Manchester Baby, photo by Parrot of Doom: source page · CC BY-SA 3.0. The Sackville Park statue, photo by David Dixon: source page · CC BY-SA 2.0. The skin of the giant pufferfish, photo by Chiswick Chap: source page · CC BY-SA 3.0. Turing’s patterns and the cheetah, figure by Philip Ball: source page · CC BY 4.0. Figures made for the site The cover, the interactive Turing machine and the reaction–diffusion simulation were made for this trail. The simulation uses the Gray–Scott model, computed on a 200 by 200 grid of cells, and contains no text, so it serves all the languages of the site. The interactive machine works in all four languages.

October 16, 2026 Read

Boole, Frege and Peano: writing down reasoning

Until the nineteenth century, mathematical notation was used to write quantities and the operations between them. There was no script for something else that mathematicians did all the time without writing it down: reasoning. “If this, then that”; “for every number”; “there is a number such that”. All of this was still said in Latin, French or German, between one formula and the next. This text tells how reasoning itself got its signs.

October 13, 2026 Read