Turing and Wiles — Computability and Fermat's Last Theorem

Geniuses Who “Solved a Concrete Problem”

The mathematicians of the earlier articles up through Noether and Germain left their mark by inventing new theories and structures. In the second half of the 20th century, another lineage of genius appears — mathematicians who, alone, solved a concrete problem left unsolved for a century or more:

• Alan Turing (1912–1954) — mechanically defined “what is computation,” founded computer science, and broke the Enigma code
• Andrew Wiles (1953–) — in 1994, proved Fermat’s Last Theorem after 358 years

Both did solitary work whose results changed their era. Years of thought maturing inside one head changed the world.

Alan Turing — Forty-one Years of Tragedy

Alan Mathison Turing (1912–1954) was a British mathematician, logician, and computer scientist. Founder of computer science and of the conceptual foundation of artificial intelligence. During World War II he worked at Bletchley Park on breaking the Enigma code and is credited with contributing decisively to the Allied victory.

After the war he was convicted of homosexuality, subjected to chemical castration, and on 7 June 1954 was found dead beside a cyanide-laced apple (the coroner ruled suicide). In 2013 he received a Royal Pardon, and in 2017 the “Alan Turing Law” passed.

Major Contributions

Result	Year	Content
Turing machine	1936	Mathematical definition of computability; undecidability of the halting problem
Universal Turing machine	1936	Conceptual ancestor of the stored-program computer
Enigma decryption	1939–1945	Bletchley Park, the Bombe
ACE design	1945	Design of one of the first British computers
Turing test	1950	A behavioral criterion for AI
Morphogenesis	1952	Reaction–diffusion equations; biological patterns

Computer science at 24, AI at 38, mathematical biology at 40 — the archetype of the genius who builds one field and then moves on.

The Turing Machine (1936) — The Birth of Computer Science

In the 1936 paper “On Computable Numbers, with an Application to the Entscheidungsproblem,” Turing defined “the mechanically computable functions” as those a Turing machine can compute.

Its components:

• An infinite tape
• A single head that reads and writes
• A finite set of states
• A finite table of rules

That is enough to express every algorithmically executable computation. Turing also constructed a Universal Turing Machine that, given the description of another Turing machine on its tape, can completely simulate that machine’s behavior.

This is the conceptual ancestor of the von Neumann (stored-program) computer. “A single paper in 1936 foretold every modern computer” — the greatest preview in the history of mathematics.

Undecidability of the Halting Problem

The same paper considered the halting problem — is there an algorithm that, given an arbitrary program, decides whether it halts in finitely many steps? — and used the diagonal method to prove no such algorithm exists.

A negative answer to Hilbert’s Entscheidungsproblem. Aligned with the direction of Gödel’s incompleteness theorems from the earlier article — “there are things mathematics cannot in principle do.”

“There are things machines cannot in principle do (deciding halting)” — the starting point of modern computational complexity theory.

The Church–Turing Thesis

Turing machines are equal in expressive power to Church’s lambda calculus. Together they give the Church–Turing thesis:

“The mechanically computable functions coincide with those computable by a Turing machine (= lambda calculus = recursive functions).”

The modern definition of “computable.” The intuitive notion of “algorithm” is given a mathematical definition. All modern programming languages are equivalent in computational power to a Turing machine.

Enigma — Shortening the War by 2–4 Years

During World War II, the German military used Enigma, a rotor-based cipher machine, to secure communications. With three or four rotors, its key space was enormous, and it was believed to be unbreakable.

Turing led the Bletchley Park team, and, building on prior Polish work by Rejewski and colleagues, designed the Bombe, an electromechanical device that rapidly searched the possible key settings and made Enigma traffic readable.

“Ultra” — Classified for 30 Years

The Allies rigorously concealed the fact that they were reading Enigma (the intelligence was called “Ultra”) and are estimated to have shortened the war by two to four years and saved millions of lives.

Turing’s wartime contribution remained classified for more than 30 years and was only revealed after his death. “He changed history, but the world did not know it in his lifetime” — a tragic double-secrecy.

The Turing Test (1950)

In “Computing Machinery and Intelligence” (1950), Turing proposed, in place of the question “can a machine think?”, an observable behavioral test:

“If a human judge, communicating only through a text terminal with a hidden partner (human or machine), cannot reliably tell which is which, the machine should be considered as thinking.”

The philosophical starting point of AI research. With the rise of large language models like ChatGPT and Claude, it is often invoked today. Turing himself predicted that “by 2000, 30% of judges would misidentify within five minutes.”

Modern LLMs are reaching regions where they are indistinguishable from humans in text conversation. Turing’s prediction has come true twenty years late.

Morphogenesis — A Pioneer of Mathematical Biology

Toward the end of his life, Turing published “The Chemical Basis of Morphogenesis” (1952), arguing that patterns on animal bodies (stripes, phyllotaxis, seashell markings) can arise spontaneously from the interplay of chemical reaction and diffusion — via reaction–diffusion equations. A pioneering work of mathematical biology.

In modern developmental biology and pattern formation theory, Turing’s ideas have been experimentally verified, and “Turing patterns” are seen in things like zebrafish stripes.

The last original contribution of a 40-year-old mathematician who moved from computing → AI → biology.

Chemical Castration and Death

In 1952, British law (Section 11 of the Labouchere Amendment, 1885) still made homosexual acts a crime. Turing, in the course of a burglary investigation, spoke about a same-sex relationship too openly, and was convicted. Given the choice between prison and chemical castration, he chose the latter so that he could continue his research.

Estrogen injections caused breast growth and deep psychological damage. On 7 June 1954, he was found dead beside a cyanide-laced apple. The coroner ruled suicide. Motives have been debated, but the background includes the humiliation of chemical castration, the erosion of research opportunities, and the burden of secrecy obligations.

The urban legend that the Apple logo (a bitten apple) is an homage to Turing is denied by Apple.

“The genius who saved the war and invented the computer bit into a poisoned apple at 41” — one of the greatest mathematical tragedies of the 20th century.

Posthumous Pardon and Recognition

• 2009 — Prime Minister Brown issues an apology to Turing
• 2013 — Queen Elizabeth II grants a Royal Pardon
• 2017 — The “Alan Turing Law” posthumously pardons about 49,000 men convicted under the same statute
• 2021 — Turing is featured on the £50 note

An institution apologizing more than half a century later — a symbolic moment for LGBT rights.

Andrew Wiles — Solving a 358-Year-Old Homework Problem

Andrew John Wiles (1953–) is a British mathematician. In 1994 he proved Fermat’s Last Theorem — the greatest homework problem in the history of mathematics, unsolved for 358 years, solved by one person.

As covered in the earlier article, Fermat scribbled in the margin of the Arithmetica around 1637:

“It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general any power beyond the second into two like powers. I have discovered a truly marvelous proof, which this margin is too narrow to contain.”

Wiles proved it 358 years later.

Wiles’s Life

• 11 April 1953 — Born in Cambridge; his father was a theologian
• At 10 — Reads about Fermat’s Last Theorem in a local library. Decides: “This is my life’s goal.”
• 1974 — Graduates from Oxford
• 1980 — PhD from Cambridge (elliptic curves)
• 1982 — Professor at Princeton
• 1986 — Learns of the link between the Taniyama–Shimura conjecture and Fermat’s Last Theorem
• 1986–1993 — Seven years of secret work in his attic
• 23 June 1993 — Announces the proof at Cambridge
• December 1993 — A serious gap is discovered
• September 1994 — Fixes the gap together with his student Taylor
• 1995 — Publishes the final proof in the Annals of Mathematics
• 1998 — Special tribute at the ICM (over the Fields Medal age limit)
• 2016 — Abel Prize

A Ten-Year-Old’s Vow

At 10, Wiles read Eric Temple Bell’s The Last Problem in a local library and learned about Fermat’s Last Theorem:

“Even a ten-year-old can understand the statement. And yet no one has solved it in hundreds of years. This is my life’s goal.”

The starting point of his mathematical career was a single book at 10. One of the longest “consistent goals” in the history of mathematics.

Seven Years in the Attic

In 1986, Frey and Ribet showed that Fermat’s Last Theorem would follow from the Taniyama–Shimura conjecture (see the earlier article). Wiles’s 10-year-old dream was rekindled, and he set out to prove the Taniyama–Shimura conjecture for semistable elliptic curves.

Seven Secret Years

In his Princeton attic, Wiles worked on Fermat’s Last Theorem from 1986 to 1993, seven years, entirely in secret except for his family and a single colleague:

• Handled his ordinary university duties (lectures, papers) while
• Working intensively in the attic
• Constructing an enormous theoretical apparatus in complete solitude
• Telling his wife Nada only that “I am working on an important problem”

“Seven years alone with the greatest homework problem in mathematics” — the longest solitary effort in late-20th-century mathematics.

June 1993 — The Announcement

On 23 June 1993, at the Isaac Newton Institute in Cambridge, on the last day of a three-day lecture series, Wiles wrote Fermat’s Last Theorem on the blackboard:

“There are no natural-number solutions to this equation. This is …”

After a short pause:

“I think I’ll stop there.”

Thunderous applause. Mathematicians around the world shared news by phone and fax. The story made the front page of the New York Times.

December 1993 — The Gap

But in December 1993, during peer review, Nick Katz found a serious gap in the proof. Wiles went from mathematical hero to standing in front of a proof that might not close.

• For months, alone, he tried and failed to patch it
• He reached a mental limit and almost gave up
• In August 1994, he rejoined the effort with his student Richard Taylor

September 1994 — The Fix

On 19 September 1994, Wiles had a miraculous insight. By combining Iwasawa theory and Euler systems — approaches he had earlier discarded — he fixed the gap:

“It was the most beautiful moment of my mathematical life.”

In 1995 the final proof appeared in the Annals of Mathematics — a 130-page paper using every advanced tool of modern number theory: elliptic curves, modular forms, Galois representations. Modern methods Fermat could not have known.

The Fields Medal and a Special Tribute

The Fields Medal is awarded to mathematicians 40 or younger. When the proof was completed in 1995, Wiles was 41 — outside the age limit. In 1998, the International Congress of Mathematicians presented a special tribute — an unprecedented exception recognizing “work that stands in the history of mathematics even if outside the medal age.”

In 2016 he received the Abel Prize — named for Abel of the earlier article, one of the top honors in mathematics.

Simon Singh’s Fermat’s Last Theorem

Wiles’s proof reached general readers through Simon Singh’s Fermat’s Last Theorem (1997) — one of the most-read non-fiction books in the history of mathematics.

The BBC documentary Fermat’s Last Theorem (1996) is famous for the scene in which Wiles weeps while recounting the moment of the fix. A rare filmed record of “a mathematician showing emotion.”

Modern Impact

Turing → today

• Computer science — theoretical basis of every programming language
• AI — the Turing test, philosophical foundation of machine learning
• Cryptography — conceptual source of public-key and quantum cryptography
• Biology — pattern formation, developmental biology
• LGBT rights — Turing Law, memorials, rehabilitation

Wiles → today

• Number theory — the full proof of Taniyama–Shimura (2001, Breuil–Conrad–Diamond–Taylor)
• The Langlands program — Wiles’s methods are central tools
• Cryptography — theoretical basis of elliptic-curve cryptography (ECC)
• Popularization — Fermat’s Last Theorem is the mathematical problem most known to the general public

The 20th Century of “Solitary Work”

What Turing and Wiles have in common is solitary work:

• Turing — the secrecy of Bletchley Park, post-war secrecy obligations, isolation after chemical castration
• Wiles — seven years in the attic, hidden even from all but his family and one colleague

“Years of thought maturing in one head change the world” — a pattern of the late 20th century that carries into the 21st, in Perelman (next article).

From This Article to the Next

We have traced the late 20th century — Turing and Wiles. In the final article of the Lineage of Mathematics series, we turn to contemporary mathematics in the 21st century — Perelman (1966–), his proof of the Poincaré conjecture and his refusal of the Fields Medal, Mochizuki Shinichi (1969–), his IUT theory and the ABC conjecture, Terence Tao (1975–), the modern universal mathematician, and the state of the Millennium Problems. Twenty-five centuries end there.